From fe9d2c91f89cadc35442cd1e9ef9e9e46bf595e0 Mon Sep 17 00:00:00 2001 From: yomguy Date: Wed, 3 Oct 2012 00:40:06 +0200 Subject: [PATCH 1/1] init 0.5.1 --- Abstract.aux | 31 + Abstract.log | 505 + Abstract.tex | 16 + Abstract.tex~ | 16 + Abstracts.aux | 31 + Biblio.bib | 74 + Biblio.bib~ | 74 + Biblio.log | 998 + Biblio.tex | 80 + Biblio.tex.backup | 19 + Biblio.tex~ | 71 + Fiche_TZ.aux | 25 + Fiche_TZ.log | 509 + Fiche_TZ.out | 0 Fiche_TZ.pdf | Bin 0 -> 327251 bytes Fiche_TZ.tex | 48 + Fiche_TZ.tex.backup | 61 + Fiche_TZ.tex~ | 46 + IntroMethFreq.log | 16555 ++++++ IntroMethFreq.tex | 380 + IntroMethFreq.tex.backup | 386 + IntroMethFreq.tex~ | 380 + Poly2005-Meava.doc | Bin 0 -> 8496128 bytes Poly2005.odt | Bin 0 -> 7189391 bytes Poly2005.pdf | Bin 0 -> 3239010 bytes ReprSignaux.tex | 126 + ReprSignaux.tex.backup | 125 + ReprSignaux.tex~ | 126 + Signal-CPDA-CNAM.aux | 157 + Signal-CPDA-CNAM.bbl | 50 + Signal-CPDA-CNAM.blg | 49 + Signal-CPDA-CNAM.brf | 0 Signal-CPDA-CNAM.log | 776 + 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Emergency stop. +<*> Abstract.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 7 strings out of 94436 + 158 string characters out of 1174733 + 48348 words of memory out of 1000000 + 3281 multiletter control sequences out of 10000+50000 + 3640 words of font info for 14 fonts, out of 500000 for 2000 + 625 hyphenation exceptions out of 8191 + 6i,0n,6p,705b,19s stack positions out of 1500i,500n,5000p,200000b,5000s +PDF statistics: + 0 PDF objects out of 300000 + 0 named destinations out of 131072 + 1 words of extra memory for PDF output out of 65536 +No pages of output. diff --git a/Abstract.tex b/Abstract.tex new file mode 100644 index 0000000..11eb418 --- /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) 2006-2011 Guillaume Pellerin} +\end{figure} diff --git a/Abstract.tex~ b/Abstract.tex~ new file mode 100644 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 index 0000000..61404db --- /dev/null +++ b/Abstracts.aux @@ -0,0 +1,31 @@ +\relax +\@setckpt{Abstracts}{ +\setcounter{page}{3} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{part}{0} +\setcounter{chapter}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{figure}{0} +\setcounter{table}{0} +\setcounter{float@type}{4} +\setcounter{parentequation}{0} +\setcounter{note}{0} +\setcounter{bex}{0} +\setcounter{subfigure}{0} +\setcounter{lofdepth}{1} +\setcounter{subtable}{0} +\setcounter{lotdepth}{1} +\setcounter{Item}{0} +\setcounter{Hfootnote}{0} +\setcounter{section@level}{0} +} diff --git a/Biblio.bib b/Biblio.bib new file mode 100644 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. 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Emergency stop. +<*> Biblio.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 5 strings out of 93945 + 118 string characters out of 1162060 + 47725 words of memory out of 1500000 + 3383 multiletter control sequences out of 10000+50000 + 3640 words of font info for 14 fonts, out of 1200000 for 2000 + 743 hyphenation exceptions out of 8191 + 5i,0n,4p,229b,16s stack positions out of 5000i,500n,6000p,200000b,5000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/Biblio.tex b/Biblio.tex new file mode 100644 index 0000000..af34a19 --- /dev/null +++ b/Biblio.tex @@ -0,0 +1,80 @@ + + + +\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 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 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. 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\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 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} + diff --git a/IntroMethFreq.log b/IntroMethFreq.log new file mode 100644 index 0000000..5add037 --- /dev/null +++ b/IntroMethFreq.log @@ -0,0 +1,16555 @@ +This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2007.5.31) 4 JUN 2007 00:25 +entering extended mode + %&-line parsing enabled. +**IntroMethFreq.tex +(./IntroMethFreq.tex +LaTeX2e <2005/12/01> +Babel and hyphenation patterns for english, usenglishmax, dumylang, noh +yphenation, croatian, bulgarian, russian, ukrainian, czech, slovak, danish, dut +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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+! Undefined control sequence. + ...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'). 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LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.7 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.7 \begin{figure}[ + h] +You're in trouble here. Try typing to proceed. +If that doesn't work, type X 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'). 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Missing $ inserted. + + $ +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 for immediate help. + ... + +l.10 \caption + {Exemple d'une réprésentation temporelle d'un signal musical.} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.12 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.12 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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. + + } +l.12 \end{figure} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H 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. 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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 for immediate help. + ... + +l.19 \subsection{C + lassification dimensionnelle} +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.21 O + n peut différentier les signaux selon leur nombre de variables libres.... + +You're in trouble here. 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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 for immediate help. + ... + +l.23 \subsection{C + lassification phénoménologique} +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.25 L + a première distinction porte sur notre capacité à prédire l'évolution ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X 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! +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 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 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 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 h 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 m 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 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 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 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 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 d in font nullfont! +Missing character: There is no u 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 ( 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 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 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 ) 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 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 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 p in font nullfont! 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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! +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 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! +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 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! +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 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! +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 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 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 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 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 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 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! +Missing character: There is no r 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 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! +Missing character: There is no r 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 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 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 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 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 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 . 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 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 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 , 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 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 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 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! +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 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 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 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 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 d 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 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 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 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 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 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 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 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 h 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 j 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 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 , 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 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 e 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 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! +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 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 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 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 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 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 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 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 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 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! +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 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 q 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 i 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 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 c 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 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! +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 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 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 d 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 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 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 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 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! +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 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! 