From e5a00441e4159bb608c16e3e40b36994ee8e3c2b Mon Sep 17 00:00:00 2001 From: Klagarge Date: Sun, 4 Feb 2024 16:49:33 +0100 Subject: [PATCH] refactor and add some miss equations --- infotro/Eln/ADDA.md | 115 ++++++++++++++++-- infotro/Eln/README.md | 3 +- infotro/Eln/Sigma-Delta.md | 77 ------------ infotro/Eln/SignauxEtEchantillonnage.md | 42 ++++++- .../Eln/{filtres_numeriques.md => filtres.md} | 66 +++++++--- infotro/Eln/medias/filters-h0.png | Bin 0 -> 38888 bytes infotro/Eln/medias/filters-h1.png | Bin 0 -> 47050 bytes infotro/Eln/medias/filters-h2.png | Bin 0 -> 50674 bytes infotro/Eln/medias/filters-h3.png | Bin 0 -> 49427 bytes infotro/Eln/medias/filters-h4.png | Bin 0 -> 49452 bytes infotro/Eln/medias/filters-h5-h6.png | Bin 0 -> 146115 bytes infotro/Eln/medias/filters-orther.png | Bin 0 -> 117761 bytes infotro/README.md | 4 +- infotro/_navbar.md | 4 +- 14 files changed, 198 insertions(+), 113 deletions(-) delete mode 100644 infotro/Eln/Sigma-Delta.md rename infotro/Eln/{filtres_numeriques.md => filtres.md} (91%) create mode 100644 infotro/Eln/medias/filters-h0.png create mode 100644 infotro/Eln/medias/filters-h1.png create mode 100644 infotro/Eln/medias/filters-h2.png create mode 100644 infotro/Eln/medias/filters-h3.png create mode 100644 infotro/Eln/medias/filters-h4.png create mode 100644 infotro/Eln/medias/filters-h5-h6.png create mode 100644 infotro/Eln/medias/filters-orther.png diff --git a/infotro/Eln/ADDA.md b/infotro/Eln/ADDA.md index 1d11084..56a054b 100644 --- a/infotro/Eln/ADDA.md +++ b/infotro/Eln/ADDA.md @@ -1,8 +1,7 @@ -
-

AD/DA

-
+

+AD/DA +

-# AD Il existe 4 méthodes principales pour une conversion Analogique vers Digital: - Parallèle - Par pesée @@ -15,13 +14,29 @@ Il existe 4 méthodes principales pour une conversion Analogique vers Digital: | **Pesée** | Taille du circuit ~$n$ | | | **Par comptage** | 1 seul switch | Nécessite d'aller très vite | +$$SNR [dB]=6,02\cdot n+1,78$$ +$n$ : nombre de bits du convertisseur + +$$OSR = \frac{f_{s}}{2f_{0}}$$ +$OSR$ : OverSampling Ratio
+$f_{s}$ : Sampling Frequency
+$f_{0}$ : Entry Fréquency
+ +$$n=\frac{SNR}{20\log_{10}(2\cdot OSR)}$$ +$n$ : Ordre du filtre
+$SNR$ : Signal sur bruit
+$OSR$ : Oversampling ratio
-## Parallèle + + +# AD - Parallèle ![AD Parallèle](medias/parallele.png ':size=200') -## Par comptage +---------- + +# AD - Par comptage Un seul switch mais avec une fréquence de commutation très rapide $$ V_{a} = \frac{t_{on}}{T}\cdot V_{ref} @@ -29,7 +44,9 @@ $$ ![AD par comptage](medias/comptage.png ':size=300') ![Graph AD par comptage](medias/comptage-graph.png ':size=200') -## Par pesée +---------- + +# AD - Par pesée Un switch par bit. ![AD par pesée](medias/pesee.png ':size=300') @@ -47,8 +64,9 @@ Pour un $V_{red} = 1[V]$ on a: | 1 | 1 | 0 | 0,750 | | 1 | 1 | 1 | 0,875 | -# DA -## Échantillonneur avec maintien +---------- + +# DA - Échantillonneur avec maintien # Caractéristique de convertisseurs @@ -59,3 +77,82 @@ $$ nBits = Ceiling\left[log_2\cdot \frac{plage}{pas}\right] $$ +---------- + +# Sigma / Delta + +[sigma-delta.drawio](medias/sigma-delta.drawio ':include :type=code') + + +## Principe +Un convertisseur analogique-numérique (AD) ou numérique-analogique (DA) delta-sigma se compose toujours d'un modulateur +delta-sigma produisant un "flux binaire" ou "signal binaire séquentiel" et d'un filtre passe-bas. + +Le modulateur peut être mis en œuvre comme