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Titre: Le Traitement du Signal aléatoire

Auteurs: P.Simard

Ecole: Université de technologie de Compiègne

Résumé: Le cadre naturel des signaux est aléatoire Dans la pratique du traitement du signal, il existe de nombreux cas ou l’application d’un raisonnement utilisant la notion de signal aléatoire permet de définir des méthodes puissantes d’analyse. Il est même possible d’éviter de commettre des erreurs indécelables dans le cadre déterministe. Parmi les méthodes qui ont fortement marqué le traitement du signal au cours des 20 dernières années, il faut citer les modèles par filtrage de bruit blanc (autorégressif et moyenne mobile) qui ont permis de développer le traitement de la parole et de développer la téléphonie à bas débit.

Parmi les erreurs indécelables dans le cadre déterministe, on peut citer les propriétés utilisant l’ergodicité, ou qualité d’une mesure à renseigner proprement du premier coup sur le processus qui lui a donné naissance. Par ailleurs, le cadre aléatoire est assez naturel. En effet, il est rare que l’étude d’un signal n’ait pas pour objet d’estimer des paramètres caractéristiques dont on ignore à peu prés tout avant l’étude. Ce peut être par exemple l’instant d’arrivée d’un signal dans le cas des transmissions, ou de la détection radar. Ce peut être l’amplitude d’un signal acoustique amorti par sa propagation dans un milieu particulier.

De plus, la généralisation des capteurs et moyen de mesure place de plus en plus souvent l’ingénieur ou le scientifique en face de signaux difficiles à décrire, présentant des fluctuations imprévisibles (au moins à première vue) et en général non répétables : dans ce cas, la recherche de paramètres communs dans les différentes mesures se fait plus facilement dans le cadre aléatoire (par exemple en calculant des moyennes).

Extrait du sommaire:

1 Besoins de modèles aléatoires pour les signaux 5
2 Principaux résultats de la théorie des probabilités 7
2.1 Formalisation des résultats d’une expérience aléatoire
2.1.1 Définitions
2.1.2 Notion d’événement
2.1.3 Définitions de la mesure de probabilité
2.1.4 Quelques lois et concepts utiles
2.2 Variables Aléatoires : Généralités
2.2.1 Définition d’une Variable Aléatoire
2.2.2 Loi de probabilité d’une variable aléatoire
2.2.3 Moments d’une variable aléatoire
2.3 Etude conjointe de Variables Aléatoires :
2.3.1 Etude Conjointe de deux Variables Aléatoires
2.4 Etude conjointe de Variables Aléatoires :
2.4.1 Etude Conjointe de n Variables Aléatoires
3 Notions générales sur l’estimation ponctuelle : 21
3.1 Position du problème
3.1.1 Cadre général de l’estimation ponctuelle
4 Le signal Aléatoire 23
4.1 Le signal Aléatoire : définitions et propriétés :
4.1.1 Modèle du Signal aléatoire
4.1.2 Description d’un signal aléatoire : Stationnarité
4.1.3 Description d’un signal aléatoire : ergodisme
4.2 Exemple de signal aléatoire : le bruit blanc
4.2.1 Définition
4.2.2 Comportement temporel
4.2.3 Comportement fréquentiel
5 Analyse spectrale des signaux aléatoires : 29
5.1 Eléments de théorie de l’analyse en fréquences
5.1.1 Théorème de Wiener-Khintchine
5.1.2 Compléments : Signaux à puissance moyennée
5.1.3 Spectre d’un signal aléatoire
5.2 Notions d’estimation spectrale :
5.2.1 Perspective historique
5.2.2 Théorie de l’analyse fréquentielle des signaux
5.2.3 Rappel : représentation de Fourier des signaux discrets de longueur _nie
5.2.4 Eléments d’estimation spectrale
6 Le Filtrage des signaux aléatoires : 39
6.1 Rappel : filtrage d’un signal déterministe
6.1.1 Signal à temps continu
6.1.2 Signal à temps discret
6.2 Filtrage d’un signal aléatoire stationnaire
6.2.1 Espérance mathématique du signal aléatoire filtré
6.2.2 Covariance du signal aléatoire filtré
6.3 Introduction aux processus ARMA
6.3.1 Définition des signaux autorégressifs (AR) :
6.3.2 Définition des signaux de moyenne mobile (MA)
6.3.3 Définition des signaux autorégressifs et de moyenne mobile (ARMA)
6.3.4 Exemple d’un signal AR(2) :
7 Quelques références bibliographiques pour aller plus loin : 49
8 Sujets des TD de la partie II 51
8.1 Variables aléatoires et Moments
8.2 Probabilité conditionnelle et Transmission d’un signal binaire
8.3 Stationnarité et Ergodisme
8.4 Bruit Blanc
9 Matlab : Tutorial pour débuter 57
9.1 Vecteurs
9.2 Fonctions
9.3 Les _gures et les tracés
9.4 les M-_les
10 UV SY06 – Les Règles d’Or du calcul 61
10.1 Règle n_1 : Le carré d’une somme n’est pas la somme des carrés
10.2 Outils du traitement du signal
10.3 Trigonométrie
10.4 Quelques définitions utiles

Formation-Traitement-du-signal-cours 13

Télécharger le fichiers PDF: Le Traitement du Signal aléatoire

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