From 7cb0973700436011a6b13664b946c9283680fb63 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?R=C3=A9mi=20Heredero?= <63239207+Klagarge@users.noreply.github.com> Date: Sat, 3 Feb 2024 17:33:54 +0100 Subject: [PATCH] Infotro/eln/signaux echantillonnage (#6) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit * add IIR filters * fix signaux et echant. * typo * add filtres numériques --- infotro/Eln/README.md | 3 +- infotro/Eln/Sigma-Delta.md | 11 +- infotro/Eln/SignauxEtEchantillonnage.md | 14 +- infotro/Eln/filtres_numeriques.md | 305 ++++++++++++++++++ .../filtre_numerique_passe_bas_2-bode.png | Bin 0 -> 85582 bytes .../filtre_numerique_passe_bas_2-reponse.png | Bin 0 -> 66900 bytes .../Eln/medias/specification_de_filtre.svg | 45 +++ infotro/README.md | 2 +- infotro/_navbar.md | 1 + 9 files changed, 372 insertions(+), 9 deletions(-) create mode 100644 infotro/Eln/filtres_numeriques.md create mode 100644 infotro/Eln/medias/filtre_numerique_passe_bas_2-bode.png create mode 100644 infotro/Eln/medias/filtre_numerique_passe_bas_2-reponse.png create mode 100644 infotro/Eln/medias/specification_de_filtre.svg diff --git a/infotro/Eln/README.md b/infotro/Eln/README.md index 5069aa6..632d9d5 100644 --- a/infotro/Eln/README.md +++ b/infotro/Eln/README.md @@ -6,4 +6,5 @@ - [AD/DA](/infotro/Eln/ADDA) - [Sigma / Delta](/infotro/Eln/Sigma-Delta) - [Transformée en Z](/infotro/Eln/Ztransform) -- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage) \ No newline at end of file +- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage) +- [Filtres numériques](/infotro/Eln/filtres_numeriques) \ No newline at end of file diff --git a/infotro/Eln/Sigma-Delta.md b/infotro/Eln/Sigma-Delta.md index b5a4567..f915995 100644 --- a/infotro/Eln/Sigma-Delta.md +++ b/infotro/Eln/Sigma-Delta.md @@ -16,18 +16,18 @@ Cette structure permet d'explorer le flux binaire et le filtre passe-bas avant d modulateur delta-sigma. ## Le flux binaire -Le flux binaire peut être interprété comme un signal digital ou analogique. +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. +- **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 digitale:** Dans le cas d'une sortie digitale, un niveau "haut" (ou "bas") dans le flux binaire représente -la valeur maximale (ou minimale) de la plage de valeurs digitales. Par exemple, dans un système à 8 bits, un niveau +- **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 @@ -69,10 +69,9 @@ 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 analogique-numérique en termes de résolution et de +convertisseurs analogiques-numérique en termes de résolution et de précision. -cccccckrhltvbdnvjilbnrchidectlggreetcdrehlet ![img.png](medias/snr_by_osr.png) diff --git a/infotro/Eln/SignauxEtEchantillonnage.md b/infotro/Eln/SignauxEtEchantillonnage.md index d0d3a74..a00eb6d 100644 --- a/infotro/Eln/SignauxEtEchantillonnage.md +++ b/infotro/Eln/SignauxEtEchantillonnage.md @@ -67,11 +67,18 @@ Après l'impulsion, la réponse impulsionnelle $h$ est la réponse du système $$ h[k] = \mathcal{H} \{ δ[k] \} $$ -$\mathcal{H}$ : Heaviside
