From 1092fb86ccf0b6b7099c8bfb3e5d750089655e9d Mon Sep 17 00:00:00 2001
From: Klagarge <remi@heredero.ch>
Date: Fri, 2 Feb 2024 18:01:26 +0100
Subject: [PATCH] =?UTF-8?q?add=20signaux=20et=20=C3=A9chantillonnage?=
MIME-Version: 1.0
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---
 infotro/Eln/DFT.md                            |  33 -
 infotro/Eln/README.md                         |   3 +-
 infotro/Eln/Reponse_impulsionelle.md          |  10 -
 infotro/Eln/SignauxEtEchantillonnage.md       | 183 ++++++
 infotro/Eln/Ztransform.md                     |  23 +-
 .../Eln/medias/fourier_cerlce_unitaire.png    | Bin 0 -> 10155 bytes
 infotro/Eln/medias/lit_recusrive_example.png  | Bin 0 -> 11760 bytes
 ...reponse_impulsionelle-continue_discret.png | Bin 0 -> 39919 bytes
 infotro/Eln/medias/schema_bloc_fir.png        | Bin 0 -> 20419 bytes
 infotro/Eln/medias/schema_bloc_fir.svg        | 589 ++++++++++++++++++
 infotro/README.md                             |   3 +-
 infotro/_navbar.md                            |   2 +
 12 files changed, 781 insertions(+), 65 deletions(-)
 delete mode 100644 infotro/Eln/DFT.md
 delete mode 100644 infotro/Eln/Reponse_impulsionelle.md
 create mode 100644 infotro/Eln/SignauxEtEchantillonnage.md
 create mode 100644 infotro/Eln/medias/fourier_cerlce_unitaire.png
 create mode 100644 infotro/Eln/medias/lit_recusrive_example.png
 create mode 100644 infotro/Eln/medias/reponse_impulsionelle-continue_discret.png
 create mode 100644 infotro/Eln/medias/schema_bloc_fir.png
 create mode 100644 infotro/Eln/medias/schema_bloc_fir.svg

diff --git a/infotro/Eln/DFT.md b/infotro/Eln/DFT.md
deleted file mode 100644
index 669609e..0000000
--- a/infotro/Eln/DFT.md
+++ /dev/null
@@ -1,33 +0,0 @@
-<header><h1>
-DFT
-</h1></header>
-
------------
-
-# Transformée de Fourier
-La transformée de Fourier est une technique de traitement du signal utilisée pour analyser et manipuler les signaux continus.
-Elle permet d'exprimer un signal dans le temps en domaine fréquentiel.
-
-La transformée de Fourier est définie par la formule suivante :
-
-![Transformée de fourier](transforme_fourier.png)
-
-# Transformée de Fourier discrète
-
-La transformée de Fourier discrète (DFT - Discret Fourier Transform) est une transformation discrète utilisée pour analyser le spectre fréquentiel
-d'un signal discret.
-Elle est l'équivalent discret de la transformée de Fourier, qui est utilisée pour analyser les signaux continus.
-
-![img.png](transfrom_fourrier_discret.png)
-
-
-
-
-La DFT est utilisée dans de nombreux domaines, notamment en traitement du signal, en télécommunications, en traitement d'images
-et en compression de données.
-
-> [!NOTE|label:FFT]
-> La transformée de Fourier rapide (FFT) est un algorithme efficace pour calculer la DFT. Elle est largement utilisée dans
-> de nombreuses applications, notamment en traitement du signal et en compression de données.
-
- 
diff --git a/infotro/Eln/README.md b/infotro/Eln/README.md
index e726937..5069aa6 100644
--- a/infotro/Eln/README.md
+++ b/infotro/Eln/README.md
@@ -6,5 +6,4 @@
 - [AD/DA](/infotro/Eln/ADDA)
 - [Sigma / Delta](/infotro/Eln/Sigma-Delta)
 - [Transformée en Z](/infotro/Eln/Ztransform)
-- [DFT](/infotro/Eln/DFT)
-- [Réponse impulsionnelle](/infotro/Eln/Reponse_impulsionelle)
\ No newline at end of file
+- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage)
\ No newline at end of file
diff --git a/infotro/Eln/Reponse_impulsionelle.md b/infotro/Eln/Reponse_impulsionelle.md
deleted file mode 100644
index 1bbcff2..0000000
--- a/infotro/Eln/Reponse_impulsionelle.md
+++ /dev/null
@@ -1,10 +0,0 @@
-<header><h1>
-Réponse impulsionnelle
-</h1></header>
-
------------
-
-La réponse impulsionnelle $h[k]$ d'un système discret LIT est sa réponse à l'impulsion unitaire à l'origine $δ[k]$.
