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theorie-signal/main.tex
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@ -830,4 +830,59 @@
\end{tikzpicture}
\end{center}
\section{Échantillonnage}
\subsection{Formalisme d'échantillonnage}
\begin{center}
\begin{tikzpicture}
\draw[help lines, dashed] (-1,-1) grid (5,3);
\draw[-latex] (-0.5,0) -- (5,0) node[right]{$t$};
\draw[-latex] (0,-0.5) -- (0,3) node[left]{$A$};
\draw[thick,orange,smooth]
plot[domain=0:5] ({\x}, {sqrt(\x)})
;
\node at (2.5,-1) {$x(t), t\in\mathbb{R}$};
\end{tikzpicture}
$\implies$
\begin{tikzpicture}
\draw[help lines, dashed] (-1,-1) grid (5,3);
\draw[-latex] (-0.5,0) -- (5,0) node[right]{$t$};
\draw[-latex] (0,-0.5) -- (0,3) node[left]{$A$};
\draw[thick,orange,smooth]
plot[domain=0:0.05] ({\x}, {sqrt(\x)})
plot[domain=0.95:1.05] ({\x}, {sqrt(\x)})
plot[domain=1.95:2.05] ({\x}, {sqrt(\x)})
plot[domain=2.95:3.05] ({\x}, {sqrt(\x)})
plot[domain=3.95:4.05] ({\x}, {sqrt(\x)})
;
\node at (1,-0.3) {$T_e$};
\node at (2,-0.3) {$2T_e$};
\node at (3,-0.3) {$3T_e$};
\node at (4,-0.3) {$4T_e$};
\node at (2.5,-1) {$x_e(t) = x(nT_e), n\in\mathbb{N}$};
\end{tikzpicture}
\end{center}
C'est un interrupteur qui réalise l'échantillonnage.
Il se ferme tous les $nT_e$.
Il reste fermé pendant $\tau < T_e$
\paragraph{Modélisation mathématique de l'interrupteur}
\begin{align*}
p_{\tau}(t) =
\left\{
\begin{array}{l}
\frac{1}{\tau}, t \in \left[\frac{-\tau}{2};\frac{\tau}{2}\right] \\
0 \text{ sinon} \\
\end{array}
\right.
\end{align*}
La valeur de l'échantillon du signal $x$ en $nT_e$ est la \emph{valeur moyenne} de $x$ pendant $\tau$, le temps de fermeture de l'interrupteur.
\begin{align*}
x(nT_e) = \frac{1}{\tau} \int_{nT_e-\frac{\tau}{2}}^{nT_e+\frac{\tau}{2}} x(t) \dif t \\
\boxed{x(nT_e) = \int_{-\infty}^{+\infty} p_{\tau}(t - nT_e) x(t) \dif t}
\end{align*}
\end{document}