+$\mathcal{H}$ : Heaviside (peut aussi se noter $\varepsilon$ pour un heaviside en discret)
$y[k]$ : réponse du système
$x[k]$ : signal d'entrée
$h[k]$ : réponse impulsionnelle
+$$ +\forall x \in \mathbb{R}, \mathcal{H}(x)=\begin{cases} +0 &\text{ si } x<0 \\ +1 &\text{ si } x \geq 0 +\end{cases} +$$ + Pour un système causal $$\begin{align} @@ -192,6 +199,11 @@ $\ln\left( \frac{1+x}{1-x} \right) \approx 2x$ ## Autres méthodes d'approximation d'un système continu LIT par un syst. discret On détermine la réponse impulsionnelle du système discret LIT à partir de celle du système continu en appliquant la relation : $$h[k] = T_{s}\cdot(kT_{s})$$ +$h(kT_s)$ : Réponse impulsionnelle en continu
+ +On peut du coup approximer l'équation différentielle par une équation aux différences. +Par exemple pour la dérivée première de $x(t)$, on a +$$x'[k]= \frac{x[k]-x[k-1]}{T_{s}}$$ diff --git a/infotro/Eln/filtres_numeriques.md b/infotro/Eln/filtres_numeriques.md new file mode 100644 index 0000000..f09f763 --- /dev/null +++ b/infotro/Eln/filtres_numeriques.md @@ -0,0 +1,305 @@ +

+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. +Les filtres récursifs sont utilisés pour des applications où la phase du filtre n'est pas critique. +Ils sont utilisés dans les télécommunications et la régulation. + +Les intérêts des filtres récursifs : +- Leur faible coût en calcul +- Faible retard (télécommunication et la régulation) + +Les inconvénients des filtres récursifs : +- Ils possèdent une phase non-linéaire +- Ils peuvent avoir une instabilité numérique + +***Fonction de transfert en $z$ :*** +$$F(z) = \frac{Y(z)}{U(z)} = \frac{b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-1}}{1+a_{1}z^{-1}+\dots+a_{n}z^{-n}}$$ +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 +$$y[k]= b_{n}u[k-n]+ \dots + b_{1}u[k-1]+b_{0}u[k]-a_{n}y[k-n]-\dots-a_{1}y[k-1]$$ + +Si on souhaite connaître la réponse fréquentielle d'un filtre, il suffit de remplacer l'opérateur *z* par sa transformée de Fourier : +$$a\cdot z^{n}=a\cdot \exp(n\cdot s\cdot T_{s})=a\cdot \exp(n\cdot \mathbf{j}\omega T_{s})$$ +$T_{s}$ : La période d'échantillonnage + +Comme l’exponentielle imaginaire est périodique, la réponse fréquentielle d’un filtre numérique est également périodique +de fréquence, et sa représentation se fait dans le domaine de fréquences allant de $0$ à $fs/2$.
+Le gain statique d’un filtre peut être calculé selon : +$$\text{Gain statique d'un filtre} = \lim_{ s \to 0 } F(s) = \lim_{ z \to 1 } F(z)$$ + +> [!NOTE|label:Passage pôle et zéro s ↔ z] +> $$\begin{align} +z&=\exp(s\cdot T_{s}) \\ +s&=\frac{1}{T_{s}\ln(z)} +\end{align}$$ + +![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 +La transformation bilinéaire : transformation de Tustin ou approximation de Padé de premier ordre +$$s = \frac{2}{T_{s}} \frac{z-1}{z+1} \qquad \qquad z = \frac{1+\frac{1}{2}sT_{s}}{1-\frac{1}{2}sT_{s}}$$ + +***Pour un filtre analogique de premier ordre :*** +$$F(s)=\frac{d_{0}+d_{1}s}{c_{0}+c_{1}s} \to F(z) = \frac{b_{1}z^{-1}+b_{0}}{a_{1}z^{-1}+1} $$ +$$ +a_{1} = \frac{c_{0}-\frac{2c_{1}}{T_{s}}}{c_{0}+\frac{2c_{1}}{T_{s}}} +\qquad \qquad +b_{0} = \frac{d_{0}+\frac{2d_{1}}{T_{s}}}{c_{0}+\frac{2c_{1}}{T_{s}}} +\qquad \qquad +b_{1} = \frac{d_{0}-\frac{2d_{1}}{T_{s}}}{c_{0}+\frac{2c_{1}}{T_{s}}} +$$ + +***Pour un filtre analogique de second ordre :*** +$$F(s)=\frac{Y(s)}{U(s)}=\frac{d_{0}+d_{1}s+d_{2}s^{2}}{c_{0}+c_{1}s+c_{2}s^{2}} +\to F(z) = \frac{Y(z)}{U(z)}=\frac{b_{2}z^{-2}+b_{1}z^{-1}+b_{0}}{a_{2}z^{-2+}a_{1}z^{-1}+1} $$ +$$\begin{gather} +C = c_{0}+\frac{2c_{1}}{T_{s}}+\frac{4c_{2}}{T_{s}^{2}} \\ +\\ +a_{1} = \frac{2\left( c_{0}-\frac{4c_{2}}{T_{s^{2}}} \right)}{C} +\qquad \qquad +a_{2} = \frac{ c_{0}-\frac{2c_{1}}{T_{s}}+\frac{4c_{2}}{T_{s^{2}}} }{C} +\qquad \qquad +b_{0} = \frac{d_{0}+\frac{2d_{1}}{T_{s}}+\frac{4d_{2}}{T_{s}^{2}}}{C} \\ +\\ +b_{1} = \frac{2\left( d_{0}-\frac{4d_{2}}{T_{s^{2}}} \right)}{C} +\qquad \qquad +b_{2} = \frac{d_{0}-\frac{2d_{1}}{T_{s}}+\frac{4d_{2}}{T_{s^{2}}}}{C} +\end{gather}$$ + +# 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. +Les filtres non-récursifs sont utilisés pour des applications où la phase du filtre est critique. +Ils sont utilisés dans le traitement du signal et la conception de filtres. + +Les intérêts des filtres non-récursifs : +- Les fonctions de transfert ne possèdent que des coefficients $b_i$. ($\forall a_{i} = 0$) +- Ils possèdent une phase linéaire exacte +- Ils sont stables + +Les inconvénients des filtres non-récursifs : +- Ils sont plus coûteux en calcul +- Ils nécessitent plus de matériel + +Coefficients symétrique: pairs $b_{n-i} = b_{i}$, impairs $b_{n-i} = -b_{i}$, $b_{n/2}=0$
+ +**Symétrie paire:** +$$H(\mathbf{j}\omega) += \sum^{n}_{i=o}b_{i}e^{-\frac{\mathbf{j}i\omega}{f_{s}}} += e^{\frac{-\mathbf{j}n\omega}{2f_{s}}} \sum^{n/2}_{i=0}b_{i}\cos\left[ \frac{\left( i-\frac{n}{2} \right)\omega}{f_{s}} \right]$$ + +**Symétrie impaire:** +$$H(\mathbf{j}\omega)=\mathbf{j}e^{\frac{-\mathbf{j}n\omega}{2f_{s}}}\sum^{n/2}_{i=0}b_{i}\sin\left( \frac{(i-\frac{n}{2})\omega}{f_{s}} \right)$$ + +Les filtres FIR à coefficients symétriques ont une phase linéaire. +$$ +\text{Symétrie paire : }\varphi(\omega)=-\frac{n\omega}{2f_{s}} +\qquad \qquad +\text{Symétrie impaire : } \varphi(\omega)= -\frac{n\omega}{2f_{s}}+\frac{\pi}{2} +$$ +$$\text{Temps de propagation : }t_{gr}=-\frac{d\varphi}{d\omega}=\frac{n}{2f_{s}}=\frac{n}{2}T_{s}$$ +Il y a donc 4 types de filtres FIR à phase linéaire : n resp. symétrie paire/impaire + + +***Fonction de transfert en $z$ :*** +$$F(z) = \frac{Y(z)}{U(z)} = b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-1}$$ + +***Équation aux différences :*** +$$y[k]= b_{n}u[k-n]+ \dots + b_{1}u[k-1]+b_{0}u[k]$$ + +***Réponse en fréquence :*** +$$z=\exp(\mathbf{j}\omega\cdot T_{s})$$ + +## Filtres FIR à symétrie paire : +$$b_{n-k} = b_{k}$$ + +Exemple : FIR à symétrie paire d'ordre 6 +$$\begin{align} +F(z) &= \frac{Y(z)}{U(z)} \\ +&= 1+2z^{-1}+3z^{-2}+4z^{-3}+3z^{-4}+2z^{-5}+z^{-6} \\ +&= \frac{z^{6}+2z^{5}+3z^{4}+4z^{3}+3z^{2}+2z^{1}+1}{z^{6}} +\end{align}$$ +Les filtres FIR à symétrie paire sont toujours de type passe-bas ou coupe-bande.
