Factoring the spectral matrix

Abstract

This paper presents a complete solution for the optimum linear system which operates on n stationary and correlated random processes so as to minimize error variance in filtering or prediction. A simple closed-form answer results if the matrix \Phi(s) of spectra of the input signals can be factored such that \Phi(s) = G(-s)G^{T}(s) where G(s) and G^{1}(s) represent matrices of stable transforms in the Laplace variables. A general factoring procedure for rational matrices is presented. G(s) can be viewed as the system which would reproduce signals with the spectrum of \Phi(s) when excited by n uncorrelated unit-density white-noise sources. In the case of a multidimensional filter, when G(s) is separated by partial fractions into two terms, S(s) + N(s) , having 1hp poles from the signal and noise spectra, respectively, the optimum unity-feedback filter is shown to have a forward-loop transference of S(s)N^{-1}(s) .