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# Convertisseurs
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- [AD/DA](/infotro/Eln/ADDA)
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- [Sigma / Delta](/infotro/Eln/Sigma-Delta)
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- [Transformée en Z](/infotro/Eln/Ztransform)
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- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage)
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<header><h1>
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Signaux et échantillonnage
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</h1></header>
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-----------
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# Fourier
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Pour comprendre les signaux, il est important de comprendre la transformée de Fourier.
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La transformée de Fourier est une technique de traitement du signal utilisée pour analyser et manipuler les signaux continus.
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Elle permet d'exprimer un signal dans le temps en domaine fréquentiel.
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## Transformée de Fourier
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La transformée de Fourier est une technique de traitement du signal utilisée pour analyser et manipuler les signaux continus.
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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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## Transformée de Fourier discrète
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La transformée de Fourier discrète (DFT - Discret Fourier Transform) est une transformation discrète utilisée pour analyser le spectre fréquentiel
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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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$$\mathcal{F\{x[k]\} = \mathcal{Z}\{x[k]\}}|_{z=e^{j\Omega}}$$
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$$x[k]=\frac{1}{2\pi} \int^{+\pi}_{-\pi}X(\Omega)e^{jk\Omega} d\Omega$$
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La DFT est utilisée dans de nombreux domaines, notamment en traitement du signal, en télécommunications, en traitement d'images
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et en compression de données.
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> [!NOTE|label:FFT]
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> La transformée de Fourier rapide (FFT) est un algorithme efficace pour calculer la DFT. Elle est largement utilisée dans
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> de nombreuses applications, notamment en traitement du signal et en compression de données.
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# Réponse impulsionnelle
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La réponse impulsionnelle $h[k]$ d'un système discret LIT est sa réponse à l'impulsion unitaire à l'origine $δ[k]$.
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Avant l'impulsion, la réponse impulsionnelle $h$ est nulle.
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Après l'impulsion, la réponse impulsionnelle $h$ est la réponse du système à l'impulsion unitaire à l'origine.
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*Causal* = le futur n'a pas d'influence sur le passé.
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$$ h[k] = \mathcal{H} \{ δ[k] \} $$
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$\mathcal{H}$ : Heaviside <br>
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$y[k]$ : réponse du système <br>
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$x[k]$ : signal d'entrée <br>
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$h[k]$ : réponse impulsionnelle <br>
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Pour un système causal
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$$\begin{align}
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y[k] &= \sum^{k}_{n=-\infty}(x[n]\cdot h[k-n]) \\
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\\
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y[k] &= \sum^{+\infty}_{n=0}(x[k-n]\cdot h[n])
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\end{align}$$
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La réponse impulsionnelle caractérise <u>ENTIÈREMENT</u> un système LIT.
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## Système LIT stable
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Un système LIT (Linéaire et Invariant dans le Temps) est stable si et seulement si sa réponse impulsionnelle est sommable.
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> [!NOTE]
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> $\sum^{+\infty}_{k=-\infty}|h[k]|<+\infty$
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## Système LIT causal à réponse impulsionnelle finie
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Un système LIT (Linéaire et Invariant dans le Temps) est causal à réponse impulsionnelle finie si et seulement si sa
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réponse impulsionnelle est nulle pour $k<0$ et sommable pour $k\geq0$.
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> [!NOTE]
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> fini: $h(k) |_{k<\infty} = 0$
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$$\begin{align}
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y[k] &= \sum^{N}_{n=0}x[k-n]\cdot h[n] \\
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&= b_{0}x[k]+b_{1}x[k-1]+\dots+b_{N}x[k-N]
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\end{align}$$
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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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La somme décomposée est appelée *équation aux différences*.
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# Transformée en Z
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La transformée en Z, également connue sous le nom de transformée en fréquence Z, est une technique de traitement du
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signal utilisée pour analyser et manipuler les signaux numériques.
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Elle permet de décomposer un signal en fonction de la fréquence et de fournir des informations sur la nature du signal.
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La principale différence entre la transformée en Z et la transformation de Laplace (ou S) est que la transformée en Z
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est adaptée aux signaux discrets, c'est-à-dire aux valeurs qui ne varient pas continuellement dans le temps.
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En revanche, la transformation de Laplace est utilisée pour les signaux continus et permet d'étudier leur réponse en
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fréquence et en phase.
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En résumé, la transformée en Z est conçue pour les signaux discrets et les systèmes numériques, tandis que la
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transformation de Laplace s'applique aux signaux continus et aux systèmes analogiques.
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[Tableau des transformées en Z](/infotro/Eln/Ztransform)
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> [!NOTE|label:S]
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> Domaine fréquentiel continue avec Laplace.
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> [!NOTE|label:Z]
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> Domaine fréquentiel discret.
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Avec $z:=e^{sT_{s}}$ on a
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$$\mathcal{Z}\{x[k]\} = X(z) \quad := \quad \sum^{+\infty}_{k=-\infty}(x[k]\cdot z^{-k})=\mathcal{L\{x_{s}(t)\}}$$
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qui est la transformée en Z de $x[k]$.<br>
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$T_{s}$ : Période d'échantillonnage.
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On a aussi la propriété suivante :
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$$\mathcal{Z}\{x[k-n]\}=z^{-n}\cdot X(z)$$
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## Marche à suivre
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1. Avoir une équation aux différences.