+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 : +$$b_{n-k} = -b_{k}$$ +Exemple : FIR à symétrie impaire d'ordre 6 +$$\begin{align} +F(z) &= \frac{Y(z)}{U(z)} \\ +&= -1-2z^{-1}-3z^{-2}+0z^{-3}+3z^{-4}+2z^{-5}+z^{-6} \\ +&= \frac{z^{6}+2z^{5}+3z^{4}+0z^{3}-3z^{2}-2z^{1}-1}{z^{8}} +\end{align}$$ +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.
+ +# Filtres numériques FIR idéaux +$$\omega_{c_{d}} = 2\pi \frac{f_{c_{a}}}{f_{s}}$$ + +## Filtre passe-bas idéal +$$H(\mathbf{j}\omega)=\begin{cases} +1 &\text{ si } |\omega| \leq \omega_{c} \\ +0 &\text{ si } |\omega| > \omega_{c} +\end{cases} $$ +La transformée de Fourier inverse donne la réponse impulsionnelle d’un filtre passe-bas idéal. + +$$h[n] = \begin{cases} +\frac{\omega_{c}}{\pi} \frac{\sin((n-N_{s})\omega_{c})}{(n-N_{s})\omega_{c}} &\text{ si } 0\leq n \leq N \\ +\\ +\frac{\omega_{c}}{\pi} &\text{ si } n=N_{s} \\ +\\ +0 &\text{ sinon} +\end{cases}$$ + +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 +$$H(\mathbf{j}\omega)=\begin{cases} +0 &\text{ si } |\omega| \leq \omega_{c} \\ +1 &\text{ si } |\omega| > \omega_{c} +\end{cases}$$ +La transformée de Fourier inverse donne la réponse impulsionnelle d’un filtre passe-bas idéal. + +$$h[n] = \begin{cases} +-\frac{\omega_{c}}{\pi} \frac{\sin((n-N_{s})\omega_{c})}{(n-N_{s})\omega_{c}} &\text{ si } 0\leq n \leq N \\ +\\ +1-\frac{\omega_{c}}{\pi} &\text{ si } n=N_{s} \\ +\\ +0 &\text{ sinon} +\end{cases}$$ + +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 +$$H(\mathbf{j}\omega)=\begin{cases} +1 &\text{ si } \omega_{c_{1}} \leq \omega \leq \omega_{c_{2}} \\ +0 &\text{ si } \omega \leq \omega_{c_{1}} \quad et \quad \omega\geq \omega_{c_{2}} +\end{cases}$$ +La transformée de Fourrier inverse donne la réponse impulsionnelle d’un filtre passe-bande idéal. + +$$h[n] = \begin{cases} +\frac{\omega_{c_{2}}}{\pi} \frac{\sin((n-N_{s})\omega_{c_{2}})}{(n-N_{s})\omega_{c_{2}}}-\frac{\omega_{c_{1}}}{\pi} \frac{\sin((n-N_{s})\omega_{c_{1}})}{(n-N_{s})\omega_{c_{1}}} &\text{ si } 0\leq n \leq N \\ +\\ +\frac{\omega_{c_{2}}-\omega_{c_{1}}}{\pi} &\text{ si } n=N_{s}\\ +\\ +0 &\text{ sinon} +\end{cases}$$ + +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 +$$H(\mathbf{j}\omega)=\begin{cases} +0 &\text{ si } \omega_{c_{1}} \leq \omega \leq \omega_{c_{2}} \\ +1 &\text{ si } \omega \leq \omega_{c_{1}} \quad et \quad \omega\geq \omega_{c_{2}} +\end{cases}$$ +La transformée de Fourrier inverse donne la réponse impulsionnelle d’un filtre coupe-bande idéal. + +$$h[n] = \begin{cases} +\frac{\omega_{c_{1}}}{\pi} \frac{\sin((n-N_{s})\omega_{c_{1}})}{(n-N_{s})\omega_{c_{1}}} +-\frac{\omega_{c_{2}}}{\pi} \frac{\sin((n-N_{s})\omega_{c_{2}})}{(n-N_{s})\omega_{c_{2}}} +&\text{ si } 0\leq n \leq N \\ +\\ +1-\frac{\omega_{c_{2}}-\omega_{c_{1}}}{\pi} &\text{ si } n=N_{s}\\ +\\ +0 &\text{ sinon} +\end{cases}$$ + +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 +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 +- **Fenêtre rectangulaire :** $$W[n] = \begin{cases} + 1 &\text{ si } 0\leq n \leq N \\ + 0 &\text{ sinon} + \end{cases}$$ +- **Fenêtre triangulaire (Bartlett) :** $$W[n] = \begin{cases} + 2 \frac{n}{N} &\text{ si } 0\leq n \leq \frac{N}{2} \\ + 2-2 \frac{n}{N} &\text{ si } \frac{N}{2}\leq n \leq N \\ + 0 &\text{ sinon} + \end{cases}$$ +- **Fenêtre sinusoïdale (Hann) :** $$W[n] = \begin{cases} + 0,5\left( 1-\cos\left( \frac{2\pi n}{N} \right) \right) &\text{ si } 0\leq n\leq N \\ + 0 &\text{ sinon} + \end{cases}$$ +- **Fenêtre de Hamming :** $$W[n] = \begin{cases} + 0,54-0,46\cos\left( \frac{2\pi n}{N} \right) &\text{ si } 0\leq n\leq N \\ + 0 &\text{ sinon} + \end{cases}$$ +- **Fenêtre de Blackman :** $$W[n] = \begin{cases} + 0,42-0,5\cos\left( \frac{2\pi n}{N} \right)+0,08\cos\left( \frac{4\pi n}{N} \right) &\text{ si } 0\leq n\leq N \\ + 0 &\text{ sinon} + \end{cases}$$ +- **Fenêtre de Kaiser-Bessel :** $$W[n] = \begin{cases} + \frac{I_{0}\left( \beta\sqrt{1-\left( \frac{1-n}{N_{s}} \right)^{2}} \right)}{I_{0}(\beta)} &\text{ si } 0\leq n\leq N \\ + 0 &\text{ sinon} + \end{cases}$$ + $I_{0}$ : Fonction de Bessel modifiée d’ordre 0
+ $\beta$ : Paramètre de forme + +***Caractéristiques spectrales de quelques fenêtres*** + +| Fenêtres | Largeur du 1er lobe | Atténuation du 1er lobe [dB] | Décroissance des lobes suivants | Ondulation Rp [dB] | Atténuation max [dB] | Bande de transition
[rad/sec]
$\Delta\omega$ | +|---------------------|---------------------|------------------------------|---------------------------------|--------------------|----------------------|----------------------------------------------------| +| Rectangulaire | $4\pi/N$ | $-13$ | $20 dB/dec$ | $0,74$ | $21$ | $1,8\pi/N$ | +| Triangulaire | $8\pi/N$ | $-27$ | $40 dB/dec$ | $0$ | $25$ | $6,1\pi/N$ | +| Cosinus | $8\pi/N$ | $-32$ | $60 dB/dec$ | $0,055$ | $44$ | $6,2\pi/N$ | +| Hamming | $8\pi/N$ | $-43$ | $20 dB/dec$ | $0,014$ | $53$ | $6,6\pi/N$ | +| Blackman | $12\pi/N$ | $-58$ | $60 dB/dec$ | $0,0017$ | $74$ | $11\pi/N$ | +| Kaiser $\beta=4,54$ | $7,2\pi/N$ | $-30$ | $20 dB/dec$ | $0,025$ | $50$ | $5,8\pi/N$ | +| Kaiser $\beta=5,66$ | $8,4\pi/N$ | $-42$ | $20 dB/dec$ | $0,009$ | $60$ | $7,2\pi/N$ | + + + +## Comment calculer son filtre + +![specification_de_filtre.svg](medias%2Fspecification_de_filtre.svg ':size=50%') + +$H(\mathbf{j}\omega)$ : Gain