un circuit numérique pour un signal numérique ou comme un circuit analogique +pour un signal analogique. De même, le filtre passe-bas peut être analogique pour générer un signal de sortie +analogique ou numérique, réalisé soit par un circuit numérique soit par un algorithme dans un processeur de signal. +Cette structure permet d'explorer le flux binaire et le filtre passe-bas avant d'approfondir l'étude du +modulateur delta-sigma. + +### Le flux binaire +Le flux binaire peut être interprété comme un signal numérique ou analogique. +Il s'agit d'un signal séquentiel à un bit avec une fréquence d'horloge significativement plus élevée que le débit des +données numériques de l'ADC ou du DAC. Sa caractéristique essentielle est que sa moyenne correspond à la moyenne du +signal d'entrée. Un niveau logique "haut" représente la valeur maximale de sortie, tandis qu'un niveau logique "bas" +représente la valeur minimale de sortie. + +- **Sortie analogique :** Le flux binaire est converti par un DA à un bit en un signal analogique. + Les valeurs d'entrée "bas" et "haut" se traduisent respectivement par deux tensions analogiques précises, par exemple, + 1V et +1V. + +- **Sortie numérique :** Dans le cas d'une sortie numérique, un niveau "haut" (ou "bas") dans le flux binaire représente + la valeur maximale (ou minimale) de la plage de valeurs numériques. Par exemple, dans un système à 8 bits, un niveau + "haut" pourrait correspondre à FF en hexadécimal, tandis qu'un niveau "bas" pourrait correspondre à 00 en hexadécimal. + +Un flux binaire similaire existe dans un signal modulé en largeur d‘impulsions (PWM), mais il présente quelques +désavantages par rapport au flux binaire d'un convertisseur delta-sigma, également appelé signal modulé en rapport +d’impulsions (PPM). La transmission en série d'une valeur numérique, telle que la sortie en série d'un ADC conventionnel +par pesée, est quant à elle appelée modulation par codage d'impulsions (PCM). + +---------- + +# + +Un convertisseur sigma-delta est un type de convertisseur +analogique-numérique (CAN) qui utilise une boucle de contre-réaction pour +échantillonner et quantifier un signal analogique en un signal numérique. +Le principe de fonctionnement d'un convertisseur sigma-delta peut être +expliqué en plusieurs étapes : + +1. Échantillonnage : Tout d'abord, le signal analogique est échantillonné + à une fréquence d'échantillonnage beaucoup plus élevée que la bande + passante du signal. Cette haute fréquence d'échantillonnage permet de + réduire considérablement les effets des distorsions et du bruit dans le + domaine fréquentiel. +2. Quantification : Le signal échantillonné est ensuite quantifié en un + signal numérique à l'aide d'un comparateur qui compare le signal avec une + référence de tension fixe. La sortie du comparateur est une suite de bits + représentant la valeur du signal analogique à chaque instant. +3. Contre-réaction : Le signal quantifié est ensuite utilisé pour régler + un oscillateur numérique qui produit un signal d'erreur. Ce signal + d'erreur est ajouté au signal d'entrée dans la boucle de contre-réaction, + ce qui permet de compenser les erreurs de quantification et de réduire le + bruit dans le signal quantifié. +4. Suréchantillonnage : Le signal numérique produit par le convertisseur + sigma-delta est suréchantillonné à une fréquence beaucoup plus élevée que + la fréquence