-
-$$ h[k] = \mathcal{H} \{ δ[k] \} $$
-
diff --git a/infotro/Eln/SignauxEtEchantillonnage.md b/infotro/Eln/SignauxEtEchantillonnage.md
new file mode 100644
index 0000000..2105a5b
--- /dev/null
+++ b/infotro/Eln/SignauxEtEchantillonnage.md
@@ -0,0 +1,183 @@
+<header><h1>
+Signaux et échantillonnage
+</h1></header>
+
+-----------
+
+# Fourier
+Pour comprendre les signaux, il est important de comprendre la transformée de Fourier.
+La transformée de Fourier est une technique de traitement du signal utilisée pour analyser et manipuler les signaux continus.
+Elle permet d'exprimer un signal dans le temps en domaine fréquentiel.
+
+## Transformée de Fourier
+La transformée de Fourier est une technique de traitement du signal utilisée pour analyser et manipuler les signaux continus.
+Elle permet d'exprimer un signal dans le temps en domaine fréquentiel.
+
+La transformée de Fourier est définie par la formule suivante :
+
+![Transformée de fourier](transforme_fourier.png)
+
+## Transformée de Fourier discrète
+
+La transformée de Fourier discrète (DFT - Discret Fourier Transform) est une transformation discrète utilisée pour analyser le spectre fréquentiel
+d'un signal discret.
+Elle est l'équivalent discret de la transformée de Fourier, qui est utilisée pour analyser les signaux continus.
+
+![img.png](transfrom_fourrier_discret.png)
+
+$$\mathcal{F\{x[k]\} = \mathcal{Z}\{x[k]\}}|_{z=e^{j\Omega}}$$
+![Fourier - cercle unitaire](medias/fourier_cerlce_unitaire.png)
+$$x[k]=\frac{1}{2\pi} \int^{+\pi}_{-\pi}X(\Omega)e^{jk\Omega} d\Omega$$
+
+
+
+
+La DFT est utilisée dans de nombreux domaines, notamment en traitement du signal, en télécommunications, en traitement d'images
+et en compression de données.
+
+> [!NOTE|label:FFT]
+> La transformée de Fourier rapide (FFT) est un algorithme efficace pour calculer la DFT. Elle est largement utilisée dans
+> de nombreuses applications, notamment en traitement du signal et en compression de données.
+
+
+# Réponse impulsionnelle
+La réponse impulsionnelle $h[k]$ d'un système discret LIT est sa réponse à l'impulsion unitaire à l'origine $δ[k]$.
+
+![Réponse impulsionelle - Continu/discret](medias/reponse_impulsionelle-continue_discret.png)
+
+Avant l'impulsion, la réponse impulsionnelle $h$ est nulle.
+Après l'impulsion, la réponse impulsionnelle $h$ est la réponse du système à l'impulsion unitaire à l'origine.
+
+*Causal* = le futur n'a pas d'influence sur le passé.
+
+$$ h[k] = \mathcal{H} \{ δ[k] \} $$
+
+$\mathcal{H}$ : Heaviside <br>
+$y[k]$ : réponse du système <br>
+$x[k]$ : signal d'entrée <br>
+$h[k]$ : réponse impulsionnelle <br>
+
+Pour un système causal
+
+$$\begin{align}
+y[k] &= \sum^{k}_{n=-\infty}(x[n]\cdot h[k-n]) \\
+\\
+y[k] &= \sum^{+\infty}_{n=0}(x[k-n]\cdot h[n])
+\end{align}$$
+
+La réponse impulsionnelle caractérise <u>ENTIÈREMENT</u> un système LIT.
+
+## Système LIT stable
+Un système LIT (Linéaire et Invariant dans le Temps) est stable si et seulement si sa réponse impulsionnelle est sommable.
+
+> [!NOTE]
+> $\sum^{+\infty}_{k=-\infty}|h[k]|<+\infty$
+
+## Système LIT causal à réponse impulsionnelle finie
+Un système LIT (Linéaire et Invariant dans le Temps) est causal à réponse impulsionnelle finie si et seulement si sa
+réponse impulsionnelle est nulle pour $k<0$ et sommable pour $k\geq0$.
+
+> [!NOTE]
+> fini: $h(k) |_{k<\infty} = 0$
+
+$$\begin{align}
+y[k] &= \sum^{N}_{n=0}x[k-n]\cdot h[n] \\
+&= b_{0}x[k]+b_{1}x[k-1]+\dots+b_{N}x[k-N]
+\end{align}$$
+$N$ : Limite de notre réponse impulsionnelle.
