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diff --git a/Abstract.tex b/Abstract.tex
new file mode 100644 (file)
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@@ -0,0 +1,16 @@
+Ce cours enseigné au Conservatoire National des Arts et Métiers (CNAM) de Paris est destiné à introduire les notions théoriques et pratiques du traitement du signal à un niveau Bac +2 ou +3.
+
+Trois parties se distinguent pour séparer tout d'abord les éléments théoriques qui fondent la \textbf{théorie du traitement de signal} par les méthodes de Fourier, puis une synthèse des différents \textbf{modes de représentation des signaux} notamment celui du spectrogramme, et enfin une introduction à la \textbf{théorie du filtrage numérique}.\newline
+%
+% Comme vous le verrez, cette première version tente un parcours non-exhaustif des fondements de l'acoustique architecturale. La plupart des éléments de ce polycopié ont été tirés des cours personnels de \textbf{Manuel Melon} (Maître de Conférences au CNAM) \cite{melon} pour les deux premiers chapîtres, de \textbf{Jean-Dominique Polack} (Professeur des Universités et Directeur du Laboratoire d'Acoustique Musicale de Paris VI) \cite{polack_salles} pour les deux chapitres suivants et du livre de \textbf{Jacques Jouhaneau} (Professeur titulaire de la Chaire d'Acoustique du CNAM) \cite{jouhaneau} pour de très nombreuses références. Je les remercie tous sincèrement de leur aide.\newline
+
+Ce document est en constante évolution. Certaines parties seront revues ou corrigées dans les versions ultérieures. Les mises à jour seront disponibles sur l'adresse web indiquée en page de garde et je vous invite donc à la consulter régulièrement.
+
+Il est publié selon les termes de la licence Creative Commons by-nc-sa 2.0 France :
+
+\url{http://creativecommons.org/licenses/by-nc-sa/2.0/fr/}
+
+\begin{figure}[b]
+  \includegraphics[width=4cm]{img/Logo_cnam_new.jpg}
+  \large{Copyright (C) 2006-2011 Guillaume Pellerin}
+\end{figure}

diff --git a/Abstract.tex~ b/Abstract.tex~
new file mode 100644 (file)
index 0000000..46cbdcc
--- /dev/null
+++ b/Abstract.tex~
@@ -0,0 +1,16 @@
+Ce cours enseigné au Conservatoire National des Arts et Métiers (CNAM) de Paris est destiné à introduire les notions théoriques et pratiques du traitement du signal à un niveau Bac +2 ou +3.
+
+Trois parties se distinguent pour séparer tout d'abord les éléments théoriques qui fondent la \textbf{théorie du traitement de signal} par les méthodes de Fourier, puis une synthèse des différents \textbf{modes de représentation des signaux} notamment celui du spectrogramme, et enfin une introduction à la \textbf{théorie du filtrage numérique}.\newline
+%
+% Comme vous le verrez, cette première version tente un parcours non-exhaustif des fondements de l'acoustique architecturale. La plupart des éléments de ce polycopié ont été tirés des cours personnels de \textbf{Manuel Melon} (Maître de Conférences au CNAM) \cite{melon} pour les deux premiers chapîtres, de \textbf{Jean-Dominique Polack} (Professeur des Universités et Directeur du Laboratoire d'Acoustique Musicale de Paris VI) \cite{polack_salles} pour les deux chapitres suivants et du livre de \textbf{Jacques Jouhaneau} (Professeur titulaire de la Chaire d'Acoustique du CNAM) \cite{jouhaneau} pour de très nombreuses références. Je les remercie tous sincèrement de leur aide.\newline
+
+Ce document est en constante évolution. Certaines parties seront revues ou corrigées dans les versions ultérieures. Les mises à jour seront disponibles sur l'adresse web indiquée en page de garde et je vous invite donc à la consulter régulièrement.
+
+Il est publié selon les termes de la licence Creative Commons by-nc-sa 2.0 France :
+
+\url{http://creativecommons.org/licenses/by-nc-sa/2.0/fr/}
+
+\begin{figure}[b]
+  \includegraphics[width=4cm]{img/Logo_cnam_new.jpg}
+  \large{Copyright (C) 2006-2010 Guillaume Pellerin}
+\end{figure}

diff --git a/Abstracts.aux b/Abstracts.aux
new file mode 100644 (file)
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diff --git a/Biblio.bib b/Biblio.bib
new file mode 100644 (file)
index 0000000..0e1799b
--- /dev/null
+++ b/Biblio.bib
@@ -0,0 +1,74 @@
+
+
+@PhdThesis{henrich,
+author = {N. Henrich},
+title = {Etude de la source glottique en voix parlée et chantée},
+school = {Université Pierre et Marie Curie},
+year = {2001},
+}
+
+@InProceedings{pelorson,
+author = {X. Pelerson},
+title = {Cordes vocales et lèvres vibrantes},
+year = {2004},
+address = {Paris},
+month = {juillet},
+organization = {Conférence SFA },
+}
+
+@InProceedings{garnier1,
+author = {M. Garnier and D. Dubois and J. Poitevineau and N. Henrich and M. Castellengo},
+title = {Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive},
+year = {2004},
+address = {Fez},
+organization = {Journées d'Études sur la Parole},
+}
+
+@Booklet{charbit,
+title = {Bases de Traitement du signal},
+author = {M. Charbit},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {2004},
+}
+
+@Booklet{laroche,
+title = {Traitement des Signaux Audio-Fréquences},
+author = {J. Laroche},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {1995},
+}
+
+@Booklet{heiser,
+title = {Traitement du signal},
+author = {T. Heiser},
+howpublished = {Support de cours},
+address = {UFR Sciences Physiques, Université Louis Pasteur, Strasbourg},
+}
+
+@Booklet{feng,
+title = {Traitement du signal},
+author = {Feng},
+howpublished = {Support de cours},
+address = {ENSERG. INP Grenoble},
+year = {2000},
+}
+
+@Booklet{fallon,
+title = {Traitement du signal},
+author = {J. Fallon},
+howpublished = {Support de cours},
+address = {ENSAM, Lille},
+year = {1997},
+}
+
+@Booklet{matignon,
+title = {Traitement du signal},
+author = {D. Matignon},
+howpublished = {Support de cours},
+address = {ENST, Paris},
+year = {2000},
+}
+
+

diff --git a/Biblio.bib~ b/Biblio.bib~
new file mode 100644 (file)
index 0000000..0e1799b
--- /dev/null
+++ b/Biblio.bib~
@@ -0,0 +1,74 @@
+
+
+@PhdThesis{henrich,
+author = {N. Henrich},
+title = {Etude de la source glottique en voix parlée et chantée},
+school = {Université Pierre et Marie Curie},
+year = {2001},
+}
+
+@InProceedings{pelorson,
+author = {X. Pelerson},
+title = {Cordes vocales et lèvres vibrantes},
+year = {2004},
+address = {Paris},
+month = {juillet},
+organization = {Conférence SFA },
+}
+
+@InProceedings{garnier1,
+author = {M. Garnier and D. Dubois and J. Poitevineau and N. Henrich and M. Castellengo},
+title = {Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive},
+year = {2004},
+address = {Fez},
+organization = {Journées d'Études sur la Parole},
+}
+
+@Booklet{charbit,
+title = {Bases de Traitement du signal},
+author = {M. Charbit},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {2004},
+}
+
+@Booklet{laroche,
+title = {Traitement des Signaux Audio-Fréquences},
+author = {J. Laroche},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {1995},
+}
+
+@Booklet{heiser,
+title = {Traitement du signal},
+author = {T. Heiser},
+howpublished = {Support de cours},
+address = {UFR Sciences Physiques, Université Louis Pasteur, Strasbourg},
+}
+
+@Booklet{feng,
+title = {Traitement du signal},
+author = {Feng},
+howpublished = {Support de cours},
+address = {ENSERG. INP Grenoble},
+year = {2000},
+}
+
+@Booklet{fallon,
+title = {Traitement du signal},
+author = {J. Fallon},
+howpublished = {Support de cours},
+address = {ENSAM, Lille},
+year = {1997},
+}
+
+@Booklet{matignon,
+title = {Traitement du signal},
+author = {D. Matignon},
+howpublished = {Support de cours},
+address = {ENST, Paris},
+year = {2000},
+}
+
+

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+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.10 P
+      elorson X. Conférence SFA « Cordes vocales et lèvres vibrantes »...
+
+You're in trouble here.  Try typing  <return>  to proceed.
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+
+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.12 C
+      harbit M. Polycopié ENST. Bases de Traitement du signal. 2004.
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.14 L
+      Aroche J.. Polycopié ENST. Traitement des Signaux Audio-Fréquences. ...
+
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+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.16 H
+      eiser T. Traitement du signal, support de cours. UFR Sciences Physique...
+
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+
+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.18 F
+      eng. Traitement du signal, support de cours. ENSERG. INP Grenoble. 2000.
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+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.20 D
+      utoit T.  Introduction au Traitement Automatique de la Parole Faculté...
+
+You're in trouble here.  Try typing  <return>  to proceed.
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diff --git a/Biblio.tex b/Biblio.tex
new file mode 100644 (file)
index 0000000..af34a19
--- /dev/null
+++ b/Biblio.tex
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+
+
+
+\noindent David B. Conférence SFA sur le vibrato. Décembre 2003. Le mans.
+
+\noindent Garnier M., Dubois D., Poitevineau J., Henrich N., Castellengo M. Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive. Journées d'Études sur la Parole, Fez, 2004.
+
+@PhdThesis{henrich,
+author = {N. Henrich},
+title = {Etude de la source glottique en voix parlée et chantée},
+school = {Université Pierre et Marie Curie},
+year = {2001},
+}
+
+
+@InProceedings{pelorson,
+author = {X. Pelerson},
+title = {Cordes vocales et lèvres vibrantes},
+year = {2004},
+address = {Paris},
+month = {juillet},
+organization = {Conférence SFA },
+}
+
+@InProceedings{garnier1,
+author = {M. Garnier and D. Dubois and J. Poitevineau and N. Henrich and M. Castellengo},
+title = {Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive},
+year = {2004},
+address = {Fez},
+organization = {Journées d'Études sur la Parole},
+}
+
+@Booklet{charbit,
+title = {Bases de Traitement du signal},
+author = {M. Charbit},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {2004},
+}
+
+Booklet{laroche,
+title = {Traitement des Signaux Audio-Fréquences},
+author = {J. Laroche},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {1995},
+}
+
+Booklet{heiser,
+title = {Traitement du signal},
+author = {T. Heiser},
+howpublished = {Support de cours},
+address = {UFR Sciences Physiques, Université Louis Pasteur, Strasbourg},
+}
+
+Booklet{feng,
+title = {Traitement du signal},
+author = {Feng},
+howpublished = {Support de cours},
+address = {ENSERG. INP Grenoble},
+year = {2000},
+}
+
+Booklet{fallon,
+title = {Traitement du signal},
+author = {J. Fallon},
+howpublished = {Support de cours},
+address = {ENSAM, Lille},
+year = {1997},
+}
+
+Booklet{matignon,
+title = {Traitement du signal},
+author = {D. Matignon},
+howpublished = {Support de cours},
+address = {ENST, Paris},
+year = {2000},
+}
+
+

diff --git a/Biblio.tex.backup b/Biblio.tex.backup
new file mode 100644 (file)
index 0000000..d731e5a
--- /dev/null
+++ b/Biblio.tex.backup
@@ -0,0 +1,19 @@
+
+\noindent David B. Conférence SFA sur le vibrato. Décembre 2003. Le mans.
+
+\noindent Garnier M., Dubois D., Poitevineau J., Henrich N., Castellengo M. Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive. Journées d'Études sur la Parole, Fez, 2004.
+
+\noindent Henrich N. Etude de la source glottique en voix parlée et chantée. Thèse de doctorat, UPMC. 2001.
+
+\noindent Pelorson X. Conférence SFA « Cordes vocales et lèvres vibrantes ». Juillet 2004. Paris
+
+\noindent Charbit M. Polycopié ENST. Bases de Traitement du signal. 2004.
+
+\noindent Laroche J. Polycopié ENST. Traitement des Signaux Audio-Fréquences. 1995.
+
+\noindent Heiser T. Traitement du signal, support de cours. UFR Sciences Physiques. université Louis Pasteur, Strasbourg
+
+\noindent Feng. Traitement du signal, support de cours. ENSERG. INP Grenoble. 2000.
+
+\noindent Dutoit T.  Introduction au Traitement Automatique de la Parole Faculté Polytechnique
+de Mons. 2004

diff --git a/Biblio.tex~ b/Biblio.tex~
new file mode 100644 (file)
index 0000000..88ad23f
--- /dev/null
+++ b/Biblio.tex~
@@ -0,0 +1,71 @@
+
+
+
+\noindent David B. Conférence SFA sur le vibrato. Décembre 2003. Le mans.
+
+\noindent Garnier M., Dubois D., Poitevineau J., Henrich N., Castellengo M. Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive. Journées d'Études sur la Parole, Fez, 2004.
+
+@PhdThesis{henrich,
+author = {N. Henrich},
+title = {Etude de la source glottique en voix parlée et chantée},
+school = {Université Pierre et Marie Curie},
+year = {2001},
+}
+
+
+@InProceedings{pelorson,
+author = {X. Pelerson},
+title = {Cordes vocales et lèvres vibrantes},
+year = {2004},
+address = {Paris},
+month = {juillet},
+organization = {Conférence SFA },
+}
+
+@InProceedings{garnier1,
+author = {M. Garnier and D. Dubois and J. Poitevineau and N. Henrich and M. Castellengo},
+title = {Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive},
+year = {2004},
+address = {Fez},
+organization = {Journées d'Études sur la Parole},
+}
+
+@Booklet{charbit,
+title = {Bases de Traitement du signal},
+author = {M. Charbit},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {2004},
+}
+
+Booklet{laroche,
+title = {Traitement des Signaux Audio-Fréquences},
+author = {J. Laroche},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {1995},
+}
+
+Booklet{heiser,
+title = {Traitement du signal},
+author = {T. Heiser},
+howpublished = {Support de cours},
+address = {UFR Sciences Physiques, Université Louis Pasteur, Strasbourg},
+}
+
+Booklet{feng,
+title = {Traitement du signal},
+author = {Feng},
+howpublished = {Support de cours},
+address = {ENSERG. INP Grenoble},
+year = {2000},
+}
+
+Booklet{fallon,
+title = {Traitement du signal},
+author = {J. Fallon},
+howpublished = {Support de cours},
+address = {ENSAM, Lille},
+year = {1997},
+}
+

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diff --git a/Fiche_TZ.out b/Fiche_TZ.out
new file mode 100644 (file)
index 0000000..e69de29

diff --git a/Fiche_TZ.pdf b/Fiche_TZ.pdf
new file mode 100644 (file)
index 0000000..ae846ab
Binary files /dev/null and b/Fiche_TZ.pdf differ

diff --git a/Fiche_TZ.tex b/Fiche_TZ.tex
new file mode 100644 (file)
index 0000000..43f1ea5
--- /dev/null
+++ b/Fiche_TZ.tex
@@ -0,0 +1,48 @@
+\NeedsTeXFormat{LaTeX2e}
+%\usepackage{tex4ht}
+\documentclass[10pt,a4paper]{article}
+
+\usepackage{./exam_profil}
+\setlinespacing{1.0}
+%\graphicspath{{/figures}}
+
+\begin{document}
+
+%\renewenvironment{theindex}{\section{\indexname}
+%\begin{itemize}}{\end{itemize}}
+
+
+\begin{floatingfigure}[l]{3cm}
+ \includegraphics[width=3cm]{img/logo_cnam}
+\end{floatingfigure}
+
+\rule{13cm}{0.5mm}\\
+\vspace{-0.5cm}
+\begin{center}
+\Large{\bf{C.P.D.A.}}\\
+\Large{Centre de Préparation au}\\
+\Large{Diplôme d'Etat d'Audioprothésiste}\\
+\smallskip
+\large{Cours de $3^{e}$ année - 2006/2007}\\
+\smallskip
+\large{\bf{Traitement du signal - Fiche transformée en Z}}\\
+\large{G. Pellerin}\\
+\end{center}
+\vspace{-4mm}
+\rule{13cm}{0.5mm}
+
+\vspace{1cm}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=15cm]{img/TZ_sign_elem}
+\end{figure}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=15cm]{img/TZ_prop}
+\end{figure}
+
+
+\end{document}
+

diff --git a/Fiche_TZ.tex.backup b/Fiche_TZ.tex.backup
new file mode 100644 (file)
index 0000000..9544ccc
--- /dev/null
+++ b/Fiche_TZ.tex.backup
@@ -0,0 +1,61 @@
+\documentclass[pdftex,11pt,a4paper,twoside]{report}
+
+\usepackage{pdf_course_profile}
+
+%\title{Cours d'électricité}
+
+\title{\begin{flushright}
+    \rule{17cm}{1mm}\\
+    \vspace{8mm}
+    \huge{\bf{Traitement du signal}}\\
+       \vspace{4mm}
+    \rule{17cm}{1mm}\\
+    \large{Cours du CNAM - CPDA - 2006}\\
+    \large{modifié le }\today\\
+    \large{Version 0.3.2}
+    \end{flushright}
+}
+
+\author{\begin{flushright}
+       \textbf{Guillaume Pellerin$^1$ et Maëva Garnier$^2$}\\
+       \end{flushright}
+    \vspace{10cm}
+       \begin{center}
+    \small{
+    $^1$ Laboratoire d'Acoustique, Conservatoire National des Arts et Métiers\\
+    5 rue du Vert Bois F-75003 Paris\\ <\href{mailto:guillaume.pellerin@cnam.fr}{guillaume.pellerin@cnam.fr}>\\
+     \vspace{0.5cm}
+    %$^2$ Equipe design sonore - IRCAM\\ 1 place Igor-Stravinsky F-75004 Paris\\ <\href{mailto:Nicolas.Misdariis@ircam.fr}{Nicolas.Misdariis@ircam.fr}>\\
+    % \vspace{0.5cm}
+    $^2$ Equipe Lutherie, Acoustique et Musique, Institut Jean Le Rond d'Alembert,\\
+    Universit\'{e} Pierre et Marie Curie, 11 rue de Lourmel F-75015 Paris\\ <\href{mailto:garnier@lam.jussieu.fr}{garnier@lam.jussieu.fr}>
+    }
+    \end{center}}
+
+\begin{document}
+
+\maketitle
+
+\newpage
+\include{Abstract}
+
+\tableofcontents
+\cleardoublepage
+\renewcommand{\labelitemi}{$\bullet$}
+
+\chapter{Introduction aux méthodes fréquentielles de traitement du signal}
+\input{IntroMethFreq}
+
+\chapter{Systèmes linéaires et filtrage}
+\input{SystLinFilt}
+
+\chapter{Modes de représentation des signaux}
+\input{ReprSignaux}
+
+\newpage
+\bibliographystyle{plain}
+\nocite{*}
+\bibliography{biblio}
+
+\end{document}
+

diff --git a/Fiche_TZ.tex~ b/Fiche_TZ.tex~
new file mode 100644 (file)
index 0000000..760f388
--- /dev/null
+++ b/Fiche_TZ.tex~
@@ -0,0 +1,46 @@
+\NeedsTeXFormat{LaTeX2e}
+%\usepackage{tex4ht}
+\documentclass[10pt,a4paper]{article}
+
+\usepackage{./exam_profil}
+\setlinespacing{1.0}
+%\graphicspath{{/figures}}
+
+\begin{document}
+
+%\renewenvironment{theindex}{\section{\indexname}
+%\begin{itemize}}{\end{itemize}}
+
+
+\begin{floatingfigure}[l]{3cm}
+ \includegraphics[width=3cm]{img/logo_cnam}
+\end{floatingfigure}
+
+\rule{13cm}{0.5mm}\\
+\vspace{-0.5cm}
+\begin{center}
+\Large{\bf{C.P.D.A.}}\\
+\Large{Centre de Préparation au}\\
+\Large{Diplôme d'Etat d'Audioprothésiste}\\
+\smallskip
+\large{Cours de $3^{e}$ année - 2006/2007}\\
+\smallskip
+\large{\bf{Traitement du signal - Fiche transformée en Z}}\\
+\large{G. Pellerin}\\
+\end{center}
+\vspace{-4mm}
+\rule{13cm}{0.5mm}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=15cm]{img/TZ_sign_elem}
+\end{figure}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=15cm]{img/TZ_prop}
+\end{figure}
+
+
+\end{document}
+

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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.5 L
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+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no j in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+! Undefined control sequence.
+<write> ...g: Reference `ex_sig' on page \thepage 
+                                                  \space undefined\on@line .
+l.5 ...vant à la modéliser. Le figure \ref{ex_sig}
+                                                   donne un aperçu de la rep...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `ex_sig' on page  undefined on input line 5.
+
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no ç in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 5--6
+[]
+ []
+
+
+Overfull \hbox (3.61111pt too wide) in paragraph at lines 5--6
+\OML/cmm/m/it/10 t$
+ []
+
+
+Overfull \hbox (10.86105pt too wide) in paragraph at lines 5--6
+[]
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.7 \begin{figure}
+                  [h]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.7 \begin{figure}[
+                   h]
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.9     \includegraphics
+                        [width=12cm]{img/ex_sig}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no = in font nullfont!
+Missing character: There is no 1 in font nullfont!
+Missing character: There is no 2 in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ] in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.9     \includegraphics[width=12cm]{img/ex_
+                                            sig}
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Extra }, or forgotten $.
+l.9     \includegraphics[width=12cm]{img/ex_sig}
+                                                
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.10     \caption
+                 {Exemple d'une réprésentation temporelle d'un signal musical.}
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+! LaTeX Error: \begin{document} ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.12 \end{figure}
+                 
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+! Missing $ inserted.
+<inserted text> 
+                $
+l.12 \end{figure}
+                 
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
+! Missing } inserted.
+<inserted text> 
+                }
+l.12 \end{figure}
+                 
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 7--13
+[]
+ []
+
+
+Overfull \hbox (267.04398pt too wide) in paragraph at lines 7--13
+[]\OML/cmm/m/it/10 ig[][]$ 
+ []
+
+! Undefined control sequence.
+l.15 \section
+             {Classification des signaux}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.15 \section{C
+               lassification des signaux}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no C in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 15--16
+[]
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.17 E
+      n théorie du signal, on a l'habitude d'envisager plusieurs types de si...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no E in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no P in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no x in font nullfont!
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 17--18
+[]
+ []
+
+! Undefined control sequence.
+l.19 \subsection
+                {Classification dimensionnelle}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.19 \subsection{C
+                  lassification dimensionnelle}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no C in font nullfont!
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+Missing character: There is no i in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 19--20
+[]
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.21 O
+      n peut différentier les signaux selon leur nombre de variables libres....
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no O in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no f in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no v in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no P in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no U in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no b in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no E in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 21--22
+[]
+ []
+
+
+Overfull \hbox (19.44447pt too wide) in paragraph at lines 21--22
+\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$
+ []
+
+! Undefined control sequence.
+l.23 \subsection
+                {Classification phénoménologique}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.23 \subsection{C
+                  lassification phénoménologique}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no C in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 23--24
+[]
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.25 L
+      a première distinction porte sur notre capacité à prédire l'évolution ...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no è in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no S in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ï in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no U in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no è in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font cmbx10!
+Missing character: There is no . in font nullfont!
+Missing character: There is no I in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ù in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no ç in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no C in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
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+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.27 P
+      our simplifier, on peut dire que les signaux déterministes sont des ca...
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+You're in trouble here.  Try typing  <return>  to proceed.
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+Missing character: There is no x in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 27--28
+[]
+ []
+
+! Undefined control sequence.
+l.29 \subsection
+                {Classification morphologique}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.29 \subsection{C
+                  lassification morphologique}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no C in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 29--30
+[]
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.31 S
+      i, comme c'est le cas pour le signal $x(t) = Acos(2\pi f_0 t)$, où $t\...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no S in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
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+Missing character: There is no c in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ù in font nullfont!
+! Undefined control sequence.
+l.31 ...$x(t) = Acos(2\pi f_0 t)$, où $t\in\mathbb
+                                                  {R}$ et $A\in\mathbb{R}$, ...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+! Undefined control sequence.
+l.31 ...0 t)$, où $t\in\mathbb{R}$ et $A\in\mathbb
+                                                  {R}$, on dit que le signal...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no . in font nullfont!
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+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no v in font nullfont!
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+Missing character: There is no u in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ù in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no . in font nullfont!
+Missing character: There is no O in font nullfont!
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+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no c in font nullfont!
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+Missing character: There is no o in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no , in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no f in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no u in font nullfont!
+Missing character: There is no ' in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
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+Missing character: There is no c in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no L in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no à in font nullfont!
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+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no j in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no j in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no c in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no . in font nullfont!
+Missing character: There is no C in font nullfont!
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+Missing character: There is no u in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
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+Missing character: There is no c in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 31--32
+[]
+ []
+
+
+Overfull \hbox (27.6597pt too wide) in paragraph at lines 31--32
+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) =
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+)$
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+
+
+Overfull \hbox (13.05551pt too wide) in paragraph at lines 31--32
+\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2
+ []
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+
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+\OML/cmm/m/it/10 R$
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+
+Overfull \hbox (16.94441pt too wide) in paragraph at lines 31--32
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+\OML/cmm/m/it/10 t[]$
+ []
+
+
+Overfull \hbox (6.00235pt too wide) in paragraph at lines 31--32
+\OML/cmm/m/it/10 n$
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.33 I
+      l est également possible de discrétiser le signal en amplitude. On app...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no I in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no é in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no a in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 33--34
+[]
+ []
+
+
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+\OT1/cmr/bx/n/10 quan-
+ []
+
+
+Overfull \hbox (17.8888pt too wide) in paragraph at lines 33--34
+\OT1/cmr/bx/n/10 tifi-
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+
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+\OT1/cmr/bx/n/10 ca-
+ []
+
+
+Overfull \hbox (19.80545pt too wide) in paragraph at lines 33--34
+\OT1/cmr/bx/n/10 tion
+ []
+
+! Undefined control sequence.
+l.36 \subsection
+                {Classification énergétique}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.36 \subsection{C
+                  lassification énergétique}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 36--37
+[]
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.38 O
+      n distingue en particulier deux types de signaux :
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no O in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no u in font nullfont!
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+Missing character: There is no   in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 38--39
+[]
+ []
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.40 \begin{itemize}
+                    
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 40--40
+[] 
+ []
+
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
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+Missing character: There is no y in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ï in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no . in font nullfont!
+! Undefined control sequence.
+l.46 \section
+             {Signaux et distributions classiques}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no S in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 46--47
+[]
+ []
+
+! Undefined control sequence.
+l.48 \subsection
+                {Fenêtre rectangulaire ou Porte}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no F in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no P in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 48--49
+[]
+ []
+
+Missing character: There is no U in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+! Undefined control sequence.
+l.50 ... se note $\Pi_{2a}(t)$ avec $t \in \mathbb
+                                                  {R}$ et sa représentation ...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+! Undefined control sequence.
+<write> ...Reference `sig_porte' on page \thepage 
+                                                  \space undefined\on@line .
+l.50 ...ion est donnée à la figure \ref{sig_porte}
+                                                  .
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `sig_porte' on page  undefined on input line 50.
+
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 50--51
+[]
+ []
+
+
+Overfull \hbox (27.7127pt too wide) in paragraph at lines 50--51
+\OT1/cmr/m/n/10 ^^E[](\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$
+ []
+
+
+Overfull \hbox (13.05551pt too wide) in paragraph at lines 50--51
+\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2
+ []
+
+
+Overfull \hbox (7.67015pt too wide) in paragraph at lines 50--51
+\OML/cmm/m/it/10 R$
+ []
+
+
+Overfull \hbox (10.86105pt too wide) in paragraph at lines 50--51
+[] 
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.52 \begin{figure}
+                   [h]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.54     \includegraphics
+                         [width=8cm]{img/sig_porte}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no = in font nullfont!
+Missing character: There is no 8 in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ] in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.54     \includegraphics[width=8cm]{img/sig_
+                                             porte}
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Extra }, or forgotten $.
+l.54 ...\includegraphics[width=8cm]{img/sig_porte}
+                                                  
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.55     \caption
+                 {Réprésentation temporelle d'un signal rectangulaire ou << ...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+! LaTeX Error: \begin{document} ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.57 \end{figure}
+                 
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+! Missing $ inserted.
+<inserted text> 
+                $
+l.57 \end{figure}
+                 
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
+! Missing } inserted.
+<inserted text> 
+                }
+l.57 \end{figure}
+                 
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 52--58
+[]
+ []
+
+
+Overfull \hbox (316.412pt too wide) in paragraph at lines 52--58
+[]\OML/cmm/m/it/10 orte[][]$ 
+ []
+
+! Undefined control sequence.
+l.60 \subsection
+                {Impulsion de Dirac}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no I in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no D in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 60--61
+[]
+ []
+
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no D in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no 0 in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no C in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+! Undefined control sequence.
+l.62 ... noté $\delta(t-t_0)$ (avec $t \in \mathbb
+                                                  {R}$) est une impulsion si...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no ) in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no o in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+! Undefined control sequence.
+<write> ...Reference `sig_dirac' on page \thepage 
+                                                  \space undefined\on@line .
+l.62 ...ion est donnée à la figure \ref{sig_dirac}
+                                                  .
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `sig_dirac' on page  undefined on input line 62.
+
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 62--63
+[]
+ []
+
+
+Overfull \hbox (20.10072pt too wide) in paragraph at lines 62--63
+\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@
+ []
+
+
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+ []
+
+
+Overfull \hbox (13.05551pt too wide) in paragraph at lines 62--63
+\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2
+ []
+
+
+Overfull \hbox (7.67015pt too wide) in paragraph at lines 62--63
+\OML/cmm/m/it/10 R$
+ []
+
+
+Overfull \hbox (8.09724pt too wide) in paragraph at lines 62--63
+\OML/cmm/m/it/10 t[]$
+ []
+
+
+Overfull \hbox (10.86105pt too wide) in paragraph at lines 62--63
+[] 
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.64 \begin{figure}
+                   [h]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.66     \includegraphics
+                         [width=6cm]{img/sig_dirac}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no = in font nullfont!
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+Missing character: There is no c in font nullfont!
+Missing character: There is no m in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
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+Missing character: There is no / in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.66     \includegraphics[width=6cm]{img/sig_
+                                             dirac}
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Extra }, or forgotten $.
+l.66 ...\includegraphics[width=6cm]{img/sig_dirac}
+                                                  
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.67     \caption
+                 {Réprésentation temporelle d'une impulsion de Dirac.}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+! LaTeX Error: \begin{document} ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.69 \end{figure}
+                 
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+! Missing $ inserted.
+<inserted text> 
+                $
+l.69 \end{figure}
+                 
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
+! Missing } inserted.
+<inserted text> 
+                }
+l.69 \end{figure}
+                 
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 64--70
+[]
+ []
+
+
+Overfull \hbox (238.12086pt too wide) in paragraph at lines 64--70
+[]\OML/cmm/m/it/10 irac[][]$ 
+ []
+
+! Undefined control sequence.
+l.71 \subsection
+                {<< Peigne >> de Dirac}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no < in font nullfont!
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+Missing character: There is no c in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ù in font nullfont!
+! Undefined control sequence.
+l.72 ...t) = \Sigma \delta (t-nT)$ où $t\in\mathbb
+                                                  {R}$ correspond à un pério...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
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+
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+
+
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+ []
+
+
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+ []
+
+
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+
+
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+ []
+
+
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+ []
+
+
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+ []
+
+
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+ []
+
+
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+ []
+
+
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+ []
+
+
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+ []
+
+
+Overfull \hbox (14.16663pt too wide) in paragraph at lines 71--73
+\OML/cmm/m/it/10 t >
+ []
+
+
+Overfull \hbox (5.00002pt too wide) in paragraph at lines 71--73
+\OT1/cmr/m/n/10 0$ 
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.74 \begin{figure}
+                   [h]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.76     \includegraphics
+                         [width=8cm]{img/sig_train_dirac}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no = in font nullfont!
+Missing character: There is no 8 in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ] in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.76     \includegraphics[width=8cm]{img/sig_
+                                             train_dirac}
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Extra }, or forgotten $.
+l.76 ...degraphics[width=8cm]{img/sig_train_dirac}
+                                                  
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.77     \caption
+                 {Réprésentation temporelle d'un peigne de Dirac.}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+! LaTeX Error: \begin{document} ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.79 \end{figure}
+                 
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
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+! Missing $ inserted.
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+l.79 \end{figure}
+                 
+I've inserted something that you may have forgotten.
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+                 
+I've inserted something that you may have forgotten.
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+
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+ []
+
+! Undefined control sequence.
+l.82 \subsection
+                {Sinus cardinal}
+The control sequence at the end of the top line
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+                                                  {R}\backslash \lbrace 0 \r...
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+                                                  .
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+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.86 \begin{figure}
+                   [h]
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+! Undefined control sequence.
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+                         [width=8cm]{img/sinc}
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+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.89     \caption
+                 {Réprésentation temporelle du sinus cardinal : $y=\frac{\si...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
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+
+! LaTeX Error: \begin{document} ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.91 \end{figure}
+                 
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
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+
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+l.94 \section
+             {Transformée de Fourier}
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+
+! Undefined control sequence.
+l.96 \subsection
+                {Rappels sur la décomposition en série de Fourier des signau...
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+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no S in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no ç in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no ) in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no P in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no j in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no < in font nullfont!
+Missing character: There is no < in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no > in font nullfont!
+Missing character: There is no > in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no ) in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no C in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no ) in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no l in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no f in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ï in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
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+Missing character: There is no u in font nullfont!
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+Missing character: There is no o in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no l in font nullfont!
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+Missing character: There is no d in font nullfont!
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+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no l in font nullfont!
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+Missing character: There is no ê in font nullfont!
+Missing character: There is no m in font nullfont!
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+Missing character: There is no é in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Overfull \hbox (20.0pt too wide) in paragraph at lines 104--105
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+Overfull \hbox (9.38199pt too wide) in paragraph at lines 104--105
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+
+! Undefined control sequence.
+l.107 \subsection
+                 {Définition de la Transformée de Fourier}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 107--108
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+
+! Undefined control sequence.
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+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.112 ...{-\infty}^{+\infty}x(t)e^{-i2\pi f t} \ud
+                                                   t}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
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+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 115--116
+[]
+ []
+
+
+Overfull \hbox (25.78825pt too wide) in paragraph at lines 115--116
+\OML/cmm/m/it/10 TF[]$
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 117--117
+[] 
+ []
+
+! Undefined control sequence.
+l.118  \boxeq
+             {x(t)=TF^{-1}(X(f))=\int_{-\infty}^{+\infty}X(f)e^{i2\pi f t} \...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.118 ..._{-\infty}^{+\infty}X(f)e^{i2\pi f t} \ud
+                                                   f}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (179.8298pt too wide) detected at line 119
+[][]
+ []
+
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+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no m in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
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+Missing character: There is no o in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no é in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no â in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no p in font nullfont!
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+Missing character: There is no l in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no : in font nullfont!
+Missing character: There is no H in font nullfont!
+Missing character: There is no z in font nullfont!
+Missing character: There is no ) in font nullfont!
+Missing character: There is no . in font nullfont!
+
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+ []
+
+
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+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$
+ []
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+\OT1/cmr/bx/n/10 mod-
+ []
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+
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+\OT1/cmr/bx/n/10 ule
+ []
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+
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+\OT1/cmr/bx/n/10 ar-
+ []
+
+
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+\OT1/cmr/bx/n/10 gu-
+ []
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+
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+\OT1/cmr/bx/n/10 ment
+ []
+
+
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+\OT1/cmr/bx/n/10 tre
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+
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+\OT1/cmr/bx/n/10 d'amplitude
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+
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+ []
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+
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+[]\OT1/cmr/bx/n/10 Remarques
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+\OT1/cmr/bx/n/10 im-
+ []
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+
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+\OT1/cmr/bx/n/10 tantes: 
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+ []
+
+
+! LaTeX Error: The font size command \normalsize is not defined:
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+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.127 \end{itemize}
+                   
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}
+! Undefined control sequence.
+<write> \newlabel{ex_sig}{{}{\thepage 
+                                      }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+\newlabel{ex_sig}{{}{}}
+! Undefined control sequence.
+<write> \newlabel{sig_porte}{{}{\thepage 
+                                         }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+\newlabel{sig_porte}{{}{}}
+! Undefined control sequence.
+<write> \newlabel{sig_dirac}{{}{\thepage 
+                                         }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+\newlabel{sig_dirac}{{}{}}
+! Undefined control sequence.
+<write> \newlabel{sig_porte}{{}{\thepage 
+                                         }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+\newlabel{sig_porte}{{}{}}
+! Undefined control sequence.
+<write> \newlabel{sin_card}{{}{\thepage 
+                                        }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
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+! Undefined control sequence.
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+                                           }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+\newlabel{DecompFour}{{1}{}}
+! Undefined control sequence.
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+                                        }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+\newlabel{fourier}{{2}{}}
+! Undefined control sequence.
+<write> \newlabel{fourier_inv}{{3}{\thepage 
+                                            }}
+l.127 \end{itemize}
+                   
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
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+! Undefined control sequence.
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+                 {Le produit de convolution}
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+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.139 ...) = \int_{-\infty}^{+\infty}x(u)y(t-u)\ud
+                                                   u = \int_{-\infty}^{+\inf...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.139 ...u = \int_{-\infty}^{+\infty}x(t-u)y(u)\ud
+                                                   u}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
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+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no : in font nullfont!
+Missing character: There is no   in font nullfont!
+! Undefined control sequence.
+<recently read> \url 
+                     
+l.175 \url
+          {http://www.jhu.edu/~signals/convolve/index.html} \\
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no h in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no : in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no j in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no l in font nullfont!
+! Undefined control sequence.
+<recently read> \url 
+                     
+l.176 \url
+          {http://www.jhu.edu/~signals/discreteconv2/index.html}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no h in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no : in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no j in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no 2 in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no l in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 174--177
+[]
+ []
+
+! Undefined control sequence.
+l.182 \subsection
+                 {Propriétés de la transformée de Fourier}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no P in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 182--183
+[]
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 184--184
+[] 
+ []
+
+Missing character: There is no é in font cmbx10!
+Missing character: There is no é in font cmbx10!
+Missing character: There is no   in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (34.49286pt too wide) in paragraph at lines 185--186
+\OT1/cmr/bx/n/10 Linarit
+ []
+
+
+Overfull \hbox (11.65967pt too wide) in paragraph at lines 185--186
+\OT1/cmr/bx/n/10 de
+ []
+
+
+Overfull \hbox (8.78465pt too wide) in paragraph at lines 185--186
+\OT1/cmr/bx/n/10 la
+ []
+
+
+Overfull \hbox (15.23602pt too wide) in paragraph at lines 185--186
+\OT1/cmr/bx/n/10 TF 
+ []
+
+! Undefined control sequence.
+l.188  \boxeq
+             {a x(t)+b y(t) \Leftrightarrow a X(f)+b Y(f)}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (137.50684pt too wide) detected at line 189
+[]
+ []
+
+Missing character: There is no   in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (39.11092pt too wide) in paragraph at lines 191--192
+\OT1/cmr/bx/n/10 Produit
+ []
+
+
+Overfull \hbox (11.65967pt too wide) in paragraph at lines 191--192
+\OT1/cmr/bx/n/10 de
+ []
+
+
+Overfull \hbox (21.08322pt too wide) in paragraph at lines 191--192
+\OT1/cmr/bx/n/10 con-
+ []
+
+
+Overfull \hbox (15.33325pt too wide) in paragraph at lines 191--192
+\OT1/cmr/bx/n/10 vo-
+ []
+
+
+Overfull \hbox (13.4166pt too wide) in paragraph at lines 191--192
+\OT1/cmr/bx/n/10 lu-
+ []
+
+
+Overfull \hbox (19.80545pt too wide) in paragraph at lines 191--192
+\OT1/cmr/bx/n/10 tion 
+ []
+
+! Undefined control sequence.
+l.194  \boxeq
+             {x(t).y(t) \Leftrightarrow   X(f) \star Y(f)}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (106.12961pt too wide) detected at line 195
+[]
+ []
+
+! Undefined control sequence.
+l.198  \boxeq
+             {x(t) \star y(t)  \Leftrightarrow  X(f) . Y(f)}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (106.12961pt too wide) detected at line 199
+[]
+ []
+
+Missing character: There is no U in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (35.08315pt too wide) in paragraph at lines 203--204
+\OT1/cmr/bx/n/10 Retard 
+ []
+
+! Undefined control sequence.
+l.206  \boxeq
+             {x(t-t_0)   \Leftrightarrow  X(f) e^{-2i\pi f t_0}}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (110.55542pt too wide) detected at line 207
+[]
+ []
+
+! Undefined control sequence.
+l.210  \boxeq
+             {x(t) \cdot e^{2\pi f_0 t} \Leftrightarrow  X(f-f_0)}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (109.19826pt too wide) detected at line 211
+[]
+ []
+
+Missing character: There is no U in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no é in font cmbx10!
+Missing character: There is no   in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (63.08994pt too wide) in paragraph at lines 215--216
+\OT1/cmr/bx/n/10 Changement
+ []
+
+
+Overfull \hbox (37.69426pt too wide) in paragraph at lines 215--216
+\OT1/cmr/bx/n/10 d'chelle 
+ []
+
+! Undefined control sequence.
+l.218 \boxeq
+            {x(at) \Leftrightarrow \frac{1}{\vert a \vert} X \left(\frac{f}{...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (85.01756pt too wide) detected at line 219
+[]
+ []
+
+Missing character: There is no é in font cmbx10!
+Missing character: There is no S in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+! Undefined control sequence.
+l.221 ...fonctions temporelles avec $t \in \mathbb
+                                                  {R}$ telles que $y(t)=x(at...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+! Undefined control sequence.
+l.221 ...lles que $y(t)=x(at)$ avec $a \in \mathbb
+                                                  {R}$. Leur transformées de...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no O in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (72.87598pt too wide) in paragraph at lines 221--222
+[]\OT1/cmr/bx/n/10 Dmonstration.
+ []
+
+
+Overfull \hbox (17.10419pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$
+ []
+
+
+Overfull \hbox (16.65051pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$
+ []
+
+
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+\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2
+ []
+
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+ []
+
+
+Overfull \hbox (27.20602pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) =
+ []
+
+
+Overfull \hbox (22.39008pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 at\OT1/cmr/m/n/10 )$
+ []
+
+
+Overfull \hbox (14.73029pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 a \OMS/cmsy/m/n/10 2
+ []
+
+
+Overfull \hbox (7.67015pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 R$
+ []
+
+
+Overfull \hbox (22.81949pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$
+ []
+
+
+Overfull \hbox (21.77785pt too wide) in paragraph at lines 221--222
+\OML/cmm/m/it/10 Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$
+ []
+
+! Undefined control sequence.
+l.224 ...-\infty}^{+\infty} y(t)e^{-2j\pi f t} \ud
+                                                   t\\
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.225 ...\infty}^{+\infty} x(at)e^{-2j\pi f t} \ud
+                                                   t \label{dem_ft_ech}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (113.57301pt too wide) in alignment at lines 223--226
+ [][][] []
+ []
+
+Missing character: There is no E in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no - in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no - in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+! Undefined control sequence.
+<recently read> \ud 
+                    
+l.228 ... variable $u=at$, c'est-à-dire aussi $\ud
+                                                   u=a \ud t$, qui revient i...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.228 ...e $u=at$, c'est-à-dire aussi $\ud u=a \ud
+                                                   t$, qui revient ici à un ...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no , in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+! Undefined control sequence.
+<write> ...eference `dem_ft_ech' on page \thepage 
+                                                  \space undefined\on@line .
+l.228 ...nt d'échelle. L'équation \ref{dem_ft_ech}
+                                                   devient :
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `dem_ft_ech' on page  undefined on input line 228.
+
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (16.28009pt too wide) in paragraph at lines 228--229
+\OML/cmm/m/it/10 u \OT1/cmr/m/n/10 =
+ []
+
+
+Overfull \hbox (8.897pt too wide) in paragraph at lines 228--229
+\OML/cmm/m/it/10 at$
+ []
+
+
+Overfull \hbox (16.28009pt too wide) in paragraph at lines 228--229
+\OML/cmm/m/it/10 u \OT1/cmr/m/n/10 =
+ []
+
+
+Overfull \hbox (8.897pt too wide) in paragraph at lines 228--229
+\OML/cmm/m/it/10 at$
+ []
+
+
+Overfull \hbox (10.86105pt too wide) in paragraph at lines 228--229
+[]
+ []
+
+! Undefined control sequence.
+<argument> \ud 
+               u
+l.231 ...)e^{-\frac{2j\pi f u}{a}} \frac{\ud u}{a}
+                                                  \\
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.232 ...infty} x(u) e^{-2j\pi f \frac{f}{a}u} \ud
+                                                   u\\
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (135.5398pt too wide) in alignment at lines 230--234
+ [][][] []
+ []
+
+Missing character: There is no C in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no ) in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no P in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ï in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ù in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+! Undefined control sequence.
+<write> ...Reference `sinus_ech' on page \thepage 
+                                                  \space undefined\on@line .
+l.236 ...ù $f_1=a f_0$ (cf. figure \ref{sinus_ech}
+                                                  ). Le facteur supplémentai...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `sinus_ech' on page  undefined on input line 236.
+
+Missing character: There is no ) in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
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+Missing character: There is no n in font nullfont!
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+ []
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+Overfull \hbox (19.9375pt too wide) in paragraph at lines 236--237
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+ []
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+[]
+ []
+
+
+Overfull \hbox (11.48764pt too wide) in paragraph at lines 236--237
+[]$
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.238 \begin{figure}
+                    [tp]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.240     \subfigure
+                    [Représentation temporelle]{\includegraphics[width=8cm]{...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no R in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
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+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.240 ...présentation temporelle]{\includegraphics
+                                                  [width=8cm]{img/sinus_ech}}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.240 ...e]{\includegraphics[width=8cm]{img/sinus_
+                                                  ech}}
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Extra }, or forgotten $.
+l.240 ...includegraphics[width=8cm]{img/sinus_ech}
+                                                  }
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+! Extra }, or forgotten $.
+l.240 ...ncludegraphics[width=8cm]{img/sinus_ech}}
+                                                  
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+! Undefined control sequence.
+l.241     \subfigure
+                    [Représentation fréquentielle]{\includegraphics[width=8c...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.241 ...sentation fréquentielle]{\includegraphics
+                                                  [width=8cm]{img/sinus_ech_...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.242     \caption
+                  {Exemple d'application d'un facteur d'échelle $a=2$ sur un...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+! Missing } inserted.
+<inserted text> 
+                }
+l.242 ...le d'application d'un facteur d'échelle $
+                                                  a=2$ sur un signal sinusoï...
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
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+Missing character: There is no ) in font nullfont!
+Missing character: There is no é in font cmr10!
+Missing character: There is no f in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.242 ...signal sinusoïdal $x(t)$ de fréquence $f_
+                                                  0=50\ \mathrm{Hz}$ tel que...
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+
+! LaTeX Error: \begin{itemize} on input line 184 ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.244 \end{figure}
+                  
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+! Missing } inserted.
+<inserted text> 
+                }
+l.244 \end{figure}
+                  
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+
+Overfull \hbox (620.39685pt too wide) in paragraph at lines 238--245
+[]\OML/cmm/m/it/10 ch\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 Reprsentationfrquentiell
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+
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+
+
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+ []
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+
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+\OT1/cmr/bx/n/10 Thorme
+ []
+
+
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+\OT1/cmr/bx/n/10 de
+ []
+
+
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+ []
+
+
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+\OT1/cmr/bx/n/10 se-
+ []
+
+
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+ []
+
+
+Overfull \hbox (7.95831pt too wide) in paragraph at lines 246--248
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+ []
+
+! Undefined control sequence.
+l.250  \boxeq
+             {E = \int_{-\infty}^{+\infty} \vert x(t) \vert^2 \ud t   =   \i...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.250 ...-\infty}^{+\infty} \vert x(t) \vert^2 \ud
+                                                   t   =   \int_{-\infty}^{+...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.250 ...nfty}^{+\infty}    \vert X(f) \vert^2 \ud
+                                                   f}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (156.77081pt too wide) detected at line 251
+[]
+ []
+
+! Undefined control sequence.
+l.256 \subsection
+                 {Transformées de Fourier des signaux courants}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no T in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
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+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 256--257
+[]
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 258--258
+[] 
+ []
+
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+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
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+Missing character: There is no l in font nullfont!
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+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+
+Overfull \hbox (28.98955pt too wide) in paragraph at lines 259--260
+\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/c
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+ []
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+Missing character: There is no l in font nullfont!
+
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+ []
+
+
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+,
+ []
+
+
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+
+
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+ []
+
+
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+ []
+
+! Undefined control sequence.
+l.264 \section
+              {Evaluation de la similitude entre deux signaux}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 264--265
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+
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+[]
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+
+
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+! Undefined control sequence.
+l.274 \subsection
+                 {Les fonctions d'inter-corrélation et d'auto-corrélation}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
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+
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+ []
+
+! Undefined control sequence.
+l.279 ...t_{-\infty}^{+\infty}x^*(t) y(t+\tau) \ud
+                                                   t
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
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+ []
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+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 290--291
+[]
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 292--292
+[] 
+ []
+
+! Undefined control sequence.
+l.293 ...\infty}^{+\infty} \vert x(t) \vert ^2 \ud
+                                                   t = E
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (116.01378pt too wide) detected at line 294
+[]\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (0) = [][] \OMS/cmsy/m/n/10 j\OML/cmm/m/i
+t/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j[]\
+OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = \OML/cmm/m/it/10 E
+ []
+
+! Undefined control sequence.
+l.297 \subsection
+                 {La fonction de densité spectrale d'énergie}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 297--298
+[]
+ []
+
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no E in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 299--300
+[]
+ []
+
+
+Overfull \hbox (15.05211pt too wide) in paragraph at lines 299--300
+\OML/cmm/m/it/10 TF$
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 302--302
+[] 
+ []
+
+
+Overfull \hbox (133.22224pt too wide) detected at line 304
+[]\OT1/cmr/m/n/10 ^^H[](\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 
+TF\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^\\OT
+1/cmr/m/n/10 )) = \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm
+/m/it/10 f\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j[]
+ []
+
+! Undefined control sequence.
+l.307 \section
+              {Les signaux numériques}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 307--308
+[]
+ []
+
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font cmbx10!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+! Undefined control sequence.
+l.309 ...une suite d'instants $t_n$, $n\in \mathbb
+                                                  {Z}$. Dans la suite nous n...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no . in font nullfont!
+Missing character: There is no D in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no , in font nullfont!
+! Undefined control sequence.
+l.309 ... que l'échantillonnage est régulier, \cad
+                                                   que $t_n$ = $nT$.
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no = in font nullfont!
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 309--310
+[]
+ []
+
+
+Overfull \hbox (26.99289pt too wide) in paragraph at lines 309--310
+\OT1/cmr/bx/n/10 chan-
+ []
+
+
+Overfull \hbox (14.69437pt too wide) in paragraph at lines 309--310
+\OT1/cmr/bx/n/10 til-
+ []
+
+
+Overfull \hbox (19.16656pt too wide) in paragraph at lines 309--310
+\OT1/cmr/bx/n/10 lon-
+ []
+
+
+Overfull \hbox (22.99986pt too wide) in paragraph at lines 309--310
+\OT1/cmr/bx/n/10 nage
+ []
+
+
+Overfull \hbox (9.05444pt too wide) in paragraph at lines 309--310
+\OML/cmm/m/it/10 t[]$
+ []
+
+
+Overfull \hbox (15.44675pt too wide) in paragraph at lines 309--310
+\OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 2
+ []
+
+
+Overfull \hbox (7.54167pt too wide) in paragraph at lines 309--310
+\OML/cmm/m/it/10 Z$
+ []
+
+
+Overfull \hbox (9.05444pt too wide) in paragraph at lines 309--310
+\OML/cmm/m/it/10 t[]$
+ []
+
+
+Overfull \hbox (13.235pt too wide) in paragraph at lines 309--310
+\OML/cmm/m/it/10 nT$ 
+ []
+
+Missing character: There is no A in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no j in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no E in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no y in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no . in font nullfont!
+Missing character: There is no P in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no ê in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
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+Missing character: There is no a in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no q in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no s in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no e in font nullfont!
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+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
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+Missing character: There is no t in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 311--312
+[]
+ []
+
+! Undefined control sequence.
+l.314 \subsection
+                 {Définition de l'échantillonnage}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no D in font nullfont!
+Missing character: There is no é in font nullfont!
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+Missing character: There is no i in font nullfont!
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+Missing character: There is no é in font nullfont!
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+Missing character: There is no n in font nullfont!
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+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 314--315
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+Missing character: There is no é in font nullfont!
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+! Undefined control sequence.
+l.316 ...analogique) peut être échantillonné (\cad
+                                                   discrétisé) à une \textbf...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no d in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
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+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
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+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no ù in font nullfont!
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+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
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+Missing character: There is no n in font nullfont!
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+
+! Undefined control sequence.
+l.331 \subsection
+                 {Définition et propriétés de la transformée de Fourier d'un...
+The control sequence at the end of the top line
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+/it/10 e[]
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+
+Missing character: There is no L in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no è in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 339--340
+[]
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 341--341
+[] 
+ []
+
+! Undefined control sequence.
+l.342 ...{1}{2}}^{\frac{1}{2}}X(f)e^{2j\pi n f}\ud
+                                                   f
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+Overfull \hbox (108.80627pt too wide) detected at line 343
+[]\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) = [][]
+ \OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )\OML/cmm
+/m/it/10 e[]f
+ []
+
+Missing character: There is no D in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ê in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no ç in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no T in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 345--346
+[]
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 347--347
+[] 
+ []
+
+Missing character: There is no R in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (23.38432pt too wide) in paragraph at lines 348--349
+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ^^@
+ []
+
+
+Overfull \hbox (27.1551pt too wide) in paragraph at lines 348--349
+\OML/cmm/m/it/10 n[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 ,
+ []
+
+
+Overfull \hbox (60.38318pt too wide) in paragraph at lines 348--349
+\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )\OML/cmm/
+m/it/10 e[]$ 
+ []
+
+Missing character: There is no P in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (24.49544pt too wide) in paragraph at lines 349--351
+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) \OML/cmm
+/m/it/10 ?
+ []
+
+
+Overfull \hbox (31.81947pt too wide) in paragraph at lines 349--351
+\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) \OMS/cms
+y/m/n/10 ,
+ []
+
+
+Overfull \hbox (47.37512pt too wide) in paragraph at lines 349--351
+\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )\OML/cmm/
+m/it/10 :Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$ 
+ []
+
+
+Overfull \hbox (54.09268pt too wide) in paragraph at lines 349--351
+\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 )\OML/cmm/
+m/it/10 :y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10
+ ,
+ []
+
+
+Overfull \hbox (27.8195pt too wide) in paragraph at lines 349--351
+\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 ) \OML/cmm
+/m/it/10 ?
+ []
+
+
+Overfull \hbox (21.77785pt too wide) in paragraph at lines 349--351
+\OML/cmm/m/it/10 Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$ 
+ []
+
+Missing character: There is no a in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no , in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 353--354
+[] 
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 355--355
+[] 
+ []
+
+
+Overfull \hbox (205.13445pt too wide) detected at line 357
+[]\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) \OML/c
+mm/m/it/10 ? y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 ) = [] \OML/cm
+m/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 k\OT1/cmr/m/n/10 )\OML/cmm/m/it/10
+ y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 k\O
+T1/cmr/m/n/10 ) = [] \OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n \OMS
+/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 k\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 y\OT1/cmr/
+m/n/10 (\OML/cmm/m/it/10 k\OT1/cmr/m/n/10 )
+ []
+
+! Undefined control sequence.
+l.360 \subsection
+                 {Conséquences de l'échantillonnage du point de vue fréquent...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no C in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 360--361
+[]
+ []
+
+Missing character: There is no E in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no x in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no F in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no : in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 362--363
+[]
+ []
+
+
+Overfull \hbox (31.95381pt too wide) in paragraph at lines 362--363
+\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) =
+ []
+
+
+Overfull \hbox (58.70918pt too wide) in paragraph at lines 362--363
+\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\OML/cm
+m/m/it/10 : [] ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@
+ []
+
+
+Overfull \hbox (20.02911pt too wide) in paragraph at lines 362--363
+\OML/cmm/m/it/10 nT[]\OT1/cmr/m/n/10 )$
+ []
+
+
+Overfull \hbox (15.05211pt too wide) in paragraph at lines 362--363
+\OML/cmm/m/it/10 TF$
+ []
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 364--364
+[] 
+ []
+
+
+Overfull \hbox (111.35538pt too wide) detected at line 366
+\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 ) = [] [
+] \OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f \OMS/cmsy/m/n/10 ^^@ 
+[]\OT1/cmr/m/n/10 )
+ []
+
+Missing character: There is no O in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no à in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no ( in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no ) in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+! Undefined control sequence.
+<write> ...: Reference `sch_ech' on page \thepage 
+                                                  \space undefined\on@line .
+l.368 ...mme le schématise la figure \ref{sch_ech}
+                                                  .
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `sch_ech' on page  undefined on input line 368.
+
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 368--369
+[]
+ []
+
+
+Overfull \hbox (21.28018pt too wide) in paragraph at lines 368--369
+\OML/cmm/m/it/10 F[] \OT1/cmr/m/n/10 =
+ []
+
+
+Overfull \hbox (17.23268pt too wide) in paragraph at lines 368--369
+\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =T$
+ []
+
+
+Overfull \hbox (16.52777pt too wide) in paragraph at lines 368--369
+\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 =
+ []
+
+
+Overfull \hbox (18.23502pt too wide) in paragraph at lines 368--369
+\OML/cmm/m/it/10 n=T$
+ []
+
+
+Overfull \hbox (6.00235pt too wide) in paragraph at lines 368--369
+\OML/cmm/m/it/10 n$
+ []
+
+
+Overfull \hbox (10.86105pt too wide) in paragraph at lines 368--369
+[] 
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.370 \begin{figure}
+                    [h]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.372     \includegraphics
+                          [width=12cm]{img/cons_ech_01}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no = in font nullfont!
+Missing character: There is no 1 in font nullfont!
+Missing character: There is no 2 in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ] in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no s in font nullfont!
+! Missing $ inserted.
+<inserted text> 
+                $
+l.372     \includegraphics[width=12cm]{img/cons_
+                                                ech_01}
+I've inserted a begin-math/end-math symbol since I think
+you left one out. Proceed, with fingers crossed.
+
+! Extra }, or forgotten $.
+l.372 ...ludegraphics[width=12cm]{img/cons_ech_01}
+                                                  
+I've deleted a group-closing symbol because it seems to be
+spurious, as in `$x}$'. But perhaps the } is legitimate and
+you forgot something else, as in `\hbox{$x}'. In such cases
+the way to recover is to insert both the forgotten and the
+deleted material, e.g., by typing `I$}'.
+
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.373     \caption
+                  {Lien entre la fréquence d'échantillonnage d'un signal et ...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+! LaTeX Error: \begin{document} ended by \end{figure}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.375 \end{figure}
+                  
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+! Missing $ inserted.
+<inserted text> 
+                $
+l.375 \end{figure}
+                  
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
+Missing character: There is no é in font cmr10!
+! Missing } inserted.
+<inserted text> 
+                }
+l.375 \end{figure}
+                  
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 370--376
+[]
+ []
+
+
+Overfull \hbox (371.41843pt too wide) in paragraph at lines 370--376
+[]\OML/cmm/m/it/10 ch[]\OT1/cmr/m/n/10 1[][]$ 
+ []
+
+Missing character: There is no I in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no b in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no è in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no ' in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no « in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no   in font nullfont!
+Missing character: There is no » in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no q in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no v in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no s in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no a in font nullfont!
+Missing character: There is no f in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no u in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+! Undefined control sequence.
+<write> ...Reference `sch_repli' on page \thepage 
+                                                  \space undefined\on@line .
+l.378 ...comme le montre la figure \ref{sch_repli}
+                                                  .
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+LaTeX Warning: Reference `sch_repli' on page  undefined on input line 378.
+
+Missing character: There is no . in font nullfont!
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 378--379
+[]
+ []
+
+
+Overfull \hbox (10.72467pt too wide) in paragraph at lines 378--379
+\OML/cmm/m/it/10 F[]$
+ []
+
+
+Overfull \hbox (10.86105pt too wide) in paragraph at lines 378--379
+[] 
+ []
+
+
+! LaTeX Error: Environment figure undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.382 \begin{figure}
+                    [h]
+Your command was ignored.
+Type  I <command> <return>  to replace it with another command,
+or  <return>  to continue without it.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.384      \includegraphics
+                           [width=12cm]{img/repliement}
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no w in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no = in font nullfont!
+Missing character: There is no 1 in font nullfont!
+Missing character: There is no 2 in font nullfont!
+Missing character: There is no c in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no ] in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no g in font nullfont!
+Missing character: There is no / in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+
+! LaTeX Error: \caption outside float.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.385      \caption
+                   {Phénomène de repliement.}
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no P in font nullfont!
+Missing character: There is no h in font nullfont!
+Missing character: There is no é in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no o in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no è in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no d in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no r in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no p in font nullfont!
+Missing character: There is no l in font nullfont!
+Missing character: There is no i in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no m in font nullfont!
+Missing character: There is no e in font nullfont!
+Missing character: There is no n in font nullfont!
+Missing character: There is no t in font nullfont!
+Missing character: There is no . in font nullfont!
+
+! LaTeX Error: \begin{document} ended by \end{figure}.
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diff --git a/IntroMethFreq.tex b/IntroMethFreq.tex
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@@ -0,0 +1,380 @@
+\section{Le produit de convolution}
+
+On appelle produit de convolution \textbf{convolution} de $x(t)$ par $y(t)$ l'opération notée $x(t) \star y(t)$ et définie par :
+
+\begin{equation}\label{conv}
+ \boxeq{x(t)\star y(t) = \int_{-\infty}^{+\infty}x(u)y(t-u)\ud u = \int_{-\infty}^{+\infty}x(t-u)y(u)\ud u}
+\end{equation}
+
+L'impulsion de Dirac est l'élément neutre de la convolution. En effet :
+
+\begin{equation}
+x(t) \star \delta (t) = x(t)
+\end{equation}
+
+Lorsque l'on convolue un signal $x(t)$ à un Dirac situé à un temps $t_0$, cela revient à retarder le signal $x(t)$ de $t_0$ :
+
+\begin{equation}
+x(t) \star \delta (t-t_0) = x(t-t_0)
+\end{equation}
+
+Par ailleurs, si l'on multiplie un signal $x(t)$ par un Dirac situé à un temps $t_0$, cela revient à connaître la valeur que prend $x(t)$ en $t_0$ (comme si l'on relevait l'ordonnée d'un point particulier d'une courbe)
+
+\begin{equation}
+x(t) \cdot \delta  (t-t_0) = x(t_0)
+\end{equation}
+
+De même, lorsque l'on convolue un signal $x(t)$ à un peigne de Dirac (de période $T$), cela revient à << périodiser >>  le signal $x(t)$ tous les $nT$ : on retarde le signal $x(t)$ de $T$, de $2T$, de $3T$, etc...
+
+\begin{equation}
+x(t) \star \Psi_T (t)  =  \Sigma x(t-nT)
+\end{equation}
+
+De façon plus générale, la convolution telle qu'elle est définie par sa formule mathématique, revient à retourner temporellement un des deux signaux (par exemple $x(t)$)  puis à le déplacer sur tout l'axe du temps  et à sommer toutes les multiplications de ce signal au deuxième signal $y(t)$.
+
+Pour des exemples animé de l'effet du produit de convolution (très pédagogique !), voir : \\
+\url{http://www.jhu.edu/~signals/convolve/index.html} \\
+\url{http://www.jhu.edu/~signals/discreteconv2/index.html}
+
+%Retournement temporel $t$ est un instant qui varie du signal $x$      dans le temps $u$ de  $-$  à  $+$.
+%On fait « glisser » le signal $x(-u)$ selon l'axe du temps.
+%On multiplie $y(u)$ par $x(t-u)$  pour chaque position de $t$ sur l'axe du temps et on somme tous les produits effectués.
+
+
+\section{Rappels sur la décomposition en série de Fourier des signaux périodiques}
+
+Pour pouvoir analyser les signaux quelque soit leur nature, il est intéressant de chercher des transformations qui mettent en évidence les particularités de leur contenu temporel mais aussi fréquentiel. Joseph Fourier a démontré que tout signal \textbf{périodique} peut être décomposé en une somme de sinus et de cosinus élémentaires telle que :
+
+\begin{equation}\label{DecompFour}
+ \boxed{x(t)=a_0 + \sum_{n=1}^{+\infty}a_n\cos(2\pi n f_0 t)+\sum_{n=1}^{+\infty}b_n\sin(2\pi n f_0 t)}
+\end{equation}
+
+où $f_0$ est appelée la fréquence \textbf{fondamentale}. Sauf effet particulier ou paradoxal, elle correspond à la hauteur principale perçue d'un son. Les autres composantes sont des fréquences supérieures à la fondamentale ($n>1$) et sont appelées \textbf{harmoniques} du signal. Par exemple, dans le cas d'un son émis par un instrument de musique, la fondamentale va correspondre à la note jouée et les autres composantes vont correspondre à la << série harmonique >> de cette note (octave, puis quinte, puis octave, puis tierce, etc...). Cette série correspond à des rapports de fréquences (fondamental $f_0$, octave $= 2*f_0$, octave $+$ quinte $= 3*f_0$, etc...). Les harmoniques ne vont pas directement contribuer à la perception de la hauteur mais plutôt à la richesse du timbre du son. C'est par exemple la présence d'harmoniques, entre autres, qui permet de différentier une sinusoïde à $f_0$ d'une note de violon dont la fondamentale est à la même fréquence.
+
+
+\section{La transformée de Fourier}
+
+\subsection{Définition de la transformée de Fourier}
+
+La transformée de Fourier est une généralisation de la décomposition précedente aux signaux non-périodiques. Soit $x(t)$ un signal quelconque, on note $X(f)$ ou $TF(x(t))$ sa transformée de Fourier telle que :
+
+\begin{equation}\label{fourier}
+ \boxeq{X(f)=TF(x(t))=\int_{-\infty}^{+\infty}x(t)e^{-i2\pi f t} \ud t}
+\end{equation}
+
+Inversement, on peut définir une transformée de Fourier inverse $TF^{-1}$ telle que :
+
+\begin{equation}\label{fourier_inv}
+ \boxeq{x(t)=TF^{-1}(X(f))=\int_{-\infty}^{+\infty}X(f)e^{i2\pi f t} \ud f}
+\end{equation}
+
+$X(f)$ est une fonction complexe même si $x(t)$ est réel. La transformée de Fourier contient donc une partie réelle et une partie imaginaire et est représentée facilement grâce à son \textbf{module} et à son \textbf{argument} : $\vert X(f) \vert$ est appelé \textbf{spectre d'amplitude} et $\mathrm{arg}(X(f))$ le \textbf{spectre de phase} du signal. La variable $f$ s'appelle la fréquence dont l'unité est le Hertz (en abrégé : Hz).\\
+
+\textbf{Remarques importantes:}
+\begin{itemize}
+ \item La représentation complète d'une transformée de Fourier nécessite 2 graphiques : le module et la phase, ou bien la partie réelle et le partie imaginaire.
+ \item Pour représenter les transformées de Fourier de signaux, il est communément utilisé l'échelle logarithmique. Pour un signal acoustique, par exemple, on calcule $20\log(\vert X(f) \vert / 2.10^{-5})$ et $\mathrm{arg}(X(f))$.\\
+\end{itemize}
+
+Ainsi, la transformée de Fourier est un opérateur mathématique qui permet d'analyser et de représenter un signal dans le domaine fréquentiel. La $TF$ ne modifie pas le signal mais permet seulement de l'observer selon différents points de vue (temporel ou fréquentiel). Il est important de retenir que $x(t)$ et $X(f)$ sont deux descriptions équivalentes du même signal. Ces deux fonctions contiennent la même information il s'agit juste de deux descriptions dans des domaines différents.
+
+$X(f)$ apporte des informations sur le système physique à l'origine du signal. Elle permet par exemple de différentier un son de trompette d'un son trombone, ou bien encore différentes ondes cérébrales, plus facilement qu'en observant le signal dans le domaine temporel. Le \textbf{contenu spectral} d'un signal est en effet assimilable à sa << carte d'identité >>.
+
+
+\subsection{Propriétés de la transformée de Fourier}
+
+\begin{itemize}
+ \item \textbf{Linéarité} :
+
+\begin{equation}
+ \boxeq{a x(t)+b y(t) \Leftrightarrow a X(f)+b Y(f)}
+\end{equation}
+
+\item \textbf{Produit de convolution} :
+
+\begin{equation}
+ \boxeq{x(t).y(t) \Leftrightarrow   X(f) \star Y(f)}
+\end{equation}
+
+\begin{equation}
+ \boxeq{x(t) \star y(t)  \Leftrightarrow  X(f) . Y(f)}
+\end{equation}
+
+Une multiplication dans un domaine correspond ainsi à un produit de convolution dans l'autre.\\
+
+\item \textbf{Retard} :
+
+\begin{equation}
+ \boxeq{x(t-t_0)   \Leftrightarrow  X(f) e^{-2i\pi f t_0}}
+\end{equation}
+
+\begin{equation}
+ \boxeq{x(t) \cdot e^{2\pi f_0 t} \Leftrightarrow  X(f-f_0)}
+\end{equation}
+
+Un retard temporel correspond ainsi à un déphasage au niveau fréquentiel, et inversement.\\
+
+\item \textbf{Changement d'échelle} :
+
+\begin{equation}
+\boxeq{x(at) \Leftrightarrow \frac{1}{\vert a \vert} X \left(\frac{f}{a} \right)}
+\end{equation}
+
+\textbf{Démonstration.} Soient $x(t)$ et $y(t)$ deux fonctions temporelles avec $t \in \mathbb{R}$ telles que $y(t)=x(at)$ avec $a \in \mathbb{R}$. Leur transformées de Fourier dans l'espace des fréquences sont notées respectivement $X(f)$ et $Y(f)$. On a alors :
+
+\begin{eqnarray}
+ Y(f) & = & \int_{-\infty}^{+\infty} y(t)e^{-2j\pi f t} \ud t\\
+ & = & \int_{-\infty}^{+\infty} x(at)e^{-2j\pi f t} \ud t \label{dem_ft_ech}
+\end{eqnarray}
+
+Effectuons le changement de variable $u=at$, c'est-à-dire aussi $\ud u=a \ud t$, qui revient ici à un changement d'échelle. L'équation \ref{dem_ft_ech} devient :
+
+\begin{eqnarray}
+ Y(f) & = & \int_{-\infty}^{+\infty} x(u)e^{-\frac{2j\pi f u}{a}} \frac{\ud u}{a}\\
+ & = & \frac{1}{\vert a \vert} \int_{-\infty}^{+\infty} x(u) e^{-2j\pi \frac{f}{a}u} \ud u\\
+ & = & \frac{1}{\vert a \vert}X\left(\frac{f}{a}\right) \label{dem_ft_ech_2}
+\end{eqnarray}
+
+Cette loi montre que lorsqu'on diminue l'échelle temporelle d'un signal ($a>1$), l'échelle fréquentielle augmente. Par exemple, si $x(t)$ est une sinusoïde de fréquence $f_0$ telle que $x(t)=\sin(2\pi f_0 t)$, alors $X(f)=\delta(f-f_0)$, $y(t)=\sin(2\pi a f_0 t)$ et $Y(f)=\frac{1}{\vert a \vert}\delta(f-f_1)$ où $f_1=a f_0$ (cf. figure \ref{sinus_ech}). Le facteur supplémentaire $\frac{1}{\vert a \vert}$ provient du principe de conservation d'énergie appliqué dans le domaine fréquentiel.\\
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Représentation temporelle]{\includegraphics[width=8cm]{img/sinus_ech}}
+    \subfigure[Représentation fréquentielle]{\includegraphics[width=8cm]{img/sinus_ech_ft}}
+    \caption{Exemple d'application d'un facteur d'échelle $a=2$ sur un signal sinusoïdal $x(t)$ de fréquence $f_0=50\ \mathrm{Hz}$ tel que $x(t)=\sin(2\pi f_0 t)$.}
+    \label{sinus_ech}
+\end{figure}
+
+\item \textbf{Théorème de Parseval} :\\
+Soit $E$ l'énergie du signal. On peut démontrer que :
+
+\begin{equation}
+ \boxeq{E = \int_{-\infty}^{+\infty} \vert x(t) \vert^2 \ud t   =   \int_{-\infty}^{+\infty}    \vert X(f) \vert^2 \ud f}
+\end{equation}
+
+\end{itemize}
+
+
+\subsection{Les transformées de Fourier de signaux courants}
+
+\begin{itemize}
+\item Dirac $\delta(t) \Leftrightarrow $ signal unité
+\item Signal porte  $\Leftrightarrow$ sinus cardinal
+\item $\sin(2\pi f_0 t)  \Leftrightarrow  \frac{1}{2}[\delta(f-f_0)  + \delta(f+f_0)]$
+\item la suite en cours...
+\end{itemize}
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La fonction de densité spectrale d'énergie est la $TF$ de la fonction d'autocorrélation vue au paragraphe \ref{intercorr}. Elle décrit la répartition de l'énergie en fréquence et on a :
+
+
+\begin{equation}\label{PSD}
+\Phi_{x}(f)= TF (\varphi_x(\tau))=\vert X(f) \vert ^2
+\end{equation}
+
+
+\section{L'échantillonnage}\label{signum}
+
+Les signaux numériques dérivent d 'un signal analogique par \textbf{échantillonnage}. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants $t_n$, $n\in \mathbb{Z}$. Dans la suite nous n'envisagerons que l'échantillonnage est régulier, \cad que $t_n$ = $nT$.
+
+Aujourd'hui l'intérêt grandissant pour la conversion de signaux analogiques en signaux numériques tient au fait qu'il devient de plus en plus simple d'appliquer des algorithmes d'analyse complexe avec l'aide des processeurs numériques. En sciences physiques par exemple, il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}\label{sampling}
+
+Le signal continu (analogique) peut être échantillonné (\cad discrétisé) à une \textbf{fréquence d'échantillonnage} $f_e$ inverse de la \textbf{période d'échantillonnage} $T_e$ telle que $f_e= 1/T_e$. Cet échantillonnage revient à multiplier le signal analogique par un peigne de Dirac de période $T_e$.
+
+Le signal échantillonné peut alors s'écrire :
+
+\begin{equation}
+ x_e(t)=x_a(t) \cdot \sum_{n=-\infty}^{+\infty} \delta (t-nT_e).
+\end{equation}
+
+$x_e(t)$ est donc nul partout sauf en $t= nTe$, $n$ variant de $-\infty$ à $+\infty$. C'est pourquoi par la suite, on ne simplifie cette notation en réduisant $x_e(t)$ non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets $x(n)$, où $n$ correspond au numéro de l'échantillon.
+
+\begin{equation}
+ x(n) = x_e ( n T_e )
+\end{equation}
+
+\subsection{La quantification}
+
+L'échantillonnage complet d'un signal nécessite de discrétiser également l'amplitude de ce signal. Quelque soit la résolution binaire donnée, cette opération suppose d'opérer un arrondi sur chaque échantillon de sorte que la valeur soit codable sur une échelle binaire. Les valeurs courantes de résolution de quantification en audio sont : 8, 16, 24 et 32 bits. De nombreux détails, exemples et schémas sont proposés en séance de cours.
+
+
+\section{La transformée de Fourier à temps discret}
+
+Dans le cas des signaux échantillonnés où le temps est discrétisé, il n'est plus nécessaire d'utiliser des intégrales continues pour sommer les valeurs de $x(t)$ sur tout l'axe des temps, puisqu'un signal échantillonné peut être assimilé à une suite contenant un nombre \textbf{fini} d'éléments . On peut donc définir la transformée de Fourier d'un signal numérique par :
+
+\begin{equation}\label{TFD}
+ X(f)=\sum_{n=-\infty}^{+\infty}x(n)e^{-2j\pi n f}
+\end{equation}
+
+La transformée de Fourier inverse se retrouve de la même manière que dans le cas continu :
+
+\begin{equation}\label{TFDinv}
+ x(n)=\int_{-\frac{1}{2}}^{\frac{1}{2}}X(f)e^{2j\pi n f}\ud f
+\end{equation}
+
+De la même façon qu'en continu, la TF présente certaines propriétés intéressantes :
+
+\begin{itemize}
+ \item \textbf{Retard} : $$x(n-n_0) \Leftrightarrow  X(f) e^{-2j\pi n_0 f}$$\\
+ \item \textbf{Produit de convolution} :
+ $$x(n)\star y(n) \Leftrightarrow X(f) . Y(f)$$\\
+ et \\
+ $$x(n).y(n) \Leftrightarrow X(f) \star Y(f)$$\\
+ \end{itemize}
+
+ avec,
+
+ \begin{equation}\label{conv_disc}
+  x(n) \star y(n)  =  \sum_{-\infty}^{+\infty}x(k) y(n-k) = \sum_{-\infty}^{+\infty}x(n-k) y(k)
+ \end{equation}
+
+
+\section{Le théorème de Shannon}
+
+La discrétisation temporelle du signal analogique détaillée à la section \ref{signum} n'est pas sans conséquences vis-à-vis des propriétés spectrales du signal échantillonné. Ainsi une mauvaise adéquation entre les fréquences contenues dans le signal et la fréquence d'échantillonnage peut être destructrice.
+
+En posant $x_e(t) = x_a(t).\sum \delta(t- nT_e)$ le signal échantillonné du signal analogique $x_a(t)$ et $X_e(f)$ sa transformée de Fourier, on obtient :
+
+\begin{equation}
+ X_e(f)=\frac{1}{T}\sum_{-\infty}^{+\infty}X_a(f-\frac{n}{T})
+\end{equation}
+
+Comme le schématise la figure \ref{sch_ech}, l'échantillonnage d'un signal analogique à la fréquence d'échantillonnage $F_e = 1/T$ induit une périodisation de son spectre dans le domaine fréquentiel (tous les $f =n/T$, $n$ étant entier).
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=12cm]{img/cons_ech_01}
+    \caption{Lien entre la fréquence d'échantillonnage d'un signal et la périodisation de son spectre.}
+    \label{sch_ech}
+\end{figure}
+
+
+Il peut survenir un problème si la fréquence d'échantillonnage $F_e$ est trop petite car les « répliques » périodiques du spectre  peuvent se superposer partiellement comme le montre la figure \ref{sch_repli}.
+
+\vspace{5mm}
+
+\begin{figure}[h]
+     \centering
+     \includegraphics[width=12cm]{img/repliement}
+     \caption{Phénomène de repliement.}
+     \label{sch_repli}
+\end{figure}
+
+
+Cela arrive si la borne supérieure d'un élément de $X_a(f)$ est plus grande que la borne inférieure de l'élement suivant, autrement dit si $B < \frac{1}{T} - B$ où $B$ est la fréquence maximale contenue dans le signal (cf. fig. \ref{sch_repli}).
+
+Ainsi, pour que le spectre $X_a(f$) ne soit pas « déformé » lors de sa périodisation, il faut donc que :
+
+\begin{equation}\label{shannon}
+ \boxeq{Fe > 2 B}
+\end{equation}
+
+Cette condition constitue le \textbf{théorème de Shannon} énoncé ainsi : <<~la fréquence d'échantillonnage d'un signal doit être égale ou supérieure au double de la fréquence maximale contenue dans ce signal~>>.
+
+
+\section{La transformée en Z}
+
+\subsection{Définition de la transformée en Z}
+
+Pour décrire un filtre dans le domaine numérique, il est pratique de définir une transformée dont la variable a la même nature que celle du signal discrétisé. En effet, un signal numérique - discontinu de nature - ne comporte qu'un nombre fini de valeurs et on a besoin d'une transformée à temps discret pour décrire les filtres : la transformée en $Z$.
+
+Soit $x(n)$ un signal discret quelconque. Sa transformée en Z s'écrit :
+
+\begin{equation}
+X(z) = \mathcal{Z}\{x(n)\} =\sum_{n=-\infty}^{+\infty}x(n)z^{-n},\quad z \in \{z\in\mathbb{C}|\sum_{n=-\infty}^{+\infty}x(n)z^{-n} \quad converge\}
+\end{equation}
+
+Remarque : on retrouve la définition de la transformée de Fourier en posant $z = e^{j2\pi f}$.
+
+\subsection{Existence de la transformée en Z}
+
+Le domaine de convergence est le sous-ensemble de $\mathbb{C}$ dans lequel la série converge.
+Autrement dit, le domaine de convergence de la transformée en z de la suite ($x_{n})_{n\in\mathbb{Z}}$ est l'ensemble :
+
+\begin{equation}
+\left\{z\in\mathbb{C} | \sum_{n=-\infty}^{\infty}x_{n}z^{-n} \quad\mathrm{existe}\right \}
+\end{equation}
+
+On l'appelle également couronne de convergence. En effet, en posant $z=\rho e^{i\theta}~$, il vient :
+
+\begin{equation}
+|X(z)|=\left| \sum_{n=-\infty}^{\infty}x_{n}z^{-n}\right|\leq \sum_{n=-\infty}^{\infty}\left|x_{n}\right|\rho^{-n}
+\end{equation}
+
+Donc $X(z)$ existe si $x(n)$ a une croissance au plus exponentielle, auquel cas le domaine de convergence est compris dans une couronne :
+
+\begin{itemize}
+\item de petit rayon le majorant de la base du côté des n négatifs
+\item de grand rayon le majorant de la base du côté des n positifs
+\end{itemize}
+
+Dans toute la suite de l'article, les transformées en Z ne seront valables que dans ce domaine de convergence sans que cela soit reprécisé.
+
+\subsection{Propriétés de la transformée en Z}
+
+\begin{enumerate}
+\item\textbf{ Linéarité}
+La transformée en Z d'une combinaison linéaire de deux signaux est la combinaison linéaire des transformées en Z de chaque signal.
+
+\begin{equation}
+  \mathcal{Z}\{a_1 x_1(n) + a_2 x_2(n)\} = a_1 \mathcal{Z}\{x_1(n)\} + a_2 \mathcal{Z}\{x_2(n)\}
+\end{equation}
+
+\item \textbf{Décalage temporel}
+
+Le décalage temporel d'un signal de k échantillons se traduit par la multiplication de la transformée en Z du signal par $z^k$.
+
+\begin{equation}
+\mathcal{Z}\{x(n-k)\} = z^{-k}\mathcal{Z}\{x(n)\}
+\end{equation}
+
+\item \textbf{Convolution}
+
+La transformée en Z d'un produit de convolution est le produit des transformées en Z
+
+\begin{equation}
+\mathcal{Z}\{x(n) \star y(n)\} = \mathcal{Z}\{x(n)\} \mathcal{Z}\{y(n)\} \                                                                      \end{equation}
+
+\item \textbf{Multiplication par une exponentielle}
+
+\begin{equation}
+\mathcal{Z}\{a^{n}x(n)\} = X\left(\frac{z}{a}\right)
+\end{equation}
+
+\item \textbf{Multiplication par la variable d'évolution}
+
+De façon générale :
+
+\begin{equation}
+\mathcal{Z}\{n^{k}x(n)\} = \left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}\                                               \end{equation}
+
+où $\textstyle\left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}$ signifie que l'on applique $k$ fois à $\mathcal{Z}\{x(n)\} l'opérateur \textstyle -z\frac{\mathrm{d} }{\mathrm{d}z}$
+
+Si l'on écrit cette formule au rang k=1, on obtient la formule de dérivation :
+
+\begin{equation}
+\mathcal{Z}\{nx(n)\} = -z \frac{\mathrm{d} }{\mathrm{d}z}X(z)\
+\end{equation}
+
+\item \textbf{Théorème de la valeur initiale}
+
+Soit $x(n)$\, un signal causal et $X(z)$\, sa transformée en Z. Alors :
+
+\begin{equation}
+x(0) = \lim_{n \to 0}x(n)=\lim_{z \to +\infty}X(z)
+\end{equation}
+
+\item \textbf{Théorème de la valeur finale}
+
+Soit $x(n)$, un signal causal et $X(z)$, sa transformée en Z. Alors lorsque la limite existe, on peut écrire :
+
+\begin{equation}
+\lim_{n \to +\infty}x(n)=\lim_{z \to 1}(z-1)X(z)
+\end{equation}
+
+\end{enumerate}

