Finish cours produit de convolution
This commit is contained in:
parent
b88bdc3a13
commit
454b7e8dc5
1 changed files with 107 additions and 22 deletions
|
|
@ -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}
|
||||
|
|
|
|||
Loading…
Reference in a new issue