Factoring the spectral matrix

This paper presents a complete solution for the optimum linear system which operates onnstationary 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)whereG(s)andG^{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 bynuncorrelated unit-density white-noise sources. In the case of a multidimensional filter, whenG(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 ofS(s)N^{-1}(s).