d'échantillonnage initiale, ce qui permet de réduire encore + davantage les effets du bruit dans le signal. +5. Décimation : Enfin, le signal suréchantillonné est décimé à la + fréquence d'échantillonnage désirée en utilisant un filtre passe-bas + numérique qui élimine les composantes de hautes fréquences indésirables. + +Le résultat final est un signal numérique représentant fidèlement le +signal analogique d'origine, avec une faible distorsion et un faible +bruit. La boucle de contre-réaction permet au convertisseur sigma-delta +d'atteindre des performances supérieures à celles des autres types de +convertisseurs analogiques-numérique en termes de résolution et de +précision. + +![img.png](medias/snr_by_osr.png) + +---------- \ No newline at end of file diff --git a/infotro/Eln/README.md b/infotro/Eln/README.md index 632d9d5..b0b561a 100644 --- a/infotro/Eln/README.md +++ b/infotro/Eln/README.md @@ -4,7 +4,6 @@ # Convertisseurs - [AD/DA](/infotro/Eln/ADDA) -- [Sigma / Delta](/infotro/Eln/Sigma-Delta) - [Transformée en Z](/infotro/Eln/Ztransform) - [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage) -- [Filtres numériques](/infotro/Eln/filtres_numeriques) \ No newline at end of file +- [Filtres](/infotro/Eln/Filtres) \ No newline at end of file diff --git a/infotro/Eln/Sigma-Delta.md b/infotro/Eln/Sigma-Delta.md deleted file mode 100644 index f915995..0000000 --- a/infotro/Eln/Sigma-Delta.md +++ /dev/null @@ -1,77 +0,0 @@ -
-

Sigma / Delta

-
- -[sigma-delta.drawio](medias/sigma-delta.drawio ':include :type=code') - - -# Principe -Un convertisseur analogique-numérique (AD) ou numérique-analogique (DA) delta-sigma se compose toujours d'un modulateur -delta-sigma produisant un "flux binaire" ou "signal binaire séquentiel" et d'un filtre passe-bas. - -Le modulateur peut être mis en œuvre comme un circuit numérique pour un signal numérique ou comme un circuit analogique -pour un signal analogique. De même, le filtre passe-bas peut être analogique pour générer un signal de sortie -analogique ou numérique, réalisé soit par un circuit numérique soit par un algorithme dans un processeur de signal. -Cette structure permet d'explorer le flux binaire et le filtre passe-bas avant d'approfondir l'étude du -modulateur delta-sigma. - -## Le flux binaire -Le flux binaire peut être interprété comme un signal numérique ou analogique. -Il s'agit d'un signal séquentiel à un bit avec une fréquence d'horloge significativement plus élevée que le débit des -données numériques de l'ADC ou du DAC. Sa caractéristique essentielle est que sa moyenne correspond à la moyenne du -signal d'entrée. Un niveau logique "haut" représente la valeur maximale de sortie, tandis qu'un niveau logique "bas" -représente la valeur minimale de sortie. - -- **Sortie analogique :** Le flux binaire est converti par un DA à un bit en un signal analogique. -Les valeurs d'entrée "bas" et "haut" se traduisent respectivement par deux tensions analogiques précises, par exemple, -1V et +1V. - -- **Sortie numérique :** Dans le cas d'une sortie numérique, un niveau "haut" (ou "bas") dans le flux binaire représente -la valeur maximale (ou minimale) de la plage de valeurs numériques. Par exemple, dans un système à 8 bits, un niveau -"haut" pourrait correspondre à FF en hexadécimal, tandis qu'un niveau "bas" pourrait correspondre à 00 en hexadécimal. - -Un flux binaire similaire existe dans un signal