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.33 I + l est également possible de discrétiser le signal en amplitude. On app... + +You're in trouble here. 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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 for immediate help. + ... + +l.36 \subsection{C + lassification énergétique} +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.38 O + n distingue en particulier deux types de signaux : +You're in trouble here. 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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.40 \begin{itemize} + +You're in trouble here. 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+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'). 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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'). 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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. + ...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 for immediate help. + ... + +l.52 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or 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. + + $ +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 for immediate help. + ... + +l.55 \caption + {Réprésentation temporelle d'un signal rectangulaire ou << ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.57 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.57 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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. + + } +l.57 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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! +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 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 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 é 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 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 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 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 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 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 é 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 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 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 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 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 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 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 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 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 y 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 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 . 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. + ...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 ^^@ + [] + + +Overfull \hbox (11.98615pt too wide) in paragraph at lines 62--63 +\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 )$ + [] + + +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 for immediate help. + ... + +l.64 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or 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! +Missing character: There is no 6 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. + + $ +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 for immediate help. + ... + +l.67 \caption + {Réprésentation temporelle d'une impulsion de Dirac.} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.69 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.69 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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. + + } +l.69 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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! +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 e 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   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 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 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 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 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 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 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 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 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 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 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 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 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! +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 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 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 ' 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 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 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 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 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 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 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 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 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 ( 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 b 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 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 71--73 +[] + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 71--73 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (6.00235pt too wide) in paragraph at lines 71--73 +\OML/cmm/m/it/10 n$ + [] + + +Overfull \hbox (36.02188pt too wide) in paragraph at lines 71--73 +\OT1/cmr/m/n/10 [](\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (27.32295pt too wide) in paragraph at lines 71--73 +\OT1/cmr/m/n/10 ^^F\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (17.1239pt too wide) in paragraph at lines 71--73 +\OML/cmm/m/it/10 nT\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (13.05551pt too wide) in paragraph at lines 71--73 +\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (7.67015pt too wide) in paragraph at lines 71--73 +\OML/cmm/m/it/10 R$ + [] + + +Overfull \hbox (13.235pt too wide) in paragraph at lines 71--73 +\OML/cmm/m/it/10 nT$ + [] + + +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$ + [] + + +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 for immediate help. + ... + +l.74 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or 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. + + $ +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 for immediate help. + ... + +l.77 \caption + {Réprésentation temporelle d'un peigne de Dirac.} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.79 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.79 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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. + + } +l.79 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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 74--80 +[] + [] + + +Overfull \hbox (240.28403pt too wide) in paragraph at lines 74--80 +[]\OML/cmm/m/it/10 rain[]irac[][]$ + [] + +! Undefined control sequence. +l.82 \subsection + {Sinus cardinal} +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 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 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 d 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 l in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 82--83 +[] + [] + +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 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 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 d 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 l 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 é 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 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 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 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   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.84 ...(t) = \frac{\sin(t)}{t}$ où $t \in \mathbb + {R}\backslash \lbrace 0 \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 . 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 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. + ... Reference `sin_card' on page \thepage + \space undefined\on@line . +l.84 ...tion est donnée à la figure \ref{sin_card} + . +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 `sin_card' on page undefined on input line 84. + +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 84--85 +[] + [] + + +Overfull \hbox (38.66667pt too wide) in paragraph at lines 84--85 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (21.51953pt too wide) in paragraph at lines 84--85 +[]$ + [] + + +Overfull \hbox (13.05551pt too wide) in paragraph at lines 84--85 +\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (27.67021pt too wide) in paragraph at lines 84--85 +\OML/cmm/m/it/10 R\OMS/cmsy/m/n/10 nf\OT1/cmr/m/n/10 0\OMS/cmsy/m/n/10 g$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 84--85 +[] + [] + + +! LaTeX Error: Environment figure undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.86 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or 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.88 \includegraphics + [width=8cm]{img/sinc} +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 n in font nullfont! +Missing character: There is no c in font nullfont! + +! LaTeX Error: \caption outside float. