All-Pass Functions for Mirroring Pairs of Complex-Conjugated Roots of Rational Matrix Functions

TL;DR

This paper develops methods to construct rational all-pass matrix functions with real coefficients that mirror pairs of complex-conjugated roots, addressing a key challenge in spectral factorization and VARMA models.

Contribution

It provides a novel construction for all-pass matrix functions with real coefficients for mirroring conjugate roots, ensuring real-valued parameter spaces for estimation.

Findings

Constructed all-pass functions with real coefficients for conjugate root mirroring.

Addressed the challenge of maintaining real-valued coefficients in all-pass functions.

Enhanced the theoretical foundation for spectral factorization and VARMA modeling.

Abstract

We construct rational all-pass matrix functions with real-valued coefficients for mirroring pairs of complex-conjugated determinantal roots of a rational matrix. This problem appears, for example, when proving the spectral factorization theorem, or, more recently, in the literature on possibly non-invertible or possibly non-causal vector autoregressive moving average (VARMA) models. In general, it is not obvious whether the all-pass matrix function (and as a consequence the all-pass transformed rational matrix with initally real-valued coefficients) which mirrors complex-conjugated roots at the unit circle has real-valued coefficients. Naive constructions result in all-pass functions with complex-valued coefficients which implies that the real-valued parameter space (usually relevant for estimation) is left.