Th`Ese Est Une Tˆache Ardue

No d’ordre : 3373 THESE` PRESENT´ EE´ A` L’UNIVERSITE´ BORDEAUX I ECOLE´ DOCTORALE DE MATHEMATIQUES´ ET D’INFORMATIQUE Par Martin Raspaud POUR OBTENIR LE GRADE DE DOCTEUR SPECIALIT´ E´ : INFORMATIQUE Modèles spectraux hiérarchiques pour les sons et applications Soutenue le : 24 Mai 2007 Après avis des rapporteurs : Daniel Arfib ....................... Directeur de Recherche Philippe Depalle .................. Professeur Devant la commission d’examen composée de : Christophe Schlick ............... Professeur .................... Président et Rapporteur Daniel Arfib ....................... Directeur de Recherche Examinateur Philippe Depalle .................. Professeur .................... Examinateur Myriam Desainte–Catherine Professeur .................... Examinateur Laurent Girin ...................... Maˆıtre de Conférences . Examinateur Sylvain Marchand ................ Maˆıtre de Conférences . Examinateur 2007 Contents English Introduction 1 Introduction en Fran¸cais 7 1 Temporal Modelling 13 1.1 Temporal model ................................................... ....................................... 13 1.2 Time-scaling in the temporal model ................................................... .......... 16 Varying the playing speed ................................................... ........................... 16 Overlap-Add (OLA) ................................................... .................................. 17 SOLA and TD-PSOLA ................................................... ............................... 17 1.3 Conclusion ................................................... ............................................... 17 2 Spectral Modelling 21 2.1 The Fourier transform ................................................... .............................. 21 2.2 Discrete case ................................................... ............................................ 24 2.3 Short-time Fourier transform ................................................... .................... 26 Convolution ................................................... ............................................. 28 Analysis windows ................................................... ...................................... 28 2.4 Time-scaling using the spectral model ................................................... ....... 32 The phase vocoder ................................................... .................................... 32 Time-scaling using the phase vocoder ................................................... ............. 34 2.5 Conclusion ................................................... ............................................... 34 3 Sinusoidal Modelling 35 3.1 Description of the sinusoidal analysis process ............................................... 37 3.2 Enhancing the precision of the discrete Fourier transform ............................. 39 3.3 Enhancing the partial tracking ................................................... ................. 41 Partial trajectories as control signals ................................................... .............. 42 Predictable control signals ................................................... ........................... 43 3.4 Synthesis of sounds from the sinusoidal model .............................................. 52 3.5 Time-scaling with the classic sinusoidal model.............................................. 54 Resampling the parameters ................................................... .......................... 54 Non-uniform time scaling ................................................... ............................ 67 3.6 Conclusion ................................................... ............................................... 68 i ii CONTENTS 4 Order-2 Spectral Modelling 69 4.1 Modelling the parameters ................................................... ......................... 70 Modelling the parameters as polynomials ................................................... ........ 70 Modelling the parameters as sums of sines ................................................... ....... 73 Modelling the parameters using sums of sines and polynomials ................................ 74 High-resolution estimation method ................................................... ................ 76 4.2 Predicting partial tracks using the new model .............................................. 79 4.3 Inaudible control signals ................................................... .......................... 84 4.4 Classification of partials ................................................... ........................... 91 Context ................................................... .................................................. 91 Mid-level representation of polyphonic sounds ................................................... .. 91 The common variation cue ................................................... .......................... 92 Proposed metric ................................................... ....................................... 95 Evaluation................................................... ............................................... 98 Results ................................................... ................................................... 101 A new tool to gather partials? ................................................... ...................... 103 4.5 Considerations on the link of tremolo and vibrato ........................................ 104 4.6 Conclusion ................................................... ............................................... 105 5 Order-2 Sinusoidal Modelling 107 5.1 Previous work on vibrato and tremolo ................................................... ...... 107 5.2 Hierarchical modelling ................................................... .............................. 108 Work on frequency and amplitude control parameters and enhanced time scaling.......... 110 Work on phase and amplitude control parameters ................................................. 115 5.3 Non-uniform time scaling at order 2................................................... .......... 120 5.4 Conclusion ................................................... ............................................... 120 6 Higher Order Modelling 121 6.1 Musical modelling ................................................... .................................... 121 6.2 Finding base functions ................................................... ............................. 122 Box functions ................................................... ........................................... 124 Impulse train functions ................................................... ............................... 126 Box train functions ................................................... .................................... 126 Linearly independent system ................................................... ........................ 127 6.3 Conclusion ................................................... ............................................... 128 7 The Clicks Software 133 7.1 Constraints and general choices ................................................... ................ 133 7.2 Organisation of the program directory ................................................... ...... 134 7.3 How to install and launch Clicks ................................................... ............... 135 7.4 Basic tools ................................................... ............................................... 135 Vector operations ................................................... ...................................... 135 Matrix operations ................................................... ..................................... 137 CONTENTS iii Linear prediction ................................................... ...................................... 138 Least-square estimation ................................................... .............................. 139 Fast Fourier transform................................................... ................................ 139 Resampling ................................................... ............................................. 142 Windows................................................... ................................................. 143 7.5 Sound modelling ................................................... ...................................... 144 General sound interface ................................................... .............................. 144 PCM sounds ................................................... ............................................ 145 Spectral peaks ................................................... .......................................... 145 Spectral sounds ................................................... ........................................ 146 Spectral peak estimation ................................................... ............................. 147 Partials ................................................... .................................................. 147 Sinusoidal sounds ................................................... ...................................... 149 Partial tracking ................................................... ........................................ 149 Time-scaling of sounds ................................................... ............................... 150 7.6 Design and other features

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