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.89 \caption + {Réprésentation temporelle du sinus cardinal : $y=\frac{\si... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + +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 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 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 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 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 d 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 l in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no ( in font nullfont! +! Undefined control sequence. +l.89 ...frac{\sin(x)}{x}$ ($x\texttt{} \in \mathbb + {R}\backslash \lbrace 0 \r... +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no ) in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.91 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 86--92 +[] + [] + + +Overfull \hbox (15.81711pt too wide) in paragraph at lines 86--92 +\OML/cmm/m/it/10 y \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (23.03342pt too wide) in paragraph at lines 86--92 +[]$ + [] + + +Overfull \hbox (15.15967pt too wide) in paragraph at lines 86--92 +\OML/cmm/m/it/10 x[] \OMS/cmsy/m/n/10 2 + [] + + +Overfull \hbox (27.67021pt too wide) in paragraph at lines 86--92 +\OML/cmm/m/it/10 R\OMS/cmsy/m/n/10 nf\OT1/cmr/m/n/10 0\OMS/cmsy/m/n/10 g$ + [] + +! Undefined control sequence. +l.94 \section + {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 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 94--95 +[] + [] + +! Undefined control sequence. +l.96 \subsection + {Rappels sur la décomposition en série de Fourier des signau... +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 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 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 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 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 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 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 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 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 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 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 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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 96--97 +[] + [] + +Missing character: There is no J 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 p in font nullfont! +Missing character: There is no h 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 a 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 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 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 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 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 cmbx10! +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 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 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 é 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 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 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 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 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 c 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 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 é 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 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 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 98--99 +[] + [] + + +Overfull \hbox (18.1527pt too wide) in paragraph at lines 98--99 +\OT1/cmr/bx/n/10 pri- + [] + + +Overfull \hbox (33.38177pt too wide) in paragraph at lines 98--99 +\OT1/cmr/bx/n/10 odique + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 100--100 +[] + [] + + +Overfull \hbox (214.36383pt too wide) detected at line 102 +[]\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = \OML +/cmm/m/it/10 a[] \OT1/cmr/m/n/10 + [] \OML/cmm/m/it/10 a[] []\OT1/cmr/m/n/10 (2 +\OML/cmm/m/it/10 ^^Ynf[]t\OT1/cmr/m/n/10 ) + [] \OML/cmm/m/it/10 b[] []\OT1/cmr +/m/n/10 (2\OML/cmm/m/it/10 ^^Ynf[]t\OT1/cmr/m/n/10 ) + [] + +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 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! 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+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! +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 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 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 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 t in font nullfont! +Missing character: There is no ô 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 r 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 h 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 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 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 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 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 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 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 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 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 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 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 , 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 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 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 , 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 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 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 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 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 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 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 à 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 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 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 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 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 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 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 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 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 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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 104--105 +[] + [] + + +Overfull \hbox (9.38199pt too wide) in paragraph at lines 104--105 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (19.48601pt too wide) in paragraph at lines 104--105 +\OT1/cmr/bx/n/10 fon- + [] + + +Overfull \hbox (15.8124pt too wide) in paragraph at lines 104--105 +\OT1/cmr/bx/n/10 da- + [] + + +Overfull \hbox (25.07626pt too wide) in paragraph at lines 104--105 +\OT1/cmr/bx/n/10 men- + [] + + +Overfull \hbox (18.52766pt too wide) in paragraph at lines 104--105 +\OT1/cmr/bx/n/10 tale + [] + + +Overfull \hbox (16.55786pt too wide) in paragraph at lines 104--105 +\OML/cmm/m/it/10 n > + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 104--105 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (20.5485pt too wide) in paragraph at lines 104--105 +\OT1/cmr/bx/n/10 har- + [] + + +Overfull \hbox (47.18169pt too wide) in paragraph at lines 104--105 +\OT1/cmr/bx/n/10 moniques + [] + + +Overfull \hbox (9.38199pt too wide) in paragraph at lines 104--105 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 104--105 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 104--105 +\OT1/cmr/m/n/10 2 \OMS/cmsy/m/n/10 ^^C + [] + + +Overfull \hbox (9.38199pt too wide) in paragraph at lines 104--105 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 104--105 +\OT1/cmr/m/n/10 +$ + [] + + +Overfull \hbox (7.7778pt too wide) in paragraph at lines 104--105 +\OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 104--105 +\OT1/cmr/m/n/10 3 \OMS/cmsy/m/n/10 ^^C + [] + + +Overfull \hbox (9.38199pt too wide) in paragraph at lines 104--105 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (9.38199pt too wide) in paragraph at lines 104--105 +\OML/cmm/m/it/10 f[]$ + [] + +! 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. + +Missing character: There is no D in font nullfont! +Missing character: There is no é 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 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 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! 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+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 . 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 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 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 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 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 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 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 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 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 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 : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 109--110 +[] + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 109--110 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (22.81949pt too wide) in paragraph at lines 109--110 +\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (39.9341pt too wide) in paragraph at lines 109--110 +\OML/cmm/m/it/10 TF\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm +/m/it/10 t\OT1/cmr/m/n/10 ))$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 111--111 +[] + [] + +! Undefined control sequence. +l.112 \boxeq + {X(f)=TF(x(t))=\int_{-\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. + +! 