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Titre: Méthodes mathématiques pour le Traitement du Signal

Auteurs: M. Bergounioux

Ecole: Université d’Orléans

Résumé: Ce cours est issu d’un enseignement donné depuis 5 ans au sein du MASTER de Mathématiques d’Orléans et s’adresse à des mathématiciens désireux de connaitre les techniques de base de traitement du signal. A contrario, il peut aussi intéresser des spécialistes de traitement du signal qui souhaitent avoir un point de vue mathématique sur les outils qu’ils utilisent fréquemment. Cet ouvrage se veut donc une introduction à la discipline plus qu’un ouvrage pointu destiné à des spécialistes du domaine. Pour le lecteur qui souhaite en savoir davantage nous renvoyons à la bibliographie qui permet d’approfondir les différents sujets. Nous avons souhaité donné de nombreuses applications tout au long de l’ouvrage et proposons comme tout livre de cours qui se respecte quelques exercices ou sujets de travaux pratiques. Le signal 1D le plus accessible par excellence étant le signal sonore nous avons consacré un court chapitre à l’analyse vocale (ou traitement de la parole), là aussi sous forme d’introduction.

Pour le traitement du son musical nous renvoyons à l’ouvrage de P. Guillaume [8] qui fourmille d’exemples en liaison directe avec la musique.

Extrait du sommaire:

