On Minimal Spectral Factors with Zeroes and Poles lying on Prescribed Region

TL;DR

This paper proves that imposing stochastic minimality ensures the uniqueness of spectral factors with zeroes and poles in prescribed regions for discrete-time rational matrix functions, extending spectral factorization theory.

Contribution

It confirms the conjecture that stochastic minimality guarantees uniqueness in spectral factorization with prescribed zeroes and poles.

Findings

Proved the conjecture on uniqueness under stochastic minimality.

Extended spectral factorization results to prescribed regions with symplectic symmetry.

Established conditions for existence and uniqueness of spectral factors.

Abstract

In this paper, we consider a general discrete-time spectral factorization problem for rational matrix-valued functions. We build on a recent result establishing existence of a spectral factor whose zeroes and poles lie in any pair of prescribed regions of the complex plane featuring a geometry compatible with symplectic symmetry. In this general setting, uniqueness of the spectral factor is not guaranteed. It was, however, conjectured that if we further impose stochastic minimality, uniqueness can be recovered. The main result of his paper is a proof of this conjecture.