Analyse de Signaux Multicomposantes : Contributions à la Décomposition Modale Empirique, aux Représentations temps-fréquence et au Synchrosqueezing

Many signals from the physical world can be modeled accurately as a superposition of amplitude-and frequency-modulated waves. This includes audio signals (speech, music), medical data (ECG) aswell as temporal series (temperature or electric consumption). This thesis deals with the analysis of suchsignals, called multicomponent because they contain several modes. The techniques involved allow for thedetection of the different modes, their demodulation (ie, determination of their instantaneous amplitudeand frequency) and reconstruction. The thesis uses the well-known framework of time-frequency andtime-scale analysis through the use of the short-time Fourier and the continuous wavelet transforms.We will also consider a more recent algorithmic method based on the symmetry of the enveloppes : theempirical mode decomposition.The first contribution proposes a new way to avoid the iterative “Sifting Process” in the empiricalmode decomposition, whose convergence and stability are not guaranteed. Instead, one uses a constrainedoptimization step together with an enhanced detection of the local extrema of the high-frequency mode.The second contribution analyses multicomponent signals through the short-time Fourier transformand the continuous wavelet transform, taking advantage of the “ridge” structure of such signals in thetime-frequency or time-scale planes. More precisely, we propose a new reconstruction method basedon local integration, adapted to the local frequency modulation. Some theoretical guarantees for thisreconstruction are provided, as well as an application to multicomponent signal denoising. The thirdcontribution deals with the quality of the time-frequency representation, using the reassignment methodand the synchrosqueezing transform : we propose two extensions of the synchrosqueezing, that enablemode reconstruction while remaining efficient for strongly modulated waves. A generalization of thesynchrosqueezing in dimension 2 is also proposed, based on the so-called monogenic wavelet transform.