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+\section{Le produit de convolution}
+
+On appelle produit de convolution \textbf{convolution} de $x(t)$ par $y(t)$ l'opération notée $x(t) \star y(t)$ et définie par :
+
+\begin{equation}\label{conv}
+ \boxeq{x(t)\star y(t) = \int_{-\infty}^{+\infty}x(u)y(t-u)\ud u = \int_{-\infty}^{+\infty}x(t-u)y(u)\ud u}
+\end{equation}
+
+L'impulsion de Dirac est l'élément neutre de la convolution. En effet :
+
+\begin{equation}
+x(t) \star \delta (t) = x(t)
+\end{equation}
+
+Lorsque l'on convolue un signal $x(t)$ à un Dirac situé à un temps $t_0$, cela revient à retarder le signal $x(t)$ de $t_0$ :
+
+\begin{equation}
+x(t) \star \delta (t-t_0) = x(t-t_0)
+\end{equation}
+
+Par ailleurs, si l'on multiplie un signal $x(t)$ par un Dirac situé à un temps $t_0$, cela revient à connaître la valeur que prend $x(t)$ en $t_0$ (comme si l'on relevait l'ordonnée d'un point particulier d'une courbe)
+
+\begin{equation}
+x(t) \cdot \delta  (t-t_0) = x(t_0)
+\end{equation}
+
+De même, lorsque l'on convolue un signal $x(t)$ à un peigne de Dirac (de période $T$), cela revient à << périodiser >>  le signal $x(t)$ tous les $nT$ : on retarde le signal $x(t)$ de $T$, de $2T$, de $3T$, etc...
+
+\begin{equation}
+x(t) \star \Psi_T (t)  =  \Sigma x(t-nT)
+\end{equation}
+
+Tandis que si on multiplie le signal $x(t)$ par un peigne de Dirac, on prélève des échantillons du signal $x(t)$ tous les $nT$. Comme vu au paragraphe \ref{sampling} cette opération correspond à l'\textbf{échantillonnage} du signal $x(t)$ telle que :
+
+\begin{equation}
+x(t) \cdot \Psi_T  (t)  =  \Sigma x(nT)
+\end{equation}
+
+De façon plus générale, la convolution telle qu'elle est définie par sa formule mathématique, revient à retourner temporellement un des deux signaux (par exemple $x(t)$)  puis à le déplacer sur tout l'axe du temps  et à sommer toutes les multiplications de ce signal au deuxième signal $y(t)$.
+
+Pour des exemples animé de l'effet du produit de convolution (très pédagogique !), voir : \\
+\url{http://www.jhu.edu/~signals/convolve/index.html} \\
+\url{http://www.jhu.edu/~signals/discreteconv2/index.html}
+
+%Retournement temporel $t$ est un instant qui varie du signal $x$      dans le temps $u$ de  $-$  à  $+$.
+%On fait « glisser » le signal $x(-u)$ selon l'axe du temps.
+%On multiplie $y(u)$ par $x(t-u)$  pour chaque position de $t$ sur l'axe du temps et on somme tous les produits effectués.
+
+
+\section{Rappels sur la décomposition en série de Fourier des signaux périodiques}
+
+Pour pouvoir analyser les signaux quelque soit leur nature, il est intéressant de chercher des transformations qui mettent en évidence les particularités de leur contenu temporel mais aussi fréquentiel. Joseph Fourier a démontré que tout signal \textbf{périodique} peut être décomposé en une somme de sinus et de cosinus élémentaires telle que :
+
+\begin{equation}\label{DecompFour}
+ \boxed{x(t)=a_0 + \sum_{n=1}^{+\infty}a_n\cos(2\pi n f_0 t)+\sum_{n=1}^{+\infty}b_n\sin(2\pi n f_0 t)}
+\end{equation}
+
+où $f_0$ est appelée la fréquence \textbf{fondamentale}. Sauf effet particulier ou paradoxal, elle correspond à la hauteur principale perçue d'un son. Les autres composantes sont des fréquences supérieures à la fondamentale ($n>1$) et sont appelées \textbf{harmoniques} du signal. Par exemple, dans le cas d'un son émis par un instrument de musique, la fondamentale va correspondre à la note jouée et les autres composantes vont correspondre à la << série harmonique >> de cette note (octave, puis quinte, puis octave, puis tierce, etc...). Cette série correspond à des rapports de fréquences (fondamental $f_0$, octave $= 2*f_0$, octave $+$ quinte $= 3*f_0$, etc...). Les harmoniques ne vont pas directement contribuer à la perception de la hauteur mais plutôt à la richesse du timbre du son. C'est par exemple la présence d'harmoniques, entre autres, qui permet de différentier une sinusoïde à $f_0$ d'une note de violon dont la fondamentale est à la même fréquence.
+
+
+\section{La transformée de Fourier}
+
+\subsection{Définition de la transformée de Fourier}
+
+La transformée de Fourier est une généralisation de la décomposition précedente aux signaux non-périodiques. Soit $x(t)$ un signal quelconque, on note $X(f)$ ou $TF(x(t))$ sa transformée de Fourier telle que :
+
+\begin{equation}\label{fourier}
+ \boxeq{X(f)=TF(x(t))=\int_{-\infty}^{+\infty}x(t)e^{-i2\pi f t} \ud t}
+\end{equation}
+
+Inversement, on peut définir une transformée de Fourier inverse $TF^{-1}$ telle que :
+
+\begin{equation}\label{fourier_inv}
+ \boxeq{x(t)=TF^{-1}(X(f))=\int_{-\infty}^{+\infty}X(f)e^{i2\pi f t} \ud f}
+\end{equation}
+
+$X(f)$ est une fonction complexe même si $x(t)$ est réel. La transformée de Fourier contient donc une partie réelle et une partie imaginaire et est représentée facilement grâce à son \textbf{module} et à son \textbf{argument} : $\vert X(f) \vert$ est appelé \textbf{spectre d'amplitude} et $\mathrm{arg}(X(f))$ le \textbf{spectre de phase} du signal. La variable $f$ s'appelle la fréquence dont l'unité est le Hertz (en abrégé : Hz).\\
+
+\textbf{Remarques importantes:}
+\begin{itemize}
+ \item La représentation complète d'une transformée de Fourier nécessite 2 graphiques : le module et la phase, ou bien la partie réelle et le partie imaginaire.
+ \item Pour représenter les transformées de Fourier de signaux, il est communément utilisé l'échelle logarithmique. Pour un signal acoustique, par exemple, on calcule $20\log(\vert X(f) \vert / 2.10^{-5})$ et $\mathrm{arg}(X(f))$.\\
+\end{itemize}
+
+Ainsi, la transformée de Fourier est un opérateur mathématique qui permet d'analyser et de représenter un signal dans le domaine fréquentiel. La $TF$ ne modifie pas le signal mais permet seulement de l'observer selon différents points de vue (temporel ou fréquentiel). Il est important de retenir que $x(t)$ et $X(f)$ sont deux descriptions équivalentes du même signal. Ces deux fonctions contiennent la même information il s'agit juste de deux descriptions dans des domaines différents.
+
+$X(f)$ apporte des informations sur le système physique à l'origine du signal. Elle permet par exemple de différentier un son de trompette d'un son trombone, ou bien encore différentes ondes cérébrales, plus facilement qu'en observant le signal dans le domaine temporel. Le \textbf{contenu spectral} d'un signal est en effet assimilable à sa << carte d'identité >>.
+
+
+\subsection{Propriétés de la transformée de Fourier}
+
+\begin{itemize}
+ \item \textbf{Linéarité} :
+
+\begin{equation}
+ \boxeq{a x(t)+b y(t) \Leftrightarrow a X(f)+b Y(f)}
+\end{equation}
+
+\item \textbf{Produit de convolution} :
+
+\begin{equation}
+ \boxeq{x(t).y(t) \Leftrightarrow   X(f) \star Y(f)}
+\end{equation}
+
+\begin{equation}
+ \boxeq{x(t) \star y(t)  \Leftrightarrow  X(f) . Y(f)}
+\end{equation}
+
+Une multiplication dans un domaine correspond ainsi à un produit de convolution dans l'autre.\\
+
+\item \textbf{Retard} :
+
+\begin{equation}
+ \boxeq{x(t-t_0)   \Leftrightarrow  X(f) e^{-2i\pi f t_0}}
+\end{equation}
+
+\begin{equation}
+ \boxeq{x(t) \cdot e^{2\pi f_0 t} \Leftrightarrow  X(f-f_0)}
+\end{equation}
+
+Un retard temporel correspond ainsi à un déphasage au niveau fréquentiel, et inversement.\\
+
+\item \textbf{Changement d'échelle} :
+
+\begin{equation}
+\boxeq{x(at) \Leftrightarrow \frac{1}{\vert a \vert} X \left(\frac{f}{a} \right)}
+\end{equation}
+
+\textbf{Démonstration.} Soient $x(t)$ et $y(t)$ deux fonctions temporelles avec $t \in \mathbb{R}$ telles que $y(t)=x(at)$ avec $a \in \mathbb{R}$. Leur transformées de Fourier dans l'espace des fréquences sont notées respectivement $X(f)$ et $Y(f)$. On a alors :
+
+\begin{eqnarray}
+ Y(f) & = & \int_{-\infty}^{+\infty} y(t)e^{-2j\pi f t} \ud t\\
+ & = & \int_{-\infty}^{+\infty} x(at)e^{-2j\pi f t} \ud t \label{dem_ft_ech}
+\end{eqnarray}
+
+Effectuons le changement de variable $u=at$, c'est-à-dire aussi $\ud u=a \ud t$, qui revient ici à un changement d'échelle. L'équation \ref{dem_ft_ech} devient :
+
+\begin{eqnarray}
+ Y(f) & = & \int_{-\infty}^{+\infty} x(u)e^{-\frac{2j\pi f u}{a}} \frac{\ud u}{a}\\
+ & = & \frac{1}{\vert a \vert} \int_{-\infty}^{+\infty} x(u) e^{-2j\pi \frac{f}{a}u} \ud u\\
+ & = & \frac{1}{\vert a \vert}X\left(\frac{f}{a}\right) \label{dem_ft_ech_2}
+\end{eqnarray}
+
+Cette loi montre que lorsqu'on diminue l'échelle temporelle d'un signal ($a>1$), l'échelle fréquentielle augmente. Par exemple, si $x(t)$ est une sinusoïde de fréquence $f_0$ telle que $x(t)=\sin(2\pi f_0 t)$, alors $X(f)=\delta(f_0)$, $y(t)=\sin(2\pi a f_0 t)$ et $Y(f)=\frac{1}{\vert a \vert}\delta(f_1)$ où $f_1=a f_0$ (cf. figure \ref{sinus_ech}). Le facteur supplémentaire $\frac{1}{\vert a \vert}$ provient du principe de conservation d'énergie appliqué dans le domaine fréquentiel.\\
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Représentation temporelle]{\includegraphics[width=8cm]{img/sinus_ech}}
+    \subfigure[Représentation fréquentielle]{\includegraphics[width=8cm]{img/sinus_ech_ft}}
+    \caption{Exemple d'application d'un facteur d'échelle $a=2$ sur un signal sinusoïdal $x(t)$ de fréquence $f_0=50\ \mathrm{Hz}$ tel que $x(t)=\sin(2\pi f_0 t)$.}
+    \label{sinus_ech}
+\end{figure}
+
+\item \textbf{Théorème de Parseval} :\\
+Soit $E$ l'énergie du signal. On peut démontrer que :
+
+\begin{equation}
+ \boxeq{E = \int_{-\infty}^{+\infty} \vert x(t) \vert^2 \ud t   =   \int_{-\infty}^{+\infty}    \vert X(f) \vert^2 \ud f}
+\end{equation}
+
+\end{itemize}
+
+
+\subsection{Les transformées de Fourier de signaux courants}
+
+\begin{itemize}
+\item Dirac $\delta(t) \Leftrightarrow $ signal unité
+\item Signal porte  $\Leftrightarrow$ sinus cardinal
+\item $\sin(2\pi f_0 t)  \Leftrightarrow  \frac{1}{2}[\delta(f-f_0)  + \delta(f+f_0)]$
+\item la suite en cours...
+\end{itemize}
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La fonction de densité spectrale d'énergie est la $TF$ de la fonction d'autocorrélation vue au paragraphe \ref{intercorr}. Elle décrit la répartition de l'énergie en fréquence et on a :
+
+
+\begin{equation}\label{PSD}
+\Phi_{x}(f)= TF (\varphi_x(\tau))=\vert X(f) \vert ^2
+\end{equation}
+
+
+\section{L'échantillonnage}\label{signum}
+
+Les signaux numériques dérivent d 'un signal analogique par \textbf{échantillonnage}. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants $t_n$, $n\in \mathbb{Z}$. Dans la suite nous n'envisagerons que l'échantillonnage est régulier, \cad que $t_n$ = $nT$.
+
+Aujourd'hui l'intérêt grandissant pour la conversion de signaux analogiques en signaux numériques tient au fait qu'il devient de plus en plus simple d'appliquer des algorithmes d'analyse complexe avec l'aide des processeurs numériques. En sciences physiques par exemple, il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}\label{sampling}
+
+Le signal continu (analogique) peut être échantillonné (\cad discrétisé) à une \textbf{fréquence d'échantillonnage} $f_e$ inverse de la \textbf{période d'échantillonnage} $T_e$ telle que $f_e= 1/T_e$. Cette échantillonage revient à multiplier le signal analogique par un peigne de Dirac de période $T_e$.
+
+Le signal échantillonné peut alors s'écrire :
+
+\begin{equation}
+ x_e(t)=x_a(t) \cdot \sum_{n=-\infty}^{+\infty} \delta (t-nT_e).
+\end{equation}
+
+$x_e(t)$ est donc nul partout sauf en $t= nTe$, $n$ variant de $-\infty$ à $+\infty$. C'est pourquoi par la suite, on ne simplifie cette notation en réduisant $x_e(t)$ non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets $x(n)$, où $n$ correspond au numéro de l'échantillon.
+
+\begin{equation}
+ x(n) = x_e ( n T_e )
+\end{equation}
+
+\subsection{La quantification}
+
+Voir cours.
+
+
+\section{La transformée de Fourier à temps discret}
+
+Dans le cas des signaux échantillonnés où le temps est discrétisé, il n'est plus nécessaire d'utiliser des intégrales continues pour sommer les valeurs de $x(t)$ sur tout l'axe des temps, puisqu'un signal échantillonné peut être assimilé à une suite contenant un nombre \textbf{fini} d'éléments . On peut donc définir la transformée de Fourier d'un signal numérique par :
+
+\begin{equation}\label{TFD}
+ X(f)=\sum_{n=-\infty}^{+\infty}x(n)e^{-2j\pi n f}
+\end{equation}
+
+La transformée de Fourier inverse se retrouve de la même manière que dans le cas continu :
+
+\begin{equation}\label{TFDinv}
+ x(n)=\int_{-\frac{1}{2}}^{\frac{1}{2}}X(f)e^{2j\pi n f}\ud f
+\end{equation}
+
+De la même façon qu'en continu, la TF présente certaines propriétés intéressantes :
+
+\begin{itemize}
+ \item \textbf{Retard} : $$x(n-n_0) \Leftrightarrow  X(f) e^{-2j\pi n_0 f}$$\\
+ \item \textbf{Produit de convolution} :
+ $$x(n)\star y(n) \Leftrightarrow X(f) . Y(f)$$\\
+ et \\
+ $$x(n).y(n) \Leftrightarrow X(f) \star Y(f)$$\\
+ \end{itemize}
+
+ avec,
+
+ \begin{equation}\label{conv_disc}
+  x(n) \star y(n)  =  \sum_{-\infty}^{+\infty}x(k) y(n-k) = \sum_{-\infty}^{+\infty}x(n-k) y(k)
+ \end{equation}
+
+
+\section{Le théorème de Shannon}
+
+La discrétisation temporelle du signal analogique détaillée à la section \ref{signum} n'est pas sans conséquences vis-à-vis des propriétés spectrales du signal échantillonné. Ainsi une mauvaise adéquation entre les fréquences contenues dans le signal et la fréquence d'échantillonnage peut être destructrice.
+
+En posant $x_e(t) = x_a(t).\sum \delta(t- nT_e)$ le signal échantillonné du signal analogique $x_a(t)$ et $X_e(f)$ sa transformée de Fourier, on obtient :
+
+\begin{equation}
+ X_e(f)=\frac{1}{T}\sum_{-\infty}^{+\infty}X_a(f-\frac{n}{T})
+\end{equation}
+
+Comme le schématise la figure \ref{sch_ech}, l'échantillonnage d'un signal analogique à la fréquence d'échantillonnage $F_e = 1/T$ induit une périodisation de son spectre dans le domaine fréquentiel (tous les $f =n/T$, $n$ étant entier).
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=12cm]{img/cons_ech_01}
+    \caption{Lien entre la fréquence d'échantillonnage d'un signal et la périodisation de son spectre.}
+    \label{sch_ech}
+\end{figure}
+
+
+Il peut survenir un problème si la fréquence d'échantillonnage $F_e$ est trop petite car les « répliques » périodiques du spectre  peuvent se superposer partiellement comme le montre la figure \ref{sch_repli}.
+
+\vspace{5mm}
+
+\begin{figure}[h]
+     \centering
+     \includegraphics[width=12cm]{img/repliement}
+     \caption{Phénomène de repliement.}
+     \label{sch_repli}
+\end{figure}
+
+
+Cela arrive si la borne supérieure d'un élément de $X_a(f)$ est plus grande que la borne inférieure de l'élement suivant, autrement dit si $B > \frac{1}{T} - B$ où $B$ est la fréquence maximale contenue dans le signal (cf. fig. \ref{sch_repli}).
+
+Ainsi, pour que le spectre $X_a(f$) ne soit pas « déformé » lors de sa périodisation, il faut donc que :
+
+\begin{equation}\label{shannon}
+ \boxeq{Fe < 2 B}
+\end{equation}
+
+Cette condition constitue le \textbf{théorème de Shannon} énoncé ainsi : <<~la fréquence d'échantillonnage d'un signal doit être égale ou supérieure au double de la fréquence maximale contenue dans ce signal~>>.
+
+
+\section{La transformée en Z}
+
+\subsection{Définition de la transformée en Z}
+
+Pour décrire un filtre dans le domaine numérique, il est pratique de définir une transformée dont la variable a la même nature que celle du signal discrétisé. En effet, un signal numérique - discontinu de nature - ne comporte qu'un nombre fini de valeurs et on a besoin d'une transformée à temps discret pour décrire les filtres : la transformée en $Z$.
+
+Soit $x(n)$ un signal discret quelconque. Sa transformée en Z s'écrit :
+
+\begin{equation}
+X(z) = \mathcal{Z}\{x(n)\} =\sum_{n=-\infty}^{+\infty}x(n)z^{-n},\quad z \in \{z\in\mathbb{C}|\sum_{n=-\infty}^{+\infty}x(n)z^{-n} \quad converge\}
+\end{equation}
+
+Remarque : on retrouve la définition de la transformée de Fourier en posant $z = e^{j2\pi f}$.
+
+\subsection{Existence de la transformée en Z}
+
+Le domaine de convergence est le sous-ensemble de $\mathbb{C}$ dans lequel la série converge.
+Autrement dit, le domaine de convergence de la transformée en z de la suite ($x_{n})_{n\in\mathbb{Z}}$ est l'ensemble :
+
+\begin{equation}
+\left\{z\in\mathbb{C} | \sum_{n=-\infty}^{\infty}x_{n}z^{-n} \quad\mathrm{existe}\right \}
+\end{equation}
+
+On l'appelle également couronne de convergence. En effet, en posant $z=\rho e^{i\theta}~$, il vient :
+
+\begin{equation}
+|X(z)|=\left| \sum_{n=-\infty}^{\infty}x_{n}z^{-n}\right|\leq \sum_{n=-\infty}^{\infty}\left|x_{n}\right|\rho^{-n}
+\end{equation}
+
+Donc $X(z)$ existe si $x(n)$ a une croissance au plus exponentielle, auquel cas le domaine de convergence est compris dans une couronne :
+
+\begin{itemize}
+\item de petit rayon le majorant de la base du côté des n négatifs
+\item de grand rayon le majorant de la base du côté des n positifs
+\end{itemize}
+
+Dans toute la suite de l'article, les transformées en Z ne seront valables que dans ce domaine de convergence sans que cela soit reprécisé.
+
+\subsection{Propriétés de la transformée en Z}
+
+\begin{enumerate}
+\item\textbf{ Linéarité}
+La transformée en Z d'une combinaison linéaire de deux signaux est la combinaison linéaire des transformées en Z de chaque signal.
+
+\begin{equation}
+  \mathcal{Z}\{a_1 x_1(n) + a_2 x_2(n)\} = a_1 \mathcal{Z}\{x_1(n)\} + a_2 \mathcal{Z}\{x_2(n)\}
+\end{equation}
+
+\item \textbf{Décalage temporel}
+
+Le décalage temporel d'un signal de k échantillons se traduit par la multiplication de la transformée en Z du signal par $z^k$.
+
+\begin{equation}
+\mathcal{Z}\{x(n-k)\} = z^{-k}\mathcal{Z}\{x(n)\}
+\end{equation}
+
+\item \textbf{Convolution}
+
+La transformée en Z d'un produit de convolution est le produit des transformées en Z
+
+\begin{equation}
+\mathcal{Z}\{x(n) \star y(n)\} = \mathcal{Z}\{x(n)\} \mathcal{Z}\{y(n)\} \                                                                      \end{equation}
+
+\item \textbf{Multiplication par une exponentielle}
+
+\begin{equation}
+\mathcal{Z}\{a^{n}x(n)\} = X\left(\frac{z}{a}\right)
+\end{equation}
+
+\item \textbf{Multiplication par la variable d'évolution}
+
+De façon générale :
+
+\begin{equation}
+\mathcal{Z}\{n^{k}x(n)\} = \left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}\                                               \end{equation}
+
+où $\textstyle\left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}$ signifie que l'on applique $k$ fois à $\mathcal{Z}\{x(n)\} l'opérateur \textstyle -z\frac{\mathrm{d} }{\mathrm{d}z}$
+
+Si l'on écrit cette formule au rang k=1, on obtient la formule de dérivation :
+
+\begin{equation}
+\mathcal{Z}\{nx(n)\} = -z \frac{\mathrm{d} }{\mathrm{d}z}X(z)\
+\end{equation}
+
+\item \textbf{Théorème de la valeur initiale}
+
+Soit $x(n)$\, un signal causal et $X(z)$\, sa transformée en Z. Alors :
+
+\begin{equation}
+x(0) = \lim_{n \to 0}x(n)=\lim_{z \to +\infty}X(z)
+\end{equation}
+
+\item \textbf{Théorème de la valeur finale}
+
+Soit $x(n)$, un signal causal et $X(z)$, sa transformée en Z. Alors lorsque la limite existe, on peut écrire :
+
+\begin{equation}
+\lim_{n \to +\infty}x(n)=\lim_{z \to 1}(z-1)X(z)
+\end{equation}
+
+\end{enumerate}

diff --git a/IntroMethFreq.tex~ b/IntroMethFreq.tex~
new file mode 100644 (file)
index 0000000..611352a
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+++ b/IntroMethFreq.tex~
@@ -0,0 +1,380 @@
+\section{Le produit de convolution}
+
+On appelle produit de convolution \textbf{convolution} de $x(t)$ par $y(t)$ l'opération notée $x(t) \star y(t)$ et définie par :
+
+\begin{equation}\label{conv}
+ \boxeq{x(t)\star y(t) = \int_{-\infty}^{+\infty}x(u)y(t-u)\ud u = \int_{-\infty}^{+\infty}x(t-u)y(u)\ud u}
+\end{equation}
+
+L'impulsion de Dirac est l'élément neutre de la convolution. En effet :
+
+\begin{equation}
+x(t) \star \delta (t) = x(t)
+\end{equation}
+
+Lorsque l'on convolue un signal $x(t)$ à un Dirac situé à un temps $t_0$, cela revient à retarder le signal $x(t)$ de $t_0$ :
+
+\begin{equation}
+x(t) \star \delta (t-t_0) = x(t-t_0)
+\end{equation}
+
+Par ailleurs, si l'on multiplie un signal $x(t)$ par un Dirac situé à un temps $t_0$, cela revient à connaître la valeur que prend $x(t)$ en $t_0$ (comme si l'on relevait l'ordonnée d'un point particulier d'une courbe)
+
+\begin{equation}
+x(t) \cdot \delta  (t-t_0) = x(t_0)
+\end{equation}
+
+De même, lorsque l'on convolue un signal $x(t)$ à un peigne de Dirac (de période $T$), cela revient à << périodiser >>  le signal $x(t)$ tous les $nT$ : on retarde le signal $x(t)$ de $T$, de $2T$, de $3T$, etc...
+
+\begin{equation}
+x(t) \star \Psi_T (t)  =  \Sigma x(t-nT)
+\end{equation}
+
+De façon plus générale, la convolution telle qu'elle est définie par sa formule mathématique, revient à retourner temporellement un des deux signaux (par exemple $x(t)$)  puis à le déplacer sur tout l'axe du temps  et à sommer toutes les multiplications de ce signal au deuxième signal $y(t)$.
+
+Pour des exemples animé de l'effet du produit de convolution (très pédagogique !), voir : \\
+\url{http://www.jhu.edu/~signals/convolve/index.html} \\
+\url{http://www.jhu.edu/~signals/discreteconv2/index.html}
+
+%Retournement temporel $t$ est un instant qui varie du signal $x$      dans le temps $u$ de  $-$  à  $+$.
+%On fait « glisser » le signal $x(-u)$ selon l'axe du temps.
+%On multiplie $y(u)$ par $x(t-u)$  pour chaque position de $t$ sur l'axe du temps et on somme tous les produits effectués.
+
+
+\section{Rappels sur la décomposition en série de Fourier des signaux périodiques}
+
+Pour pouvoir analyser les signaux quelque soit leur nature, il est intéressant de chercher des transformations qui mettent en évidence les particularités de leur contenu temporel mais aussi fréquentiel. Joseph Fourier a démontré que tout signal \textbf{périodique} peut être décomposé en une somme de sinus et de cosinus élémentaires telle que :
+
+\begin{equation}\label{DecompFour}
+ \boxed{x(t)=a_0 + \sum_{n=1}^{+\infty}a_n\cos(2\pi n f_0 t)+\sum_{n=1}^{+\infty}b_n\sin(2\pi n f_0 t)}
+\end{equation}
+
+où $f_0$ est appelée la fréquence \textbf{fondamentale}. Sauf effet particulier ou paradoxal, elle correspond à la hauteur principale perçue d'un son. Les autres composantes sont des fréquences supérieures à la fondamentale ($n>1$) et sont appelées \textbf{harmoniques} du signal. Par exemple, dans le cas d'un son émis par un instrument de musique, la fondamentale va correspondre à la note jouée et les autres composantes vont correspondre à la << série harmonique >> de cette note (octave, puis quinte, puis octave, puis tierce, etc...). Cette série correspond à des rapports de fréquences (fondamental $f_0$, octave $= 2*f_0$, octave $+$ quinte $= 3*f_0$, etc...). Les harmoniques ne vont pas directement contribuer à la perception de la hauteur mais plutôt à la richesse du timbre du son. C'est par exemple la présence d'harmoniques, entre autres, qui permet de différentier une sinusoïde à $f_0$ d'une note de violon dont la fondamentale est à la même fréquence.
+
+
+\section{La transformée de Fourier}
+
+\subsection{Définition de la transformée de Fourier}
+
+La transformée de Fourier est une généralisation de la décomposition précedente aux signaux non-périodiques. Soit $x(t)$ un signal quelconque, on note $X(f)$ ou $TF(x(t))$ sa transformée de Fourier telle que :
+
+\begin{equation}\label{fourier}
+ \boxeq{X(f)=TF(x(t))=\int_{-\infty}^{+\infty}x(t)e^{-i2\pi f t} \ud t}
+\end{equation}
+
+Inversement, on peut définir une transformée de Fourier inverse $TF^{-1}$ telle que :
+
+\begin{equation}\label{fourier_inv}
+ \boxeq{x(t)=TF^{-1}(X(f))=\int_{-\infty}^{+\infty}X(f)e^{i2\pi f t} \ud f}
+\end{equation}
+
+$X(f)$ est une fonction complexe même si $x(t)$ est réel. La transformée de Fourier contient donc une partie réelle et une partie imaginaire et est représentée facilement grâce à son \textbf{module} et à son \textbf{argument} : $\vert X(f) \vert$ est appelé \textbf{spectre d'amplitude} et $\mathrm{arg}(X(f))$ le \textbf{spectre de phase} du signal. La variable $f$ s'appelle la fréquence dont l'unité est le Hertz (en abrégé : Hz).\\
+
+\textbf{Remarques importantes:}
+\begin{itemize}
+ \item La représentation complète d'une transformée de Fourier nécessite 2 graphiques : le module et la phase, ou bien la partie réelle et le partie imaginaire.
+ \item Pour représenter les transformées de Fourier de signaux, il est communément utilisé l'échelle logarithmique. Pour un signal acoustique, par exemple, on calcule $20\log(\vert X(f) \vert / 2.10^{-5})$ et $\mathrm{arg}(X(f))$.\\
+\end{itemize}
+
+Ainsi, la transformée de Fourier est un opérateur mathématique qui permet d'analyser et de représenter un signal dans le domaine fréquentiel. La $TF$ ne modifie pas le signal mais permet seulement de l'observer selon différents points de vue (temporel ou fréquentiel). Il est important de retenir que $x(t)$ et $X(f)$ sont deux descriptions équivalentes du même signal. Ces deux fonctions contiennent la même information il s'agit juste de deux descriptions dans des domaines différents.
+
+$X(f)$ apporte des informations sur le système physique à l'origine du signal. Elle permet par exemple de différentier un son de trompette d'un son trombone, ou bien encore différentes ondes cérébrales, plus facilement qu'en observant le signal dans le domaine temporel. Le \textbf{contenu spectral} d'un signal est en effet assimilable à sa << carte d'identité >>.
+
+
+\subsection{Propriétés de la transformée de Fourier}
+
+\begin{itemize}
+ \item \textbf{Linéarité} :
+
+\begin{equation}
+ \boxeq{a x(t)+b y(t) \Leftrightarrow a X(f)+b Y(f)}
+\end{equation}
+
+\item \textbf{Produit de convolution} :
+
+\begin{equation}
+ \boxeq{x(t).y(t) \Leftrightarrow   X(f) \star Y(f)}
+\end{equation}
+
+\begin{equation}
+ \boxeq{x(t) \star y(t)  \Leftrightarrow  X(f) . Y(f)}
+\end{equation}
+
+Une multiplication dans un domaine correspond ainsi à un produit de convolution dans l'autre.\\
+
+\item \textbf{Retard} :
+
+\begin{equation}
+ \boxeq{x(t-t_0)   \Leftrightarrow  X(f) e^{-2i\pi f t_0}}
+\end{equation}
+
+\begin{equation}
+ \boxeq{x(t) \cdot e^{2\pi f_0 t} \Leftrightarrow  X(f-f_0)}
+\end{equation}
+
+Un retard temporel correspond ainsi à un déphasage au niveau fréquentiel, et inversement.\\
+
+\item \textbf{Changement d'échelle} :
+
+\begin{equation}
+\boxeq{x(at) \Leftrightarrow \frac{1}{\vert a \vert} X \left(\frac{f}{a} \right)}
+\end{equation}
+
+\textbf{Démonstration.} Soient $x(t)$ et $y(t)$ deux fonctions temporelles avec $t \in \mathbb{R}$ telles que $y(t)=x(at)$ avec $a \in \mathbb{R}$. Leur transformées de Fourier dans l'espace des fréquences sont notées respectivement $X(f)$ et $Y(f)$. On a alors :
+
+\begin{eqnarray}
+ Y(f) & = & \int_{-\infty}^{+\infty} y(t)e^{-2j\pi f t} \ud t\\
+ & = & \int_{-\infty}^{+\infty} x(at)e^{-2j\pi f t} \ud t \label{dem_ft_ech}
+\end{eqnarray}
+
+Effectuons le changement de variable $u=at$, c'est-à-dire aussi $\ud u=a \ud t$, qui revient ici à un changement d'échelle. L'équation \ref{dem_ft_ech} devient :
+
+\begin{eqnarray}
+ Y(f) & = & \int_{-\infty}^{+\infty} x(u)e^{-\frac{2j\pi f u}{a}} \frac{\ud u}{a}\\
+ & = & \frac{1}{\vert a \vert} \int_{-\infty}^{+\infty} x(u) e^{-2j\pi \frac{f}{a}u} \ud u\\
+ & = & \frac{1}{\vert a \vert}X\left(\frac{f}{a}\right) \label{dem_ft_ech_2}
+\end{eqnarray}
+
+Cette loi montre que lorsqu'on diminue l'échelle temporelle d'un signal ($a>1$), l'échelle fréquentielle augmente. Par exemple, si $x(t)$ est une sinusoïde de fréquence $f_0$ telle que $x(t)=\sin(2\pi f_0 t)$, alors $X(f)=\delta(f-f_0)$, $y(t)=\sin(2\pi a f_0 t)$ et $Y(f)=\frac{1}{\vert a \vert}\delta(f-f_1)$ où $f_1=a f_0$ (cf. figure \ref{sinus_ech}). Le facteur supplémentaire $\frac{1}{\vert a \vert}$ provient du principe de conservation d'énergie appliqué dans le domaine fréquentiel.\\
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Représentation temporelle]{\includegraphics[width=8cm]{img/sinus_ech}}
+    \subfigure[Représentation fréquentielle]{\includegraphics[width=8cm]{img/sinus_ech_ft}}
+    \caption{Exemple d'application d'un facteur d'échelle $a=2$ sur un signal sinusoïdal $x(t)$ de fréquence $f_0=50\ \mathrm{Hz}$ tel que $x(t)=\sin(2\pi f_0 t)$.}
+    \label{sinus_ech}
+\end{figure}
+
+\item \textbf{Théorème de Parseval} :\\
+Soit $E$ l'énergie du signal. On peut démontrer que :
+
+\begin{equation}
+ \boxeq{E = \int_{-\infty}^{+\infty} \vert x(t) \vert^2 \ud t   =   \int_{-\infty}^{+\infty}    \vert X(f) \vert^2 \ud f}
+\end{equation}
+
+\end{itemize}
+
+
+\subsection{Les transformées de Fourier de signaux courants}
+
+\begin{itemize}
+\item Dirac $\delta(t) \Leftrightarrow $ signal unité
+\item Signal porte  $\Leftrightarrow$ sinus cardinal
+\item $\sin(2\pi f_0 t)  \Leftrightarrow  \frac{1}{2}[\delta(f-f_0)  + \delta(f+f_0)]$
+\item la suite en cours...
+\end{itemize}
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La fonction de densité spectrale d'énergie est la $TF$ de la fonction d'autocorrélation vue au paragraphe \ref{intercorr}. Elle décrit la répartition de l'énergie en fréquence et on a :
+
+
+\begin{equation}\label{PSD}
+\Phi_{x}(f)= TF (\varphi_x(\tau))=\vert X(f) \vert ^2
+\end{equation}
+
+
+\section{L'échantillonnage}\label{signum}
+
+Les signaux numériques dérivent d 'un signal analogique par \textbf{échantillonnage}. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants $t_n$, $n\in \mathbb{Z}$. Dans la suite nous n'envisagerons que l'échantillonnage est régulier, \cad que $t_n$ = $nT$.
+
+Aujourd'hui l'intérêt grandissant pour la conversion de signaux analogiques en signaux numériques tient au fait qu'il devient de plus en plus simple d'appliquer des algorithmes d'analyse complexe avec l'aide des processeurs numériques. En sciences physiques par exemple, il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}\label{sampling}
+
+Le signal continu (analogique) peut être échantillonné (\cad discrétisé) à une \textbf{fréquence d'échantillonnage} $f_e$ inverse de la \textbf{période d'échantillonnage} $T_e$ telle que $f_e= 1/T_e$. Cet échantillonnage revient à multiplier le signal analogique par un peigne de Dirac de période $T_e$.
+
+Le signal échantillonné peut alors s'écrire :
+
+\begin{equation}
+ x_e(t)=x_a(t) \cdot \sum_{n=-\infty}^{+\infty} \delta (t-nT_e).
+\end{equation}
+
+$x_e(t)$ est donc nul partout sauf en $t= nTe$, $n$ variant de $-\infty$ à $+\infty$. C'est pourquoi par la suite, on ne simplifie cette notation en réduisant $x_e(t)$ non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets $x(n)$, où $n$ correspond au numéro de l'échantillon.
+
+\begin{equation}
+ x(n) = x_e ( n T_e )
+\end{equation}
+
+\subsection{La quantification}
+
+L'échantillonnage complet d'un signal nécessite de discrétiser également l'amplitude de ce signal. Quelque soit la résolution binaire donnée, cette opération suppose d'opérer un arrondi sur chaque échantillon de sorte que la valeur soit codable sur une échelle binaire. Les valeurs courantes de résolution de quantification en audio sont : 8, 16, 24 et 32 bits. De nombreux détails, exemples et schémas sont proposés en séance de cours.
+
+
+\section{La transformée de Fourier à temps discret}
+
+Dans le cas des signaux échantillonnés où le temps est discrétisé, il n'est plus nécessaire d'utiliser des intégrales continues pour sommer les valeurs de $x(t)$ sur tout l'axe des temps, puisqu'un signal échantillonné peut être assimilé à une suite contenant un nombre \textbf{fini} d'éléments . On peut donc définir la transformée de Fourier d'un signal numérique par :
+
+\begin{equation}\label{TFD}
+ X(f)=\sum_{n=-\infty}^{+\infty}x(n)e^{-2j\pi n f}
+\end{equation}
+
+La transformée de Fourier inverse se retrouve de la même manière que dans le cas continu :
+
+\begin{equation}\label{TFDinv}
+ x(n)=\int_{-\frac{1}{2}}^{\frac{1}{2}}X(f)e^{2j\pi n f}\ud f
+\end{equation}
+
+De la même façon qu'en continu, la TF présente certaines propriétés intéressantes :
+
+\begin{itemize}
+ \item \textbf{Retard} : $$x(n-n_0) \Leftrightarrow  X(f) e^{-2j\pi n_0 f}$$\\
+ \item \textbf{Produit de convolution} :
+ $$x(n)\star y(n) \Leftrightarrow X(f) . Y(f)$$\\
+ et \\
+ $$x(n).y(n) \Leftrightarrow X(f) \star Y(f)$$\\
+ \end{itemize}
+
+ avec,
+
+ \begin{equation}\label{conv_disc}
+  x(n) \star y(n)  =  \sum_{-\infty}^{+\infty}x(k) y(n-k) = \sum_{-\infty}^{+\infty}x(n-k) y(k)
+ \end{equation}
+
+
+\section{Le théorème de Shannon}
+
+La discrétisation temporelle du signal analogique détaillée à la section \ref{signum} n'est pas sans conséquences vis-à-vis des propriétés spectrales du signal échantillonné. Ainsi une mauvaise adéquation entre les fréquences contenues dans le signal et la fréquence d'échantillonnage peut être destructrice.
+
+En posant $x_e(t) = x_a(t).\sum \delta(t- nT_e)$ le signal échantillonné du signal analogique $x_a(t)$ et $X_e(f)$ sa transformée de Fourier, on obtient :
+
+\begin{equation}
+ X_e(f)=\frac{1}{T}\sum_{-\infty}^{+\infty}X_a(f-\frac{n}{T})
+\end{equation}
+
+Comme le schématise la figure \ref{sch_ech}, l'échantillonnage d'un signal analogique à la fréquence d'échantillonnage $F_e = 1/T$ induit une périodisation de son spectre dans le domaine fréquentiel (tous les $f =n/T$, $n$ étant entier).
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=12cm]{img/cons_ech_01}
+    \caption{Lien entre la fréquence d'échantillonnage d'un signal et la périodisation de son spectre.}
+    \label{sch_ech}
+\end{figure}
+
+
+Il peut survenir un problème si la fréquence d'échantillonnage $F_e$ est trop petite car les « répliques » périodiques du spectre  peuvent se superposer partiellement comme le montre la figure \ref{sch_repli}.
+
+\vspace{5mm}
+
+\begin{figure}[h]
+     \centering
+     \includegraphics[width=12cm]{img/repliement}
+     \caption{Phénomène de repliement.}
+     \label{sch_repli}
+\end{figure}
+
+
+Cela arrive si la borne supérieure d'un élément de $X_a(f)$ est plus grande que la borne inférieure de l'élement suivant, autrement dit si $B > \frac{1}{T} - B$ où $B$ est la fréquence maximale contenue dans le signal (cf. fig. \ref{sch_repli}).
+
+Ainsi, pour que le spectre $X_a(f$) ne soit pas « déformé » lors de sa périodisation, il faut donc que :
+
+\begin{equation}\label{shannon}
+ \boxeq{Fe < 2 B}
+\end{equation}
+
+Cette condition constitue le \textbf{théorème de Shannon} énoncé ainsi : <<~la fréquence d'échantillonnage d'un signal doit être égale ou supérieure au double de la fréquence maximale contenue dans ce signal~>>.
+
+
+\section{La transformée en Z}
+
+\subsection{Définition de la transformée en Z}
+
+Pour décrire un filtre dans le domaine numérique, il est pratique de définir une transformée dont la variable a la même nature que celle du signal discrétisé. En effet, un signal numérique - discontinu de nature - ne comporte qu'un nombre fini de valeurs et on a besoin d'une transformée à temps discret pour décrire les filtres : la transformée en $Z$.
+
+Soit $x(n)$ un signal discret quelconque. Sa transformée en Z s'écrit :
+
+\begin{equation}
+X(z) = \mathcal{Z}\{x(n)\} =\sum_{n=-\infty}^{+\infty}x(n)z^{-n},\quad z \in \{z\in\mathbb{C}|\sum_{n=-\infty}^{+\infty}x(n)z^{-n} \quad converge\}
+\end{equation}
+
+Remarque : on retrouve la définition de la transformée de Fourier en posant $z = e^{j2\pi f}$.
+
+\subsection{Existence de la transformée en Z}
+
+Le domaine de convergence est le sous-ensemble de $\mathbb{C}$ dans lequel la série converge.
+Autrement dit, le domaine de convergence de la transformée en z de la suite ($x_{n})_{n\in\mathbb{Z}}$ est l'ensemble :
+
+\begin{equation}
+\left\{z\in\mathbb{C} | \sum_{n=-\infty}^{\infty}x_{n}z^{-n} \quad\mathrm{existe}\right \}
+\end{equation}
+
+On l'appelle également couronne de convergence. En effet, en posant $z=\rho e^{i\theta}~$, il vient :
+
+\begin{equation}
+|X(z)|=\left| \sum_{n=-\infty}^{\infty}x_{n}z^{-n}\right|\leq \sum_{n=-\infty}^{\infty}\left|x_{n}\right|\rho^{-n}
+\end{equation}
+
+Donc $X(z)$ existe si $x(n)$ a une croissance au plus exponentielle, auquel cas le domaine de convergence est compris dans une couronne :
+
+\begin{itemize}
+\item de petit rayon le majorant de la base du côté des n négatifs
+\item de grand rayon le majorant de la base du côté des n positifs
+\end{itemize}
+
+Dans toute la suite de l'article, les transformées en Z ne seront valables que dans ce domaine de convergence sans que cela soit reprécisé.
+
+\subsection{Propriétés de la transformée en Z}
+
+\begin{enumerate}
+\item\textbf{ Linéarité}
+La transformée en Z d'une combinaison linéaire de deux signaux est la combinaison linéaire des transformées en Z de chaque signal.
+
+\begin{equation}
+  \mathcal{Z}\{a_1 x_1(n) + a_2 x_2(n)\} = a_1 \mathcal{Z}\{x_1(n)\} + a_2 \mathcal{Z}\{x_2(n)\}
+\end{equation}
+
+\item \textbf{Décalage temporel}
+
+Le décalage temporel d'un signal de k échantillons se traduit par la multiplication de la transformée en Z du signal par $z^k$.
+
+\begin{equation}
+\mathcal{Z}\{x(n-k)\} = z^{-k}\mathcal{Z}\{x(n)\}
+\end{equation}
+
+\item \textbf{Convolution}
+
+La transformée en Z d'un produit de convolution est le produit des transformées en Z
+
+\begin{equation}
+\mathcal{Z}\{x(n) \star y(n)\} = \mathcal{Z}\{x(n)\} \mathcal{Z}\{y(n)\} \                                                                      \end{equation}
+
+\item \textbf{Multiplication par une exponentielle}
+
+\begin{equation}
+\mathcal{Z}\{a^{n}x(n)\} = X\left(\frac{z}{a}\right)
+\end{equation}
+
+\item \textbf{Multiplication par la variable d'évolution}
+
+De façon générale :
+
+\begin{equation}
+\mathcal{Z}\{n^{k}x(n)\} = \left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}\                                               \end{equation}
+
+où $\textstyle\left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}$ signifie que l'on applique $k$ fois à $\mathcal{Z}\{x(n)\} l'opérateur \textstyle -z\frac{\mathrm{d} }{\mathrm{d}z}$
+
+Si l'on écrit cette formule au rang k=1, on obtient la formule de dérivation :
+
+\begin{equation}
+\mathcal{Z}\{nx(n)\} = -z \frac{\mathrm{d} }{\mathrm{d}z}X(z)\
+\end{equation}
+
+\item \textbf{Théorème de la valeur initiale}
+
+Soit $x(n)$\, un signal causal et $X(z)$\, sa transformée en Z. Alors :
+
+\begin{equation}
+x(0) = \lim_{n \to 0}x(n)=\lim_{z \to +\infty}X(z)
+\end{equation}
+
+\item \textbf{Théorème de la valeur finale}
+
+Soit $x(n)$, un signal causal et $X(z)$, sa transformée en Z. Alors lorsque la limite existe, on peut écrire :
+
+\begin{equation}
+\lim_{n \to +\infty}x(n)=\lim_{z \to 1}(z-1)X(z)
+\end{equation}
+
+\end{enumerate}

