Analyse de signaux multicomposantes : contributions à la décomposition modale Empirique, aux représentations temps-fréquence et au Synchrosqueezing

摘要

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) as well as temporal series (temperature or electric consumption). This thesis deals with the analysis of such signals, called multicomponent because they contain several modes. The techniques involved allow for the detection of the different modes, their demodulation (ie, determination of their instantaneous amplitude and frequency) and reconstruction. The thesis uses the well-known framework of time-frequency and time-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 : the empirical mode decomposition. The first contribution proposes a new way to avoid the iterative ``Sifting Process'' in the empirical mode decomposition, whose convergence and stability are not guaranteed. Instead, one uses a constrained optimization 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 transform and the continuous wavelet transform, taking advantage of the ``ridge'' structure of such signals in the time-frequency or time-scale planes. More precisely, we propose a new reconstruction method based on local integration, adapted to the local frequency modulation. Some theoretical guarantees for this reconstruction are provided, as well as an application to multicomponent signal denoising. The third contribution deals with the quality of the time-frequency representation, using the reassignment method and the synchrosqueezing transform: we propose two extensions of the synchrosqueezing, that enable mode reconstruction while remaining efficient for strongly modulated waves. A generalization of the synchrosqueezing in dimension 2 is also proposed, based on the so-called monogenic wavelet transform.