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. + + +Overfull \hbox (167.26723pt too wide) detected at line 113 +[][] + [] + +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 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 d in font nullfont! +Missing character: There is no é 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 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 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 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 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 +[][] + [] + +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 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! +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 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! +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 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 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 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! +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 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! +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 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 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! +Missing character: There is no c 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 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 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! +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 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 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 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 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! +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 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 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 ' 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 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 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 H 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 z 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 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 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! + +Overfull \hbox (42.81949pt too wide) in paragraph at lines 121--122 +[]$\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 121--122 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (25.87486pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 mod- + [] + + +Overfull \hbox (14.8541pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 ule + [] + + +Overfull \hbox (14.15964pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 ar- + [] + + +Overfull \hbox (15.97214pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 gu- + [] + + +Overfull \hbox (25.3957pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 ment + [] + + +Overfull \hbox (28.37506pt too wide) in paragraph at lines 121--122 +\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$ + [] + + +Overfull \hbox (25.45958pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 spec- + [] + + +Overfull \hbox (14.47911pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 tre + [] + + +Overfull \hbox (60.05522pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 d'amplitude + [] + + +Overfull \hbox (44.65286pt too wide) in paragraph at lines 121--122 +[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cm +r/m/n/10 ))$ + [] + + +Overfull \hbox (25.45958pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 spec- + [] + + +Overfull \hbox (14.47911pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 tre + [] + + +Overfull \hbox (11.65967pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 de + [] + + +Overfull \hbox (28.17482pt too wide) in paragraph at lines 121--122 +\OT1/cmr/bx/n/10 phase + [] + + +Overfull \hbox (5.97226pt too wide) in paragraph at lines 121--122 +\OML/cmm/m/it/10 f$ + [] + + +Overfull \hbox (76.07053pt too wide) in paragraph at lines 123--124 +[]\OT1/cmr/bx/n/10 Remarques + [] + + +Overfull \hbox (16.61102pt too wide) in paragraph at lines 123--124 +\OT1/cmr/bx/n/10 im- + [] + + +Overfull \hbox (21.02768pt too wide) in paragraph at lines 123--124 +\OT1/cmr/bx/n/10 por- + [] + + +Overfull \hbox (33.60535pt too wide) in paragraph at lines 123--124 +\OT1/cmr/bx/n/10 tantes: + [] + +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 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 è 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 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 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 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 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 e in font nullfont! +Missing character: There is no 2 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 p in font nullfont! +Missing character: There is no h 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 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 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 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 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 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 u 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 e 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 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 r 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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LaTeX Error: The font size command \normalsize is not defined: + there is probably something wrong with the class file. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.127 \end{itemize} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map} +! Undefined control sequence. + \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. + \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. + \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. + \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. + \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. + +\newlabel{sin_card}{{}{}} +! Undefined control sequence. + \newlabel{DecompFour}{{1}{\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{DecompFour}{{1}{}} +! Undefined control sequence. + \newlabel{fourier}{{2}{\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{fourier}{{2}{}} +! Undefined control sequence. + \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. + +\newlabel{fourier_inv}{{3}{}} +] +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 , 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! 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+Missing character: There is no s 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 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 . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 130--131 +[] + [] + + +Overfull \hbox (15.05211pt too wide) in paragraph at lines 130--131 +\OML/cmm/m/it/10 TF$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 130--131 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (22.81949pt too wide) in paragraph at lines 130--131 +\OML/cmm/m/it/10 X\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 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 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! 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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 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 c 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 b 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 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 l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! 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+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 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 . 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! 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+Missing character: There is no m 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 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 à 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 < 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 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 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 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 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 > in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (42.81949pt too wide) in paragraph at lines 132--133 +[]$\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (21.08322pt too wide) in paragraph at lines 132--133 +\OT1/cmr/bx/n/10 con- + [] + + +Overfull \hbox (22.20128pt too wide) in paragraph at lines 132--133 +\OT1/cmr/bx/n/10 tenu + [] + + +Overfull \hbox (25.45958pt too wide) in paragraph at lines 132--133 +\OT1/cmr/bx/n/10 spec- + [] + + +Overfull \hbox (17.99295pt too wide) in paragraph at lines 132--133 +\OT1/cmr/bx/n/10 tral + [] + +! Undefined control sequence. +l.134 \subsection + {Le produit de convolution} +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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 134--135 +[] + [] + +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 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 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 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 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 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 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 é 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 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 : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 136--137 +[] + [] + + +Overfull \hbox (21.08322pt too wide) in paragraph at lines 136--137 +\OT1/cmr/bx/n/10 con- + [] + + +Overfull \hbox (15.33325pt too wide) in paragraph at lines 136--137 +\OT1/cmr/bx/n/10 vo- + [] + + +Overfull \hbox (13.4166pt too wide) in paragraph at lines 136--137 +\OT1/cmr/bx/n/10 lu- + [] + + +Overfull \hbox (19.80545pt too wide) in paragraph at lines 136--137 +\OT1/cmr/bx/n/10 tion + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 136--137 +\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 136--137 +\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (22.1042pt too wide) in paragraph at lines 136--137 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OML/cmm +/m/it/10 ? + [] + + +Overfull \hbox (16.65051pt too wide) in paragraph at lines 136--137 +\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 138--138 +[] + [] + +! Undefined control sequence. +l.139 \boxeq + {x(t)\star y(t) = \int_{-\infty}^{+\infty}x(u)y(t-u)\ud u = \in... +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 ...) = \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. + + +Overfull \hbox (241.50002pt too wide) detected at line 140 +[][] + [] + +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 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 ' 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 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 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 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 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 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! +Missing character: There is no E 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 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 : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 142--143 +[] + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 144--144 +[] + [] + + +Overfull \hbox (68.75343pt too wide) detected at line 146 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OML/cmm +/m/it/10 ? ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = \OML/cmm/m +/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) + [] + +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 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 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 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 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 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 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 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 é 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 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 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 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 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 à 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 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 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 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 : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 148--149 +[] + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 148--149 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 148--149 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 148--149 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 148--149 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 150--150 +[] + [] + + +Overfull \hbox (100.50352pt too wide) detected at line 152 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OML/cmm +/m/it/10 ? ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@ \OML/cmm +/m/it/10 t[]\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/i +t/10 t \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) + [] + +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 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 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 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 ' 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 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 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 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 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 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 é 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 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 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 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 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 à 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 n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no î 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 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 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 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 e 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 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 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 ' 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 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 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 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 ' 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 ' 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 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 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 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 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 b 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 154--155 +[] + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 154--155 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 154--155 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 154--155 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 154--155 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 156--156 +[] + [] + + +Overfull \hbox (86.89238pt too wide) detected at line 158 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cms +y/m/n/10 ^^A \OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/ +m/n/10 ^^@ \OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 x\OT1/cmr/m +/n/10 (\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 ) + [] + +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 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 , 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 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 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 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 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 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 e 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 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 ( 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 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 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 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 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 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 à 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 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 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 > 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 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 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 : 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 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 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 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 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 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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 160--161 +[] + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 160--161 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 160--161 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 160--161 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (13.235pt too wide) in paragraph at lines 160--161 +\OML/cmm/m/it/10 nT$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 160--161 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 160--161 +\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (12.23267pt too wide) in paragraph at lines 160--161 +\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (12.23267pt too wide) in paragraph at lines 160--161 +\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 T$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 162--162 +[] + [] + + +Overfull \hbox (106.24301pt too wide) detected at line 164 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OML/cmm +/m/it/10 ? \OT1/cmr/m/n/10 [](\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = ^^F\OML/cm +m/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@ \OML/cmm/m/ +it/10 nT\OT1/cmr/m/n/10 ) + [] + +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! +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 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 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! 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+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! +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. + \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. + \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 )$ + [] + + +Overfull \hbox (13.05551pt too wide) in paragraph at lines 221--222 +\OML/cmm/m/it/10 t \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 (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. + \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. + ...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. + \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. + ...