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+\section{Fenêtrage des signaux}
+
+Nous avons vu dans le chapitre précédent que la Transformée de Fourier permet de représenter le spectre d'un signal dans le domaine fréquentiel. Pour une sinusoïde infinie, toute l'énergie du spectre est concentrée à une fréquence donnée, c'est à dire la fréquence de la sinusoïde (cf. fig. \ref{ft_sin}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin}
+    \caption{Représentation temporelle et spectre d'une sinusoïde dit << son pur >>.}
+    \label{ft_sin}
+\end{figure}
+
+Si on veut observer ce sinus sur un temps fini, on va être obligé de le tronquer. L'énergie va alors d'une part se répartir autour de la fréquence de la sinusoïde. C'est ce qu'on appelle l'\textbf{étalement spectral}. D'autre part, on observe la présence d'énergie dans toutes les fréquences. Ce phénomène est appelé \textbf{fuite spectrale} (cf. fig. \ref{ft_sin_01}). L'étalement et fuite sont directement liés à la troncature.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin_01}
+    \caption{Représentation temporelle et spectre d'une sinusoïde tronquée.}
+    \label{ft_sin_01}
+\end{figure}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin_fen}
+    \caption{Représentation temporelle et spectre d'une sinusoïde modulée par une fenêtre de Hanning.}
+    \label{ft_sin_fen}
+\end{figure}
+
+Si on réalise la troncature de façon non rectangulaire mais en « fenêtrant » le signal - par une fenêtre de Hanning par exemple, (cf.  fig. \ref{ft_sin_fen}) - les transitions dans le signal sont alors plus douces. La fuite spectrale est alors limitée mais l'étalement en fréquence est toujours présent. Ce point est important pour comprendre le rôle des \textbf{fenêtres d'analyse}. Si la fenêtre a des discontinuités fortes, les fuites spectrales vont être importantes, mais l'étalement moindre. Si on prend une fenêtre de discontinuité plus douce, on va au contraire obtenir un étalement plus grand, mais moins de fuites spectrales.
+
+
+\section{Le spectrogramme}
+
+L'intérêt du spectrogramme est de pouvoir représenter le spectre en évoluant dans le temps. Le nom scientifique de la fonction mathématique associée à cet outil, plus communément appelé « spectrogramme », est la Transformée de Fourier à Court Terme (TFCT). Ce nom provient de l'analyse effectuée sur des fenêtres de support temporel fini. Une autre dénomination de cette représentation est « sonagramme ». Il s'agit d'une marque déposée Kay Electronics.
+
+Le principe du spectrogramme est de « découper » le son en trames. Pour chacune de ces trames on calcule une transformée de Fourier comme le schématise la figure \ref{tfct}. Ce spectre est alors représenté à un temps correspondant à celui du centre de la fenêtre, sous forme d'un code de couleur.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/tfct}
+    \caption{Description schématique de l'analyse temps/fréquence par la FFT.}
+    \label{tfct}
+\end{figure}
+
+La figure \ref{spectr_voix1} montre un exemple de spectrogramme d'un échantillon sonore de voix chantée. Il s'agit d'un glissando C5-E5 réalisé par une soprano. L'analyse a été effectuée avec une fenêtre de Hanning de longueur 23 ms. Le jaune correspond aux amplitudes les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a ainsi une idée de l'aspect du spectre au temps t. A chaque calcul du spectre, le signal est fenêtré de façon à pouvoir régler à la fois la fuite et l'étalement spectral. On observe facilement le glissando et le vibrato de la chanteuse.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectr_voix1}
+    \caption{Spectrogramme d'un glissando C5-E5 réalisé par une soprano (fenêtre : Hanning, 6 ms de largeur).}
+    \label{spectr_voix1}
+\end{figure}
+
+Pour mesurer le vibrato par exemple, on serait tenté de réduire la longueur de la fenêtre dans le temps pour gagner en précision et suivre au mieux les variations du spectre. En réalité, si on réduit la longueur des fenêtres (cf. fig. \ref{spectr_fen}), l'étalement spectral augmente, par conséquent la largeur des raies sur le spectrogramme aussi, ce qui perturbe finalement la mesure, car on ne distingue plus distinctement les différentes trajectoires dans le spectrogramme.
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Fenêtre de 100 ms]{\includegraphics[width=8cm]{img/spectrogram_01.png}}
+    \subfigure[Fenêtre de 20 ms]{\includegraphics[width=8cm]{img/spectrogram_02.png}}
+    \caption{Effet de la modification de la largeur d'une fenêtre temporelle lors du calcul d'un spectrogramme sur un signal contenant un vibrato.}
+    \label{spectr_fen}
+\end{figure}
+
+Si on revient à la même longueur de fenêtre que dans le premier exemple, tout en utilisant une fenêtre rectangulaire au lieu de la fenêtre « douce » de Hanning, l'étalement spectral est plus faible et les lignes sur le spectrogramme plus fines. Dans ce cas, les fuites spectrales sont beaucoup plus importantes et caractérisées par un manque de contraste dans la représentation du spectrogramme (cf. fig. \ref{spectr_fen_2}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectrogram_03.png}
+    \caption{Spectrogramme d'un glissando C5-E5 réalisé par une soprano (fenêtre : rectangulaire, 23 ms de largeur).}
+    \label{spectr_fen_2}
+\end{figure}
+
+Les points importants quant à l'utilisation du spectrogramme sont donc:
+\begin{itemize}
+ \item la longueur de fenêtre pour ajuster la précision temporelle, au prix d'un étalement spectral qui peut devenir rédhibitoire,
+ \item le choix de la fenêtre qui va conditionner le contraste du spectrogramme, pour une longueur de fenêtre donnée.
+\end{itemize}
+
+
+\section{Pertinence et choix d'un mode de représentation}
+
+La représentation du signal en temporel est particulièrement adapté à l'analyse de ses variations temporelles, par exemple lors de l'attaque ou de l'extinction d'un son. Au contraire, cette représentation apporte peu d'information lorsqu'on s'intéresse plutôt au contenu spectral du signal. Sur la figure \ref{spectr_voix2}, on voit que la représentation spectrale de ces deux signaux de parole permet assez bien de différentier la voix « tendue »  de la voix « relâchée ».
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Voix relâchée]{\includegraphics[width=8cm]{img/spectro_vibrato.png}}
+    \subfigure[Voix tendue]{\includegraphics[width=8cm]{img/spectro_vibrato_2.png}}
+    \caption{Spectrogrammes d'un son de voix relâchée (A) et d'une voix tendue (B).}
+    \label{spectr_voix2}
+\end{figure}
+
+Un autre intérêt de la représentation du signal à l'aide d'un spectrogramme est de pouvoir séparer des sources sonores, \cad des structures, ou, tout du moins, de pouvoir repérer des événements comme le montre la figure \ref{spectr_sep} où l'on distingue deux structures spectrales distinctes.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectr_sep.pdf}
+    \caption{Identification de structures sonores à l'aide d'un spectrogramme.}
+    \label{spectr_sep}
+\end{figure}
+
+
+Pour des sons très harmoniques, une voix chantée par exemple, le spectrogramme permet de repérer très facilement la fondamentale ainsi que la série harmonique. La figure \ref{spectro_harm} représente le spectrogramme d'un chanteur effectuant un glissando du bas au haut de sa tessiture puis l'inverse. On distingue facilement l'évolution temporelle de chaque harmonique contenu dans le signal.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectro_harm.png}
+    \caption{Identification de l'évolution temporelle des harmoniques à l'aide d'un spectrogramme.}
+    \label{spectro_harm}
+\end{figure}
+
+Dans le cas de la voix parlée, on peut repérer les parties voisée - harmoniques ou non - et même identifier quels sont les phonèmes prononcés avec un peu d'expérience. Cette technique permet par exemple d'améliorer les algorithme de reconnaissance vocale (cf. fig. \ref{spec_lux}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spec_lux.png}
+    \caption{Identification de mots à l'aide d'un spectrogramme. Phrase : << C'est une maison qui s'trouve au Luxembourg >>.}
+    \label{spec_lux}
+\end{figure}
+
+
+\section{Le spectre moyenné (Long Term Average Spectrum LTAS)}
+
+Dans certains cas, il est également très utile de représenter une partie du signal sous forme de spectre moyenné. Le spectre moyenné consiste à faire la moyenne du spectrogramme sur l'ensemble des instants d'un intervalle de temps. Cette opération peut être effectuée sur le signal tout entier comme sur un morceau particulier.
+
+Pour examiner une voyelle en particulier dans un signal de parole par exemple, il est tout à fait approprié de moyenner le spectrogramme sur l'intervalle de temps de cette voyelle, de façon à pouvoir estimer les formants ou bien le contenu spectral de cette voyelle. Ainsi, sur la figure suivante, on peut examiner le spectre moyenné de la production vocale d'un chanteur sur un /a/. Cette représentation permet de caractériser ses premiers formants vocaliques, ainsi que le formant du chanteur (renforcement énergétique dans la zone 2000-4000 Hz) caractéristique des chanteurs lyriques et des acteurs de théâtre.
+
+

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index 0000000..27eb19e
--- /dev/null
+++ b/ReprSignaux.tex.backup
@@ -0,0 +1,125 @@
+\section{Fenêtrage des signaux}
+
+Nous avons vu dans le chapitre précédent que la Transformée de Fourier permet de représenter le spectre d'un signal dans le domaine fréquentiel. Pour une sinusoïde infinie, toute l'énergie du spectre est concentrée à une fréquence donnée, c'est à dire la fréquence de la sinusoïde (cf. fig. \ref{ft_sin}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin}
+    \caption{Représentation temporelle et spectre d'une sinusoïde dit << son pur >>.}
+    \label{ft_sin}
+\end{figure}
+
+Si on veut observer ce sinus sur un temps fini, on va être obligé de le tronquer. L'énergie va alors d'une part se répartir autour de la fréquence de la sinusoïde. C'est ce qu'on appelle l'\textbf{étalement spectral}. D'autre part, on observe la présence d'énergie dans toutes les fréquences. Ce phénomène est appelé \textbf{fuite spectrale} (cf. fig. \ref{ft_sin_01}). L'étalement et fuite sont directement liés à la troncature.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin_01}
+    \caption{Représentation temporelle et spectre d'une sinusoïde tronquée.}
+    \label{ft_sin_01}
+\end{figure}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin_fen}
+    \caption{Représentation temporelle et spectre d'une sinusoïde modulée par une fenêtre de Hanning.}
+    \label{ft_sin_fen}
+\end{figure}
+
+Si on réalise la troncature de façon non rectangulaire mais en « fenêtrant » le signal - par une fenêtre de Hanning par exemple, (cf.  fig. \ref{ft_sin_fen}) - les transitions dans le signal sont alors plus douces. La fuite spectrale est alors limitée mais l'étalement en fréquence est toujours présent. Ce point est important pour comprendre le rôle des \textbf{fenêtres d'analyse}. Si la fenêtre a des discontinuités fortes, les fuites spectrales vont être importantes, mais l'étalement moindre. Si on prend une fenêtre de discontinuité plus douce, on va au contraire obtenir un étalement plus grand, mais moins de fuites spectrales.
+
+
+\section{Le spectrogramme}
+
+L'intérêt du spectrogramme est de pouvoir représenter le spectre en évoluant dans le temps. Le nom scientifique de la fonction mathématique associée à cet outil, plus communément appelé « spectrogramme », est la Transformée de Fourier à Court Terme (TFCT). Ce nom provient de l'analyse effectuée sur des fenêtres de support temporel fini. Une autre dénomination de cette représentation est « sonagramme ». Il s'agit d'une marque déposée Kay Electronics.
+
+Le principe du spectrogramme est de « découper » le son en tranches ou trames qui se recouvrent. Pour chacune de ces tranches on calcule une transformée de Fourier comme le schématise la figure \ref{tfct}. Ce spectre est alors représenté à un temps correspondant à celui du centre de la fenêtre, sous forme d'un code de couleur.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/tfct}
+    \caption{Description schématique de l'analyse temps/fréquence par la FFT.}
+    \label{tfct}
+\end{figure}
+
+La figure \ref{spectr_voix1} montre un exemple de spectrogramme d'un échantillon sonore de voix chantée. Il s'agit d'un glissando C5-E5 réalisé par une soprano. L'analyse a été effectuée avec une fenêtre de Hanning de longueur 23 ms. Le jaune correspond aux amplitudes les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a ainsi une idée de l'aspect du spectre au temps t. A chaque calcul du spectre, le signal est fenêtré de façon à pouvoir régler à la fois la fuite et l'étalement spectral. On observe facilement le glissando et le vibrato de la chanteuse.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectr_voix1}
+    \caption{Spectrogramme d'un glissando C5-E5 réalisé par une soprano (fenêtre : Hanning, 6 ms de largeur).}
+    \label{spectr_voix1}
+\end{figure}
+
+Pour mesurer le vibrato par exemple, on serait tenté de réduire la longueur de la fenêtre dans le temps pour gagner en précision et suivre au mieux les variations du spectre. En réalité, si on réduit la longueur des fenêtres (cf. fig. \ref{spectr_fen}), l'étalement spectral augmente, par conséquent la largeur des raies sur le spectrogramme aussi, ce qui perturbe finalement la mesure, car on ne distingue plus distinctement les différentes trajectoires dans le spectrogramme.
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Fenêtre de 100 ms]{\includegraphics[width=8cm]{img/spectrogram_01.png}}
+    \subfigure[Fenêtre de 20 ms]{\includegraphics[width=8cm]{img/spectrogram_02.png}}
+    \caption{Effet de la modification de la largeur d'une fenêtre temporelle lors du calcul d'un spectrogramme sur un signal contenant un vibrato.}
+    \label{spectr_fen}
+\end{figure}
+
+Si on revient à la même longueur de fenêtre que dans le premier exemple, tout en utilisant une fenêtre rectangulaire au lieu de la fenêtre « douce » de Hanning, l'étalement spectral est plus faible et les lignes sur le spectrogramme plus fines, mais on a par contre des fuites spectrales beaucoup plus importantes, caractérisées par un manque de contraste dans la représentation du spectrogramme (cf. fig. \ref{spectr_fen_2}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectrogram_03.png}
+    \caption{Spectrogramme d'un glissando C5-E5 réalisé par une soprano (fenêtre : rectangulaire, 23 ms de largeur).}
+    \label{spectr_fen_2}
+\end{figure}   
+
+Les points importants quant à l'utilisation du spectrogramme sont donc:
+\begin{itemize}
+ \item la longueur de fenêtre pour ajuster la précision temporelle, au prix d'un étalement spectral qui peut devenir rédhibitoire,
+ \item le choix de la fenêtre qui va conditionner le contraste du spectrogramme, pour une longueur de fenêtre donnée.
+\end{itemize}
+
+
+\section{Pertinence et choix d'un mode de représentation}
+
+La représentation du signal en temporel est particulièrement adapté à l'analyse de ses variations temporelles, par exemple lors de l'attaque ou de l'extinction d'un son. Au contraire, cette représentation apporte peu d'information lorsqu'on s'intéresse plutôt au contenu spectral du signal. Sur la figure \ref{spectr_voix2}, on voit que la représentation spectrale de ces deux signaux de parole permet assez bien de différentier la voix « tendue »  de la voix « relâchée ».
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Voix relâchée]{\includegraphics[width=8cm]{img/spectro_vibrato.png}}
+    \subfigure[Voix tendue]{\includegraphics[width=8cm]{img/spectro_vibrato_2.png}}
+    \caption{Spectrogrammes d'un son de voix relâchée (A) et d'une voix tendue (B).}
+    \label{spectr_voix2}
+\end{figure}
+
+Un autre intérêt de la représentation du signal à l'aide d'un spectrogramme est de pouvoir séparer des sources sonores, \cad des structures, ou, tout du moins, de pouvoir repérer des événements comme le montre la figure \ref{spectr_sep} où l'on distingue deux structures spectrales distinctes.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectr_sep.pdf}
+    \caption{Identification de structures sonores à l'aide d'un spectrogramme.}
+    \label{spectr_sep}
+\end{figure}
+
+
+Pour des sons très harmoniques, une voix chantée par exemple, le spectrogramme permet de repérer très facilement la fondamentale ainsi que la série harmonique. La figure \ref{spectro_harm} représente le spectrogramme d'un chanteur effectuant un glissando du bas au haut de sa tessiture puis l'inverse. On distingue facilement l'évolution temporelle de chaque harmonique contenu dans le signal.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectro_harm.png}
+    \caption{Identification de l'évolution temporelle des harmoniques à l'aide d'un spectrogramme.}
+    \label{spectro_harm}
+\end{figure}
+
+Dans le cas de la voix parlée, on peut repérer les parties voisée - harmoniques ou non - et même identifier quels sont les phonèmes prononcés avec un peu d'expérience. Cette technique permet par exemple d'améliorer les algorithme de reconnaissance vocale (cf. fig. \ref{spec_lux}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spec_lux.png}
+    \caption{Identification de mots à l'aide d'un spectrogramme. Phrase : << C'est une maison qui s'trouve au Luxembourg >>.}
+    \label{spec_lux}
+\end{figure}
+
+
+\section{Le spectre moyenné (Long Term Average Spectrum   LTAS)}
+
+Il est également très utile dans certains cas de représenter une partie du signal sous forme de spectre moyenné. Le spectre moyenné consiste à faire la moyenne du spectrogramme sur l'ensemble des instants d'un intervalle de temps. Cette opération peut être effectuée sur le signal tout entier comme sur un morceau particulier.
+
+Pour examiner une voyelle en particulier dans un signal de parole par exemple, il est tout à fait approprié de moyenner le spectrogramme sur l'intervalle de temps de cette voyelle, de façon à pouvoir estimer les formants ou bien le contenu spectral de cette voyelle. Ainsi, sur la figure suivante, on peut examiner le spectre moyenné de la production vocale d'un chanteur sur un /a/. Cette représentation permet de caractériser ses premiers formants vocaliques, ainsi que le formant du chanteur (renforcement énergétique dans la zone 2000-4000 Hz) caractéristique des chanteurs lyriques et des acteurs de théâtre.
+

diff --git a/ReprSignaux.tex~ b/ReprSignaux.tex~
new file mode 100644 (file)
index 0000000..a45066c
--- /dev/null
+++ b/ReprSignaux.tex~
@@ -0,0 +1,126 @@
+\section{Fenêtrage des signaux}
+
+Nous avons vu dans le chapitre précédent que la Transformée de Fourier permet de représenter le spectre d'un signal dans le domaine fréquentiel. Pour une sinusoïde infinie, toute l'énergie du spectre est concentrée à une fréquence donnée, c'est à dire la fréquence de la sinusoïde (cf. fig. \ref{ft_sin}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin}
+    \caption{Représentation temporelle et spectre d'une sinusoïde dit << son pur >>.}
+    \label{ft_sin}
+\end{figure}
+
+Si on veut observer ce sinus sur un temps fini, on va être obligé de le tronquer. L'énergie va alors d'une part se répartir autour de la fréquence de la sinusoïde. C'est ce qu'on appelle l'\textbf{étalement spectral}. D'autre part, on observe la présence d'énergie dans toutes les fréquences. Ce phénomène est appelé \textbf{fuite spectrale} (cf. fig. \ref{ft_sin_01}). L'étalement et fuite sont directement liés à la troncature.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin_01}
+    \caption{Représentation temporelle et spectre d'une sinusoïde tronquée.}
+    \label{ft_sin_01}
+\end{figure}
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/ft_sin_fen}
+    \caption{Représentation temporelle et spectre d'une sinusoïde modulée par une fenêtre de Hanning.}
+    \label{ft_sin_fen}
+\end{figure}
+
+Si on réalise la troncature de façon non rectangulaire mais en « fenêtrant » le signal - par une fenêtre de Hanning par exemple, (cf.  fig. \ref{ft_sin_fen}) - les transitions dans le signal sont alors plus douces. La fuite spectrale est alors limitée mais l'étalement en fréquence est toujours présent. Ce point est important pour comprendre le rôle des \textbf{fenêtres d'analyse}. Si la fenêtre a des discontinuités fortes, les fuites spectrales vont être importantes, mais l'étalement moindre. Si on prend une fenêtre de discontinuité plus douce, on va au contraire obtenir un étalement plus grand, mais moins de fuites spectrales.
+
+
+\section{Le spectrogramme}
+
+L'intérêt du spectrogramme est de pouvoir représenter le spectre en évoluant dans le temps. Le nom scientifique de la fonction mathématique associée à cet outil, plus communément appelé « spectrogramme », est la Transformée de Fourier à Court Terme (TFCT). Ce nom provient de l'analyse effectuée sur des fenêtres de support temporel fini. Une autre dénomination de cette représentation est « sonagramme ». Il s'agit d'une marque déposée Kay Electronics.
+
+Le principe du spectrogramme est de « découper » le son en tranches ou trames qui se recouvrent. Pour chacune de ces tranches on calcule une transformée de Fourier comme le schématise la figure \ref{tfct}. Ce spectre est alors représenté à un temps correspondant à celui du centre de la fenêtre, sous forme d'un code de couleur.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/tfct}
+    \caption{Description schématique de l'analyse temps/fréquence par la FFT.}
+    \label{tfct}
+\end{figure}
+
+La figure \ref{spectr_voix1} montre un exemple de spectrogramme d'un échantillon sonore de voix chantée. Il s'agit d'un glissando C5-E5 réalisé par une soprano. L'analyse a été effectuée avec une fenêtre de Hanning de longueur 23 ms. Le jaune correspond aux amplitudes les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a ainsi une idée de l'aspect du spectre au temps t. A chaque calcul du spectre, le signal est fenêtré de façon à pouvoir régler à la fois la fuite et l'étalement spectral. On observe facilement le glissando et le vibrato de la chanteuse.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectr_voix1}
+    \caption{Spectrogramme d'un glissando C5-E5 réalisé par une soprano (fenêtre : Hanning, 6 ms de largeur).}
+    \label{spectr_voix1}
+\end{figure}
+
+Pour mesurer le vibrato par exemple, on serait tenté de réduire la longueur de la fenêtre dans le temps pour gagner en précision et suivre au mieux les variations du spectre. En réalité, si on réduit la longueur des fenêtres (cf. fig. \ref{spectr_fen}), l'étalement spectral augmente, par conséquent la largeur des raies sur le spectrogramme aussi, ce qui perturbe finalement la mesure, car on ne distingue plus distinctement les différentes trajectoires dans le spectrogramme.
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Fenêtre de 100 ms]{\includegraphics[width=8cm]{img/spectrogram_01.png}}
+    \subfigure[Fenêtre de 20 ms]{\includegraphics[width=8cm]{img/spectrogram_02.png}}
+    \caption{Effet de la modification de la largeur d'une fenêtre temporelle lors du calcul d'un spectrogramme sur un signal contenant un vibrato.}
+    \label{spectr_fen}
+\end{figure}
+
+Si on revient à la même longueur de fenêtre que dans le premier exemple, tout en utilisant une fenêtre rectangulaire au lieu de la fenêtre « douce » de Hanning, l'étalement spectral est plus faible et les lignes sur le spectrogramme plus fines. Dans ce cas, les fuites spectrales sont beaucoup plus importantes et caractérisées par un manque de contraste dans la représentation du spectrogramme (cf. fig. \ref{spectr_fen_2}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectrogram_03.png}
+    \caption{Spectrogramme d'un glissando C5-E5 réalisé par une soprano (fenêtre : rectangulaire, 23 ms de largeur).}
+    \label{spectr_fen_2}
+\end{figure}
+
+Les points importants quant à l'utilisation du spectrogramme sont donc:
+\begin{itemize}
+ \item la longueur de fenêtre pour ajuster la précision temporelle, au prix d'un étalement spectral qui peut devenir rédhibitoire,
+ \item le choix de la fenêtre qui va conditionner le contraste du spectrogramme, pour une longueur de fenêtre donnée.
+\end{itemize}
+
+
+\section{Pertinence et choix d'un mode de représentation}
+
+La représentation du signal en temporel est particulièrement adapté à l'analyse de ses variations temporelles, par exemple lors de l'attaque ou de l'extinction d'un son. Au contraire, cette représentation apporte peu d'information lorsqu'on s'intéresse plutôt au contenu spectral du signal. Sur la figure \ref{spectr_voix2}, on voit que la représentation spectrale de ces deux signaux de parole permet assez bien de différentier la voix « tendue »  de la voix « relâchée ».
+
+\begin{figure}[tp]
+    \centering
+    \subfigure[Voix relâchée]{\includegraphics[width=8cm]{img/spectro_vibrato.png}}
+    \subfigure[Voix tendue]{\includegraphics[width=8cm]{img/spectro_vibrato_2.png}}
+    \caption{Spectrogrammes d'un son de voix relâchée (A) et d'une voix tendue (B).}
+    \label{spectr_voix2}
+\end{figure}
+
+Un autre intérêt de la représentation du signal à l'aide d'un spectrogramme est de pouvoir séparer des sources sonores, \cad des structures, ou, tout du moins, de pouvoir repérer des événements comme le montre la figure \ref{spectr_sep} où l'on distingue deux structures spectrales distinctes.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectr_sep.pdf}
+    \caption{Identification de structures sonores à l'aide d'un spectrogramme.}
+    \label{spectr_sep}
+\end{figure}
+
+
+Pour des sons très harmoniques, une voix chantée par exemple, le spectrogramme permet de repérer très facilement la fondamentale ainsi que la série harmonique. La figure \ref{spectro_harm} représente le spectrogramme d'un chanteur effectuant un glissando du bas au haut de sa tessiture puis l'inverse. On distingue facilement l'évolution temporelle de chaque harmonique contenu dans le signal.
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spectro_harm.png}
+    \caption{Identification de l'évolution temporelle des harmoniques à l'aide d'un spectrogramme.}
+    \label{spectro_harm}
+\end{figure}
+
+Dans le cas de la voix parlée, on peut repérer les parties voisée - harmoniques ou non - et même identifier quels sont les phonèmes prononcés avec un peu d'expérience. Cette technique permet par exemple d'améliorer les algorithme de reconnaissance vocale (cf. fig. \ref{spec_lux}).
+
+\begin{figure}[htbp]
+    \centering
+    \includegraphics[width=12cm]{img/spec_lux.png}
+    \caption{Identification de mots à l'aide d'un spectrogramme. Phrase : << C'est une maison qui s'trouve au Luxembourg >>.}
+    \label{spec_lux}
+\end{figure}
+
+
+\section{Le spectre moyenné (Long Term Average Spectrum LTAS)}
+
+Dans certains cas, il est également très utile de représenter une partie du signal sous forme de spectre moyenné. Le spectre moyenné consiste à faire la moyenne du spectrogramme sur l'ensemble des instants d'un intervalle de temps. Cette opération peut être effectuée sur le signal tout entier comme sur un morceau particulier.
+
+Pour examiner une voyelle en particulier dans un signal de parole par exemple, il est tout à fait approprié de moyenner le spectrogramme sur l'intervalle de temps de cette voyelle, de façon à pouvoir estimer les formants ou bien le contenu spectral de cette voyelle. Ainsi, sur la figure suivante, on peut examiner le spectre moyenné de la production vocale d'un chanteur sur un /a/. Cette représentation permet de caractériser ses premiers formants vocaliques, ainsi que le formant du chanteur (renforcement énergétique dans la zone 2000-4000 Hz) caractéristique des chanteurs lyriques et des acteurs de théâtre.
+
+

diff --git a/Signal-CPDA-CNAM.aux b/Signal-CPDA-CNAM.aux
new file mode 100644 (file)
index 0000000..1d00b47
--- /dev/null
+++ b/Signal-CPDA-CNAM.aux
@@ -0,0 +1,157 @@
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+\@writefile{toc}{\contentsline {section}{\numberline {1.2}Classification des signaux}{6}{section.1.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Classification dimensionnelle}{6}{subsection.1.2.1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Classification ph\'enom\'enologique}{6}{subsection.1.2.2}}
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+\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.4}Classification \'energ\'etique}{7}{subsection.1.2.4}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.3}Signaux et distributions classiques}{7}{section.1.3}}
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+\@writefile{toc}{\contentsline {section}{\numberline {1.4}Evaluation de la similitude entre deux signaux}{8}{section.1.4}}
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+\citation{*}
+\bibdata{biblio}
+\bibcite{charbit}{1}
+\bibcite{fallon}{2}
+\bibcite{feng}{3}
+\bibcite{garnier1}{4}
+\bibcite{heiser}{5}
+\bibcite{henrich}{6}
+\bibcite{laroche}{7}
+\bibcite{matignon}{8}
+\bibcite{pelorson}{9}

diff --git a/Signal-CPDA-CNAM.bbl b/Signal-CPDA-CNAM.bbl
new file mode 100644 (file)
index 0000000..0de7f85
--- /dev/null
+++ b/Signal-CPDA-CNAM.bbl
@@ -0,0 +1,50 @@
+\begin{thebibliography}{1}
+
+\bibitem{charbit}
+M.~Charbit.
+\newblock Bases de traitement du signal.
+\newblock Cours de DEA, ENST, Paris, 2004.
+
+\bibitem{fallon}
+M.~Fallon.
+\newblock Traitement du signal.
+\newblock Support de cours, ENSAM, Lille, 1997.
+
+\bibitem{feng}
+M.~Feng.
+\newblock Traitement du signal.
+\newblock Support de cours, ENSERG. INP Grenoble, 2000.
+
+\bibitem{garnier1}
+M.~Garnier, D.~Dubois, J.~Poitevineau, N.~Henrich, and M.~Castellengo.
+\newblock Perception et description verbale de la qualité vocale dans le chant
+  lyrique : une approche cognitive.
+\newblock Fez, 2004. Journées d'Études sur la Parole.
+
+\bibitem{heiser}
+T.~Heiser.
+\newblock Traitement du signal.
+\newblock Support de cours, UFR Sciences Physiques, Université Louis Pasteur,
+  Strasbourg.
+
+\bibitem{henrich}
+N.~Henrich.
+\newblock {\em Etude de la source glottique en voix parlée et chantée}.
+\newblock PhD thesis, Université Pierre et Marie Curie, 2001.
+
+\bibitem{laroche}
+J.~Laroche.
+\newblock Traitement des signaux audio-fréquences.
+\newblock Cours de DEA, ENST, Paris, 1995.
+
+\bibitem{matignon}
+D.~Matignon.
+\newblock Traitement du signal.
+\newblock Support de cours, ENST, Paris, 2000.
+
+\bibitem{pelorson}
+X.~Pelerson.
+\newblock Cordes vocales et lèvres vibrantes.
+\newblock Paris, juillet 2004. Conférence SFA.
+
+\end{thebibliography}

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diff --git a/Signal-CPDA-CNAM.out b/Signal-CPDA-CNAM.out
new file mode 100644 (file)
index 0000000..327c521
--- /dev/null
+++ b/Signal-CPDA-CNAM.out
@@ -0,0 +1,47 @@
+\BOOKMARK [0][]{chapter.1}{1 Les signaux}{}
+\BOOKMARK [1][]{section.1.1}{1.1 Pr\351ambule}{chapter.1}
+\BOOKMARK [1][]{section.1.2}{1.2 Classification des signaux}{chapter.1}
+\BOOKMARK [2][]{subsection.1.2.1}{1.2.1 Classification dimensionnelle}{section.1.2}
+\BOOKMARK [2][]{subsection.1.2.2}{1.2.2 Classification ph\351nom\351nologique}{section.1.2}
+\BOOKMARK [2][]{subsection.1.2.3}{1.2.3 Classification morphologique}{section.1.2}
+\BOOKMARK [2][]{subsection.1.2.4}{1.2.4 Classification \351nerg\351tique}{section.1.2}
+\BOOKMARK [1][]{section.1.3}{1.3 Signaux et distributions classiques}{chapter.1}
+\BOOKMARK [2][]{subsection.1.3.1}{1.3.1 Fen\352tre rectangulaire ou Porte}{section.1.3}
+\BOOKMARK [2][]{subsection.1.3.2}{1.3.2 Impulsion de Dirac}{section.1.3}
+\BOOKMARK [2][]{subsection.1.3.3}{1.3.3 << Peigne >> de Dirac}{section.1.3}
+\BOOKMARK [2][]{subsection.1.3.4}{1.3.4 Sinus cardinal}{section.1.3}
+\BOOKMARK [1][]{section.1.4}{1.4 Evaluation de la similitude entre deux signaux}{chapter.1}
+\BOOKMARK [2][]{subsection.1.4.1}{1.4.1 Les fonctions d'inter-corr\351lation et d'auto-corr\351lation}{section.1.4}
+\BOOKMARK [0][]{chapter.2}{2 Transform\351es de signaux}{}
+\BOOKMARK [1][]{section.2.1}{2.1 Le produit de convolution}{chapter.2}
+\BOOKMARK [1][]{section.2.2}{2.2 Rappels sur la d\351composition en s\351rie de Fourier des signaux p\351riodiques}{chapter.2}
+\BOOKMARK [1][]{section.2.3}{2.3 La transform\351e de Fourier}{chapter.2}
+\BOOKMARK [2][]{subsection.2.3.1}{2.3.1 D\351finition de la transform\351e de Fourier}{section.2.3}
+\BOOKMARK [2][]{subsection.2.3.2}{2.3.2 Propri\351t\351s de la transform\351e de Fourier}{section.2.3}
+\BOOKMARK [2][]{subsection.2.3.3}{2.3.3 Les transform\351es de Fourier de signaux courants}{section.2.3}
+\BOOKMARK [2][]{subsection.2.3.4}{2.3.4 La fonction de densit\351 spectrale d'\351nergie}{section.2.3}
+\BOOKMARK [1][]{section.2.4}{2.4 L'\351chantillonnage}{chapter.2}
+\BOOKMARK [2][]{subsection.2.4.1}{2.4.1 D\351finition de l'\351chantillonnage}{section.2.4}
+\BOOKMARK [2][]{subsection.2.4.2}{2.4.2 La quantification}{section.2.4}
+\BOOKMARK [1][]{section.2.5}{2.5 La transform\351e de Fourier \340 temps discret}{chapter.2}
+\BOOKMARK [1][]{section.2.6}{2.6 Le th\351or\350me de Shannon}{chapter.2}
+\BOOKMARK [1][]{section.2.7}{2.7 La transform\351e en Z}{chapter.2}
+\BOOKMARK [2][]{subsection.2.7.1}{2.7.1 D\351finition de la transform\351e en Z}{section.2.7}
+\BOOKMARK [2][]{subsection.2.7.2}{2.7.2 Existence de la transform\351e en Z}{section.2.7}
+\BOOKMARK [2][]{subsection.2.7.3}{2.7.3 Propri\351t\351s de la transform\351e en Z}{section.2.7}
+\BOOKMARK [0][]{chapter.3}{3 Syst\350mes lin\351aires et filtrage}{}
+\BOOKMARK [1][]{section.3.1}{3.1 Introduction}{chapter.3}
+\BOOKMARK [1][]{section.3.2}{3.2 R\351ponse impulsionnelle d'un syst\350me}{chapter.3}
+\BOOKMARK [1][]{section.3.3}{3.3 R\351ponse fr\351quentielle d'un syst\350me}{chapter.3}
+\BOOKMARK [1][]{section.3.4}{3.4 Filtrage}{chapter.3}
+\BOOKMARK [2][]{subsection.3.4.1}{3.4.1 Filtres classiques - vocabulaire}{section.3.4}
+\BOOKMARK [2][]{subsection.3.4.2}{3.4.2 Filtres num\351riques}{section.3.4}
+\BOOKMARK [2][]{subsection.3.4.3}{3.4.3 Exemples de filtres}{section.3.4}
+\BOOKMARK [3][]{subsubsection.3.4.3.1}{3.4.3.1 Le filtre moyenneur lisseur}{subsection.3.4.3}
+\BOOKMARK [3][]{subsubsection.3.4.3.2}{3.4.3.2 Le filtre passe-bas}{subsection.3.4.3}
+\BOOKMARK [3][]{subsubsection.3.4.3.3}{3.4.3.3 Le filtre vocal}{subsection.3.4.3}
+\BOOKMARK [0][]{chapter.4}{4 Modes de repr\351sentation des signaux}{}
+\BOOKMARK [1][]{section.4.1}{4.1 Fen\352trage des signaux}{chapter.4}
+\BOOKMARK [1][]{section.4.2}{4.2 Le spectrogramme}{chapter.4}
+\BOOKMARK [1][]{section.4.3}{4.3 Pertinence et choix d'un mode de repr\351sentation}{chapter.4}
+\BOOKMARK [1][]{section.4.4}{4.4 Le spectre moyenn\351 \(Long Term Average Spectrum LTAS\)}{chapter.4}

diff --git a/Signal-CPDA-CNAM.pdf b/Signal-CPDA-CNAM.pdf
new file mode 100644 (file)
index 0000000..7a18324
Binary files /dev/null and b/Signal-CPDA-CNAM.pdf differ

diff --git a/Signal-CPDA-CNAM.tex b/Signal-CPDA-CNAM.tex
new file mode 100644 (file)
index 0000000..3779e3c
--- /dev/null
+++ b/Signal-CPDA-CNAM.tex
@@ -0,0 +1,60 @@
+\documentclass[pdftex,11pt,a4paper,twoside]{report}
+
+\usepackage{pdf_course_profile}
+
+%\title{Cours d'électricité}
+
+\title{\begin{flushright}
+    \rule{17cm}{1mm}\\
+    \vspace{8mm}
+    \huge{\bf{Traitement du signal}}\\
+       \vspace{4mm}
+    \rule{17cm}{1mm}\\
+    \large{Cours du CNAM - CPDA - 2012}\\
+    \large{modifié le }\today\\
+    \large{Version 0.5.1}
+    \end{flushright}
+}
+
+\author{\begin{flushright}
+       \textbf{Guillaume Pellerin$^1$}\\
+       \end{flushright}
+    \vspace{12cm}
+       \begin{center}
+    \small{
+    $^1$ Laboratoire d'Acoustique, Conservatoire National des Arts et Métiers\\
+    5 rue du Vert Bois F-75003 Paris\\ <\href{mailto:guillaume.pellerin@parisson.com}{guillaume.pellerin@parisson.com}>\\
+     \vspace{0.5cm}
+    }
+    \end{center}}
+
+\begin{document}
+
+\maketitle
+
+\newpage
+\include{Abstract}
+
+\tableofcontents
+\cleardoublepage
+\renewcommand{\labelitemi}{$\bullet$}
+
+\chapter{Les signaux}
+\input{Signaux}
+
+\chapter{Transformées de signaux}
+\input{IntroMethFreq}
+
+\chapter{Systèmes linéaires et filtrage}
+\input{SystLinFilt}
+
+\chapter{Modes de représentation des signaux}
+\input{ReprSignaux}
+
+\newpage
+\bibliographystyle{plain}
+\nocite{*}
+\bibliography{biblio}
+
+\end{document}
+

diff --git a/Signal-CPDA-CNAM.tex~ b/Signal-CPDA-CNAM.tex~
new file mode 100644 (file)
index 0000000..26ab320
--- /dev/null
+++ b/Signal-CPDA-CNAM.tex~
@@ -0,0 +1,60 @@
+\documentclass[pdftex,11pt,a4paper,twoside]{report}
+
+\usepackage{pdf_course_profile}
+
+%\title{Cours d'électricité}
+
+\title{\begin{flushright}
+    \rule{17cm}{1mm}\\
+    \vspace{8mm}
+    \huge{\bf{Traitement du signal}}\\
+       \vspace{4mm}
+    \rule{17cm}{1mm}\\
+    \large{Cours du CNAM - CPDA - 2011}\\
+    \large{modifié le }\today\\
+    \large{Version 0.5.1}
+    \end{flushright}
+}
+
+\author{\begin{flushright}
+       \textbf{Guillaume Pellerin$^1$}\\
+       \end{flushright}
+    \vspace{12cm}
+       \begin{center}
+    \small{
+    $^1$ Laboratoire d'Acoustique, Conservatoire National des Arts et Métiers\\
+    5 rue du Vert Bois F-75003 Paris\\ <\href{mailto:guillaume.pellerin@parisson.com}{guillaume.pellerin@parisson.com}>\\
+     \vspace{0.5cm}
+    }
+    \end{center}}
+
+\begin{document}
+
+\maketitle
+
+\newpage
+\include{Abstract}
+
+\tableofcontents
+\cleardoublepage
+\renewcommand{\labelitemi}{$\bullet$}
+
+\chapter{Les signaux}
+\input{Signaux}
+
+\chapter{Transformées de signaux}
+\input{IntroMethFreq}
+
+\chapter{Systèmes linéaires et filtrage}
+\input{SystLinFilt}
+
+\chapter{Modes de représentation des signaux}
+\input{ReprSignaux}
+
+\newpage
+\bibliographystyle{plain}
+\nocite{*}
+\bibliography{biblio}
+
+\end{document}
+

diff --git a/Signal-CPDA-CNAM.toc b/Signal-CPDA-CNAM.toc
new file mode 100644 (file)
index 0000000..c11837f
--- /dev/null
+++ b/Signal-CPDA-CNAM.toc
@@ -0,0 +1,48 @@
+\select@language {french}
+\contentsline {chapter}{\numberline {1}Les signaux}{5}{chapter.1}
+\contentsline {section}{\numberline {1.1}Pr\'eambule}{5}{section.1.1}
+\contentsline {section}{\numberline {1.2}Classification des signaux}{6}{section.1.2}
+\contentsline {subsection}{\numberline {1.2.1}Classification dimensionnelle}{6}{subsection.1.2.1}
+\contentsline {subsection}{\numberline {1.2.2}Classification ph\'enom\'enologique}{6}{subsection.1.2.2}
+\contentsline {subsection}{\numberline {1.2.3}Classification morphologique}{6}{subsection.1.2.3}
+\contentsline {subsection}{\numberline {1.2.4}Classification \'energ\'etique}{7}{subsection.1.2.4}
+\contentsline {section}{\numberline {1.3}Signaux et distributions classiques}{7}{section.1.3}
+\contentsline {subsection}{\numberline {1.3.1}Fen\^etre rectangulaire ou Porte}{7}{subsection.1.3.1}
+\contentsline {subsection}{\numberline {1.3.2}Impulsion de Dirac}{7}{subsection.1.3.2}
+\contentsline {subsection}{\numberline {1.3.3}<<\IeC {\nobreakspace }Peigne\IeC {\nobreakspace }>> de Dirac}{8}{subsection.1.3.3}
+\contentsline {subsection}{\numberline {1.3.4}Sinus cardinal}{8}{subsection.1.3.4}
+\contentsline {section}{\numberline {1.4}Evaluation de la similitude entre deux signaux}{8}{section.1.4}
+\contentsline {subsection}{\numberline {1.4.1}Les fonctions d'inter-corr\'elation et d'auto-corr\'elation}{8}{subsection.1.4.1}
+\contentsline {chapter}{\numberline {2}Transform\'ees de signaux}{10}{chapter.2}
+\contentsline {section}{\numberline {2.1}Le produit de convolution}{10}{section.2.1}
+\contentsline {section}{\numberline {2.2}Rappels sur la d\'ecomposition en s\'erie de Fourier des signaux p\'eriodiques}{11}{section.2.2}
+\contentsline {section}{\numberline {2.3}La transform\'ee de Fourier}{11}{section.2.3}
+\contentsline {subsection}{\numberline {2.3.1}D\'efinition de la transform\'ee de Fourier}{11}{subsection.2.3.1}
+\contentsline {subsection}{\numberline {2.3.2}Propri\'et\'es de la transform\'ee de Fourier}{12}{subsection.2.3.2}
+\contentsline {subsection}{\numberline {2.3.3}Les transform\'ees de Fourier de signaux courants}{13}{subsection.2.3.3}
+\contentsline {subsection}{\numberline {2.3.4}La fonction de densit\'e spectrale d'\'energie}{13}{subsection.2.3.4}
+\contentsline {section}{\numberline {2.4}L'\'echantillonnage}{14}{section.2.4}
+\contentsline {subsection}{\numberline {2.4.1}D\'efinition de l'\'echantillonnage}{14}{subsection.2.4.1}
+\contentsline {subsection}{\numberline {2.4.2}La quantification}{14}{subsection.2.4.2}
+\contentsline {section}{\numberline {2.5}La transform\'ee de Fourier \`a temps discret}{14}{section.2.5}
+\contentsline {section}{\numberline {2.6}Le th\'eor\`eme de Shannon}{15}{section.2.6}
+\contentsline {section}{\numberline {2.7}La transform\'ee en Z}{16}{section.2.7}
+\contentsline {subsection}{\numberline {2.7.1}D\'efinition de la transform\'ee en Z}{16}{subsection.2.7.1}
+\contentsline {subsection}{\numberline {2.7.2}Existence de la transform\'ee en Z}{16}{subsection.2.7.2}
+\contentsline {subsection}{\numberline {2.7.3}Propri\'et\'es de la transform\'ee en Z}{17}{subsection.2.7.3}
+\contentsline {chapter}{\numberline {3}Syst\`emes lin\'eaires et filtrage}{19}{chapter.3}
+\contentsline {section}{\numberline {3.1}Introduction}{19}{section.3.1}
+\contentsline {section}{\numberline {3.2}R\'eponse impulsionnelle d'un syst\`eme}{19}{section.3.2}
+\contentsline {section}{\numberline {3.3}R\'eponse fr\'equentielle d'un syst\`eme}{20}{section.3.3}
+\contentsline {section}{\numberline {3.4}Filtrage}{20}{section.3.4}
+\contentsline {subsection}{\numberline {3.4.1}Filtres classiques - vocabulaire}{21}{subsection.3.4.1}
+\contentsline {subsection}{\numberline {3.4.2}Filtres num\'eriques}{21}{subsection.3.4.2}
+\contentsline {subsection}{\numberline {3.4.3}Exemples de filtres}{22}{subsection.3.4.3}
+\contentsline {subsubsection}{\numberline {3.4.3.1}Le filtre moyenneur lisseur}{22}{subsubsection.3.4.3.1}
+\contentsline {subsubsection}{\numberline {3.4.3.2}Le filtre passe-bas}{22}{subsubsection.3.4.3.2}
+\contentsline {subsubsection}{\numberline {3.4.3.3}Le filtre vocal}{25}{subsubsection.3.4.3.3}
+\contentsline {chapter}{\numberline {4}Modes de repr\'esentation des signaux}{27}{chapter.4}
+\contentsline {section}{\numberline {4.1}Fen\^etrage des signaux}{27}{section.4.1}
+\contentsline {section}{\numberline {4.2}Le spectrogramme}{29}{section.4.2}
+\contentsline {section}{\numberline {4.3}Pertinence et choix d'un mode de repr\'esentation}{31}{section.4.3}
+\contentsline {section}{\numberline {4.4}Le spectre moyenn\'e (Long Term Average Spectrum LTAS)}{34}{section.4.4}

diff --git a/Signaux.tex b/Signaux.tex
new file mode 100644 (file)
index 0000000..c7193ee
--- /dev/null
+++ b/Signaux.tex
@@ -0,0 +1,150 @@
+\section{Préambule}
+
+L'information, qui se présente à nous lors de l'observation d'un phénomène, se manifeste sous forme d'une ou plusieurs grandeurs physiques qui se déroulent à la fois dans le temps et dans l'espace. Dans les problèmes rencontrés en pratique, on est souvent amené à s'intéresser plus particulièrement à l'un ou l'autre de ces deux aspects. On désigne alors par le terme de \textbf{signal} l'évolution temporelle, tandis que l'on désigne par le terme d'\textbf{image} l'évolution spatiale.
+
+Les différentes opérations que l'on fait subir aux signaux nécessitent des outils mathématiques appropriés qui sont à la base du traitement du signal. Dans ce cours nous envisageons l'étude de certains de ces outils. La théorie du signal propose de modéliser la grandeur physique observée par un objet mathématique dépendant de la variable réelle $t$ représentant le temps. Dans la suite, le  signal désignera indifféremment la grandeur physique observée ou l'objet mathématique servant à la modéliser. Le figure \ref{ex_sig} donne un aperçu de la représentation temporelle de la forme d'onde d'un signal musical.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=12cm]{img/ex_sig}
+    \caption{Exemple d'une réprésentation temporelle d'un signal musical combinant un son de grosse caisse et celui d'une caisse claire.}
+    \label{ex_sig}
+\end{figure}
+
+
+\section{Classification des signaux}
+
+En théorie du signal, on a l'habitude d'envisager plusieurs types de signaux. Pour rendre la suite de ce cours plus compréhensible, nous présentons dans cette section les différents modes de classification des signaux.
+
+\subsection{Classification dimensionnelle}
+
+On peut différentier les signaux selon leur nombre de variables libres. Par exemple un signal électrique $V(t)$ est un signal unidimensionnel dépendant du temps. Une image statique en noir et blanc peut être modélisée par un signal bidimensionnel correspondant à la brillance en fonction de la largeur et de la hauteur de l'image. Enfin, un film en noir et blanc peut être considéré comme une image fonction du temps, autrement dit modélisé par un signal tridimensionnel dépendant de l'abscisse et de l'ordonnée de l'image mais également du temps.
+
+\subsection{Classification phénoménologique}
+
+La première distinction porte sur notre capacité à prédire l'évolution temporelle de la grandeur observée. Dans le cas d'un oscillateur sinusoïdal d'amplitude $A$ et de fréquence $f_0$ par exemple, moyennant la mesure de $A$ et de $f_0$, on peut prédire la valeur de l'amplitude, à tout instant, par l'expression $x(t) = A\cos(2 \pi f_0 t)$. Un tel modèle de signal est dit \textbf{déterministe}. Il existe cependant des situations où il n'est pas concevable d'expliciter, de cette façon, la forme de l'évolution temporelle du signal. C'est le cas, par exemple, pour la tension engendrée à la sortie d'un microphone ou encore pour le courant électrique produit par l'agitation thermique des particules dans un conducteur (bruit de fond). On ne peut pas dire avec certitude combien vaudra $x(t)$ à l'instant $t$, mais on pourra éventuellement supposer que cette grandeur est distribuée suivant une certaine loi de probabilité. On dit alors que le signal est \textbf{aléatoire}.
+
+Pour simplifier, on peut dire que les signaux déterministes sont des cas relativement théoriques et idéaux, ou bien modélisent des signaux observés expérimentalement. Car en fait, tout signal physique réel comporte une composante aléatoire correspondant à la perturbation externe.
+
+\subsection{Classification morphologique}
+
+Si, comme c'est le cas pour le signal $x(t) = Acos(2\pi f_0 t)$, où $t\in\mathbb{R}$ et $A\in\mathbb{R}$, on dit que le signal est à temps continu. Toutefois on rencontre aussi en traitement du signal des grandeurs qui évoluent uniquement à des instants discrets $t_n$ où $n$ est un entier. On parle alors de signal à temps discret ou encore de signal numérique. En terme mathématique, un signal à temps continu est une fonction du temps tandis qu'un signal à temps discret est une suite. Le développement et l'essor des techniques numériques ont fait que les problèmes portant sur le traitements des signaux à temps discret ont pris une place majeure aujourd'hui, comparée à celle qu'occupent les traitements portant sur les signaux à temps continu. C'est pourquoi ce cours est centre avant tout sur les problèmes de temps discret et sur le passage du temps continu au temps discret (principe de l'échantillonnage).
+
+Il est également possible de discrétiser le signal en amplitude. On appelle cela la \textbf{quantification} du signal.
+
+\newpage
+\subsection{Classification énergétique}
+
+On distingue en particulier deux types de signaux :
+
+\begin{itemize}
+\item les signaux à énergie finie comme tous les signaux physiques,
+\item les signaux à puissance moyenne finie comme par exemple un signal sinusoïdal.
+\end{itemize}
+
+
+\section{Signaux et distributions classiques}
+
+\subsection{Fenêtre rectangulaire ou Porte}
+
+Soit $t \in \mathbb{R}$. Un signal rectangulaire s'écrit :
+
+\begin{equation}
+\boxed{\Pi_{2a}(t)}
+\end{equation}
+
+La représentation graphique d'un signal porte est donnée à la figure \ref{sig_porte}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=8cm]{img/sig_porte}
+    \caption{Réprésentation temporelle d'un signal rectangulaire ou << porte >>.}
+    \label{sig_porte}
+\end{figure}
+
+
+\subsection{Impulsion de Dirac}
+
+Les propriétés de l'impulsion de Dirac sont identiques à celles d'un signal rectangulaire dont la largeur tend vers 0 et la hauteur vers l'infini, à surface constante. Soit $t \in \mathbb{R}$ et $t_0 \in \mathbb{R}$. L'impulsion située au temps $t_0$ s'écrit :
+
+\begin{equation}
+\boxed{\delta(t-t_0)}
+\end{equation}
+
+Il est souvent utilisé pour déterminer la réponse impulsionnelle d'un système dynamique. Une représentation est donnée à la figure \ref{sig_dirac}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=6cm]{img/sig_dirac}
+    \caption{Réprésentation temporelle d'une impulsion de Dirac.}
+    \label{sig_dirac}
+\end{figure}
+
+\subsection{<< Peigne >> de Dirac}
+Soit $T$ une période en secondes, $t\in\mathbb{R}$, et $i\in\mathbb{N}$ et $n\in\mathbb{N}$. Le peigne de Dirac correspond à une périodisation d'impulsions de Dirac tous les $iT$ sur tout l'axe temporel (aussi bien $t<0$  que $t> 0$). Il s'écrit :
+
+\begin{equation}
+\boxed{\Psi_T(t) = \sum_{i=-\infty}^{+\infty} \delta (t-iT)}
+\end{equation}
+
+Une représentation dun peigne de Dirac est donnée à la figure \ref{sig_peigne}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=8cm]{img/sig_train_dirac}
+    \caption{Réprésentation temporelle d'un peigne de Dirac.}
+    \label{sig_peigne}
+\end{figure}
+
+
+\subsection{Sinus cardinal}
+
+Soit $t \in \mathbb{R}\backslash \lbrace 0 \rbrace$. Le sinus cardinal se définit par la formule :
+
+\begin{equation}
+\boxed{\mathrm{sinc} (t) = \frac{\sin(t)}{t}}
+\end{equation}
+
+Une représentation est donnée à la figure \ref{sin_card}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=10cm]{img/sinc}
+    \caption{Réprésentation temporelle du sinus cardinal : $y=\frac{\sin(x)}{x}$ ($x\texttt{} \in \mathbb{R}\backslash \lbrace 0 \rbrace$).}
+    \label{sin_card}
+\end{figure}
+
+\section{Evaluation de la similitude entre deux signaux}
+
+Il existe en traitement du signal des outils mathématiques qui permettent d'évaluer la similitude entre deux signaux :
+
+\begin{itemize}
+\item au niveau de leur évolution temporelle grâce  à la fonction d'inter ou d'auto-corrélation.
+\item au niveau de leur composition spectrale grâce à la fonction de densité spectrale.
+\end{itemize}
+
+
+\subsection{Les fonctions d'inter-corrélation et d'auto-corrélation}\label{intercorr}
+
+La fonction d'\textbf{intercorrélation} donne une quantité liée à la similitude entre deux signaux. Elle se définit par la formule suivante :
+
+\begin{equation}\label{inter_cor}
+\varphi_{xy}(\tau)=\int_{-\infty}^{+\infty}x^*(t) y(t+\tau) \ud t
+\end{equation}
+
+où $x^*(t)$ est le conjugué de $x(t)$. Cette fonction renvoie un maximum lorsque les deux fonctions deviennent les plus similaires à t donnée. La fonction d'\textbf{autocorrélation} est un cas particulier de la fonction d'intercorrélation pour laquelle $y(t) = x(t)$. Elle s'écrit donc :
+
+\begin{equation}\label{auto_cor}
+\varphi_{x}(\tau)=\int_{-\infty}^{+\infty}x^*(t) x(t+\tau) \ud t
+\end{equation}
+
+La fonction d'autocorrélation mesure ainsi la similitude de $x(t)$ avec une version décallée de $x(t)$. Elle atteint un maximum pour le temps $t_0$ auquel $x(t-t_0)$ ressemble le plus à $x(t)$.  C'est le cas particulièrement pour les signaux périodiques qui reprennent la même valeur à chaque période $T$. La fonction d'autocorrélation permet ainsi d'estimer la périodicité d'un signal semi-périodique en repérant le temps pour lequel elle atteint son maximum.
+
+La fonction d'autocorrélation permet également de calculer l'énergie du signal puisque:
+
+\begin{equation}\label{inter_cor2}
+\varphi_{x}(0)=\int_{-\infty}^{+\infty} \vert x(t) \vert ^2 \ud t = E
+\end{equation}
+
+
+

diff --git a/Signaux.tex.backup b/Signaux.tex.backup
new file mode 100644 (file)
index 0000000..4c3342e
--- /dev/null
+++ b/Signaux.tex.backup
@@ -0,0 +1,181 @@
+\section{Préambule}
+
+L'information, qui se présente à nous lors de l'observation d'un phénomène, se manifeste sous forme d'une ou plusieurs grandeurs physiques qui se déroulent à la fois dans le temps et dans l'espace. Dans les problèmes rencontrés en pratique, on est souvent amené à s'intéresser plus particulièrement à l'un ou l'autre de ces deux aspects. On désigne alors par le terme de \textbf{signal} l'évolution temporelle, tandis que l'on désigne par le terme d'\textbf{image} l'évolution spatiale.
+
+Les différentes opérations que l'on fait subir aux signaux nécessitent des outils mathématiques appropriés qui sont à la base du traitement du signal. Dans ce cours nous envisageons l'étude de certains de ces outils. La théorie du signal propose de modéliser la grandeur physique observée par un objet mathématique dépendant de la variable réelle $t$ représentant le temps. Dans la suite, le  signal désignera indifféremment la grandeur physique observée ou l'objet mathématique servant à la modéliser. Le figure \ref{ex_sig} donne un aperçu de la représentation temporelle de la forme d'onde d'un signal musical.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=12cm]{img/ex_sig}
+    \caption{Exemple d'une réprésentation temporelle d'un signal musical combinant un son de grosse caisse et celui d'une caisse claire.}
+    \label{ex_sig}
+\end{figure}
+
+
+\section{Classification des signaux}
+
+En théorie du signal, on a l'habitude d'envisager plusieurs types de signaux. Pour rendre la suite de ce cours plus compréhensible, nous présentons dans cette section les différents modes de classification des signaux.
+
+\subsection{Classification dimensionnelle}
+
+On peut différentier les signaux selon leur nombre de variables libres. Par exemple un signal électrique $V(t)$ est un signal unidimensionnel dépendant du temps. Une image statique en noir et blanc peut être modélisée par un signal bidimensionnel correspondant à la brillance en fonction de la largeur et de la hauteur de l'image. Enfin, un film en noir et blanc peut être considéré comme une image fonction du temps, autrement dit modélisé par un signal tridimensionnel dépendant de l'abscisse et de l'ordonnée de l'image mais également du temps.
+
+\subsection{Classification phénoménologique}
+
+La première distinction porte sur notre capacité à prédire l'évolution temporelle de la grandeur observée. Dans le cas d'un oscillateur sinusoïdal d'amplitude $A$ et de fréquence $f_0$ par exemple, moyennant la mesure de $A$ et de $f_0$, on peut prédire la valeur de l'amplitude, à tout instant, par l'expression $x(t) = A\cos(2 \pi f_0 t)$. Un tel modèle de signal est dit \textbf{déterministe}. Il existe cependant des situations où il n'est pas concevable d'expliciter, de cette façon, la forme de l'évolution temporelle du signal. C'est le cas, par exemple, pour la tension engendrée à la sortie d'un microphone ou encore pour le courant électrique produit par l'agitation thermique des particules dans un conducteur (bruit de fond). On ne peut pas dire avec certitude combien vaudra $x(t)$ à l'instant $t$, mais on pourra éventuellement supposer que cette grandeur est distribuée suivant une certaine loi de probabilité. On dit alors que le signal est \textbf{aléatoire}.
+
+Pour simplifier, on peut dire que les signaux déterministes sont des cas relativement théoriques et idéaux, ou bien modélisent des signaux observés expérimentalement. Car en fait, tout signal physique réel comporte une composante aléatoire correspondant à la perturbation externe.
+
+\subsection{Classification morphologique}
+
+Si, comme c'est le cas pour le signal $x(t) = Acos(2\pi f_0 t)$, où $t\in\mathbb{R}$ et $A\in\mathbb{R}$, on dit que le signal est à temps continu. Toutefois on rencontre aussi en traitement du signal des grandeurs qui évoluent uniquement à des instants discrets $t_n$ où $n$ est un entier. On parle alors de signal à temps discret ou encore de signal numérique. En terme mathématique, un signal à temps continu est une fonction du temps tandis qu'un signal à temps discret est une suite. Le développement et l'essor des techniques numériques ont fait que les problèmes portant sur le traitements des signaux à temps discret ont pris une place majeure aujourd'hui, comparée à celle qu'occupent les traitements portant sur les signaux à temps continu. C'est pourquoi ce cours est centre avant tout sur les problèmes de temps discret et sur le passage du temps continu au temps discret (principe de l'échantillonnage).
+
+Il est également possible de discrétiser le signal en amplitude. On appelle cela la \textbf{quantification} du signal.
+
+\newpage
+\subsection{Classification énergétique}
+
+On distingue en particulier deux types de signaux :
+
+\begin{itemize}
+\item les signaux à énergie finie comme tous les signaux physiques,
+\item les signaux à puissance moyenne finie comme par exemple un signal sinusoïdal.
+\end{itemize}
+
+
+\section{Signaux et distributions classiques}
+
+\subsection{Fenêtre rectangulaire ou Porte}
+
+Soit $t \in \mathbb{R}$. Un signal rectangulaire s'écrit :
+
+\begin{equation}
+\boxed{\Pi_{2a}(t)}
+\end{equation}
+
+La représentation graphique d'un signal porte est donnée à la figure \ref{sig_porte}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=8cm]{img/sig_porte}
+    \caption{Réprésentation temporelle d'un signal rectangulaire ou << porte >>.}
+    \label{sig_porte}
+\end{figure}
+
+
+\subsection{Impulsion de Dirac}
+
+Les propriétés de l'impulsion de Dirac sont identiques à celles d'un signal rectangulaire dont la largeur tend vers 0 et la hauteur vers l'infini, à surface constante. Soit $t \in \mathbb{R}$ et $t_0 \in \mathbb{R}$. L'impulsion située au temps $t_0$ s'écrit :
+
+\begin{equation}
+\boxed{\delta(t-t_0)}
+\end{equation}
+
+Il est souvent utilisé pour déterminer la réponse impulsionnelle d'un système dynamique. Une représentation est donnée à la figure \ref{sig_dirac}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=6cm]{img/sig_dirac}
+    \caption{Réprésentation temporelle d'une impulsion de Dirac.}
+    \label{sig_dirac}
+\end{figure}
+
+\subsection{<< Peigne >> de Dirac}
+Soit $T$ une période en secondes, $t\in\mathbb{R}$, et $i\in\mathbb{N}$ et $n\in\mathbb{N}$. Le peigne de Dirac correspond à une périodisation d'impulsions de Dirac tous les $iT$ sur tout l'axe temporel (aussi bien $t<0$  que $t> 0$). Il s'écrit :
+
+\begin{equation}
+\boxed{\Psi_T(t) = \Sigma_{i=1}^n \delta (t-iT)}
+\end{equation}
+
+Une représentation dun peigne de Dirac est donnée à la figure \ref{sig_peigne}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=8cm]{img/sig_train_dirac}
+    \caption{Réprésentation temporelle d'un peigne de Dirac.}
+    \label{sig_peigne}
+\end{figure}
+
+
+\subsection{Sinus cardinal}
+
+Soit $t \in \mathbb{R}\backslash \lbrace 0 \rbrace$. Le sinus cardinal se définit par la formule :
+
+\begin{equation}
+\boxed{\mathrm{sinc} (t) = \frac{\sin(t)}{t}}
+\end{equation}
+
+Une représentation est donnée à la figure \ref{sin_card}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=10cm]{img/sinc}
+    \caption{Réprésentation temporelle du sinus cardinal : $y=\frac{\sin(x)}{x}$ ($x\texttt{} \in \mathbb{R}\backslash \lbrace 0 \rbrace$).}
+    \label{sin_card}
+\end{figure}
+
+\section{Evaluation de la similitude entre deux signaux}
+
+Il existe en traitement du signal des outils mathématiques qui permettent d'évaluer la similitude entre deux signaux :
+
+\begin{itemize}
+\item au niveau de leur évolution temporelle grâce  à la fonction d'inter ou d'auto-corrélation.
+\item au niveau de leur composition spectrale grâce à la fonction de densité spectrale.
+\end{itemize}
+
+
+\subsection{Les fonctions d'inter-corrélation et d'auto-corrélation}
+
+La fonction d'\textbf{intercorrélation} donne une quantité liée à la similitude entre deux signaux. Elle se définit par la formule suivante :
+
+\begin{equation}\label{inter_cor}
+\varphi_{xy}(\tau)=\int_{-\infty}^{+\infty}x^*(t) y(t+\tau) \ud t
+\end{equation}
+
+où $x^*(t)$ est le conjugué de $x(t)$. Cette fonction renvoie un maximum lorsque les deux fonctions deviennent les plus similaires à t donnée. La fonction d'\textbf{autocorrélation} est un cas particulier de la fonction d'intercorrélation pour laquelle $y(t) = x(t)$. Elle s'écrit donc :
+
+\begin{equation}\label{auto_cor}
+\varphi_{x}(\tau)=\int_{-\infty}^{+\infty}x^*(t) x(t+\tau) \ud t
+\end{equation}
+
+La fonction d'autocorrélation mesure ainsi la similitude de $x(t)$ avec une version décallée de $x(t)$. Elle atteint un maximum pour le temps $t_0$ auquel $x(t-t_0)$ ressemble le plus à $x(t)$.  C'est le cas particulièrement pour les signaux périodiques qui reprennent la même valeur à chaque période $T$. La fonction d'autocorrélation permet ainsi d'estimer la périodicité d'un signal semi-périodique en repérant le temps pour lequel elle atteint son maximum.
+
+La fonction d'autocorrélation permet également de calculer l'énergie du signal puisque:
+
+\begin{equation}\label{inter_cor2}
+\varphi_{x}(0)=\int_{-\infty}^{+\infty} \vert x(t) \vert ^2 \ud t = E
+\end{equation}
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La fonction de densité spectrale d'énergie est la $TF$ de la fonction d'autocorrélation. Elle décrit la répartition de l'énergie en fréquence et on a :
+
+
+\begin{equation}\label{PSD}
+\Phi_{x}(f)= TF (\varphi_x(\tau))=\vert X(f) \vert ^2
+\end{equation}
+
+\section{Les signaux numériques}\label{signum}
+
+Les signaux numériques dérivent d 'un signal analogique par \textbf{échantillonnage}. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants $t_n$, $n\in \mathbb{Z}$. Dans la suite nous n'envisagerons que l'échantillonnage est régulier, \cad que $t_n$ = $nT$.
+
+Aujourd'hui l'intérêt grandissant pour la conversion de signaux analogiques en signaux numériques tient au fait qu'il devient de plus en plus simple d'appliquer des algorithmes d'analyse complexe avec l'aide des processeurs numériques. En sciences physiques par exemple, il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}\label{sampling}
+
+Le signal continu (analogique) peut être échantillonné (\cad discrétisé) à une \textbf{fréquence d'échantillonnage} $f_e$ inverse de la \textbf{période d'échantillonnage} $T_e$ telle que $f_e= 1/T_e$. Cette échantillonage revient à multiplier le signal analogique par un peigne de Dirac de période $T_e$.
+
+Le signal échantillonné peut alors s'écrire :
+
+\begin{equation}
+ x_e(t)=x_a(t) \cdot \sum_{n=-\infty}^{+\infty} \delta (t-nT_e).
+\end{equation}
+
+$x_e(t)$ est donc nul partout sauf en $t= nTe$, $n$ variant de $-\infty$ à $+\infty$. C'est pourquoi par la suite, on ne simplifie cette notation en réduisant $x_e(t)$ non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets $x(n)$, où $n$ correspond au numéro de l'échantillon.
+
+\begin{equation}
+ x(n) = x_e ( n T_e )
+\end{equation}
+

