Parametrization of Minimal Spectral Factors of Discrete-Time Rational Spectral Densities

TL;DR

This paper presents the first complete parametrization of minimal spectral factors for discrete-time rational spectral densities using all-pass divisors, resolving a longstanding conjecture without additional assumptions.

Contribution

It provides a comprehensive parametrization framework for minimal spectral factors in discrete-time, extending previous partial results mainly in continuous-time cases.

Findings

Complete parametrization established

Resolves a conjecture from 2016

Applicable to general discrete-time spectral densities

Abstract

In this paper, the problem of providing a complete parametrization of the minimal spectral factors of a discrete-time rational spectral density is considered. The desired parametrization, given in terms of the all-pass divisors of a certain all-pass function, is established in the most general setting: after several partial results, mostly in the continuous-time case, this is indeed the first complete parametrization obtained without resorting to any facilitating assumption. This result provides a positive answer to a conjecture raised in [Baggio and Ferrante, 2016].