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<header><h1>
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AD/DA
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</h1></header>
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Il existe 4 méthodes principales pour une conversion Analogique vers Digital:
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- Parallèle
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- Par pesée
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- Par comptage
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- Sigma-delta
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| | + (Avantage) | - (Inconvénient) |
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|-------------------------|------------------------|-----------------------------|
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| **Parallèle<br>Unaire** | Simplicité<br>Monotone | Taille du circuit ~$2^{n}$ |
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| **Pesée** | Taille du circuit ~$n$ | |
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| **Par comptage** | 1 seul switch | Nécessite d'aller très vite |
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$$SNR [dB]=6,02\cdot n+1,78$$
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$n$ : nombre de bits du convertisseur
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$$OSR = \frac{f_{s}}{2f_{0}}$$
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$OSR$ : OverSampling Ratio <br>
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$f_{s}$ : Sampling Frequency <br>
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$f_{0}$ : Entry Fréquency <br>
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$$n=\frac{SNR}{20\log_{10}(2\cdot OSR)}$$
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$n$ : Ordre du filtre <br>
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$SNR$ : Signal sur bruit <br>
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$OSR$ : Oversampling ratio <br>
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# AD - Parallèle
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----------
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# AD - Par comptage
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Un seul switch mais avec une fréquence de commutation très rapide
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$$
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V_{a} = \frac{t_{on}}{T}\cdot V_{ref}
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$$
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----------
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# AD - Par pesée
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Un switch par bit.
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Pour un $V_{red} = 1[V]$ on a:
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| z2 | z1 | z0 | Va |
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|----|----|----|-------|
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| 0 | 0 | 0 | 0,000 |
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| 0 | 0 | 1 | 0,125 |
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| 0 | 1 | 0 | 0,250 |
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| 0 | 1 | 1 | 0,375 |
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| 1 | 0 | 0 | 0,500 |
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| 1 | 0 | 1 | 0,625 |
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| 1 | 1 | 0 | 0,750 |
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| 1 | 1 | 1 | 0,875 |
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----------
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# DA - Échantillonneur avec maintien
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# Caractéristique de convertisseurs
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## Digits et affichage
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$$
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nBits = Ceiling\left[log_2\cdot \frac{plage}{pas}\right]
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$$
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----------
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# Sigma / Delta
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[sigma-delta.drawio](medias/sigma-delta.drawio ':include :type=code')
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## Principe
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Un convertisseur analogique-numérique (AD) ou numérique-analogique (DA) delta-sigma se compose toujours d'un modulateur
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delta-sigma produisant un "flux binaire" ou "signal binaire séquentiel" et d'un filtre passe-bas.
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Le modulateur peut être mis en œuvre comme un circuit numérique pour un signal numérique ou comme un circuit analogique
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pour un signal analogique. De même, le filtre passe-bas peut être analogique pour générer un signal de sortie
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analogique ou numérique, réalisé soit par un circuit numérique soit par un algorithme dans un processeur de signal.
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Cette structure permet d'explorer le flux binaire et le filtre passe-bas avant d'approfondir l'étude du
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modulateur delta-sigma.
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### Le flux binaire
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Le flux binaire peut être interprété comme un signal numérique ou analogique.
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Il s'agit d'un signal séquentiel à un bit avec une fréquence d'horloge significativement plus élevée que le débit des
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données numériques de l'ADC ou du DAC. Sa caractéristique essentielle est que sa moyenne correspond à la moyenne du
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signal d'entrée. Un niveau logique "haut" représente la valeur maximale de sortie, tandis qu'un niveau logique "bas"
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représente la valeur minimale de sortie.
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- **Sortie analogique :** Le flux binaire est converti par un DA à un bit en un signal analogique.
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Les valeurs d'entrée "bas" et "haut" se traduisent respectivement par deux tensions analogiques précises, par exemple,
