Factoring the spectral matrix

Abstract

This paper presents a complete solution for the optimum linear system which operates on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> stationary and correlated random processes so as to minimize error variance in filtering or prediction. A simple closed-form answer results if the matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(s)</tex> of spectra of the input signals can be factored such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(s) = G(-s)G^{T}(s)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G^{1}(s)</tex> represent matrices of stable transforms in the Laplace variables. A general factoring procedure for rational matrices is presented. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)</tex> can be viewed as the system which would reproduce signals with the spectrum of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(s)</tex> when excited by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> uncorrelated unit-density white-noise sources. In the case of a multidimensional filter, when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G(s)</tex> is separated by partial fractions into two terms, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S(s) + N(s)</tex> , having 1hp poles from the signal and noise spectra, respectively, the optimum unity-feedback filter is shown to have a forward-loop transference of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S(s)N^{-1}(s)</tex> .