diff --git a/Signaux.tex~ b/Signaux.tex~
new file mode 100644 (file)
index 0000000..0b84ea6
--- /dev/null
+++ b/Signaux.tex~
@@ -0,0 +1,150 @@
+\section{Préambule}
+
+L'information, qui se présente à nous lors de l'observation d'un phénomène, se manifeste sous forme d'une ou plusieurs grandeurs physiques qui se déroulent à la fois dans le temps et dans l'espace. Dans les problèmes rencontrés en pratique, on est souvent amené à s'intéresser plus particulièrement à l'un ou l'autre de ces deux aspects. On désigne alors par le terme de \textbf{signal} l'évolution temporelle, tandis que l'on désigne par le terme d'\textbf{image} l'évolution spatiale.
+
+Les différentes opérations que l'on fait subir aux signaux nécessitent des outils mathématiques appropriés qui sont à la base du traitement du signal. Dans ce cours nous envisageons l'étude de certains de ces outils. La théorie du signal propose de modéliser la grandeur physique observée par un objet mathématique dépendant de la variable réelle $t$ représentant le temps. Dans la suite, le  signal désignera indifféremment la grandeur physique observée ou l'objet mathématique servant à la modéliser. Le figure \ref{ex_sig} donne un aperçu de la représentation temporelle de la forme d'onde d'un signal musical.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=12cm]{img/ex_sig}
+    \caption{Exemple d'une réprésentation temporelle d'un signal musical combinant un son de grosse caisse et celui d'une caisse claire.}
+    \label{ex_sig}
+\end{figure}
+
+
+\section{Classification des signaux}
+
+En théorie du signal, on a l'habitude d'envisager plusieurs types de signaux. Pour rendre la suite de ce cours plus compréhensible, nous présentons dans cette section les différents modes de classification des signaux.
+
+\subsection{Classification dimensionnelle}
+
+On peut différentier les signaux selon leur nombre de variables libres. Par exemple un signal électrique $V(t)$ est un signal unidimensionnel dépendant du temps. Une image statique en noir et blanc peut être modélisée par un signal bidimensionnel correspondant à la brillance en fonction de la largeur et de la hauteur de l'image. Enfin, un film en noir et blanc peut être considéré comme une image fonction du temps, autrement dit modélisé par un signal tridimensionnel dépendant de l'abscisse et de l'ordonnée de l'image mais également du temps.
+
+\subsection{Classification phénoménologique}
+
+La première distinction porte sur notre capacité à prédire l'évolution temporelle de la grandeur observée. Dans le cas d'un oscillateur sinusoïdal d'amplitude $A$ et de fréquence $f_0$ par exemple, moyennant la mesure de $A$ et de $f_0$, on peut prédire la valeur de l'amplitude, à tout instant, par l'expression $x(t) = A\cos(2 \pi f_0 t)$. Un tel modèle de signal est dit \textbf{déterministe}. Il existe cependant des situations où il n'est pas concevable d'expliciter, de cette façon, la forme de l'évolution temporelle du signal. C'est le cas, par exemple, pour la tension engendrée à la sortie d'un microphone ou encore pour le courant électrique produit par l'agitation thermique des particules dans un conducteur (bruit de fond). On ne peut pas dire avec certitude combien vaudra $x(t)$ à l'instant $t$, mais on pourra éventuellement supposer que cette grandeur est distribuée suivant une certaine loi de probabilité. On dit alors que le signal est \textbf{aléatoire}.
+
+Pour simplifier, on peut dire que les signaux déterministes sont des cas relativement théoriques et idéaux, ou bien modélisent des signaux observés expérimentalement. Car en fait, tout signal physique réel comporte une composante aléatoire correspondant à la perturbation externe.
+
+\subsection{Classification morphologique}
+
+Si, comme c'est le cas pour le signal $x(t) = Acos(2\pi f_0 t)$, où $t\in\mathbb{R}$ et $A\in\mathbb{R}$, on dit que le signal est à temps continu. Toutefois on rencontre aussi en traitement du signal des grandeurs qui évoluent uniquement à des instants discrets $t_n$ où $n$ est un entier. On parle alors de signal à temps discret ou encore de signal numérique. En terme mathématique, un signal à temps continu est une fonction du temps tandis qu'un signal à temps discret est une suite. Le développement et l'essor des techniques numériques ont fait que les problèmes portant sur le traitements des signaux à temps discret ont pris une place majeure aujourd'hui, comparée à celle qu'occupent les traitements portant sur les signaux à temps continu. C'est pourquoi ce cours est centre avant tout sur les problèmes de temps discret et sur le passage du temps continu au temps discret (principe de l'échantillonnage).
+
+Il est également possible de discrétiser le signal en amplitude. On appelle cela la \textbf{quantification} du signal.
+
+\newpage
+\subsection{Classification énergétique}
+
+On distingue en particulier deux types de signaux :
+
+\begin{itemize}
+\item les signaux à énergie finie comme tous les signaux physiques,
+\item les signaux à puissance moyenne finie comme par exemple un signal sinusoïdal.
+\end{itemize}
+
+
+\section{Signaux et distributions classiques}
+
+\subsection{Fenêtre rectangulaire ou Porte}
+
+Soit $t \in \mathbb{R}$. Un signal rectangulaire s'écrit :
+
+\begin{equation}
+\boxed{\Pi_{2a}(t)}
+\end{equation}
+
+La représentation graphique d'un signal porte est donnée à la figure \ref{sig_porte}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=8cm]{img/sig_porte}
+    \caption{Réprésentation temporelle d'un signal rectangulaire ou << porte >>.}
+    \label{sig_porte}
+\end{figure}
+
+
+\subsection{Impulsion de Dirac}
+
+Les propriétés de l'impulsion de Dirac sont identiques à celles d'un signal rectangulaire dont la largeur tend vers 0 et la hauteur vers l'infini, à surface constante. Soit $t \in \mathbb{R}$ et $t_0 \in \mathbb{R}$. L'impulsion située au temps $t_0$ s'écrit :
+
+\begin{equation}
+\boxed{\delta(t-t_0)}
+\end{equation}
+
+Il est souvent utilisé pour déterminer la réponse impulsionnelle d'un système dynamique. Une représentation est donnée à la figure \ref{sig_dirac}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=6cm]{img/sig_dirac}
+    \caption{Réprésentation temporelle d'une impulsion de Dirac.}
+    \label{sig_dirac}
+\end{figure}
+
+\subsection{<< Peigne >> de Dirac}
+Soit $T$ une période en secondes, $t\in\mathbb{R}$, et $i\in\mathbb{N}$ et $n\in\mathbb{N}$. Le peigne de Dirac correspond à une périodisation d'impulsions de Dirac tous les $iT$ sur tout l'axe temporel (aussi bien $t<0$  que $t> 0$). Il s'écrit :
+
+\begin{equation}
+\boxed{\Psi_T(t) = \Sigma_{i=-\infty}^{+\infty} \delta (t-iT)}
+\end{equation}
+
+Une représentation dun peigne de Dirac est donnée à la figure \ref{sig_peigne}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=8cm]{img/sig_train_dirac}
+    \caption{Réprésentation temporelle d'un peigne de Dirac.}
+    \label{sig_peigne}
+\end{figure}
+
+
+\subsection{Sinus cardinal}
+
+Soit $t \in \mathbb{R}\backslash \lbrace 0 \rbrace$. Le sinus cardinal se définit par la formule :
+
+\begin{equation}
+\boxed{\mathrm{sinc} (t) = \frac{\sin(t)}{t}}
+\end{equation}
+
+Une représentation est donnée à la figure \ref{sin_card}.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=10cm]{img/sinc}
+    \caption{Réprésentation temporelle du sinus cardinal : $y=\frac{\sin(x)}{x}$ ($x\texttt{} \in \mathbb{R}\backslash \lbrace 0 \rbrace$).}
+    \label{sin_card}
+\end{figure}
+
+\section{Evaluation de la similitude entre deux signaux}
+
+Il existe en traitement du signal des outils mathématiques qui permettent d'évaluer la similitude entre deux signaux :
+
+\begin{itemize}
+\item au niveau de leur évolution temporelle grâce  à la fonction d'inter ou d'auto-corrélation.
+\item au niveau de leur composition spectrale grâce à la fonction de densité spectrale.
+\end{itemize}
+
+
+\subsection{Les fonctions d'inter-corrélation et d'auto-corrélation}\label{intercorr}
+
+La fonction d'\textbf{intercorrélation} donne une quantité liée à la similitude entre deux signaux. Elle se définit par la formule suivante :
+
+\begin{equation}\label{inter_cor}
+\varphi_{xy}(\tau)=\int_{-\infty}^{+\infty}x^*(t) y(t+\tau) \ud t
+\end{equation}
+
+où $x^*(t)$ est le conjugué de $x(t)$. Cette fonction renvoie un maximum lorsque les deux fonctions deviennent les plus similaires à t donnée. La fonction d'\textbf{autocorrélation} est un cas particulier de la fonction d'intercorrélation pour laquelle $y(t) = x(t)$. Elle s'écrit donc :
+
+\begin{equation}\label{auto_cor}
+\varphi_{x}(\tau)=\int_{-\infty}^{+\infty}x^*(t) x(t+\tau) \ud t
+\end{equation}
+
+La fonction d'autocorrélation mesure ainsi la similitude de $x(t)$ avec une version décallée de $x(t)$. Elle atteint un maximum pour le temps $t_0$ auquel $x(t-t_0)$ ressemble le plus à $x(t)$.  C'est le cas particulièrement pour les signaux périodiques qui reprennent la même valeur à chaque période $T$. La fonction d'autocorrélation permet ainsi d'estimer la périodicité d'un signal semi-périodique en repérant le temps pour lequel elle atteint son maximum.
+
+La fonction d'autocorrélation permet également de calculer l'énergie du signal puisque:
+
+\begin{equation}\label{inter_cor2}
+\varphi_{x}(0)=\int_{-\infty}^{+\infty} \vert x(t) \vert ^2 \ud t = E
+\end{equation}
+
+
+

diff --git a/SystLinFilt.tex b/SystLinFilt.tex
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+\section{Introduction}
+
+Pour modifier l'évolution temporelle ou fréquentielle d'un signal déterminé, on a généralement recours à des \textit{fonctions} temporelles ou fréquentielles qui s'appliquent aux valeurs du signal. Ces fonctions peuvent être regroupées au sein d'un système filtrant $\mathbb{S}$ nommé \textit{filtre} délivrant un signal $y(t)$ en réponse à une stimulation  $x(t)$. On peut donc écrire, si $S$ est la fonction du filtre :
+
+\begin{equation}
+ y(t)=S[x(t)]
+\end{equation}
+
+D'un point de vue très général, on peut considérer que tout signal transitant dans une chaîne de transmission est soumis à une opération de filtrage. Voici quelques exemples de filtres :
+
+\begin{itemize}
+ \item un système de mesure conçu pour détecter la présence d'un signal de forme particulière,
+ \item un filtre électronique, un amplificateur, un convertisseur A/D,
+ \item un algorithme informatique agissant sur un signal numérique.
+\end{itemize}
+
+Le présent chapitre constitue une synthèse de la théorie des filtres analogiques et numériques qui tente de donner les éléments de description des transformations temporelles ou fréquentielles - comme par exemple la réponse impulsionnelle - qu'opère $\mathbb{S}$ sur les signaux réels. L'objectif est également de comprendre les méthode de conception de la fonction de filtrage $S$ pour réaliser une opération particulière. Enfin, il sera donné quelques méthodes pour déterminer les propriétés de la fonction $S$ en observant les réponses du système à des simulations extérieures.
+
+
+\section{Réponse impulsionnelle d'un système}
+
+Il est possible de décrire un système $\mathbb{S}$ par l'intermédiaire de sa réponse à des signaux particulier. Mathématiquement, une méthode efficace est de déterminer la réponse de $\mathbb{S}$ à une impulsion de Dirac, \cad une la \textbf{réponse impulsionnelle} telle que :
+
+\begin{equation}
+ s_{\delta}(t) = S(\delta(t))
+\end{equation}
+
+Le signal $s_{\delta}(t)$ récupéré constitue une signature caractéristique du filtre. En effet, la transformée de Fourier d'une impulsion étant une constante (dans le l'espace des fréquences) la transformée de Fourier de la réponse impusionnelle donne la réponse fréquentielle du filtre pour toutes les fréquences. Ainsi, il est possible de mesurer rapidement le comportement de n'importe quel filtre. La réponse d'une salle acoustique, par exemple, peut être évaluée avec un explosif ou un autre son bref.
+
+Dans la nature, tous les signaux sont \textbf{causaux}, \cad que les éléments du signal $y(t)$ ne peuvent exister avant ceux de $x(y)$. En d'autres termes, la causalité impose qu'un signal ne peut précéder celui qui lui a donner naissance\footnote{En mathématique, il est possible de définir des filtres non-causaux, mais cela n'est pas l'objet ici.}. Ainsi les système causaux ont une réponse impulsionnelle nulle avant l'instant d'impulsion, soit $y(t<0) = 0$. Par ailleurs, pour un système causal, le signal de sortie à l'instant $t$ dépend du signal d'entrée aux instants $t'< t$. La durée de la réponse impulsionnelle $s_{\delta}(t)$ correspond au temps de réponse du système.
+
+Une fois la réponse impulsionnelle connue, on peut prédire la réponse du filtre $y(t)$ issue de n'importe quel signal d'entrée $x(t)$ grâce au produit de convolution :
+
+\begin{equation}
+y(t) = x(t) \star s_{\delta}(t)
+\end{equation}
+
+
+\section{Réponse fréquentielle d'un système}
+
+La réponse en fréquence d'un système correspond à la transformée de Fourier de la réponse impulsionnelle du système.
+
+$G(f)$ décrit comment la distribution spectrale d'un signal est modifiée ou "filtrée" par le système $\mathbb{S}$. Il est important de noter que le système peut seulement modifier des composantes spectrales mais ne peut en aucun cas en créer de nouvelles. $\vert G(f) \vert$  est le gain du système, c'est à dire la façon dont il modifie les amplitudes de chaque composantes spectrales. $Arg[G(f)]$ représente le déphasage causé par le système, c'est à dire le << retard >> ou l'<< avance >> qu'il impose à certaines composantes spectrales.
+
+La réponse en fréquence, comme la réponse impulsionnelle permet de décrire complètement le système et de prédire la réponse du système à n'importe quelle entrée. Nous retrouvons l'équivalence entre le produit de convolution dans le domaine temporel et le produit scalire dans le domaine fréquentiel :
+
+\begin{equation}
+y(t) = x(t) \star g(t) \Leftrightarrow Y(f)  = X(f) . G(f)
+\end{equation}
+
+Ainsi, la fonction de transfert $G(f)$ d'un système constitue le rapport entre signal reçu et le signal émis dans le domaine fréquentiel tel que :
+
+\begin{equation}
+ G(f) = \frac{Y(f)}{X(f)}
+\end{equation}
+
+
+\section{Filtrage}
+
+Le filtrage linéaire est une opération utile pour sélectionner une partie du spectre d'un signal ou en atténuer une partie, pour
+modifier le contenu spectral d'un signal  ou encore modéliser la transformation d'un signal dans un conduit propagatif, dans une salle.
+
+Pour tous les filtres, on peut déterminer une fonction de transfert égale au rapport entre spectre du signal de sortie et celui du signal d'entrée :
+
+\begin{equation}
+ G(f) = \frac{Y(f)}{X(f)}
+\end{equation}
+
+Cette fonction de transfert peut aussi s'écrire dans l'<< espace Z >> :
+
+\begin{equation}
+ G(z) = \frac{Y(z)}{X(z)}
+\end{equation}
+
+Ainsi la connaissance de la fonction de transfert d'un filtre nous renseigne complètement sur sa nature, quelque soit l'espace de représentation. La fonction de transfert d'un filtre numérique est donc définie comme la transformée en $Z$ de la réponse impulsionnelle du filtre.
+
+Nous n'aborderons pas dans ce cours la description de la transformée de Laplace\footnote{\url{http://fr.wikipedia.org/wiki/Transformée_de_Laplace}} qui est le pendant de la transformée en Z dans le domaine analogique, c'est-à-dire temps continu. Mais il est essentiel de bien la connaître pour concevoir des filtres analogiques.
+
+\subsection{Filtres classiques - vocabulaire}
+
+On définit ici 4 types de filtres les plus classiques :
+
+\begin{itemize}
+\item les filtres \textbf{passe-bas} qui laissent intact les basses fréquences d'un signal et en atténuent les hautes fréquences,
+\item les filtres \textbf{passe-haut} qui laissent intact les hautes fréquences d'un signal et en atténuent les basses fréquences,
+\item les filtres \textbf{passe-bande} qui sélectionnent une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande,
+\item les filtres \textbf{coupe-bande}, qui atténuent fortement une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.                                                                                                                     \end{itemize}
+
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=8cm]{img/filtres_ideaux}
+    \caption{Les filtres classiques. L'axe horizontal représente la dimension fréquentielle, l'axe vertical le module de la fonction de filtrage $|G(f)|$.}
+    \label{filtres_ideaux}
+\end{figure}
+
+Pour tous ces filtres, on définit des \textbf{fréquences de coupure}, c'est à dire les fréquences pour lesquelles le spectre du signal d'entrée va être atténué d'un facteur $\sqrt{2}$ par rapport à la valeur maximale du spectre d'amplitude. Cette variation équivaut à une variation de 3 dB dans l'échelle logarithmique.
+
+Le filtre  passe-bas à donc une fréquence de coupure dans les médiums / hautes fréquences, le passe haut une fréquence de coupure dans les médiums / basses fréquences. Les passe bande et coupe bande possèdent deux fréquences de coupure autour de la fréquence centrale sur laquelle ils se centrent. On spécifie également la pente de l'atténuation de ces filtres, en dB par octave qui apporte une information sur la \textbf{sélectivité} du filtre. Enfin, pour les filtres passe-bande et coupe-bande, on détermine leur largeur de bande, c'est à dire la différence entre leurs deux fréquences de coupure qui renseigne aussi sur sa sélectivité.
+
+\subsection{Filtres numériques}
+
+Avant de déterminer sa fonction de transfert, un filtre numérique peut-être défini par une équation aux différences, c'est-à-dire l'opération mathématique du filtre dans le domaine temporel discret entre le signal d'entrée $x(n)$ quelconque et le signal filtré de sortie $y(n)$. D'un point de vue général, en respectant le principe de causalité, tout signal filtré $y(n)$ s'écrit comme un composition linéaire des échantillons passés de $x(n)$ et $y(n)$ tel que :
+
+\begin{equation}
+y(n) = {\sum_{k=0}^N} b_k  x(n-k) - {\sum_{k=1}^M} a_k y(n-k)\end{equation}
+
+En calculant les transformée en Z de chaque membre, on calcule aisément la fonction de transfert $H(z)$ dans le domaine z :
+
+\begin{equation}
+H(z) = \frac {Y(z)} {X(z)} = \frac {{\sum_{k=0}^N} b_k z^{-k}} {1 + {\sum_{k=1}^M} a_k z^{-k}}
+\end{equation}
+
+Les valeurs des coefficients $a_k$ et $b_k$ fixeront le type de filtre (passe-bas, passe-haut, et...).
+
+
+\subsection{Exemples de filtres}
+
+\subsubsection{Le filtre moyenneur lisseur}
+
+Soit un signal numérique $y(n)$ avec $n \in \mathbb{N}$ issu d'un signal $x(n)$ tel que :
+
+\begin{equation}
+ y(n)=\frac{x(n)+x(n-1)+x(n-2)+...+x(n-N+1)}{N}
+\end{equation}
+
+La transformée en $Z$ de $y(n)$ s'écrit alors :
+
+\begin{eqnarray}
+ Y(z) & = & \sum_{p=0}^{N-1} X(z)\frac{z^{-p}}{N} \\
+      & = & X(z) \sum_{p=0}^{N-1}\frac{z^{-p}}{N}
+\end{eqnarray}
+
+On en déduit la fonction de transfert $H(z)$ du filtre équivalent :
+
+\begin{equation}
+ H(z)=\frac{Y(z)}{X(z)} = \sum_{p=0}^{N-1}\frac{z^{-p}}{N}
+\end{equation}
+
+\subsubsection{Le filtre passe-bas}
+
+Soit $x(n)$ un signal numérique quelconque de fréquence d'échantillonnage $f_e$. La loi récursive qui produit un filtrage passe-bas de fréquence de coupure $f_c$ pour obtenir le signal filtré $y(n)$ s'écrit :
+
+\begin{eqnarray}
+ y(n) & = & y(n-1) + a [x(n)- y(n-1)] \\
+ y(n) & = & a x(n)+[1-a] y(n-1) \\
+ y(n) + [a-1] y(n-1) & = & a x(n) \label{filtre_pb}
+\end{eqnarray}
+
+en ayant posé :
+
+\begin{equation*}
+ a = \frac{1}{1+\frac{f_e}{2 \pi f_c}}
+\end{equation*}
+
+En effectuant la transformée de Z de chaque membre de l'équation \ref{filtre_pb}, on obtient :
+
+\begin{eqnarray}
+  \sum_{n=-\infty}^{+\infty}y(n)z^{-n} + [a-1]  \sum_{n=-\infty}^{+\infty}y(n-1)z^{-n} & = & a \sum_{n=-\infty}^{+\infty}x(n)z^{-n} \\
+  \sum_{n=-\infty}^{+\infty}y(n)z^{-n} + [a-1]  \sum_{n=-\infty}^{+\infty}y(n)z^{-n-1} & = & a \sum_{n=-\infty}^{+\infty}x(n)z^{-n} \\
+  Y(z) + [a-1]  Y(z) . z^{-1} & = & a X(z) \\
+  \frac{Y(z)}{X(z)} = \frac{a}{1+(a-1)z^{-1}}
+\end{eqnarray}
+
+D'un point de vue pratique, c'est cette fonction qui permet d'implémenter - c'est à dire de mettre en oeuvre sous la forme d'un programme - la fonction de filtrage de type passe-bas dans un programme informatique. Dans le logiciel libre Octave\footnote{\url{http://www.octave.org}}, le code suivant permet par exemple de tester ce filtrage sur des signaux aléatoires à partir de la loi récursive mais aussi en utilisant la fonction \verb|filter| :
+
+\begin{verbatim}
+% Script Matlab / Octave pour tester une fonction de filtrage passe-bas.
+
+f0 = 20;
+f1 = 22050;
+t1 = 1;
+fe = 44100;
+fc = 3000;
+a = 1/(1+fe/(2*pi*fc));
+
+t = [0:1/fe:t1];
+n = length(t);
+N = 10;
+
+for k=1:N
+
+    s = randn(1,n);
+    %sweep = sin(2*pi*(f0*t1)/log(f1/f0) * (exp(t'/t1*log(f1/f0))-1))';
+    sf(1) = s(1);
+
+    % Avec une boucle
+    tic;
+    for i=2:n
+        sf(i)=a*s(i)+(1-a)*sf(i-1);
+    end
+    t_boucle(k) = toc;
+
+    % Avec filter
+    tic;
+    sff = filter(a,[1 a-1],s);
+    t_filter(k) = toc;
+
+    % Fonction de transfert
+    nfft = 16384;
+    fft_s = fft(s,nfft);
+    fft_sf = fft(sf,nfft);
+    fft_sff = fft(sff,nfft);
+    f = [0:fe/nfft:fe/2];
+    f(1) = [];
+
+    mod_FT(k,:) = abs(fft_sf)./abs(fft_s);
+    pha_FT(k,:) = angle(fft_sf);
+    mod_FTF(k,:) = abs(fft_sff)./abs(fft_s);
+    pha_FTF(k,:) = angle(fft_sff);
+
+end
+
+mod_FT = mean(mod_FT);
+figure(1)
+semilogx(f,20*log10(mod_FT(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la methode recursive (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FT = mean(pha_FT);
+figure(2)
+semilogx(f,unwrap(pha_FT(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la méthode recursive']);
+
+mod_FTF = mean(mod_FTF);
+figure(3)
+semilogx(f,20*log10(mod_FTF(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la fonction filter (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FTF = mean(pha_FTF);
+figure(4)
+semilogx(f,unwrap(pha_FTF(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la fonction filter']);
+
+figure(5)
+plot(t_boucle);
+title(['Temps d execution (s)']);
+hold on
+plot(t_filter,'r');
+
+pause;
+
+\end{verbatim}
+
+Pour plus d'exemple sur les fonctions numérique de filtrage, voir \url{http://math.fullerton.edu/mathews/c2003/ZTransformFilterMod.html}.
+
+\subsubsection{Le filtre vocal}
+
+On peut modéliser la sensation d'intensité auditive par un filtre. L'oreille humaine est particulièrement sensible entre 3 et 4 kHz.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/sens_oreille}
+    \caption{Diagramme de sensibilité de l'oreille humaine.}
+    \label{sens_oreille}
+\end{figure}
+
+On peut également modéliser la production phonatoire par un modèle source-filtre, où le rôle de la source est joué par les cordes vocales qui produisent un son harmonique avec une distribution de l'énergie assez plate en fréquence. Le conduit vocal, les fosses nasales ainsi que la place des articulateurs (langue, mâchoire, lèvres) peuvent être modélisés par un filtre qui modifie le son glottique pour produire le son tel que nous le percevons à la sortie des lèvres d'un locuteur.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/sf_voix}
+    \caption{Modèle source/filtre de la voix.}
+    \label{sf_voix}
+\end{figure}
+

diff --git a/SystLinFilt.tex.backup b/SystLinFilt.tex.backup
new file mode 100644 (file)
index 0000000..8bf2432
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+++ b/SystLinFilt.tex.backup
@@ -0,0 +1,387 @@
+\section{Introduction}
+
+Pour modifier l'évolution temporelle ou fréquentielle d'un signal déterminé, on a généralement recours à des \textit{fonctions} temporelles ou fréquentielles qui s'appliquent aux valeurs du signal. Ces fonctions peuvent être regroupées au sein d'un système filtrant $\mathbb{S}$ nommé \textit{filtre} délivrant un signal $y(t)$ en réponse à une stimulation  $x(t)$. On peut donc écrire, si $S$ est la fonction du filtre :
+
+\begin{equation}
+ y(t)=S[x(t)]
+\end{equation}
+
+D'un point de vue très général, on peut considérer que tout signal transitant dans une chaîne de transmission est soumis à une opération de filtrage. Voici quelques exemples de filtres :
+
+\begin{itemize}
+ \item un système de mesure conçu pour détecter la présence d'un signal de forme particulière,
+ \item un filtre électronique, un amplificateur, un convertisseur A/D,
+ \item un algorithme informatique agissant sur un signal numérique.
+\end{itemize}
+
+Le présent chapitre constitue une synthèse de la théorie des filtres analogiques et numériques qui tente de donner les éléments de description des transformations temporelles ou fréquentielles - comme par exemple la réponse impulsionnelle - qu'opère $\mathbb{S}$ sur les signaux réels. L'objectif est également de comprendre les méthode de conception de la fonction de filtrage $S$ pour réaliser une opération particulière. Enfin, il sera donné quelques méthodes pour déterminer les propriétés de la fonction $S$ en observant les réponses du système à des simulations extérieures.
+
+
+\section{Réponse impulsionnelle d'un système}
+
+Il est possible de décrire un système $\mathbb{S}$ par l'intermédiaire de sa réponse à des signaux particulier. Mathématiquement, une méthode efficace est de déterminer la réponse de $\mathbb{S}$ à une impulsion de Dirac, \cad une la \textbf{réponse impulsionnelle} telle que :
+
+\begin{equation}
+ s_{\delta}(t) = S(\delta(t))
+\end{equation}
+
+Le signal $s_{\delta}(t)$ récupéré constitue une signature caractéristique du filtre. En effet, la transformée de Fourier d'une impulsion étant une constante (dans le l'espace des fréquences) la transformée de Fourier de la réponse impusionnelle donne la réponse fréquentielle du filtre pour toutes les fréquences. Ainsi, il est possible de mesurer rapidement le comportement de n'importe quel filtre. La réponse d'une salle acoustique, par exemple, peut être évaluée avec un explosif ou un autre son bref.
+
+Dans la nature, tous les signaux sont \textbf{causaux}, \cad que les éléments du signal $y(t)$ ne peuvent exister avant ceux de $x(y)$. En d'autres termes, la causalité impose qu'un signal ne peut précéder celui qui lui a donner naissance\footnote{En mathématique, il est possible de définir des filtres non-causaux, mais cela n'est pas l'objet ici.}. Ainsi les système causaux ont une réponse impulsionnelle nulle avant l'instant d'impulsion, soit $y(t<0) = 0$. Par ailleurs, pour un système causal, le signal de sortie à l'instant $t$ dépend du signal d'entrée aux instants $t'< t$. La durée de la réponse impulsionnelle $s_{\delta}(t)$ correspond au temps de réponse du système.
+
+Une fois la réponse impulsionnelle connue, on peut prédire la réponse du filtre $y(t)$ issue de n'importe quel signal d'entrée $x(t)$ grâce au produit de convolution :
+
+\begin{equation}
+y(t) = x(t) \star s_{\delta}(t)
+\end{equation}
+
+
+
+\section{Réponse en fréquence d'un système}
+
+La réponse en fréquence d'un système correspond à la transformée de Fourier de la réponse impulsionnelle du système.
+
+$G(f)$ décrit comment la distribution spectrale d'un signal est modifiée ou "filtrée" par le système $\mathbb{S}$. Il est important de noter que le système peut seulement modifier des composantes spectrales mais ne peut en aucun cas en créer de nouvelles. $\vert G(f) \vert$  est le gain du système, c'est à dire la façon dont il modifie les amplitudes de chaque composantes spectrales. $Arg[G(f)]$ représente le déphasage causé par le système, c'est à dire le << retard >> ou l'<< avance >> qu'il impose à certaines composantes spectrales.
+
+La réponse en fréquence, comme la réponse impulsionnelle permet de décrire complètement le système et de prédire la réponse du système à n'importe quelle entrée. Nous retrouvons l'équivalence entre le produit de convolution dans le domaine temporel et le produit scalire dans le domaine fréquentiel :
+
+\begin{equation}
+y(t) = x(t) \star g(t) \Leftrightarrow Y(f)  = X(f) . G(f)
+\end{equation}
+
+Ainsi, la fonction de transfert $G(f)$ d'un système constitue le rapport entre signal reçu et le signal émis dans le domaine fréquentiel tel que :
+
+\begin{equation}
+ G(f) = \frac{Y(f)}{X(f)}
+\end{equation}
+
+
+
+
+\section{Transformée en Z}
+
+\subsection{Définition}
+Pour décrire un filtre dans le domaine numérique, il est pratique de définir une transformée dont la variable a la même nature que celle du signal discrétisé. En effet, un signal numérique - discontinu de nature - ne comporte qu'un nombre fini de valeurs et on a besoin d'une transformée à temps discret pour décrire les filtres : la transformée en $Z$. La fonction de transfert d'un filtre numérique est alors défini comme la transformée en $Z$ de la réponse impulsionnelle du filtre.
+
+Soit $x(n)$ un signal discret quelconque. Sa transformée en Z s'écrit :
+
+\begin{equation}
+X(z) = \mathcal{Z}\{x(n)\} =\sum_{n=-\infty}^{+\infty}x(n)z^{-n},\quad z \in \{z\in\mathbb{C}|\sum_{n=-\infty}^{+\infty}x(n)z^{-n} \quad converge\}
+\end{equation}
+
+Remarque : on retrouve la définition de la transformée de Fourier en posant $z = e^{j2\pi f}$.
+
+\subsection{Existence de la transformée en Z}
+
+Le domaine de convergence est le sous-ensemble de $\mathbb{C}$ dans lequel la série converge.
+Autrement dit, le domaine de convergence de la transformée en z de la suite ($x_{n})_{n\in\mathbb{Z}}$ est l'ensemble :
+
+\begin{equation}
+\left\{z\in\mathbb{C} | \sum_{n=-\infty}^{\infty}x_{n}z^{-n} \quad\mathrm{existe}\right \}
+\end{equation}
+
+On l'appelle également couronne de convergence. En effet, en posant $z=\rho e^{i\theta}~$, il vient :
+
+\begin{equation}
+|X(z)|=\left| \sum_{n=-\infty}^{\infty}x_{n}z^{-n}\right|\leq \sum_{n=-\infty}^{\infty}\left|x_{n}\right|\rho^{-n}
+\end{equation}
+
+Donc $X(z)$ existe si $x(n)$ a une croissance au plus exponentielle, auquel cas le domaine de convergence est compris dans une couronne :
+
+\begin{itemize}
+\item de petit rayon le majorant de la base du côté des n négatifs
+\item de grand rayon le majorant de la base du côté des n positifs
+\end{itemize}
+
+Dans toute la suite de l'article, les transformées en Z ne seront valables que dans ce domaine de convergence sans que cela soit reprécisé.
+
+\subsection{Propriétés}
+
+\begin{enumerate}
+\item\textbf{ Linéarité}
+La transformée en Z d'une combinaison linéaire de deux signaux est la combinaison linéaire des transformées en Z de chaque signal.
+
+\begin{equation}
+  \mathcal{Z}\{a_1 x_1(n) + a_2 x_2(n)\} = a_1 \mathcal{Z}\{x_1(n)\} + a_2 \mathcal{Z}\{x_2(n)\}
+\end{equation}
+
+\item \textbf{Décalage temporel}
+
+Le décalage temporel d'un signal de k échantillons se traduit par la multiplication de la transformée en Z du signal par $z^k$.
+
+\begin{equation}
+\mathcal{Z}\{x(n-k)\} = z^{-k}\mathcal{Z}\{x(n)\}
+\end{equation}
+
+\item \textbf{Convolution}
+
+La transformée en Z d'un produit de convolution est le produit des transformées en Z
+
+\begin{equation}
+\mathcal{Z}\{x(n) \star y(n)\} = \mathcal{Z}\{x(n)\} \mathcal{Z}\{y(n)\} \                                                                      \end{equation}
+
+\item \textbf{Multiplication par une exponentielle}
+
+\begin{equation}
+\mathcal{Z}\{a^{n}x(n)\} = X\left(\frac{z}{a}\right)
+\end{equation}
+
+\item \textbf{Multiplication par la variable d'évolution}
+
+De façon générale :
+
+\begin{equation}
+\mathcal{Z}\{n^{k}x(n)\} = \left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}\                                               \end{equation}
+
+où $\textstyle\left(-z \frac{\mathrm{d} }{\mathrm{d}z}\right)^{k}\mathcal{Z}\{x(n)\}$ signifie que l'on applique $k$ fois à $\mathcal{Z}\{x(n)\} l'opérateur \textstyle -z\frac{\mathrm{d} }{\mathrm{d}z}$
+
+Si l'on écrit cette formule au rang k=1, on obtient la formule de dérivation :
+
+\begin{equation}
+\mathcal{Z}\{nx(n)\} = -z \frac{\mathrm{d} }{\mathrm{d}z}X(z)\
+\end{equation}
+
+\item \textbf{Théorème de la valeur initiale}
+
+Soit $x(n)$\, un signal causal et $X(z)$\, sa transformée en Z. Alors :
+
+\begin{equation}
+x(0) = \lim_{n \to 0}x(n)=\lim_{z \to +\infty}X(z)
+\end{equation}
+
+\item \textbf{Théorème de la valeur finale}
+
+Soit $x(n)$, un signal causal et $X(z)$, sa transformée en Z. Alors lorsque la limite existe, on peut écrire :
+
+\begin{equation}
+\lim_{n \to +\infty}x(n)=\lim_{z \to 1}(z-1)X(z)
+\end{equation}
+
+\end{enumerate}
+
+
+\section{Filtrage}
+
+Le filtrage linéaire est une opération utile pour sélectionner une partie du spectre d'un signal ou en atténuer une partie, pour
+modifier le contenu spectral d'un signal  ou encore modéliser la transformation d'un signal dans un conduit propagatif, dans une salle.
+
+Pour tous les filtres, on peut déterminer une fonction de transfert égale au rapport entre spectre du signal de sortie et celui du signal d'entrée :
+
+\begin{equation}
+ G(f) = \frac{Y(f)}{X(f)}
+\end{equation}
+
+Cette fonction de transfert peut aussi s'écrire dans l'<< espace Z >> :
+
+\begin{equation}
+ G(z) = \frac{Y(z)}{X(z)}
+\end{equation}
+
+Ainsi la connaissance de la fonction de transfert d'un filtre nous renseigne complètement sur sa nature, quelque soit l'espace de représentation.
+
+Nous n'aborderons pas dans ce cours la description de la transformée de Laplace\footnote{\url{http://fr.wikipedia.org/wiki/Transformée_de_Laplace}} qui est le pendant de la transformée en Z dans le domaine analogique, c'est-à-dire temps continu. Mais il est essentiel de bien la connaître pour concevoir des filtres analogiques.
+
+\subsection{Filtres classiques - vocabulaire}
+
+On définit ici 4 types de filtres les plus classiques :
+
+\begin{itemize}
+\item les filtres \textbf{passe-bas} qui laissent intact les basses fréquences d'un signal et en atténuent les hautes fréquences,
+\item les filtres \textbf{passe-haut} qui laissent intact les hautes fréquences d'un signal et en atténuent les basses fréquences,
+\item les filtres \textbf{passe-bande} qui sélectionnent une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande,
+\item les filtres \textbf{coupe-bande}, qui atténuent fortement une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.                                                                                                                     \end{itemize}
+
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=8cm]{img/filtres_ideaux}
+    \caption{Les filtres classiques. L'axe horizontal représente la dimension fréquentielle, l'axe vertical le module de la fonction de filtrage $|G(f)|$.}
+    \label{filtres_ideaux}
+\end{figure}
+
+Pour tous ces filtres, on définit des \textbf{fréquences de coupure}, c'est à dire les fréquences pour lesquelles le spectre du signal d'entrée va être atténué d'un facteur $\sqrt{2}$ par rapport à la valeur maximale du spectre d'amplitude. Cette variation équivaut à une variation de 3 dB dans l'échelle logarithmique.
+
+Le filtre  passe-bas à donc une fréquence de coupure dans les médiums / hautes fréquences, le passe haut une fréquence de coupure dans les médiums / basses fréquences. Les passe bande et coupe bande possèdent deux fréquences de coupure autour de la fréquence centrale sur laquelle ils se centrent. On spécifie également la pente de l'atténuation de ces filtres, en dB par octave qui apporte une information sur la \textbf{sélectivité} du filtre. Enfin, pour les filtres passe-bande et coupe-bande, on détermine leur largeur de bande, c'est à dire la différence entre leurs deux fréquences de coupure qui renseigne aussi sur sa sélectivité.
+
+\subsection{Filtres numériques}
+
+Avant de déterminer sa fonction de transfert, un filtre numérique peut-être défini par une équation aux différences, c'est-à-dire l'opération mathématique du filtre dans le domaine temporel discret entre le signal d'entrée $x(n)$ quelconque et le signal filtré de sortie $y(n)$. D'un point de vue général, en respectant le principe de causalité, tout signal filtré $y(n)$ s'écrit comme un composition linéaire des échantillons passés de $x(n)$ et $y(n)$ tel que :
+
+\begin{equation}
+y(n) = {\sum_{k=0}^N} b_k  x(n-k) - {\sum_{k=1}^M} a_k y(n-k)\end{equation}
+
+En calculant les transformée en Z de chaque membre, on calcule aisément la fonction de transfert $H(z)$ dans le domaine z :
+
+\begin{equation}
+H(z) = \frac {Y(z)} {X(z)} = \frac {{\sum_{k=0}^N} b_k z^{-k}} {1 + {\sum_{k=1}^M} a_k z^{-k}}
+\end{equation}
+
+Les valeurs des coefficients $a_k$ et $b_k$ fixeront le type de filtre (passe-bas, passe-haut, et...).
+
+
+\subsection{Exemples de filtres}
+
+\subsubsection{Le filtre moyenneur lisseur}
+
+Soit un signal numérique $y(n)$ avec $n \in \mathbb{N}$ issu d'un signal $x(n)$ tel que :
+
+\begin{equation}
+ y(n)=\frac{x(n)+x(n-1)+x(n-2)+...+x(n-N+1)}{N}
+\end{equation}
+
+La transformée en $Z$ de $y(n)$ s'écrit alors :
+
+\begin{eqnarray}
+ Y(z) & = & \sum_{p=0}^{N-1} X(z)\frac{z^{-p}}{N} \\
+      & = & X(z) \sum_{p=0}^{N-1}\frac{z^{-p}}{N}
+\end{eqnarray}
+
+On en déduit la fonction de transfert $H(z)$ du filtre équivalent :
+
+\begin{equation}
+ H(z)=\frac{Y(z)}{X(z)} = \sum_{p=0}^{N-1}\frac{z^{-p}}{N}
+\end{equation}
+
+\subsubsection{Le filtre passe-bas}
+
+Soit $x(n)$ un signal numérique quelconque de fréquence d'échantillonnage $f_e$. La loi récursive qui produit un filtrage passe-bas de fréquence de coupure $f_c$ pour obtenir le signal filtré $y(n)$ s'écrit :
+
+\begin{eqnarray}
+ y(n) & = & y(n-1) + a [x(n)- y(n-1)] \\
+ y(n) & = & a x(n)+[1-a] y(n-1) \\
+ y(n) + [a-1] y(n-1) & = & a x(n) \label{filtre_pb}
+\end{eqnarray}
+
+avec
+
+\begin{equation*}
+ a = \frac{1}{1+\frac{f_e}{2 \pi f_c}}
+\end{equation*}
+
+En effectuant la transformée de Z de chaque membre de l'équation \ref{filtre_pb}, on obtient :
+
+\begin{eqnarray}
+  \sum_{n=-\infty}^{+\infty}y(n)z^{-n} + [a-1]  \sum_{n=-\infty}^{+\infty}y(n-1)z^{-n} & = & a \sum_{n=-\infty}^{+\infty}x(n)z^{-n} \\
+  \sum_{n=-\infty}^{+\infty}y(n)z^{-n} + [a-1]  \sum_{n=-\infty}^{+\infty}y(n)z^{-n-1} & = & a \sum_{n=-\infty}^{+\infty}x(n)z^{-n} \\
+  Y(z) + [a-1]  Y(z) . z^{-1} & = & a X(z) \\
+  \frac{Y(z)}{X(z)} = \frac{a}{1+(a-1)z^{-1}}
+\end{eqnarray}
+
+D'un point de vue pratique, c'est cette fonction qui permet d'implémenter - c'est à dire de mettre en oeuvre sous la forme d'un programme - la fonction de filtrage de type passe-bas dans un programme informatique. Dans le logiciel libre Octave\footnote{\url{http://www.octave.org}}, le code suivant permet par exemple de tester ce filtrage sur des signaux aléatoires à partir de la loi récursive mais aussi en utilisant la fonction \verb|filter| :
+
+\begin{verbatim}
+% Script Matlab / Octave pour tester une fonction de filtrage passe-bas.
+
+f0 = 20;
+f1 = 22050;
+t1 = 1;
+fe = 44100;
+fc = 3000;
+a = 1/(1+fe/(2*pi*fc));
+
+t = [0:1/fe:t1];
+n = length(t);
+N = 10;
+
+for k=1:N
+
+    s = randn(1,n);
+    %sweep = sin(2*pi*(f0*t1)/log(f1/f0) * (exp(t'/t1*log(f1/f0))-1))';
+    sf(1) = s(1);
+
+    % Avec une boucle
+    tic;
+    for i=2:n
+        sf(i)=a*s(i)+(1-a)*sf(i-1);
+    end
+    t_boucle(k) = toc;
+
+    % Avec filter
+    tic;
+    sff = filter(a,[1 a-1],s);
+    t_filter(k) = toc;
+
+    % Fonction de transfert
+    nfft = 16384;
+    fft_s = fft(s,nfft);
+    fft_sf = fft(sf,nfft);
+    fft_sff = fft(sff,nfft);
+    f = [0:fe/nfft:fe/2];
+    f(1) = [];
+
+    mod_FT(k,:) = abs(fft_sf)./abs(fft_s);
+    pha_FT(k,:) = angle(fft_sf);
+    mod_FTF(k,:) = abs(fft_sff)./abs(fft_s);
+    pha_FTF(k,:) = angle(fft_sff);
+
+end
+
+mod_FT = mean(mod_FT);
+figure(1)
+semilogx(f,20*log10(mod_FT(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la methode recursive (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FT = mean(pha_FT);
+figure(2)
+semilogx(f,unwrap(pha_FT(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la méthode recursive']);
+
+mod_FTF = mean(mod_FTF);
+figure(3)
+semilogx(f,20*log10(mod_FTF(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la fonction filter (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FTF = mean(pha_FTF);
+figure(4)
+semilogx(f,unwrap(pha_FTF(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la fonction filter']);
+
+figure(5)
+plot(t_boucle);
+title(['Temps d execution (s)']);
+hold on
+plot(t_filter,'r');
+
+pause;
+
+\end{verbatim}
+
+Pour plus d'exemple sur les fonctions numérique de filtrage, voir \url{http://math.fullerton.edu/mathews/c2003/ZTransformFilterMod.html}.
+
+\subsubsection{Le filtre vocal}
+
+On peut modéliser la sensation d'intensité auditive par un filtre. L'oreille humaine est particulièrement sensible entre 3 et 4 kHz.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/sens_oreille}
+    \caption{Diagramme de sensibilité de l'oreille humaine.}
+    \label{sens_oreille}
+\end{figure}
+
+On peut également modéliser la production phonatoire par un modèle source-filtre, où le rôle de la source est joué par les cordes vocales qui produisent un son harmonique avec une distribution de l'énergie assez plate en fréquence. Le conduit vocal, les fosses nasales ainsi que la place des articulateurs (langue, mâchoire, lèvres) peuvent être modélisés par un filtre qui modifie le son glottique pour produire le son tel que nous le percevons à la sortie des lèvres d'un locuteur.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/sf_voix}
+    \caption{Modèle source/filtre de la voix.}
+    \label{sf_voix}
+\end{figure}
+
+
+
+
+
+

diff --git a/SystLinFilt.tex~ b/SystLinFilt.tex~
new file mode 100644 (file)
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+\section{Introduction}
+
+Pour modifier l'évolution temporelle ou fréquentielle d'un signal déterminé, on a généralement recours à des \textit{fonctions} temporelles ou fréquentielles qui s'appliquent aux valeurs du signal. Ces fonctions peuvent être regroupées au sein d'un système filtrant $\mathbb{S}$ nommé \textit{filtre} délivrant un signal $y(t)$ en réponse à une stimulation  $x(t)$. On peut donc écrire, si $S$ est la fonction du filtre :
+
+\begin{equation}
+ y(t)=S[x(t)]
+\end{equation}
+
+D'un point de vue très général, on peut considérer que tout signal transitant dans une chaîne de transmission est soumis à une opération de filtrage. Voici quelques exemples de filtres :
+
+\begin{itemize}
+ \item un système de mesure conçu pour détecter la présence d'un signal de forme particulière,
+ \item un filtre électronique, un amplificateur, un convertisseur A/D,
+ \item un algorithme informatique agissant sur un signal numérique.
+\end{itemize}
+
+Le présent chapitre constitue une synthèse de la théorie des filtres analogiques et numériques qui tente de donner les éléments de description des transformations temporelles ou fréquentielles - comme par exemple la réponse impulsionnelle - qu'opère $\mathbb{S}$ sur les signaux réels. L'objectif est également de comprendre les méthode de conception de la fonction de filtrage $S$ pour réaliser une opération particulière. Enfin, il sera donné quelques méthodes pour déterminer les propriétés de la fonction $S$ en observant les réponses du système à des simulations extérieures.
+
+
+\section{Réponse impulsionnelle d'un système}
+
+Il est possible de décrire un système $\mathbb{S}$ par l'intermédiaire de sa réponse à des signaux particulier. Mathématiquement, une méthode efficace est de déterminer la réponse de $\mathbb{S}$ à une impulsion de Dirac, \cad une la \textbf{réponse impulsionnelle} telle que :
+
+\begin{equation}
+ s_{\delta}(t) = S(\delta(t))
+\end{equation}
+
+Le signal $s_{\delta}(t)$ récupéré constitue une signature caractéristique du filtre. En effet, la transformée de Fourier d'une impulsion étant une constante (dans le l'espace des fréquences) la transformée de Fourier de la réponse impusionnelle donne la réponse fréquentielle du filtre pour toutes les fréquences. Ainsi, il est possible de mesurer rapidement le comportement de n'importe quel filtre. La réponse d'une salle acoustique, par exemple, peut être évaluée avec un explosif ou un autre son bref.
+
+Dans la nature, tous les signaux sont \textbf{causaux}, \cad que les éléments du signal $y(t)$ ne peuvent exister avant ceux de $x(y)$. En d'autres termes, la causalité impose qu'un signal ne peut précéder celui qui lui a donner naissance\footnote{En mathématique, il est possible de définir des filtres non-causaux, mais cela n'est pas l'objet ici.}. Ainsi les système causaux ont une réponse impulsionnelle nulle avant l'instant d'impulsion, soit $y(t<0) = 0$. Par ailleurs, pour un système causal, le signal de sortie à l'instant $t$ dépend du signal d'entrée aux instants $t'< t$. La durée de la réponse impulsionnelle $s_{\delta}(t)$ correspond au temps de réponse du système.
+
+Une fois la réponse impulsionnelle connue, on peut prédire la réponse du filtre $y(t)$ issue de n'importe quel signal d'entrée $x(t)$ grâce au produit de convolution :
+
+\begin{equation}
+y(t) = x(t) \star s_{\delta}(t)
+\end{equation}
+
+
+\section{Réponse fréquentielle d'un système}
+
+La réponse en fréquence d'un système correspond à la transformée de Fourier de la réponse impulsionnelle du système.
+
+$G(f)$ décrit comment la distribution spectrale d'un signal est modifiée ou "filtrée" par le système $\mathbb{S}$. Il est important de noter que le système peut seulement modifier des composantes spectrales mais ne peut en aucun cas en créer de nouvelles. $\vert G(f) \vert$  est le gain du système, c'est à dire la façon dont il modifie les amplitudes de chaque composantes spectrales. $Arg[G(f)]$ représente le déphasage causé par le système, c'est à dire le << retard >> ou l'<< avance >> qu'il impose à certaines composantes spectrales.
+
+La réponse en fréquence, comme la réponse impulsionnelle permet de décrire complètement le système et de prédire la réponse du système à n'importe quelle entrée. Nous retrouvons l'équivalence entre le produit de convolution dans le domaine temporel et le produit scalire dans le domaine fréquentiel :
+
+\begin{equation}
+y(t) = x(t) \star g(t) \Leftrightarrow Y(f)  = X(f) . G(f)
+\end{equation}
+
+Ainsi, la fonction de transfert $G(f)$ d'un système constitue le rapport entre signal reçu et le signal émis dans le domaine fréquentiel tel que :
+
+\begin{equation}
+ G(f) = \frac{Y(f)}{X(f)}
+\end{equation}
+
+
+\section{Filtrage}
+
+Le filtrage linéaire est une opération utile pour sélectionner une partie du spectre d'un signal ou en atténuer une partie, pour
+modifier le contenu spectral d'un signal  ou encore modéliser la transformation d'un signal dans un conduit propagatif, dans une salle.
+
+Pour tous les filtres, on peut déterminer une fonction de transfert égale au rapport entre spectre du signal de sortie et celui du signal d'entrée :
+
+\begin{equation}
+ G(f) = \frac{Y(f)}{X(f)}
+\end{equation}
+
+Cette fonction de transfert peut aussi s'écrire dans l'<< espace Z >> :
+
+\begin{equation}
+ G(z) = \frac{Y(z)}{X(z)}
+\end{equation}
+
+Ainsi la connaissance de la fonction de transfert d'un filtre nous renseigne complètement sur sa nature, quelque soit l'espace de représentation. La fonction de transfert d'un filtre numérique est donc définie comme la transformée en $Z$ de la réponse impulsionnelle du filtre.
+
+Nous n'aborderons pas dans ce cours la description de la transformée de Laplace\footnote{\url{http://fr.wikipedia.org/wiki/Transformée_de_Laplace}} qui est le pendant de la transformée en Z dans le domaine analogique, c'est-à-dire temps continu. Mais il est essentiel de bien la connaître pour concevoir des filtres analogiques.
+
+\subsection{Filtres classiques - vocabulaire}
+
+On définit ici 4 types de filtres les plus classiques :
+
+\begin{itemize}
+\item les filtres \textbf{passe-bas} qui laissent intact les basses fréquences d'un signal et en atténuent les hautes fréquences,
+\item les filtres \textbf{passe-haut} qui laissent intact les hautes fréquences d'un signal et en atténuent les basses fréquences,
+\item les filtres \textbf{passe-bande} qui sélectionnent une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande,
+\item les filtres \textbf{coupe-bande}, qui atténuent fortement une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.                                                                                                                     \end{itemize}
+
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=8cm]{img/filtres_ideaux}
+    \caption{Les filtres classiques. L'axe horizontal représente la dimension fréquentielle, l'axe vertical le module de la fonction de filtrage $|G(f)|$.}
+    \label{filtres_ideaux}
+\end{figure}
+
+Pour tous ces filtres, on définit des \textbf{fréquences de coupure}, c'est à dire les fréquences pour lesquelles le spectre du signal d'entrée va être atténué d'un facteur $\sqrt{2}$ par rapport à la valeur maximale du spectre d'amplitude. Cette variation équivaut à une variation de 3 dB dans l'échelle logarithmique.
+
+Le filtre  passe-bas à donc une fréquence de coupure dans les médiums / hautes fréquences, le passe haut une fréquence de coupure dans les médiums / basses fréquences. Les passe bande et coupe bande possèdent deux fréquences de coupure autour de la fréquence centrale sur laquelle ils se centrent. On spécifie également la pente de l'atténuation de ces filtres, en dB par octave qui apporte une information sur la \textbf{sélectivité} du filtre. Enfin, pour les filtres passe-bande et coupe-bande, on détermine leur largeur de bande, c'est à dire la différence entre leurs deux fréquences de coupure qui renseigne aussi sur sa sélectivité.
+
+\subsection{Filtres numériques}
+
+Avant de déterminer sa fonction de transfert, un filtre numérique peut-être défini par une équation aux différences, c'est-à-dire l'opération mathématique du filtre dans le domaine temporel discret entre le signal d'entrée $x(n)$ quelconque et le signal filtré de sortie $y(n)$. D'un point de vue général, en respectant le principe de causalité, tout signal filtré $y(n)$ s'écrit comme un composition linéaire des échantillons passés de $x(n)$ et $y(n)$ tel que :
+
+\begin{equation}
+y(n) = {\sum_{k=0}^N} b_k  x(n-k) - {\sum_{k=1}^M} a_k y(n-k)\end{equation}
+
+En calculant les transformée en Z de chaque membre, on calcule aisément la fonction de transfert $H(z)$ dans le domaine z :
+
+\begin{equation}
+H(z) = \frac {Y(z)} {X(z)} = \frac {{\sum_{k=0}^N} b_k z^{-k}} {1 + {\sum_{k=1}^M} a_k z^{-k}}
+\end{equation}
+
+Les valeurs des coefficients $a_k$ et $b_k$ fixeront le type de filtre (passe-bas, passe-haut, et...).
+
+
+\subsection{Exemples de filtres}
+
+\subsubsection{Le filtre moyenneur lisseur}
+
+Soit un signal numérique $y(n)$ avec $n \in \mathbb{N}$ issu d'un signal $x(n)$ tel que :
+
+\begin{equation}
+ y(n)=\frac{x(n)+x(n-1)+x(n-2)+...+x(n-N+1)}{N}
+\end{equation}
+
+La transformée en $Z$ de $y(n)$ s'écrit alors :
+
+\begin{eqnarray}
+ Y(z) & = & \sum_{p=0}^{N-1} X(z)\frac{z^{-p}}{N} \\
+      & = & X(z) \sum_{p=0}^{N-1}\frac{z^{-p}}{N}
+\end{eqnarray}
+
+On en déduit la fonction de transfert $H(z)$ du filtre équivalent :
+
+\begin{equation}
+ H(z)=\frac{Y(z)}{X(z)} = \sum_{p=0}^{N-1}\frac{z^{-p}}{N}
+\end{equation}
+
+\subsubsection{Le filtre passe-bas}
+
+Soit $x(n)$ un signal numérique quelconque de fréquence d'échantillonnage $f_e$. La loi récursive qui produit un filtrage passe-bas de fréquence de coupure $f_c$ pour obtenir le signal filtré $y(n)$ s'écrit :
+
+\begin{eqnarray}
+ y(n) & = & y(n-1) + a [x(n)- y(n-1)] \\
+ y(n) & = & a x(n)+[1-a] y(n-1) \\
+ y(n) + [a-1] y(n-1) & = & a x(n) \label{filtre_pb}
+\end{eqnarray}
+
+en ayant posé :
+
+\begin{equation*}
+ a = \frac{1}{1+\frac{f_e}{2 \pi f_c}}
+\end{equation*}
+
+En effectuant la transformée de Z de chaque membre de l'équation \ref{filtre_pb}, on obtient :
+
+\begin{eqnarray}
+  \sum_{n=-\infty}^{+\infty}y(n)z^{-n} + [a-1]  \sum_{n=-\infty}^{+\infty}y(n-1)z^{-n} & = & a \sum_{n=-\infty}^{+\infty}x(n)z^{-n} \\
+  \sum_{n=-\infty}^{+\infty}y(n)z^{-n} + [a-1]  \sum_{n=-\infty}^{+\infty}y(n)z^{-n-1} & = & a \sum_{n=-\infty}^{+\infty}x(n)z^{-n} \\
+  Y(z) + [a-1]  Y(z) . z^{-1} & = & a X(z) \\
+  \frac{Y(z)}{X(z)} = \frac{a}{1+(a-1)z^{-1}}
+\end{eqnarray}
+
+D'un point de vue pratique, c'est cette fonction qui permet d'implémenter - c'est à dire de mettre en oeuvre sous la forme d'un programme - la fonction de filtrage de type passe-bas dans un programme informatique. Dans le logiciel libre Octave\footnote{\url{http://www.octave.org}}, le code suivant permet par exemple de tester ce filtrage sur des signaux aléatoires à partir de la loi récursive mais aussi en utilisant la fonction \verb|filter| :
+
+\begin{verbatim}
+% Script Matlab / Octave pour tester une fonction de filtrage passe-bas.
+
+f0 = 20;
+f1 = 22050;
+t1 = 1;
+fe = 44100;
+fc = 3000;
+a = 1/(1+fe/(2*pi*fc));
+
+t = [0:1/fe:t1];
+n = length(t);
+N = 10;
+
+for k=1:N
+
+    s = randn(1,n);
+    %sweep = sin(2*pi*(f0*t1)/log(f1/f0) * (exp(t'/t1*log(f1/f0))-1))';
+    sf(1) = s(1);
+
+    % Avec une boucle
+    tic;
+    for i=2:n
+        sf(i)=a*s(i)+(1-a)*sf(i-1);
+    end
+    t_boucle(k) = toc;
+
+    % Avec filter
+    tic;
+    sff = filter(a,[1 a-1],s);
+    t_filter(k) = toc;
+
+    % Fonction de transfert
+    nfft = 16384;
+    fft_s = fft(s,nfft);
+    fft_sf = fft(sf,nfft);
+    fft_sff = fft(sff,nfft);
+    f = [0:fe/nfft:fe/2];
+    f(1) = [];
+
+    mod_FT(k,:) = abs(fft_sf)./abs(fft_s);
+    pha_FT(k,:) = angle(fft_sf);
+    mod_FTF(k,:) = abs(fft_sff)./abs(fft_s);
+    pha_FTF(k,:) = angle(fft_sff);
+
+end
+
+mod_FT = mean(mod_FT);
+figure(1)
+semilogx(f,20*log10(mod_FT(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la methode recursive (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FT = mean(pha_FT);
+figure(2)
+semilogx(f,unwrap(pha_FT(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la méthode recursive']);
+
+mod_FTF = mean(mod_FTF);
+figure(3)
+semilogx(f,20*log10(mod_FTF(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la fonction filter (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FTF = mean(pha_FTF);
+figure(4)
+semilogx(f,unwrap(pha_FTF(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la fonction filter']);
+
+figure(5)
+plot(t_boucle);
+title(['Temps d execution (s)']);
+hold on
+plot(t_filter,'r');
+
+pause;
+
+\end{verbatim}
+
+Pour plus d'exemple sur les fonctions numérique de filtrage, voir \url{http://math.fullerton.edu/mathews/c2003/ZTransformFilterMod.html}.
+
+\subsubsection{Le filtre vocal}
+
+On peut modéliser la sensation d'intensité auditive par un filtre. L'oreille humaine est particulièrement sensible entre 3 et 4 kHz.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/sens_oreille}
+    \caption{Diagramme de sensibilité de l'oreille humaine.}
+    \label{sens_oreille}
+\end{figure}
+
+On peut également modéliser la production phonatoire par un modèle source-filtre, où le rôle de la source est joué par les cordes vocales qui produisent un son harmonique avec une distribution de l'énergie assez plate en fréquence. Le conduit vocal, les fosses nasales ainsi que la place des articulateurs (langue, mâchoire, lèvres) peuvent être modélisés par un filtre qui modifie le son glottique pour produire le son tel que nous le percevons à la sortie des lèvres d'un locuteur.
+
+\begin{figure}[htp]
+    \centering
+    \includegraphics[width=12cm]{img/sf_voix}
+    \caption{Modèle source/filtre de la voix.}
+    \label{sf_voix}
+\end{figure}
+

diff --git a/biblio.bib b/biblio.bib
new file mode 100644 (file)
index 0000000..a6369bd
--- /dev/null
+++ b/biblio.bib
@@ -0,0 +1,74 @@
+
+
+@PhdThesis{henrich,
+author = {N. Henrich},
+title = {Etude de la source glottique en voix parlée et chantée},
+school = {Université Pierre et Marie Curie},
+year = {2001},
+}
+
+@InProceedings{pelorson,
+author = {X. Pelerson},
+title = {Cordes vocales et lèvres vibrantes},
+year = {2004},
+address = {Paris},
+month = {juillet},
+organization = {Conférence SFA },
+}
+
+@InProceedings{garnier1,
+author = {M. Garnier and D. Dubois and J. Poitevineau and N. Henrich and M. Castellengo},
+title = {Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive},
+year = {2004},
+address = {Fez},
+organization = {Journées d'Études sur la Parole},
+}
+
+@Booklet{charbit,
+title = {Bases de Traitement du signal},
+author = {M. Charbit},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {2004},
+}
+
+@Booklet{laroche,
+title = {Traitement des Signaux Audio-Fréquences},
+author = {J. Laroche},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {1995},
+}
+
+@Booklet{heiser,
+title = {Traitement du signal},
+author = {T. Heiser},
+howpublished = {Support de cours},
+address = {UFR Sciences Physiques, Université Louis Pasteur, Strasbourg},
+}
+
+@Booklet{feng,
+title = {Traitement du signal},
+author = {M. Feng},
+howpublished = {Support de cours},
+address = {ENSERG. INP Grenoble},
+year = {2000},
+}
+
+@Booklet{fallon,
+title = {Traitement du signal},
+author = {M. Fallon},
+howpublished = {Support de cours},
+address = {ENSAM, Lille},
+year = {1997},
+}
+
+@Booklet{matignon,
+title = {Traitement du signal},
+author = {D. Matignon},
+howpublished = {Support de cours},
+address = {ENST, Paris},
+year = {2000},
+}
+
+

diff --git a/biblio.bib~ b/biblio.bib~
new file mode 100644 (file)
index 0000000..02c7120
--- /dev/null
+++ b/biblio.bib~
@@ -0,0 +1,74 @@
+
+
+@PhdThesis{henrich,
+author = {N. Henrich},
+title = {Etude de la source glottique en voix parlée et chantée},
+school = {Université Pierre et Marie Curie},
+year = {2001},
+}
+
+@InProceedings{pelorson,
+author = {X. Pelerson},
+title = {Cordes vocales et lèvres vibrantes},
+year = {2004},
+address = {Paris},
+month = {juillet},
+organization = {Conférence SFA },
+}
+
+@InProceedings{garnier1,
+author = {M. Garnier and D. Dubois and J. Poitevineau and N. Henrich and M. Castellengo},
+title = {Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive},
+year = {2004},
+address = {Fez},
+organization = {Journées d'Études sur la Parole},
+}
+
+@Booklet{charbit,
+title = {Bases de Traitement du signal},
+author = {M. Charbit},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {2004},
+}
+
+@Booklet{laroche,
+title = {Traitement des Signaux Audio-Fréquences},
+author = {J. Laroche},
+howpublished = {Cours de DEA},
+address = {ENST, Paris},
+year = {1995},
+}
+
+@Booklet{heiser,
+title = {Traitement du signal},
+author = {T. Heiser},
+howpublished = {Support de cours},
+address = {UFR Sciences Physiques, Université Louis Pasteur, Strasbourg},
+}
+
+@Booklet{feng,
+title = {Traitement du signal},
+author = {Feng},
+howpublished = {Support de cours},
+address = {ENSERG. INP Grenoble},
+year = {2000},
+}
+
+@Booklet{fallon,
+title = {Traitement du signal},
+author = {M. Fallon},
+howpublished = {Support de cours},
+address = {ENSAM, Lille},
+year = {1997},
+}
+
+@Booklet{matignon,
+title = {Traitement du signal},
+author = {D. Matignon},
+howpublished = {Support de cours},
+address = {ENST, Paris},
+year = {2000},
+}
+
+

diff --git a/biblio.log b/biblio.log
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+                                                  
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+ ...                                              
+                                                  
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+ ...                                              
+                                                  
+l.35 @
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+ ...                                              
+                                                  
+l.43 @
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+ ...                                              
+                                                  
+l.50 @
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+ ...                                              
+                                                  
+l.58 @
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+
+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.66 @
+      Booklet{matignon,
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+
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+
+)
+! Emergency stop.
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+              
+*** (job aborted, no legal \end found)
+
+ 
+Here is how much of TeX's memory you used:
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diff --git a/exam_profil.sty b/exam_profil.sty
new file mode 100644 (file)
index 0000000..b467695
--- /dev/null
+++ b/exam_profil.sty
@@ -0,0 +1,145 @@
+\usepackage{times}\r
+\usepackage{mltex}\r
+\usepackage{amssymb}\r
+\usepackage[francais]{babel}\r
+\usepackage[centertags]{amsmath}\r
+\usepackage{amsfonts}\r
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+\usepackage{amsthm}\r
+\usepackage{newlfont}\r
+%\usepackage{xthesis} %DAL Thesis Style\r
+%\usepackage{xtocinc} %Include Table of Contents as the first entry in TOC\r
+%                     Faculty of Grad Studies insists on this!?\r
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+\usepackage[pdftex,bookmarksopen,bookmarksnumbered,colorlinks,pagebackref,breaklinks,citecolor={blue},linkcolor={black}]{hyperref}\r
+%\usepackage[bookmarksopen,colorlinks,pagebackref,linktocpage]{hyperref}\r
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+%       the#1\endcsname.\hspace*{6pt}}}}%\r
+%\makeatother\r
+\r
+\usepackage{url}\r
+\urlstyle{small}\r
+\r
+\renewcommand{\labelitemi}{--}\r
+\newcommand\address[1]{}\r
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+%\else\r
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+%\pdftrue \fi\r
+%\r
+%\ifpdf\r
+\usepackage[pdftex]{graphicx}\r
+%\else\r
+%\usepackage{graphicx}\r
+%\fi\r
+\r
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+% Fuzz -------------------------------------------------------------------\r
+\hfuzz2pt % Don't bother to report over-full boxes if over-edge is < 2pt\r
+\r
+% Line spacing -----------------------------------------------------------\r
+\newlength{\defbaselineskip}\r
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+           {\setlength{\baselineskip}{#1 \defbaselineskip}}\r
+\newcommand{\doublespacing}{\setlength{\baselineskip}%\r
+                           {2.0 \defbaselineskip}}\r
+\newcommand{\singlespacing}{\setlength{\baselineskip}{\defbaselineskip}}\r
+\r
+% MATH -------------------------------------------------------------------\r
+\newcommand{\A}{{\cal A}}\r
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+\newcommand{\IM}{\operatorname{Im}}\r
+\newcommand{\Poly}{{\cal{P}}(E)}\r
+\newcommand{\EssD}{{\cal{D}}}\r
+\r
+% THEOREMS ---------------------------------------------------------------\r
+\theoremstyle{plain}\r
+\newtheorem{thm}{Theorem}[section]\r
+\newtheorem{cor}[thm]{Corollary}\r
+\newtheorem{lem}[thm]{Lemma}\r
+\newtheorem{prop}[thm]{Proposition}\r
+\theoremstyle{definition}\r
+\newtheorem{defn}{Definition}[section]\r
+\theoremstyle{remark}\r
+\newtheorem{rem}{Remark}[section]\r
+\numberwithin{equation}{section}\r
+\renewcommand{\theequation}{\thesection.\arabic{equation}}\r
+\r
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+% spécification de la mise en page\r
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+\r
+\renewcommand{\labelitemi}{--}\r
+\r
+\newtheorem{theorem}{Theorem}\r
+\newtheorem{corollary}[theorem]{Corollary}\r
+\newtheorem{definition}{Definition}\r
+\r
+\setlength{\parindent}{0pt}\r
+\r
+%\setlength{\parskip}{0cm plus4mm minus3mm}\r
+\r
+%-------------------------------------------------------------\r
+% GRAPHICS\r
+\graphicspath{{figures/}}\r
+\r
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+%\ifpdf\r
+\DeclareGraphicsExtensions{.pdf,.jpg,.tif,.png}\r
+%\else\r
+%\DeclareGraphicsExtensions{.eps, .jpg}\r
+%\fi\r