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! +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 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 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 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 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 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 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! + +Overfull \hbox (15.8414pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 a > + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 236--237 +\OT1/cmr/m/n/10 1$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (9.38199pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (27.6597pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (44.10777pt too wide) in paragraph at lines 236--237 +[]\OT1/cmr/m/n/10 (2\OML/cmm/m/it/10 ^^Yf[]t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (33.375pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 X\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (21.9827pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (27.20602pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (49.39366pt too wide) in paragraph at lines 236--237 +[]\OT1/cmr/m/n/10 (2\OML/cmm/m/it/10 ^^Yaf[]t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (32.33336pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 Y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (33.47034pt too wide) in paragraph at lines 236--237 +[]\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (19.9375pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 f[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (14.66788pt too wide) in paragraph at lines 236--237 +\OML/cmm/m/it/10 af[]$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 236--237 +[] + [] + + +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 for immediate help. + ... + +l.238 \begin{figure} + [tp] +Your command was ignored. +Type I to replace it with another command, +or 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! +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 ] 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! +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 n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +! Missing $ inserted. + + $ +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 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 to proceed. +If that doesn't work, type X to quit. + +! Missing } inserted. + + } +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 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 character: There is no a in font nullfont! +Missing character: There is no = in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no ï in font cmr10! +Missing character: There is no x 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 ) in font nullfont! +Missing character: There is no é in font cmr10! +Missing character: There is no f in font nullfont! +! Missing $ inserted. + + $ +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 for immediate help. + ... + +l.244 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing } inserted. + + } +l.244 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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 +e\OT1/cmr/m/n/10 ][][]$$\OML/cmm/m/it/10 surunsignalsinusodal$$defrquence$$[] \ +OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (21.94449pt too wide) in paragraph at lines 238--245 +\OT1/cmr/m/n/10 50 []$ + [] + + +Overfull \hbox (27.6597pt too wide) in paragraph at lines 238--245 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (44.10777pt too wide) in paragraph at lines 238--245 +[]\OT1/cmr/m/n/10 (2\OML/cmm/m/it/10 ^^Yf[]t\OT1/cmr/m/n/10 )$ + [] + +Missing character: There is no é in font cmbx10! +Missing character: There is no è in font cmbx10! +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 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 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 . 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 d 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 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 r 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 (39.72897pt too wide) in paragraph at lines 246--248 +\OT1/cmr/bx/n/10 Thorme + [] + + +Overfull \hbox (11.65967pt too wide) in paragraph at lines 246--248 +\OT1/cmr/bx/n/10 de + [] + + +Overfull \hbox (21.70126pt too wide) in paragraph at lines 246--248 +\OT1/cmr/bx/n/10 Par- + [] + + +Overfull \hbox (13.6402pt too wide) in paragraph at lines 246--248 +\OT1/cmr/bx/n/10 se- + [] + + +Overfull \hbox (14.21518pt too wide) in paragraph at lines 246--248 +\OT1/cmr/bx/n/10 val + [] + + +Overfull \hbox (7.95831pt too wide) in paragraph at lines 246--248 +\OML/cmm/m/it/10 E$ + [] + +! 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! +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 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 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 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 +[] + [] + +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 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 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 +msy/m/n/10 ,$ + [] + +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 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 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 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 d 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 l in font nullfont! + +Overfull \hbox (10.00002pt too wide) in paragraph at lines 260--261 +\OMS/cmsy/m/n/10 ,$ + [] + + +Overfull \hbox (56.8855pt too wide) in paragraph at lines 261--262 +[]\OT1/cmr/m/n/10 (2\OML/cmm/m/it/10 ^^Yf[]t\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 +, + [] + + +Overfull \hbox (31.62578pt too wide) in paragraph at lines 261--262 +[]\OT1/cmr/m/n/10 [\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f \OMS +/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (21.04869pt too wide) in paragraph at lines 261--262 +\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 ) + + [] + + +Overfull \hbox (22.46187pt too wide) in paragraph at lines 261--262 +\OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f \OT1/cmr/m/n/10 + + [] + + +Overfull \hbox (16.04868pt too wide) in paragraph at lines 261--262 +\OML/cmm/m/it/10 f[]\OT1/cmr/m/n/10 )]$ + [] + +! 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. + +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 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 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 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 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 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 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 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 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 264--265 +[] + [] + +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 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 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 l 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 266--267 +[] + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 268--268 +[] + [] + +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 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 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 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 g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no â 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 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 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 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 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 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 - 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 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 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 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 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 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 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 g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no â 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 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 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 . in font nullfont! +! 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. + +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 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 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 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 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 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 ' 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 - 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 274--275 +[] + [] + +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 é in font cmbx10! +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 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 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 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 i 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 276--277 +[] + [] + + +Overfull \hbox (13.4166pt too wide) in paragraph at lines 276--277 +\OT1/cmr/bx/n/10 in- + [] + + +Overfull \hbox (18.31242pt too wide) in paragraph at lines 276--277 +\OT1/cmr/bx/n/10 ter- + [] + + +Overfull \hbox (19.43047pt too wide) in paragraph at lines 276--277 +\OT1/cmr/bx/n/10 cor- + [] + + +Overfull \hbox (17.35406pt too wide) in paragraph at lines 276--277 +\OT1/cmr/bx/n/10 rla- + [] + + +Overfull \hbox (19.80545pt too wide) in paragraph at lines 276--277 +\OT1/cmr/bx/n/10 tion + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 278--278 +[] + [] + +! 