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1V et +1V.
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- **Sortie numérique :** Dans le cas d'une sortie numérique, un niveau "haut" (ou "bas") dans le flux binaire représente
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la valeur maximale (ou minimale) de la plage de valeurs numériques. Par exemple, dans un système à 8 bits, un niveau
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"haut" pourrait correspondre à FF en hexadécimal, tandis qu'un niveau "bas" pourrait correspondre à 00 en hexadécimal.
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Un flux binaire similaire existe dans un signal modulé en largeur d‘impulsions (PWM), mais il présente quelques
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désavantages par rapport au flux binaire d'un convertisseur delta-sigma, également appelé signal modulé en rapport
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d’impulsions (PPM). La transmission en série d'une valeur numérique, telle que la sortie en série d'un ADC conventionnel
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par pesée, est quant à elle appelée modulation par codage d'impulsions (PCM).
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----------
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#
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Un convertisseur sigma-delta est un type de convertisseur
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analogique-numérique (CAN) qui utilise une boucle de contre-réaction pour
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échantillonner et quantifier un signal analogique en un signal numérique.
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Le principe de fonctionnement d'un convertisseur sigma-delta peut être
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expliqué en plusieurs étapes :
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1. Échantillonnage : Tout d'abord, le signal analogique est échantillonné
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à une fréquence d'échantillonnage beaucoup plus élevée que la bande
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passante du signal. Cette haute fréquence d'échantillonnage permet de
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réduire considérablement les effets des distorsions et du bruit dans le
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domaine fréquentiel.
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2. Quantification : Le signal échantillonné est ensuite quantifié en un
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signal numérique à l'aide d'un comparateur qui compare le signal avec une
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référence de tension fixe. La sortie du comparateur est une suite de bits
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représentant la valeur du signal analogique à chaque instant.
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3. Contre-réaction : Le signal quantifié est ensuite utilisé pour régler
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un oscillateur numérique qui produit un signal d'erreur. Ce signal
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d'erreur est ajouté au signal d'entrée dans la boucle de contre-réaction,
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ce qui permet de compenser les erreurs de quantification et de réduire le
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bruit dans le signal quantifié.
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4. Suréchantillonnage : Le signal numérique produit par le convertisseur
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sigma-delta est suréchantillonné à une fréquence beaucoup plus élevée que
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la fréquence d'échantillonnage initiale, ce qui permet de réduire encore
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davantage les effets du bruit dans le signal.
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5. Décimation : Enfin, le signal suréchantillonné est décimé à la
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fréquence d'échantillonnage désirée en utilisant un filtre passe-bas
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numérique qui élimine les composantes de hautes fréquences indésirables.
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Le résultat final est un signal numérique représentant fidèlement le
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signal analogique d'origine, avec une faible distorsion et un faible
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bruit. La boucle de contre-réaction permet au convertisseur sigma-delta
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d'atteindre des performances supérieures à celles des autres types de
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convertisseurs analogiques-numérique en termes de résolution et de
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précision.
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----------
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@@ -0,0 +1,339 @@
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<header><h1>
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Filtres
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</h1></header>
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-----------
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# Diagramme de Bode
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## H0 - Gain statique
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$$H_{0}(s)=A$$
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## H1 - Coupe bas, amplis haut
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$$H_{1}(s)=\frac{s}{\omega_{0}}$$
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## H2 - Ampli haut, coupe bas
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$$H_{2}(s)=\frac{\omega_{0}}{s}$$
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## H3 - Ampli haut
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$$H_{3}(s)=\frac{\omega_{0}+1}{\omega_{0}} = 1+ \frac{s}{\omega_{0}}$$
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## H4 - Coupe haut
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$$H_{4}(s)=\frac{\omega_{0}}{\omega_{0}+s} = \frac{1}{1+\frac{s}{\omega_{0}}}$$
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## Résonance
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----------
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# Filtres numériques
