Filtering of infinite sets of stochastic signals: An approach based on interpolation techniques

Abstract

We propose an approach to the filtering of infinite sets of stochastic signals, K Y and K X . The known Wiener-type approach cannot be applied to infinite sets of signals. Even in the case when K Y and K X are finite sets, the computational work associated with the Wiener approach becomes unreasonably hard. To avoid such difficulties, a new theory is studied. The problem addressed is as follows. Given two infinite sets of stochastic signals, K Y and K X , find a single filter F : K Y → K X that estimates signals from K Y with a controlled associated error. Our approach is based on exploiting a signal interpolation idea. The proposed filter F is represented in the form of a sum of p terms, F ( y ) = ∑ j = 1 p T j R j Q j ( y ) . Each term is derived from three operations presented by matrices, Q i , R i and T i with i=1,…, p. Each operation is a special stage of the filtering aimed at facilitating the associated numerical work. In particular, Q 1 , … , Q p are used to transform an observable signal y ∈ K Y to p different signals. Matrices R 1,…, R p reduce a set of related matrix equations to p independent equations. Their solution requires much less computational effort than would be required with the full set of matrix equations. Matrices T i ,…, T p are determined from interpolation conditions. We show that the proposed filter is asymptotically optimal. Moreover, the filter model is determined in terms of pseudo-inverse matrices and, therefore, it always exists.