diff --git a/html/Poly2005.html b/html/Poly2005.html
new file mode 100644 (file)
index 0000000..daaeb29
--- /dev/null
+++ b/html/Poly2005.html
@@ -0,0 +1,2835 @@
+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
+<HTML>
+<HEAD>
+       <META HTTP-EQUIV="CONTENT-TYPE" CONTENT="text/html; charset=iso-8859-15">
+       <TITLE></TITLE>
+       <META NAME="GENERATOR" CONTENT="OpenOffice.org 2.0  (Linux)">
+       <META NAME="AUTHOR" CONTENT="garnier">
+       <META NAME="CREATED" CONTENT="20051114;16020000">
+       <META NAME="CHANGEDBY" CONTENT="LAM">
+       <META NAME="CHANGED" CONTENT="20060926;16550000">
+       <STYLE TYPE="text/css">
+       <!--
+               @page { size: 21cm 29.7cm; margin-right: 2.5cm; margin-top: 2.5cm; margin-bottom: 1.25cm }
+               P { margin-bottom: 0cm; direction: ltr; color: #000000; text-align: justify; widows: 2; orphans: 2 }
+               P.western { font-family: "Arial", sans-serif; font-size: 12pt; so-language: fr-FR }
+               P.cjk { font-family: "Times New Roman", serif; font-size: 12pt }
+               P.ctl { font-family: "Arial", sans-serif; font-size: 10pt; so-language: ar-SA }
+               H1 { margin-top: 0cm; margin-bottom: 0cm; direction: ltr; color: #000000; text-align: right; widows: 2; orphans: 2 }
+               H1.western { font-family: "Arial", sans-serif; font-size: 20pt; so-language: fr-FR }
+               H1.cjk { font-family: "Times New Roman", serif; font-size: 20pt }
+               H1.ctl { font-family: "Arial", sans-serif; font-size: 12pt; so-language: ar-SA }
+               H2 { margin-top: 0cm; margin-bottom: 0cm; direction: ltr; color: #000000; widows: 2; orphans: 2 }
+               H2.western { font-family: "Arial", sans-serif; font-size: 14pt; so-language: fr-FR }
+               H2.cjk { font-family: "Times New Roman", serif; font-size: 14pt }
+               H2.ctl { font-family: "Arial", sans-serif; font-size: 12pt; so-language: ar-SA }
+               H3 { margin-top: 0cm; margin-bottom: 0cm; direction: ltr; color: #000000; widows: 2; orphans: 2; text-decoration: underline }
+               H3.western { font-family: "Arial", sans-serif; font-size: 12pt; so-language: fr-FR; font-weight: medium }
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+               H3.ctl { font-family: "Arial", sans-serif; font-size: 12pt; so-language: ar-SA; font-weight: medium }
+               H5 { margin-top: 0cm; margin-bottom: 0cm; direction: ltr; color: #000000; text-align: right; widows: 2; orphans: 2 }
+               H5.western { font-family: "Arial", sans-serif; font-size: 24pt; so-language: fr-FR; font-weight: medium }
+               H5.cjk { font-family: "Times New Roman", serif; font-size: 24pt; font-weight: medium }
+               H5.ctl { font-family: "Arial", sans-serif; font-size: 12pt; so-language: ar-SA; font-weight: medium }
+               H6 { margin-top: 0cm; margin-bottom: 0cm; direction: ltr; color: #000000; text-align: right; widows: 2; orphans: 2 }
+               H6.western { font-family: "Arial", sans-serif; font-size: 18pt; so-language: fr-FR; font-weight: medium }
+               H6.cjk { font-family: "Times New Roman", serif; font-size: 18pt; font-weight: medium }
+               H6.ctl { font-family: "Arial", sans-serif; font-size: 12pt; so-language: ar-SA; font-weight: medium }
+               A:link { color: #0000ff }
+               A:visited { color: #800080 }
+       -->
+       </STYLE>
+</HEAD>
+<BODY LANG="fr-FR" TEXT="#000000" LINK="#0000ff" VLINK="#800080" DIR="LTR">
+<P LANG="" CLASS="western" ALIGN=RIGHT><IMG SRC="Poly2005_html_46a534a4.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<H5 CLASS="western"><B>B<IMG SRC="Poly2005_html_46a534a4.gif" ALIGN=LEFT HSPACE=12>ases
+de traitement du signal</B></H5>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<H6 CLASS="western">Cours CNAM &ndash; CPDA &ndash; 2005</H6>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=RIGHT><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_46a534a4.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Arial, sans-serif">Ma&euml;va
+Garnier</FONT></P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Arial, sans-serif">Laboratoire
+d&rsquo;Acoustique musicale</FONT></P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Arial, sans-serif">11 rue
+de Lourmel. 75015 Paris</FONT></P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><FONT COLOR="#0000ff"><U><A HREF="mailto:garnier@lam.jussieu.fr"><FONT FACE="Arial, sans-serif">garnier@lam.jussieu.fr</FONT></A></U></FONT></FONT></P>
+<P ALIGN=CENTER STYLE="page-break-after: avoid"><FONT SIZE=6>Table
+des mati&egrave;res</FONT></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<DIV ID="Table of Contents1" DIR="LTR">
+       <P ALIGN=CENTER><A HREF="#1.Chapitre 1&nbsp;: Concepts essentiels de traitement du signal|outline">Chapitre
+       1&nbsp;: Concepts essentiels de traitement du signal    3</A></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.1.1.1. Introduction|outline">1.1.
+       Introduction    3</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.2.1.2. Classification des signaux|outline">1.2.
+       Classification des signaux      4</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.2.1.1.2.1. Classification dimensionnelle|outline">1.2.1.
+       Classification dimensionnelle   4</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.2.2.1.2.2. Classification ph&eacute;nom&eacute;nologique|outline">1.2.2.
+       Classification ph&eacute;nom&eacute;nologique   4</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.2.3.1.2.3. Classification morphologique|outline">1.2.3.
+       Classification morphologique    4</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.2.6.1.2.4. Classification &eacute;nerg&eacute;tique|outline">1.2.4.
+       Classification &eacute;nerg&eacute;tique        5</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.4.1.3. Signaux et distributions classiques|outline">1.3.
+       Signaux et distributions classiques     5</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.4.1.1.3.1. Fen&ecirc;tre rectangulaire ou Porte|outline">1.3.1.
+       Fen&ecirc;tre rectangulaire ou Porte    5</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.4.5.1.3.2. Impulsion de Dirac|outline">1.3.2.
+       Impulsion de Dirac      5</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.4.6.1.3.3. &laquo;&nbsp;Peigne&nbsp;&raquo; de Dirac|outline">1.3.3.
+       &laquo;&nbsp;Peigne&nbsp;&raquo; de Dirac       6</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.4.7.1.3.4.  Sinus cardinal|outline">1.3.4.
+        Sinus cardinal 6</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.1.4. Transform&eacute;e de Fourier|outline">1.4.
+       Transform&eacute;e de Fourier   6</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.1.1.4.1.  Rappels sur la d&eacute;composition en s&eacute;rie de Fourier des signaux p&eacute;riodiques|outline">1.4.1.
+        Rappels sur la d&eacute;composition en s&eacute;rie de Fourier des
+       signaux p&eacute;riodiques      6</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.2.1.4.2.  D&eacute;finition de la Transform&eacute;e de Fourier|outline">1.4.2.
+        D&eacute;finition de la Transform&eacute;e de Fourier  7</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.3.1.4.3. Le produit de convolution|outline">1.4.3.
+       Le produit de convolution       8</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.5.L&rsquo;impulsion de Dirac est l&rsquo;&eacute;l&eacute;ment neutre de la convolution, c&rsquo;est &agrave; dire que |outline">L&rsquo;impulsion
+       de Dirac est l&rsquo;&eacute;l&eacute;ment neutre de la convolution,
+       c&rsquo;est &agrave; dire que   8</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.8.1.4.4. Propri&eacute;t&eacute;s de la transform&eacute;e de Fourier|outline">1.4.4.
+       Propri&eacute;t&eacute;s de la transform&eacute;e de Fourier    9</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.5.9.1.4.5. Transform&eacute;es de Fourier des signaux courants|outline">1.4.5.
+       Transform&eacute;es de Fourier des signaux courants     10</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.6.1.5. Evaluation de la similitude entre deux signaux|outline">1.5.
+       Evaluation de la similitude entre deux signaux  10</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.6.1.1.5.1. Les fonctions d&rsquo;inter- et d&rsquo;auto-corr&eacute;lation|outline">1.5.1.
+       Les fonctions d&rsquo;inter- et d&rsquo;auto-corr&eacute;lation 11</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.6.2.1.5.2. La fonction de densit&eacute; spectrale d&rsquo;&eacute;nergie|outline">1.5.2.
+       La fonction de densit&eacute; spectrale d&rsquo;&eacute;nergie  11</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.7.1.6. Les signaux num&eacute;riques|outline">1.6.
+       Les signaux num&eacute;riques   12</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.7.1.1.6.1. D&eacute;finition de l&rsquo;&eacute;chantillonnage|outline">1.6.1.
+       D&eacute;finition de l&rsquo;&eacute;chantillonnage     12</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.7.2.1.6.2. D&eacute;finition et propri&eacute;t&eacute;s de la transform&eacute;e de Fourier d&rsquo;un signal &eacute;chantillonn&eacute;|outline">1.6.2.
+       D&eacute;finition et propri&eacute;t&eacute;s de la transform&eacute;e
+       de Fourier d&rsquo;un signal &eacute;chantillonn&eacute;        12</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#1.7.3.1.6.2. Cons&eacute;quences de l&rsquo;&eacute;chantillonnage du point de vue fr&eacute;quentiel |outline">1.6.2.
+       Cons&eacute;quences de l&rsquo;&eacute;chantillonnage du point de
+       vue fr&eacute;quentiel  13</A></FONT></P>
+       <P ALIGN=CENTER><A HREF="#2.Chapitre 2&nbsp;: Modes de repr&eacute;sentation des signaux|outline">Chapitre
+       2&nbsp;: Modes de repr&eacute;sentation des signaux     15</A></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#2.1.2.1. Fen&ecirc;trage des signaux |outline">2.1.
+       Fen&ecirc;trage des signaux     15</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#2.2.2.2. Le spectrogramme|outline">2.2.
+       Le spectrogramme        16</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#2.4.2.3. Pertinence et choix d&rsquo;un mode de repr&eacute;sentation&nbsp;|outline">2.3.
+       Pertinence et choix d&rsquo;un mode de repr&eacute;sentation&nbsp;      18</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#2.5.2.4. Le spectre moyenn&eacute; (Long Term Average Spectrum   LTAS)|outline">2.4.
+       Le spectre moyenn&eacute; (Long Term Average Spectrum   LTAS)   20</A></FONT></P>
+       <P ALIGN=CENTER><A HREF="#3.Chapitre 3&nbsp;: Syst&egrave;mes lin&eacute;aires et filtrage|outline">Chapitre
+       3&nbsp;: Syst&egrave;mes lin&eacute;aires et filtrage   22</A></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#3.1.3.1. R&eacute;ponse impulsionnelle d&rsquo;un syst&egrave;me|outline">3.1.
+       R&eacute;ponse impulsionnelle d&rsquo;un syst&egrave;me 22</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#3.2.3.2. R&eacute;ponse en fr&eacute;quence d&rsquo;un syst&egrave;me|outline">3.2.
+       R&eacute;ponse en fr&eacute;quence d&rsquo;un syst&egrave;me    23</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#5.13.3.3. Fonction de transfert d&rsquo;un syst&egrave;me (et d&eacute;finition de la Transform&eacute;e en Z)|outline">3.3.
+       Fonction de transfert d&rsquo;un syst&egrave;me (et d&eacute;finition
+       de la Transform&eacute;e en Z)  24</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.42cm"><FONT FACE="Times New Roman, serif"><A HREF="#5.16.3.4. Filtrage|outline">3.4.
+       Filtrage        25</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#5.16.1.3.4.1. Filtre id&eacute;aux|outline">3.4.1.
+       Filtre id&eacute;aux    25</A></FONT></P>
+       <P ALIGN=LEFT STYLE="margin-left: 0.85cm"><FONT FACE="Times New Roman, serif"><A HREF="#5.16.3.3.4.2. Exemples de filtres naturels |outline">3.4.2.
+       Exemples de filtres naturels    26</A></FONT></P>
+       <P ALIGN=CENTER><A HREF="#6.Bibliographie|outline">Bibliographie        28</A></P>
+</DIV>
+<H1 CLASS="western" ALIGN=LEFT STYLE="page-break-before: always"><A NAME="1.Chapitre 1&nbsp;: Concepts essentiels de traitement du signal|outline"></A>
+Chapitre 1&nbsp;: Concepts essentiels de traitement du signal</H1>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H2 CLASS="western">1.1. Introduction</H2>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western">L'information, qui se pr&eacute;sente &agrave;
+nous lors de l'observation d'un ph&eacute;nom&egrave;ne, se manifeste
+sous forme d'une ou plusieurs grandeurs physiques qui se d&eacute;roulent
+&agrave; la fois dans le temps et dans l'espace. Dans les probl&egrave;mes
+rencontr&eacute;s en pratique, on est souvent amen&eacute; &agrave;
+s'int&eacute;resser plus particuli&egrave;rement &agrave; l'un ou
+l'autre de ces deux aspects. On d&eacute;signe alors par le terme de
+signal l'&eacute;volution temporelle, tandis que l'on d&eacute;signe
+par le terme d'image l'&eacute;volution spatiale.</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Les
+diff&eacute;rents traitements, que l'on fait subir aux signaux,
+n&eacute;cessitent pour &ecirc;tre &eacute;tudi&eacute;s des outils
+math&eacute;matiques appropries, qui sont &agrave; la base du
+traitement du signal. Dans ce cours nous envisageons l'&eacute;tude
+de certains de ces outils.</FONT></P>
+<P CLASS="western">En signal on mod&eacute;lise la grandeur physique
+observ&eacute;e par un objet math&eacute;matique d&eacute;pendant de
+la variable r&eacute;elle t repr&eacute;sentant le temps. Dans la
+suite, le mot signal d&eacute;signera indiff&eacute;remment la
+grandeur physique observ&eacute;e ou l'objet math&eacute;matique
+servant &agrave; la mod&eacute;liser.</P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_263b3b56.png" NAME="graphics1" ALIGN=BOTTOM WIDTH=421 HEIGHT=253 BORDER=0></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_m3608984.png" NAME="graphics2" ALIGN=BOTTOM WIDTH=478 HEIGHT=151 BORDER=0></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<H2 CLASS="western">1.2. Classification des signaux</H2>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western">En th&eacute;orie du signal, on a l&rsquo;habitude
+d&rsquo;envisager plusieurs types de signaux. Pour rendre la suite de
+ce cours plus compr&eacute;hensible, nous pr&eacute;sentons dans ce
+paragraphe les diff&eacute;rents modes de classification des signaux
+. 
+</P>
+<P CLASS="western"><BR>
+</P>
+<H3 CLASS="western">1.2.1. Classification dimensionnelle</H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">On peut
+diff&eacute;rentier les signaux selon leur nombre de variables
+libres. Par exemple un signal &eacute;lectrique V(t) est un signal
+unidimensionnel d&eacute;pendant du temps. Une image statique en noir
+et blanc peut &ecirc;tre mod&eacute;lis&eacute;e par un signal
+bidimensionnel correspondant &agrave; la brillance en fonction de la
+largeur et de la hauteur de l&rsquo;image. Enfin, un film en noir et
+blanc peut &ecirc;tre consid&eacute;r&eacute; comme une image
+fonction du temps, autrement dit mod&eacute;lis&eacute; par un signal
+tridimensionnel d&eacute;pendant le l&rsquo;abscisse et de l&rsquo;ordonn&eacute;e
+de l&rsquo;image mais &eacute;galement du temps.</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.2.2. Classification ph&eacute;nom&eacute;nologique</H3>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">la
+premi&egrave;re distinction porte sur notre capacit&eacute; &agrave;
+pr&eacute;dire l'&eacute;volution temporelle de la grandeur observ&eacute;e.
+Si on prend, par exemple, un oscillateur sinuso&iuml;dal d'amplitude
+A et de fr&eacute;quence f0, moyennant la mesure de A et de f0, on
+peut alors pr&eacute;dire la valeur de l'amplitude, &agrave; tout
+instant, par l'expression x(t) = Acos(2&frac14;f0t). Un tel mod&egrave;le
+de signal est dit d&eacute;terministe. Il existe cependant des
+situations o&ugrave; il n'est pas concevable d'expliciter, de cette
+fa&ccedil;on, la forme de l'&eacute;volution temporelle du signal.
+C'est le cas, par exemple, pour la tension engendr&eacute;e &agrave;
+la sortie d'un microphone ou encore pour le courant &eacute;lectrique
+produit par l'agitation thermique des particules dans un conducteur
+(bruit de fond). On ne peut pas dire avec certitude combien vaudra
+x(t) &agrave; l'instant t, mais on pourra &eacute;ventuellement
+supposer que cette grandeur est distribu&eacute;e suivant une
+certaine loi de probabilit&eacute;. On dit alors que le signal est
+al&eacute;atoire.</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Pour
+simplifier, on peut dire que les signaux d&eacute;terministes sont
+des cas relativement th&eacute;oriques et id&eacute;aux, ou bien
+mod&eacute;lisent des signaux observ&eacute;s exp&eacute;rimentalement.
+Car en fait, tout signal physique r&eacute;el comporte une composante
+al&eacute;atoire correspondant &agrave; la perturbation externe.</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_m48aec8df.png" NAME="graphics3" ALIGN=BOTTOM WIDTH=516 HEIGHT=200 BORDER=0></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.2.3. Classification morphologique</H3>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT SIZE=2 STYLE="font-size: 11pt"><FONT FACE="Arial, sans-serif">Si,
+comme c'est le cas pour le signal x(t) = Acos(2&frac14;f0t), le temps
+t prend ses valeurs dans R, on dit que le signal est &agrave; temps
+continu. Toutefois on rencontre aussi en traitement du signal des
+grandeurs qui &eacute;voluent uniquement &agrave; des instants
+discrets tn o&ugrave; n est un entier. On parle alors de signal &agrave;
+temps discret ou encore de signal num&eacute;rique. En terme
+math&eacute;matique, un signal &agrave; temps continu est une
+fonction du temps tandis qu'un signal &agrave; temps discret est une
+suite. Le d&eacute;veloppement et l'essor des techniques num&eacute;riques
+ont fait que les probl&egrave;mes portant sur le traitements des
+signaux &agrave; temps discret ont pris une place majeure
+aujourd'hui, compar&eacute;e &agrave; celle qu'occupent les
+traitements portant sur les signaux &agrave; temps continu. C'est
+pourquoi ce cours est centre avant tout sur les probl&egrave;mes de
+temps discret et sur le passage du temps continu au temps discret
+(th&eacute;or&egrave;me d'&eacute;chantillonnage). </FONT></FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif"><FONT SIZE=2 STYLE="font-size: 11pt">Il
+est &eacute;galement possible de discr&eacute;tiser le signal en
+amplitude. On appelle cela la quantification du signal. </FONT></FONT>
+</P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><FONT COLOR="#000000"><IMG SRC="Poly2005_html_m46aac255.png" NAME="graphics4" ALIGN=LEFT HSPACE=12 WIDTH=376 HEIGHT=296 BORDER=1></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<H3 CLASS="western"></H3>
+<H3 CLASS="western"></H3>
+<H3 CLASS="western">1.2.4. Classification &eacute;nerg&eacute;tique</H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">On
+distingue en particulier deux types de signaux&nbsp;:</FONT></P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">- Les
+signaux &agrave; &eacute;nergie finie     (Tout signal physique)</FONT></P>
+<P CLASS="western" ALIGN=LEFT> 
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">- Les
+signaux &agrave; puissance moyenne finie    (Id&eacute;alisation.
+exemple:  signal sinuso&iuml;dal)</FONT></P>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western">1.3. Signaux et distributions classiques</H2>
+<H3 CLASS="western">1.3.1. Fen&ecirc;tre rectangulaire ou Porte</H3>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Notation&nbsp;:
+       &Pi;<SUB>2a</SUB> (t) </FONT></FONT>
+</P>
+<H3 CLASS="western"></H3>
+<H3 LANG="" CLASS="western"><IMG SRC="Poly2005_html_d6ba553.png" NAME="graphics5" ALIGN=LEFT HSPACE=12 WIDTH=252 HEIGHT=108 BORDER=0></H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Repr&eacute;sentation&nbsp;:
+</FONT>
+</P>
+<H3 CLASS="western"></H3>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.3.2. Impulsion de Dirac</H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Les
+propri&eacute;t&eacute;s de l&rsquo;impulsion de Dirac sont
+identiques &agrave; celles d&rsquo;un signal rectangulaire dont la
+largeur tend vers 0 et la hauteur vers l&rsquo;infini, &agrave;
+surface constante. La r&eacute;ponse d&rsquo;un syst&egrave;me &agrave;
+un signal &ldquo;extr&ecirc;mement bref&rdquo; sera d&eacute;crite
+par sa r&eacute;ponse &agrave; l&rsquo;impulsion de Dirac.</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_d6ba553.png" NAME="graphics6" ALIGN=BOTTOM WIDTH=229 HEIGHT=114 BORDER=0></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<H3 CLASS="western">1.3.3. &laquo;&nbsp;Peigne&nbsp;&raquo; de Dirac</H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Le
+peigne de Dirac correspond &agrave; un p&eacute;riodisation
+d&rsquo;impulsions de Dirac tous les nT  sur tout l&rsquo;axe de des
+temps (aussi bien t&lt;0 que t&gt; 0).</FONT></P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_m6e06f742.png" NAME="graphics7" ALIGN=LEFT HSPACE=12 WIDTH=237 HEIGHT=125 BORDER=0><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">&#1064;T
+ (t) = &Sigma; <I>&delta;</I>(t-nT)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<H3 CLASS="western">1.3.4.  Sinus cardinal</H3>
+<P ALIGN=CENTER>Le sinus cardinal se d&eacute;finit par la formule&nbsp;:
+Sinc (t) = sin(t) / t  . 
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_5d24021a.png" NAME="graphics8" ALIGN=BOTTOM WIDTH=349 HEIGHT=180 BORDER=0></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<H2 CLASS="western">1.4. Transform&eacute;e de Fourier</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.4.1.  Rappels sur la d&eacute;composition en
+s&eacute;rie de Fourier des signaux p&eacute;riodiques</H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Tout
+signal p&eacute;riodique est d&eacute;composable en une somme de
+sinus et de cosinus. </FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=CENTER><IMG SRC="Poly2005_html_5108fbfc.png" NAME="graphics9" ALIGN=BOTTOM WIDTH=593 HEIGHT=262 BORDER=0></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">La
+composante continue de cette d&eacute;composition, aussi appel&eacute;e
+fondamentale, correspond &agrave; la hauteur per&ccedil;ue d&rsquo;un
+son par exemple. Cette composante est &agrave; la fr&eacute;quence
+inverse de la p&eacute;riode du signal. </FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Les
+autres composantes sont de fr&eacute;quences sup&eacute;rieures &agrave;
+la fondamentale et sont appel&eacute;es harmoniques du signal.
+Toujours dans le cas d&rsquo;un son harmonique , jou&eacute; par
+exemple par un instrument de musique, la fondamentale va correspondre
+&agrave; la note jou&eacute;e et les autres composantes vont
+correspondre &agrave; la &laquo;&nbsp;s&eacute;rie harmonique&nbsp;&raquo;
+de cette note (octave, puis quinte, puis octave, puis tierce&hellip;etc).
+Cette s&eacute;rie correspond &agrave; des rapports de fr&eacute;quences
+(fondamental fo, octave = fo*2, octave + quinte = fo*3&hellip;etc).
+Les harmoniques ne vont pas directement contribuer &agrave; la
+perception de la hauteur mais plut&ocirc;t &agrave; la richesse du
+timbre du son. C&rsquo;est par exemple la pr&eacute;sence
+d&rsquo;harmoniques -entre autres- qui permet de diff&eacute;rentier
+une sinuso&iuml;de &agrave; fo (seulement un fondamental =&gt; son
+proche d&rsquo;un buzzer &eacute;lectronique) d&rsquo;une note de
+violon jou&eacute;e &agrave; la m&ecirc;me fr&eacute;quence.</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.4.2.  D&eacute;finition de la Transform&eacute;e
+de Fourier</H3>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">La
+transform&eacute;e de Fourier est une &laquo;&nbsp;g&eacute;n&eacute;ralisation&nbsp;&raquo;
+de cette d&eacute;composition aux signaux non p&eacute;riodiques.</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">On
+d&eacute;finit la transform&eacute;e de Fourier comme&nbsp;:</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_mb5958d.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame90" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61563;</FONT></P>
+</SPAN>
+<SPAN ID="Frame89" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61565;</FONT></P>
+</SPAN>
+<SPAN ID="Frame88" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame87" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame86" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61480;</FONT></P>
+</SPAN>
+<SPAN ID="Frame85" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61481;</FONT></P>
+</SPAN>
+<SPAN ID="Frame84" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61480;</FONT></P>
+</SPAN>
+<SPAN ID="Frame83" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61481;</FONT></P>
+</SPAN>
+<SPAN ID="Frame82" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61480;</FONT></P>
+</SPAN>
+<SPAN ID="Frame81" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61481;</FONT></P>
+</SPAN>
+<SPAN ID="Frame80" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame79" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame78" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61480;</FONT></P>
+</SPAN>
+<SPAN ID="Frame77" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61481;</FONT></P>
+</SPAN>
+<SPAN ID="Frame76" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61563;</FONT></P>
+</SPAN>
+<SPAN ID="Frame75" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61565;</FONT></P>
+</SPAN>
+<SPAN ID="Frame74" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61480;</FONT></P>
+</SPAN>
+<SPAN ID="Frame73" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61481;</FONT></P>
+</SPAN>
+<SPAN ID="Frame72" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61480;</FONT></P>
+</SPAN>
+<SPAN ID="Frame71" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61481;</FONT></P>
+</SPAN>
+<SPAN ID="Frame70" DIR="LTR" STYLE="float: left; width: 2.1cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>inverse&quot;</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame69" DIR="LTR" STYLE="float: left; width: 2.03cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>Fourier
+       </FONT></FONT></FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame68" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame67" DIR="LTR" STYLE="float: left; width: 0.6cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>de</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame66" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame65" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>e</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame64" DIR="LTR" STYLE="float: left; width: 2.96cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>Transform&eacute;</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame63" DIR="LTR" STYLE="float: left; width: 0.26cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>&quot;</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame62" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame61" DIR="LTR" STYLE="float: left; width: 0.32cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">  </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame60" DIR="LTR" STYLE="float: left; width: 0.32cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>2</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame59" DIR="LTR" STYLE="float: left; width: 0.92cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>exp</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame58" DIR="LTR" STYLE="float: left; width: 0.46cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>et</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame57" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame5" DIR="LTR" STYLE="float: left; width: 0.87cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>dee</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame56" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame55" DIR="LTR" STYLE="float: left; width: 2.13cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>Fourier&quot;</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame54" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame53" DIR="LTR" STYLE="float: left; width: 0.6cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>de</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame52" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame51" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>e</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame50" DIR="LTR" STYLE="float: left; width: 2.97cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>Transform&eacute;</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame49" DIR="LTR" STYLE="float: left; width: 0.26cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>&quot;</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame48" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame47" DIR="LTR" STYLE="float: left; width: 0.48cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">   </FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame46" DIR="LTR" STYLE="float: left; width: 0.32cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>2</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame45" DIR="LTR" STYLE="float: left; width: 0.92cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>exp</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame44" DIR="LTR" STYLE="float: left; width: 0.21cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>)</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame43" DIR="LTR" STYLE="float: left; width: 0.21cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>(</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame42" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=4 STYLE="font-size: 16pt">1</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame41" DIR="LTR" STYLE="float: left; width: 0.14cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=4 STYLE="font-size: 16pt">.</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame40" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61501;</FONT></P>
+</SPAN>
+<SPAN ID="Frame39" DIR="LTR" STYLE="float: left; width: 0.17cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61682;</FONT></P>
+</SPAN>
+<SPAN ID="Frame38" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61655;</FONT></P>
+</SPAN>
+<SPAN ID="Frame37" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61501;</FONT></P>
+</SPAN>
+<SPAN ID="Frame36" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61501;</FONT></P>
+</SPAN>
+<SPAN ID="Frame35" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61501;</FONT></P>
+</SPAN>
+<SPAN ID="Frame34" DIR="LTR" STYLE="float: left; width: 0.17cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61682;</FONT></P>
+</SPAN>
+<SPAN ID="Frame33" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61655;</FONT></P>
+</SPAN>
+<SPAN ID="Frame32" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61485;</FONT></P>
+</SPAN>
+<SPAN ID="Frame31" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61501;</FONT></P>
+</SPAN>
+<SPAN ID="Frame30" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61501;</FONT></P>
+</SPAN>
+<SPAN ID="Frame29" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61605;</FONT></P>
+</SPAN>
+<SPAN ID="Frame28" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61483;</FONT></P>
+</SPAN>
+<SPAN ID="Frame27" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61605;</FONT></P>
+</SPAN>
+<SPAN ID="Frame26" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61485;</FONT></P>
+</SPAN>
+<SPAN ID="Frame25" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61485;</FONT></P>
+</SPAN>
+<SPAN ID="Frame24" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61605;</FONT></P>
+</SPAN>
+<SPAN ID="Frame23" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61483;</FONT></P>
+</SPAN>
+<SPAN ID="Frame22" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61605;</FONT></P>
+</SPAN>
+<SPAN ID="Frame21" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61485;</FONT></P>
+</SPAN>
+<SPAN ID="Frame20" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>i</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame19" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>X</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame18" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>X</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame17" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I><B>t</B></I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame16" DIR="LTR" STYLE="float: left; width: 0.32cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I><B>x</B></I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame15" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame14" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame2" DIR="LTR" STYLE="float: left; width: 0.89cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>dt</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame1" DIR="LTR" STYLE="float: left; width: 0.49cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>ft</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame13" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>i</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame12" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame11" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame4" DIR="LTR" STYLE="float: left; width: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT> <FONT FACE="Times New Roman, serif"><FONT SIZE=4><I><B>f</B></I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame10" DIR="LTR" STYLE="float: left; width: 0.43cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I><B>X</B></I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame9" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame8" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame3" DIR="LTR" STYLE="float: left; width: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>TF</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame7" DIR="LTR" STYLE="float: left; width: 0.69cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=4 STYLE="font-size: 16pt"><I>d&eacute;f</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame6" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61552;</FONT></P>
+</SPAN>
+<SPAN ID="Frame91" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61552;</FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=CENTER><SPAN ID="Frame96" DIR="LTR" STYLE="float: left; width: 0.9cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>df</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame92" DIR="LTR" STYLE="float: left; width: 0.5cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>ft</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame95" DIR="LTR" STYLE="float: left; width: 0.5cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>f</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame93" DIR="LTR" STYLE="float: left; width: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>f</I></FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame94" DIR="LTR" STYLE="float: left; width: 0.96cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: -1.25cm">
+       <FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>TF</I></FONT></FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><I>X(f)</I>
+est une fonction <I>complexe</I> m&ecirc;me si <I>x(t) </I>est r&eacute;el.
+</FONT><FONT FACE="Times New Roman, serif">|</FONT><I><FONT FACE="Arial, sans-serif">X(f)</FONT></I><FONT FACE="Times New Roman, serif">|</FONT><FONT FACE="Arial, sans-serif">
+est appel&eacute; &ldquo;spectre d&rsquo;amplitude&rdquo; et
+<I>Arg[X(f)] </I>=&ldquo;spectre de phase&rdquo; du signal. </FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">La
+variable f s'appelle la fr&eacute;quence. Son unit&eacute; est le
+Hertz (en abr&eacute;g&eacute; : Hz).</FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">La
+Transform&eacute;e de Fourier est un op&eacute;rateur math&eacute;matique
+qui permet  <U>d&rsquo;analyser</U> et de <U>repr&eacute;senter</U>
+un signal dans le domaine fr&eacute;quentiel. La TF ne modifie pas le
+signal mais permet seulement de l&rsquo;observer selon diff&eacute;rents
+points de vue.(temporel ou fr&eacute;quentiel). Il est important de
+retenir que x(t) et X(f) sont deux descriptions <U>&eacute;quivalentes</U>
+du m&ecirc;me signal. Ces deux fonctions contiennent la m&ecirc;me
+information. Il s&rsquo;agit juste de deux descriptions dans des
+domaines diff&eacute;rents (temporel et fr&eacute;quentiel).</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><I>X(f)
+<SUB> </SUB></I>apporte des information sur le syst&egrave;me
+physique &agrave; l&rsquo;origine du signal. Elle permet par exemple
+de diff&eacute;rentier un son de trompette d&rsquo;un son trombone,
+ou bien encore diff&eacute;rentes ondes c&eacute;r&eacute;brales ,
+mieux qu&rsquo;en observant le signal dans le domaine temporel.</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_m49351038.png" NAME="graphics10" ALIGN=BOTTOM WIDTH=493 HEIGHT=240 BORDER=0></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=CENTER><BR>
+</P>
+<P ALIGN=CENTER><BR>
+</P>
+<H3 CLASS="western">1.4.3. Le produit de convolution</H3>
+<H3 CLASS="western"></H3>
+<P ALIGN=LEFT>On appelle convolution de x(t) par y(t) l'op&eacute;ration
+not&eacute;e x(t) * y(t) et d&eacute;finie par :</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_ff104a0.png" NAME="graphics11" ALIGN=BOTTOM WIDTH=509 HEIGHT=80 BORDER=0></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<H3 CLASS="western"><A NAME="1.5.5.L&rsquo;impulsion de Dirac est l&rsquo;&eacute;l&eacute;ment neutre de la convolution, c&rsquo;est &agrave; dire que |outline"></A>
+L&rsquo;impulsion de Dirac est l&rsquo;&eacute;l&eacute;ment neutre
+de la convolution, c&rsquo;est &agrave; dire que 
+</H3>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">x(t)
+<B>*</B> &delta; (t) = x(t).</FONT></FONT></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western">Lorsque l&rsquo;on convolue un signal x(t) &agrave;
+un Dirac en to par exemple, cela revient &agrave; retarder le signal
+x(t) de t<SUB>o</SUB>&nbsp;:           x(t) <B>*</B> &delta; (t-t<SUB>o</SUB>)
+= x(t-t<SUB>o</SUB>).</P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western">Alors que si l&rsquo;on multiplie un signal x(t)
+par un Dirac en t<SUB>o</SUB>, cela revient &agrave; conna&icirc;tre
+la valeur que prend x(t) en t<SUB>o</SUB> (comme si l&rsquo;on
+relevait l&rsquo;ordonn&eacute;e d&rsquo;un point particulier d&rsquo;une
+courbe)                         x(t) <B>.</B> &delta; (t-t<SUB>o</SUB>) = x(t<SUB>o</SUB>).            </P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">De
+m&ecirc;me, lorsque l&rsquo;on convolue un signal x(t) &agrave; un
+peigne de Dirac (de p&eacute;riode T), cela revient &agrave;
+&laquo;&nbsp;p&eacute;riodiser&nbsp;&raquo;  le signal x(t) tous les
+nT&nbsp;: on retarde le signal x(t) de T , de 2T, de 3T&hellip;.etc  
+                        x(t) <B>*</B> &#1064;T  (t)  =  &Sigma; x(t-nT)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">Tandis que si on multiplie le signal x(t) par un
+peigne de Dirac, on pr&eacute;l&egrave;ve des &eacute;chantillons du
+signal x(t) tous les nT. Nous verrons plus loin que cette op&eacute;ration
+correspond &agrave; l&rsquo; &laquo;&eacute;chantillonnage&raquo;  du
+signal x(t).</P>
+<P CLASS="western" ALIGN=LEFT> <FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">                                     x(t)
+<B>*</B> &#1064;T  (t)  =  &Sigma; x(nT)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">De fa&ccedil;on plus g&eacute;n&eacute;rale, la
+convolution telle qu&rsquo;elle est d&eacute;finie par sa formule
+math&eacute;matique, revient &agrave; retourner temporellement un des
+deux signaux (par exemple x(t) )  puis &agrave; le d&eacute;placer
+sur tout l&rsquo;axe du temps  et &agrave; sommer toutes les
+multiplications de ce signal retourn&eacute; d&eacute;plac&eacute; au
+deuxi&egrave;me signal ( y(t) )&nbsp;.</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_6e4f9962.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame101" DIR="LTR" STYLE="float: left; width: 1.27cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(u)</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m33e81f2e.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame99" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(-u)</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_21babc37.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame97" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(t-u)</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_m33e81f2e.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m27cc59a0.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m3617dad0.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame100" DIR="LTR" STYLE="float: left; width: 2.22cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">Temps
+       u</FONT></P>
+</SPAN>
+<SPAN ID="Frame98" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">u</FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame102" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">u</FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 6.24cm; text-indent: -1.25cm">
+<FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Retournement
+temporel              t est un instant qui varie du signal x               dans le
+temps u de  <SUP>-</SUP>&infin;  &agrave;  <SUP>+</SUP>&infin;.</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 10.57cm">  <FONT FACE="Arial, sans-serif">On
+fait &laquo;&nbsp;glisser&nbsp;&raquo; le signal </FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 10.57cm">  <FONT FACE="Arial, sans-serif">x(-u)
+selon l&rsquo;axe du temps.</FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 10.57cm"><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 10.57cm"><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 10.57cm"><FONT FACE="Arial, sans-serif">     
+</FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame110" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(t-u)</FONT></P>
+</SPAN>
+<SPAN ID="Frame109" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=2><B>y(u)</B></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m4ab5e85d.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame108" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(t-u)</FONT></P>
+</SPAN>
+<SPAN ID="Frame106" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">u</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m4c734faf.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_7fa8a812.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_7fa8a812.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame107" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=2><B>y(u)</B></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_m15a6fd57.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame104" DIR="LTR" STYLE="float: left; width: 0.74cm; height: 0.53cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=1 STYLE="font-size: 8pt"><B>x</B></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame105" DIR="LTR" STYLE="float: left; width: 0.74cm; height: 0.53cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=1 STYLE="font-size: 8pt"><B>x</B></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame103" DIR="LTR" STYLE="float: left; width: 0.74cm; height: 0.53cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=1 STYLE="font-size: 8pt"><B>x</B></FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame115" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(t-u)</FONT></P>
+</SPAN>
+<SPAN ID="Frame114" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">u</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m4c734faf.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m289c9194.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m79adcff2.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame113" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=2><B>y(u)</B></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_m27cc59a0.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame112" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5 STYLE="font-size: 20pt"><B>+</B></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame111" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5 STYLE="font-size: 20pt"><B>+</B></FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m4c734faf.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_m289c9194.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m79adcff2.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m33e81f2e.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame116" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">u</FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">On multiplie y(u) par x(t-u)  pour chaque position
+de t sur l&rsquo;axe du temps et on somme tous les produits
+effectu&eacute;s.  
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_6e4f9962.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame118" DIR="LTR" STYLE="float: left; width: 1.27cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">x(u)</FONT></P>
+</SPAN>
+<SPAN ID="Frame117" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5 STYLE="font-size: 20pt"><B>*</B></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_1779af18.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_672a464b.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m7d2040c2.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame119" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">u</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_781c2f4d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m4c734faf.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_37042a56.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m79adcff2.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame120" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=2><B>y(u)</B></FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame121" DIR="LTR" STYLE="float: left; width: 2.22cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">Temps
+       u</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_m2a542914.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_37042a56.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m79adcff2.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western"></H3>
+<H3 CLASS="western"></H3>
+<H3 CLASS="western">1.4.4. Propri&eacute;t&eacute;s de la transform&eacute;e
+de Fourier</H3>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Lin&eacute;arit&eacute;
+       de la TF&nbsp;:  a x(t)+b y(t)     </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+         a X(f)+b Y(f)</FONT></FONT></P>
+</UL>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 0.64cm"><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">P<IMG SRC="Poly2005_html_4bc2ff5a.gif" ALIGN=LEFT HSPACE=12>roduit
+       de convolution&nbsp;:</FONT></P>
+</UL>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">x(t)<SUP>.</SUP>y(t)
+    </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+  X(f) * Y(f)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">x(t)
+* y(t)   </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+  X(f) . Y(f)  </FONT><FONT SIZE=2><FONT COLOR="#545472"> </FONT></FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P STYLE="margin-left: 2.5cm">Une multiplication dans un domaine
+correspond &agrave; un produit de convolution dans l&rsquo;autre. 
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Retard&nbsp;:
+       </FONT>
+       </P>
+</UL>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY STYLE="margin-left: 2.5cm"> <IMG SRC="Poly2005_html_4bc2ff5a.gif" ALIGN=LEFT HSPACE=12>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY STYLE="margin-left: 3.77cm"><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=JUSTIFY></P>
+       <LI><P CLASS="western" ALIGN=JUSTIFY></P>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><SPAN LANG="de-DE"><FONT FACE="Arial, sans-serif">x(t-t<SUB>o</SUB><SUB>)
+          </SUB></FONT><SUB><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+       </FONT></SUB><FONT FACE="Arial, sans-serif">   X(f) e<SUP>-i2pft</SUP><SUB>o</SUB></FONT></SPAN></FONT></P>
+</UL>
+<P LANG="de-DE" CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Times New Roman, serif"><SPAN LANG="de-DE"><FONT FACE="Arial, sans-serif">x(t)<SUP>.</SUP>e<SUP>i2pfot
+    </SUP> </FONT><SUB><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+</FONT></SUB><FONT FACE="Arial, sans-serif"> X(f-f<SUB>o</SUB>)</FONT></SPAN></FONT></P>
+<P LANG="de-DE" CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif">          </FONT></FONT></P>
+<P LANG="de-DE" CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<UL>
+       <LI><P LANG="de-DE" CLASS="western" ALIGN=JUSTIFY></P>
+</UL>
+<P CLASS="western" ALIGN=JUSTIFY STYLE="margin-left: 0.64cm"><BR>
+</P>
+<P STYLE="margin-left: 1.25cm; page-break-after: avoid">Un retard
+UU&lt;u</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 2.5cm"><FONT FACE="Arial, sans-serif">Un
+retard temporel correspond &agrave; un d&eacute;phasage au niveau
+fr&eacute;quentiel, et inversement.</FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 2.5cm"><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Changement
+       d&rsquo;&eacute;chelle&nbsp;:   x(at)   </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+        1/a  *  X(f/a)    </FONT></FONT>
+       </P>
+</UL>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Th&eacute;or&egrave;me
+       de Parseval&nbsp;:</FONT></P>
+</UL>
+<P LANG="" CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<SPAN ID="Frame131" DIR="LTR" STYLE="float: left; width: 0.17cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61682;</FONT></P>
+</SPAN>
+<SPAN ID="Frame130" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame129" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61483;</FONT></P>
+</SPAN>
+<SPAN ID="Frame128" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame127" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61485;</FONT></P>
+</SPAN>
+<SPAN ID="Frame126" DIR="LTR" STYLE="float: left; width: 0.17cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61682;</P>
+</SPAN>
+<SPAN ID="Frame125" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame123" DIR="LTR" STYLE="float: left; width: 0.4cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame124" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61483;</FONT></P>
+</SPAN>
+<SPAN ID="Frame122" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">&#61485;</FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_6a13cd96.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Arial, sans-serif">&Eacute;nergie      =   |x(t)|&sup2;
+dt   =       |X(f)|&sup2; df</FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<H3 CLASS="western">1.4.5. Transform&eacute;es de Fourier des signaux
+courants</H3>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Dirac
+(t)   </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+ signal unit&eacute;</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Signal
+porte </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+ sinus cardinal</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Sin(2
+Pi fo t)   </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+  &frac12; ( dirac (f-fo)  + dirac (f+fo)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Arial, sans-serif">A compl&egrave;ter avec vos notes de
+cours&hellip;.&nbsp;</FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<H2 CLASS="western">1.5. Evaluation de la similitude entre deux
+signaux</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Il
+existe en traitement du signal des outils math&eacute;matiques qui
+permettent d&rsquo;&eacute;valuer la similitude entre deux signaux, </FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">-au
+niveau de leur &eacute;volution temporelle gr&acirc;ce  &agrave; la
+fonction d&rsquo;inter- ou d&rsquo;auto-corr&eacute;lation.</FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">-au
+niveau de leur composition spectrale gr&acirc;ce &agrave; la fonction
+de densit&eacute; spectrale. </FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.5.1. Les fonctions d&rsquo;inter- et
+d&rsquo;auto-corr&eacute;lation</H3>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">La 
+fonction d&rsquo;intercorr&eacute;lation se d&eacute;finit par la
+formule suivante&nbsp;:</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_18cba134.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame147" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame146" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame145" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame144" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame143" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame142" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame141" DIR="LTR" STYLE="float: left; width: 0.86cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61682;</P>
+</SPAN>
+<SPAN ID="Frame140" DIR="LTR" STYLE="float: left; width: 0.65cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame139" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61483;</P>
+</SPAN>
+<SPAN ID="Frame138" DIR="LTR" STYLE="float: left; width: 0.65cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame137" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61485;</P>
+</SPAN>
+<SPAN ID="Frame136" DIR="LTR" STYLE="float: left; width: 0.73cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61483;</P>
+</SPAN>
+<SPAN ID="Frame135" DIR="LTR" STYLE="float: left; width: 0.73cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame134" DIR="LTR" STYLE="float: left; width: 0.44cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4>*</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame149" DIR="LTR" STYLE="float: left; width: 0.71cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>dt</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame133" DIR="LTR" STYLE="float: left; width: 0.43cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame132" DIR="LTR" STYLE="float: left; width: 0.54cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>y</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame154" DIR="LTR" STYLE="float: left; width: 0.43cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame153" DIR="LTR" STYLE="float: left; width: 0.54cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame152" DIR="LTR" STYLE="float: left; width: 0.44cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>xy</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame151" DIR="LTR" STYLE="float: left; width: 0.66cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61556;</P>
+</SPAN>
+<SPAN ID="Frame150" DIR="LTR" STYLE="float: left; width: 0.66cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61556;</P>
+</SPAN>
+<SPAN ID="Frame148" DIR="LTR" STYLE="float: left; width: 0.77cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61546;</P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">La
+fonction d&rsquo;autocorr&eacute;lation est un cas particulier de la
+fonction d&rsquo;intercorr&eacute;lation pour y(t) = x(t). Elle se
+d&eacute;finit par&nbsp;:</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_7aa4aa7.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame176" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame175" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame174" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame173" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame172" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame171" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame170" DIR="LTR" STYLE="float: left; width: 0.86cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61682;</P>
+</SPAN>
+<SPAN ID="Frame169" DIR="LTR" STYLE="float: left; width: 0.65cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame168" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61483;</P>
+</SPAN>
+<SPAN ID="Frame167" DIR="LTR" STYLE="float: left; width: 0.65cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame166" DIR="LTR" STYLE="float: left; width: 0.57cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61485;</P>
+</SPAN>
+<SPAN ID="Frame165" DIR="LTR" STYLE="float: left; width: 0.73cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61483;</P>
+</SPAN>
+<SPAN ID="Frame164" DIR="LTR" STYLE="float: left; width: 0.73cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame163" DIR="LTR" STYLE="float: left; width: 1.21cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>      
+       </P>
+</SPAN>
+<SPAN ID="Frame162" DIR="LTR" STYLE="float: left; width: 0.45cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4>*</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame161" DIR="LTR" STYLE="float: left; width: 0.43cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame160" DIR="LTR" STYLE="float: left; width: 0.54cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame159" DIR="LTR" STYLE="float: left; width: 0.43cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame158" DIR="LTR" STYLE="float: left; width: 0.54cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame157" DIR="LTR" STYLE="float: left; width: 0.42cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame156" DIR="LTR" STYLE="float: left; width: 0.66cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61556;</P>
+</SPAN>
+<SPAN ID="Frame155" DIR="LTR" STYLE="float: left; width: 0.66cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61556;</P>
+</SPAN>
+<SPAN ID="Frame177" DIR="LTR" STYLE="float: left; width: 0.77cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61546;</P>
+</SPAN><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame178" DIR="LTR" STYLE="float: left; width: 0.65cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>dt</I></FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">La
+fonction d&rsquo;autocorr&eacute;lation mesure la similitude de x(t)
+avec une version d&eacute;call&eacute;e de x(t). Elle atteint un
+maximum pour le temps t<SUB>o</SUB></FONT><FONT SIZE=2><FONT COLOR="#545472">
+</FONT></FONT><FONT FACE="Arial, sans-serif">auquel x(t-to) ressemble
+le plus &agrave; x(t).  C&rsquo;est le cas particuli&egrave;rement
+pour les signaux p&eacute;riodiques qui reprennent la m&ecirc;me
+valeur &agrave; chaque p&eacute;riode T. La fonction
+d&rsquo;autocorr&eacute;lation permet ainsi d&rsquo;estimer la
+p&eacute;riode d&rsquo;un signal p&eacute;riodique en rep&eacute;rant
+le temps pour lequel elle atteint son maximum.</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">La
+fonction d&rsquo;autocorr&eacute;lation permet &eacute;galement de
+calculer l&rsquo;&eacute;nergie du signal  puisque </FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><SPAN ID="Frame196" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame195" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame194" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame193" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<IMG SRC="Poly2005_html_m11477c31.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m11477c31.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame192" DIR="LTR" STYLE="float: left; width: 0.22cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame191" DIR="LTR" STYLE="float: left; width: 1.1cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>dt</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame179" DIR="LTR" STYLE="float: left; width: 0.51cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame190" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame189" DIR="LTR" STYLE="float: left; width: 0.25cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4>2</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame188" DIR="LTR" STYLE="float: left; width: 0.32cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5>0</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame187" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame186" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61682;</P>
+</SPAN>
+<SPAN ID="Frame185" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame184" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61483;</P>
+</SPAN>
+<SPAN ID="Frame180" DIR="LTR" STYLE="float: left; width: 0.73cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61546;</P>
+</SPAN>
+<SPAN ID="Frame183" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61485;</P>
+</SPAN>
+<SPAN ID="Frame182" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame181" DIR="LTR" STYLE="float: left; width: 2.76cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>=
+        &Eacute;nergie</I></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_m69497d7c.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<H3 CLASS="western">1.5.2. La fonction de densit&eacute; spectrale
+d&rsquo;&eacute;nergie</H3>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">La
+Fonction de densit&eacute; spectrale d&rsquo;<U>&eacute;nergie</U>
+est la TF de la fonction d&rsquo;autocorr&eacute;lation. Elle d&eacute;crit
+la r&eacute;partition de l&rsquo;&eacute;nergie en fr&eacute;quence.</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame213" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame212" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<IMG SRC="Poly2005_html_m35327156.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m35327156.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame211" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame210" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame209" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT> 
+       </P>
+</SPAN>
+<SPAN ID="Frame208" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT> 
+       </P>
+</SPAN>
+<SPAN ID="Frame207" DIR="LTR" STYLE="float: left; width: 0.25cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4>2</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame198" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>f</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame197" DIR="LTR" STYLE="float: left; width: 0.54cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>f</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame206" DIR="LTR" STYLE="float: left; width: 0.39cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>X</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame205" DIR="LTR" STYLE="float: left; width: 1.11cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>TF
+       (</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame204" DIR="LTR" STYLE="float: left; width: 0.22cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame203" DIR="LTR" STYLE="float: left; width: 0.33cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61542;</P>
+</SPAN>
+<SPAN ID="Frame202" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame201" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame200" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame199" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame217" DIR="LTR" STYLE="float: left; width: 0.22cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame216" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61556;</P>
+</SPAN>
+<SPAN ID="Frame215" DIR="LTR" STYLE="float: left; width: 0.38cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61546;</P>
+</SPAN>
+<SPAN ID="Frame214" DIR="LTR" STYLE="float: left; width: 0.21cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>)</I></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_m69497d7c.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H2 CLASS="western">1.6. Les signaux num&eacute;riques</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">Les signaux num&eacute;riques d&eacute;rivent
+d&nbsp;&rsquo;un signal analogique par &eacute;chantillonnage.
+<FONT SIZE=2><FONT FACE="CMR10">L</FONT></FONT>'&eacute;chantillonnage
+est une op&eacute;ration qui consiste &agrave; pr&eacute;lever sur un
+signal &agrave; temps continu une suite de valeurs, prises en une
+suite d'instants tn, n &eacute;tant entier. Dans la suite nous
+n'envisagerons que l'&eacute;chantillonnage dit r&eacute;gulier o&ugrave;
+tn = nT. L'int&eacute;r&ecirc;t porte aux probl&egrave;mes de
+l'&eacute;chantillonnage tient dans le d&eacute;veloppement des
+techniques num&eacute;riques de traitement du signal. En Physique il
+est fr&eacute;quent que les r&eacute;sultats exp&eacute;rimentaux
+soient sous forme d&rsquo;une suite de valeurs num&eacute;riques. Les
+possibilit&eacute;s d&rsquo;application du traitement num&eacute;rique
+des signaux sont d&rsquo;autant plus nombreuses que la vitesse de
+calcul des microprocesseurs est &eacute;lev&eacute;e. Pour
+caract&eacute;riser et traiter un tel signal num&eacute;rique on peut
+utiliser les m&ecirc;me concepts math&eacute;matiques que pour les
+signaux continus, &agrave; condition de les adapter aux contexte
+num&eacute;rique.</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<H3 CLASS="western">1.6.1. D&eacute;finition de l&rsquo;&eacute;chantillonnage</H3>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Le
+signal continu (analogique) peut &ecirc;tre &eacute;chantillonn&eacute;
+(ou discr&eacute;tis&eacute;) &agrave; une fr&eacute;quence
+d&rsquo;&eacute;chantillonnage Fe  (= 1/Te). </FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Cette
+&eacute;chantillonage revient &agrave; multiplier le signal
+analogique par un peigne de Dirac de p&eacute;riode Te.</FONT></P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><FONT COLOR="#000000"><IMG SRC="Poly2005_html_2ab0b7f8.png" NAME="graphics12" ALIGN=LEFT HSPACE=12 WIDTH=392 HEIGHT=141 BORDER=1></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Le
+signal &eacute;chantillonn&eacute; peut alors s&rsquo;&eacute;crire&nbsp;:</FONT></P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame241" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame240" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame239" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61480;</P>
+</SPAN>
+<SPAN ID="Frame238" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61481;</P>
+</SPAN>
+<SPAN ID="Frame237" DIR="LTR" STYLE="float: left; width: 0.71cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61669;</P>
+</SPAN>
+<SPAN ID="Frame236" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN>
+<SPAN ID="Frame235" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61483;</P>
+</SPAN>
+<SPAN ID="Frame234" DIR="LTR" STYLE="float: left; width: 0.62cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61485;&#61605;</P>
+</SPAN>
+<SPAN ID="Frame233" DIR="LTR" STYLE="float: left; width: 0.27cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame232" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61485;</P>
+</SPAN>
+<SPAN ID="Frame231" DIR="LTR" STYLE="float: left; width: 0.16cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61655;</P>
+</SPAN>
+<SPAN ID="Frame230" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61501;</P>
+</SPAN>
+<SPAN ID="Frame229" DIR="LTR" STYLE="float: left; width: 0.25cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>n</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame228" DIR="LTR" STYLE="float: left; width: 0.25cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>a</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame227" DIR="LTR" STYLE="float: left; width: 0.22cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=4><I>e</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame226" DIR="LTR" STYLE="float: left; width: 0.89cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><I><FONT COLOR="#000000"><FONT SIZE=5>nT</FONT><FONT SIZE=4>e</FONT></FONT></I></FONT></P>
+</SPAN>
+<SPAN ID="Frame225" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame224" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame223" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame222" DIR="LTR" STYLE="float: left; width: 0.18cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>t</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame221" DIR="LTR" STYLE="float: left; width: 0.28cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>x</I></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame220" DIR="LTR" STYLE="float: left; width: 0.21cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5>)</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame219" DIR="LTR" STYLE="float: left; width: 0.21cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5>(</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame218" DIR="LTR" STYLE="float: left; width: 0.31cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61540;</P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame242" DIR="LTR" STYLE="float: left; width: 0.35cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61605;</P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">                           </FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">x<SUB>e</SUB>(t)
+est donc nul partout sauf en t= nTe, n variant de -</FONT><FONT COLOR="#000000"><FONT SIZE=4><FONT FACE="Symbol, serif">&#61605;&#61472;</FONT></FONT><FONT FACE="Arial, sans-serif">&agrave;</FONT><FONT SIZE=4><FONT FACE="Symbol, serif">&#61472;&#61483;&#61605;&#61472;&#61486;</FONT></FONT></FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">C&rsquo;est
+pourquoi par la suite, on ne simplifie cette notation en r&eacute;duisant
+x<SUB>e</SUB>(t) non plus &agrave; un signal continu poss&eacute;dant
+une infinit&eacute; de valeurs mais &agrave; une suite d&rsquo;&eacute;chantillons
+discrets x(n), o&ugrave; n correspond au num&eacute;ro de
+l&rsquo;&eacute;chantillon.</FONT></FONT></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><I><SPAN LANG="de-DE"><FONT FACE="Arial, sans-serif"><FONT COLOR="#000000">x(n)
+= x<SUB>e </SUB>( n T<SUB>e </SUB>)</FONT></FONT></SPAN></I></FONT></P>
+<P LANG="de-DE" CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P LANG="de-DE" CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">1.6.2. D&eacute;finition et propri&eacute;t&eacute;s
+de la transform&eacute;e de Fourier d&rsquo;un signal &eacute;chantillonn&eacute;</H3>
+<P ALIGN=JUSTIFY>Dans le cas des signaux &eacute;chantillonn&eacute;s,
+il n&rsquo;est plus n&eacute;cessaire d&rsquo;utiliser des int&eacute;grales
+pour sommer les valeurs de x(t) sur tout l&rsquo;axe des temps,
+puisqu&rsquo;un signal &eacute;chantillonn&eacute; peut &ecirc;tre
+assimil&eacute; &agrave; une suite d&rsquo;&eacute;l&eacute;ments
+discrets.</P>
+<P ALIGN=JUSTIFY>O<IMG SRC="Poly2005_html_m157a3a50.gif" NAME="Object1" ALIGN=LEFT HSPACE=12 WIDTH=177 HEIGHT=47><BR CLEAR=LEFT><IMG SRC="Poly2005_html_m45d233f0.gif" NAME="Object2" ALIGN=LEFT HSPACE=12 WIDTH=167 HEIGHT=74><BR CLEAR=LEFT><IMG SRC="Poly2005_html_42aa381a.gif" ALIGN=LEFT HSPACE=12>n
+peut donc d&eacute;finir la transform&eacute;e de Fourier d&rsquo;un
+signal num&eacute;rique par&nbsp;:</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=LEFT>De la m&ecirc;me fa&ccedil;on qu&rsquo;en continu, la
+TF pr&eacute;sente certaines propri&eacute;t&eacute;s int&eacute;ressantes&nbsp;:</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><FONT COLOR="#000000">retard:
+         </FONT>x(n-n<SUB>o</SUB>)  </FONT><SPAN LANG="de-DE"><FONT FACE="Wingdings">&#61683;</FONT></SPAN><FONT FACE="Arial, sans-serif">
+         X(f)   e<SUP>-j 2<SPAN LANG="de-DE">&pi;</SPAN>fno</SUP></FONT></FONT></P>
+       <LI><P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">produit
+       de convolution:   x(n) * y(n)  </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+         X(f) . Y(f) </FONT></FONT>
+       </P>
+</UL>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 5cm">        <FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">x(n)
+. y(n)  </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+  X(f) * Y(f) </FONT></FONT>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT STYLE="margin-left: 0.64cm"><IMG SRC="Poly2005_html_37506943.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 0.64cm"><FONT FACE="Times New Roman, serif"><FONT COLOR="#000000"><FONT FACE="Arial, sans-serif">    
+avec     x(n)  * y(n)  =  </FONT><FONT FACE="Symbol, serif">&#61669;&#61472;&#61472;</FONT><FONT FACE="Arial, sans-serif">x(k)
+y(n-k)    =     </FONT><FONT FACE="Symbol, serif">&#61669;&#61472;&#61472;</FONT><FONT FACE="Arial, sans-serif">x(n-k)
+y(k)     .</FONT></FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 0.64cm"><BR>
+</P>
+<H3 CLASS="western">1.6.2. Cons&eacute;quences de l&rsquo;&eacute;chantillonnage
+du point de vue fr&eacute;quentiel 
+</H3>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">En
+&eacute;crivant  x<SUB>e</SUB>(t) = x<SUB>a</SUB>(t) . </FONT><FONT COLOR="#000000"><FONT FACE="Symbol, serif">&#61669;&#61472;<I>&#61540;&#61472;</I></FONT><FONT FACE="Arial, sans-serif">(t-
+nT<SUB>e</SUB>), on peut alors exprimer la transform&eacute;e de
+Fourier du signal &eacute;chantillonn&eacute; &agrave; partir de la
+TF du signal analogique.</FONT></FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_m3eca2753.gif" NAME="Object3" ALIGN=LEFT HSPACE=12 WIDTH=182 HEIGHT=47><BR CLEAR=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">On
+voit que l&rsquo;&eacute;chantillonnage d&rsquo;un signal analogique
+(&agrave; la fr&eacute;quence d&rsquo;&eacute;chantillonnage F<SUB>e</SUB>
+= 1/T) induit une p&eacute;riodisation de son spectre dans le domaine
+fr&eacute;quentiel, tous les f =n/ T, n &eacute;tant entier.</FONT></FONT></P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_m27831f9b.png" NAME="graphics13" ALIGN=BOTTOM WIDTH=457 HEIGHT=209 BORDER=0><SPAN ID="Frame246" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Xa(f)</FONT></P>
+</SPAN>
+<SPAN ID="Frame245" DIR="LTR" STYLE="float: left; width: 1.91cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=RIGHT><FONT FACE="Arial, sans-serif">Xe(f)
+       =</FONT></P>
+</SPAN>
+<SPAN ID="Frame244" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=1 STYLE="font-size: 8pt">B</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame243" DIR="LTR" STYLE="float: left; width: 0.95cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=1 STYLE="font-size: 8pt">-B</FONT></FONT></P>
+</SPAN>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Il peut
+survenir un probl&egrave;me si la fr&eacute;quence d&rsquo;&eacute;chantillonnage
+Fe = 1/T est trop petite car les &laquo;&nbsp;r&eacute;pliques&nbsp;&raquo;
+p&eacute;riodiques du spectre  peuvent se superposer partiellement .</FONT></P>
+<P LANG="" CLASS="western" ALIGN=LEFT><SPAN ID="Frame248" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#ff0000"><FONT FACE="Arial, sans-serif"><B>Xe(f)
+       </B></FONT></FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame247" DIR="LTR" STYLE="float: left; width: 1.59cm; height: 0.74cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Xa(f)</FONT></P>
+</SPAN><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_7aaf54a5.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_m6691bec4.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m74220b0e.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m28359ac8.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_251d386e.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_251d386e.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m15d74430.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m15d74430.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame252" DIR="LTR" STYLE="float: left; width: 0.89cm; height: 0.91cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> <FONT FACE="Times New Roman, serif"><I>f</I></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_35556691.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame251" DIR="LTR" STYLE="float: left; width: 1.51cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><I>1/T
+       </I></FONT></FONT>
+       </P>
+</SPAN>
+<IMG SRC="Poly2005_html_m28359ac8.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m28359ac8.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame250" DIR="LTR" STYLE="float: left; width: 1.51cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><I>-B
+       </I></FONT></FONT>
+       </P>
+</SPAN>
+<SPAN ID="Frame249" DIR="LTR" STYLE="float: left; width: 1.51cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><I>B
+       </I></FONT></FONT>
+       </P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Cela
+arrive si   borne max (Xa(f))  &gt;    borne inf  (1ere r&eacute;plique
+de Xa(f) en f =1/T)</FONT></P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Autrement
+dit si                 B        &gt;   1/T  - B</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Pour que
+le spectre Xa(f) ne soit pas &laquo;&nbsp;d&eacute;form&eacute;&nbsp;&raquo;
+lors de sa p&eacute;riodisation, il faut donc que&nbsp;:</FONT></P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_208ff5d.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 2.5cm"><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">           
+       <B>Fe &lt; 2 B</B></FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT> 
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Cette
+condition est appel&eacute;e <U>Th&eacute;or&egrave;me de Shannon</U>.</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H1 CLASS="western" ALIGN=LEFT STYLE="page-break-before: always"><A NAME="2.Chapitre 2&nbsp;: Modes de repr&eacute;sentation des signaux|outline"></A>
+Chapitre 2&nbsp;: Modes de repr&eacute;sentation des signaux</H1>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H2 CLASS="western">2.1. Fen&ecirc;trage des signaux 
+</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Nous
+avons vu dans le chapitre pr&eacute;c&eacute;dent que la Transform&eacute;e
+de Fourier permet de repr&eacute;senter le spectre d&rsquo;un signal
+dans le domaine fr&eacute;quentiel. Pour une sinuso&iuml;de infinie,
+toute l&rsquo;&eacute;nergie du spectre est concentr&eacute;e &agrave;
+une fr&eacute;quence donn&eacute;e, c&rsquo;est &agrave; dire la
+fr&eacute;quence de la sinuso&iuml;de. </FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_a8d0a10.png" NAME="graphics14" ALIGN=BOTTOM WIDTH=327 HEIGHT=214 BORDER=0></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P><FONT SIZE=2><FONT SIZE=3>Si on veut observer ce sinus sur un
+temps fini, on va &ecirc;tre oblig&eacute; de le tronquer. L&rsquo;&eacute;nergie
+va alors d&rsquo;une part se r&eacute;partir autour de la fr&eacute;quence
+de la sinuso&iuml;de (c&rsquo;est ce qu&rsquo;on appelle l&rsquo;&eacute;talement
+spectral), d&rsquo;autre part on observe la pr&eacute;sence d&rsquo;&eacute;nergie
+dans toutes les fr&eacute;quences (Ce ph&eacute;nom&egrave;ne est
+appel&eacute; fuite spectrale). Etalement et fuite sont li&eacute;s &agrave;
+la troncature.</FONT></FONT></P>
+<P><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_67aaf31.png" NAME="graphics15" ALIGN=BOTTOM WIDTH=457 HEIGHT=300 BORDER=0><IMG SRC="Poly2005_html_726c6201.gif" ALIGN=LEFT><IMG SRC="Poly2005_html_m2dc3f000.gif" ALIGN=LEFT><IMG SRC="Poly2005_html_726c6201.gif" ALIGN=LEFT><IMG SRC="Poly2005_html_413a4208.gif" ALIGN=LEFT><IMG SRC="Poly2005_html_7ee3462f.gif" ALIGN=LEFT></P>
+<P ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><FONT COLOR="#000000">Si
+on r&eacute;alise la troncature de fa&ccedil;on non rectangulaire,
+mais en &laquo;&nbsp;fen&ecirc;trant&nbsp;&raquo; le signal par une
+fen&ecirc;tre de Hanning par exemple, les transitions dans le signal
+sont alors plus douces. Cela limite la fuite spectrale mais en
+revanche augmente l&rsquo;&eacute;talement en fr&eacute;quence. Ce
+point est important pour comprendre le r&ocirc;le des fen&ecirc;tres
+d&rsquo;analyse. Si la fen&ecirc;tre a des discontinuit&eacute;s
+fortes, les fuites spectrales vont &ecirc;tre importantes, mais
+l'&eacute;talement moindre. Si on prend un fen&ecirc;tre de
+discontinuit&eacute; plus douce, on va au contraire obtenir un
+&eacute;talement plus grand, mais moins de fuites spectrales.</FONT></FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_60b899b9.png" NAME="graphics16" ALIGN=BOTTOM WIDTH=457 HEIGHT=300 BORDER=0><IMG SRC="Poly2005_html_71a40959.gif" ALIGN=LEFT><IMG SRC="Poly2005_html_7ee3462f.gif" ALIGN=LEFT><IMG SRC="Poly2005_html_m3351203.gif" ALIGN=LEFT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H2 CLASS="western">2.2. Le spectrogramme</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">L&rsquo;int&eacute;r&ecirc;t
+du spectrogramme est de pouvoir repr&eacute;senter le spectre en
+&eacute;voluant dans le temps. Le nom scientifique de la fonction
+math&eacute;matique associ&eacute;e &agrave; cet outil, plus
+commun&eacute;ment appel&eacute; &laquo;&nbsp;spectrogramme&nbsp;&raquo;,
+est la Transform&eacute;e de Fourier &agrave; Court Terme (TFCT). Ce
+nom provient de l'analyse effectu&eacute;e sur des fen&ecirc;tres de
+support temporel fini. Une autre d&eacute;nomination de cette
+repr&eacute;sentation est &laquo;&nbsp;sonagramme&nbsp;&raquo;. Il
+s&rsquo;agit d&rsquo;une marque d&eacute;pos&eacute;e Kay
+Electronics.</FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=JUSTIFY>Le principe du spectrogramme est de &laquo;&nbsp;d&eacute;couper&nbsp;&raquo;
+le son en tranches ou trames qui se recouvrent. Pour chacune de ces
+tranches on calcule une transform&eacute;e de Fourier. Ce spectre est
+alors repr&eacute;sent&eacute; &agrave; un temps correspondant &agrave;
+celui du centre de la fen&ecirc;tre, sous forme d&rsquo;un code de
+couleur. Sur l'exemple suivant, le jaune correspond aux amplitudes
+les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a
+ainsi une id&eacute;e de l&rsquo;aspect du spectre au temps t. A
+chaque calcul du spectre, le signal est fen&ecirc;tr&eacute; de fa&ccedil;on
+&agrave; pouvoir r&eacute;gler &agrave; la fois la fuite et
+l&rsquo;&eacute;talement spectral. 
+</P>
+<P ALIGN=CENTER><IMG SRC="Poly2005_html_m2d92858a.png" NAME="graphics17" ALIGN=BOTTOM WIDTH=384 HEIGHT=251 BORDER=0></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><FONT COLOR="#000000">Voici
+un exemple de voix chant&eacute;e. Il s&rsquo;agit d&rsquo;un
+</FONT>glissando C5-E5 r&eacute;alis&eacute; par une soprano.
+L'analyse a &eacute;t&eacute; effectu&eacute;e avec une fen&ecirc;tre
+de Hanning de longueur 23 ms. Sur le spectrogramme, on observe bien
+le glissando et le vibrato de la chanteuse.</FONT></FONT></P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_m5853ee65.png" NAME="graphics18" ALIGN=BOTTOM WIDTH=381 HEIGHT=243 BORDER=0></P>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">P<FONT COLOR="#000000">our
+mesurer le vibrato par exemple, on serait tent&eacute; de r&eacute;duire
+la longueur de la fen&ecirc;tre dans le temps pour gagner en
+pr&eacute;cision et suivre au mieux les variations du spectre. En
+r&eacute;alit&eacute;, si on r&eacute;duit la longueur des fen&ecirc;tres,
+l&rsquo;&eacute;talement spectral augmente, par cons&eacute;quent la
+largeur des raies sur le spectrogramme aussi, ce qui perturbe
+finalement la mesure, car on ne distingue plus distinctement les
+diff&eacute;rentes trajectoires dans le spectrogramme.</FONT></FONT></FONT></P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><FONT COLOR="#000000"><IMG SRC="Poly2005_html_6e4c6078.png" NAME="graphics20" ALIGN=LEFT HSPACE=12 WIDTH=229 HEIGHT=146 BORDER=1></FONT><FONT COLOR="#000000"><IMG SRC="Poly2005_html_4ed905b1.png" NAME="graphics19" ALIGN=LEFT HSPACE=12 WIDTH=229 HEIGHT=143 BORDER=1></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><SPAN ID="Frame254" DIR="LTR" STYLE="float: left; width: 3.81cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=CENTER><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif">Fen&ecirc;tre
+       de 100 ms</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame253" DIR="LTR" STYLE="float: left; width: 4.76cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif">Fen&ecirc;tre
+       de  20 ms</FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><FONT COLOR="#000000">Si
+on revient &agrave; la m&ecirc;me longueur de fen&ecirc;tre que dans
+le premier exemple, tout en utilisant une fen&ecirc;tre rectangulaire
+au lieu de la fen&ecirc;tre &laquo;&nbsp;douce&nbsp;&raquo; de
+Hanning, l&rsquo;&eacute;talement spectral est plus faible et les
+lignes sur le spectrogramme plus fines, mais on a par contre des
+fuites spectrales beaucoup plus importantes, caract&eacute;ris&eacute;es
+par un manque de contraste dans la repr&eacute;sentation du
+spectrogramme. </FONT></FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_m6121ad8d.png" NAME="graphics21" ALIGN=BOTTOM WIDTH=287 HEIGHT=190 BORDER=0></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Les
+points importants quant &agrave; l&rsquo;utilisation du
+spectrogramme&nbsp;sont donc: </FONT>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">la
+       longueur de fen&ecirc;tre pour ajuster la pr&eacute;cision
+       temporelle, au prix d&rsquo;un &eacute;talement spectral qui peut
+       devenir r&eacute;dhibitoire</FONT></P>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><FONT COLOR="#000000">le
+       choix de la fen&ecirc;tre qui va conditionner le contraste du
+       spectrogramme, pour une longueur de fen&ecirc;tre donn&eacute;e.</FONT></FONT></FONT></P>
+</UL>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western">2.3. Pertinence et choix d&rsquo;un mode de
+repr&eacute;sentation&nbsp;</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">La
+repr&eacute;sentation du signal en temporel est particuli&egrave;rement
+adapt&eacute; &agrave; l&rsquo;analyse de ses variations temporelles 
+(par exemple l&rsquo;attaque ou l&rsquo;extinction d&rsquo;un son).</FONT></P>
+<P LANG="" CLASS="western" ALIGN=LEFT><FONT COLOR="#000000"><IMG SRC="Poly2005_html_m20aeb917.png" NAME="graphics22" ALIGN=LEFT HSPACE=12 WIDTH=386 HEIGHT=240 BORDER=1></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">Au
+contraire, cette repr&eacute;sentation apporte peu d&rsquo;information
+lorsqu&rsquo;on s&rsquo;int&eacute;resse plut&ocirc;t au contenu
+spectral du signal. Sur la figure suivante, on voit que la
+repr&eacute;sentation spectrale de ces deux signaux de parole permet
+assez bien de diff&eacute;rentier la voix &laquo;&nbsp;tendue&nbsp;&raquo;
+ de la voix &laquo;&nbsp;rel&acirc;ch&eacute;e&nbsp;&raquo;.</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><FONT COLOR="#000000"><IMG SRC="Poly2005_html_mf0072c2.png" NAME="graphics24" ALIGN=LEFT HSPACE=12 WIDTH=360 HEIGHT=189 BORDER=1></FONT><FONT COLOR="#000000"><IMG SRC="Poly2005_html_4cd30576.png" NAME="graphics23" ALIGN=LEFT HSPACE=12 WIDTH=348 HEIGHT=190 BORDER=1></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<FONT FACE="Arial, sans-serif">Voix rel&acirc;ch&eacute;e                                              Voix
+tendue</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">U<IMG SRC="Poly2005_html_43f5087c.png" NAME="graphics26" ALIGN=LEFT WIDTH=592 HEIGHT=101 BORDER=0><IMG SRC="Poly2005_html_m2fe19920.png" NAME="graphics25" ALIGN=LEFT WIDTH=600 HEIGHT=163 BORDER=0><IMG SRC="Poly2005_html_m26e75a3.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_12342a36.gif" ALIGN=LEFT HSPACE=12>n
+autre int&eacute;r&ecirc;t de la repr&eacute;sentation du signal &agrave;
+l&rsquo;aide d&rsquo;un spectrogramme est de pouvoir s&eacute;parer
+des sources sonores, ou du moins de pouvoir rep&eacute;rer des
+&eacute;v&eacute;nements, l&agrave; o&ugrave; le signal temporel
+englobe tout de fa&ccedil;on indiff&eacute;renci&eacute;e.</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Pour
+des sons tr&egrave;s harmoniques (par exemple voix chant&eacute;e),
+le spectrogramme permet de rep&eacute;rer tr&egrave;s facilement la
+fondamentale ainsi que la s&eacute;rie harmonique.</FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Sur
+la figure ci-dessous est repr&eacute;sent&eacute; le spectrogramme
+d&rsquo;un chanteur effectuant un glissando du bas au haut de sa
+tessiture puis l&rsquo;inverse.</FONT></P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><IMG SRC="Poly2005_html_m6bdd0d87.png" NAME="graphics27" ALIGN=LEFT WIDTH=424 HEIGHT=173 BORDER=0><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Dans
+le cas de la voix parl&eacute;e, on peut rep&eacute;rer les parties
+vois&eacute;e (harmoniques) ou non, et m&ecirc;me &laquo;&nbsp;lire&nbsp;&raquo;
+quels sont les phon&egrave;mes prononc&eacute;s avec un peu
+d&rsquo;exp&eacute;rience. Nous reviendrons l&agrave; dessus dans le
+chapitre 5.</FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><IMG SRC="Poly2005_html_6291535b.png" NAME="graphics28" ALIGN=LEFT WIDTH=400 HEIGHT=157 BORDER=0><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><IMG SRC="Poly2005_html_m3a01ea24.png" NAME="graphics29" ALIGN=LEFT WIDTH=408 HEIGHT=217 BORDER=0><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=CENTER>   &laquo;&nbsp;C&rsquo;est une maison qui s&rsquo;trouve
+au Luxembourg&nbsp;&raquo;</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H2 CLASS="western">2.4. Le spectre moyenn&eacute; (Long Term Average
+Spectrum   LTAS)</H2>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western">Il est &eacute;galement tr&egrave;s utile dans
+certains cas de repr&eacute;senter une partie du signal sous forme de
+spectre moyenn&eacute;. Le spectre moyenn&eacute;, comme son nom
+l&rsquo;indique, consiste &agrave; faire la moyenne du spectrogramme
+sur l&rsquo;ensemble des instants d&rsquo;un intervalle de temps plus
+ou moins long . Ce peut &ecirc;tre sur le signal tout entier comme
+sur un morceau particulier.</P>
+<P CLASS="western"><IMG SRC="Poly2005_html_m3a01ea24.png" NAME="graphics30" ALIGN=LEFT WIDTH=204 HEIGHT=177 BORDER=0><BR CLEAR=LEFT><IMG SRC="Poly2005_html_9c20788.gif" ALIGN=LEFT><BR CLEAR=LEFT><BR>
+</P>
+<P CLASS="western">Pour examiner une voyelle en particulier dans un
+signal de parole par exemple, il est tout &agrave; fait appropri&eacute;
+de moyenner le spectrogramme sur l&rsquo;intervalle de temps de cette
+voyelle, de fa&ccedil;on &agrave; pouvoir estimer les formants ou
+bien le contenu spectral de cette voyelle. Ainsi, sur la figure
+suivante, on peut examiner le spectre moyenn&eacute; de la production
+vocale d&rsquo;un chanteur sur un /a/. Cette repr&eacute;sentation
+permet de caract&eacute;riser ses premiers formants vocaliques, ainsi
+que son <I>formant du chanteur</I> (renforcement &eacute;nerg&eacute;tique
+dans la zone 2000-4000 Hz), caract&eacute;ristique des chanteurs
+lyriques et des acteurs de th&eacute;&acirc;tre.</P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><FONT COLOR="#000000"><IMG SRC="Poly2005_html_m2b5c76b.jpg" NAME="graphics31" ALIGN=LEFT HSPACE=12 WIDTH=280 HEIGHT=229 BORDER=1></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<H1 CLASS="western" ALIGN=LEFT STYLE="page-break-before: always"><A NAME="3.Chapitre 3&nbsp;: Syst&egrave;mes lin&eacute;aires et filtrage|outline"></A>
+Chapitre 3&nbsp;: Syst&egrave;mes lin&eacute;aires et filtrage</H1>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_m1f3f0d38.png" NAME="graphics32" ALIGN=BOTTOM WIDTH=604 HEIGHT=339 BORDER=0></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H2 CLASS="western">3.1. R&eacute;ponse impulsionnelle d&rsquo;un
+syst&egrave;me</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">Il est possible de d&eacute;crire un syst&egrave;me
+S par l'interm&eacute;diaire de sa r&eacute;ponse &agrave; un signal
+extr&ecirc;mement bref. Math&eacute;matiquement, il s&rsquo;agit de
+mod&eacute;liser sa r&eacute;ponse &agrave; une impulsion de Dirac.</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">La
+r&eacute;ponse impulsionnelle d&rsquo;un syst&egrave;me est donc
+d&eacute;finie comme la sortie d&rsquo;un syst&egrave;me lorsque
+l&rsquo;entr&eacute;e est une impulsion de Dirac.</FONT></P>
+<P LANG="" CLASS="western" ALIGN=JUSTIFY><IMG SRC="Poly2005_html_7e586537.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><I>g(t)
+= </I><B>S</B>{<I>d(t)</I>}</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western">Lorsque nous claquons des mains dans une pi&egrave;ce
+pour &eacute;valuer son acoustique, son temps de r&eacute;verb&eacute;ration,
+etc, nous ne faisons pas moins en fait qu&rsquo;&eacute;valuer la
+r&eacute;ponse impulsionnelle de ce syst&egrave;me pi&egrave;ce. 
+</P>
+<P CLASS="western">Cela semble physiquement logique d&rsquo;entendre
+du son dans la pi&egrave;ce au moment o&ugrave; nous claquons dans
+nos mains, voire apr&egrave;s, mais certainement pas avant .
+Pourtant, en math&eacute;matique, on con&ccedil;oit que la r&eacute;ponse
+impulsionnelle puisse &ecirc;tre non nulle avant l&rsquo;instant de
+l&rsquo;impulsion d&rsquo;entr&eacute;e.</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">On
+d&eacute;finit ainsi les syst&egrave;me <U>causaux</U> comme les
+syst&egrave;mes dont la r&eacute;ponse impulsionnelle est nulle avant
+l&rsquo;instant d&rsquo;impulsion, autrement dit      <I>g(t&lt;0)</I>
+= <I>0  .</I></FONT></FONT></P>
+<P CLASS="western">Pour un syst&egrave;me causal, le signal de sortie
+&agrave; l'instant t d&eacute;pend du signal d'entr&eacute;e aux
+instants t'&lt; t. La dur&eacute;e de la r&eacute;ponse
+impulsionnelle g(t) correspond au <U>temps de r&eacute;ponse</U> du
+syst&egrave;me.</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western">Nous avons vu que la r&eacute;ponse impulsionnelle
+correspond &agrave; la sortie y(t) du syst&egrave;me lorsque l&rsquo;entr&eacute;e
+x(t) = <I><FONT FACE="Symbol, serif">&#61540;</FONT>(t)</I> .</P>
+<P CLASS="western">Connaissant la r&eacute;ponse impulsionnelle g(t)
+d&rsquo;un syst&egrave;me, on peut ensuite pr&eacute;dire sa r&eacute;ponse
+y(t) &agrave; n&rsquo;importe quel signal d&rsquo;entr&eacute;e x(t)
+gr&acirc;ce &agrave; la formule&nbsp;:</P>
+<P LANG="" CLASS="western"><IMG SRC="Poly2005_html_m680d51f4.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western"> <FONT SIZE=4 STYLE="font-size: 16pt">                                  
+y(t) = x(t) * g(t)</FONT></P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western"><BR>
+</P>
+<P CLASS="western"><BR>
+</P>
+<P LANG="" CLASS="western"><SPAN ID="Frame255" DIR="LTR" STYLE="float: left; width: 2.78cm; border: 1px solid #545472; padding: 0.14cm 0.26cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">    <FONT FACE="Times New Roman, serif"><FONT SIZE=6>S</FONT></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_2fa12638.gif" ALIGN=LEFT>
+<IMG SRC="Poly2005_html_188332fa.gif" ALIGN=LEFT>
+<SPAN ID="Frame259" DIR="LTR" STYLE="float: left; width: 1.69cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT>&#61540;<FONT COLOR="#545472"> <FONT FACE="Times New Roman, serif"><FONT SIZE=5>(t)</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame260" DIR="LTR" STYLE="float: left; width: 1.48cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>g(t)</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame257" DIR="LTR" STYLE="float: left; width: 2.78cm; border: 1px solid #545472; padding: 0.14cm 0.26cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472">    <FONT FACE="Times New Roman, serif"><FONT SIZE=6>S</FONT></FONT></FONT></P>
+</SPAN>
+<IMG SRC="Poly2005_html_2fa12638.gif" ALIGN=LEFT>
+<IMG SRC="Poly2005_html_188332fa.gif" ALIGN=LEFT>
+<SPAN ID="Frame256" DIR="LTR" STYLE="float: left; width: 1.69cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT SIZE=5><I><FONT FACE="Arial, sans-serif">x</FONT></I><FONT COLOR="#545472">(t)</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame258" DIR="LTR" STYLE="float: left; width: 4.76cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>y(t)
+       = x(t) * g(t)</FONT></FONT></FONT></P>
+</SPAN><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><BR>
+</P>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<H2 CLASS="western">3.2. R&eacute;ponse en fr&eacute;quence d&rsquo;un
+syst&egrave;me</H2>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">La r&eacute;ponse en
+fr&eacute;quence d&rsquo;un syst&egrave;me correspond &agrave; la
+transform&eacute;e de Fourier de la r&eacute;ponse impulsionnelle du
+syst&egrave;me.</FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<H1 CLASS="western" ALIGN=JUSTIFY></H1>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">G(f)
+d&eacute;crit comment la distribution spectrale d'un signal est
+modifi&eacute;e ou &quot;filtr&eacute;e&quot; par le syst&egrave;me
+S.<B> </B>Il est important de noter que le syst&egrave;me peut
+seulement modifier des composantes spectrales mais ne peut en aucun
+cas en cr&eacute;er de nouvelles.</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">|<I>G(f)</I>|
+ est le gain du syst&egrave;me, c&rsquo;est &agrave; dire la fa&ccedil;on
+dont il modifie les amplitudes de chaque composantes spectrales.</FONT></FONT></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">A<IMG SRC="Poly2005_html_m2d0c09f0.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_5296f0a4.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m7aabbb1b.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame261" DIR="LTR" STYLE="float: left; width: 0.82cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>1</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame263" DIR="LTR" STYLE="float: left; width: 1.06cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> <FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>f
+       </I></FONT></FONT></FONT>
+       </P>
+</SPAN>
+<IMG SRC="Poly2005_html_m77b353f8.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m3633fa7d.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m200c7b20.gif" NAME="Object4" ALIGN=LEFT HSPACE=12 WIDTH=49 HEIGHT=20><BR CLEAR=LEFT>
+<IMG SRC="Poly2005_html_m2d0c09f0.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_5296f0a4.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame264" DIR="LTR" STYLE="float: left; width: 0.82cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"><FONT FACE="Times New Roman, serif"><FONT SIZE=5>0</FONT></FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame262" DIR="LTR" STYLE="float: left; width: 1.2cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT COLOR="#545472"> <FONT FACE="Times New Roman, serif"><FONT SIZE=5><I>f
+       </I></FONT></FONT></FONT>
+       </P>
+</SPAN>
+<IMG SRC="Poly2005_html_389f6b57.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_5040c209.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_5b286f73.gif" ALIGN=LEFT HSPACE=12>
+<IMG SRC="Poly2005_html_m6078bfca.gif" NAME="Object7" ALIGN=LEFT HSPACE=12 WIDTH=20 HEIGHT=39><BR CLEAR=LEFT>
+<IMG SRC="Poly2005_html_m5ab862b9.gif" NAME="Object6" ALIGN=LEFT HSPACE=12 WIDTH=33 HEIGHT=39><BR CLEAR=LEFT>
+<IMG SRC="Poly2005_html_1542f785.gif" NAME="Object5" ALIGN=LEFT HSPACE=12 WIDTH=78 HEIGHT=20><BR CLEAR=LEFT>rg[<I>G(f)</I>]
+repr&eacute;sente le d&eacute;phasage caus&eacute; par le syst&egrave;me,
+c&rsquo;est &agrave; dire le &laquo;&nbsp;retard&nbsp;&raquo; qu&rsquo;il
+impose &agrave; certaines composantes spectrales.</FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><BR>
+</P>
+<H1 CLASS="western" ALIGN=JUSTIFY></H1>
+<H2 CLASS="western" ALIGN=JUSTIFY></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<P CLASS="western">La r&eacute;ponse en fr&eacute;quence, comme la
+r&eacute;ponse impulsionnelle permet de d&eacute;crire compl&egrave;tement
+le syst&egrave;me et de pr&eacute;dire la r&eacute;ponse du syst&egrave;me
+&agrave; n&rsquo;importe quelle entr&eacute;e. 
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_a5ed3e.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">y(t)
+= x(t) * g(t)    </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+    Y(f)  = X(f) . G(f)</FONT></FONT></P>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western"></H2>
+<P CLASS="western" ALIGN=LEFT STYLE="margin-left: 1.25cm; text-indent: 1.25cm">
+<BR>
+</P>
+<H2 CLASS="western">3.3. Fonction de transfert d&rsquo;un syst&egrave;me
+(et d&eacute;finition de la Transform&eacute;e en Z)</H2>
+<H2 CLASS="western"></H2>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">La
+fonction de transfert d&rsquo;un syst&egrave;me est d&eacute;fini
+comme la <U>transform&eacute;e en Z de la r&eacute;ponse
+impulsionnelle du syst&egrave;me</U>.</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">La
+Transform&eacute;e en Z est tr&egrave;s proche de la Transform&eacute;e
+de Fourier et se d&eacute;finit par&nbsp;:</FONT></P>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_537efe78.png" NAME="graphics33" ALIGN=BOTTOM WIDTH=312 HEIGHT=78 BORDER=0><IMG SRC="Poly2005_html_63b11438.gif" ALIGN=LEFT HSPACE=12></P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">La
+TZ est un<SUB> </SUB>outil de caract&eacute;risation des syst&egrave;mes
+de traitement num&eacute;rique des signaux. </FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">On
+retrouve la d&eacute;finition de la TF en posant z = <I>e<SUP>i2</SUP></I></FONT><I><SUP><FONT FACE="Symbol, serif">&#61552;</FONT><FONT FACE="Arial, sans-serif">
+f.</FONT></SUP></I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">La TZ
+pr&eacute;sente aussi comme propri&eacute;t&eacute;&nbsp;:</FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><SPAN LANG="de-DE"><FONT FACE="Arial, sans-serif">*
+Retard&nbsp;:                  x(k &ndash;ko)     </FONT></SPAN><FONT FACE="Wingdings">&#61683;</FONT><SPAN LANG="de-DE"><FONT FACE="Arial, sans-serif">
+    X(z) z-ko</FONT></SPAN></FONT></P>
+<P LANG="de-DE" CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">*Convolution
+ :     x(k)*y(k)    </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+    X(z).Y(z)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">                      x(k).y(k)
+   </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+    X(z)*Y(z)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif">                  </FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western">D<IMG SRC="Poly2005_html_537efe78.png" NAME="graphics34" ALIGN=LEFT HSPACE=12 WIDTH=673 HEIGHT=168 BORDER=0><BR CLEAR=LEFT>onc
+la r&eacute;ponse impulsionnelle g(n) et la fonction de transfert
+G(z) permettent de d&eacute;crire un syst&egrave;me num&eacute;rique.</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_1aa1671d.gif" ALIGN=LEFT HSPACE=12><BR>
+</P>
+<P CLASS="western" ALIGN=CENTER><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">y(n)
+= x(n) * g(n)    </FONT><FONT FACE="Wingdings">&#61683;</FONT><FONT FACE="Arial, sans-serif">
+    Y(z)  = X(z) . G(z)</FONT></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<H2 CLASS="western"></H2>
+<H2 CLASS="western">3.4. Filtrage</H2>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Le
+filtrage lin&eacute;aire est une op&eacute;ration utile pour
+s&eacute;lectionner une partie du spectre d&rsquo;un signal ou en
+att&eacute;nuer une partie, pour </FONT>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">modifier
+le contenu spectral d&rsquo;un signal  ou encore pour m</FONT></P>
+<P ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">od&eacute;liser la
+transformation d&rsquo;un signal dans un conduit, dans une pi&egrave;ce&hellip;
+</FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<H3 CLASS="western">3.4.1. Filtre id&eacute;aux</H3>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">On
+d&eacute;finit classiquement 4 types de filtres id&eacute;aux&nbsp;: </FONT>
+</P>
+<UL>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">les
+       filtres passe-bas qui laissent intact les basses fr&eacute;quences
+       d&rsquo;un signal et en att&eacute;nuent les hautes fr&eacute;quences.</FONT></P>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">les
+       filtres passe-haut qui laissent intact les hautes fr&eacute;quences
+       d&rsquo;un signal et en att&eacute;nuent les basses fr&eacute;quences.</FONT></P>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Les
+       filtres passe-bande qui s&eacute;lectionnent une partie du spectre
+       d&rsquo;un signal autour d&rsquo;une fr&eacute;quence sp&eacute;cifi&eacute;e,
+       avec une largeur plus ou moins grande.</FONT></P>
+       <LI><P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">Les
+       filtres coupe-bande, qui att&eacute;nuent fortement une partie du
+       spectre d&rsquo;un signal autour d&rsquo;une fr&eacute;quence
+       sp&eacute;cifi&eacute;e, avec une largeur plus ou moins grande.</FONT></P>
+</UL>
+<P CLASS="western" ALIGN=CENTER><IMG SRC="Poly2005_html_1694ead1.png" NAME="graphics35" ALIGN=BOTTOM WIDTH=360 HEIGHT=233 BORDER=0></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Pour
+tous ces filtres, on d&eacute;finit des <U>fr&eacute;quences de
+coupure</U>, c&rsquo;est &agrave; dire les fr&eacute;quences pour
+lesquelles le spectre du signal d&rsquo;entr&eacute;e va &ecirc;tre
+att&eacute;nu&eacute; d&rsquo;un facteur &radic;2. </FONT></FONT>
+</P>
+<P ALIGN=CENTER><IMG SRC="Poly2005_html_1694ead1.png" NAME="graphics36" ALIGN=BOTTOM WIDTH=241 HEIGHT=143 BORDER=0><IMG SRC="Poly2005_html_779fb13.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame267" DIR="LTR" STYLE="float: left; width: 1.91cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=2>A
+       max</FONT></FONT></P>
+</SPAN>
+<SPAN ID="Frame266" DIR="LTR" STYLE="float: left; width: 2.54cm; height: 0.64cm; border: none; padding: 0cm; background: #ffffff">
+       <P ALIGN=LEFT><FONT FACE="Arial, sans-serif"><FONT SIZE=2>A max/ &radic;2
+        </FONT></FONT>
+       </P>
+</SPAN>
+<IMG SRC="Poly2005_html_7f7761cb.gif" ALIGN=LEFT HSPACE=12>
+<SPAN ID="Frame265" DIR="LTR" STYLE="float: left; width: 1.27cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">fc</FONT></P>
+</SPAN>
+</P>
+<P ALIGN=CENTER><BR>
+</P>
+<P ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Le
+passe bas &agrave; donc une fr&eacute;quence de coupure dans les
+m&eacute;diums / hautes fr&eacute;quences, le passe haut une
+fr&eacute;quence de coupure dans les m&eacute;diums / basses
+fr&eacute;quences. Les passe bande et coupe bande poss&egrave;dent
+deux fr&eacute;quences de coupure autour de la fr&eacute;quence
+centrale sur laquelle ils se centrent. </FONT></FONT>
+</P>
+<P ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">On
+sp&eacute;cifie &eacute;galement la <U>pente de l&rsquo;att&eacute;nuation</U>
+de ces filtres, en dB par octave. Cela apporte une information sur la
+s&eacute;lectivit&eacute; du filtre.</FONT></FONT></P>
+<P ALIGN=JUSTIFY><BR>
+</P>
+<P ALIGN=JUSTIFY><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif">Enfin,
+pour les filtres passe-bande et coupe-bande, on d&eacute;termine leur
+<U>largeur de bande</U>, c&rsquo;est &agrave; dire la diff&eacute;rence
+entre leurs deux fr&eacute;quences de coupure, ce qui renseigne sur
+le caract&egrave;re plus ou moins &laquo;&nbsp;aigu&nbsp;&raquo; du
+filtre.</FONT></FONT></P>
+<P ALIGN=CENTER><IMG SRC="Poly2005_html_1694ead1.png" NAME="graphics37" ALIGN=BOTTOM WIDTH=226 HEIGHT=160 BORDER=0><IMG SRC="Poly2005_html_6049656c.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_7f7761cb.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_7f7761cb.gif" ALIGN=LEFT HSPACE=12><IMG SRC="Poly2005_html_m64fd60d7.gif" ALIGN=LEFT HSPACE=12><SPAN ID="Frame268" DIR="LTR" STYLE="float: left; width: 2.22cm; height: 0.95cm; border: none; padding: 0cm; background: #ffffff">
+       <P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif">fc<SUB>2</SUB>
+       &ndash; fc<SUB>1</SUB></FONT></P>
+</SPAN>
+</P>
+<H3 CLASS="western"></H3>
+<H3 CLASS="western">3.4.2. Exemples de filtres naturels 
+</H3>
+<H3 CLASS="western"></H3>
+<P ALIGN=LEFT><FONT FACE="Arial, sans-serif">On peut mod&eacute;liser
+la sensation d&rsquo;intensit&eacute; auditive par un filtre.
+L&rsquo;oreille humaine est particuli&egrave;rement sensible entre 3
+et 4 kHz </FONT>
+</P>
+<P LANG="" CLASS="western" ALIGN=LEFT><FONT COLOR="#000000"><IMG SRC="Poly2005_html_5836fa0f.jpg" NAME="graphics38" ALIGN=LEFT HSPACE=12 WIDTH=480 HEIGHT=296 BORDER=1><BR CLEAR=LEFT></FONT><BR>
+</P>
+<P CLASS="western" ALIGN=JUSTIFY><FONT FACE="Arial, sans-serif">On
+peut &eacute;galement mod&eacute;liser la production phonatoire par
+un mod&egrave;le source-filtre, o&ugrave; le r&ocirc;le de la source
+est jou&eacute; par les cordes vocales qui produisent un son
+harmonique avec une distribution de l&rsquo;&eacute;nergie assez
+plate en fr&eacute;quence. Le conduit vocal, les fosses nasales ainsi
+que la place des articulateurs (langue, m&acirc;choire, l&egrave;vres)
+peuvent &ecirc;tre mod&eacute;lis&eacute;s par un filtre qui modifie
+le son glottique pour produire le son tel que nous le percevons &agrave;
+la sortie des l&egrave;vres d&rsquo;un locuteur.</FONT></P>
+<P ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><IMG SRC="Poly2005_html_5a346f7a.png" NAME="graphics39" ALIGN=BOTTOM WIDTH=505 HEIGHT=254 BORDER=0></P>
+<H1 CLASS="western" ALIGN=LEFT STYLE="page-break-before: always"><A NAME="6.Bibliographie|outline"></A>
+Bibliographie</H1>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=LEFT><FONT FACE="Arial, sans-serif"><I>David B. Conf&eacute;rence
+SFA sur le vibrato. D&eacute;cembre 2003. Le mans.</I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><FONT FACE="Arial, sans-serif"><FONT SIZE=2 STYLE="font-size: 11pt"><I>Garnier
+M, Dubois D, Poitevineau J, Henrich N, Castellengo M. Perception et
+description verbale de la qualit&eacute; vocale dans le chant
+lyrique&nbsp;: une approche cognitive. </I>Journ&eacute;es d&rsquo;&Eacute;tudes
+sur la Parole, <I>Fez, 2004</I>.<I> </I></FONT>. </FONT></FONT>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><I>Henrich
+N. Etude de la source glottique en voix parl&eacute;e et chant&eacute;e.
+Th&egrave;se de doctorat, UPMC. 2001.</I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><I>Pelorson
+X. Conf&eacute;rence SFA &laquo;&nbsp;cordes vocales et l&egrave;vres
+vibrantes&nbsp;&raquo;. Juillet 2004. Paris</I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P ALIGN=LEFT STYLE="page-break-after: avoid"><I>Charbit M. Polycopi&eacute;
+ENST. Bases de Traitement du signal. 2004</I></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><I><FONT FACE="Arial, sans-serif">Heiser
+T. Traitement du signal, support de cours. UFR Sciences Physiques.
+universit&eacute; Louis Pasteur, Strasbourg</FONT></I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><I>Feng.
+Traitement du signal, support de cours. ENSERG. INP Grenoble. 2000.</I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Times New Roman, serif"><I><FONT FACE="Arial, sans-serif">Dutoit
+T.  Introduction au Traitement Automatique de la Parole Facult&eacute;
+Polytechnique</FONT></I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><FONT FACE="Arial, sans-serif"><I>de
+Mons. 2004</I></FONT></P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
+<P CLASS="western" ALIGN=LEFT><BR>
+</P>
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+</DIV>
+</BODY>
+</HTML>
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+<html><head><title>The Joy of Convolution</title></head><body bgcolor="#f0f0f0"><img src="index_fichiers/demo6.gif" height="40" width="405"><br><br><br><b><font size="+1"><a href="http://mfile.akamai.com/7111/rm/www.jhu.edu/signals/convolve/welcome.ram">Welcome!</a></font></b><font size="+1"><img src="index_fichiers/rafile.gif" height="25" width="46"></font><p>The behavior of a linear, continuous-time, time-invariant system with input signal<i>x(t)</i> and output signal <i>y(t)</i> is described by the <i>convolution</i> integral</p><p></p><center><img src="index_fichiers/conv.gif"></center>The signal <i>h(t)</i>, assumed known, is the response of thesystem to a unit impulse input.<p>To compute the output <i>y(t)</i> at a specified <i>t</i>, first theintegrand <i>h(v) x(t - v)</i> is computed as a function of <i>v</i>.Then integration with respect to <i>v</i> is performed, resulting in<i>y(t)</i>.</p><p>These mathematical operations have simple graphical interpretations.First, plot <i>h(v)</i> and the "flipped and shifted" <i>x(t - v)</i>on the <i>v</i> axis, where <i>t</i> is fixed.  Second, multiply thetwo signals and compute the signed area of the resulting function of<i>v</i> to obtain <i>y(t)</i>.  These operations can be repeatedfor every value of <i>t</i> of interest.</p><p>To explore graphical convolution, selectsignals <i>x(t)</i> and <i>h(t)</i>
+from the provided examples below,or use the mouse to draw your own
+signal or to modify a selectedsignal. Then click at a desired value of <i>t</i> on the first<i>v</i> axis.  After a moment, <i>h(v)</i> and <i>x(t - v)</i> will appear.  Drag the <i>t</i> symbol along the <i>v</i> axis to change thevalue of <i>t</i>.  For each <i>t</i>, the corresponding integrand<i>h(v) x(t - v)</i> and the output value <i>y(t)</i> will bedisplayed in their respective windows.</p><p></p><p><b><font size="+1"><a href="http://mfile.akamai.com/7111/rm/www.jhu.edu/signals/convolve/exercises.ram">Exercises</a></font></b><font size="+1"><img src="index_fichiers/rafile.gif" height="25" width="46"></font><br><br></p><center><table bgcolor="#c0c0c0" border="2" cellpadding="0" cellspacing="1" height="600" width="480"><tbody><tr><td><div id="adblock-frame-n5" adblockframe="true" style="margin: 0px; padding: 0px; overflow: visible; width: 480px; display: block;"><div style="overflow: visible; height: 0px; width: 100%;" align="right"><div style="border-style: ridge ridge none; border-width: 2px 2px 0px; padding: 1px; overflow: visible; vertical-align: bottom; -moz-border-radius-topleft: 10px; -moz-border-radius-topright: 10px; opacity: 0.5; background-color: white; position: relative; top: -19px; left: -5px; z-index: 900; width: 48px; height: 15px; cursor: pointer;" align="center"><span style="font-family: Arial,Helvetica,Sans-serif; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; line-height: 140%; text-align: right; text-decoration: none; opacity: 1.5; color: black;">Adblock</span></div></div></div><applet adblockframename="adblock-frame-n5" adblockframedobject="true" code="index_fichiers/Convolve.bin" height="600" width="480">You need a Java-compatible browser to see the demo.</applet></td></tr></tbody></table></center><p> <b>
+<font size="+1"><a href="http://www.ece.jhu.edu/%7Erugh/quiz/takeQuiz.php?quiz_id=2" target="_blank">Quiz</a></font></b>
+<br><br><br></p><center><a href="http://www.jhu.edu/%7Esignals/index.html">    <img src="index_fichiers/return_arrow.gif" border="0" height="13" width="30">   return to demonstrations page</a><br><br><br><table bgcolor="#e0e0e0" cellpadding="4">  <tbody><tr>             <td>                    <font size="-1">                        <a href="http://www.jhu.edu/%7Esignals/discreteconv/index.html">Discrete-time version</a>                       <br>                    Applet by <a href="mailto:crutch@stanford.edu">Steve Crutchfield</a>                    <br>                                                                    </font>         </td>   </tr></tbody></table></center></body></html>
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diff --git a/html/index2.html b/html/index2.html
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+<html><head><title>Joy of Convolution (Discrete Time)</title></head><body bgcolor="#f0f0f0"><img src="index2_fichiers/demo6.gif" height="40" width="405"><br><br><br><p><b><font size="+1"><a href="http://mfile.akamai.com/7111/rm/www.jhu.edu/signals/discreteconv2/welcome.ram">Welcome!</a></font></b><font size="+1"><img src="index2_fichiers/rafile.gif" height="25" width="46"></font></p><p>The behavior of a linear, time-invariant discrete-time system with input signal<i>x[n]</i> and output signal <i>y[n]</i> is described by the <i>convolution sum</i> </p><p></p><center><img src="index2_fichiers/conv.gif" height="44" width="153"></center>The signal <i>h[n]</i>, assumed known, is the response of thesystem to a unit-pulse input.<p>The convolution summation has a simple graphical interpretation.First, plot <i>h[k]</i> and the "flipped and shifted" <i>x[n - k]</i>on the <i>k</i> axis, where <i>n</i> is fixed.  Second, multiply thetwo signals to obtain a plot of the summand sequence indexed by <i>k</i>.Summing the values of this sequence with respect to <i>k</i>yields <i>y[n]</i>.  These operations can be repeatedfor every value of <i>n</i> of interest.</p><p>To explore graphical convolution, selectsignals <i>x[n]</i> and <i>h[n]</i>
+from the provided examples below,or use the mouse to draw your own
+signal or to modify a selectedsignal. Then click at a desired value of <i>n</i> on the first<i>k</i> axis.  After a moment, <i>h[k]</i> and <i>x[n - k]</i> will appear.  Drag the <i>n</i> symbol along the <i>k</i> axis to change thevalue of <i>n</i>.  For each <i>n</i>, the corresponding summand<i>h[k]x[n - k]</i> and output value <i>y[n]</i> will bedisplayed in their respective windows.</p><p></p><p><b><font size="+1"><a href="http://mfile.akamai.com/7111/rm/www.jhu.edu/signals/discreteconv2/exercises.ram">Exercises.</a></font></b><font size="+1"><img src="index2_fichiers/rafile.gif" height="25" width="46"></font>
+</p><p>
+
+</p><center><object classid="clsid:8AD9C840-044E-11D1-B3E9-00805F499D93" codebase="http://java.sun.com/products/plugin/1.3/jinstall-13-win32.cab#Version=1,3,0,0" align="middle" height="600" width="600">
+<param name="code" value="DTConvolve.DTConvolve.class">
+<param name="archive" value="DTConvolve.jar">
+<param name="type" value="application/x-java-applet;version=1.3">
+<param name="scriptable" value="false">
+<comment>
+    </comment><embed type="application/x-java-applet;version=1.3" code="DTConvolve.DTConvolve.class" archive="DTConvolve.jar" pluginspage="http://java.sun.com/products/plugin/1.3/plugin-install.html" align="middle" height="600" width="600">
+    <noembed>
+    &lt;/COMMENT&gt;
+           No Java 2 SDK, Standard Edition v 1.3 support for APPLET!!
+    </noembed>
+</object>
+</center>
+
+
+
+<p> <b>
+<font size="+1"><a href="http://www.ece.jhu.edu/%7Erugh/quiz/takeQuiz.php?quiz_id=3" target="_blank">Quiz</a></font></b>
+<br><br><br></p><center><a href="http://www.jhu.edu/%7Esignals/index.html">    <img src="index2_fichiers/return_arrow.gif" border="0" height="13" width="30">  return to demonstrations page</a><br><br><br><table bgcolor="#e0e0e0" cellpadding="4">  <tbody><tr>             <td>                    <font size="-1">                        <a href="http://www.jhu.edu/%7Esignals/convolve/index.html">Continuous-time version.</a>                        <br>    
+               Original applet by Steve Crutchfield. Update by Michael Ross.                   </font>         </td>   </tr></tbody></table></center></body></html>
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+<html><head><title>The Joy of Convolution</title></head><body bgcolor="#f0f0f0"><img src="joy_of_convolution_1_fichiers/demo6.gif" height="40" width="405"><br><br><br><b><font size="+1"><a href="http://mfile.akamai.com/7111/rm/www.jhu.edu/signals/convolve/welcome.ram">Welcome!</a></font></b><font size="+1"><img src="joy_of_convolution_1_fichiers/rafile.gif" height="25" width="46"></font><p>The behavior of a linear, continuous-time, time-invariant system with input signal<i>x(t)</i> and output signal <i>y(t)</i> is described by the <i>convolution</i> integral</p><p></p><center><img src="joy_of_convolution_1_fichiers/conv.gif"></center>The signal <i>h(t)</i>, assumed known, is the response of thesystem to a unit impulse input.<p>To compute the output <i>y(t)</i> at a specified <i>t</i>, first theintegrand <i>h(v) x(t - v)</i> is computed as a function of <i>v</i>.Then integration with respect to <i>v</i> is performed, resulting in<i>y(t)</i>.</p><p>These mathematical operations have simple graphical interpretations.First, plot <i>h(v)</i> and the "flipped and shifted" <i>x(t - v)</i>on the <i>v</i> axis, where <i>t</i> is fixed.  Second, multiply thetwo signals and compute the signed area of the resulting function of<i>v</i> to obtain <i>y(t)</i>.  These operations can be repeatedfor every value of <i>t</i> of interest.</p><p>To explore graphical convolution, selectsignals <i>x(t)</i> and <i>h(t)</i>
+from the provided examples below,or use the mouse to draw your own
+signal or to modify a selectedsignal. Then click at a desired value of <i>t</i> on the first<i>v</i> axis.  After a moment, <i>h(v)</i> and <i>x(t - v)</i> will appear.  Drag the <i>t</i> symbol along the <i>v</i> axis to change thevalue of <i>t</i>.  For each <i>t</i>, the corresponding integrand<i>h(v) x(t - v)</i> and the output value <i>y(t)</i> will bedisplayed in their respective windows.</p><p></p><p><b><font size="+1"><a href="http://mfile.akamai.com/7111/rm/www.jhu.edu/signals/convolve/exercises.ram">Exercises</a></font></b><font size="+1"><img src="joy_of_convolution_1_fichiers/rafile.gif" height="25" width="46"></font><br><br></p><center><table bgcolor="#c0c0c0" border="2" cellpadding="0" cellspacing="1" height="600" width="480"><tbody><tr><td><applet code="joy_of_convolution_1_fichiers/Convolve.bin" height="600" width="480">You need a Java-compatible browser to see the demo.</applet></td></tr></tbody></table></center><p> <b>
+<font size="+1"><a href="http://www.ece.jhu.edu/%7Erugh/quiz/takeQuiz.php?quiz_id=2" target="_blank">Quiz</a></font></b>
+<br><br><br></p><center><a href="http://www.jhu.edu/%7Esignals/index.html">    <img src="joy_of_convolution_1_fichiers/return_arrow.gif" border="0" height="13" width="30">    return to demonstrations page</a><br><br><br><table bgcolor="#e0e0e0" cellpadding="4">  <tbody><tr>             <td>                    <font size="-1">                        <a href="http://www.jhu.edu/%7Esignals/discreteconv/index.html">Discrete-time version</a>                       <br>                    Applet by <a href="mailto:crutch@stanford.edu">Steve Crutchfield</a>                    <br>                                                                    </font>         </td>   </tr></tbody></table></center></body></html>
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diff --git a/html/joy_of_convolution_2.html b/html/joy_of_convolution_2.html
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+<html><head><title>Joy of Convolution (Discrete Time)</title></head><body bgcolor="#f0f0f0"><img src="joy_of_convolution_2_fichiers/demo6.gif" height="40" width="405"><br><br><br><p><b><font size="+1"><a href="http://www.jhu.edu:8080/ramgen/signals/discreteconv/welcome.rm">Welcome!</a></font></b><font size="+1"><img src="joy_of_convolution_2_fichiers/rafile.gif" height="25" width="46"></font></p><p>The behavior of a linear, time-invariant discrete-time system with input signal<i>x[n]</i> and output signal <i>y[n]</i> is described by the <i>convolution sum</i> </p><p></p><center><img src="joy_of_convolution_2_fichiers/conv.gif" height="44" width="153"></center>The signal <i>h[n]</i>, assumed known, is the response of thesystem to a unit-pulse input.<p>The convolution summation has a simple graphical interpretation.First, plot <i>h[k]</i> and the "flipped and shifted" <i>x[n - k]</i>on the <i>k</i> axis, where <i>n</i> is fixed.  Second, multiply thetwo signals to obtain a plot of the summand sequence indexed by <i>k</i>.Summing the values of this sequence with respect to <i>k</i>yields <i>y[n]</i>.  These operations can be repeatedfor every value of <i>n</i> of interest.</p><p>To explore graphical convolution, selectsignals <i>x[n]</i> and <i>h[n]</i>
+from the provided examples below,or use the mouse to draw your own
+signal or to modify a selectedsignal. Then click at a desired value of <i>n</i> on the first<i>k</i> axis.  After a moment, <i>h[k]</i> and <i>x[n - k]</i> will appear.  Drag the <i>n</i> symbol along the <i>k</i> axis to change thevalue of <i>n</i>.  For each <i>n</i>, the corresponding summand<i>h[k]x[n - k]</i> and output value <i>y[n]</i> will bedisplayed in their respective windows.</p><p></p><p><b><font size="+1"><a href="http://www.jhu.edu:8080/ramgen/signals/discreteconv/exercises.rm">Exercises.</a></font></b><font size="+1"><img src="joy_of_convolution_2_fichiers/rafile.gif" height="25" width="46"></font></p><p></p><center><table bgcolor="#c0c0c0" border="2" cellpadding="0" cellspacing="1" height="600" width="480"><tbody><tr><td><applet code="joy_of_convolution_2_fichiers/Convolve.bin" height="600" width="480">You need a Java-compatible browser to see the demo.</applet></td></tr></tbody></table></center>
+<p> <b>
+<font size="+1"><a href="http://www.ece.jhu.edu/%7Erugh/quiz/takeQuiz.php?quiz_id=3" target="_blank">Quiz</a></font></b>
+<br><br><br></p><center><a href="http://www.jhu.edu/%7Esignals/index.html">    <img src="joy_of_convolution_2_fichiers/return_arrow.gif" border="0" height="13" width="30">    return to demonstrations page</a><br><br><br><table bgcolor="#e0e0e0" cellpadding="4">  <tbody><tr>             <td>                    <font size="-1">                        <a href="http://www.jhu.edu/%7Esignals/convolve/index.html">Continuous-time version.</a>                        <br>                    Applet by <a href="mailto:crutch@stanford.edu">Steve Crutchfield</a>                    </font>         </td>   </tr></tbody></table></center></body></html>
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+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
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+       <g transform="translate(60.2,550.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-60</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,470.4 L774.4,470.4 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,470.4 L77.5,470.4 M774.4,470.4 L765.4,470.4 '></path>
+       <g transform="translate(60.2,474.9)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-50</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,394.7 L774.4,394.7 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,394.7 L77.5,394.7 M774.4,394.7 L765.4,394.7 '></path>
+       <g transform="translate(60.2,399.2)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-40</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,319.1 L774.4,319.1 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,319.1 L77.5,319.1 M774.4,319.1 L765.4,319.1 '></path>
+       <g transform="translate(60.2,323.6)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-30</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,243.5 L774.4,243.5 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,243.5 L77.5,243.5 M774.4,243.5 L765.4,243.5 '></path>
+       <g transform="translate(60.2,248.0)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-20</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,167.9 L774.4,167.9 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,167.9 L77.5,167.9 M774.4,167.9 L765.4,167.9 '></path>
+       <g transform="translate(60.2,172.4)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-10</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,92.2 L774.4,92.2 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,92.2 L77.5,92.2 M774.4,92.2 L765.4,92.2 '></path>
+       <g transform="translate(60.2,96.7)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan> 0</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,16.6 L774.4,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,16.6 L77.5,16.6 M774.4,16.6 L765.4,16.6 '></path>
+       <g transform="translate(60.2,21.1)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan> 10</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,546.0 L68.5,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,546.0 L68.5,537.0 M68.5,16.6 L68.5,25.6 '></path>
+       <g transform="translate(68.5,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 0.001</tspan>
+               </text>
+       </g>
+       <path d='M68.5,546.0 L68.5,541.5 M68.5,16.6 L68.5,21.1 M103.9,546.0 L103.9,541.5 M103.9,16.6 L103.9,21.1 
+               M124.6,546.0 L124.6,541.5 M124.6,16.6 L124.6,21.1 M139.3,546.0 L139.3,541.5 M139.3,16.6 L139.3,21.1 
+               M150.7,546.0 L150.7,541.5 M150.7,16.6 L150.7,21.1 M160.0,546.0 L160.0,541.5 M160.0,16.6 L160.0,21.1 
+               M167.9,546.0 L167.9,541.5 M167.9,16.6 L167.9,21.1 M174.7,546.0 L174.7,541.5 M174.7,16.6 L174.7,21.1 
+               M180.8,546.0 L180.8,541.5 M180.8,16.6 L180.8,21.1 '></path>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M186.1,546.0 L186.1,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M186.1,546.0 L186.1,537.0 M186.1,16.6 L186.1,25.6 '></path>
+       <g transform="translate(186.1,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 0.01</tspan>
+               </text>
+       </g>
+       <path d='M186.1,546.0 L186.1,541.5 M186.1,16.6 L186.1,21.1 M221.6,546.0 L221.6,541.5 M221.6,16.6 L221.6,21.1 
+               M242.3,546.0 L242.3,541.5 M242.3,16.6 L242.3,21.1 M257.0,546.0 L257.0,541.5 M257.0,16.6 L257.0,21.1 
+               M268.4,546.0 L268.4,541.5 M268.4,16.6 L268.4,21.1 M277.7,546.0 L277.7,541.5 M277.7,16.6 L277.7,21.1 
+               M285.6,546.0 L285.6,541.5 M285.6,16.6 L285.6,21.1 M292.4,546.0 L292.4,541.5 M292.4,16.6 L292.4,21.1 
+               M298.4,546.0 L298.4,541.5 M298.4,16.6 L298.4,21.1 '></path>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M303.8,546.0 L303.8,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M303.8,546.0 L303.8,537.0 M303.8,16.6 L303.8,25.6 '></path>
+       <g transform="translate(303.8,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 0.1</tspan>
+               </text>
+       </g>
+       <path d='M303.8,546.0 L303.8,541.5 M303.8,16.6 L303.8,21.1 M339.2,546.0 L339.2,541.5 M339.2,16.6 L339.2,21.1 
+               M359.9,546.0 L359.9,541.5 M359.9,16.6 L359.9,21.1 M374.6,546.0 L374.6,541.5 M374.6,16.6 L374.6,21.1 
+               M386.0,546.0 L386.0,541.5 M386.0,16.6 L386.0,21.1 M395.3,546.0 L395.3,541.5 M395.3,16.6 L395.3,21.1 
+               M403.2,546.0 L403.2,541.5 M403.2,16.6 L403.2,21.1 M410.0,546.0 L410.0,541.5 M410.0,16.6 L410.0,21.1 
+               M416.1,546.0 L416.1,541.5 M416.1,16.6 L416.1,21.1 '></path>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M421.5,546.0 L421.5,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M421.5,546.0 L421.5,537.0 M421.5,16.6 L421.5,25.6 '></path>
+       <g transform="translate(421.5,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 1</tspan>
+               </text>
+       </g>
+       <path d='M421.5,546.0 L421.5,541.5 M421.5,16.6 L421.5,21.1 M456.9,546.0 L456.9,541.5 M456.9,16.6 L456.9,21.1 
+               M477.6,546.0 L477.6,541.5 M477.6,16.6 L477.6,21.1 M492.3,546.0 L492.3,541.5 M492.3,16.6 L492.3,21.1 
+               M503.7,546.0 L503.7,541.5 M503.7,16.6 L503.7,21.1 M513.0,546.0 L513.0,541.5 M513.0,16.6 L513.0,21.1 
+               M520.9,546.0 L520.9,541.5 M520.9,16.6 L520.9,21.1 M527.7,546.0 L527.7,541.5 M527.7,16.6 L527.7,21.1 
+               M533.7,546.0 L533.7,541.5 M533.7,16.6 L533.7,21.1 '></path>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M539.1,546.0 L539.1,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M539.1,546.0 L539.1,537.0 M539.1,16.6 L539.1,25.6 '></path>
+       <g transform="translate(539.1,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 10</tspan>
+               </text>
+       </g>
+       <path d='M539.1,546.0 L539.1,541.5 M539.1,16.6 L539.1,21.1 M574.5,546.0 L574.5,541.5 M574.5,16.6 L574.5,21.1 
+               M595.2,546.0 L595.2,541.5 M595.2,16.6 L595.2,21.1 M609.9,546.0 L609.9,541.5 M609.9,16.6 L609.9,21.1 
+               M621.3,546.0 L621.3,541.5 M621.3,16.6 L621.3,21.1 M630.6,546.0 L630.6,541.5 M630.6,16.6 L630.6,21.1 
+               M638.5,546.0 L638.5,541.5 M638.5,16.6 L638.5,21.1 M645.3,546.0 L645.3,541.5 M645.3,16.6 L645.3,21.1 
+               M651.4,546.0 L651.4,541.5 M651.4,16.6 L651.4,21.1 '></path>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M656.8,546.0 L656.8,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M656.8,546.0 L656.8,537.0 M656.8,16.6 L656.8,25.6 '></path>
+       <g transform="translate(656.8,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 100</tspan>
+               </text>
+       </g>
+       <path d='M656.8,546.0 L656.8,541.5 M656.8,16.6 L656.8,21.1 M692.2,546.0 L692.2,541.5 M692.2,16.6 L692.2,21.1 
+               M712.9,546.0 L712.9,541.5 M712.9,16.6 L712.9,21.1 M727.6,546.0 L727.6,541.5 M727.6,16.6 L727.6,21.1 
+               M739.0,546.0 L739.0,541.5 M739.0,16.6 L739.0,21.1 M748.3,546.0 L748.3,541.5 M748.3,16.6 L748.3,21.1 
+               M756.2,546.0 L756.2,541.5 M756.2,16.6 L756.2,21.1 M763.0,546.0 L763.0,541.5 M763.0,16.6 L763.0,21.1 
+               M769.0,546.0 L769.0,541.5 M769.0,16.6 L769.0,21.1 '></path>
+</g>
+<g style="fill:none; color:gray; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M774.4,546.0 L774.4,16.6 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M774.4,546.0 L774.4,537.0 M774.4,16.6 L774.4,25.6 '></path>
+       <g transform="translate(774.4,568.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan> 1000</tspan>
+               </text>
+       </g>
+       <path d='M774.4,546.0 L774.4,541.5 M774.4,16.6 L774.4,21.1 M68.5,16.6 L68.5,546.0 L774.4,546.0 L774.4,16.6 
+               L68.5,16.6 Z '></path>
+       <g transform="translate(22.5,281.3) rotate(-90)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan>Gain (dB)</tspan>
+               </text>
+       </g>
+       <g transform="translate(421.4,595.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan>Angular frequency (rad/s)</tspan>
+               </text>
+       </g>
+       <g transform="translate(245.0,497.6)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan>Passband</tspan>
+               </text>
+       </g>
+       <g transform="translate(597.9,497.6)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan>Stopband</tspan>
+               </text>
+       </g>
+       <g transform="translate(456.9,66.5)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:start">
+               <text><tspan>Cutoff frequency</tspan>
+               </text>
+       </g>
+       <g transform="translate(386.0,149.7)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:end">
+               <text><tspan>-3.01 dB</tspan>
+               </text>
+       </g>
+       <g transform="translate(656.8,232.9)" style="stroke:none; fill:black; font-family:Arial; font-size:12.00pt; text-anchor:middle">
+               <text><tspan>Slope: -20 dB/decade</tspan>
+               </text>
+       </g>
+</g>
+<g style="fill:none; color:blue; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M421.5,92.2 L774.4,546.0 M68.5,92.2 L421.5,92.2 '></path>
+</g>
+<g style="fill:none; color:green; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M421.5,546.0 L421.5,16.6 '></path>
+</g>
+<g style="fill:none; color:cyan; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <g style = 'stroke:none; shape-rendering:crispEdges'>
+               <polygon fill = 'currentColor' points = '431.0,86.1 436.7,87.4 421.5,92.2 432.2,80.4 431.0,86.1 '/>
+       </g>
+       <path d='M431.0,86.1 L436.7,87.4 L421.5,92.2 L432.2,80.4 L431.0,86.1 M456.9,69.5 L431.0,86.1 '></path>
+       <g style = 'stroke:none; shape-rendering:crispEdges'>
+               <polygon fill = 'currentColor' points = '412.0,120.9 406.3,119.7 421.5,114.9 410.8,126.7 412.0,120.9 '/>
+       </g>
+       <path d='M412.0,120.9 L406.3,119.7 L421.5,114.9 L410.8,126.7 L412.0,120.9 M386.0,137.6 L412.0,120.9 '></path>
+</g>
+<g style="fill:none; color:blue; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M656.8,243.5 L548.4,243.5 M656.8,243.5 L656.8,387.2 '></path>
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:1.00; stroke-linecap:butt; stroke-linejoin:miter">
+</g>
+<g style="fill:none; color:black; stroke:currentColor; stroke-width:2.00; stroke-linecap:butt; stroke-linejoin:miter">
+</g>
+<g style="fill:none; color:red; stroke:currentColor; stroke-width:2.00; stroke-linecap:butt; stroke-linejoin:miter">
+       <path d='M68.5,92.2 L68.9,92.2 L69.2,92.2 L69.6,92.2 L69.9,92.2 L70.3,92.2 L70.6,92.2 L71.0,92.2 
+               L71.3,92.2 L71.7,92.2 L72.0,92.2 L72.4,92.2 L72.7,92.2 L73.1,92.2 L73.4,92.2 L73.8,92.2 
+               L74.1,92.2 L74.5,92.2 L74.9,92.2 L75.2,92.2 L75.6,92.2 L75.9,92.2 L76.3,92.2 L76.6,92.2 
+               L77.0,92.2 L77.3,92.2 L77.7,92.2 L78.0,92.2 L78.4,92.2 L78.7,92.2 L79.1,92.2 L79.4,92.2 
+               L79.8,92.2 L80.1,92.2 L80.5,92.2 L80.9,92.2 L81.2,92.2 L81.6,92.2 L81.9,92.2 L82.3,92.2 
+               L82.6,92.2 L83.0,92.2 L83.3,92.2 L83.7,92.2 L84.0,92.2 L84.4,92.2 L84.7,92.2 L85.1,92.2 
+               L85.4,92.2 L85.8,92.2 L86.1,92.2 L86.5,92.2 L86.9,92.2 L87.2,92.2 L87.6,92.2 L87.9,92.2 
+               L88.3,92.2 L88.6,92.2 L89.0,92.2 L89.3,92.2 L89.7,92.2 L90.0,92.2 L90.4,92.2 L90.7,92.2 
+               L91.1,92.2 L91.4,92.2 L91.8,92.2 L92.1,92.2 L92.5,92.2 L92.9,92.2 L93.2,92.2 L93.6,92.2 
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+               L99.6,92.2 L99.9,92.2 L100.3,92.2 L100.6,92.2 L101.0,92.2 L101.3,92.2 L101.7,92.2 L102.0,92.2 
+               L102.4,92.2 L102.7,92.2 L103.1,92.2 L103.4,92.2 L103.8,92.2 L104.1,92.2 L104.5,92.2 L104.9,92.2 
+               L105.2,92.2 L105.6,92.2 L105.9,92.2 L106.3,92.2 L106.6,92.2 L107.0,92.2 L107.3,92.2 L107.7,92.2 
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+%%Trailer
+%%DocumentFonts: Helvetica