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## Filtres numériques récursifs (IIR)
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Les filtres numériques récursifs (IIR) sont des filtres numériques qui utilisent des rétroactions.
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Ils sont basés sur des équations aux différences.
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Les filtres récursifs sont utilisés pour des applications où la phase du filtre n'est pas critique.
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Ils sont utilisés dans les télécommunications et la régulation.
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Les intérêts des filtres récursifs :
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- Leur faible coût en calcul
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- Faible retard (télécommunication et la régulation)
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Les inconvénients des filtres récursifs :
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- Ils possèdent une phase non-linéaire
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- Ils peuvent avoir une instabilité numérique
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***Fonction de transfert en $z$ :***
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$$F(z) = \frac{Y(z)}{U(z)} = \frac{b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-n}}{1+a_{1}z^{-1}+\dots+a_{n}z^{-n}}$$
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Numérisation par conversion DA et AD d'un filtre analogique :
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$$F(z) = (1-z^{-1})Z \left\{ L^{-1}\left[ \frac{F(s)}{s} \right] \right\} $$
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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
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$$y[k]= b_{n}u[k-n]+ \dots + b_{1}u[k-1]+b_{0}u[k]-a_{n}y[k-n]-\dots-a_{1}y[k-1]$$
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Si on souhaite connaître la réponse fréquentielle d'un filtre, il suffit de remplacer l'opérateur *z* par sa transformée de Fourier :
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$$a\cdot z^{n}=a\cdot \exp(n\cdot s\cdot T_{s})=a\cdot \exp(n\cdot \mathbf{j}\omega T_{s})$$
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$T_{s}$ : La période d'échantillonnage
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Comme l’exponentielle imaginaire est périodique, la réponse fréquentielle d’un filtre numérique est également périodique
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de fréquence, et sa représentation se fait dans le domaine de fréquences allant de $0$ à $fs/2$. <br>
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Le gain statique d’un filtre peut être calculé selon :
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$$\text{Gain statique d'un filtre} = \lim_{ s \to 0 } F(s) = \lim_{ z \to 1 } F(z)$$
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> [!NOTE|label:Passage pôle et zéro s ↔ z]
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> $$\begin{align}
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z&=\exp(s\cdot T_{s}) \\
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s&=\frac{1}{T_{s}\ln(z)}
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\end{align}$$
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### Transformation analogique-numérique
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La transformation bilinéaire : transformation de Tustin ou approximation de Padé de premier ordre
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$$s = \frac{2}{T_{s}} \frac{z-1}{z+1} \qquad \qquad z = \frac{1+\frac{1}{2}sT_{s}}{1-\frac{1}{2}sT_{s}}$$
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***Pour un filtre analogique de premier ordre :***
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$$F(s)=\frac{d_{0}+d_{1}s}{c_{0}+c_{1}s} \to F(z) = \frac{b_{1}z^{-1}+b_{0}}{a_{1}z^{-1}+1} $$
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$$
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a_{1} = \frac{c_{0}-\frac{2c_{1}}{T_{s}}}{c_{0}+\frac{2c_{1}}{T_{s}}}
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\qquad \qquad
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b_{0} = \frac{d_{0}+\frac{2d_{1}}{T_{s}}}{c_{0}+\frac{2c_{1}}{T_{s}}}
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\qquad \qquad
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b_{1} = \frac{d_{0}-\frac{2d_{1}}{T_{s}}}{c_{0}+\frac{2c_{1}}{T_{s}}}
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$$
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***Pour un filtre analogique de second ordre :***
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$$F(s)=\frac{Y(s)}{U(s)}=\frac{d_{0}+d_{1}s+d_{2}s^{2}}{c_{0}+c_{1}s+c_{2}s^{2}}
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\to F(z) = \frac{Y(z)}{U(z)}=\frac{b_{2}z^{-2}+b_{1}z^{-1}+b_{0}}{a_{2}z^{-2+}a_{1}z^{-1}+1} $$
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$$\begin{gather}
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C = c_{0}+\frac{2c_{1}}{T_{s}}+\frac{4c_{2}}{T_{s}^{2}} \\
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\\
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a_{1} = \frac{2\left( c_{0}-\frac{4c_{2}}{T_{s}^{2}} \right)}{C}
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\qquad \qquad
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a_{2} = \frac{ c_{0}-\frac{2c_{1}}{T_{s}}+\frac{4c_{2}}{T_{s}^{2}} }{C}
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\qquad \qquad
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b_{0} = \frac{d_{0}+\frac{2d_{1}}{T_{s}}+\frac{4d_{2}}{T_{s}^{2}}}{C} \\
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\\
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b_{1} = \frac{2\left( d_{0}-\frac{4d_{2}}{T_{s}^{2}} \right)}{C}
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\qquad \qquad
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b_{2} = \frac{d_{0}-\frac{2d_{1}}{T_{s}}+\frac{4d_{2}}{T_{s}^{2}}}{C}
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\end{gather}$$
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## Filtre numérique non-récursif (FIR)
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Les filtres numériques non-récursifs (FIR) sont des filtres numériques qui n'utilisent pas de rétroactions.
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Ils sont basés sur des équations aux différences.
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Les filtres non-récursifs sont utilisés pour des applications où la phase du filtre est critique.