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. + + +Overfull \hbox (124.01978pt too wide) detected at line 280 +[]\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^\\OT1/cmr/m/n/10 ) = +[][] \OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\O +ML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 + \OML/cmm/ +m/it/10 ^^\\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 t + [] + +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 é in font cmbx10! +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 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 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 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 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 r 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 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 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 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 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 ' 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 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   in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 282--283 +[] + [] + + +Overfull \hbox (15.8124pt too wide) in paragraph at lines 282--283 +\OT1/cmr/bx/n/10 au- + [] + + +Overfull \hbox (14.05548pt too wide) in paragraph at lines 282--283 +\OT1/cmr/bx/n/10 to- + [] + + +Overfull \hbox (19.43047pt too wide) in paragraph at lines 282--283 +\OT1/cmr/bx/n/10 cor- + [] + + +Overfull \hbox (17.35406pt too wide) in paragraph at lines 282--283 +\OT1/cmr/bx/n/10 rla- + [] + + +Overfull \hbox (19.80545pt too wide) in paragraph at lines 282--283 +\OT1/cmr/bx/n/10 tion + [] + + +Overfull \hbox (27.20602pt too wide) in paragraph at lines 282--283 +\OML/cmm/m/it/10 y\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 282--283 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 284--284 +[] + [] + +! Undefined control sequence. +l.285 ...t_{-\infty}^{+\infty}x^*(t) x(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. + + +Overfull \hbox (120.1667pt too wide) detected at line 286 +[]\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 ^^\\OT1/cmr/m/n/10 ) = +[][] \OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )\O +ML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 + \OML/cmm/ +m/it/10 ^^\\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 t + [] + +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 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 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 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 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 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 d 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 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 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 é 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 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 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 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 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 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 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 a 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 m 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 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 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 a in font nullfont! +Missing character: There is no u 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 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 e 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 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 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 à 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 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 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 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 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 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 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 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 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 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 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 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 à 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 i 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 e 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 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 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 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 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! +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 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 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 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 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 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 m in font nullfont! +Missing character: There is no a 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 m 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 . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 288--289 +[] + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (8.09724pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 t[]$ + [] + + +Overfull \hbox (20.99309pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (11.98615pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 t[]\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (7.23265pt too wide) in paragraph at lines 288--289 +\OML/cmm/m/it/10 T$ + [] + +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 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 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! +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 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 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 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 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 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 p 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 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 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 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 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 c 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 s 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 a 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 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 u in font nullfont! +Missing character: There is no x 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 e in font nullfont! +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 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 . in font nullfont! + +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! +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 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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 314--315 +[] + [] + +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 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 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 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 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 ( in font nullfont! +! 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! +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 é 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 é 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 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 é in font cmbx10! +Missing character: There is no é in font cmbx10! +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 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 é in font cmbx10! +Missing character: There is no é in font cmbx10! +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 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 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 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 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 à 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 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 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 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 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 e 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 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 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 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 e in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 316--317 +[] + [] + + +Overfull \hbox (42.7498pt too wide) in paragraph at lines 316--317 +\OT1/cmr/bx/n/10 frquence + [] + + +Overfull \hbox (81.61758pt too wide) in paragraph at lines 316--317 +\OT1/cmr/bx/n/10 d'chantillonnage + [] + + +Overfull \hbox (9.18997pt too wide) in paragraph at lines 316--317 +\OML/cmm/m/it/10 f[]$ + [] + + +Overfull \hbox (18.1527pt too wide) in paragraph at lines 316--317 +\OT1/cmr/bx/n/10 pri- + [] + + +Overfull \hbox (17.72908pt too wide) in paragraph at lines 316--317 +\OT1/cmr/bx/n/10 ode + [] + + +Overfull \hbox (81.61758pt too wide) in paragraph at lines 316--317 +\OT1/cmr/bx/n/10 d'chantillonnage + [] + + +Overfull \hbox (10.13786pt too wide) in paragraph at lines 316--317 +\OML/cmm/m/it/10 T[]$ + [] + + +Overfull \hbox (19.74548pt too wide) in paragraph at lines 316--317 +\OML/cmm/m/it/10 f[] \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (20.1379pt too wide) in paragraph at lines 316--317 +\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =T[]$ + [] + + +Overfull \hbox (10.13786pt too wide) in paragraph at lines 316--317 +\OML/cmm/m/it/10 T[]$ + [] + +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 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 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 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 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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 318--319 +[] + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 320--320 +[] + [] + + +Overfull \hbox (129.3299pt too wide) detected at line 322 +\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) = \OML +/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 ) \OMS/cmsy/ +m/n/10 ^^A [] \OML/cmm/m/it/10 ^^N\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t \OMS/cmsy +/m/n/10 ^^@ \OML/cmm/m/it/10 nT[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 : + [] + +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 c 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 l 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 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 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 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 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 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 à 