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Filtres lineaires stationnaires
TODO
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La causalite
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La stabilite
Types de filtres
- Filtre a reponse impulsionnelle finie (RIF)
- Filtres a reponse impulsionelle infinie (RII)
- Filtre causal
- Filtre non causal
La TF joue une role fondamentale dans l'analyse des operateurs stationnaires
2 effets:
- amplitude: amplification ou attenuation
- phase: decalage et deformation de
Exemple
Soit
Filtre moyenneur:
Retour aux types:
- filtres a phase nulle
- Filtres a phase lineaire (implique un decalage en temps)
- Filtre a phase non lineaire
On classe surtout les filtres en fonction des plages de frequences qu'ils laissent passer:
- Passe-bas
- Passe-haut
- Passe-bande
Passe ideaux
Passe-bas ideal
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TODO
Reponse impulsionnelle d'un filtre passe-bas ideal
TDtd inverse
Inconvenients:
- Reponses impulsionnelle a support infini
- Oscillation TODO
Passe-haut ideal
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Passe-bande ideal
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Construire un filtre
1ere idee: approximer un filtre ideal
- On tronque la reponse impulsionnelle d'un filtre ideal
le filtre tronque de longueur
TODO
Lobe principal: pente de coupure (trnsition abrupte au niveau de la coupure)
Lobe secondaire: effets non souhaitable
On souhaite:
- Lobe principal etroit
- Lobes secondaires faibles
- Fenetre avec un peu d'echantillons
- Pente raide
Equations recurrentes a coefficients constants
On souhaite resoudre une equation de forme:
- equivalent a construire un filtre en passant par l'espace de Fourier
- controle l'entree pour eviter les effets indesirables
Comment calculer ces coefficients ?
Et la fonction de tranfert de filtre ?
Fourier ?
Mais pour des systemes complexes, ca devient vite delicat
La TZ d'un signal discret
Avec la TFtd
La TFtd est egale a la TZ sur le cercle unite
- Pour nos filtres, il faut donc que le cercle unite appartienne au domaine de convergence
Si on reprend l'equation de depart, sa TZ:
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Le domaine de convergence correspond a la couronne du plan ne contenant aucun pole et le cercle unite
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Domaine de convergence
- Les points du domaine de convergence ou , s'appelle les zeros
- Les points du domaine de convergence ou , s'appelle les pole
Le domaine de convergence ne peut inclure les poles
Si est a support fini alors le domaine de convergence est le plan tout entier (sauf pour et )
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