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Ils sont utilisés dans le traitement du signal et la conception de filtres.
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Les intérêts des filtres non-récursifs :
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- Les fonctions de transfert ne possèdent que des coefficients $b_i$. ($\forall a_{i} = 0$)
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- Ils possèdent une phase linéaire exacte
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- Ils sont stables
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Les inconvénients des filtres non-récursifs :
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- Ils sont plus coûteux en calcul
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- Ils nécessitent plus de matériel
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Coefficients symétrique: pairs $b_{n-i} = b_{i}$, impairs $b_{n-i} = -b_{i}$, $b_{n/2}=0$ <br>
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**Symétrie paire:**
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$$H(\mathbf{j}\omega)
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= \sum^{n}_{i=o}b_{i}e^{-\frac{\mathbf{j}i\omega}{f_{s}}}
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= e^{\frac{-\mathbf{j}n\omega}{2f_{s}}} \sum^{n/2}_{i=0}b_{i}\cos\left[ \frac{\left( i-\frac{n}{2} \right)\omega}{f_{s}} \right]$$
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**Symétrie impaire:**
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$$H(\mathbf{j}\omega)=\mathbf{j}e^{\frac{-\mathbf{j}n\omega}{2f_{s}}}\sum^{n/2}_{i=0}b_{i}\sin\left( \frac{(i-\frac{n}{2})\omega}{f_{s}} \right)$$
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Les filtres FIR à coefficients symétriques ont une phase linéaire.
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$$
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\text{Symétrie paire : }\varphi(\omega)=-\frac{n\omega}{2f_{s}}
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\qquad \qquad
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\text{Symétrie impaire : } \varphi(\omega)= -\frac{n\omega}{2f_{s}}+\frac{\pi}{2}
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$$
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$$\text{Temps de propagation : }t_{gr}=-\frac{d\varphi}{d\omega}=\frac{n}{2f_{s}}=\frac{n}{2}T_{s}$$
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Il y a donc 4 types de filtres FIR à phase linéaire : n resp. symétrie paire/impaire
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***Fonction de transfert en $z$ :***
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$$F(z) = \frac{Y(z)}{U(z)} = b_{0}+b_{1}z^{-1}+\dots+b_{n}z^{-n}$$
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***Équation aux différences :***
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$$y[k]= b_{n}u[k-n]+ \dots + b_{1}u[k-1]+b_{0}u[k]$$
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***Réponse en fréquence :***
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$$z=\exp(\mathbf{j}\omega\cdot T_{s})$$
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### Filtres FIR à symétrie paire :
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$$b_{n-k} = b_{k}$$
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Exemple : FIR à symétrie paire d'ordre 6
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$$\begin{align}
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F(z) &= \frac{Y(z)}{U(z)} \\
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&= 1+2z^{-1}+3z^{-2}+4z^{-3}+3z^{-4}+2z^{-5}+z^{-6} \\
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&= \frac{z^{6}+2z^{5}+3z^{4}+4z^{3}+3z^{2}+2z^{1}+1}{z^{6}}
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\end{align}$$
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Les filtres FIR à symétrie paire sont toujours de type passe-bas ou coupe-bande. <br>
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Ils possèdent autant de pôles que l'order du filtre. Ces derniers sont à (0,0) sur le cercle unité. <br>
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Ils possèdent autant de zéros que l'ordre du filtre. Ces derniers sont sur le cercle unité. <br>
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### Filtres FIR à symétrie impaire :
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$$b_{n-k} = -b_{k}$$
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Exemple : FIR à symétrie impaire d'ordre 6
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$$\begin{align}
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F(z) &= \frac{Y(z)}{U(z)} \\
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&= -1-2z^{-1}-3z^{-2}+0z^{-3}+3z^{-4}+2z^{-5}+z^{-6} \\
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&= \frac{z^{6}+2z^{5}+3z^{4}+0z^{3}-3z^{2}-2z^{1}-1}{z^{8}}
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\end{align}$$
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Les filtres FIR à symétrie paire sont toujours de type passe-haut ou passe-bande. <br>
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Ils possèdent autant de pôles que l'ordre du filtre. Ces derniers sont à (0,0) sur le cercle unité. <br>
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Ils possèdent autant de zéros que l'ordre du filtre. <br>
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## Filtres numériques FIR idéaux
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$$\omega_{c_{d}} = 2\pi \frac{f_{c_{a}}}{f_{s}}$$
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### Filtre passe-bas idéal
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$$H(\mathbf{j}\omega)=\begin{cases}
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1 &\text{ si } |\omega| \leq \omega_{c} \\
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0 &\text{ si } |\omega| > \omega_{c}
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\end{cases} $$
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La transformée de Fourier inverse donne la réponse impulsionnelle d’un filtre passe-bas idéal.
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$$h[n] = \begin{cases}
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\frac{\omega_{c}}{\pi} \frac{\sin((n-N_{s})\omega_{c})}{(n-N_{s})\omega_{c}} &\text{ si } 0\leq n \leq N \\
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\\
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\frac{\omega_{c}}{\pi} &\text{ si } n=N_{s} \\
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\\
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0 &\text{ sinon}
