Contribution of the bayesian method to pure line spectral analysis and high resolution goniometry.

Summary This work is entirely devoted to the solution of the classical problems of pure line search in time series spectral analysis and of goniometric analysis in antenna processing. These are formulated as inverse problems and approached in the context of Bayesian estimation. In a high resolution analysis, the estimation of the number of sources or spectral lines (detection) plays a determining role insofar as it conditions the global quality of the results. The traditional methodological approach, which consists in separating the estimation of the number of sources from their location, suffers from a number of limitations. These are listed and explained in a first step. In order to overcome them, we propose to treat these two tasks jointly. To this end, we exploit an additional information related to the structure of the solution, namely its impulsive character. In the context of Bayesian statistical estimation, such a structure is satisfactorily described by composite random processes. Two types of probabilistic models are considered, inspired by the field of impulse deconvolution: the Bernoulli-Gaussian model and its fish-Gaussian extension. On an algorithmic level, the regularized solution is obtained by optimizing a mixed criterion, composed of a term of fidelity to the observed data and a term translating the a priori introduced on the solution: a unique criterion is formed, which includes the variable dimension of the solution. We show the interest of combinatorial exploration techniques to optimize the likelihood criteria used. Finally, examples of treatments highlight the substantial improvement obtained by this new approach.