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2. Pour chaque terme de l'équation, appliquer la transformée en Z.
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Avec :
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- $f[k] => F(z)$
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- $f[k-n] => z^{-n}\cdot F(z)$
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3. Isoler $Y(z)$.
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4. $H(z) = \frac{Y(z)}{X(z)}$ est la fonction de transfert.
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> [!NOTE|label:Example]
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> L'équation aux différences est donnée par $$\tag{1} y[k] = \frac{1}{2}(x[k]+y[k-1])$$
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> devient
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> $$\begin{align}
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\tag{2} Y(z) &= \frac{1}{2}(X(z)+z^{-1}\cdot Y(z)) \\
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Y(z)-\frac{1}{2}\cdot z^{-1}\cdot Y(z) &= \frac{1}{2}\cdot X(z) \\
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\tag{3} Y(z) &= \frac{1}{2-z^{-1}}X(z) \\
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\tag{4} H(z) &= \frac{Y(z)}{X(z)} = \frac{1}{2-z^{-1}}
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\end{align}$$
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> [!TIP]
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> $h[k]\stackrel{\text{Z tabel}}{\longleftrightarrow}H(z)$
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>
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> Exemple: $h[k] = 2^{-(k+1)}$ devient
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> $$\begin{align}
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2^{-(k+1)} &= 2^{-k}\cdot 2^{-1} = \frac{1}{2}\cdot 2^{-k} = \frac{1}{2}\cdot (2^{-1})^k \\
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&\to \frac{1}{2}\cdot \frac{z}{z-\frac{1}{2}} =\frac{z}{2z-1} = \frac{1}{2-\frac{1}{z}} = \frac{1}{2-z^{-1}}
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\end{align}$$
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> [!WARNING]
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> Les propriétés de la transformée en Z se trouvent dans le [Tableau des transformées en Z](/infotro/Eln/Ztransform)
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L'équation aux différences
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$$y[k] + a_{1}y[k-1]+\dots+a_{M}y[k-M] \quad = \quad b_{0}x[k]+b_{1}x[k-1]+\dots+b_{N}x[k-N]$$
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devient par la transformation en z
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$$H(z) = \frac{Y(z)}{X(z)} = \frac{b_{0}+b_{1}z^{-1}+\dots+b_{N}z^{-1}}{1+a_{1}z^{-1}+\dots+a_{M}z^{-M}} \quad a_{m}, b_{n} \space \text{réels}$$
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Les zéros et les pôles de la fonction de transfert donnent donc :
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$$H(z) = K \frac{(z-o_{1})\cdot (z-o_{2})\cdot \space \dots \space \cdot (z-o_{N})}{(z-p_{1})\cdot (z-p_{2})\cdot \space \dots \space \cdot (z-p_{M}) }\quad o_{n}, p_{n} \space \text{complexes}$$
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# Approximation bilinéaire
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L'approximation bilinéaire est une technique utilisée pour convertir une fonction de transfert continue en une fonction
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de transfert discrète.
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Elle est utilisée dans le domaine du traitement du signal pour convertir un système continu en un système discret.
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Développement en série de la fonction $ln$ autour de 1 ($x=0$) :
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$$\ln\left( \frac{1+x}{1-x} \right) = 2\left(x + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \dots \right)$$
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donne l'approximation
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$\ln\left( \frac{1+x}{1-x} \right) \approx 2x$
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#TODO add formule from lab6-filters for bilinear approximation
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## Autres méthodes d'approximation d'un système continu LIT par un syst. discret
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On détermine la réponse impulsionnelle du système discret LIT à partir de celle du système continu en appliquant la relation :
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$$h[k] = T_{s}\cdot(kT_{s})$$
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<header><h1>
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Tableau des transformées en Z
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</h1></header>
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-----------
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| Nr. | $F_k$ | $F(z)$ | $f(t)$ | $F(s)$ |
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|-----|-------------------|-----------------|-------------|--------|
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| 1 | $\delta_{k}, 0^k$ | 1 | $\delta(t)$ | 1 |
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| 2 | $1^k$ | $\frac{z}{z-1}$ | $u(t)$ | $\frac{1}{s-a}$ |
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Dans la théorie Eln la partie transformée en Z commence à la slide 23 de "Eln2_LTI_0910_f.pdf".
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@@ -15,6 +15,8 @@
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## [Eln - Électronique](/infotro/Eln/)
|
||||
- [AD/DA](/infotro/Eln/ADDA)
|
||||
- [Sigma / Delta](/infotro/Eln/Sigma-Delta)
|
||||
- [Transformée en Z](/infotro/Eln/Ztransform)
|
||||
- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage)
|
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## [SEm - Système Embarqué](/infotro/SEm/)
|
||||
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@@ -10,6 +10,8 @@
|
||||
* [Eln - Électronique](/infotro/Eln/)
|
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* [AD/DA](/infotro/Eln/ADDA)
|
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* [Sigma / Delta](/infotro/Eln/Sigma-Delta)
|
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* [Transformée en Z](/infotro/Eln/Ztransform)
|
||||
* [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage)
|
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* [SEm - Systèmes Embarqués](/infotro/SEm/)
|
||||
* [PLA - Programmation en langage assembleur](/infotro/SEm/PLA.md)
|
||||
* [ASM - Assembleur ARM](/infotro/SEm/ASM.md)
|
||||
Reference in new issue
Block a user