On the state space and dynamics selection in linear stochastic models: a spectral factorization approach

TL;DR

This paper explores the relationship between different spectral factorization methods in linear stochastic models, focusing on the interplay of zero- and pole-structure parametrizations via Riccati equations.

Contribution

It introduces a unified framework linking two Riccati equations for spectral factors, enabling construction of spectral factors with combined zero- and pole-flipping.

Findings

Established the relation between two Riccati equation solution sets

Provided a method to construct spectral factors with combined zero- and pole-flipping

Enhanced understanding of spectral factorization in stochastic models

Abstract

Matrix spectral factorization is traditionally described as finding spectral factors having a fixed analytic pole configuration. The classification of spectral factors then involves studying the solutions of a certain algebraic Riccati equation which parametrizes their zero structure. The pole structure of the spectral factors can be also parametrized in terms of solutions of another Riccati equation. We study the relation between the solution sets of these two Riccati equations and describe the construction of general spectral factors which involve both zero- and pole-flipping on an arbitrary reference spectral factor.