+
+![schema_bloc_fir.png](medias/schema_bloc_fir.png ':size=75%')
+
+En faisant la somme, la réponse impulsionnelle se transforme en facteurs $(b_0, b_1, b_2, ..., b_n)$.
+On peut retrouver la réponse impulsionnelle à partir de ces facteurs.
+La somme décomposée est appelée *équation aux différences*.
+
+# Transformée en Z
+La transformée en Z, également connue sous le nom de transformée en fréquence Z, est une technique de traitement du
+signal utilisée pour analyser et manipuler les signaux numériques.
+Elle permet de décomposer un signal en fonction de la fréquence et de fournir des informations sur la nature du signal.
+
+La principale différence entre la transformée en Z et la transformation de Laplace (ou S) est que la transformée en Z
+est adaptée aux signaux discrets, c'est-à-dire aux valeurs qui ne varient pas continuellement dans le temps.
+En revanche, la transformation de Laplace est utilisée pour les signaux continus et permet d'étudier leur réponse en
+fréquence et en phase.
+
+En résumé, la transformée en Z est conçue pour les signaux discrets et les systèmes numériques, tandis que la
+transformation de Laplace s'applique aux signaux continus et aux systèmes analogiques.
+
+[Tableau des transformées en Z](/infotro/Eln/Ztransform)
+
+> [!NOTE|label:S]
+> Domaine fréquentiel continue avec Laplace.
+
+> [!NOTE|label:Z]
+> Domaine fréquentiel discret. 
+
+Avec $z:=e^{sT_{s}}$ on a
+$$\mathcal{Z}\{x[k]\} = X(z) \quad := \quad \sum^{+\infty}_{k=-\infty}(x[k]\cdot z^{-k})=\mathcal{L\{x_{s}(t)\}}$$
+qui est la transformée en Z de $x[k]$.<br>
+$T_{s}$ : Période d'échantillonnage.
+
+On a aussi la propriété suivante :
+$$\mathcal{Z}\{x[k-n]\}=z^{-n}\cdot X(z)$$
+
+## Marche à suivre
+1. Avoir une équation aux différences.
+2. Pour chaque terme de l'équation, appliquer la transformée en Z.
+Avec : 
+    - $f[k] => F(z)$
+    - $f[k-n] => z^{-n}\cdot F(z)$
+3. Isoler $Y(z)$.
+4. $H(z) = \frac{Y(z)}{X(z)}$ est la fonction de transfert.
+
+> [!NOTE|label:Example]
+> L'équation aux différences est donnée par $$\tag{1} y[k] = \frac{1}{2}(x[k]+y[k-1])$$
+> devient
+> $$\begin{align}
+\tag{2} Y(z) &= \frac{1}{2}(X(z)+z^{-1}\cdot Y(z)) \\
+Y(z)-\frac{1}{2}\cdot  z^{-1}\cdot Y(z) &= \frac{1}{2}\cdot X(z) \\
+\tag{3} Y(z) &= \frac{1}{2-z^{-1}}X(z) \\
+\tag{4} H(z) &= \frac{Y(z)}{X(z)} = \frac{1}{2-z^{-1}}
+\end{align}$$
+
+> [!TIP]
+> $h[k]\stackrel{\text{Z tabel}}{\longleftrightarrow}H(z)$
+> 
+> Exemple: $h[k] = 2^{-(k+1)}$ devient
+> $$\begin{align}
+2^{-(k+1)} &= 2^{-k}\cdot 2^{-1} = \frac{1}{2}\cdot 2^{-k} = \frac{1}{2}\cdot (2^{-1})^k  \\
+&\to \frac{1}{2}\cdot \frac{z}{z-\frac{1}{2}} =\frac{z}{2z-1} = \frac{1}{2-\frac{1}{z}} = \frac{1}{2-z^{-1}}
+\end{align}$$
+
+> [!WARNING]
+> Les propriétés de la transformée en Z se trouvent dans le [Tableau des transformées en Z](/infotro/Eln/Ztransform)
+
+
+L'équation aux différences
+$$y[k] + a_{1}y[k-1]+\dots+a_{M}y[k-M] \quad = \quad b_{0}x[k]+b_{1}x[k-1]+\dots+b_{N}x[k-N]$$
+devient par la transformation en z
+$$H(z) = \frac{Y(z)}{X(z)} = \frac{b_{0}+b_{1}z^{-1}+\dots+b_{N}z^{-1}}{1+a_{1}z^{-1}+\dots+a_{M}z^{-M}} \quad a_{m}, b_{n} \space \text{réels}$$
+
+Les zéros et les pôles de la fonction de transfert donnent donc :
+$$H(z) = K \frac{(z-o_{1})\cdot (z-o_{2})\cdot \space \dots \space \cdot (z-o_{N})}{(z-p_{1})\cdot (z-p_{2})\cdot \space \dots \space \cdot (z-p_{M}) }\quad o_{n}, p_{n} \space \text{complexes}$$
+
+
+# Approximation bilinéaire
+L'approximation bilinéaire est une technique utilisée pour convertir une fonction de transfert continue en une fonction
+de transfert discrète.
