Dual-mode filtering of polynomial signals in noise

A procedure is presented for finding the best (minimum mean-square error) filter out of a class of dual-mode (adaptive) filters when the input consists of a sample out of an ensemble of fixed order polynomials plus "almost white" noise. The discussion is based on sampled data filtering with the error being observed only at uniformly spaced sampling instants. The filters are non-linear, and consist of two linear discrete subfilters and an input-sensitive decision element which switches between them. The linear subfilter design procedures used here are modifications of the Zadeh-Ragazzini approach to polynomial filtering. The modification makes the resulting filters unbiased estimators over an ensemble of polynomials of a given degree rather than unbiased for each polynomial of a given degree. Simultaneous optimization of the filter functions and switching rule results in transcendental equations which must be solved by trial and error. An efficient iterative procedure for searching out the solutions is presented.