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\end{cases}$$
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On tronque à $N=ordre$ du filtre. $N_{s}=\frac{N}{2}$ <br>
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Ces coefficients ne sont rien d’autre que les coefficients de la fonction de transfert en *z* du filtre passe-bas idéal.
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### Filtre passe-haut idéal
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$$H(\mathbf{j}\omega)=\begin{cases}
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0 &\text{ si } |\omega| \leq \omega_{c} \\
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1 &\text{ si } |\omega| > \omega_{c}
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\end{cases}$$
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La transformée de Fourier inverse donne la réponse impulsionnelle d’un filtre passe-bas idéal.
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$$h[n] = \begin{cases}
|
||||
-\frac{\omega_{c}}{\pi} \frac{\sin((n-N_{s})\omega_{c})}{(n-N_{s})\omega_{c}} &\text{ si } 0\leq n \leq N \\
|
||||
\\
|
||||
1-\frac{\omega_{c}}{\pi} &\text{ si } n=N_{s} \\
|
||||
\\
|
||||
0 &\text{ sinon}
|
||||
\end{cases}$$
|
||||
|
||||
On tronque à $N=ordre$ du filtre. $N_{s}=\frac{N}{2}$ <br>
|
||||
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. $N_{s}=\frac{N}{2}$ <br>
|
||||
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. $N_{s}=\frac{N}{2}$ <br>
|
||||
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 <br>
|
||||
$\beta$ : 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<br>[rad/sec]<br>$\Delta\omega$ |
|
||||
|---------------------|---------------------|------------------------------|---------------------------------|--------------------|----------------------|----------------------------------------------------|
|
||||
| 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
|
||||
|
||||
|
||||
|
||||
$H(\mathbf{j}\omega)$ : Gain <br>
|
||||
$\delta_{1}$ : L'erreur de gain tolérable <br>
|
||||
$\delta_{2}$ : L'erreur d'atténuation tolérable <br>
|
||||
$\omega_{p}$ : Fréquence de pulsation de passage <br>
|
||||
$\omega_{a}$ : Fréquence de pulsation d'arrêt <br>
|
||||
|
||||
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 $h_{f}[n]$
|
||||
|
||||
<details>
|
||||
<summary>Démarche selon les slides</summary>
|
||||
|
||||
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
|
||||
</details>
|
||||
|
||||
|
||||
|
||||
|
||||
----------
|
||||
@@ -0,0 +1,9 @@
|
||||
<header>
|
||||
<h1>Électronique</h1>
|
||||
</header>
|
||||
|
||||
# Convertisseurs
|
||||
- [AD/DA](/infotro/Eln/ADDA)
|
||||
- [Signaux et échantillonnage](/infotro/Eln/SignauxEtEchantillonnage)
|
||||
- [Filtres](/infotro/Eln/Filtres)
|
||||
- [RF](/infotro/Eln/RF)
|
||||
@@ -0,0 +1,145 @@
|
||||
<header><h1>
|
||||
Radio Fréquence
|
||||
</h1></header>
|
||||
|
||||
----------
|
||||
|
||||
# Skin effect
|
||||
Le courant électrique circule principalement à la surface des conducteurs.
|
||||
|
||||
Pour un fil de cuivre rond, la résistance en continue est donnée par la formule suivante:
|
||||
$$R_{DC} = \rho\frac{l}{\pi r²}$$
|
||||
avec:
|
||||
- $R_{DC}$: résistance en continue
|
||||
- $\rho$: résistivité du cuivre
|
||||
- $l$: longueur du fil
|
||||
- $r$: rayon du fil
|
||||
|
||||
La résistance en courant alternatif est donnée par la formule suivante:
|
||||
$$R_{AC} = R_{DC} \frac{r}{2\delta}$$
|
||||
avec:
|
||||
- $R_{AC}$: résistance en courant alternatif
|
||||
- $R_{DC}$: résistance en continue
|
||||
- $r$: rayon du fil
|
||||
- $\delta$: profondeur de pénétration du courant
|
||||
|
||||
La profondeur de pénétration du courant est donnée par la formule suivante:
|
||||
$$\delta = \sqrt{\frac{\rho}{\pi f \mu_0}}$$
|
||||
avec:
|
||||
- $\delta$: profondeur de pénétration du courant
|
||||
- $\rho$: résistivité du cuivre
|
||||
- $f$: fréquence
|
||||
- $\mu_0$: perméabilité magnétique du vide $= 4\pi 10^{-7} H/m$
|
||||
|
||||
On peut en déduire la résistance en courant alternatif:
|
||||
$$R_{AC} = \frac{l\sqrt{\rho\pi\mu_0}}{2\pi r} \sqrt{f} \qquad \qquad R_{AC(f_{2})} = R_{AC(f_{1})} \cdot \sqrt{ \frac{f_{2}}{f_{1}} }$$
|
||||
|
||||
Pour calculer l'inductance et la capacité d'un fil, on peut utiliser les formules suivantes :
|
||||
$$\begin{align}
|
||||
L&=\frac{\mu_{0}l}{\pi}acosh\left( \frac{d}{2r} \right) \\
|
||||
C&=\frac{\pi\epsilon_{0}l}{acosh\left( \frac{d}{2r} \right)}
|
||||
\end{align}$$
|
||||
avec :
|
||||
- $\mu_{0}$: perméabilité magnétique du vide $=4\pi\cdot 10^{-7}\left[ \frac{H}{m} \right]$
|
||||
- $\epsilon_{0}$: permittivité du vide $=8.854\cdot 10^{-12}\left[ \frac{F}{m} \right]$
|
||||
|
||||
# Model à haute fréquence
|
||||
|
||||
## Inductance
|
||||
|
||||
|
||||
<u>Facteur de qualité :</u>
|
||||
$$Q(f) = \frac{X_{L}(f)}{R(f)} \qquad \qquad R_{AC}(f)= \frac{l\sqrt{ \rho \pi \mu_{0} }}{2\pi r}\sqrt{ f}$$
|
||||
|
||||
<u>Réactance & Impédance:</u>
|
||||
$$X_{L}=\omega L=2\pi fL \qquad \qquad Z_{L}=jX_{L}=j\omega L$$
|
||||
|
||||
<u>Résonance :</u> $\omega_{r} ≃\omega_{0}=\frac{1}{\sqrt{ LC }}$ <br>
|
||||
SRF: Self Resonant Frequency = $\frac{\omega_{r}}{2\pi}$
|
||||
|
||||
$$Z_{L,tot} = (j\omega L+R) \mid \mid \frac{1}{j\omega C} =\frac{sL+R}{1+sRC+s^{2}LC}$$
|
||||
|
||||
|
||||
|
||||
# Capacité
|
||||
|
||||
|
||||
<u>Facteur de qualité :</u>
|
||||
$$Q(f) = \frac{X_{C}(f)}{R} \qquad \qquad \omega_{r} ≃\omega_{0}=\frac{1}{\sqrt{ LC }}$$
|
||||
|
||||
<u>Réactance & Impédance:</u>
|
||||
$$X_{C}=\frac{1}{\omega C} \qquad \qquad Z_{C}=\frac{1}{j\omega C}$$
|
||||
|
||||
|
||||
|
||||
# Résistance
|
||||
|
||||
|
||||
|
||||
# Transmission line
|
||||
|
||||
$$ k = \frac{2\pi}{\lambda} \qquad \qquad \lambda = \frac{c_{mat}}{f} $$
|
||||
|
||||
$\Gamma$ (Gamma) : coefficient de réflexion
|
||||
|
||||
# Smith Chart
|
||||
|
||||
|
||||
## Ajouter une ligne de réflexion
|
||||
|
||||
|
||||
1. Placer $Z1$
|
||||
1. Normaliser ($\hat{Z_{1}}=\frac{Z_{1}}{Z_{0}}$)
|
||||
2. Placer le point au croisement de l'axe réel et imaginaire (Impédance = bleu)
|
||||
2. Chercher $\Gamma$ (indice de reflexion)
|
||||
1. Tracer l'angle ("Angle of reflexion coefficient in degrees")
|
||||
2. Tracer le module (cercle au compas + rapport sur les échelles du bas)
|
||||
3. Ajouter la ligne selon le sens
|
||||
- $\circlearrowright$ : toward generator
|
||||
- $\circlearrowleft$ : toward load
|
||||
4. Lire les nouvelles valeures
|
||||
1. Tracer le nouvel angle $\to \angle(\Gamma_{2})$
|
||||
2. Tracer et lire $\hat{Z_{2}}$ puis $Z_{2}$
|
||||
|
||||
|
||||
|
||||
# Impedance Matching
|