modulé en largeur d‘impulsions (PWM), mais il présente quelques -désavantages par rapport au flux binaire d'un convertisseur delta-sigma, également appelé signal modulé en rapport -d’impulsions (PPM). La transmission en série d'une valeur numérique, telle que la sortie en série d'un ADC conventionnel -par pesée, est quant à elle appelée modulation par codage d'impulsions (PCM). - -# - -Un convertisseur sigma-delta est un type de convertisseur -analogique-numérique (CAN) qui utilise une boucle de contre-réaction pour -échantillonner et quantifier un signal analogique en un signal numérique. -Le principe de fonctionnement d'un convertisseur sigma-delta peut être -expliqué en plusieurs étapes : - -1. Échantillonnage : Tout d'abord, le signal analogique est échantillonné -à une fréquence d'échantillonnage beaucoup plus élevée que la bande - passante du signal. Cette haute fréquence d'échantillonnage permet de - réduire considérablement les effets des distorsions et du bruit dans le - domaine fréquentiel. -2. Quantification : Le signal échantillonné est ensuite quantifié en un - signal numérique à l'aide d'un comparateur qui compare le signal avec une - référence de tension fixe. La sortie du comparateur est une suite de bits - représentant la valeur du signal analogique à chaque instant. -3. Contre-réaction : Le signal quantifié est ensuite utilisé pour régler - un oscillateur numérique qui produit un signal d'erreur. Ce signal - d'erreur est ajouté au signal d'entrée dans la boucle de contre-réaction, - ce qui permet de compenser les erreurs de quantification et de réduire le - bruit dans le signal quantifié. -4. Suréchantillonnage : Le signal numérique produit par le convertisseur - sigma-delta est suréchantillonné à une fréquence beaucoup plus élevée que - la fréquence d'échantillonnage initiale, ce qui permet de réduire encore - davantage les effets du bruit dans le signal. -5. Décimation : Enfin, le signal suréchantillonné est décimé à la - fréquence d'échantillonnage désirée en utilisant un filtre passe-bas - numérique qui élimine les composantes de hautes fréquences indésirables. - -Le résultat final est un signal numérique représentant fidèlement le -signal analogique d'origine, avec une faible distorsion et un faible -bruit. La boucle de contre-réaction permet au convertisseur sigma-delta -d'atteindre des performances supérieures à celles des autres types de -convertisseurs analogiques-numérique en termes de résolution et de -précision. - - - -![img.png](medias/snr_by_osr.png) diff --git a/infotro/Eln/SignauxEtEchantillonnage.md b/infotro/Eln/SignauxEtEchantillonnage.md index 5f478f2..001e310 100644 --- a/infotro/Eln/SignauxEtEchantillonnage.md +++ b/infotro/Eln/SignauxEtEchantillonnage.md @@ -54,6 +54,7 @@ et en compression de données. > 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 $δ[k]$. @@ -115,6 +116,9 @@ On peut retrouver la réponse impulsionnelle à partir de ces facteurs. La somme décomposée est appelée *équation aux différences*. Pour obtenir la réponse indicielle depuis la réponse impulsionnelle, on utilise la somme cumulée de la réponse impulsionnelle. +On peut aussi mathématiquement obtenir la réponse impulsionnelle en multipliant la réponse indicielle par Heaviside ($F(z)\cdot \varepsilon(z)$). + +---------- # Transformée en Z La transformée en Z, également connue sous le nom de transformée en fréquence Z, est une technique de traitement du @@ -147,7 +151,7 @@ $x[k]$ : Fonction du [Tableau des transformées en Z](/infotro/Eln/Ztransform) o On a aussi la propriété suivante : $$\mathcal{Z}\{x[k-n]\}=z^{-n}\cdot X(z)$$ -## Marche à suivre +## Marche à suivre - Transformée en *Z* 1. Avoir une équation aux différences. 