On the Factorization of Rational Discrete-Time Spectral Densities

In this paper, we consider an arbitrary matrix-valued, rational spectral density P(z) . We show with a constructive proof that P(z) admits a factorization of the form P(z)=W'(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.