||||
$Z_{LD} \to Z_S^*$, sans résistance
|
||||
- Capacité + inductance
|
||||
- ligne de transmission + capacité / inductance + ligne de transmission
|
||||
- ligne de transmission + stub + ligne de transmission
|
||||
-
|
||||
|
||||
|
||||
## Stub
|
||||
Pour trouver la longueur du stub :
|
||||
1. Pour un short stub, placer une inductance parallèle et pour un open stub un condensateur parallèle
|
||||
2. Trouver la valeur normalisée de cette admittance.
|
||||
3. Pour un short stub, placer le delta dans les négatifs (parti du dessus) et pour un open stub dans les positifs (partie du dessous).
|
||||
4. Regarder la longueur d'onde. Partir du côté open circuit pour un open stub et du côté short circuit pour un short stub.
|
||||
|
||||
# Filtres
|
||||
|
||||
Filter components <br>
|
||||
 <br>
|
||||
|
||||
|
||||
Normalized value for Butterworth order 1 to 10 <br>
|
||||
|
||||
|
||||
Normalized value for Chebyshev 0.5dB Ripple <br>
|
||||
|
||||
|
||||
Normalized value for Chebyshev 3dB Ripple <br>
|
||||
|
||||
|
||||
## Filtres passe-bas en microstrip
|
||||
$l$ : longueur physique en mm <br>
|
||||
$\theta$ : longueur électrique / degré <br>
|
||||
$$ l_l = g_k * \frac{\lambda_{gl} * Z_0}{2\pi *Z_{0l}} $$
|
||||
$$ l_c = g_k * \frac{\lambda_{gc} * Z_{0c}}{2\pi *Z_0} $$
|
||||
$$ \theta_l = g_k * \frac{360 * Z_0}{2\pi *Z_{0l}} $$
|
||||
$$ \theta_c = g_k * \frac{360* Z_{0c}}{2\pi *Z_0} $$
|
||||
|
||||
|
||||
|
||||
@@ -0,0 +1,256 @@
|
||||
<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 :
|
||||
|
||||
Soit $f(x)$ une fonction non périodique définie sur $\mathbb{R}$.
|
||||
$$\begin{align}
|
||||
\text{Transformée de Fourier de }f(x) &: \qquad \widehat{f}(\omega) = \int^{\infty}_{-\infty}f(x)e^{-\mathbf{i}\omega x}dx \qquad (\omega\in \mathbb{R})\\
|
||||
\text{Intégrale de Fourier de } f(x) &: \qquad \frac{1}{2\pi}\int_{-\infty}^{\infty} \widehat{f}(\omega) e^{\mathbf{i}x\omega}\, d\omega \qquad(x \in\mathbb{R}).
|
||||
\end{align}$$
|
||||
**Théorème** : Si $\int_{-\infty}^{\infty} |f(x) \, dx < \infty$ et si $f$ est une fonction de variation bornée sur tout $\mathbb{R}$, alors
|
||||
$$
|
||||
\frac{1}{2\pi}\int_{-\infty}^{\infty} \widehat{f}(\omega)e^{\mathbf{i}x\omega} \, d\omega = \begin{cases}
|
||||
f(x) &\text{si }f \text{ est continue au point }x \\
|
||||
\text{milieu du saut de } f(x) &\text{si }f \text{ a un saut en }x
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
|
||||
## 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.
|
||||
|
||||
On pose: $\Omega := \omega \cdot T_{s}$
|
||||
La transformation de Fourier de $x[k]$ est donnée par la formule suivante :
|
||||
$$\mathcal{F}\{x[k]\} = X(\Omega) \quad := \quad\sum^{+\infty}_{k=-\infty}x[k]\cdot e^{-jk\Omega}= \mathcal{F}\{x_{s}(t)\}$$
|
||||
|
||||
|
||||
$$\mathcal{F\{x[k]\} = \mathcal{Z}\{x[k]\}}|_{z=e^{j\Omega}}$$
|
||||
|
||||
$$x[k]=\frac{1}{2\pi} \int^{+\pi}_{-\pi}X(\Omega)e^{jk\Omega} d\Omega$$
|
||||
|
||||
|
||||
|
||||
|
||||
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 $δ[k]$.
|
||||
|
||||
|
||||
|
||||
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 (peut aussi se noter $\varepsilon$ pour un heaviside en discret) <br>
|
||||
$y[k]$ : réponse du système <br>
|
||||
$x[k]$ : signal d'entrée <br>
|
||||
$h[k]$ : réponse impulsionnelle <br>
|
||||
|
||||
$$
|
||||
\forall x \in \mathbb{R}, \mathcal{H}(x)=\begin{cases}
|
||||
0 &\text{ si } x<0 \\
|
||||
1 &\text{ si } x \geq 0
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
Pour un système causal
|
||||
|
||||
$$\begin{align}
|
||||
y[k] &= \sum^{k}_{n=-\infty}(x[n]\cdot h[k-n]) \\
|
||||
\\
|
||||
y[k] &= \sum^{+\infty}_{n=0}(x[k-n]\cdot h[n])
|
||||
\end{align}$$
|
||||
|
||||
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.
|
||||
|
||||
|
||||
|
||||
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*.
|
||||
|
||||
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
|
||||
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. <br>
|
||||
$x[k]$ : Fonction du [Tableau des transformées en Z](/infotro/Eln/Ztransform) ou valeur du graphe des réponses impulsionnelle<br>
|
||||
|
||||
On a aussi la propriété suivante :
|
||||
$$\mathcal{Z}\{x[k-n]\}=z^{-n}\cdot X(z)$$
|
||||
|
||||
## 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 :
|
||||
- $f[k] => F(z)$
|
||||
- $f[k-n] => z^{-n}\cdot F(z)$
|
||||
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 :
|
||||
$$H(z) = K \frac{(z-o_{1})\cdot (z-o_{2})\cdot \space \dots \space \cdot (z-o_{N})}{(z-p_{1})\cdot (z-p_{2})\cdot \space \dots \space \cdot (z-p_{M}) }\quad o_{n}, p_{n} \space \text{complexes}$$
|
||||
|
||||
|
||||
## 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.
|
||||
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
|
||||
$\ln\left( \frac{1+x}{1-x} \right) \approx 2x$
|
||||
|
||||
## 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 $\omega_o$ et vice-versa, on utilise la relation suivante :
|
||||
$$\begin{align}
|
||||
\omega_{0} &= 2f_{s}\tan\left( \frac{\omega_{0}'}{2f_{s}} \right) \\
|
||||
\\
|
||||
\omega_{0}'&=2f_{S}\cdot \arctan\left( \frac{\omega_{0}}{2fs} \right)
|
||||
\end{align}$$
|
||||
|
||||
|
||||
## 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 <br>
|
||||
|
||||
On peut du coup approximer l'équation différentielle par une équation aux différences.
|
||||
Par exemple pour la dérivée première de $x(t)$, on a
|
||||
$$x'[k]= \frac{x[k]-x[k-1]}{T_{s}}$$
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
----------
|
||||
@@ -0,0 +1,30 @@
|
||||
<header><h1>
|
||||
Tableau des transformées en Z
|
||||
</h1></header>
|
||||
|
||||
-----------
|
||||
|
||||
$$f[k] =\varepsilon[x]=\begin{cases}
|
||||
1^{x} &\text{ si } x\geq 0 \\
|
||||
0 &\text{ si } x<0
|
||||
\end{cases} \qquad \forall x=k,k-n
|
||||
$$
|
||||
$\varepsilon$ : fonction de Heaviside en discret
|
||||
|
||||
----------
|
||||
|
||||
| Nr. | $f[k]$ | $F(z)$ | $f(t)$ | $F(s)$ |
|
||||
|------|-------------------|----------------------------------------|-------------|--------------------|
|
||||
| 1 | $\delta_{k}, 0^k$ | 1 | $\delta(t)$ | 1 |
|
||||
| 2 | $1^k$ | $\frac{z}{z-1}$ | $1$ | $\frac{1}{s}$ |
|
||||
| 3 | $kT$ | $\frac{Tz}{z-1}^{2}$ | $t$ | $\frac{1}{s^{2}}$ |
|
||||
| 4 | $(kT)^{2}$ | $\frac{T^{2}z(z+1)}{(z-1)^{3}}$ | $t^{2}$ | $\frac{2!}{s^{3}}$ |
|
||||
| 5 | $(kT)^{3}$ | $\frac{T^{3}z(z^{2}+4z+1)}{(z-1)^{4}}$ | $t^{3}$ | $\frac{3!}{s^{4}}$ |
|
||||
| 6 | $e^{akT}$ | $\frac{z}{z-e^{aT}}$ | $e^{at}$ | $\frac{1}{s-a}$ |
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Dans la théorie Eln la partie transformée en Z commence à la slide 23 de "Eln2_LTI_0910_f.pdf".
