Merge pull request from infotro/Eln/signaux_echantillonnage
update pictures in signaux et echant.
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@@ -15,7 +15,19 @@ Elle permet d'exprimer un signal dans le temps en domaine fréquentiel.
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La transformée de Fourier est définie par la formule suivante :
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Soit $f(x)$ une fonction non périodique définie sur $\mathbb{R}$.
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$$\begin{align}
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\text{Transformée de Fourier de }f(x) &: \qquad \widehat{f}(\omega) = \int^{\infty}_{-\infty}f(x)e^{-\mathbf{i}\omega x}dx \qquad (\omega\in \mathbb{R})\\
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\text{Intégrale de Fourier de } f(x) &: \qquad \frac{1}{2\pi}\int_{-\infty}^{\infty} \widehat{f}(\omega) e^{\mathbf{i}x\omega}\, d\omega \qquad(x \in\mathbb{R}).
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\end{align}$$
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**Théorème** : Si $\int_{-\infty}^{\infty} |f(x) \, dx < \infty$ et si $f$ est une fonction de variation bornée sur tout $\mathbb{R}$, alors
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$$
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\frac{1}{2\pi}\int_{-\infty}^{\infty} \widehat{f}(\omega)e^{\mathbf{i}x\omega} \, d\omega = \begin{cases}
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f(x) &\text{si }f \text{ est continue au point }x \\
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\text{milieu du saut de } f(x) &\text{si }f \text{ a un saut en }x
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\end{cases}
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$$
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## Transformée de Fourier discrète
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@@ -23,10 +35,13 @@ La transformée de Fourier discrète (DFT - Discret Fourier Transform) est une t
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d'un signal discret.
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Elle est l'équivalent discret de la transformée de Fourier, qui est utilisée pour analyser les signaux continus.
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On pose: $\Omega := \omega \cdot T_{s}$
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La transformation de Fourier de $x[k]$ est donnée par la formule suivante :
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$$\mathcal{F}\{x[k]\} = X(\Omega) \quad := \quad\sum^{+\infty}_{k=-\infty}x[k]\cdot e^{-jk\Omega}= \mathcal{F}\{x_{s}(t)\}$$
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$$\mathcal{F\{x[k]\} = \mathcal{Z}\{x[k]\}}|_{z=e^{j\Omega}}$$
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$$x[k]=\frac{1}{2\pi} \int^{+\pi}_{-\pi}X(\Omega)e^{jk\Omega} d\Omega$$
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@@ -86,7 +101,7 @@ y[k] &= \sum^{N}_{n=0}x[k-n]\cdot h[n] \\
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\end{align}$$
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$N$ : Limite de notre réponse impulsionnelle.
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En faisant la somme, la réponse impulsionnelle se transforme en facteurs $(b_0, b_1, b_2, ..., b_n)$.
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On peut retrouver la réponse impulsionnelle à partir de ces facteurs.
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@@ -398,7 +396,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path14" />
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id="path14"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="M 629,40 H 729"
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fill="none"
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@@ -407,7 +406,8 @@
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stroke-miterlimit="10"
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stroke-dasharray="12, 12"
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pointer-events="stroke"
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id="path15" />
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id="path15"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 329,40 h 80 v 68.03"
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fill="none"
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@@ -415,7 +415,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="stroke"
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id="path16" />
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id="path16"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 409,115.53 -5,-10 5,2.5 5,-2.5 z"
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fill="#808080"
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@@ -423,7 +424,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path17" />
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id="path17"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 129,40 v 68.03"
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fill="none"
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@@ -431,7 +433,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="stroke"
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id="path18" />
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id="path18"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 129,115.53 -5,-10 5,2.5 5,-2.5 z"
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fill="#808080"
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@@ -439,7 +442,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path19" />
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id="path19"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 129,200 -0.08,27.79"
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fill="none"
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@@ -447,7 +451,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="stroke"
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id="path20" />
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id="path20"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 128.89,235.29 -4.97,-10.02 5,2.52 5,-2.49 z"
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fill="#808080"
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@@ -455,7 +460,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path21" />
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id="path21"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 408.66,200 v 28.03"
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fill="none"
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@@ -463,7 +469,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="stroke"
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id="path22" />
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id="path22"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 408.66,235.53 -5,-10 5,2.5 5,-2.5 z"
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fill="#808080"
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@@ -471,7 +478,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path23" />
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id="path23"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 968.66,200 v 28.03"
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fill="none"
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@@ -479,7 +487,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="stroke"
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id="path24" />
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id="path24"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 968.66,235.53 -5,-10 5,2.5 5,-2.5 z"
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fill="#808080"
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@@ -487,7 +496,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path25" />
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id="path25"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 1089,300 h 108.58"
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fill="none"
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@@ -495,7 +505,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="stroke"
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id="path26" />
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id="path26"
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style="background-color:rgb(255, 255, 255)" />
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<path
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d="m 1205.08,300 -10,5 2.5,-5 -2.5,-5 z"
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fill="#808080"
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@@ -503,7 +514,8 @@
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stroke-width="4"
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stroke-miterlimit="10"
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pointer-events="all"
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id="path27" />
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id="path27"
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style="background-color:rgb(255, 255, 255)" />
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<rect
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x="1109"
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y="261"
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@@ -514,10 +526,12 @@
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fill="none"
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stroke="none"
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pointer-events="all"
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id="rect27" />
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id="rect27"
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style="background-color:rgb(255, 255, 255)" />
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<g
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transform="translate(-0.5,-0.5)"
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id="g27">
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id="g27"
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style="background-color:rgb(255, 255, 255)"
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transform="translate(-0.5,-0.5)">
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<switch
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id="switch27">
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<foreignObject
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@@ -555,10 +569,12 @@
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fill="none"
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stroke="none"
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pointer-events="all"
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id="rect28" />
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id="rect28"
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style="background-color:rgb(255, 255, 255)" />
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<g
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transform="translate(-0.5,-0.5)"
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id="g28">
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id="g28"
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style="background-color:rgb(255, 255, 255)"
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transform="translate(-0.5,-0.5)">
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<switch
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id="switch28">
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<foreignObject
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@@ -586,4 +602,5 @@
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id="image28" />
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</switch>
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</g>
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</g>
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</svg>
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|
Before Width: | Height: | Size: 57 KiB After Width: | Height: | Size: 59 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 73 KiB |
Binary file not shown.
|
Before Width: | Height: | Size: 32 KiB |
Reference in new issue
Block a user