Factoring the spectral matrix

Author

Davis, Michal C.

Author_Institution

U.S. Navy, Bureau of Ships, Washington, D.C., USA

Volume

8

Issue

4

fYear

1963

fDate

10/1/1963 12:00:00 AM

Firstpage

296

Lastpage

305

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 $S(s)N^{-1}(s)$ .

Keywords

Linear systems, stochastic; Matrix factorization; Stochastic processes; Stochastic systems, linear; Filtering; Filters; Linear systems; Marine vehicles; Multidimensional signal processing; Multidimensional systems; Random processes; Signal processing; Transforms; Vectors;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1963.1105614

Filename

1105614