Factoring the spectral matrix

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