2. Pour chaque terme de l'équation, appliquer la transformée en Z. Avec : @@ -188,6 +192,18 @@ Les zéros et les pôles de la fonction de transfert donnent donc : $$H(z) = K \frac{(z-o_{1})\cdot (z-o_{2})\cdot \space \dots \space \cdot (z-o_{N})}{(z-p_{1})\cdot (z-p_{2})\cdot \space \dots \space \cdot (z-p_{M}) }\quad o_{n}, p_{n} \space \text{complexes}$$ +## Marche à suivre - Transformée en *Z* inverse +1. Avoir une fonction de transfert : $H(z) = \frac{Y(z)}{X(z)}$. +2. Effectuer le numérateur et le dénominateur. +3. Mettre les $z$ en puissance négative : $\cdot \frac{z^{-1}}{z^{-1}}$ +4. Effectuer le produit en croix. +5. Pour chaque terme, appliquer la transformée en Z inverse. + - $F(z) \implies f[k]$ + - $F(z)\cdot z^{-n} \implies f[k-n]$ +6. Isoler $y[k]$. Et on a notre équation aux différences. + +---------- + # Approximation bilinéaire L'approximation bilinéaire est une technique utilisée pour convertir une fonction de transfert continue en une fonction de transfert discrète. @@ -198,7 +214,26 @@ $$\ln\left( \frac{1+x}{1-x} \right) = 2\left(x + \frac{x^{3}}{3} + \frac{x^{5}}{ donne l'approximation $\ln\left( \frac{1+x}{1-x} \right) \approx 2x$ -#TODO add formule from lab6-filters for bilinear approximation +## Approximation bilinéaire - Pôles et zéros +$$p_{di} = e^{p_{ai} \cdot T} \qquad z_{di} = e^{z_{ai} \cdot T} $$ +- $k_a$, $p_{ai}$ and $z_{ai}$ sont le gain, les pôles et les zéros de notre filtre analogique + Attention, $k_a$ n'est pas le gain DC +- $k_d$, $p_{di}$ and $z_{di}$ sont le gain, des pôles et les zéros de notre filtre numérique. +Pour un filtre passe bas sans zéros, le gain numérique est: +$$k_{d} = k_{a}\cdot \prod_{i} \frac{1-e^{P_{ai}\cdot T}}{P_{ai}}$$ + +On peut appliquer la transformation bilinéaire aux pôles et zéros du filtre analogique pour obtenir les pôles et zéros du filtre numérique. +$$ +p_{di} = \frac{1+p_{ai}\cdot \frac{T}{2}}{1-p_{ai}\cdot \frac{T}{2}} +\qquad \qquad +z_{di} = \frac{1+z_{ai}\cdot \frac{T}{2}}{1-z_{ai}\cdot \frac{T}{2}} +$$ +Pour un filtre passe bas, le gain "bilinear" est : +$$k_{d} = k_{a}\cdot \frac{1}{\prod_{i}\left( \frac{2}{T}-p_{ai} \right)}$$ + +Pour passer de $\omega_o'$ à $\omega_o$, on utilise la relation suivante : +$$\omega_{0} = 2f_{s}\tan\left( \frac{\omega_{0}'}{2f_{s}} \right)$$ + ## Autres méthodes d'approximation d'un système continu LIT par un syst. discret On détermine la réponse impulsionnelle du système discret LIT à partir de celle du système continu en appliquant la relation : @@ -212,3 +247,6 @@ $$x'[k]= \frac{x[k]-x[k-1]}{T_{s}}$$ + + +---------- \ No newline at end of file diff --git a/infotro/Eln/filtres_numeriques.md b/infotro/Eln/filtres.md similarity index 91% rename from infotro/Eln/filtres_numeriques.md rename to infotro/Eln/filtres.md index f09f763..2b0117f 100644 --- a/infotro/Eln/filtres_numeriques.md +++ b/infotro/Eln/filtres.md @@ -1,10 +1,40 @@

-Filtres numériques +Filtres

