Contributions to Nonstationary Spectrum Estimation and Stationarity Tests in the Time-Frequency Plane

Abstract : The analysis and processing of nonstationary signals has already led to the development of numerous time-frequency methods and algorithms. The purpose of this Thesis is to contribute new approaches in this area. More precisely, the reported work consists in: (i) improving representation tools for exploratory signal analysis in nonstationary contexts; and (ii) testing stationarity of a signal relatively to an observation scale. Existing time-frequency tools still call for improvements in terms of non- stationary spectrum estimation, in particular in the case of chirp signals em- bedded in nonstationary noise. The first part of the Thesis addresses such an issue, with the two-fold objective of a sharp localization for chirp components and a reduced level of statistical fluctuations for the noise background. The technique consists in combining time-frequency reassignment with multita- pering, with two variations. The first one, primarily aimed at nonstationary spectrum estimation, is based on sums of reassignment-based estimates with different tapers, whereas the second one makes use of differences between the same estimates for a sake of chirp enhancement. The principle of the technique is outlined, its implementation based on Hermite functions is jus- tified and discussed, and some typical examples (including an application to the problem of Euler’s disk) are provided for supporting the efficiency of the approaches, both qualitatively and quantitatively. Turning to stationarity, the second part of the Thesis proposes to go beyond the classical definition that refers to a strict invariance of statistical properties over time, and to develop an operational framework for testing sta- tionarity, in a wide sense, relatively to an observation scale in both stochastic and deterministic contexts. The statistical test is based on a comparison between local and global time-frequency features and the originality is to make use of a family of stationary surrogates generated from the original signal un- der test for defining the null hypothesis of stationarity. Within this general framework, two different approaches are proposed. The first one takes advan- tage of suitably chosen distance measures between local and global spectra and characterizes the null hypothesis of stationarity by constructing a para- metric model that is derived from the distribution of surrogates variances. The second approach is implemented by a one-class Support Vector Machine operating on signal features extracted from the time-frequency plane, with those of its stationary surrogates serving as a learning set. The principle of the test and its two variations are presented, some results are shown on typical examples of signals (including speech) which can be considered as stationary or nonstationary, depending on the chosen observation scale.
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Jun Xiao. Contributions to Nonstationary Spectrum Estimation and Stationarity Tests in the Time-Frequency Plane. Signal and Image processing. Ecole normale supérieure de Lyon, 2008. English. ⟨tel-02167626⟩

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