1 Introduction 3
2 Analyse spectrale des signaux unidimensionnels 9
2.1 Signaux analogiques périodiques 9
2.1.1 Les séries de Fourier 9
2.1.2 Exemples et terminologie 19
2.2 La transformation de Fourier discrète (DFT) et la FFT 22
2.2.1 Calcul des coefficients de Fourier 22
2.2.2 Propriétés de la transformée de Fourier discrète 25
2.2.3 L’algorithme de FFT 28
2.2.4 Utilisation de la FFT sous SCILAB ou MATLAB 33
2.3 La transformation de Fourier 34
2.3.1 Propriétés de la transformation de Fourier 37
2.3.2 Théorème d’inversion de Fourier dans L1 R 44
2.4 La transformation de Fourier-Plancherel 48
2.4.1 Les fonctions à décroissance rapide 48
2.4.2 L’espace S R 50
2.4.3 Transformation de Fourier inverse sur S R 51
2.5 Transformation de Fourier-Plancherel 52
2.5.1 Coefficient de Fourier versus transformation de Fourier 55
2.5.2 Application : calcul de certaines transformées de Fourier 56
2.6 La transformation de Laplace 58
3 Analyse corrélative des signaux 61
3.1 Relations statistiques entre plusieurs variables 61
3.1.1 Notation -Abréviations 61
3.1.2 Matrice des covariances 62
3.2 Fonctions de corrélation de signaux 64
3.2.1 Définitions 64
3.2.2 Quelques propriétés des fonctions de corrélation dans le cas analogique 67
3.2.3 Densité spectrale de puissance 69
3.2.4 Cas d’un signal numérique 70
3.2.5 Exemples 71
3.3 Détection de signaux périodiques 73
3.3.1 Détection par autocorrélation d’un signal périodique noyé dans du bruit 74
3.3.2 Détection par intercorrélation d’un signal périodique noyé dans du bruit 75
3.4 Autres exemples d’applications 76
3.4.1 Mesure de retard entre signaux aléatoires par corrélation 76
4 Filtrage 81
4.1 Systèmes 81
4.1.1 Quelques définitions 82
4.1.2 Propriétés algébriques des systèmes 83
4.2 Filtres linéaires 86
4.2.1 Masse (impulsion) de Dirac 89
4.3 Filtrage numérique 91
4.3.1 Filtrage numérique linéaire 92
4.3.2 Filtres linéaires discrets et équations aux différences 94
4.3.3 Transformation en z 96
4.3.4 Fonction de transfert 98
4.3.5 Exemples de filtres du premier et du deuxième ordre 99
4.4 Filtrage analogique 104
4.4.1 Exemples 107
4.4.2 Filtrage et corrélation 113
5 Echantillonnage 115
5.1 Peigne et mesure de Dirac 115
5.1.1 Peigne de Dirac 116
5.1.2 Convolution entre une fonction et !a 119
5.1.3 Transformée de Fourier de !a 120
5.1.4 Développement en série de Fourier du peigne de Dirac. 121
5.1.5 Transformée de Fourier du peigne de Dirac 124
5.1.6 Transformée de Fourier d’un signal périodique 125
5.2 Formule de Poisson dans L1 R 125
5.2.1 Application à l’étude d’un signal échantillonné 127
5.3 Théorème d’échantillonnage de Shannon 129
5.3.1 Démonstration dans le cas général 136
5.3.2 Echantillonnage et calcul numérique du spectre – Aliasing . 137
6 Analyse temps-fréquence 145
6.1 Introduction 145
6.2 Principe d’incertitude 147
6.2.1 Comportements temporel et spectral d’un signal 147
6.2.2 Signaux à durée finie ou à spectre borné 149
6.2.3 Troncature du signal en temps ou en fréquence 150
6.3 Transformée de Fourier à fenêtre glissante (STFT) 150
6.3.1 Fenêtrage 150
6.3.2 Les formules de Gabor 153
6.3.3 Comparaison des méthodes de Fourier et Gabor 157
6.3.4 Méthodes d’analyse 159
6.3.5 Estimation de la densité spectrale de puissance d’un signal numérique 159
6.4 Distribution de Wigner-Ville 161
6.5 La classe de Cohen 163
6.5.1 Positivité 164
6.5.2 La classe de Cohen 164
6.6 Exemples : chirps 165
6.7 Code SCILAB sommaire pour le spectrogramme 166
7 Analyse temps-échelle – Ondelettes 169
7.1 Transformée en ondelettes 169
7.1.1 Définition des ondelettes 173
7.1.2 Ondelettes dans l’espace physique 175
7.1.3 Transformée de Fourier des ondelettes 176
7.2 Transformée en ondelettes continue : Inversion 177
7.2.1 Exemples 178
7.3 Les ondelettes orthogonales 183
7.3.1 Problème des moments de l’ondelette 187
7.4 Analyse multi-résolution dans L2 R 188
7.4.1 Exemples de fonctions d’échelle et d’ondelettes 192
7.5 Algorithme rapide de décomposition en ondelettes 193
7.6 Filtrage par ondelettes 198
266 TABLE DES MATI èRES
8 Introduction à l’analyse vocale 199
8.1 Caractéristiques physiques et perceptives des sons 199
8.1.1 Sons périodiques simples : hauteur, intensité 199
8.1.2 Sons périodiques complexes : hauteur, intensité, timbre . 201
8.1.3 Sons apériodiques 202
8.1.4 Résumé des qualités d’un son 203
8.2 Reconnaissance des sons voisés 205
8.2.1 Sons voisés, non voisés, pitch 205
8.2.2 Spectrogramme de parole 206
8.2.3 Analyse LPC – Prédiction linéaire 212
8.3 Exemple 214
8.3.1 Reconnaissance automatique d’un son voisé par corrélation à court terme et calcul de la fréquence fondamentale 214
8.3.2 Les formants 215
9 Exercices et travaux pratiques 219
9.1 Chapitre 2 : analyse spectrale 219
9.1.1 Exercices 219
9.1.2 Travaux pratiques 222
9.2 Chapitre 3 : corrélation des signaux 225
9.2.1 Exercices 225
9.2.2 Travaux pratiques 228
9.3 Chapitre 4 : filtrage 230
9.3.1 Exercices 230
9.3.2 Travaux pratiques 236
9.4 Chapitre 5 : échantillonnage 237
9.4.1 Exercices 237
9.4.2 Travaux pratiques 240
9.5 Chapitre 6 : analyse temps-fréquence 243
9.5.1 Exercices 243
9.5.2 Travaux pratiques 245
9.6 Chapitre 7 : analyse temps-échelle 246
9.6.1 Exercices 246
9.6.2 Travaux pratiques 247
9.7 Chapitre 8 : introduction à l’analyse vocale 249
9.7.1 Exercices 249
9.7.2 Travaux pratiques 251
9.8 Travaux pratiques de synthèse 253

Mathématique appliquée cours 2

Télécharger le fichier PDF: Méthodes mathématiques pour le le Traitement du Signal

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