----------- -# Filtres numériques récursifs (IIR) + +# Diagramme de Bode + +## H0 - Gain statique +$$H_{0}(s)=A$$ +![filters-h0.png](medias%2Ffilters-h0.png) + +## H1 - Coupe bas, amplis haut +$$H_{1}(s)=\frac{s}{\omega_{0}}$$ +![filters-h1.png](medias%2Ffilters-h1.png) + +## H2 - Ampli haut, coupe bas +$$H_{2}(s)=\frac{\omega_{0}}{s}$$ +![filters-h2.png](medias%2Ffilters-h2.png) + +## H3 - Ampli haut +$$H_{3}(s)=\frac{\omega_{0}+1}{\omega_{0}} = 1+ \frac{s}{\omega_{0}}$$ +![filters-h3.png](medias%2Ffilters-h3.png) + +## H4 - Coupe haut +$$H_{4}(s)=\frac{\omega_{0}}{\omega_{0}+s} = \frac{1}{1+\frac{s}{\omega_{0}}}$$ +![filters-h4.png](medias%2Ffilters-h4.png) + +## Résonance +![filters-h5-h6.png](medias%2Ffilters-h5-h6.png) + +---------- + +# Filtres numériques + +## Filtres numériques récursifs (IIR) Les filtres numériques récursifs (IIR) sont des filtres numériques qui utilisent des rétroactions. Ils sont basés sur des équations aux différences. @@ -20,7 +50,7 @@ Les inconvénients des filtres récursifs : - Ils peuvent avoir une instabilité numérique ***Fonction de transfert en $z$ :*** -$$F(z) = \frac{Y(z)}{U(z)} = \frac{b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-1}}{1+a_{1}z^{-1}+\dots+a_{n}z^{-n}}$$ +$$F(z) = \frac{Y(z)}{U(z)} = \frac{b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-n}}{1+a_{1}z^{-1}+\dots+a_{n}z^{-n}}$$ Numérisation par conversion DA et AD d'un filtre analogique : $$F(z) = (1-z^{-1})Z \left\{ L^{-1}\left[ \frac{F(s)}{s} \right] \right\} $$ La fonction de transfert en *z* permet ensuite de tirer l'équation aux différences qui servira à réaliser le filtre sous forme d'un algorithme récursif @@ -44,7 +74,7 @@ s&=\frac{1}{T_{s}\ln(z)} ![Filtre numérique passe bas ordre 2 - Réponse impulsionnel](medias/filtre_numerique_passe_bas_2-reponse.png ':size=40%') ![Filtre numérique passe bas ordre 2 - Bode](medias/filtre_numerique_passe_bas_2-bode.png ':size=40%') -## Transformation analogique-numérique +### Transformation analogique-numérique La transformation bilinéaire : transformation de Tustin ou approximation de Padé de premier ordre $$s = \frac{2}{T_{s}} \frac{z-1}{z+1} \qquad \qquad z = \frac{1+\frac{1}{2}sT_{s}}{1-\frac{1}{2}sT_{s}}$$ @@ -75,7 +105,7 @@ b_{1} = \frac{2\left( d_{0}-\frac{4d_{2}}{T_{s^{2}}} \right)}{C} b_{2} = \frac{d_{0}-\frac{2d_{1}}{T_{s}}+\frac{4d_{2}}{T_{s^{2}}}}{C} \end{gather}$$ -# Filtre numérique non-récursif (FIR) +## Filtre numérique non-récursif (FIR) Les filtres numériques non-récursifs (FIR) sont des filtres numériques qui n'utilisent pas de rétroactions. Ils sont basés sur des équations aux différences. @@ -112,7 +142,7 @@ Il y a donc 4 types de filtres FIR à phase linéaire : n resp. symétrie paire/ ***Fonction de transfert en $z$ :*** -$$F(z) = \frac{Y(z)}{U(z)} = b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-1}$$ +$$F(z) = \frac{Y(z)}{U(z)} = b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-n}$$ ***Équation aux différences :*** $$y[k]= b_{n}u[k-n]+ \dots + b_{1}u[k-1]+b_{0}u[k]$$ @@ -120,7 +150,7 @@ $$y[k]= b_{n}u[k-n]+ \dots + b_{1}u[k-1]+b_{0}u[k]$$ ***Réponse en fréquence :*** $$z=\exp(\mathbf{j}\omega\cdot T_{s})$$ -## Filtres FIR à symétrie paire : +### Filtres FIR à symétrie paire : $$b_{n-k} = b_{k}$$ Exemple : FIR à symétrie paire d'ordre 6 @@ -133,7 +163,7 @@ Les filtres FIR à symétrie paire sont toujours de type passe-bas ou coupe-band Ils possèdent autant de pôles que l'order du filtre. Ces derniers sont à (0,0) sur le cercle unité.
Ils possèdent autant de zéros que l'ordre du filtre. Ces derniers sont sur le cercle unité.
