Microstatistics in signal decomposition and the optimal filtering problem

The author introduces and analyzes a large class of nonlinear filters which are based on signal decomposition and where the estimation in the decomposed space uses linear combinations of either the observation vector, the sorted observation vector, or, in general, a nonlinear transformation of the observation vector. Thus, nonlinear filter response characteristics are achieved, but with the machinery of linear systems theory available for their optimization and design. It is shown that linear filters are a subclass of microstatistic filters and that the optimal linear filter solution is suboptimal in the decomposed signal space. The filtering problem reduces to a set of filters operating on the decomposed signals; the output is a weighted sum of the decomposed filtered signals. The optimal interconnection structure between decomposed signals has complexity O(M-1), where M is the cardinality of the decomposition. The formulation is given for a radix-1 decomposition and generalized for a radix-q decomposition. Computer simulations illustrate the performance