Décompositions Modales Empiriques. Contributions à la théorie, l'algorithmie et l'analyse de performances

Abstract : The Empirical Mode Decomposition (EMD) is a novel signal processing tool dedicated to the analysis of nonstationary and/or nonlinear signals. The EMD provides for any signal a data-driven multi-scale decomposition. The components are oscillatory signals, not necessarily harmonic, whose characteristics, waveform, amplitude and frequency may be time-varying. The EMD being rather recent, it is only defined as the output of an unusual algorithm, with many degrees of freedom and no sound theoretical basis. We first focus on the algorithm of the EMD. The questions raised by the possibilities for the degrees of freedom are studied in order to propose an implementation. We also propose some slight variations on the original algorithm and an extension to process bivariate signals. Motivated by the fact that the algorithm is initially presented in a continuous time framework, but systematically applied to sampled signals, we study the effect of sampling on the decomposition. A model of sampling effects is proposed for the simple case of a sinusoidal input signal and a bound on the magnitude of these effects is derived for arbitrary input signals. Finally the mechanism underlying the decomposition is studied by means of the analysis of two complementary situations, the decomposition of sums of two sinusoids and that of broadband noise signals. The first case allows the derivation of a simple model explaining the behavior of the EMD for a vast majority of the possibilities of sums of sinusoids. This model remains valid for slightly amplitude and frequency modulated sinusoids and also for some cases of sums of non harmonic periodic waveforms. As for broadband noise signals, we observe that the behavior of the EMD is close to that of an autosimilar filter bank, analogous to those corresponding to discrete wavelet transforms. The properties of the equivalent filter bank are studied in detail as a function of the key parameters of the EMD algorithm. A link is also established between this filter bank behavior and the model developed for the sums of sinusoids.
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Gabriel Rilling. Décompositions Modales Empiriques. Contributions à la théorie, l'algorithmie et l'analyse de performances. Traitement du signal et de l'image [eess.SP]. Ecole normale supérieure de lyon - ENS LYON, 2007. Français. ⟨tel-00442634⟩



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