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

We propose an approach to the filtering of infinite sets of stochastic signals, KY and KX. The known Wiener-type approach cannot be applied to infinite sets of signals. Even in the case when KY and KX 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, KY and KX, find a single filter F : KY-KX that estimates signals from KY 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, FðyÞ 1⁄4 Pp j 1⁄4 1 TjRjQjðyÞ. Each term is derived from three operations presented by matrices, Qi, Ri and Ti with i1⁄41,y,p. Each operation is a special stage of the filtering aimed at facilitating the associated numerical work. In particular, Q1 , . . . ,Qp are used to transform an observable signal y 2 KY to p different signals. Matrices R1,y,Rp 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 Ti,y,Tp 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. & 2011 Elsevier B.V. All rights reserved.