diff --git a/img/ex_sig.pdf b/img/ex_sig.pdf
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diff --git a/img/ex_wav.m b/img/ex_wav.m
new file mode 100644 (file)
index 0000000..f636829
--- /dev/null
+++ b/img/ex_wav.m
@@ -0,0 +1,17 @@
+% Wavread
+
+cd /home/momo/CNAM/CPDA/Signal/cours/img/;
+fic = 'regularfries.wav';
+[x,fs,bits] = wavread(fic);
+x = x(:,1);
+lx = length(x);
+t = [0:lx-1];
+t = t/fs;
+
+plot(t,x,'-;wav;')
+
+axis([0,1,-1.2,1.2]);
+xlabel('Temps (s)');
+ylabel('Amplitude');
+print('./ex_sig.eps', '-deps');
+

diff --git a/img/filtres_ideaux.png b/img/filtres_ideaux.png
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diff --git a/img/ft_sin.png b/img/ft_sin.png
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diff --git a/img/ft_sin_01.png b/img/ft_sin_01.png
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diff --git a/img/ft_sin_fen.png b/img/ft_sin_fen.png
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diff --git a/img/logo_cnam.png b/img/logo_cnam.png
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diff --git a/img/octave-core b/img/octave-core
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index 0000000..0fde110
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diff --git a/img/regularfries.wav b/img/regularfries.wav
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diff --git a/img/repliement.png b/img/repliement.png
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diff --git a/img/sch_ech.png b/img/sch_ech.png
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diff --git a/img/sch_ech_2.png b/img/sch_ech_2.png
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diff --git a/img/sens_oreille.jpg b/img/sens_oreille.jpg
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index 0000000..9366128
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diff --git a/img/sf_voix.png b/img/sf_voix.png
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index 0000000..206f8f8
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diff --git a/img/sig_cardinal.png b/img/sig_cardinal.png
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index 0000000..761ed3f
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diff --git a/img/sig_dirac.ipe b/img/sig_dirac.ipe
new file mode 100644 (file)
index 0000000..ecf70c5
--- /dev/null
+++ b/img/sig_dirac.ipe
@@ -0,0 +1,28 @@
+<ipe version="60028" creator="Ipe 6.0 preview 28">
+<info created="D:20070319172651" modified="D:20070319173636" bbox="cropbox"/>
+<page gridsize="16">
+<layer name="alpha"/>
+<layer name="beta"/>
+<view layers="alpha beta" active="beta"/>
+<path layer="alpha" stroke="black" matrix="0.521739 0 0 1 189.913 0" pen="normal" arrow="normal">
+96 192 m
+464 192 l
+</path>
+<path stroke="black" matrix="1 0 0 0.625 0 94" pen="normal" arrow="normal">
+288 80 m
+288 336 l
+</path>
+<text layer="beta" stroke="black" matrix="1 0 0 1 32 -16" pos="416 208" type="label" width="4.5523" height="7.3534" depth="0" valign="baseline" size="large" style="default">t</text>
+<text stroke="black" matrix="1 0 0 1 0 16" pos="304 256" type="label" width="5.853" height="7.7044" depth="0" valign="baseline" size="large" style="default">1</text>
+<text stroke="black" pos="272 176" type="label" valign="baseline" size="large" style="default">0</text>
+<path stroke="black" pen="fat" arrow="normal">
+352 192 m
+352 256 l
+</path>
+<text stroke="black" pos="352 176" type="label" valign="baseline" size="large" style="default">$f_0$</text>
+<path stroke="black" dash="dashed" pen="normal">
+288 256 m
+384 256 l
+</path>
+</page>
+</ipe>

diff --git a/img/sig_dirac.pdf b/img/sig_dirac.pdf
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diff --git a/img/sig_porte.ipe b/img/sig_porte.ipe
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+showpage
+%%Trailer
+%%DocumentFonts: Helvetica

diff --git a/img/sinus_ech.m b/img/sinus_ech.m
new file mode 100644 (file)
index 0000000..968787a
--- /dev/null
+++ b/img/sinus_ech.m
@@ -0,0 +1,33 @@
+% Premier test octave
+
+f0 = 50;
+f1 = 100;
+k = 1;
+a = f0/f1;
+t = [0:1/10000:1];
+f = [1:1:1000];
+
+x=k*sin(2*pi*f0*t);
+y=k*sin(2*pi*f1*t);
+
+figure(1);
+%plot(t, x , '-;sin(2*pi*50t);', t, y, '.@;sin(2*pi*100*t);');
+plot(t, x , ';sin(2*pi*50t);', t, y, ';sin(2*pi*100*t);');
+axis([0,0.1,-1.2,1.2]);
+xlabel('Temps (s)');
+ylabel('Amplitude');
+print('./sinus_ech.eps', '-deps');
+
+fx = zeros(1000,1);
+fy = zeros(1000,1);
+fx(f0) = k;
+fy(f1) = a;
+
+%[b0 b1] = bar([f0 f1],[k a]);
+
+figure(2);
+semilogx(f,fx,';sin(2*pi*50t);',f,fy,';sin(2*pi*100*t);');
+axis([10,1000,0,1.2]);
+xlabel('Frequence (Hz)');
+ylabel('Amplitude');
+print('./sinus_ech_ft.eps', '-deps');
\ No newline at end of file

