Factoring the spectral matrix

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)
.