|
||||
@@ -0,0 +1,133 @@
|
||||
<header><h1>
|
||||
Lab 06 - Filters
|
||||
</h1></header>
|
||||
|
||||
----------
|
||||
|
||||
In this laboratory on digital filters, the main objective is to understand how digital filters work.
|
||||
We will use MATLAB to simulate and implement these filters on a digital signal processor (DSP).
|
||||
|
||||
# Analog filter design
|
||||
|
||||
We must design Butterworth low-pass filter that cuts at 1kHz as shown with the following equation :
|
||||
|
||||
$$F(s) = \frac{1}{\frac{s^{2}}{\omega_c^{2}} + s\cdot\frac{1.414}{\omega_c} + 1} $$
|
||||
|
||||
In according to the sample theory, we have to choose at least 2 times the maximum frequency of the signal.
|
||||
In this case, the maximum frequency is 1 kHz, so we have to choose a sample frequency at least at 2 kHz.
|
||||
We take 10 time $f_c$ for the sample frequency, so we choose 20 kHz.
|
||||
|
||||
|
||||
## Transfer function
|
||||
For calculate the transfer function, we have to calculate the poles and zeros of the filter.
|
||||
|
||||
$$p_{di} = e^{p_{ai} \cdot T} \qquad z_{di} = e^{z_{ai} \cdot T} $$
|
||||
|
||||
- $k_a$, $p_{ai}$ and $z_{ai}$ are the gain, poles and zeros of the analog filter that the representation *zpk*.
|
||||
Be careful, $k_a$ is not the DC gain.
|
||||
- $k_d$, $p_{di}$ and $z_{di}$ are the gain, poles and zeros of the digital filter.
|
||||
|
||||
For a low pass filter without zeros, the "digital" gain is :
|
||||
$$k_{d} = k_{a}\cdot \prod_{i} \frac{1-e^{P_{ai}\cdot T}}{P_{ai}}$$
|
||||
|
||||
We can apply the bilinear transformation to the poles and zeros of the analog filter to obtain the poles and zeros of the digital filter.
|
||||
$$
|
||||
p_{di} = \frac{1+p_{ai}\cdot \frac{T}{2}}{1-p_{ai}\cdot \frac{T}{2}}
|
||||
\qquad \qquad
|
||||
z_{di} = \frac{1+z_{ai}\cdot \frac{T}{2}}{1-z_{ai}\cdot \frac{T}{2}}
|
||||
$$
|
||||
And for the low pass filter, the "bilinear" gain is :
|
||||
$$k_{d} = k_{a}\cdot \frac{1}{\prod_{i}\left( \frac{2}{T}-p_{ai} \right)}$$
|
||||
|
||||
On the following diagram, we can see the bode diagram of the analog, digital filter and the filter with the bilinear transformation.
|
||||
|
||||
|
||||
|
||||
# IIR filter design
|
||||
|
||||
We must design an IIR filter with the following specifications :
|
||||
|
||||
- $H_0 = 0 [dB]$
|
||||
- $H_1 = -3 [dB]$
|
||||
- $H_2 = -40 [dB]$
|
||||
- $f_{pass} = 7 [kHz]$
|
||||
- $f_{stop} = 9 [kHz]$
|
||||
- $f_s = 48 [kHz]$
|
||||
|
||||
We first calculate the analog filter for search the order with the function *ellipord*.
|
||||
Then, we calculate the poles and zeros with the function *ellip*.
|
||||
We apply the bilinear transformation to the poles and zeros to obtain the poles and zeros of the digital filter.
|
||||
Finally, we can plot the bode diagram of the digital filter and the filter with the bilinear transformation.
|
||||
|
||||
|
||||
|
||||
|
||||
# DSP implementation
|
||||
|
||||
We will implement the previously designed digital filter using a C program that will run in real-time on a fixed-point DSP.
|
||||
We will employ a cascade structure, consisting of 2 second-order stages.
|
||||
We will use the TMS320C6211 DSK from Texas Instruments.
|
||||
|
||||
The following equation represent an universal transfer function of a second-order filter :
|
||||
$$H(z) = \frac{b_{0}+b_{1}\cdot z^{-1}+b_{2}\cdot z^{-2}}{a_{0}+a_{1}\cdot z^{-1}+a_{2}\cdot z^{-2}}$$
|
||||
|
||||
The following figure depicts the block diagram of this implementation in the case where a0 = 1
|
||||
This structure is commonly referred to as "Direct Form I." It requires the storage of past values of both the input and output.
|
||||
|
||||
|
||||
|
||||
So we have the difference equation :
|
||||
$$y(n) = b_{0}\cdot x(n)+b_{1}\cdot x(n-1)+b_{2}\cdot x(n-2)-a_{1}\cdot y(n-1)-a_{2}\cdot y(n-2)$$
|
||||
|
||||
It exists a second structure called "Direct Form II" which is more efficient in terms of computation and memory usage.
|
||||
|
||||
|
||||
|
||||
The difference equation is :
|
||||
$$\begin{align}
|
||||
y(n) &= tmp\cdot b_{0}+w[0]\cdot b_{1}+w[1]\cdot b_{2} \\
|
||||
\\
|
||||
tmp &= x(n)-a_{1}\cdot w[0]-a_{2}\cdot w[1] \\
|
||||
\\
|
||||
w[0] &= tmp_{prev} \\
|
||||
\\
|
||||
w[1] &= w[0]_{prev}
|
||||
\end{align}$$
|
||||
|
||||
The variables can be translated into code by shifting the coefficients 14 positions to the left to enhance precision
|
||||
after the decimal point, as the DSP only works with integers.
|
||||
|
||||
Structures for the coefficients and the state variables are defined as follows :
|
||||
```c
|
||||
typedef struct {
|
||||
short b0;
|
||||
short b1;
|
||||
short b2;
|
||||
short a0;
|
||||
short a1;
|
||||
short a2;
|
||||
} coeff_sec_order;
|
||||
|
||||
short w3[2];
|
||||
short w4[2];
|
||||
|
||||
coeff_sec_order stage1 = { 469, 349, 469, 16384, -23699, 10692};
|
||||
coeff_sec_order stage2 = {16384, -11187, 16384, 16384, -19657, 14890};
|
||||
```
|
||||
|
||||
The following code is the implementation of the Direct Form II function :
|
||||
```c
|
||||
void iir_sec_order_dir_s(coeff_sec_order Coeff, short*w, short* Input, short* Output) {
|
||||
register short tmp;
|
||||
|
||||
tmp = (long)* Input - (( (Long)Coeff.a1 * (long)w[0]) >> SHIFT) - (( (long)Coeff.a2 * (long)w[1]) >> SHIFT);
|
||||
*Output = (( (long)Coeff.b0 * (long)tmp >> SHIFT) + (( (long)Coeff.b1 * (long)w[0]) >> SHIFT) + (( (long)Coeff.b2 * (long)w[1]) >> SHIFT);
|
||||
|
||||
w[1] = w[0];
|
||||
w[0] = tmp;
|
||||
}
|
||||
```
|
||||
|
||||
For a better effect of the filter, the function can be called several times.
|
||||
|
||||
|
||||
File diff suppressed because it is too large.
Load diff
|
After Width: | Height: | Size: 78 KiB |
@@ -0,0 +1,4 @@
|
||||
<?xml version="1.0" encoding="UTF-8"?>
|
||||
<!-- Do not edit this file with editors other than draw.io -->
|
||||
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
|
||||
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.1" width="840px" height="286px" viewBox="-0.5 -0.5 840 286" content="<mxfile host="app.diagrams.net" modified="2024-02-03T22:03:26.442Z" agent="Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:122.0) Gecko/20100101 Firefox/122.0" etag="Da41eLOf3AT8AQXfy7Nj" version="22.1.22"><diagram name="Page-1" id="CkdF-7UHB_tb0CRjxC_V">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</diagram></mxfile>"><defs><style type="text/css">@import url(https://fonts.googleapis.com/css2?family=Architects+Daughter:wght@400;500);
</style><style xmlns="http://www.w3.org/1999/xhtml" id="MJX-SVG-styles">
mjx-container[jax="SVG"] {
 direction: ltr;
}

