Recursive solution to the multichannel filtering problem

Abstract

Systems made up of arrays of sensors (seismographs, hydrophones, antennas, meteorological stations) receive geophysical data on a number of channels. Such multichannel data can be processed by means of a multichannel least-squares (Wiener) digital filter; the filter is determined by requiring that the sum of the mean-square errors between the desired outputs and the actual outputs for each channel be minimized. This requirement leads to a set of simultaneous linear equations, called the normal equations, where each element is itself a square matrix. The coefficients of the digital filter are given by the solution of these normal equations. An exact recursive procedure is derived for the numerical solution of the normal equations. In this recursive procedure advantage is taken of the fact that the matrix of the normal equations is an autocorrelation matrix, and hence is in the form of a Toeplitz matrix, namely one with equal elements along any diagonal. A set of auxiliary coefficients that are generated in the recursive procedure are the coefficients of matrix-valued polynomials which are the counterparts of the classical (scalar-valued) polynomials orthogonal on the unit circle. This set of auxiliary coefficients may also be used for a sideward recursion that corresponds to shifting the time index of the right-hand side of the normal equations. This sideward recursion is valuable in applications because it represents a relatively inexpensive way of determining the filter that incorporates the optimum time delay between the multichannel input signal and the multichannel output signal. The machine time required to solve the normal equations for a filter with m (matrix-valued) coefficients is proportional to m2 for the proposed recursive procedure, as compared with m3 for conventional techniques for the solution of simultaneous equations. Another advantage of using the recursive procedure is that it only requires computer storage space proportional to m, rather than to m2 as in the case of the conventional techniques.