On the multidimensional prediction and filtering problem and the factorization of spectral matrices

Since the linear prediction and filtering of multiple correlated stationary random signals involves the solution of a matrix integral equation, several methods of solution of this equation have been developed. In this paper the direct solution by complex variable techniques is extended from the one-dimensional case. The problem reduces to the factorization of a spectral density matrix and this factorization is the principal topic of the paper. Under very general conditions, a factorization procedure has been developed by Wiener and Masani (1) for discrete time series. Their technique is applied here with some modifications to the factorization of the spectral density matrices of continuous processes. A simplified procedure is developed also for the case where the elements of the spectral density matrix are rational functions of frequency. Examples illustrate the general technique and the simplified procedure. © 1961.