mjx-container[jax="SVG"] > svg {
 overflow: visible;
 min-height: 1px;
 min-width: 1px;
}

mjx-container[jax="SVG"] > svg a {
 fill: blue;
 stroke: blue;
}

mjx-container[jax="SVG"][display="true"] {
 display: block;
 text-align: center;
 margin: 1em 0;
}

mjx-container[jax="SVG"][display="true"][width="full"] {
 display: flex;
}

mjx-container[jax="SVG"][justify="left"] {
 text-align: left;
}

mjx-container[jax="SVG"][justify="right"] {
 text-align: right;
}

g[data-mml-node="merror"] > g {
 fill: red;
 stroke: red;
}

g[data-mml-node="merror"] > rect[data-background] {
 fill: yellow;
 stroke: none;
}

g[data-mml-node="mtable"] > line[data-line], svg[data-table] > g > line[data-line] {
 stroke-width: 70px;
 fill: none;
}

g[data-mml-node="mtable"] > rect[data-frame], svg[data-table] > g > rect[data-frame] {
 stroke-width: 70px;
 fill: none;
}

g[data-mml-node="mtable"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed {
 stroke-dasharray: 140;
}

g[data-mml-node="mtable"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted {
 stroke-linecap: round;
 stroke-dasharray: 0,140;
}

g[data-mml-node="mtable"] > g > svg {
 overflow: visible;
}

[jax="SVG"] mjx-tool {
 display: inline-block;
 position: relative;
 width: 0;
 height: 0;
}

[jax="SVG"] mjx-tool > mjx-tip {
 position: absolute;
 top: 0;
 left: 0;
}

mjx-tool > mjx-tip {
 display: inline-block;
 padding: .2em;
 border: 1px solid #888;
 font-size: 70%;
 background-color: #F8F8F8;
 color: black;
 box-shadow: 2px 2px 5px #AAAAAA;
}

g[data-mml-node="maction"][data-toggle] {
 cursor: pLine truncated
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|
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<?xml version="1.0" encoding="UTF-8"?>
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<!-- Do not edit this file with editors other than draw.io -->
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<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
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</style><style xmlns="http://www.w3.org/1999/xhtml" id="MJX-SVG-styles">
mjx-container[jax="SVG"] {
 direction: ltr;
}

mjx-container[jax="SVG"] > svg {
 overflow: visible;
 min-height: 1px;
 min-width: 1px;
}

mjx-container[jax="SVG"] > svg a {
 fill: blue;
 stroke: blue;
}

mjx-container[jax="SVG"][display="true"] {
 display: block;
 text-align: center;
 margin: 1em 0;
}

mjx-container[jax="SVG"][display="true"][width="full"] {
 display: flex;
}

mjx-container[jax="SVG"][justify="left"] {
 text-align: left;
}

mjx-container[jax="SVG"][justify="right"] {
 text-align: right;
}

g[data-mml-node="merror"] > g {
 fill: red;
 stroke: red;
}

g[data-mml-node="merror"] > rect[data-background] {
 fill: yellow;
 stroke: none;
}

g[data-mml-node="mtable"] > line[data-line], svg[data-table] > g > line[data-line] {
 stroke-width: 70px;
 fill: none;
}

g[data-mml-node="mtable"] > rect[data-frame], svg[data-table] > g > rect[data-frame] {
 stroke-width: 70px;
 fill: none;
}

g[data-mml-node="mtable"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed {
 stroke-dasharray: 140;
}

g[data-mml-node="mtable"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted {
 stroke-linecap: round;
 stroke-dasharray: 0,140;
}

g[data-mml-node="mtable"] > g > svg {
 overflow: visible;
}

[jax="SVG"] mjx-tool {
 display: inline-block;
 position: relative;
 width: 0;
 height: 0;
}

[jax="SVG"] mjx-tool > mjx-tip {
 position: absolute;
 top: 0;
 left: 0;
}

mjx-tool > mjx-tip {
 display: inline-block;
 padding: .2em;
 border: 1px solid #888;
 font-size: 70%;
 background-color: #F8F8F8;
 color: black;
 box-shadow: 2px 2px 5px #AAAAAA;
}

g[data-mml-node="maction"][data-toggle] {
 cursor: pLine truncated
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<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">
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 </mxCell>
 <mxCell id="hvOSe9N11qCTzQnM01ZF-8" value="$$\mathcal{Z} = e^{\mathbf{j}\omega}$$" style="text;strokeColor=none;fillColor=none;fontSize=20;fontStyle=1;verticalAlign=middle;align=center;fontColor=#808080;html=1;" vertex="1" parent="1">
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 </root>
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 </diagram>
</mxfile>
"><defs><style xmlns="http://www.w3.org/1999/xhtml" id="MJX-SVG-styles">
mjx-container[jax="SVG"] {
 direction: ltr;
}

mjx-container[jax="SVG"] > svg {
 overflow: visible;
 min-height: 1px;
 min-width: 1px;
}

mjx-container[jax="SVG"] > svg a {
 fill: blue;
 stroke: blue;
}

mjx-container[jax="SVG"][display="true"] {
 display: block;
 text-align: center;
 margin: 1em 0;
}

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<?xml version="1.0" encoding="UTF-8" standalone="no"?>
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<svg
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style=""
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version="1.1"
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width="1220px"
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height="361px"
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viewBox="-0.5 -0.5 1220 361"
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Reference in new issue
Block a user