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 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 q 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 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 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 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 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 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 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 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 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 n 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 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 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 n 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 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 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 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 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 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 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! +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 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 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! +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 , 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 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 u 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 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 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 . in font nullfont! + +Overfull \hbox (41.3983pt too wide) in paragraph at lines 324--325 +[]$\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (14.16663pt too wide) in paragraph at lines 324--325 +\OML/cmm/m/it/10 t \OT1/cmr/m/n/10 = + [] + + +Overfull \hbox (17.89127pt too wide) in paragraph at lines 324--325 +\OML/cmm/m/it/10 nTe$ + [] + + +Overfull \hbox (6.00235pt too wide) in paragraph at lines 324--325 +\OML/cmm/m/it/10 n$ + [] + + +Overfull \hbox (17.77782pt too wide) in paragraph at lines 324--325 +\OMS/cmsy/m/n/10 ^^@1$ + [] + + +Overfull \hbox (17.77782pt too wide) in paragraph at lines 324--325 +\OT1/cmr/m/n/10 +\OMS/cmsy/m/n/10 1$ + [] + + +Overfull \hbox (21.3983pt too wide) in paragraph at lines 324--325 +\OML/cmm/m/it/10 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (19.49542pt too wide) in paragraph at lines 324--325 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 n\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (6.00235pt too wide) in paragraph at lines 324--325 +\OML/cmm/m/it/10 n$ + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 326--326 +[] + [] + + +Overfull \hbox (66.75604pt too wide) detected at line 328 +\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 x[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 nT[]\OT1/cmr/m/n/10 ) + [] + +! 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 +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! +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 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 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 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! +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 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! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 331--332 +[] + [] + +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 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 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 s 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 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 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 é 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 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 ' 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 x 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 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 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 ' 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 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 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 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 m 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 é 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 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 e 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 u 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 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 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 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 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 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 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 d in font nullfont! +Missing character: There is no é 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 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 ' 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 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 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   in font nullfont! +Missing character: There is no : in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 333--334 +[] + [] + + +Overfull \hbox (17.10419pt too wide) in paragraph at lines 333--334 +\OML/cmm/m/it/10 x\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 t\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (15.97214pt too wide) in paragraph at lines 333--334 +\OT1/cmr/bx/n/10 fini + [] + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 335--335 +[] + [] + + +Overfull \hbox (116.28012pt too wide) detected at line 337 +[]\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 n\OT1/cmr/m/n/10 )\OML/cmm/m +/it/10 e[] + [] + +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! 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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! 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Undefined control sequence. + ...: 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 for immediate help. + ... + +l.370 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or 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. + + $ +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 for immediate help. + ... + +l.373 \caption + {Lien entre la fréquence d'échantillonnage d'un signal et ... + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +! LaTeX Error: \begin{document} ended by \end{figure}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.375 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Missing $ inserted. + + $ +l.375 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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. + + } +l.375 \end{figure} + +I've inserted something that you may have forgotten. +(See the 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! 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Undefined control sequence. + ...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 for immediate help. + ... + +l.382 \begin{figure} + [h] +Your command was ignored. +Type I to replace it with another command, +or 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 for immediate help. + ... + +l.385 \caption + {Phénomène de repliement.} +You're in trouble here. Try typing to proceed. +If that doesn't work, type X 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}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.387 \end{figure} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 382--388 +[] + [] + +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 a 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 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 s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! 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Undefined control sequence. + ...Reference `sch_repli' on page \thepage + \space undefined\on@line . +l.390 ... dans le signal (cf. fig. \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 390. + +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 390--391 +[] + [] + + +Overfull \hbox (26.87244pt too wide) in paragraph at lines 390--391 +\OML/cmm/m/it/10 X[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f\OT1/cmr/m/n/10 )$ + [] + + +Overfull \hbox (18.64232pt too wide) in paragraph at lines 390--391 +\OML/cmm/m/it/10 B > + [] + + +Overfull \hbox (15.97745pt too wide) in paragraph at lines 390--391 +[] \OMS/cmsy/m/n/10 ^^@ + [] + + +Overfull \hbox (8.0868pt too wide) in paragraph at lines 390--391 +\OML/cmm/m/it/10 B$ + [] + + +Overfull \hbox (8.0868pt too wide) in paragraph at lines 390--391 +\OML/cmm/m/it/10 B$ + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 390--391 +[] + [] + +Missing character: There is no A in font nullfont! 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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} diff --git a/IntroMethFreq.tex.backup b/IntroMethFreq.tex.backup new file mode 100644 index 0000000..47ab3fe --- /dev/null +++ b/IntroMethFreq.tex.backup @@ -0,0 +1,386 @@ +\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 index 0000000..611352a --- /dev/null +++ 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} diff --git a/Poly2005-Meava.doc b/Poly2005-Meava.doc new file mode 100644 index 0000000000000000000000000000000000000000..ca37a6c09d147374e790b8843fc1a8f180ffbe51 GIT binary patch literal 8496128 zcmeF434oPl+4r9TR8YqiH*k3*k^y0a0ToSDWDyY+1$SJ|%$$LPGv|zFW)N`Ss@KYt ze9O#~-0~GOQY+0;``pSjOLM{31T2&RF>=?kzTf}8pZh%bbLO0xV_1Ff`^oS>=UMOV zTJGz*?&bXEsjXLh=LZ8;21Wl440;ED`+L7&WAFM!9Dn_w=)zV(a5cxhINgr#D!#Yp zdk3&1*a-{(1HsN<7qBbX4TNBKFbI4G3$v{mok4IDCBN<-IF;ugs0!ZP3r?ea z`sbc=-%3~i4tmqxFK%B~PxBBTs_%$S+GOKi!2#ZT?tS;I^q+uxl-~U{w!JRmc~@T7 zZfYO@pL^fGR>`rek4sPK{FS_2dY}8SD#(-2-hFxnGx^?eWDwi|Bf9cbl1Hg$wvNj0 z>hlKtvN;cq+-Ksi&R+RwmH4SKht`45^l!u}+s^+%IkvEu%%{>3->#rtr|>QQ)4y-< z+(NzwZ)VT+{x;R|xy#?x)A6tSE~aPkd9Pp@srRk!dq}D+-4-^c!}feC)1GcgXWPTZ zj<78=JDX~*uCAVt>&Ul-6OJ5zMELb^!t{yb`KlW^a&K;R^RI995vjHFQsJSgd^VHL zS5HgT=klrcTqd6m8yAgl$aS=}XU^8FwZ#!nMRH+>E_yS-@QkU 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