diff --git a/img/sinus_ech.oct b/img/sinus_ech.oct
new file mode 100644 (file)
index 0000000..2a0458c
--- /dev/null
+++ b/img/sinus_ech.oct
@@ -0,0 +1,13 @@
+#premier test octave
+
+t=[0:1/10000:1];
+x=sin(2*pi*50*t);
+y=sin(2*pi*100*t);
+
+plot(t, x , '-;sin(2*pi*50t);', t, y, '.@;sin(2*pi*100*t);');
+
+axis([0,0.1,-1.2,1.2]);
+xlabel('Temps (s)');
+ylabel('Amplitude');
+print('./sinus_ech.eps', '-deps');
+

diff --git a/img/sinus_ech.pdf b/img/sinus_ech.pdf
new file mode 100644 (file)
index 0000000..433721a
Binary files /dev/null and b/img/sinus_ech.pdf differ

diff --git a/img/sinus_ech_ft.eps b/img/sinus_ech_ft.eps
new file mode 100644 (file)
index 0000000..4109fbb
--- /dev/null
+++ b/img/sinus_ech_ft.eps
@@ -0,0 +1,2574 @@
+%!PS-Adobe-2.0 EPSF-2.0
+%%Title: ./sinus_ech_ft.eps
+%%Creator: gnuplot 4.0 patchlevel 0
+%%CreationDate: Mon Mar 19 16:32:38 2007
+%%DocumentFonts: (atend)
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+%%Orientation: Portrait
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diff --git a/img/sinus_ech_ft.pdf b/img/sinus_ech_ft.pdf
new file mode 100644 (file)
index 0000000..aeaa34c
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diff --git a/img/sinusoides.fkt b/img/sinusoides.fkt
new file mode 100644 (file)
index 0000000..d0f37d5
--- /dev/null
+++ b/img/sinusoides.fkt
@@ -0,0 +1,33 @@
+<!DOCTYPE kmpdoc>
+<kmpdoc version="2" >
+    <axes width="5" tic-width="1" color="#808080" tic-legth="10" >
+        <show-axes>1</show-axes>
+        <show-arrows>1</show-arrows>
+        <show-label>1</show-label>
+        <show-frame>1</show-frame>
+        <show-extra-frame>1</show-extra-frame>
+        <xcoord>2</xcoord>
+        <ycoord>4</ycoord>
+        <ymin>-1.2</ymin>
+        <ymax>1.2</ymax>
+    </axes>
+    <grid width="1" color="#c0c0c0" >
+        <mode>0</mode>
+    </grid>
+    <scale>
+        <tic-x>8</tic-x>
+        <tic-y>8</tic-y>
+        <print-tic-x>3</print-tic-x>
+        <print-tic-y>3</print-tic-y>
+    </scale>
+    <function width="1" use-slider="-1" color="#00ff00" visible="1" >
+        <equation>f(x)=sin(x)</equation>
+    </function>
+    <function width="1" use-slider="-1" color="#0000ff" visible="1" >
+        <equation>g(x)=sin(2x)</equation>
+    </function>
+    <fonts>
+        <axes-font>Sans Serif</axes-font>
+        <header-table-font>Sans Serif</header-table-font>
+    </fonts>
+</kmpdoc>

diff --git a/img/sinusoides.png b/img/sinusoides.png
new file mode 100644 (file)
index 0000000..68b83c8
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diff --git a/img/spec_lux.png b/img/spec_lux.png
new file mode 100644 (file)
index 0000000..27873c7
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diff --git a/img/spectr_sep.pdf b/img/spectr_sep.pdf
new file mode 100644 (file)
index 0000000..36ed777
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diff --git a/img/spectr_sep.png b/img/spectr_sep.png
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index 0000000..671c812
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diff --git a/img/spectr_sep_temp.png b/img/spectr_sep_temp.png
new file mode 100644 (file)
index 0000000..df7f72e
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diff --git a/img/spectr_voix1.png b/img/spectr_voix1.png
new file mode 100644 (file)
index 0000000..11c0732
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diff --git a/img/spectro_harm.png b/img/spectro_harm.png
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index 0000000..aed5a36
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diff --git a/img/spectro_vibrato.png b/img/spectro_vibrato.png
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index 0000000..01783da
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diff --git a/img/spectro_vibrato_2.png b/img/spectro_vibrato_2.png
new file mode 100644 (file)
index 0000000..5f273e7
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diff --git a/img/spectrogram_01.png b/img/spectrogram_01.png
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index 0000000..d0f3613
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diff --git a/img/spectrogram_02.png b/img/spectrogram_02.png
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index 0000000..0aa5786
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diff --git a/img/spectrogram_03.png b/img/spectrogram_03.png
new file mode 100644 (file)
index 0000000..0d7d114
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diff --git a/img/test.tex b/img/test.tex
new file mode 100644 (file)
index 0000000..1fceb94
--- /dev/null
+++ b/img/test.tex
@@ -0,0 +1,34 @@
+\documentclass[a4paper]{minimal}
+\usepackage{pstricks}
+\usepackage{pst-plot}
+\author{Kig 0.10.6}
+\begin{document}
+\begin{pspicture*}(0,0)(15,10.2824)
+\psset{xunit=0.934763}
+\psset{yunit=0.934763}
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+\end{document}

diff --git a/img/tfct.png b/img/tfct.png
new file mode 100644 (file)
index 0000000..0b02717
Binary files /dev/null and b/img/tfct.png differ

diff --git a/mestheoremes.sty b/mestheoremes.sty
new file mode 100644 (file)
index 0000000..d98851f
--- /dev/null
+++ b/mestheoremes.sty
@@ -0,0 +1,36 @@
+\usepackage{amsthm}\r
+\r
+\newtheoremstyle{note}  % nom\r
+{16pt} % espace avant\r
+{8pt} % espace après\r
+{\itshape} % body font\r
+{} % Indent amount (empty = no indent, \parindent = para indent)\r
+{\bfseries} % note head font\r
+{~:} % punctuation après note head\r
+{ } % espace après note head " " = normal interword space ; \newline = linebreak\r
+{} % note head spec (can be left empty, meaning 'normal')\r
+\r
+\theoremstyle{note}\r
+\newtheorem{note}{Note}\r
+\r
+\newtheoremstyle{break}\r
+{16pt}         % espace avant\r
+{8pt}  % espace après\r
+{}     % body font\r
+{}             % indent amount (empty=no indent,\parindent=para indent)\r
+{\bfseries}    % break header font\r
+{~:}   % ponctuation\r
+{ }    % Space après head : \newline = \linebreak\r
+{}             % break head spec\r
+\r
+\theoremstyle{break}\r
+\newtheorem*{bnotes}{Notes}\r
+\newtheorem{bex}{Exemple}\r
+\newtheorem*{brems}{Remarques}\r
+\newtheorem*{bdefis}{Définitions}   % l'étoile supprime la numérotation\r
+\newtheorem*{bnote}{Note}           % idem \r
+\newtheorem*{rem}{Remarque}      % idem \r
+\r
+\theoremstyle{note}\r
+\newtheorem*{defi}{Définition}   % l'étoile supprime la numérotation\r
+\newtheorem*{enon}{Enoncé}      % idem \r

diff --git a/octave/filter_test.m b/octave/filter_test.m
new file mode 100644 (file)
index 0000000..0243090
--- /dev/null
+++ b/octave/filter_test.m
@@ -0,0 +1,89 @@
+close all;
+%clear all;
+
+f0 = 20;
+f1 = 22050;
+t1 = 1;
+fe = 44100;
+fc = 3000;
+a = 1/(1+fe/(2*pi*fc));
+
+t = [0:1/fe:t1];
+n = length(t);
+N = 1;
+
+for k=1:2
+
+    s = randn(1,n);
+    %sweep = sin(2*pi*(f0*t1)/log(f1/f0) * (exp(t'/t1*log(f1/f0))-1))';
+    sf(1) = s(1);
+    
+    % Avec une boucle
+    tic;
+    for i=2:n
+        sf(i)=a*s(i)+(1-a)*sf(i-1);
+    end
+    t_boucle(k) = toc;
+    
+    % Avec filter
+    tic;
+    sff = filter(a,[1 a-1],s);
+    t_filter(k) = toc;
+    
+    % Fonction de transfert
+    nfft = 16384;
+    fft_s = fft(s,nfft);
+    fft_sf = fft(sf,nfft);
+    fft_sff = fft(sff,nfft);
+    f = [0:fe/nfft:fe/2];
+    f(1) = [];
+
+    mod_FT(k,:) = abs(fft_sf)./abs(fft_s);
+    pha_FT(k,:) = angle(fft_sf);
+    mod_FTF(k,:) = abs(fft_sff)./abs(fft_s);
+    pha_FTF(k,:) = angle(fft_sff);
+
+end
+
+mod_FT = mean(mod_FT);
+figure(1)
+semilogx(f,20*log10(mod_FT(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la methode recursive (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FT = mean(pha_FT);
+figure(2)
+semilogx(f,unwrap(pha_FT(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la méthode recursive']);
+
+mod_FTF = mean(mod_FTF);
+figure(3)
+semilogx(f,20*log10(mod_FTF(1:nfft/2)));
+hold on
+semilogx(f,20*log10(abs(1./(1+j*(f/fc)))),'r');
+title(['Module de la fonction de transfert par la fonction filter (reel / theorique)']);
+xlabel(['Frequence (Hz)']);
+ylabel(['Amplitude (dB)']);
+
+pha_FTF = mean(pha_FTF);
+figure(4)
+semilogx(f,unwrap(pha_FTF(1:nfft/2)));
+xlabel(['Frequence (Hz)']);
+ylabel(['Phase (rad)']);
+title(['Phase de la fonction de transfert par la fonction filter']);
+
+'temps boucle (s): '
+t_boucle
+'temps filtre (s): '
+t_filter
+
+hold on
+plot(t_filter,'r');
+
+pause;
+

diff --git a/octave/sig_exc.wav b/octave/sig_exc.wav
new file mode 100644 (file)
index 0000000..9c2febe
Binary files /dev/null and b/octave/sig_exc.wav differ

diff --git a/octave/spectrogram2.m b/octave/spectrogram2.m
new file mode 100644 (file)
index 0000000..ffee9db
--- /dev/null
+++ b/octave/spectrogram2.m
@@ -0,0 +1,151 @@
+%% Copyright (C) 2000 Paul Kienzle
+%%
+%% This program is free software and may be used for any purpose.  This
+%% copyright notice must be maintained. Paul Kienzle is not responsible
+%% for the consequences of using this software.
+
+%% usage: [S, f, t] = spectrogram(x, Fs, window, step, maxF, shape, minE)
+%%
+%% Generate a spectrogram for the signal. This chops the signal into
+%% overlapping slices, windows each slice and applies a Fourier
+%% transform to determine the frequency components at that slice.
+%%
+%% x:      signal to analyse
+%% Fs:     sampling rate for the signal
+%% window: analysis window length (default 30 msec)
+%% step:   time between windows, start to start (default 5 ms)
+%% maxF:   maximum frequency to display (default 4000 Hz)
+%%    Alternatively, use [maxF, nF], where nF is the minimum
+%%    of frequency points to display.  If nF is greater than
+%%    what it would normally be for the given window size and
+%%    maximum displayed frequency, the FFT is zero-padded until
+%%    it at least nF points are displayed on the y axis.
+%% shape:  window analysis function (default 'hanning')
+%%    Shape is any function which takes an integer n and returns
+%%    a vector of length n.  If shape contains %d and ends with
+%%    ')', as for example '(1:%d)' or 'kaiser(%d,0.5)' do, then
+%%    %d is replaced with the desired window length, and the
+%%    expression is evaluated.
+%% minE:   noise floor (default -40dB)
+%%    Any value less than the noise floor is clipped before the
+%%    spectrogram is displayed.  This limits the dynamic range
+%%    that your spectrogram must accomodate.  Alternatively,
+%%    use [minE, maxE], where maxE is the clipping ceiling, also
+%%    in decibels.
+%%
+%% Return values
+%%    S is the spectrogram in S with linear magnitude normalized to 1.
+%%    f is the frequency indices corresponding to the rows of S.
+%%    t is the time indices corresponding to the columns of S.
+%%    If no return value is requested, the spectrogram is displayed instead.
+%%
+%% Global variables
+%%    spectrogram_{window,step,maxF,nF,shape,minE,maxE} can override
+%%    the default values with your own.
+%%
+%% To make a good spectrogram, generating spectral slices is only half
+%% the problem.  Before you generate them, you must first choose your
+%% window size, step size and FFT size.  A wide window shows more
+%% harmonic detail, a narrow window shows more formant structure.  This
+%% defines your time-frequency resolution. Step size controls the
+%% horizontal scale of the spectrogram. Decrease it to stretch, or
+%% increase it to compress. Certainly, increasing step size will reduce
+%% time resolution, but decreasing it will not improve it much beyond
+%% the limits imposed by the window size (you do gain a little bit,
+%% depending on the shape of your window, as the peak of the window
+%% slides over peaks in the signal energy).  The range 1-5 msec is good
+%% for speech. Finally, FFT length controls the vertical scale, with
+%% larger values stretching the frequency range.  Clearly, padding with
+%% zeros does not add any information to the spectrum, but it is a
+%% cheap, easy and good way to interpolate between frequency points, and
+%% can make for prettier spectrograms.
+%%
+%% After you have generated the spectral slices, there are a number of
+%% decisions for displaying them.  Firstly, the entire frequency range
+%% does not need to be displayed.  The frequency range of the FFT is
+%% determined by sampling rate.  If most of your signal is below 4 kHz
+%% (in speech for example), there is no reason to display up to the
+%% Nyquist frequency of 10 kHz for a 20 kHz sampling rate.  Next, there
+%% is the dynamic range of the signal.  Since the information in speech
+%% is well above the noise floor, it makes sense to eliminate any
+%% dynamic range at the bottom end.  This is done by taking the max of
+%% the normalized magnitude and some lower limit such as -40 dB.
+%% Similarly, there is not much information in the very top of the
+%% range, so clipping to -3 dB makes sense there. Finally, there is the
+%% choice of colormap.  A brightness varying colormap such as copper or
+%% bone gives good shape to the ridges and valleys.  A hue varying
+%% colormap such as jet or hsv gives an indication of the steepness of
+%% the slopes.
+
+%% TODO: Accept vector of frequencies at which to sample the signal.
+%% TODO: Consider accepting maxF (values > 0), shape (value is string)
+%% TODO:    and dynamic range (values <= 0) in any order.
+%% TODO: Consider defaulting step and maxF so that the spectrogram is
+%% TODO:    an appropriate size for the screen (eg, 600x100).
+%% TODO: Consider drawing in frequency/time grid;
+%% TODO:    (necessary with automatic sizing as suggested above)
+%% TODO: Consider using step vs. [nT, nF] rather than maxF vs [maxF, nF]
+%% TODO: Figure out why exist() is so slow: 50 ms vs 1 ms for lookup.
+
+function [S, f, t] = spectrogram(x, Fs, window, step, minF, maxF, shape, minE)
+%      spectrogram_window=30;
+%      spectrogram_step=5;
+%      spectrogram_maxF=4000;
+%      spectrogram_shape='hanning';
+%      spectrogram_minE=-40;
+%      spectrogram_nF=1;
+%       spectrogram_maxE=0;
+    
+  maxE=0;
+  step_n = fix(step*Fs/1000);    % one spectral slice every step ms
+    
+  %% generate window from duration and shape function name
+  win_n = fix(window*Fs/1000);
+  
+  if shape(length(shape)) == ')' 
+    shape = sprintf(shape, win_n);
+  else
+    shape = [shape '(' num2str(win_n) ')'];
+  end
+  win_vec = eval([shape]);
+  if size(win_vec,2) ~= 1
+    win_vec = win_vec'; end
+  if size(win_vec,2) ~= 1 || size(win_vec,1) ~= win_n
+    %error("spectrogram %s did not return a window of length %d", \ shape, win_n);
+  end
+         
+  %% FFT length from size of window and number of freq. pts requested
+  fft_n = 2^nextpow2(win_n);    % next highest power of 2
+  dF = Fs/fft_n;                % freq. step with current fft_n
+  nF = ceil(maxF(1)/dF);        % freq. pts with current fft_n,maxF
+  if ~isempty(minF)           % make sure there are at least n freq. pts
+    if minF > nF             % if not enough
+      dF = maxF/minF;            % figure out what freq. step we need
+      fft_n = 2^nextpow2(Fs/dF);   % figure out what fft_n this requires
+      dF = Fs/fft_n;               % freq. step with new fft_n
+      nF = ceil(maxF/dF);          % freq. pts with new fft_n,maxF
+    end
+  end
+
+  %% build matrix of windowed data slices
+  offset = 1:step_n:length(x)-win_n;
+  S = zeros (fft_n, length(offset));
+  for i=1:length(offset)
+    S(1:win_n, i) = x(offset(i):offset(i)+win_n-1) .* win_vec;
+  end
+
+  %% compute fourier transform
+  S = fft(S);
+  S = abs(S(1:nF,:));        % select the desired frequencies
+  S = S/max(S(:));           % normalize magnitude so that max is 0 dB.
+  S = max(S, 10^(minE/10));  % clip below minF dB.
+  S = min(S, 10^(maxE/10));  % clip above maxF dB.
+
+  f = [minF:Fs/fft_n:maxF];
+  t = offset/Fs;
+  if nargout==0
+    imagesc(f,t,20*log10(flipud(S)));
+  end
+
+end
+

diff --git a/pdf_course_profile.log b/pdf_course_profile.log
new file mode 100644 (file)
index 0000000..e582ea5
--- /dev/null
+++ b/pdf_course_profile.log
@@ -0,0 +1,1197 @@
+This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2007.4.29)  3 MAY 2007 14:57
+entering extended mode
+ %&-line parsing enabled.
+**pdf_course_profile.sty
+(./pdf_course_profile.sty
+LaTeX2e <2005/12/01>
+Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
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+ch, finnish, finnish, french, basque, french, german, ngerman, german, ngerman,
+ greek, monogreek, ancientgreek, ibycus, hungarian, hungarian, italian, italian
+, latin, latin, mongolian, mongolian, norsk, norsk, coptic, esperanto, estonian
+, icelandic, indonesian, interlingua, romanian, serbian, slovenian, turkish, up
+persorbian, welsh, polish, polish, portuguese, portuguese, spanish, catalan, ga
+lician, spanish, catalan, galician, swedish, swedish, loaded.
+
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+                                                  
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+Type  H <return>  for immediate help.
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+                                                  
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+                      17cm, 25cm}]{geometry}
+\usepackage may only appear in the document preamble, i.e.,
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+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                                                  
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+See the LaTeX manual or LaTeX Companion for explanation.
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+                        babel}    % typographie française
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
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+                    fontenc}      % encodage 8 bits des fontes utilisees
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.8 \usepackage{
+                verbatim}
+\usepackage may only appear in the document preamble, i.e.,
+between \documentclass and \begin{document}.
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                                                  
+l.13 \usepackage{
+                 float}                        % pour le positionnement des figures là où on veut
+\usepackage may only appear in the document preamble, i.e.,
+between \documentclass and \begin{document}.
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                 amsmath}              % mathematiques
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+! LaTeX Error: \usepackage before \documentclass.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                 amsfonts}
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                                            subfigure}
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+Type  H <return>  for immediate help.
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+                 url}
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+between \documentclass and \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                 setspace}
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+                                      color}
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+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
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+                                                  
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+Type  H <return>  for immediate help.
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+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+                                                  
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
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+
+
+! LaTeX Error: Missing \begin{document}.
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.43 ...rs de Traitement du signal - CNAM - CPDA},
+                                                  
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+Missing character: There is no e in font nullfont!
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+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 43--63
+[]
+ []
+
+! Undefined control sequence.
+<argument> \undefinedpagestyle 
+                               
+l.66 \pagestyle{fancy}
+                                             % Sets fancy header and footer
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.67 \fancyfoot
+               {}                            % Delete current footer settings
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: \chaptermark undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.68 \renewcommand{\chaptermark}
+                                [1]{         % Lower Case Chapter marker style
+Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+! Undefined control sequence.
+l.72 \fancyhead
+               [LE,RO]{\bfseries\thepage}    % Page number (boldface) in lef...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: Missing \begin{document}.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.72 \fancyhead[
+                LE,RO]{\bfseries\thepage}    % Page number (boldface) in lef...
+
+You're in trouble here.  Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no E in font nullfont!
+Missing character: There is no , in font nullfont!
+Missing character: There is no R in font nullfont!
+Missing character: There is no O in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.72 \fancyhead[LE,RO]{\bfseries\thepage
+                                        }    % Page number (boldface) in lef...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.74 \fancyhead
+               [RE]{\footnotesize{\bfseries\leftmark}}      % Chapter in the...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no R in font nullfont!
+Missing character: There is no E in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.74 \fancyhead[RE]{\footnotesize
+                                 {\bfseries\leftmark}}      % Chapter in the...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+! Undefined control sequence.
+l.75 \fancyhead
+               [LO]{\footnotesize{\bfseries\rightmark}}     % Section in the...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+Missing character: There is no [ in font nullfont!
+Missing character: There is no L in font nullfont!
+Missing character: There is no O in font nullfont!
+Missing character: There is no ] in font nullfont!
+! Undefined control sequence.
+l.75 \fancyhead[LO]{\footnotesize
+                                 {\bfseries\rightmark}}     % Section in the...
+The control sequence at the end of the top line
+of your error message was never \def'ed. If you have
+misspelled it (e.g., `\hobx'), type `I' and the correct
+spelling (e.g., `I\hbox'). Otherwise just continue,
+and I'll forget about whatever was undefined.
+
+
+! LaTeX Error: \headrulewidth undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.76 \renewcommand{\headrulewidth}
+                                  {0.3pt}    % Width of head rule
+Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+
+Overfull \hbox (20.0pt too wide) in paragraph at lines 72--77
+[]
+ []
+
+
+! LaTeX Error: \maketitle undefined.
+
+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
+ ...                                              
+                                                  
+l.86 \renewcommand{\maketitle}
+                              {
+Try typing  <return>  to proceed.
+If that doesn't work, type  X <return>  to quit.
+
+)
+! Emergency stop.
+<*> pdf_course_profile.sty
+                          
+*** (job aborted, no legal \end found)
+
+ 
+Here is how much of TeX's memory you used:
+ 34 strings out of 93945
+ 379 string characters out of 1162060
+ 47725 words of memory out of 1500000
+ 3410 multiletter control sequences out of 10000+50000
+ 3948 words of font info for 15 fonts, out of 1200000 for 2000
+ 743 hyphenation exceptions out of 8191
+ 12i,1n,9p,132b,55s stack positions out of 5000i,500n,6000p,200000b,5000s
+!  ==> Fatal error occurred, no output PDF file produced!

diff --git a/pdf_course_profile.sty b/pdf_course_profile.sty
new file mode 100644 (file)
index 0000000..a392b6a
--- /dev/null
+++ b/pdf_course_profile.sty
@@ -0,0 +1,166 @@
+\usepackage[pdftex]{graphicx}\r
+\usepackage[body={17cm, 25cm}]{geometry}\r
+\setlength{\headheight}{30pt}         % more space for the head\r
+\r
+\usepackage[french]{babel}    % typographie française\r
+\usepackage[latin1]{inputenc} % entree 8 bits iso-latin1\r
+\usepackage[T1]{fontenc}      % encodage 8 bits des fontes utilisees\r
+\usepackage{verbatim}\r
+%\usepackage{newcent}\r
+%\usepackage{fullpage}         % marges simplifiées\r
+\usepackage{footnote}\r
+\usepackage{graphicx}          % pour les figures\r
+\usepackage{float}                     % pour le positionnement des figures là où on veut\r
+\usepackage{amsmath}           % mathematiques\r
+\usepackage{amsfonts}\r
+%\usepackage{newlfont}\r
+\usepackage{mestheoremes}      % redéfinitions des théorèmes (francisés)\r
+\usepackage[tight,centerlast,sc,small]{subfigure}\r
+\usepackage{latexsym}\r
+\usepackage{url}\r
+\usepackage{setspace}\r
+\usepackage[usenames,dvipsnames]{color}\r
+\DeclareGraphicsExtensions{.jpg,.png,.tif,.tiff,.pdf}\r
+\usepackage{fancyhdr}                    % Fancy Header and Footer\r
+\r
+% style général\r
+\parskip=5pt  % indentation de 0.5 pt\r
+\setcounter{secnumdepth}{3}    % numérotation jusqu'à la subsubsection\r
+%\thispagestyle{empty}\r
+\setcounter{tocdepth}{5}\r
+\r
+% PDF\r
+% PDF Link colors\r
+\definecolor{refcolor}{rgb}{0,0,0.4}    % Color for the reference links\r
+\definecolor{anchorcolor}{rgb}{0,0,0}   % Color for anchor text\r
+\definecolor{citecolor}{rgb}{0,0.4,0.4} % Color for bibligraphical citations\r
+\definecolor{urlcolor}{rgb}{0,0,0.4}   % Color for linked URLs\r
+\definecolor{filecolor}{rgb}{0.75,0,0}  % Color for URLs which open local files\r
+\definecolor{menucolor}{rgb}{0.75,0,0}  % Color for Acrobat menu items\r
+\definecolor{pagecolor}{rgb}{0.75,0,0}  % Color for links to other pages\r
+\r
+\usepackage[pdftex,bookmarksopen,bookmarksnumbered,colorlinks,hyperindex,pagebackref,breaklinks,\r
+pdftitle={Cours de Traitement du signal - CNAM - CPDA},\r
+pdfauthor={Guillaume Pellerin and Maëva Garnier},\r
+pdfsubject={Introduction aux méthodes de traitement du signal},\r
+pdfkeywords={traitement, signal, transformée, Fourier, filtrage, représentation, CNAM, CPDA},\r
+hyperfigures=true,          % provide links to figures\r
+hyperindex=true,           % provide links from index to pages (broken?)\r
+pagebackref=true,           % provide links to pages where bib is cited from\r
+backref=true,               % provide links to sec where bib is cited from\r
+colorlinks=true,            %***\r
+linkcolor=refcolor,         %\r
+anchorcolor=anchorcolor,    %\r
+citecolor=citecolor,        % Colors are defined above;\r
+filecolor=filecolor,        % change them there\r
+menucolor=menucolor,        %\r
+pagecolor=pagecolor,        %\r
+urlcolor=urlcolor,          %***\r
+bookmarksopen=true,        % whether to open bookmarks window on open\r
+pdfpagemode=UseOutlines,    % show the bookmarks in outline form\r
+pdfstartview=FitB,          % default window sizing\r
+pdfview=FitB]{hyperref}     % default view after clicking a link\r
+\r
+    %%% Fancy Header %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
+    % Fancy Header Style Options\r
+\pagestyle{fancy}                       % Sets fancy header and footer\r
+\fancyfoot{}                            % Delete current footer settings\r
+\renewcommand{\chaptermark}[1]{         % Lower Case Chapter marker style\r
+\markboth{\chaptername\ \thechapter.\ #1}{}} %\r
+\renewcommand{\sectionmark}[1]{        % Lower case Section marker style\r
+\markright{\thesection.\ #1}}         %\r
+\fancyhead[LE,RO]{\bfseries\thepage}    % Page number (boldface) in left on even\r
+                                            % pages and right on odd pages\r
+\fancyhead[RE]{\footnotesize{\bfseries\leftmark}}      % Chapter in the right on even pages\r
+\fancyhead[LO]{\footnotesize{\bfseries\rightmark}}     % Section in the left on odd pages\r
+\renewcommand{\headrulewidth}{0.3pt}    % Width of head rule\r
+\r
+%pb odd/even pages with twosides\r
+\setlength{\oddsidemargin}{0cm}\r
+\setlength{\evensidemargin}{\paperwidth}\r
+\addtolength{\evensidemargin}{-\textwidth}\r
+\addtolength{\evensidemargin}{-2.0in}\r
+\addtolength{\evensidemargin}{-\oddsidemargin}\r
+\r
+%Titre\r
+\renewcommand{\maketitle}{\r
+    \begin{titlepage}\r
+    \vspace{5cm}\r
+       \textsf{\@title}\r
+        \centering\r
+        \vspace{1cm}\r
+        \large{\@author}\r
+        \rule{17cm}{0.5mm}\\\r
+        \small{Versions électroniques au format PDF avec hypertexte :}\\\r
+        \footnotesize{\url{http://www.cnam.fr/cpda/docscours/3A/PELLERIN/Signal-CPDA-CNAM.pdf}}\\\r
+               \footnotesize{\url{http://files.parisson.com/CNAM/Signal-CPDA-CNAM.pdf}}\r
+    \end{titlepage}\r
+}\r
+\r
+\r
+% newcommand texte\r
+\newcommand{\be}{\begin{enumerate}}  % raccourci liste numérotée début\r
+\newcommand{\ee}{\end{enumerate}}    % raccourci liste numérotée fin\r
+%\newcommand{\bold}[1]{\textbf{#1}}   % racourci texte en gras\r
+\newcommand{\cad}{c'est-à-dire }\r
+\r
+% newcommand mathématique\r
+\newcommand{\rad}{\mathop{\mathrm{rad}}}\r
+\newcommand{\radn}{\mathop{\mathrm{rad}}\nolimits}\r
+\newcommand{\ud}{\mathrm{d}}            % élément différientiel en caractères romains\r
+\newcommand{\fractext}[2]{\frac{\displaystyle #1}{\displaystyle #2}}\r
+\newcommand{\grad} {\overrightarrow{grad}}    % gradient avec une longue flèche\r
+\newcommand{\boxeq}[1]{\fbox{$\displaystyle{#1}$}}\r
+\r
+%\renewcommand{\floatpagefraction}{0.9}\r
+%\renewcommand{\textfraction}{0.1}\r
+%\setlength{\abovecaptionskip}{0.2cm}\r
+%\setlength{\belowcaptionskip}{0.2cm}\r
+% Different font in captions\r
+\newcommand{\captionfonts}{\small\it}\r
+\r
+\makeatletter  % Allow the use of @ in command names\r
+\long\def\@makecaption#1#2{%\r
+  \vskip\abovecaptionskip\r
+  \sbox\@tempboxa{{\captionfonts #1: #2}}%\r
+  \ifdim \wd\@tempboxa >\hsize\r
+    {\captionfonts #1: #2\par}\r
+  \else\r
+    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%\r
+  \fi\r
+  \vskip\belowcaptionskip}\r
+\makeatother   % Cancel the effect of \makeatletter\r
+\r
+\r
+% macros\r
+\r
+%%%% debut macro page vierge avant chapitre%%%%\r
+% whitecdp (formerly schulzrinne.sty) --provide for blank pages\r
+% between chapters\r
+% This redefinition of the \cleardoublepage command provides\r
+% for a special pagestyle for the "extra" pages which are generated\r
+% to ensure that the chapter opener is on a recto page.\r
+% The pagestyle is "chapterverso"; for many publishers, this should be\r
+% identical to "empty", so that's the default.\r
+\makeatletter\r
+\def\cleardoublepage{\clearpage\r
+ \if@twoside\r
+  \ifodd\c@page\else\r
+   \null\thispagestyle{chapterverso}\newpage\r
+   \if@twocolumn\null\newpage\fi\r
+   \fi\r
+  \fi\r
+ }%\r
+\def\ps@chapterverso{\ps@empty}%\r
+\makeatother\r
+%%%% fin macro %%%%\r
+\r
+% http://www.ctan.org/tex-archive/macros/eplain/\r
+% pour faire une jolie fraction 1/2\r
+\def\fract#1/#2{\leavevmode\r
+   \kern.1em \raise .5ex \hbox{\the\scriptfont0 #1}%\r
+   \kern-.1em $/$%\r
+   \kern-.15em \lower .25ex \hbox{\the\scriptfont0 #2}%\r
+}%\r
+\r
+\r

diff --git a/pdf_course_profile.sty~ b/pdf_course_profile.sty~
new file mode 100644 (file)
index 0000000..1964017
--- /dev/null
+++ b/pdf_course_profile.sty~
@@ -0,0 +1,166 @@
+\usepackage[pdftex]{graphicx}\r
+\usepackage[body={17cm, 25cm}]{geometry}\r
+\setlength{\headheight}{30pt}         % more space for the head\r
+\r
+\usepackage[french]{babel}    % typographie française\r
+\usepackage[latin1]{inputenc} % entree 8 bits iso-latin1\r
+\usepackage[T1]{fontenc}      % encodage 8 bits des fontes utilisees\r
+\usepackage{verbatim}\r
+%\usepackage{newcent}\r
+%\usepackage{fullpage}         % marges simplifiées\r
+\usepackage{footnote}\r
+\usepackage{graphicx}          % pour les figures\r
+\usepackage{float}                     % pour le positionnement des figures là où on veut\r
+\usepackage{amsmath}           % mathematiques\r
+\usepackage{amsfonts}\r
+%\usepackage{newlfont}\r
+\usepackage{mestheoremes}      % redéfinitions des théorèmes (francisés)\r
+\usepackage[tight,centerlast,sc,small]{subfigure}\r
+\usepackage{latexsym}\r
+\usepackage{url}\r
+\usepackage{setspace}\r
+\usepackage[usenames,dvipsnames]{color}\r
+\DeclareGraphicsExtensions{.jpg,.png,.tif,.tiff,.pdf}\r
+\usepackage{fancyhdr}                    % Fancy Header and Footer\r
+\r
+% style général\r
+\parskip=5pt  % indentation de 0.5 pt\r
+\setcounter{secnumdepth}{3}    % numérotation jusqu'à la subsubsection\r
+%\thispagestyle{empty}\r
+\setcounter{tocdepth}{5}\r
+\r
+% PDF\r
+% PDF Link colors\r
+\definecolor{refcolor}{rgb}{0,0,0.4}    % Color for the reference links\r
+\definecolor{anchorcolor}{rgb}{0,0,0}   % Color for anchor text\r
+\definecolor{citecolor}{rgb}{0,0.4,0.4} % Color for bibligraphical citations\r
+\definecolor{urlcolor}{rgb}{0,0,0.4}   % Color for linked URLs\r
+\definecolor{filecolor}{rgb}{0.75,0,0}  % Color for URLs which open local files\r
+\definecolor{menucolor}{rgb}{0.75,0,0}  % Color for Acrobat menu items\r
+\definecolor{pagecolor}{rgb}{0.75,0,0}  % Color for links to other pages\r
+\r
+\usepackage[pdftex,bookmarksopen,bookmarksnumbered,colorlinks,hyperindex,pagebackref,breaklinks,\r
+pdftitle={Cours de Traitement du signal - CNAM - CPDA},\r
+pdfauthor={Guillaume Pellerin and Maëva Garnier},\r
+pdfsubject={Introduction aux méthodes de traitement du signal},\r
+pdfkeywords={traitement, signal, transformée, Fourier, filtrage, représentation, CNAM, CPDA},\r
+hyperfigures=true,          % provide links to figures\r
+hyperindex=true,           % provide links from index to pages (broken?)\r
+pagebackref=true,           % provide links to pages where bib is cited from\r
+backref=true,               % provide links to sec where bib is cited from\r
+colorlinks=true,            %***\r
+linkcolor=refcolor,         %\r
+anchorcolor=anchorcolor,    %\r
+citecolor=citecolor,        % Colors are defined above;\r
+filecolor=filecolor,        % change them there\r
+menucolor=menucolor,        %\r
+pagecolor=pagecolor,        %\r
+urlcolor=urlcolor,          %***\r
+bookmarksopen=true,        % whether to open bookmarks window on open\r
+pdfpagemode=UseOutlines,    % show the bookmarks in outline form\r
+pdfstartview=FitB,          % default window sizing\r
+pdfview=FitB]{hyperref}     % default view after clicking a link\r
+\r
+    %%% Fancy Header %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
+    % Fancy Header Style Options\r
+\pagestyle{fancy}                       % Sets fancy header and footer\r
+\fancyfoot{}                            % Delete current footer settings\r
+\renewcommand{\chaptermark}[1]{         % Lower Case Chapter marker style\r
+\markboth{\chaptername\ \thechapter.\ #1}{}} %\r
+\renewcommand{\sectionmark}[1]{        % Lower case Section marker style\r
+\markright{\thesection.\ #1}}         %\r
+\fancyhead[LE,RO]{\bfseries\thepage}    % Page number (boldface) in left on even\r
+                                            % pages and right on odd pages\r
+\fancyhead[RE]{\footnotesize{\bfseries\leftmark}}      % Chapter in the right on even pages\r
+\fancyhead[LO]{\footnotesize{\bfseries\rightmark}}     % Section in the left on odd pages\r
+\renewcommand{\headrulewidth}{0.3pt}    % Width of head rule\r
+\r
+%pb odd/even pages with twosides\r
+\setlength{\oddsidemargin}{0cm}\r
+\setlength{\evensidemargin}{\paperwidth}\r
+\addtolength{\evensidemargin}{-\textwidth}\r
+\addtolength{\evensidemargin}{-2.0in}\r
+\addtolength{\evensidemargin}{-\oddsidemargin}\r
+\r
+%Titre\r
+\renewcommand{\maketitle}{\r
+    \begin{titlepage}\r
+    \vspace{5cm}\r
+       \textsf{\@title}\r
+        \centering\r
+        \vspace{1cm}\r
+        \large{\@author}\r
+        \rule{17cm}{0.5mm}\\\r
+        \small{Versions électroniques au format PDF avec hypertexte :}\\\r
+        \footnotesize{\url{http://www.cnam.fr/cpda/docscours/3A/PELLERIN/Signal-CPDA-CNAM.pdf}}\\\r
+               \footnotesize{\url{http://files.parisson.com/CNAM/Signal-CPDA-CNAM.pdf}\r
+    \end{titlepage}\r
+}\r
+\r
+\r
+% newcommand texte\r
+\newcommand{\be}{\begin{enumerate}}  % raccourci liste numérotée début\r
+\newcommand{\ee}{\end{enumerate}}    % raccourci liste numérotée fin\r
+%\newcommand{\bold}[1]{\textbf{#1}}   % racourci texte en gras\r
+\newcommand{\cad}{c'est-à-dire }\r
+\r
+% newcommand mathématique\r
+\newcommand{\rad}{\mathop{\mathrm{rad}}}\r
+\newcommand{\radn}{\mathop{\mathrm{rad}}\nolimits}\r
+\newcommand{\ud}{\mathrm{d}}            % élément différientiel en caractères romains\r
+\newcommand{\fractext}[2]{\frac{\displaystyle #1}{\displaystyle #2}}\r
+\newcommand{\grad} {\overrightarrow{grad}}    % gradient avec une longue flèche\r
+\newcommand{\boxeq}[1]{\fbox{$\displaystyle{#1}$}}\r
+\r
+%\renewcommand{\floatpagefraction}{0.9}\r
+%\renewcommand{\textfraction}{0.1}\r
+%\setlength{\abovecaptionskip}{0.2cm}\r
+%\setlength{\belowcaptionskip}{0.2cm}\r
+% Different font in captions\r
+\newcommand{\captionfonts}{\small\it}\r
+\r
+\makeatletter  % Allow the use of @ in command names\r
+\long\def\@makecaption#1#2{%\r
+  \vskip\abovecaptionskip\r
+  \sbox\@tempboxa{{\captionfonts #1: #2}}%\r
+  \ifdim \wd\@tempboxa >\hsize\r
+    {\captionfonts #1: #2\par}\r
+  \else\r
+    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%\r
+  \fi\r
+  \vskip\belowcaptionskip}\r
+\makeatother   % Cancel the effect of \makeatletter\r
+\r
+\r
+% macros\r
+\r
+%%%% debut macro page vierge avant chapitre%%%%\r
+% whitecdp (formerly schulzrinne.sty) --provide for blank pages\r
+% between chapters\r
+% This redefinition of the \cleardoublepage command provides\r
+% for a special pagestyle for the "extra" pages which are generated\r
+% to ensure that the chapter opener is on a recto page.\r
+% The pagestyle is "chapterverso"; for many publishers, this should be\r
+% identical to "empty", so that's the default.\r
+\makeatletter\r
+\def\cleardoublepage{\clearpage\r
+ \if@twoside\r
+  \ifodd\c@page\else\r
+   \null\thispagestyle{chapterverso}\newpage\r
+   \if@twocolumn\null\newpage\fi\r
+   \fi\r
+  \fi\r
+ }%\r
+\def\ps@chapterverso{\ps@empty}%\r
+\makeatother\r
+%%%% fin macro %%%%\r
+\r
+% http://www.ctan.org/tex-archive/macros/eplain/\r
+% pour faire une jolie fraction 1/2\r
+\def\fract#1/#2{\leavevmode\r
+   \kern.1em \raise .5ex \hbox{\the\scriptfont0 #1}%\r
+   \kern-.1em $/$%\r
+   \kern-.15em \lower .25ex \hbox{\the\scriptfont0 #2}%\r
+}%\r
+\r
+\r

diff --git a/pdf_course_profile_utf8.sty b/pdf_course_profile_utf8.sty
new file mode 100644 (file)
index 0000000..3fbc8e5
--- /dev/null
+++ b/pdf_course_profile_utf8.sty
@@ -0,0 +1,166 @@
+\usepackage[pdftex]{graphicx}\r
+\usepackage[body={17cm, 25cm}]{geometry}\r
+\setlength{\headheight}{30pt}         % more space for the head\r
+\r
+\usepackage[french]{babel}    % typographie française\r
+\usepackage[latin1]{inputenc} % entree 8 bits iso-latin1\r
+\usepackage[T1]{fontenc}      % encodage 8 bits des fontes utilisees\r
+\usepackage{verbatim}\r
+%\usepackage{newcent}\r
+%\usepackage{fullpage}         % marges simplifiées\r
+\usepackage{footnote}\r
+\usepackage{graphicx}          % pour les figures\r
+\usepackage{float}                     % pour le positionnement des figures là où on veut\r
+\usepackage{amsmath}           % mathematiques\r
+\usepackage{amsfonts}\r
+%\usepackage{newlfont}\r
+\usepackage{mestheoremes}      % redéfinitions des théorèmes (francisés)\r
+\usepackage[tight,centerlast,sc,small]{subfigure}\r
+\usepackage{latexsym}\r
+\usepackage{url}\r
+\usepackage{setspace}\r
+\usepackage[usenames,dvipsnames]{color}\r
+\DeclareGraphicsExtensions{.jpg,.png,.tif,.tiff,.pdf}\r
+\usepackage{fancyhdr}                    % Fancy Header and Footer\r
+\r
+% style général\r
+\parskip=5pt  % indentation de 0.5 pt\r
+\setcounter{secnumdepth}{3}    % numérotation jusqu'à la subsubsection\r
+%\thispagestyle{empty}\r
+\setcounter{tocdepth}{5}\r
+\r
+% PDF\r
+% PDF Link colors\r
+\definecolor{refcolor}{rgb}{0,0,0.4}    % Color for the reference links\r
+\definecolor{anchorcolor}{rgb}{0,0,0}   % Color for anchor text\r
+\definecolor{citecolor}{rgb}{0,0.4,0.4} % Color for bibligraphical citations\r
+\definecolor{urlcolor}{rgb}{0,0,0.4}   % Color for linked URLs\r
+\definecolor{filecolor}{rgb}{0.75,0,0}  % Color for URLs which open local files\r
+\definecolor{menucolor}{rgb}{0.75,0,0}  % Color for Acrobat menu items\r
+\definecolor{pagecolor}{rgb}{0.75,0,0}  % Color for links to other pages\r
+\r
+\usepackage[pdftex,bookmarksopen,bookmarksnumbered,colorlinks,hyperindex,pagebackref,breaklinks,\r
+pdftitle={Cours de Traitement du signal - CNAM - CPDA},\r
+pdfauthor={Guillaume Pellerin and Maëva Garnier},\r
+pdfsubject={Introduction aux méthodes de traitement du signal},\r
+pdfkeywords={traitement, signal, transformée, Fourier, filtrage, représentation, CNAM, CPDA},\r
+hyperfigures=true,          % provide links to figures\r
+hyperindex=true,           % provide links from index to pages (broken?)\r
+pagebackref=true,           % provide links to pages where bib is cited from\r
+backref=true,               % provide links to sec where bib is cited from\r
+colorlinks=true,            %***\r
+linkcolor=refcolor,         %\r
+anchorcolor=anchorcolor,    %\r
+citecolor=citecolor,        % Colors are defined above;\r
+filecolor=filecolor,        % change them there\r
+menucolor=menucolor,        %\r
+pagecolor=pagecolor,        %\r
+urlcolor=urlcolor,          %***\r
+bookmarksopen=true,        % whether to open bookmarks window on open\r
+pdfpagemode=UseOutlines,    % show the bookmarks in outline form\r
+pdfstartview=FitB,          % default window sizing\r
+pdfview=FitB]{hyperref}     % default view after clicking a link\r
+\r
+    %%% Fancy Header %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
+    % Fancy Header Style Options\r
+\pagestyle{fancy}                       % Sets fancy header and footer\r
+\fancyfoot{}                            % Delete current footer settings\r
+\renewcommand{\chaptermark}[1]{         % Lower Case Chapter marker style\r
+\markboth{\chaptername\ \thechapter.\ #1}{}} %\r
+\renewcommand{\sectionmark}[1]{        % Lower case Section marker style\r
+\markright{\thesection.\ #1}}         %\r
+\fancyhead[LE,RO]{\bfseries\thepage}    % Page number (boldface) in left on even\r
+                                            % pages and right on odd pages\r
+\fancyhead[RE]{\footnotesize{\bfseries\leftmark}}      % Chapter in the right on even pages\r
+\fancyhead[LO]{\footnotesize{\bfseries\rightmark}}     % Section in the left on odd pages\r
+\renewcommand{\headrulewidth}{0.3pt}    % Width of head rule\r
+\r
+%pb odd/even pages with twosides\r
+\setlength{\oddsidemargin}{0cm}\r
+\setlength{\evensidemargin}{\paperwidth}\r
+\addtolength{\evensidemargin}{-\textwidth}\r
+\addtolength{\evensidemargin}{-2.0in}\r
+\addtolength{\evensidemargin}{-\oddsidemargin}\r
+\r
+%Titre\r
+\renewcommand{\maketitle}{\r
+    \begin{titlepage}\r
+    \vspace{5cm}\r
+       \textsf{\@title}\r
+        \centering\r
+        \vspace{1cm}\r
+        \large{\@author}\r
+        \rule{17cm}{0.5mm}\\\r
+        \small{Versions électroniques au format PDF :}\\\r
+        \footnotesize{\url{http://www.cnam.fr/cpda/docscours/3A/PELLERIN/Signal-CPDA-CNAM.pdf}}\\\r
+               \footnotesize{\url{http://files.parisson.com/CNAM/Signal-CPDA-CNAM.pdf}}\r
+    \end{titlepage}\r
+}\r
+\r
+\r
+% newcommand texte\r
+\newcommand{\be}{\begin{enumerate}}  % raccourci liste numérotée début\r
+\newcommand{\ee}{\end{enumerate}}    % raccourci liste numérotée fin\r
+%\newcommand{\bold}[1]{\textbf{#1}}   % racourci texte en gras\r
+\newcommand{\cad}{c'est-à-dire }\r
+\r
+% newcommand mathématique\r
+\newcommand{\rad}{\mathop{\mathrm{rad}}}\r
+\newcommand{\radn}{\mathop{\mathrm{rad}}\nolimits}\r
+\newcommand{\ud}{\mathrm{d}}            % élément différientiel en caractères romains\r
+\newcommand{\fractext}[2]{\frac{\displaystyle #1}{\displaystyle #2}}\r
+\newcommand{\grad} {\overrightarrow{grad}}    % gradient avec une longue flèche\r
+\newcommand{\boxeq}[1]{\fbox{$\displaystyle{#1}$}}\r
+\r
+%\renewcommand{\floatpagefraction}{0.9}\r
+%\renewcommand{\textfraction}{0.1}\r
+%\setlength{\abovecaptionskip}{0.2cm}\r
+%\setlength{\belowcaptionskip}{0.2cm}\r
+% Different font in captions\r
+\newcommand{\captionfonts}{\small\it}\r
+\r
+\makeatletter  % Allow the use of @ in command names\r
+\long\def\@makecaption#1#2{%\r
+  \vskip\abovecaptionskip\r
+  \sbox\@tempboxa{{\captionfonts #1: #2}}%\r
+  \ifdim \wd\@tempboxa >\hsize\r
+    {\captionfonts #1: #2\par}\r
+  \else\r
+    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%\r
+  \fi\r
+  \vskip\belowcaptionskip}\r
+\makeatother   % Cancel the effect of \makeatletter\r
+\r
+\r
+% macros\r
+\r
+%%%% debut macro page vierge avant chapitre%%%%\r
+% whitecdp (formerly schulzrinne.sty) --provide for blank pages\r
+% between chapters\r
+% This redefinition of the \cleardoublepage command provides\r
+% for a special pagestyle for the "extra" pages which are generated\r
+% to ensure that the chapter opener is on a recto page.\r
+% The pagestyle is "chapterverso"; for many publishers, this should be\r
+% identical to "empty", so that's the default.\r
+\makeatletter\r
+\def\cleardoublepage{\clearpage\r
+ \if@twoside\r
+  \ifodd\c@page\else\r
+   \null\thispagestyle{chapterverso}\newpage\r
+   \if@twocolumn\null\newpage\fi\r
+   \fi\r
+  \fi\r
+ }%\r
+\def\ps@chapterverso{\ps@empty}%\r
+\makeatother\r
+%%%% fin macro %%%%\r
+\r
+% http://www.ctan.org/tex-archive/macros/eplain/\r
+% pour faire une jolie fraction 1/2\r
+\def\fract#1/#2{\leavevmode\r
+   \kern.1em \raise .5ex \hbox{\the\scriptfont0 #1}%\r
+   \kern-.1em $/$%\r
+   \kern-.15em \lower .25ex \hbox{\the\scriptfont0 #2}%\r
+}%\r
+\r
+\r

diff --git a/pdf_course_profile_utf8.sty~ b/pdf_course_profile_utf8.sty~
new file mode 100644 (file)
index 0000000..cebd270
--- /dev/null
+++ b/pdf_course_profile_utf8.sty~
@@ -0,0 +1,165 @@
+\usepackage[pdftex]{graphicx}\r
+\usepackage[body={17cm, 25cm}]{geometry}\r
+\setlength{\headheight}{30pt}         % more space for the head\r
+\r
+\usepackage[french]{babel}    % typographie française\r
+\usepackage[latin1]{inputenc} % entree 8 bits iso-latin1\r
+\usepackage[T1]{fontenc}      % encodage 8 bits des fontes utilisees\r
+\usepackage{verbatim}\r
+%\usepackage{newcent}\r
+%\usepackage{fullpage}         % marges simplifiées\r
+\usepackage{footnote}\r
+\usepackage{graphicx}          % pour les figures\r
+\usepackage{float}                     % pour le positionnement des figures là où on veut\r
+\usepackage{amsmath}           % mathematiques\r
+\usepackage{amsfonts}\r
+%\usepackage{newlfont}\r
+\usepackage{mestheoremes}      % redéfinitions des théorèmes (francisés)\r
+\usepackage[tight,centerlast,sc,small]{subfigure}\r
+\usepackage{latexsym}\r
+\usepackage{url}\r
+\usepackage{setspace}\r
+\usepackage[usenames,dvipsnames]{color}\r
+\DeclareGraphicsExtensions{.jpg,.png,.tif,.tiff,.pdf}\r
+\usepackage{fancyhdr}                    % Fancy Header and Footer\r
+\r
+% style général\r
+\parskip=5pt  % indentation de 0.5 pt\r
+\setcounter{secnumdepth}{3}    % numérotation jusqu'à la subsubsection\r
+%\thispagestyle{empty}\r
+\setcounter{tocdepth}{5}\r
+\r
+% PDF\r
+% PDF Link colors\r
+\definecolor{refcolor}{rgb}{0,0,0.4}    % Color for the reference links\r
+\definecolor{anchorcolor}{rgb}{0,0,0}   % Color for anchor text\r
+\definecolor{citecolor}{rgb}{0,0.4,0.4} % Color for bibligraphical citations\r
+\definecolor{urlcolor}{rgb}{0,0,0.4}   % Color for linked URLs\r
+\definecolor{filecolor}{rgb}{0.75,0,0}  % Color for URLs which open local files\r
+\definecolor{menucolor}{rgb}{0.75,0,0}  % Color for Acrobat menu items\r
+\definecolor{pagecolor}{rgb}{0.75,0,0}  % Color for links to other pages\r
+\r
+\usepackage[pdftex,bookmarksopen,bookmarksnumbered,colorlinks,hyperindex,pagebackref,breaklinks,\r
+pdftitle={Cours de Traitement du signal - CNAM - CPDA},\r
+pdfauthor={Guillaume Pellerin and Maëva Garnier},\r
+pdfsubject={Introduction aux méthodes de traitement du signal},\r
+pdfkeywords={traitement, signal, transformée, Fourier, filtrage, représentation, CNAM, CPDA},\r
+hyperfigures=true,          % provide links to figures\r
+hyperindex=true,           % provide links from index to pages (broken?)\r
+pagebackref=true,           % provide links to pages where bib is cited from\r
+backref=true,               % provide links to sec where bib is cited from\r
+colorlinks=true,            %***\r
+linkcolor=refcolor,         %\r
+anchorcolor=anchorcolor,    %\r
+citecolor=citecolor,        % Colors are defined above;\r
+filecolor=filecolor,        % change them there\r
+menucolor=menucolor,        %\r
+pagecolor=pagecolor,        %\r
+urlcolor=urlcolor,          %***\r
+bookmarksopen=true,        % whether to open bookmarks window on open\r
+pdfpagemode=UseOutlines,    % show the bookmarks in outline form\r
+pdfstartview=FitB,          % default window sizing\r
+pdfview=FitB]{hyperref}     % default view after clicking a link\r
+\r
+    %%% Fancy Header %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r
+    % Fancy Header Style Options\r
+\pagestyle{fancy}                       % Sets fancy header and footer\r
+\fancyfoot{}                            % Delete current footer settings\r
+\renewcommand{\chaptermark}[1]{         % Lower Case Chapter marker style\r
+\markboth{\chaptername\ \thechapter.\ #1}{}} %\r
+\renewcommand{\sectionmark}[1]{        % Lower case Section marker style\r
+\markright{\thesection.\ #1}}         %\r
+\fancyhead[LE,RO]{\bfseries\thepage}    % Page number (boldface) in left on even\r
+                                            % pages and right on odd pages\r
+\fancyhead[RE]{\footnotesize{\bfseries\leftmark}}      % Chapter in the right on even pages\r
+\fancyhead[LO]{\footnotesize{\bfseries\rightmark}}     % Section in the left on odd pages\r
+\renewcommand{\headrulewidth}{0.3pt}    % Width of head rule\r
+\r
+%pb odd/even pages with twosides\r
+\setlength{\oddsidemargin}{0cm}\r
+\setlength{\evensidemargin}{\paperwidth}\r
+\addtolength{\evensidemargin}{-\textwidth}\r
+\addtolength{\evensidemargin}{-2.0in}\r
+\addtolength{\evensidemargin}{-\oddsidemargin}\r
+\r
+%Titre\r
+\renewcommand{\maketitle}{\r
+    \begin{titlepage}\r
+    \vspace{5cm}\r
+       \textsf{\@title}\r
+        \centering\r
+        \vspace{1cm}\r
+        \large{\@author}\r
+        \rule{17cm}{0.5mm}\\\r
+        \small{Version électronique type PDF avec hypertexte :}\\\r
+        \footnotesize{\url{http://www.cnam.fr/cpda/docscours/3A/PELLERIN/Signal-CPDA-CNAM.pdf}}\r
+    \end{titlepage}\r
+}\r
+\r
+\r
+% newcommand texte\r
+\newcommand{\be}{\begin{enumerate}}  % raccourci liste numérotée début\r
+\newcommand{\ee}{\end{enumerate}}    % raccourci liste numérotée fin\r
+%\newcommand{\bold}[1]{\textbf{#1}}   % racourci texte en gras\r
+\newcommand{\cad}{c'est-à-dire }\r
+\r
+% newcommand mathématique\r
+\newcommand{\rad}{\mathop{\mathrm{rad}}}\r
+\newcommand{\radn}{\mathop{\mathrm{rad}}\nolimits}\r
+\newcommand{\ud}{\mathrm{d}}            % élément différientiel en caractères romains\r
+\newcommand{\fractext}[2]{\frac{\displaystyle #1}{\displaystyle #2}}\r
+\newcommand{\grad} {\overrightarrow{grad}}    % gradient avec une longue flèche\r
+\newcommand{\boxeq}[1]{\fbox{$\displaystyle{#1}$}}\r
+\r
+%\renewcommand{\floatpagefraction}{0.9}\r
+%\renewcommand{\textfraction}{0.1}\r
+%\setlength{\abovecaptionskip}{0.2cm}\r
+%\setlength{\belowcaptionskip}{0.2cm}\r
+% Different font in captions\r
+\newcommand{\captionfonts}{\small\it}\r
+\r
+\makeatletter  % Allow the use of @ in command names\r
+\long\def\@makecaption#1#2{%\r
+  \vskip\abovecaptionskip\r
+  \sbox\@tempboxa{{\captionfonts #1: #2}}%\r
+  \ifdim \wd\@tempboxa >\hsize\r
+    {\captionfonts #1: #2\par}\r
+  \else\r
+    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%\r
+  \fi\r
+  \vskip\belowcaptionskip}\r
+\makeatother   % Cancel the effect of \makeatletter\r
+\r
+\r
+% macros\r
+\r
+%%%% debut macro page vierge avant chapitre%%%%\r
+% whitecdp (formerly schulzrinne.sty) --provide for blank pages\r
+% between chapters\r
+% This redefinition of the \cleardoublepage command provides\r
+% for a special pagestyle for the "extra" pages which are generated\r
+% to ensure that the chapter opener is on a recto page.\r
+% The pagestyle is "chapterverso"; for many publishers, this should be\r
+% identical to "empty", so that's the default.\r
+\makeatletter\r
+\def\cleardoublepage{\clearpage\r
+ \if@twoside\r
+  \ifodd\c@page\else\r
+   \null\thispagestyle{chapterverso}\newpage\r
+   \if@twocolumn\null\newpage\fi\r
+   \fi\r
+  \fi\r
+ }%\r
+\def\ps@chapterverso{\ps@empty}%\r
+\makeatother\r
+%%%% fin macro %%%%\r
+\r
+% http://www.ctan.org/tex-archive/macros/eplain/\r
+% pour faire une jolie fraction 1/2\r
+\def\fract#1/#2{\leavevmode\r
+   \kern.1em \raise .5ex \hbox{\the\scriptfont0 #1}%\r
+   \kern-.1em $/$%\r
+   \kern-.15em \lower .25ex \hbox{\the\scriptfont0 #2}%\r
+}%\r
+\r
+\r

diff --git a/python/filter_ex.py b/python/filter_ex.py
new file mode 100644 (file)
index 0000000..ec9b014
--- /dev/null
+++ b/python/filter_ex.py
@@ -0,0 +1,27 @@
+# -*- coding: utf-8 -*-
+from scipy.signal import lfilter
+from numpy import *
+import matplotlib.pyplot as plt
+
+fe = 44100
+
+x = ones(64)
+
+x[:16] = 0.0
+x[48:] = 0.0
+
+
+fc = 5000
+b = 1/(1+fe/(2*pi*fc))
+a = b-1.0
+a = array([ 1.0, a ])
+
+# low pass
+b = array([ b ])
+
+y = lfilter(b, a, x)
+
+plt.plot(y, marker='o', color='r')
+plt.plot(x, marker='x')
+plt.show()
+

diff --git a/signal-cpda.kilepr b/signal-cpda.kilepr
new file mode 100644 (file)
index 0000000..e02a0d2
--- /dev/null
+++ b/signal-cpda.kilepr
@@ -0,0 +1,171 @@
+[General]
+def_graphic_ext=
+img_extIsRegExp=false
+img_extensions=.eps .jpg .jpeg .png .pdf .ps .fig .gif .dvi
+kileprversion=2
+kileversion=2.0.85
+lastDocument=SystLinFilt.tex
+masterDocument=
+name=signal-cpda
+pkg_extIsRegExp=false
+pkg_extensions=.cls .sty .bbx .cbx .lbx
+src_extIsRegExp=false
+src_extensions=.tex .ltx .latex .dtx .ins .bib .mp
+
+[Tools]
+MakeIndex=
+QuickBuild=
+
+[document-settings,item:Abstract.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[document-settings,item:Fiche_TZ.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[document-settings,item:IntroMethFreq.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[document-settings,item:ReprSignaux.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[document-settings,item:Signal-CPDA-CNAM.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[document-settings,item:Signaux.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[document-settings,item:SystLinFilt.tex]
+Bookmarks=
+Encoding=ISO-8859-15
+Highlighting=LaTeX
+Indentation Mode=python
+Mode=LaTeX
+ReadWrite=true
+
+[item:Abstract.tex]
+archive=true
+column=0
+encoding=ISO-8859-15
+highlight=LaTeX
+line=0
+mode=LaTeX
+open=true
+order=1
+
+[item:Fiche_TZ.tex]
+archive=true
+column=0
+encoding=ISO-8859-15
+highlight=LaTeX
+line=0
+mode=LaTeX
+open=true
+order=0
+
+[item:IntroMethFreq.tex]
+archive=true
+column=12
+encoding=ISO-8859-15
+highlight=LaTeX
+line=273
+mode=LaTeX
+open=true
+order=2
+
+[item:ReprSignaux.tex]
+archive=true
+column=0
+encoding=ISO-8859-15
+highlight=LaTeX
+line=0
+mode=LaTeX
+open=true
+order=3
+
+[item:Signal-CPDA-CNAM.tex]
+archive=true
+column=24
+encoding=ISO-8859-15
+highlight=LaTeX
+line=14
+mode=LaTeX
+open=true
+order=5
+
+[item:Signaux.tex]
+archive=true
+column=39
+encoding=ISO-8859-15
+highlight=LaTeX
+line=86
+mode=LaTeX
+open=true
+order=6
+
+[item:SystLinFilt.tex]
+archive=true
+column=0
+encoding=ISO-8859-15
+highlight=LaTeX
+line=142
+mode=LaTeX
+open=true
+order=4
+
+[view-settings,view=0,item:Abstract.tex]
+CursorColumn=0
+CursorLine=0
+
+[view-settings,view=0,item:Fiche_TZ.tex]
+CursorColumn=0
+CursorLine=0
+
+[view-settings,view=0,item:IntroMethFreq.tex]
+CursorColumn=12
+CursorLine=273
+
+[view-settings,view=0,item:ReprSignaux.tex]
+CursorColumn=0
+CursorLine=0
+
+[view-settings,view=0,item:Signal-CPDA-CNAM.tex]
+CursorColumn=24
+CursorLine=14
+
+[view-settings,view=0,item:Signaux.tex]
+CursorColumn=39
+CursorLine=86
+
+[view-settings,view=0,item:SystLinFilt.tex]
+CursorColumn=0
+CursorLine=142

