On the Factorization of Rational Discrete-Time Spectral Densities

TL;DR

This paper proves that any rational spectral density matrix in discrete-time can be factorized into a product involving a minimal spectral factor, with controllable analyticity properties, extending classical results to a more general setting.

Contribution

It provides a constructive proof for the factorization of rational spectral densities with specified analyticity and symplectic conditions, generalizing classical Youla factorization.

Findings

Existence of a spectral factorization for any rational spectral density

Construction of a minimal spectral factor with prescribed analyticity regions

Extension of classical factorization results to a broader class of spectral densities

Abstract

In this paper, we consider an arbitrary matrix-valued, rational spectral density $\Phi(z)$. We show with a constructive proof that $\Phi(z)$ admits a factorization of the form $\Phi(z)=W^\top (z^{-1})W(z)$, where $W(z)$ is stochastically minimal. Moreover, $W(z)$ and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic-type conditions. By suitably selecting the analyticity regions, this extremely general result particularizes into a corollary that may be viewed as the discrete-time counterpart of the matrix factorization method devised by Youla in his celebrated work (Youla, 1961).