On the Factorization of Rational Discrete-Time Spectral Densities

In this paper, we consider an arbitrary matrix-valued, rational spectral density $Φ(z)$. We show with a constructive proof that $Φ(z)$ admits a factorization of the form $Φ(z)=W^\top (z^{-1})W(z)$, 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).

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