Finish cours produit de convolution
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1 changed files with 107 additions and 22 deletions
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@ -663,85 +663,170 @@
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\subsection{Interprétation physique}
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\begin{center}
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\hfill
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\begin{tikzpicture}
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\draw[help lines, dashed] (-2,-1) grid (5,3);
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\draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$};
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\draw[thick, orange,smooth]
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(0,0) -- (0,3)
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$}
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)})
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;
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\draw[thick, teal]
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(0,0) -- (0,1.5) node[left]{$y(\tau)$}
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(0,0) -- (0,1.5)
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plot[domain=0:1]({\x}, {1.5})
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(1,1.5) -- (1,0)
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;
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\node at (0,-0.3) {0};
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\node at (1,-0.3) {$T$};
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\node[orange,anchor=east] at (5,2.5) {$x(\tau)$};
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\node[teal,anchor=east] at (5,2) {$y(t - \tau)$};
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}
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\draw[help lines, dashed] (-2,-1) grid (5,3);
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\draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$};
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\draw[thick, orange,smooth]
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(0,0) -- (0,3)
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$}
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)})
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;
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\draw[thick, teal]
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(-1.5,0) -- (-1.5,1.5) node[above]{$y(t - \tau)$}
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(-1.5,0) -- (-1.5,1.5)
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plot[domain=-1.5:-0.5]({\x}, {1.5})
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(-0.5,1.5) -- (-0.5,0)
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;
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\node at (-1.5,-0.3) {$t-T$};
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\node at (-0.5,-0.3) {$t$};
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\node[orange,anchor=east] at (5,2.5) {$x(\tau)$};
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\node[teal,anchor=east] at (5,2) {$y(t - \tau)$};
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\end{tikzpicture}
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\hfill
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\hfill
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\begin{tikzpicture}
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\draw[help lines, dashed] (-2,-1) grid (5,3);
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\fill [red!30,domain=0:0.5,variable=\x]
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(0,0)
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-- plot ({\x}, {1.5/(0.5+\x)})
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node[above right,red] {$x(\tau) \cdot y(t - \tau)$}
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-- (0.5,0)
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-- cycle;
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(0,0) -- plot ({\x}, {1.5/(0.5+\x)}) -- (0.5,0) -- cycle
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;
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\draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$};
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\draw[thick, orange,smooth]
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(0,0) -- (0,3)
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$}
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)})
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;
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\draw[thick, teal]
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(-0.5,0) -- (-0.5,1.5) node[left]{$y(t - \tau)$}
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(-0.5,0) -- (-0.5,1.5)
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plot[domain=-0.5:0.5]({\x}, {1.5})
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(0.5,1.5) -- (0.5,0)
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;
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\node at (-0.5,-0.3) {$t-T$};
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\node at (0.5,-0.3) {$t$};
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\node[orange,anchor=east] at (5,2.5) {$x(\tau)$};
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\node[teal,anchor=east] at (5,2) {$y(t - \tau)$};
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\node[red,anchor=east] at (5,1.5) {$x(\tau) \cdot y(t - \tau)$};
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\end{tikzpicture}
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\hfill
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\begin{tikzpicture}
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\draw[help lines, dashed] (-2,-1) grid (5,3);
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\fill [red!30,domain=0.5:1.5,variable=\x]
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node[above,red] {$x(\tau) \cdot y(t - \tau)$}
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(0.5,0)
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-- plot ({\x}, {1.5/(0.5+\x)})
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-- (1.5,0)
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-- cycle;
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(0.5,0) -- plot ({\x}, {1.5/(0.5+\x)}) -- (1.5,0) -- cycle
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;
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\draw[-latex] (-2,0) -- (5,0) node[below]{$\tau$};
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\draw[thick, orange,smooth]
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(0,0) -- (0,3)
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)}) node[above]{$x(\tau)$}
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plot[domain=0:5]({\x}, {1.5/(0.5+\x)})
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;
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\draw[thick, teal]
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(0.5,0) -- (0.5,1.5)
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plot[domain=0.5:1.5]({\x}, {1.5})
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(1.5,1.5)
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node[right]{$y(t - \tau)$}
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-- (1.5,0)
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(1.5,1.5) -- (1.5,0)
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;
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\node at (0.5,-0.3) {$t-T$};
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\node at (1.5,-0.3) {$t$};
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\node[orange,anchor=east] at (5,2.5) {$x(\tau)$};
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\node[teal,anchor=east] at (5,2) {$y(t - \tau)$};
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\node[red,anchor=east] at (5,1.5) {$x(\tau) \cdot y(t - \tau)$};
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\end{tikzpicture}
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\hfill
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\subsection{Propriétés}
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Soient $f$, $g$ et $h$ trois fonctions dérivables sur $\mathbb{R}$ pour presque tout $t \in \mathbb{R}$.