-## Filtres FIR à symétrie impaire : +### Filtres FIR à symétrie impaire : $$b_{n-k} = -b_{k}$$ Exemple : FIR à symétrie impaire d'ordre 6 $$\begin{align} @@ -143,12 +173,12 @@ F(z) &= \frac{Y(z)}{U(z)} \\ \end{align}$$ Les filtres FIR à symétrie paire sont toujours de type passe-haut ou passe-bande.
Ils possèdent autant de pôles que l'ordre du filtre. Ces derniers sont à (0,0) sur le cercle unité.
-Ils possèdent autant de zéros que l'ordre du filtre.
+Ils possèdent autant de zéros que l'ordre du filtre.
-# Filtres numériques FIR idéaux +## Filtres numériques FIR idéaux $$\omega_{c_{d}} = 2\pi \frac{f_{c_{a}}}{f_{s}}$$ -## Filtre passe-bas idéal +### Filtre passe-bas idéal $$H(\mathbf{j}\omega)=\begin{cases} 1 &\text{ si } |\omega| \leq \omega_{c} \\ 0 &\text{ si } |\omega| > \omega_{c} @@ -166,7 +196,7 @@ $$h[n] = \begin{cases} On tronque à $N=ordre$ du filtre. $N_{s}=\frac{N}{2}$
Ces coefficients ne sont rien d’autre que les coefficients de la fonction de transfert en *z* du filtre passe-bas idéal. -## Filtre passe-haut idéal +### Filtre passe-haut idéal $$H(\mathbf{j}\omega)=\begin{cases} 0 &\text{ si } |\omega| \leq \omega_{c} \\ 1 &\text{ si } |\omega| > \omega_{c} @@ -184,7 +214,7 @@ $$h[n] = \begin{cases} On tronque à $N=ordre$ du filtre. $N_{s}=\frac{N}{2}$
Ces coefficients ne sont rien d’autre que les coefficients de la fonction de transfert en *z* du filtre passe-haut idéal. -## Filtre passe-bande idéal +### Filtre passe-bande idéal $$H(\mathbf{j}\omega)=\begin{cases} 1 &\text{ si } \omega_{c_{1}} \leq \omega \leq \omega_{c_{2}} \\ 0 &\text{ si } \omega \leq \omega_{c_{1}} \quad et \quad \omega\geq \omega_{c_{2}} @@ -202,7 +232,7 @@ $$h[n] = \begin{cases} On tronque à $N=ordre$ du filtre. $N_{s}=\frac{N}{2}$
Ces coefficients ne sont rien d’autre que les coefficients de la fonction de transfert en *z* du filtre passe-bande idéal. -## Filtre coupe-bande idéal +### Filtre coupe-bande idéal $$H(\mathbf{j}\omega)=\begin{cases} 0 &\text{ si } \omega_{c_{1}} \leq \omega \leq \omega_{c_{2}} \\ 1 &\text{ si } \omega \leq \omega_{c_{1}} \quad et \quad \omega\geq \omega_{c_{2}} @@ -222,11 +252,11 @@ $$h[n] = \begin{cases} On tronque à $N=ordre$ du filtre. $N_{s}=\frac{N}{2}$
Ces coefficients ne sont rien d’autre que les coefficients de la fonction de transfert en *z* du filtre coupe-bande idéal. -# Synthèse par fenêtrage +## Synthèse par fenêtrage Le point de départ de la synthèse des filtres FIR est donné par la considération des réponses impulsionnelles des filtres idéaux. Comme celles-ci sont infiniment longues et donc non réalisables, on va se contenter d’une approximation de leurs réponses. -## Méthode de fenêtrage +### Méthode de fenêtrage - **Fenêtre rectangulaire :** $$W[n] = \begin{cases} 1 &\text{ si } 0\leq n \leq N \\ 0 &\text{ sinon} @@ -269,7 +299,7 @@ Comme celles-ci sont infiniment longues et donc non réalisables, on va se conte -## Comment calculer son filtre +### Comment calculer son filtre ![specification_de_filtre.svg](medias%2Fspecification_de_filtre.svg ':size=50%') @@ -303,3 +333,5 @@ $\omega_{a}$ : Fréquence de pulsation d'arrêt
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