+$\delta_{1}$ : L'erreur de gain tolérable
+$\delta_{2}$ : L'erreur d'atténuation tolérable
+$\omega_{p}$ : Fréquence de pulsation de passage
+$\omega_{a}$ : Fréquence de pulsation d'arrêt
+ +1. En fonction du gabarit, déterminer le type de filtre désiré (`passe-bas`, `passe-haut`, `passe-bande`, `coupe-bande`) +2. Calculer $f_{c} \to \omega_{c}$ et $\Delta f \to \Delta\omega$ + $$\begin{gather} + f_{c} = \frac{f_{a}+f_{p}}{2} \qquad & \qquad \Delta f = f_{a}-f_{p} \\ + \omega_{c} = 2\pi \frac{f_{c}}{f_{s}} \quad & \quad \Delta\omega = \frac{2\pi\Delta f}{f_{s}} + \end{gather}$$ +3. Choisir une fenêtre $W[n]$ satisfaisante du point de vue de l’atténuation et du reste du cahier des charges.(voir tableau ci-dessus) +4. Calculer l'ordre du filtre à l'aide de la bande de transition du filtre choisit. Arrondir au nombre pair supérieur. +5. Calculer la réponse impulsionnelle $h_{f}[n]$ du filtre avec $h[n]*W[n]$ +6. Calculer les coefficients $b_{n}$ du filtre avec $h_{f}[n]$ + +
+Démarche selon les slides + +1. Connaissant le gabarit du filtre désiré, choisir le filtre idéal correspondant +2. Calculer les pulsations caractéristiques se situant au centre des bandes de transitions +3. Rechercher la réponse impulsionnelle $h[n]$ du filtre idéal, si celle-ci n’est pas connue, on peut la calculer par transformation de Fourier inverse +4. Choisir une fenêtre $W[n]$ satisfaisante du point de vue de l’atténuation +5. Connaissant la largeur de la bande de transition , calculer l’ordre du filtre $N$ +6. Calculer les coefficients du filtre en multipliant la réponse impulsionnelle par la fenêtre choisie +
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0000000..e258574 --- /dev/null +++ b/infotro/Eln/medias/specification_de_filtre.svg @@ -0,0 +1,45 @@ + + + + diff --git a/infotro/README.md b/infotro/README.md index e9e2476..8db67e3 100644 --- a/infotro/README.md +++ b/infotro/README.md @@ -17,7 +17,7 @@ - [Sigma / Delta](/infotro/Eln/Sigma-Delta) - [Transformée en Z](/infotro/Eln/Ztransform) - [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage) - +- [Filtres numériques](/infotro/Eln/filtres_numeriques) ## [SEm - Système Embarqué](/infotro/SEm/) - [PLA - Programmation en langage assembleur](/infotro/SEm/PLA.md) diff --git a/infotro/_navbar.md b/infotro/_navbar.md index f0aad4e..aa0ae44 100644 --- a/infotro/_navbar.md +++ b/infotro/_navbar.md @@ -12,6 +12,7 @@ * [Sigma / Delta](/infotro/Eln/Sigma-Delta) * [Transformée en Z](/infotro/Eln/Ztransform) * [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage) + * [Filtres numériques](/infotro/Eln/filtres_numeriques) * [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