From 454b7e8dc5b6b3cf6912a585ebf6f9329b55173a Mon Sep 17 00:00:00 2001 From: "flyingscorpio@arch-desktop" Date: Sun, 28 Nov 2021 19:01:08 +0100 Subject: [PATCH] Finish cours produit de convolution --- theorie-signal/main.tex | 129 +++++++++++++++++++++++++++++++++------- 1 file changed, 107 insertions(+), 22 deletions(-) diff --git a/theorie-signal/main.tex b/theorie-signal/main.tex index 7a64ca5..e567036 100644 --- a/theorie-signal/main.tex +++ b/theorie-signal/main.tex @@ -663,86 +663,171 @@ \subsection{Interprétation physique} - \begin{center} + \hfill \begin{tikzpicture} \draw[help lines, dashed] (-2,-1) grid (5,3); \draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$}; \draw[thick, orange,smooth] (0,0) -- (0,3) - plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$} + plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) ; \draw[thick, teal] - (0,0) -- (0,1.5) node[left]{$y(\tau)$} + (0,0) -- (0,1.5) plot[domain=0:1]({\x}, {1.5}) (1,1.5) -- (1,0) ; \node at (0,-0.3) {0}; \node at (1,-0.3) {$T$}; + \node[orange,anchor=east] at (5,2.5) {$x(\tau)$}; + \node[teal,anchor=east] at (5,2) {$y(t - \tau)$}; \end{tikzpicture} + \hfill \begin{tikzpicture} \draw[help lines, dashed] (-2,-1) grid (5,3); \draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$}; \draw[thick, orange,smooth] (0,0) -- (0,3) - plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$} + plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) ; \draw[thick, teal] - (-1.5,0) -- (-1.5,1.5) node[above]{$y(t - \tau)$} + (-1.5,0) -- (-1.5,1.5) plot[domain=-1.5:-0.5]({\x}, {1.5}) (-0.5,1.5) -- (-0.5,0) ; \node at (-1.5,-0.3) {$t-T$}; \node at (-0.5,-0.3) {$t$}; + \node[orange,anchor=east] at (5,2.5) {$x(\tau)$}; + \node[teal,anchor=east] at (5,2) {$y(t - \tau)$}; \end{tikzpicture} + \hfill + \hfill \begin{tikzpicture} \draw[help lines, dashed] (-2,-1) grid (5,3); \fill [red!30,domain=0:0.5,variable=\x] - (0,0) - -- plot ({\x}, {1.5/(0.5+\x)}) - node[above right,red] {$x(\tau) \cdot y(t - \tau)$} - -- (0.5,0) - -- cycle; + (0,0) -- plot ({\x}, {1.5/(0.5+\x)}) -- (0.5,0) -- cycle ; \draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$}; \draw[thick, orange,smooth] (0,0) -- (0,3) - plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$} + plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) ; \draw[thick, teal] - (-0.5,0) -- (-0.5,1.5) node[left]{$y(t - \tau)$} + (-0.5,0) -- (-0.5,1.5) plot[domain=-0.5:0.5]({\x}, {1.5}) (0.5,1.5) -- (0.5,0) ; \node at (-0.5,-0.3) {$t-T$}; \node at (0.5,-0.3) {$t$}; - + \node[orange,anchor=east] at (5,2.5) {$x(\tau)$}; + \node[teal,anchor=east] at (5,2) {$y(t - \tau)$}; + \node[red,anchor=east] at (5,1.5) {$x(\tau) \cdot y(t - \tau)$}; \end{tikzpicture} + \hfill \begin{tikzpicture} \draw[help lines, dashed] (-2,-1) grid (5,3); \fill [red!30,domain=0.5:1.5,variable=\x] - node[above,red] {$x(\tau) \cdot y(t - \tau)$} - (0.5,0) - -- plot ({\x}, {1.5/(0.5+\x)}) - -- (1.5,0) - -- cycle; + (0.5,0) -- plot ({\x}, {1.5/(0.5+\x)}) -- (1.5,0) -- cycle ; \draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$}; \draw[thick, orange,smooth] (0,0) -- (0,3) - plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$} + plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) ; \draw[thick, teal] (0.5,0) -- (0.5,1.5) plot[domain=0.5:1.5]({\x}, {1.5}) - (1.5,1.5) - node[right]{$y(t - \tau)$} - -- (1.5,0) + (1.5,1.5) -- (1.5,0) ; \node at (0.5,-0.3) {$t-T$}; \node at (1.5,-0.3) {$t$}; + \node[orange,anchor=east] at (5,2.5) {$x(\tau)$}; + \node[teal,anchor=east] at (5,2) {$y(t - \tau)$}; + \node[red,anchor=east] at (5,1.5) {$x(\tau) \cdot y(t - \tau)$}; + \end{tikzpicture} + \hfill + \subsection{Propriétés} + + Soient $f$, $g$ et $h$ trois fonctions dérivables sur $\mathbb{R}$ pour presque tout $t \in \mathbb{R}$. + + \paragraph{Commutativité} + \begin{equation*} + (f*g)(t) = (g*f)(t) + \end{equation*} + + \paragraph{Distributivité} + \begin{equation*} + (f*(g+h))(t) = (f*g)(t) + (f*h)(t) + \end{equation*} + + \paragraph{Associativité} + \begin{align*} + ((f*g)*h)(t) &= (f*(g*h))(t) \\ + &= \int_{\mathbb{R}} (f*g)(t-\tau) h(\tau) \dif \tau \\ + &= \int_{\mathbb{R}} \int_{\mathbb{R}} f(t-\tau-\nu) g(\nu) h(\tau) \dif \tau + \end{align*} + + \paragraph{Parité} + + Si les fonctions $f$ et $g$ sont paires alors le produit de convolution de $f$ et $g$ est pair. + + \subsection{Produit de convolution et filtrage (SLIT)} + + \begin{center} + \begin{tikzpicture}[every text node part/.style={align=center}] + \node[rectangle,draw,minimum width=3cm,thick] (r) at (0,0) {$h(t)$ \\ SLIT}; + \draw[-latex]++(-3cm,0) -- (r.west) node[above,at start]{$x(t)$} node[below,at start]{entrée}; + \draw[-latex](r) -- ++(3cm,0) node[above]{$y(t)$} node[below]{sortie}; \end{tikzpicture} \end{center} + Un filtre peut être défini~: + \begin{itemize} + \item en fréquence~: $H(j\omega)$ (\emph{fonction de transfert}) + \item en temps~: $h(t)$ (\emph{réponse impulsionnelle}) + \end{itemize} + + Pour la réponse impulsionnelle~: + + La sortie $h(t)$ est obtenue lorsqu'en entrée du filtre on applique un \emph{signal impulsionnel}. + Ce signal impulsionnel est modélisé par l'\emph{impulsion de Dirac}, $\delta$. + + \begin{center} + \begin{tikzpicture}[every text node part/.style={align=center}] + \node[rectangle,draw,minimum width=3cm,thick] (r) at (0,0) {$h(t)$ \\ SLIT}; + \draw[-latex]++(-3cm,0) -- (r.west) node[above,at start]{$x(t) = \delta(t)$}; + \draw[-latex](r) -- ++(3cm,0) node[above]{$y(t) = h(t)$}; + \end{tikzpicture} + \end{center} + + On a donc~: + \begin{equation*} + y(t) = (h*x)(t) = h(t) + \end{equation*} + + $\delta$ est l'élément neutre du produit de convolution, de la même façon que 0 est l'élément neutre de l'addition et que 1 est l'élément neutre de la multiplication. + \begin{equation*} + (f*\delta)(t) = f(t) + \end{equation*} + + \paragraph{Retard} + \begin{equation*} + (f*\delta_{t_0})(t) = f(t - t_0) + \end{equation*} + + où $\delta_{t_0}$ est l'impulsion de Dirac retardée de $t_0$~: \quad $\delta_{t_0} = \delta(t - t_0)$. + + \begin{center} + \begin{tikzpicture} + \draw[-latex] (-0.5,0) -- (2,0) node[right]{$t$}; + \draw[-latex] (0,-0.2) -- (0,2) node[above]{$\delta(t-t_0)$}; + \draw[-latex,thick, red] + (1,0) -- (1,1) + ; + \node at (-0.2,-0.3) {0}; + \node at (1,-0.3) {$t_0$}; + \end{tikzpicture} + \end{center} + \end{document}