On non-optimal spectral factorizations

TL;DR

This paper explores the set of all polynomial spectral factors of a positive definite Laurent polynomial matrix on the unit circle, including those not invertible inside the circle, expanding understanding of spectral factorizations.

Contribution

It introduces a comprehensive analysis of spectral factorizations for Laurent polynomial matrices, including non-invertible factors inside the unit circle.

Findings

Characterization of all polynomial spectral factors of a given matrix function.

Extension of spectral factorization theory to non-invertible factors.

Insights into the structure of spectral factors on the unit circle.

Abstract

For a given Laurent polynomial matrix function $S$, which is positive definite on the unit circle in the complex plane, we consider all possible polynomial spectral factors of $S$ which are not necessarily invertible inside the unit circle.