+Elle est utilisée dans le domaine du traitement du signal pour convertir un système continu en un système discret.
+
+Développement en série de la fonction $ln$ autour de 1 ($x=0$) :
+$$\ln\left( \frac{1+x}{1-x} \right) = 2\left(x + \frac{x^{3}}{3} + \frac{x^{5}}{5} + \dots \right)$$
+donne l'approximation
+$\ln\left( \frac{1+x}{1-x} \right) \approx 2x$
+
+#TODO add formule from lab6-filters for bilinear approximation
+
+## Autres méthodes d'approximation d'un système continu LIT par un syst. discret
+On détermine la réponse impulsionnelle du système discret LIT à partir de celle du système continu en appliquant la relation :
+$$h[k] = T_{s}\cdot(kT_{s})$$
+
+
+
+
diff --git a/infotro/Eln/Ztransform.md b/infotro/Eln/Ztransform.md
index edb4b96..c03598a 100644
--- a/infotro/Eln/Ztransform.md
+++ b/infotro/Eln/Ztransform.md
@@ -1,26 +1,13 @@
 <header><h1>
-Transformée en Z
+Tableau des transformées en Z
 </h1></header>
 
 -----------
 
-La transformée en Z, également connue sous le nom de transformée en fréquence Z, est une technique de traitement du
-signal utilisée pour analyser et manipuler les signaux numériques.
-Elle permet de décomposer un signal en fonction de la fréquence et de fournir des informations sur la nature du signal.
-
-La principale différence entre la transformée en Z et la transformation de Laplace (ou S) est que la transformée en Z
-est adaptée aux signaux discrets, c'est-à-dire aux valeurs qui ne varient pas continuellement dans le temps.
-En revanche, la transformation de Laplace est utilisée pour les signaux continus et permet d'étudier leur réponse en
-fréquence et en phase.
-
-En résumé, la transformée en Z est conçue pour les signaux discrets et les systèmes numériques, tandis que la
-transformation de Laplace s'applique aux signaux continus et aux systèmes analogiques.
-
-> [!NOTE|label:S]
-> Domaine fréquentiel continue avec Laplace.
-
-> [!NOTE|label:Z]
-> Domaine fréquentiel discret. 
+| Nr. | $F_k$             | $F(z)$          | $f(t)$      | $F(s)$ |
+|-----|-------------------|-----------------|-------------|--------|
+| 1   | $\delta_{k}, 0^k$ | 1               | $\delta(t)$ | 1      |
+| 2   | $1^k$        | $\frac{z}{z-1}$ | $u(t)$ | $\frac{1}{s-a}$ |
 
 ![ztabelle01.png](ztabelle01.png)
 ![ztabelle02.png](ztabelle02.png)
diff --git a/infotro/Eln/medias/fourier_cerlce_unitaire.png b/infotro/Eln/medias/fourier_cerlce_unitaire.png
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diff --git a/infotro/Eln/medias/lit_recusrive_example.png b/infotro/Eln/medias/lit_recusrive_example.png
new file mode 100644
index 0000000000000000000000000000000000000000..35788c196149480b00fecb6ac0824f805bbfad4f
GIT binary patch
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diff --git a/infotro/README.md b/infotro/README.md
index 3784c95..e9e2476 100644
--- a/infotro/README.md
+++ b/infotro/README.md
@@ -16,8 +16,7 @@
 - [AD/DA](/infotro/Eln/ADDA)
 - [Sigma / Delta](/infotro/Eln/Sigma-Delta)
 - [Transformée en Z](/infotro/Eln/Ztransform)
-- [DFT](/infotro/Eln/DFT)
-- [Réponse impulsionnelle](/infotro/Eln/Reponse_impulsionelle)
+- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage)
 
 
 ## [SEm - Système Embarqué](/infotro/SEm/)
diff --git a/infotro/_navbar.md b/infotro/_navbar.md
index bd1e3b7..f0aad4e 100644
--- a/infotro/_navbar.md
+++ b/infotro/_navbar.md
@@ -10,6 +10,8 @@
 * [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)
 * [SEm - Systèmes Embarqués](/infotro/SEm/)
   * [PLA - Programmation en langage assembleur](/infotro/SEm/PLA.md)
   * [ASM - Assembleur ARM](/infotro/SEm/ASM.md)
\ No newline at end of file