diff --git a/tmp.tex b/tmp.tex
new file mode 100644 (file)
index 0000000..80f3513
--- /dev/null
+++ b/tmp.tex
@@ -0,0 +1,581 @@
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+X(f) est une fonction complexe même si x(t) est réel. |X(f)| est appelé 'spectre d'amplitude' et Arg[X(f)] ='spectre de phase' du signal. 
+La variable f s'appelle la fréquence. Son unité est le Hertz (en abrégé : Hz).
+
+
+La Transformée de Fourier est un opérateur mathématique qui permet  d'analyser et de représenter un signal dans le domaine fréquentiel. La TF ne modifie pas le signal mais permet seulement de l'observer selon différents points de vue.(temporel ou fréquentiel). Il est important de retenir que x(t) et X(f) sont deux descriptions équivalentes du même signal. Ces deux fonctions contiennent la même information. Il s'agit juste de deux descriptions dans des domaines différents (temporel et fréquentiel).
+
+X(f)  apporte des information sur le système physique à l'origine du signal. Elle permet par exemple de différentier un son de trompette d'un son trombone, ou bien encore différentes ondes cérébrales , mieux qu'en observant le signal dans le domaine temporel.
+
+
+
+
+
+
+
+
+\subsection{Le produit de convolution}
+
+On appelle convolution de x(t) par y(t) l'opération notée x(t) * y(t) et définie par :
+
+
+L'impulsion de Dirac est l'élément neutre de la convolution, c'est à dire que 
+x(t) * ' (t) = x(t).
+
+
+Lorsque l'on convolue un signal x(t) à un Dirac en to par exemple, cela revient à retarder le signal x(t) de to :           x(t) * ' (t-to) = x(t-to).
+
+Alors que si l'on multiplie un signal x(t) par un Dirac en to, cela revient à connaître la valeur que prend x(t) en to (comme si l'on relevait l'ordonnée d'un point particulier d'une courbe)                  x(t) . ' (t-to) = x(to).               
+
+De même, lorsque l'on convolue un signal x(t) à un peigne de Dirac (de période T), cela revient à « périodiser »  le signal x(t) tous les nT : on retarde le signal x(t) de T , de 2T, de 3T'.etc                       x(t) * 'T  (t)  =  ' x(t-nT)
+
+Tandis que si on multiplie le signal x(t) par un peigne de Dirac, on prélève des échantillons du signal x(t) tous les nT. Nous verrons plus loin que cette opération correspond à l' «échantillonnage»  du signal x(t).
+                                       x(t) * 'T  (t)  =  ' x(nT)
+
+De façon plus générale, la convolution telle qu'elle est définie par sa formule mathématique, revient à retourner temporellement un des deux signaux (par exemple x(t) )  puis à le déplacer sur tout l'axe du temps  et à sommer toutes les multiplications de ce signal retourné déplacé au deuxième signal ( y(t) ) .
+
+
+
+
+
+
+
+
+
+
+
+Retournement temporel         t est un instant qui varie du signal x               dans le temps u de  -'  à  +'.
+  On fait « glisser » le signal 
+  x(-u) selon l'axe du temps.
+
+
+        
+
+
+
+
+
+
+
+
+
+On multiplie y(u) par x(t-u)  pour chaque position de t sur l'axe du temps et on somme tous les produits effectués.  
+
+
+
+
+
+
+
+
+
+
+
+
+
+\subsection{Propriétés de la transformée de Fourier}
+
+Linéarité de la TF :  a x(t)+b y(t)     '   a X(f)+b Y(f)
+
+Produit de convolution :
+x(t).y(t)     '   X(f) * Y(f)
+x(t) * y(t)   '   X(f) . Y(f)   
+
+Une multiplication dans un domaine correspond à un produit de convolution dans l'autre. 
+
+
+
+Retard : 
+
+ 
+
+
+
+x(t-to)    '    X(f) e-i2pfto
+
+x(t).ei2pfot      '  X(f-fo)
+               
+
+
+
+Un retard UU<u
+Un retard temporel correspond à un déphasage au niveau fréquentiel, et inversement.
+
+Changement d'échelle :   x(at)   '  1/a  *  X(f/a)    
+
+
+
+Théorème de Parseval :
+
+$Énergie      =   \vert x(t) \vert^2 dt   =       \vert X(f) \vert^2 df$
+
+
+
+
+\subsection{Transformées de Fourier des signaux courants}
+
+Dirac (t)   '  signal unité
+
+Signal porte '  sinus cardinal
+
+Sin(2 Pi fo t)   '   ' ( dirac (f-fo)  + dirac (f+fo)
+
+
+A complèter avec vos notes de cours'. 
+
+
+
+
+
+
+\section{Evaluation de la similitude entre deux signaux}
+
+Il existe en traitement du signal des outils mathématiques qui permettent d'évaluer la similitude entre deux signaux, 
+-au niveau de leur évolution temporelle grâce  à la fonction d'inter- ou d'auto-corrélation.
+-au niveau de leur composition spectrale grâce à la fonction de densité spectrale. 
+
+\subsection{Les fonctions d'inter- et d'auto-corrélation}
+
+
+La  fonction d'intercorrélation se définit par la formule suivante :
+
+
+
+
+
+
+
+La fonction d'autocorrélation est un cas particulier de la fonction d'intercorrélation pour y(t) = x(t). Elle se définit par :
+
+
+
+
+
+
+La fonction d'autocorrélation mesure la similitude de x(t) avec une version décallée de x(t). Elle atteint un maximum pour le temps to auquel x(t-to) ressemble le plus à x(t).  C'est le cas particulièrement pour les signaux périodiques qui reprennent la même valeur à chaque période T. La fonction d'autocorrélation permet ainsi d'estimer la période d'un signal périodique en repérant le temps pour lequel elle atteint son maximum.
+
+La fonction d'autocorrélation permet également de calculer l'énergie du signal  puisque 
+
+
+
+
+
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La Fonction de densité spectrale d'énergie est la TF de la fonction d'autocorrélation. Elle décrit la répartition de l'énergie en fréquence.
+
+
+
+
+
+
+
+
+\section{Les signaux numériques}
+
+Les signaux numériques dérivent d 'un signal analogique par échantillonnage. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants tn, n étant entier. Dans la suite nous n'envisagerons que l'échantillonnage dit régulier où tn = nT. L'intérêt porte aux problèmes de l'échantillonnage tient dans le développement des techniques numériques de traitement du signal. En Physique il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}
+
+Le signal continu (analogique) peut être échantillonné (ou discrétisé) à une fréquence d'échantillonnage Fe  (= 1/Te). 
+Cette échantillonage revient à multiplier le signal analogique par un peigne de Dirac de période Te.
+
+
+
+
+
+
+
+
+
+Le signal échantillonné peut alors s'écrire :
+
+
+
+                                
+
+
+
+xe(t) est donc nul partout sauf en t= nTe, n variant de -''à'''''
+C'est pourquoi par la suite, on ne simplifie cette notation en réduisant xe(t) non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets x(n), où n correspond au numéro de l'échantillon.
+
+x(n) = xe ( n Te )
+
+
+\subsection{Définition et propriétés de la transformée de Fourier d'un signal échantillonné}
+Dans le cas des signaux échantillonnés, il n'est plus nécessaire d'utiliser des intégrales pour sommer les valeurs de x(t) sur tout l'axe des temps, puisqu'un signal échantillonné peut être assimilé à une suite d'éléments discrets.
+On peut donc définir la transformée de Fourier d'un signal numérique par :
+
+De la même façon qu'en continu, la TF présente certaines propriétés intéressantes :
+
+retard:   x(n-no)  '   X(f)   e-j 2'fno
+produit de convolution:   x(n) * y(n)  '   X(f) . Y(f) 
+        x(n) . y(n)  '   X(f) * Y(f) 
+
+        avec     x(n)  * y(n)  =  '''x(k) y(n-k)    =     '''x(n-k) y(k)     .
+
+\subsection{Conséquences de l'échantillonnage du point de vue fréquentiel} 
+
+En écrivant  xe(t) = xa(t) . ''''(t- nTe), on peut alors exprimer la transformée de Fourier du signal échantillonné à partir de la TF du signal analogique.
+
+
+
+On voit que l'échantillonnage d'un signal analogique (à la fréquence d'échantillonnage Fe = 1/T) induit une périodisation de son spectre dans le domaine fréquentiel, tous les f =n/ T, n étant entier.
+
+
+
+
+Il peut survenir un problème si la fréquence d'échantillonnage Fe = 1/T est trop petite car les « répliques » périodiques du spectre  peuvent se superposer partiellement .
+
+
+
+
+
+
+
+
+Cela arrive si   borne max (Xa(f))  >    borne inf  (1ere réplique de Xa(f) en f =1/T)
+Autrement dit si               B        >      1/T  - B
+
+
+Pour que le spectre Xa(f) ne soit pas « déformé » lors de sa périodisation, il faut donc que :
+
+                       Fe < 2 B
+ 
+Cette condition est appelée Théorème de Shannon.
+
+
+
+\chapter{Modes de représentation des signaux}
+
+
+\section{Fenêtrage des signaux} 
+
+Nous avons vu dans le chapitre précédent que la Transformée de Fourier permet de représenter le spectre d'un signal dans le domaine fréquentiel. Pour une sinusoïde infinie, toute l'énergie du spectre est concentrée à une fréquence donnée, c'est à dire la fréquence de la sinusoïde. 
+
+
+
+
+Si on veut observer ce sinus sur un temps fini, on va être obligé de le tronquer. L'énergie va alors d'une part se répartir autour de la fréquence de la sinusoïde (c'est ce qu'on appelle l'étalement spectral), d'autre part on observe la présence d'énergie dans toutes les fréquences (Ce phénomène est appelé fuite spectrale). Etalement et fuite sont liés à la troncature.
+
+
+
+Si on réalise la troncature de façon non rectangulaire, mais en « fenêtrant » le signal par une fenêtre de Hanning par exemple, les transitions dans le signal sont alors plus douces. Cela limite la fuite spectrale mais en revanche augmente l'étalement en fréquence. Ce point est important pour comprendre le rôle des fenêtres d'analyse. Si la fenêtre a des discontinuités fortes, les fuites spectrales vont être importantes, mais l'étalement moindre. Si on prend un fenêtre de discontinuité plus douce, on va au contraire obtenir un étalement plus grand, mais moins de fuites spectrales.
+
+
+
+\section{Le spectrogramme}
+
+L'intérêt du spectrogramme est de pouvoir représenter le spectre en évoluant dans le temps. Le nom scientifique de la fonction mathématique associée à cet outil, plus communément appelé « spectrogramme », est la Transformée de Fourier à Court Terme (TFCT). Ce nom provient de l'analyse effectuée sur des fenêtres de support temporel fini. Une autre dénomination de cette représentation est « sonagramme ». Il s'agit d'une marque déposée Kay Electronics.
+
+Le principe du spectrogramme est de « découper » le son en tranches ou trames qui se recouvrent. Pour chacune de ces tranches on calcule une transformée de Fourier. Ce spectre est alors représenté à un temps correspondant à celui du centre de la fenêtre, sous forme d'un code de couleur. Sur l'exemple suivant, le jaune correspond aux amplitudes les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a ainsi une idée de l'aspect du spectre au temps t. A chaque calcul du spectre, le signal est fenêtré de façon à pouvoir régler à la fois la fuite et l'étalement spectral. 
+
+
+Voici un exemple de voix chantée. Il s'agit d'un glissando C5-E5 réalisé par une soprano. L'analyse a été effectuée avec une fenêtre de Hanning de longueur 23 ms. Sur le spectrogramme, on observe bien le glissando et le vibrato de la chanteuse.
+
+
+Pour mesurer le vibrato par exemple, on serait tenté de réduire la longueur de la fenêtre dans le temps pour gagner en précision et suivre au mieux les variations du spectre. En réalité, si on réduit la longueur des fenêtres, l'étalement spectral augmente, par conséquent la largeur des raies sur le spectrogramme aussi, ce qui perturbe finalement la mesure, car on ne distingue plus distinctement les différentes trajectoires dans le spectrogramme.
+
+
+
+
+
+
+
+
+
+
+
+
+Si on revient à la même longueur de fenêtre que dans le premier exemple, tout en utilisant une fenêtre rectangulaire au lieu de la fenêtre « douce » de Hanning, l'étalement spectral est plus faible et les lignes sur le spectrogramme plus fines, mais on a par contre des fuites spectrales beaucoup plus importantes, caractérisées par un manque de contraste dans la représentation du spectrogramme. 
+
+
+Les points importants quant à l'utilisation du spectrogramme sont donc: 
+la longueur de fenêtre pour ajuster la précision temporelle, au prix d'un étalement spectral qui peut devenir rédhibitoire
+le choix de la fenêtre qui va conditionner le contraste du spectrogramme, pour une longueur de fenêtre donnée.
+
+\section{Pertinence et choix d'un mode de représentation}
+
+La représentation du signal en temporel est particulièrement adapté à l'analyse de ses variations temporelles  (par exemple l'attaque ou l'extinction d'un son).
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Au contraire, cette représentation apporte peu d'information lorsqu'on s'intéresse plutôt au contenu spectral du signal. Sur la figure suivante, on voit que la représentation spectrale de ces deux signaux de parole permet assez bien de différentier la voix « tendue »  de la voix « relâchée ».
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Voix relâchée                                          Voix tendue
+
+
+Un autre intérêt de la représentation du signal à l'aide d'un spectrogramme est de pouvoir séparer des sources sonores, ou du moins de pouvoir repérer des événements, là où le signal temporel englobe tout de façon indifférenciée.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Pour des sons très harmoniques (par exemple voix chantée), le spectrogramme permet de repérer très facilement la fondamentale ainsi que la série harmonique.
+Sur la figure ci-dessous est représenté le spectrogramme d'un chanteur effectuant un glissando du bas au haut de sa tessiture puis l'inverse.
+
+
+
+
+
+
+
+
+
+
+
+
+Dans le cas de la voix parlée, on peut repérer les parties voisée (harmoniques) ou non, et même « lire » quels sont les phonèmes prononcés avec un peu d'expérience. Nous reviendrons là dessus dans le chapitre 5.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+   « C'est une maison qui s'trouve au Luxembourg »
+
+
+\section{Le spectre moyenné (Long Term Average Spectrum   LTAS)}
+
+Il est également très utile dans certains cas de représenter une partie du signal sous forme de spectre moyenné. Le spectre moyenné, comme son nom l'indique, consiste à faire la moyenne du spectrogramme sur l'ensemble des instants d'un intervalle de temps plus ou moins long . Ce peut être sur le signal tout entier comme sur un morceau particulier.
+
+Pour examiner une voyelle en particulier dans un signal de parole par exemple, il est tout à fait approprié de moyenner le spectrogramme sur l'intervalle de temps de cette voyelle, de façon à pouvoir estimer les formants ou bien le contenu spectral de cette voyelle. Ainsi, sur la figure suivante, on peut examiner le spectre moyenné de la production vocale d'un chanteur sur un /a/. Cette représentation permet de caractériser ses premiers formants vocaliques, ainsi que son formant du chanteur (renforcement énergétique dans la zone 2000-4000 Hz), caractéristique des chanteurs lyriques et des acteurs de théâtre.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\chapter{Systèmes linéaires et filtrage}
+
+
+
+
+
+\section{Réponse impulsionnelle d'un système}
+
+Il est possible de décrire un système S par l'intermédiaire de sa réponse à un signal extrêmement bref. Mathématiquement, il s'agit de modéliser sa réponse à une impulsion de Dirac.
+
+La réponse impulsionnelle d'un système est donc définie comme la sortie d'un système lorsque l'entrée est une impulsion de Dirac.
+
+g(t) = S{d(t)}
+
+Lorsque nous claquons des mains dans une pièce pour évaluer son acoustique, son temps de réverbération, etc, nous ne faisons pas moins en fait qu'évaluer la réponse impulsionnelle de ce système pièce. 
+Cela semble physiquement logique d'entendre du son dans la pièce au moment où nous claquons dans nos mains, voire après, mais certainement pas avant . Pourtant, en mathématique, on conçoit que la réponse impulsionnelle puisse être non nulle avant l'instant de l'impulsion d'entrée.
+On définit ainsi les système causaux comme les systèmes dont la réponse impulsionnelle est nulle avant l'instant d'impulsion, autrement dit      g(t<0) = 0  .
+Pour un système causal, le signal de sortie à l'instant t dépend du signal d'entrée aux instants t'< t. La durée de la réponse impulsionnelle g(t) correspond au temps de réponse du système.
+
+
+
+Nous avons vu que la réponse impulsionnelle correspond à la sortie y(t) du système lorsque l'entrée x(t) = '(t) .
+Connaissant la réponse impulsionnelle g(t) d'un système, on peut ensuite prédire sa réponse y(t) à n'importe quel signal d'entrée x(t) grâce à la formule :
+
+                                    y(t) = x(t) * g(t)
+
+
+
+
+
+
+
+
+
+
+
+
+\section{Réponse en fréquence d'un système}
+
+La réponse en fréquence d'un système correspond à la transformée de Fourier de la réponse impulsionnelle du système.
+
+
+G(f) décrit comment la distribution spectrale d'un signal est modifiée ou "filtrée" par le système S. Il est important de noter que le système peut seulement modifier des composantes spectrales mais ne peut en aucun cas en créer de nouvelles.
+|G(f)|  est le gain du système, c'est à dire la façon dont il modifie les amplitudes de chaque composantes spectrales.
+Arg[G(f)] représente le déphasage causé par le système, c'est à dire le « retard » qu'il impose à certaines composantes spectrales.
+
+
+
+
+
+
+
+
+
+
+
+
+
+La réponse en fréquence, comme la réponse impulsionnelle permet de décrire complètement le système et de prédire la réponse du système à n'importe quelle entrée. 
+
+y(t) = x(t) * g(t)    '     Y(f)  = X(f) . G(f)
+
+
+
+\section{Fonction de transfert d'un système (et définition de la Transformée en Z)}
+
+La fonction de transfert d'un système est défini comme la transformée en Z de la réponse impulsionnelle du système.
+
+
+La Transformée en Z est très proche de la Transformée de Fourier et se définit par :
+
+La TZ est un outil de caractérisation des systèmes de traitement numérique des signaux. 
+On retrouve la définition de la TF en posant z = ei2' f.
+
+La TZ présente aussi comme propriété :
+
+* Retard :                     x(k 'ko)     '     X(z) z-ko
+
+*Convolution  :        x(k)*y(k)    '     X(z).Y(z)
+                       x(k).y(k)    '     X(z)*Y(z)
+                       
+
+Donc la réponse impulsionnelle g(n) et la fonction de transfert G(z) permettent de décrire un système numérique.
+
+y(n) = x(n) * g(n)    '     Y(z)  = X(z) . G(z)
+
+
+
+
+
+
+\section{Filtrage}
+
+Le filtrage linéaire est une opération utile pour sélectionner une partie du spectre d'un signal ou en atténuer une partie, pour 
+modifier le contenu spectral d'un signal  ou encore pour m
+odéliser la transformation d'un signal dans un conduit, dans une pièce' 
+
+\subsection{Filtre idéaux}
+On définit classiquement 4 types de filtres idéaux : 
+les filtres passe-bas qui laissent intact les basses fréquences d'un signal et en atténuent les hautes fréquences.
+les filtres passe-haut qui laissent intact les hautes fréquences d'un signal et en atténuent les basses fréquences.
+Les filtres passe-bande qui sélectionnent une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.
+Les filtres coupe-bande, qui atténuent fortement une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.
+
+
+Pour tous ces filtres, on définit des fréquences de coupure, c'est à dire les fréquences pour lesquelles le spectre du signal d'entrée va être atténué d'un facteur '2. 
+
+
+Le passe bas à donc une fréquence de coupure dans les médiums / hautes fréquences, le passe haut une fréquence de coupure dans les médiums / basses fréquences. Les passe bande et coupe bande possèdent deux fréquences de coupure autour de la fréquence centrale sur laquelle ils se centrent. 
+On spécifie également la pente de l'atténuation de ces filtres, en dB par octave. Cela apporte une information sur la sélectivité du filtre.
+
+Enfin, pour les filtres passe-bande et coupe-bande, on détermine leur largeur de bande, c'est à dire la différence entre leurs deux fréquences de coupure, ce qui renseigne sur le caractère plus ou moins « aigu » du filtre.
+
+
+\subsection{Exemples de filtres naturels} 
+
+On peut modéliser la sensation d'intensité auditive par un filtre. L'oreille humaine est particulièrement sensible entre 3 et 4 kHz 
+
+On peut également modéliser la production phonatoire par un modèle source-filtre, où le rôle de la source est joué par les cordes vocales qui produisent un son harmonique avec une distribution de l'énergie assez plate en fréquence. Le conduit vocal, les fosses nasales ainsi que la place des articulateurs (langue, mâchoire, lèvres) peuvent être modélisés par un filtre qui modifie le son glottique pour produire le son tel que nous le percevons à la sortie des lèvres d'un locuteur.
+
+
+Bibliographie
+
+David B. Conférence SFA sur le vibrato. Décembre 2003. Le mans.
+
+Garnier M, Dubois D, Poitevineau J, Henrich N, Castellengo M. Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive. Journées d'Études sur la Parole, Fez, 2004. . 
+
+Henrich N. Etude de la source glottique en voix parlée et chantée. Thèse de doctorat, UPMC. 2001.
+
+Pelorson X. Conférence SFA « cordes vocales et lèvres vibrantes ». Juillet 2004. Paris
+
+Charbit M. Polycopié ENST. Bases de Traitement du signal. 2004
+
+Heiser T. Traitement du signal, support de cours. UFR Sciences Physiques. université Louis Pasteur, Strasbourg
+
+Feng. Traitement du signal, support de cours. ENSERG. INP Grenoble. 2000.
+
+Dutoit T.  Introduction au Traitement Automatique de la Parole Faculté Polytechnique
+de Mons. 2004
+

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+
+
+
+
+
+
+
+
+
+\subsection{Classification énergétique}
+On distingue en particulier deux types de signaux :
+- Les signaux à énergie finie     (Tout signal physique)
+ 
+
+- Les signaux à puissance moyenne finie    (Idéalisation. exemple:  signal sinusoïdal)
+
+\section{Signaux et distributions classiques}
+\subsection{Fenêtre rectangulaire ou Porte}
+
+Notation :     '2a (t) 
+
+
+Représentation : 
+
+
+
+
+
+
+\subsection{Impulsion de Dirac}
+Les propriétés de l'impulsion de Dirac sont identiques à celles d'un signal rectangulaire dont la largeur tend vers 0 et la hauteur vers l'infini, à surface constante. La réponse d'un système à un signal 'extrêmement bref' sera décrite par sa réponse à l'impulsion de Dirac.
+
+
+
+\subsection{« Peigne » de Dirac}
+Le peigne de Dirac correspond à un périodisation d'impulsions de Dirac tous les nT  sur tout l'axe de des temps (aussi bien t<0 que t> 0).
+
+
+'T  (t) = ' '(t-nT)
+
+
+
+
+
+
+
+
+\subsection{Sinus cardinal}
+Le sinus cardinal se définit par la formule : Sinc (t) = sin(t) / t  . 
+
+
+
+
+\section{Transformée de Fourier}
+
+\subsection{Rappels sur la décomposition en série de Fourier des signaux périodiques}
+
+Tout signal périodique est décomposable en une somme de sinus et de cosinus. 
+
+
+
+
+La composante continue de cette décomposition, aussi appelée fondamentale, correspond à la hauteur perçue d'un son par exemple. Cette composante est à la fréquence inverse de la période du signal. 
+Les autres composantes sont de fréquences supérieures à la fondamentale et sont appelées harmoniques du signal. Toujours dans le cas d'un son harmonique , joué par exemple par un instrument de musique, la fondamentale va correspondre à la note jouée et les autres composantes vont correspondre à la « série harmonique » de cette note (octave, puis quinte, puis octave, puis tierce'etc). Cette série correspond à des rapports de fréquences (fondamental fo, octave = fo*2, octave + quinte = fo*3'etc). Les harmoniques ne vont pas directement contribuer à la perception de la hauteur mais plutôt à la richesse du timbre du son. C'est par exemple la présence d'harmoniques -entre autres- qui permet de différentier une sinusoïde à fo (seulement un fondamental => son proche d'un buzzer électronique) d'une note de violon jouée à la même fréquence.
+
+
+\subsection{Définition de la Transformée de Fourier]
+La transformée de Fourier est une « généralisation » de cette décomposition aux signaux non périodiques.
+
+On définit la transformée de Fourier comme :
+
+
+
+
+
+
+
+
+
+
+
+
+X(f) est une fonction complexe même si x(t) est réel. |X(f)| est appelé 'spectre d'amplitude' et Arg[X(f)] ='spectre de phase' du signal. 
+La variable f s'appelle la fréquence. Son unité est le Hertz (en abrégé : Hz).
+
+
+La Transformée de Fourier est un opérateur mathématique qui permet  d'analyser et de représenter un signal dans le domaine fréquentiel. La TF ne modifie pas le signal mais permet seulement de l'observer selon différents points de vue.(temporel ou fréquentiel). Il est important de retenir que x(t) et X(f) sont deux descriptions équivalentes du même signal. Ces deux fonctions contiennent la même information. Il s'agit juste de deux descriptions dans des domaines différents (temporel et fréquentiel).
+
+X(f)  apporte des information sur le système physique à l'origine du signal. Elle permet par exemple de différentier un son de trompette d'un son trombone, ou bien encore différentes ondes cérébrales , mieux qu'en observant le signal dans le domaine temporel.
+
+
+
+
+
+
+
+
+\subsection{Le produit de convolution}
+
+On appelle convolution de x(t) par y(t) l'opération notée x(t) * y(t) et définie par :
+
+
+L'impulsion de Dirac est l'élément neutre de la convolution, c'est à dire que 
+x(t) * ' (t) = x(t).
+
+
+Lorsque l'on convolue un signal x(t) à un Dirac en to par exemple, cela revient à retarder le signal x(t) de to :           x(t) * ' (t-to) = x(t-to).
+
+Alors que si l'on multiplie un signal x(t) par un Dirac en to, cela revient à connaître la valeur que prend x(t) en to (comme si l'on relevait l'ordonnée d'un point particulier d'une courbe)                  x(t) . ' (t-to) = x(to).               
+
+De même, lorsque l'on convolue un signal x(t) à un peigne de Dirac (de période T), cela revient à « périodiser »  le signal x(t) tous les nT : on retarde le signal x(t) de T , de 2T, de 3T'.etc                       x(t) * 'T  (t)  =  ' x(t-nT)
+
+Tandis que si on multiplie le signal x(t) par un peigne de Dirac, on prélève des échantillons du signal x(t) tous les nT. Nous verrons plus loin que cette opération correspond à l' «échantillonnage»  du signal x(t).
+                                       x(t) * 'T  (t)  =  ' x(nT)
+
+De façon plus générale, la convolution telle qu'elle est définie par sa formule mathématique, revient à retourner temporellement un des deux signaux (par exemple x(t) )  puis à le déplacer sur tout l'axe du temps  et à sommer toutes les multiplications de ce signal retourné déplacé au deuxième signal ( y(t) ) .
+
+
+
+
+
+
+
+
+
+
+
+Retournement temporel         t est un instant qui varie du signal x               dans le temps u de  -'  à  +'.
+  On fait « glisser » le signal 
+  x(-u) selon l'axe du temps.
+
+
+        
+
+
+
+
+
+
+
+
+
+On multiplie y(u) par x(t-u)  pour chaque position de t sur l'axe du temps et on somme tous les produits effectués.  
+
+
+
+
+
+
+
+
+
+
+
+
+
+\subsection{Propriétés de la transformée de Fourier}
+
+Linéarité de la TF :  a x(t)+b y(t)     '   a X(f)+b Y(f)
+
+Produit de convolution :
+x(t).y(t)     '   X(f) * Y(f)
+x(t) * y(t)   '   X(f) . Y(f)   
+
+Une multiplication dans un domaine correspond à un produit de convolution dans l'autre. 
+
+
+
+Retard : 
+
+ 
+
+
+
+x(t-to)    '    X(f) e-i2pfto
+
+x(t).ei2pfot      '  X(f-fo)
+               
+
+
+
+Un retard UU<u
+Un retard temporel correspond à un déphasage au niveau fréquentiel, et inversement.
+
+Changement d'échelle :   x(at)   '  1/a  *  X(f/a)    
+
+
+
+Théorème de Parseval :
+
+$Énergie      =   \vert x(t) \vert^2 dt   =       \vert X(f) \vert^2 df$
+
+
+
+
+\subsection{Transformées de Fourier des signaux courants}
+
+Dirac (t)   '  signal unité
+
+Signal porte '  sinus cardinal
+
+Sin(2 Pi fo t)   '   ' ( dirac (f-fo)  + dirac (f+fo)
+
+
+A complèter avec vos notes de cours'. 
+
+
+
+
+
+
+\section{Evaluation de la similitude entre deux signaux}
+
+Il existe en traitement du signal des outils mathématiques qui permettent d'évaluer la similitude entre deux signaux, 
+-au niveau de leur évolution temporelle grâce  à la fonction d'inter- ou d'auto-corrélation.
+-au niveau de leur composition spectrale grâce à la fonction de densité spectrale. 
+
+\subsection{Les fonctions d'inter- et d'auto-corrélation}
+
+
+La  fonction d'intercorrélation se définit par la formule suivante :
+
+
+
+
+
+
+
+La fonction d'autocorrélation est un cas particulier de la fonction d'intercorrélation pour y(t) = x(t). Elle se définit par :
+
+
+
+
+
+
+La fonction d'autocorrélation mesure la similitude de x(t) avec une version décallée de x(t). Elle atteint un maximum pour le temps to auquel x(t-to) ressemble le plus à x(t).  C'est le cas particulièrement pour les signaux périodiques qui reprennent la même valeur à chaque période T. La fonction d'autocorrélation permet ainsi d'estimer la période d'un signal périodique en repérant le temps pour lequel elle atteint son maximum.
+
+La fonction d'autocorrélation permet également de calculer l'énergie du signal  puisque 
+
+
+
+
+
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La Fonction de densité spectrale d'énergie est la TF de la fonction d'autocorrélation. Elle décrit la répartition de l'énergie en fréquence.
+
+
+
+
+
+
+
+
+\section{Les signaux numériques}
+
+Les signaux numériques dérivent d 'un signal analogique par échantillonnage. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants tn, n étant entier. Dans la suite nous n'envisagerons que l'échantillonnage dit régulier où tn = nT. L'intérêt porte aux problèmes de l'échantillonnage tient dans le développement des techniques numériques de traitement du signal. En Physique il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}
+
+Le signal continu (analogique) peut être échantillonné (ou discrétisé) à une fréquence d'échantillonnage Fe  (= 1/Te). 
+Cette échantillonage revient à multiplier le signal analogique par un peigne de Dirac de période Te.
+
+
+
+
+
+
+
+
+
+Le signal échantillonné peut alors s'écrire :
+
+
+
+                                
+
+
+
+xe(t) est donc nul partout sauf en t= nTe, n variant de -''à'''''
+C'est pourquoi par la suite, on ne simplifie cette notation en réduisant xe(t) non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets x(n), où n correspond au numéro de l'échantillon.
+
+x(n) = xe ( n Te )
+
+
+\subsection{Définition et propriétés de la transformée de Fourier d'un signal échantillonné}
+Dans le cas des signaux échantillonnés, il n'est plus nécessaire d'utiliser des intégrales pour sommer les valeurs de x(t) sur tout l'axe des temps, puisqu'un signal échantillonné peut être assimilé à une suite d'éléments discrets.
+On peut donc définir la transformée de Fourier d'un signal numérique par :
+
+De la même façon qu'en continu, la TF présente certaines propriétés intéressantes :
+
+retard:   x(n-no)  '   X(f)   e-j 2'fno
+produit de convolution:   x(n) * y(n)  '   X(f) . Y(f) 
+        x(n) . y(n)  '   X(f) * Y(f) 
+
+        avec     x(n)  * y(n)  =  '''x(k) y(n-k)    =     '''x(n-k) y(k)     .
+
+\subsection{Conséquences de l'échantillonnage du point de vue fréquentiel} 
+
+En écrivant  xe(t) = xa(t) . ''''(t- nTe), on peut alors exprimer la transformée de Fourier du signal échantillonné à partir de la TF du signal analogique.
+
+
+
+On voit que l'échantillonnage d'un signal analogique (à la fréquence d'échantillonnage Fe = 1/T) induit une périodisation de son spectre dans le domaine fréquentiel, tous les f =n/ T, n étant entier.
+
+
+
+
+Il peut survenir un problème si la fréquence d'échantillonnage Fe = 1/T est trop petite car les « répliques » périodiques du spectre  peuvent se superposer partiellement .
+
+
+
+
+
+
+
+
+Cela arrive si   borne max (Xa(f))  >    borne inf  (1ere réplique de Xa(f) en f =1/T)
+Autrement dit si               B        >      1/T  - B
+
+
+Pour que le spectre Xa(f) ne soit pas « déformé » lors de sa périodisation, il faut donc que :
+
+                       Fe < 2 B
+ 
+Cette condition est appelée Théorème de Shannon.
+
+
+
+\chapter{Modes de représentation des signaux}
+
+
+\section{Fenêtrage des signaux} 
+
+Nous avons vu dans le chapitre précédent que la Transformée de Fourier permet de représenter le spectre d'un signal dans le domaine fréquentiel. Pour une sinusoïde infinie, toute l'énergie du spectre est concentrée à une fréquence donnée, c'est à dire la fréquence de la sinusoïde. 
+
+
+
+
+Si on veut observer ce sinus sur un temps fini, on va être obligé de le tronquer. L'énergie va alors d'une part se répartir autour de la fréquence de la sinusoïde (c'est ce qu'on appelle l'étalement spectral), d'autre part on observe la présence d'énergie dans toutes les fréquences (Ce phénomène est appelé fuite spectrale). Etalement et fuite sont liés à la troncature.
+
+
+
+Si on réalise la troncature de façon non rectangulaire, mais en « fenêtrant » le signal par une fenêtre de Hanning par exemple, les transitions dans le signal sont alors plus douces. Cela limite la fuite spectrale mais en revanche augmente l'étalement en fréquence. Ce point est important pour comprendre le rôle des fenêtres d'analyse. Si la fenêtre a des discontinuités fortes, les fuites spectrales vont être importantes, mais l'étalement moindre. Si on prend un fenêtre de discontinuité plus douce, on va au contraire obtenir un étalement plus grand, mais moins de fuites spectrales.
+
+
+
+\section{Le spectrogramme}
+
+L'intérêt du spectrogramme est de pouvoir représenter le spectre en évoluant dans le temps. Le nom scientifique de la fonction mathématique associée à cet outil, plus communément appelé « spectrogramme », est la Transformée de Fourier à Court Terme (TFCT). Ce nom provient de l'analyse effectuée sur des fenêtres de support temporel fini. Une autre dénomination de cette représentation est « sonagramme ». Il s'agit d'une marque déposée Kay Electronics.
+
+Le principe du spectrogramme est de « découper » le son en tranches ou trames qui se recouvrent. Pour chacune de ces tranches on calcule une transformée de Fourier. Ce spectre est alors représenté à un temps correspondant à celui du centre de la fenêtre, sous forme d'un code de couleur. Sur l'exemple suivant, le jaune correspond aux amplitudes les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a ainsi une idée de l'aspect du spectre au temps t. A chaque calcul du spectre, le signal est fenêtré de façon à pouvoir régler à la fois la fuite et l'étalement spectral. 
+
+
+Voici un exemple de voix chantée. Il s'agit d'un glissando C5-E5 réalisé par une soprano. L'analyse a été effectuée avec une fenêtre de Hanning de longueur 23 ms. Sur le spectrogramme, on observe bien le glissando et le vibrato de la chanteuse.
+
+
+Pour mesurer le vibrato par exemple, on serait tenté de réduire la longueur de la fenêtre dans le temps pour gagner en précision et suivre au mieux les variations du spectre. En réalité, si on réduit la longueur des fenêtres, l'étalement spectral augmente, par conséquent la largeur des raies sur le spectrogramme aussi, ce qui perturbe finalement la mesure, car on ne distingue plus distinctement les différentes trajectoires dans le spectrogramme.
+
+
+
+
+
+
+
+
+
+
+
+
+Si on revient à la même longueur de fenêtre que dans le premier exemple, tout en utilisant une fenêtre rectangulaire au lieu de la fenêtre « douce » de Hanning, l'étalement spectral est plus faible et les lignes sur le spectrogramme plus fines, mais on a par contre des fuites spectrales beaucoup plus importantes, caractérisées par un manque de contraste dans la représentation du spectrogramme. 
+
+
+Les points importants quant à l'utilisation du spectrogramme sont donc: 
+la longueur de fenêtre pour ajuster la précision temporelle, au prix d'un étalement spectral qui peut devenir rédhibitoire
+le choix de la fenêtre qui va conditionner le contraste du spectrogramme, pour une longueur de fenêtre donnée.
+
+\section{Pertinence et choix d'un mode de représentation}
+
+La représentation du signal en temporel est particulièrement adapté à l'analyse de ses variations temporelles  (par exemple l'attaque ou l'extinction d'un son).
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Au contraire, cette représentation apporte peu d'information lorsqu'on s'intéresse plutôt au contenu spectral du signal. Sur la figure suivante, on voit que la représentation spectrale de ces deux signaux de parole permet assez bien de différentier la voix « tendue »  de la voix « relâchée ».
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Voix relâchée                                          Voix tendue
+
+
+Un autre intérêt de la représentation du signal à l'aide d'un spectrogramme est de pouvoir séparer des sources sonores, ou du moins de pouvoir repérer des événements, là où le signal temporel englobe tout de façon indifférenciée.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Pour des sons très harmoniques (par exemple voix chantée), le spectrogramme permet de repérer très facilement la fondamentale ainsi que la série harmonique.
+Sur la figure ci-dessous est représenté le spectrogramme d'un chanteur effectuant un glissando du bas au haut de sa tessiture puis l'inverse.
+
+
+
+
+
+
+
+
+
+
+
+
+Dans le cas de la voix parlée, on peut repérer les parties voisée (harmoniques) ou non, et même « lire » quels sont les phonèmes prononcés avec un peu d'expérience. Nous reviendrons là dessus dans le chapitre 5.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+   « C'est une maison qui s'trouve au Luxembourg »
+
+
+\section{Le spectre moyenné (Long Term Average Spectrum   LTAS)}
+
+Il est également très utile dans certains cas de représenter une partie du signal sous forme de spectre moyenné. Le spectre moyenné, comme son nom l'indique, consiste à faire la moyenne du spectrogramme sur l'ensemble des instants d'un intervalle de temps plus ou moins long . Ce peut être sur le signal tout entier comme sur un morceau particulier.
+
+Pour examiner une voyelle en particulier dans un signal de parole par exemple, il est tout à fait approprié de moyenner le spectrogramme sur l'intervalle de temps de cette voyelle, de façon à pouvoir estimer les formants ou bien le contenu spectral de cette voyelle. Ainsi, sur la figure suivante, on peut examiner le spectre moyenné de la production vocale d'un chanteur sur un /a/. Cette représentation permet de caractériser ses premiers formants vocaliques, ainsi que son formant du chanteur (renforcement énergétique dans la zone 2000-4000 Hz), caractéristique des chanteurs lyriques et des acteurs de théâtre.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\chapter{Systèmes linéaires et filtrage}
+
+
+
+
+
+\section{Réponse impulsionnelle d'un système}
+
+Il est possible de décrire un système S par l'intermédiaire de sa réponse à un signal extrêmement bref. Mathématiquement, il s'agit de modéliser sa réponse à une impulsion de Dirac.
+
+La réponse impulsionnelle d'un système est donc définie comme la sortie d'un système lorsque l'entrée est une impulsion de Dirac.
+
+g(t) = S{d(t)}
+
+Lorsque nous claquons des mains dans une pièce pour évaluer son acoustique, son temps de réverbération, etc, nous ne faisons pas moins en fait qu'évaluer la réponse impulsionnelle de ce système pièce. 
+Cela semble physiquement logique d'entendre du son dans la pièce au moment où nous claquons dans nos mains, voire après, mais certainement pas avant . Pourtant, en mathématique, on conçoit que la réponse impulsionnelle puisse être non nulle avant l'instant de l'impulsion d'entrée.
+On définit ainsi les système causaux comme les systèmes dont la réponse impulsionnelle est nulle avant l'instant d'impulsion, autrement dit      g(t<0) = 0  .
+Pour un système causal, le signal de sortie à l'instant t dépend du signal d'entrée aux instants t'< t. La durée de la réponse impulsionnelle g(t) correspond au temps de réponse du système.
+
+
+
+Nous avons vu que la réponse impulsionnelle correspond à la sortie y(t) du système lorsque l'entrée x(t) = '(t) .
+Connaissant la réponse impulsionnelle g(t) d'un système, on peut ensuite prédire sa réponse y(t) à n'importe quel signal d'entrée x(t) grâce à la formule :
+
+                                    y(t) = x(t) * g(t)
+
+
+
+
+
+
+
+
+
+
+
+
+\section{Réponse en fréquence d'un système}
+
+La réponse en fréquence d'un système correspond à la transformée de Fourier de la réponse impulsionnelle du système.
+
+
+G(f) décrit comment la distribution spectrale d'un signal est modifiée ou "filtrée" par le système S. Il est important de noter que le système peut seulement modifier des composantes spectrales mais ne peut en aucun cas en créer de nouvelles.
+|G(f)|  est le gain du système, c'est à dire la façon dont il modifie les amplitudes de chaque composantes spectrales.
+Arg[G(f)] représente le déphasage causé par le système, c'est à dire le « retard » qu'il impose à certaines composantes spectrales.
+
+
+
+
+
+
+
+
+
+
+
+
+
+La réponse en fréquence, comme la réponse impulsionnelle permet de décrire complètement le système et de prédire la réponse du système à n'importe quelle entrée. 
+
+y(t) = x(t) * g(t)    '     Y(f)  = X(f) . G(f)
+
+
+
+\section{Fonction de transfert d'un système (et définition de la Transformée en Z)}
+
+La fonction de transfert d'un système est défini comme la transformée en Z de la réponse impulsionnelle du système.
+
+
+La Transformée en Z est très proche de la Transformée de Fourier et se définit par :
+
+La TZ est un outil de caractérisation des systèmes de traitement numérique des signaux. 
+On retrouve la définition de la TF en posant z = ei2' f.
+
+La TZ présente aussi comme propriété :
+
+* Retard :                     x(k 'ko)     '     X(z) z-ko
+
+*Convolution  :        x(k)*y(k)    '     X(z).Y(z)
+                       x(k).y(k)    '     X(z)*Y(z)
+                       
+
+Donc la réponse impulsionnelle g(n) et la fonction de transfert G(z) permettent de décrire un système numérique.
+
+y(n) = x(n) * g(n)    '     Y(z)  = X(z) . G(z)
+
+
+
+
+
+
+\section{Filtrage}
+
+Le filtrage linéaire est une opération utile pour sélectionner une partie du spectre d'un signal ou en atténuer une partie, pour 
+modifier le contenu spectral d'un signal  ou encore pour m
+odéliser la transformation d'un signal dans un conduit, dans une pièce' 
+
+\subsection{Filtre idéaux}
+On définit classiquement 4 types de filtres idéaux : 
+les filtres passe-bas qui laissent intact les basses fréquences d'un signal et en atténuent les hautes fréquences.
+les filtres passe-haut qui laissent intact les hautes fréquences d'un signal et en atténuent les basses fréquences.
+Les filtres passe-bande qui sélectionnent une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.
+Les filtres coupe-bande, qui atténuent fortement une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.
+
+
+Pour tous ces filtres, on définit des fréquences de coupure, c'est à dire les fréquences pour lesquelles le spectre du signal d'entrée va être atténué d'un facteur '2. 
+
+
+Le passe bas à donc une fréquence de coupure dans les médiums / hautes fréquences, le passe haut une fréquence de coupure dans les médiums / basses fréquences. Les passe bande et coupe bande possèdent deux fréquences de coupure autour de la fréquence centrale sur laquelle ils se centrent. 
+On spécifie également la pente de l'atténuation de ces filtres, en dB par octave. Cela apporte une information sur la sélectivité du filtre.
+
+Enfin, pour les filtres passe-bande et coupe-bande, on détermine leur largeur de bande, c'est à dire la différence entre leurs deux fréquences de coupure, ce qui renseigne sur le caractère plus ou moins « aigu » du filtre.
+
+
+\subsection{Exemples de filtres naturels} 
+
+On peut modéliser la sensation d'intensité auditive par un filtre. L'oreille humaine est particulièrement sensible entre 3 et 4 kHz 
+
+On peut également modéliser la production phonatoire par un modèle source-filtre, où le rôle de la source est joué par les cordes vocales qui produisent un son harmonique avec une distribution de l'énergie assez plate en fréquence. Le conduit vocal, les fosses nasales ainsi que la place des articulateurs (langue, mâchoire, lèvres) peuvent être modélisés par un filtre qui modifie le son glottique pour produire le son tel que nous le percevons à la sortie des lèvres d'un locuteur.
+
+
+Bibliographie
+
+David B. Conférence SFA sur le vibrato. Décembre 2003. Le mans.
+
+Garnier M, Dubois D, Poitevineau J, Henrich N, Castellengo M. Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive. Journées d'Études sur la Parole, Fez, 2004. . 
+
+Henrich N. Etude de la source glottique en voix parlée et chantée. Thèse de doctorat, UPMC. 2001.
+
+Pelorson X. Conférence SFA « cordes vocales et lèvres vibrantes ». Juillet 2004. Paris
+
+Charbit M. Polycopié ENST. Bases de Traitement du signal. 2004
+
+Heiser T. Traitement du signal, support de cours. UFR Sciences Physiques. université Louis Pasteur, Strasbourg
+
+Feng. Traitement du signal, support de cours. ENSERG. INP Grenoble. 2000.
+
+Dutoit T.  Introduction au Traitement Automatique de la Parole Faculté Polytechnique
+de Mons. 2004
+

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+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\subsection{Impulsion de Dirac}
+Les propriétés de l'impulsion de Dirac sont identiques à celles d'un signal rectangulaire dont la largeur tend vers 0 et la hauteur vers l'infini, à surface constante. La réponse d'un système à un signal 'extrêmement bref' sera décrite par sa réponse à l'impulsion de Dirac.
+
+
+
+\subsection{« Peigne » de Dirac}
+Le peigne de Dirac correspond à un périodisation d'impulsions de Dirac tous les nT  sur tout l'axe de des temps (aussi bien t<0 que t> 0).
+
+
+'T  (t) = ' '(t-nT)
+
+
+
+
+
+
+
+
+\subsection{Sinus cardinal}
+Le sinus cardinal se définit par la formule : Sinc (t) = sin(t) / t  . 
+
+
+
+
+\section{Transformée de Fourier}
+
+\subsection{Rappels sur la décomposition en série de Fourier des signaux périodiques}
+
+Tout signal périodique est décomposable en une somme de sinus et de cosinus. 
+
+
+
+
+La composante continue de cette décomposition, aussi appelée fondamentale, correspond à la hauteur perçue d'un son par exemple. Cette composante est à la fréquence inverse de la période du signal. 
+Les autres composantes sont de fréquences supérieures à la fondamentale et sont appelées harmoniques du signal. Toujours dans le cas d'un son harmonique , joué par exemple par un instrument de musique, la fondamentale va correspondre à la note jouée et les autres composantes vont correspondre à la « série harmonique » de cette note (octave, puis quinte, puis octave, puis tierce'etc). Cette série correspond à des rapports de fréquences (fondamental fo, octave = fo*2, octave + quinte = fo*3'etc). Les harmoniques ne vont pas directement contribuer à la perception de la hauteur mais plutôt à la richesse du timbre du son. C'est par exemple la présence d'harmoniques -entre autres- qui permet de différentier une sinusoïde à fo (seulement un fondamental => son proche d'un buzzer électronique) d'une note de violon jouée à la même fréquence.
+
+
+\subsection{Définition de la Transformée de Fourier]
+La transformée de Fourier est une « généralisation » de cette décomposition aux signaux non périodiques.
+
+On définit la transformée de Fourier comme :
+
+
+
+
+
+
+
+
+
+
+
+
+X(f) est une fonction complexe même si x(t) est réel. |X(f)| est appelé 'spectre d'amplitude' et Arg[X(f)] ='spectre de phase' du signal. 
+La variable f s'appelle la fréquence. Son unité est le Hertz (en abrégé : Hz).
+
+
+La Transformée de Fourier est un opérateur mathématique qui permet  d'analyser et de représenter un signal dans le domaine fréquentiel. La TF ne modifie pas le signal mais permet seulement de l'observer selon différents points de vue.(temporel ou fréquentiel). Il est important de retenir que x(t) et X(f) sont deux descriptions équivalentes du même signal. Ces deux fonctions contiennent la même information. Il s'agit juste de deux descriptions dans des domaines différents (temporel et fréquentiel).
+
+X(f)  apporte des information sur le système physique à l'origine du signal. Elle permet par exemple de différentier un son de trompette d'un son trombone, ou bien encore différentes ondes cérébrales , mieux qu'en observant le signal dans le domaine temporel.
+
+
+
+
+
+
+
+
+\subsection{Le produit de convolution}
+
+On appelle convolution de x(t) par y(t) l'opération notée x(t) * y(t) et définie par :
+
+
+L'impulsion de Dirac est l'élément neutre de la convolution, c'est à dire que 
+x(t) * ' (t) = x(t).
+
+
+Lorsque l'on convolue un signal x(t) à un Dirac en to par exemple, cela revient à retarder le signal x(t) de to :           x(t) * ' (t-to) = x(t-to).
+
+Alors que si l'on multiplie un signal x(t) par un Dirac en to, cela revient à connaître la valeur que prend x(t) en to (comme si l'on relevait l'ordonnée d'un point particulier d'une courbe)                  x(t) . ' (t-to) = x(to).               
+
+De même, lorsque l'on convolue un signal x(t) à un peigne de Dirac (de période T), cela revient à « périodiser »  le signal x(t) tous les nT : on retarde le signal x(t) de T , de 2T, de 3T'.etc                       x(t) * 'T  (t)  =  ' x(t-nT)
+
+Tandis que si on multiplie le signal x(t) par un peigne de Dirac, on prélève des échantillons du signal x(t) tous les nT. Nous verrons plus loin que cette opération correspond à l' «échantillonnage»  du signal x(t).
+                                       x(t) * 'T  (t)  =  ' x(nT)
+
+De façon plus générale, la convolution telle qu'elle est définie par sa formule mathématique, revient à retourner temporellement un des deux signaux (par exemple x(t) )  puis à le déplacer sur tout l'axe du temps  et à sommer toutes les multiplications de ce signal retourné déplacé au deuxième signal ( y(t) ) .
+
+
+
+
+
+
+
+
+
+
+
+Retournement temporel         t est un instant qui varie du signal x               dans le temps u de  -'  à  +'.
+  On fait « glisser » le signal 
+  x(-u) selon l'axe du temps.
+
+
+        
+
+
+
+
+
+
+
+
+
+On multiplie y(u) par x(t-u)  pour chaque position de t sur l'axe du temps et on somme tous les produits effectués.  
+
+
+
+
+
+
+
+
+
+
+
+
+
+\subsection{Propriétés de la transformée de Fourier}
+
+Linéarité de la TF :  a x(t)+b y(t)     '   a X(f)+b Y(f)
+
+Produit de convolution :
+x(t).y(t)     '   X(f) * Y(f)
+x(t) * y(t)   '   X(f) . Y(f)   
+
+Une multiplication dans un domaine correspond à un produit de convolution dans l'autre. 
+
+
+
+Retard : 
+
+ 
+
+
+
+x(t-to)    '    X(f) e-i2pfto
+
+x(t).ei2pfot      '  X(f-fo)
+               
+
+
+
+Un retard UU<u
+Un retard temporel correspond à un déphasage au niveau fréquentiel, et inversement.
+
+Changement d'échelle :   x(at)   '  1/a  *  X(f/a)    
+
+
+
+Théorème de Parseval :
+
+$Énergie      =   \vert x(t) \vert^2 dt   =       \vert X(f) \vert^2 df$
+
+
+
+
+\subsection{Transformées de Fourier des signaux courants}
+
+Dirac (t)   '  signal unité
+
+Signal porte '  sinus cardinal
+
+Sin(2 Pi fo t)   '   ' ( dirac (f-fo)  + dirac (f+fo)
+
+
+A complèter avec vos notes de cours'. 
+
+
+
+
+
+
+\section{Evaluation de la similitude entre deux signaux}
+
+Il existe en traitement du signal des outils mathématiques qui permettent d'évaluer la similitude entre deux signaux, 
+-au niveau de leur évolution temporelle grâce  à la fonction d'inter- ou d'auto-corrélation.
+-au niveau de leur composition spectrale grâce à la fonction de densité spectrale. 
+
+\subsection{Les fonctions d'inter- et d'auto-corrélation}
+
+
+La  fonction d'intercorrélation se définit par la formule suivante :
+
+
+
+
+
+
+
+La fonction d'autocorrélation est un cas particulier de la fonction d'intercorrélation pour y(t) = x(t). Elle se définit par :
+
+
+
+
+
+
+La fonction d'autocorrélation mesure la similitude de x(t) avec une version décallée de x(t). Elle atteint un maximum pour le temps to auquel x(t-to) ressemble le plus à x(t).  C'est le cas particulièrement pour les signaux périodiques qui reprennent la même valeur à chaque période T. La fonction d'autocorrélation permet ainsi d'estimer la période d'un signal périodique en repérant le temps pour lequel elle atteint son maximum.
+
+La fonction d'autocorrélation permet également de calculer l'énergie du signal  puisque 
+
+
+
+
+
+
+
+\subsection{La fonction de densité spectrale d'énergie}
+
+La Fonction de densité spectrale d'énergie est la TF de la fonction d'autocorrélation. Elle décrit la répartition de l'énergie en fréquence.
+
+
+
+
+
+
+
+
+\section{Les signaux numériques}
+
+Les signaux numériques dérivent d 'un signal analogique par échantillonnage. L'échantillonnage est une opération qui consiste à prélever sur un signal à temps continu une suite de valeurs, prises en une suite d'instants tn, n étant entier. Dans la suite nous n'envisagerons que l'échantillonnage dit régulier où tn = nT. L'intérêt porte aux problèmes de l'échantillonnage tient dans le développement des techniques numériques de traitement du signal. En Physique il est fréquent que les résultats expérimentaux soient sous forme d'une suite de valeurs numériques. Les possibilités d'application du traitement numérique des signaux sont d'autant plus nombreuses que la vitesse de calcul des microprocesseurs est élevée. Pour caractériser et traiter un tel signal numérique on peut utiliser les même concepts mathématiques que pour les signaux continus, à condition de les adapter aux contexte numérique.
+
+
+\subsection{Définition de l'échantillonnage}
+
+Le signal continu (analogique) peut être échantillonné (ou discrétisé) à une fréquence d'échantillonnage Fe  (= 1/Te). 
+Cette échantillonage revient à multiplier le signal analogique par un peigne de Dirac de période Te.
+
+
+
+
+
+
+
+
+
+Le signal échantillonné peut alors s'écrire :
+
+
+
+                                
+
+
+
+xe(t) est donc nul partout sauf en t= nTe, n variant de -''à'''''
+C'est pourquoi par la suite, on ne simplifie cette notation en réduisant xe(t) non plus à un signal continu possédant une infinité de valeurs mais à une suite d'échantillons discrets x(n), où n correspond au numéro de l'échantillon.
+
+x(n) = xe ( n Te )
+
+
+\subsection{Définition et propriétés de la transformée de Fourier d'un signal échantillonné}
+Dans le cas des signaux échantillonnés, il n'est plus nécessaire d'utiliser des intégrales pour sommer les valeurs de x(t) sur tout l'axe des temps, puisqu'un signal échantillonné peut être assimilé à une suite d'éléments discrets.
+On peut donc définir la transformée de Fourier d'un signal numérique par :
+
+De la même façon qu'en continu, la TF présente certaines propriétés intéressantes :
+
+retard:   x(n-no)  '   X(f)   e-j 2'fno
+produit de convolution:   x(n) * y(n)  '   X(f) . Y(f) 
+        x(n) . y(n)  '   X(f) * Y(f) 
+
+        avec     x(n)  * y(n)  =  '''x(k) y(n-k)    =     '''x(n-k) y(k)     .
+
+\subsection{Conséquences de l'échantillonnage du point de vue fréquentiel} 
+
+En écrivant  xe(t) = xa(t) . ''''(t- nTe), on peut alors exprimer la transformée de Fourier du signal échantillonné à partir de la TF du signal analogique.
+
+
+
+On voit que l'échantillonnage d'un signal analogique (à la fréquence d'échantillonnage Fe = 1/T) induit une périodisation de son spectre dans le domaine fréquentiel, tous les f =n/ T, n étant entier.
+
+
+
+
+Il peut survenir un problème si la fréquence d'échantillonnage Fe = 1/T est trop petite car les « répliques » périodiques du spectre  peuvent se superposer partiellement .
+
+
+
+
+
+
+
+
+Cela arrive si   borne max (Xa(f))  >    borne inf  (1ere réplique de Xa(f) en f =1/T)
+Autrement dit si               B        >      1/T  - B
+
+
+Pour que le spectre Xa(f) ne soit pas « déformé » lors de sa périodisation, il faut donc que :
+
+                       Fe < 2 B
+ 
+Cette condition est appelée Théorème de Shannon.
+
+
+
+\chapter{Modes de représentation des signaux}
+
+
+\section{Fenêtrage des signaux} 
+
+Nous avons vu dans le chapitre précédent que la Transformée de Fourier permet de représenter le spectre d'un signal dans le domaine fréquentiel. Pour une sinusoïde infinie, toute l'énergie du spectre est concentrée à une fréquence donnée, c'est à dire la fréquence de la sinusoïde. 
+
+
+
+
+Si on veut observer ce sinus sur un temps fini, on va être obligé de le tronquer. L'énergie va alors d'une part se répartir autour de la fréquence de la sinusoïde (c'est ce qu'on appelle l'étalement spectral), d'autre part on observe la présence d'énergie dans toutes les fréquences (Ce phénomène est appelé fuite spectrale). Etalement et fuite sont liés à la troncature.
+
+
+
+Si on réalise la troncature de façon non rectangulaire, mais en « fenêtrant » le signal par une fenêtre de Hanning par exemple, les transitions dans le signal sont alors plus douces. Cela limite la fuite spectrale mais en revanche augmente l'étalement en fréquence. Ce point est important pour comprendre le rôle des fenêtres d'analyse. Si la fenêtre a des discontinuités fortes, les fuites spectrales vont être importantes, mais l'étalement moindre. Si on prend un fenêtre de discontinuité plus douce, on va au contraire obtenir un étalement plus grand, mais moins de fuites spectrales.
+
+
+
+\section{Le spectrogramme}
+
+L'intérêt du spectrogramme est de pouvoir représenter le spectre en évoluant dans le temps. Le nom scientifique de la fonction mathématique associée à cet outil, plus communément appelé « spectrogramme », est la Transformée de Fourier à Court Terme (TFCT). Ce nom provient de l'analyse effectuée sur des fenêtres de support temporel fini. Une autre dénomination de cette représentation est « sonagramme ». Il s'agit d'une marque déposée Kay Electronics.
+
+Le principe du spectrogramme est de « découper » le son en tranches ou trames qui se recouvrent. Pour chacune de ces tranches on calcule une transformée de Fourier. Ce spectre est alors représenté à un temps correspondant à celui du centre de la fenêtre, sous forme d'un code de couleur. Sur l'exemple suivant, le jaune correspond aux amplitudes les plus fortes, le bleu/violet aux amplitudes les plus faibles. On a ainsi une idée de l'aspect du spectre au temps t. A chaque calcul du spectre, le signal est fenêtré de façon à pouvoir régler à la fois la fuite et l'étalement spectral. 
+
+
+Voici un exemple de voix chantée. Il s'agit d'un glissando C5-E5 réalisé par une soprano. L'analyse a été effectuée avec une fenêtre de Hanning de longueur 23 ms. Sur le spectrogramme, on observe bien le glissando et le vibrato de la chanteuse.
+
+
+Pour mesurer le vibrato par exemple, on serait tenté de réduire la longueur de la fenêtre dans le temps pour gagner en précision et suivre au mieux les variations du spectre. En réalité, si on réduit la longueur des fenêtres, l'étalement spectral augmente, par conséquent la largeur des raies sur le spectrogramme aussi, ce qui perturbe finalement la mesure, car on ne distingue plus distinctement les différentes trajectoires dans le spectrogramme.
+
+
+
+
+
+
+
+
+
+
+
+
+Si on revient à la même longueur de fenêtre que dans le premier exemple, tout en utilisant une fenêtre rectangulaire au lieu de la fenêtre « douce » de Hanning, l'étalement spectral est plus faible et les lignes sur le spectrogramme plus fines, mais on a par contre des fuites spectrales beaucoup plus importantes, caractérisées par un manque de contraste dans la représentation du spectrogramme. 
+
+
+Les points importants quant à l'utilisation du spectrogramme sont donc: 
+la longueur de fenêtre pour ajuster la précision temporelle, au prix d'un étalement spectral qui peut devenir rédhibitoire
+le choix de la fenêtre qui va conditionner le contraste du spectrogramme, pour une longueur de fenêtre donnée.
+
+\section{Pertinence et choix d'un mode de représentation}
+
+La représentation du signal en temporel est particulièrement adapté à l'analyse de ses variations temporelles  (par exemple l'attaque ou l'extinction d'un son).
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Au contraire, cette représentation apporte peu d'information lorsqu'on s'intéresse plutôt au contenu spectral du signal. Sur la figure suivante, on voit que la représentation spectrale de ces deux signaux de parole permet assez bien de différentier la voix « tendue »  de la voix « relâchée ».
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Voix relâchée                                          Voix tendue
+
+
+Un autre intérêt de la représentation du signal à l'aide d'un spectrogramme est de pouvoir séparer des sources sonores, ou du moins de pouvoir repérer des événements, là où le signal temporel englobe tout de façon indifférenciée.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Pour des sons très harmoniques (par exemple voix chantée), le spectrogramme permet de repérer très facilement la fondamentale ainsi que la série harmonique.
+Sur la figure ci-dessous est représenté le spectrogramme d'un chanteur effectuant un glissando du bas au haut de sa tessiture puis l'inverse.
+
+
+
+
+
+
+
+
+
+
+
+
+Dans le cas de la voix parlée, on peut repérer les parties voisée (harmoniques) ou non, et même « lire » quels sont les phonèmes prononcés avec un peu d'expérience. Nous reviendrons là dessus dans le chapitre 5.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+   « C'est une maison qui s'trouve au Luxembourg »
+
+
+\section{Le spectre moyenné (Long Term Average Spectrum   LTAS)}
+
+Il est également très utile dans certains cas de représenter une partie du signal sous forme de spectre moyenné. Le spectre moyenné, comme son nom l'indique, consiste à faire la moyenne du spectrogramme sur l'ensemble des instants d'un intervalle de temps plus ou moins long . Ce peut être sur le signal tout entier comme sur un morceau particulier.
+
+Pour examiner une voyelle en particulier dans un signal de parole par exemple, il est tout à fait approprié de moyenner le spectrogramme sur l'intervalle de temps de cette voyelle, de façon à pouvoir estimer les formants ou bien le contenu spectral de cette voyelle. Ainsi, sur la figure suivante, on peut examiner le spectre moyenné de la production vocale d'un chanteur sur un /a/. Cette représentation permet de caractériser ses premiers formants vocaliques, ainsi que son formant du chanteur (renforcement énergétique dans la zone 2000-4000 Hz), caractéristique des chanteurs lyriques et des acteurs de théâtre.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+\chapter{Systèmes linéaires et filtrage}
+
+
+
+
+
+\section{Réponse impulsionnelle d'un système}
+
+Il est possible de décrire un système S par l'intermédiaire de sa réponse à un signal extrêmement bref. Mathématiquement, il s'agit de modéliser sa réponse à une impulsion de Dirac.
+
+La réponse impulsionnelle d'un système est donc définie comme la sortie d'un système lorsque l'entrée est une impulsion de Dirac.
+
+g(t) = S{d(t)}
+
+Lorsque nous claquons des mains dans une pièce pour évaluer son acoustique, son temps de réverbération, etc, nous ne faisons pas moins en fait qu'évaluer la réponse impulsionnelle de ce système pièce. 
+Cela semble physiquement logique d'entendre du son dans la pièce au moment où nous claquons dans nos mains, voire après, mais certainement pas avant . Pourtant, en mathématique, on conçoit que la réponse impulsionnelle puisse être non nulle avant l'instant de l'impulsion d'entrée.
+On définit ainsi les système causaux comme les systèmes dont la réponse impulsionnelle est nulle avant l'instant d'impulsion, autrement dit      g(t<0) = 0  .
+Pour un système causal, le signal de sortie à l'instant t dépend du signal d'entrée aux instants t'< t. La durée de la réponse impulsionnelle g(t) correspond au temps de réponse du système.
+
+
+
+Nous avons vu que la réponse impulsionnelle correspond à la sortie y(t) du système lorsque l'entrée x(t) = '(t) .
+Connaissant la réponse impulsionnelle g(t) d'un système, on peut ensuite prédire sa réponse y(t) à n'importe quel signal d'entrée x(t) grâce à la formule :
+
+                                    y(t) = x(t) * g(t)
+
+
+
+
+
+
+
+
+
+
+
+
+\section{Réponse en fréquence d'un système}
+
+La réponse en fréquence d'un système correspond à la transformée de Fourier de la réponse impulsionnelle du système.
+
+
+G(f) décrit comment la distribution spectrale d'un signal est modifiée ou "filtrée" par le système S. Il est important de noter que le système peut seulement modifier des composantes spectrales mais ne peut en aucun cas en créer de nouvelles.
+|G(f)|  est le gain du système, c'est à dire la façon dont il modifie les amplitudes de chaque composantes spectrales.
+Arg[G(f)] représente le déphasage causé par le système, c'est à dire le « retard » qu'il impose à certaines composantes spectrales.
+
+
+
+
+
+
+
+
+
+
+
+
+
+La réponse en fréquence, comme la réponse impulsionnelle permet de décrire complètement le système et de prédire la réponse du système à n'importe quelle entrée. 
+
+y(t) = x(t) * g(t)    '     Y(f)  = X(f) . G(f)
+
+
+
+\section{Fonction de transfert d'un système (et définition de la Transformée en Z)}
+
+La fonction de transfert d'un système est défini comme la transformée en Z de la réponse impulsionnelle du système.
+
+
+La Transformée en Z est très proche de la Transformée de Fourier et se définit par :
+
+La TZ est un outil de caractérisation des systèmes de traitement numérique des signaux. 
+On retrouve la définition de la TF en posant z = ei2' f.
+
+La TZ présente aussi comme propriété :
+
+* Retard :                     x(k 'ko)     '     X(z) z-ko
+
+*Convolution  :        x(k)*y(k)    '     X(z).Y(z)
+                       x(k).y(k)    '     X(z)*Y(z)
+                       
+
+Donc la réponse impulsionnelle g(n) et la fonction de transfert G(z) permettent de décrire un système numérique.
+
+y(n) = x(n) * g(n)    '     Y(z)  = X(z) . G(z)
+
+
+
+
+
+
+\section{Filtrage}
+
+Le filtrage linéaire est une opération utile pour sélectionner une partie du spectre d'un signal ou en atténuer une partie, pour 
+modifier le contenu spectral d'un signal  ou encore pour m
+odéliser la transformation d'un signal dans un conduit, dans une pièce' 
+
+\subsection{Filtre idéaux}
+On définit classiquement 4 types de filtres idéaux : 
+les filtres passe-bas qui laissent intact les basses fréquences d'un signal et en atténuent les hautes fréquences.
+les filtres passe-haut qui laissent intact les hautes fréquences d'un signal et en atténuent les basses fréquences.
+Les filtres passe-bande qui sélectionnent une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.
+Les filtres coupe-bande, qui atténuent fortement une partie du spectre d'un signal autour d'une fréquence spécifiée, avec une largeur plus ou moins grande.
+
+
+Pour tous ces filtres, on définit des fréquences de coupure, c'est à dire les fréquences pour lesquelles le spectre du signal d'entrée va être atténué d'un facteur '2. 
+
+
+Le passe bas à donc une fréquence de coupure dans les médiums / hautes fréquences, le passe haut une fréquence de coupure dans les médiums / basses fréquences. Les passe bande et coupe bande possèdent deux fréquences de coupure autour de la fréquence centrale sur laquelle ils se centrent. 
+On spécifie également la pente de l'atténuation de ces filtres, en dB par octave. Cela apporte une information sur la sélectivité du filtre.
+
+Enfin, pour les filtres passe-bande et coupe-bande, on détermine leur largeur de bande, c'est à dire la différence entre leurs deux fréquences de coupure, ce qui renseigne sur le caractère plus ou moins « aigu » du filtre.
+
+
+\subsection{Exemples de filtres naturels} 
+
+On peut modéliser la sensation d'intensité auditive par un filtre. L'oreille humaine est particulièrement sensible entre 3 et 4 kHz 
+
+On peut également modéliser la production phonatoire par un modèle source-filtre, où le rôle de la source est joué par les cordes vocales qui produisent un son harmonique avec une distribution de l'énergie assez plate en fréquence. Le conduit vocal, les fosses nasales ainsi que la place des articulateurs (langue, mâchoire, lèvres) peuvent être modélisés par un filtre qui modifie le son glottique pour produire le son tel que nous le percevons à la sortie des lèvres d'un locuteur.
+
+
+Bibliographie
+
+David B. Conférence SFA sur le vibrato. Décembre 2003. Le mans.
+
+Garnier M, Dubois D, Poitevineau J, Henrich N, Castellengo M. Perception et description verbale de la qualité vocale dans le chant lyrique : une approche cognitive. Journées d'Études sur la Parole, Fez, 2004. . 
+
+Henrich N. Etude de la source glottique en voix parlée et chantée. Thèse de doctorat, UPMC. 2001.
+
+Pelorson X. Conférence SFA « cordes vocales et lèvres vibrantes ». Juillet 2004. Paris
+
+Charbit M. Polycopié ENST. Bases de Traitement du signal. 2004
+
+Heiser T. Traitement du signal, support de cours. UFR Sciences Physiques. université Louis Pasteur, Strasbourg
+
+Feng. Traitement du signal, support de cours. ENSERG. INP Grenoble. 2000.
+
+Dutoit T.  Introduction au Traitement Automatique de la Parole Faculté Polytechnique
+de Mons. 2004
+