Micro Statistics in Signal Decomposition and the Optimal Filtering Problem

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Optimal Filtering is a statistical signal processing problem of major engineering importance. Historically, the most frequently used signal processing tools have been linear in nature, but despite rich linear systems theories, many signal processing problems have not been satisfactorily addressed through the use of linear schemes. This research aims at the theoretical development of a large class of non-linear filters which are based in combination of either the observation vector, the sorted observation vector, or in general a non-linear transformation of the observation vector. Thus, we achieve non-linear filter response characteristics but with the machinery of linear systems theory available for their optimization and design. The optimal solution requires the statistical characterization of the set of decomposed signals (micro-statistics). The filtering problem reduces to a set of filters operating on the decomposed signals (micro-statistic filters) where the output is a weighted sum of the decomposed filtered signals. In this work, a theory will be developed for robust micro-statistic filters in which linear operations are executed in a sorted observation vector. For environments with unknown or non-stationary characteristics, we develop adaptive micro-statistic filters and address issues such as convergence, lattice structures, fast algorithms, and complexity.