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\paragraph{Commutativité}
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\begin{equation*}
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(f*g)(t) = (g*f)(t)
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\end{equation*}
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\paragraph{Distributivité}
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\begin{equation*}
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(f*(g+h))(t) = (f*g)(t) + (f*h)(t)
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\end{equation*}
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\paragraph{Associativité}
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\begin{align*}
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((f*g)*h)(t) &= (f*(g*h))(t) \\
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&= \int_{\mathbb{R}} (f*g)(t-\tau) h(\tau) \dif \tau \\
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&= \int_{\mathbb{R}} \int_{\mathbb{R}} f(t-\tau-\nu) g(\nu) h(\tau) \dif \tau
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\end{align*}
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\paragraph{Parité}
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Si les fonctions $f$ et $g$ sont paires alors le produit de convolution de $f$ et $g$ est pair.
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\subsection{Produit de convolution et filtrage (SLIT)}
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\begin{center}
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\begin{tikzpicture}[every text node part/.style={align=center}]
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\node[rectangle,draw,minimum width=3cm,thick] (r) at (0,0) {$h(t)$ \\ SLIT};
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\draw[-latex]++(-3cm,0) -- (r.west) node[above,at start]{$x(t)$} node[below,at start]{entrée};
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\draw[-latex](r) -- ++(3cm,0) node[above]{$y(t)$} node[below]{sortie};
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\end{tikzpicture}
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\end{center}
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Un filtre peut être défini~:
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\begin{itemize}
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\item en fréquence~: $H(j\omega)$ (\emph{fonction de transfert})
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\item en temps~: $h(t)$ (\emph{réponse impulsionnelle})
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\end{itemize}
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Pour la réponse impulsionnelle~:
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La sortie $h(t)$ est obtenue lorsqu'en entrée du filtre on applique un \emph{signal impulsionnel}.
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Ce signal impulsionnel est modélisé par l'\emph{impulsion de Dirac}, $\delta$.
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\begin{center}
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\begin{tikzpicture}[every text node part/.style={align=center}]
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\node[rectangle,draw,minimum width=3cm,thick] (r) at (0,0) {$h(t)$ \\ SLIT};
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\draw[-latex]++(-3cm,0) -- (r.west) node[above,at start]{$x(t) = \delta(t)$};
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\draw[-latex](r) -- ++(3cm,0) node[above]{$y(t) = h(t)$};
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\end{tikzpicture}
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\end{center}
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On a donc~:
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\begin{equation*}
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y(t) = (h*x)(t) = h(t)
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\end{equation*}
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$\delta$ 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.
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\begin{equation*}
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(f*\delta)(t) = f(t)
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\end{equation*}
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\paragraph{Retard}
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\begin{equation*}
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(f*\delta_{t_0})(t) = f(t - t_0)
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\end{equation*}
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où $\delta_{t_0}$ est l'impulsion de Dirac retardée de $t_0$~: \quad $\delta_{t_0} = \delta(t - t_0)$.
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\begin{center}
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\begin{tikzpicture}
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\draw[-latex] (-0.5,0) -- (2,0) node[right]{$t$};
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\draw[-latex] (0,-0.2) -- (0,2) node[above]{$\delta(t-t_0)$};
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\draw[-latex,thick, red]
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(1,0) -- (1,1)
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;
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\node at (-0.2,-0.3) {0};
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\node at (1,-0.3) {$t_0$};
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\end{tikzpicture}
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\end{center}
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