General spectral factorization with respect to the disk

In this paper we present the solution to the spectral factorization problem formulated for a completely general linear time-invariant discrete-time system. The resulting spectral factor is given by a state-space formula, and the underlying algorithm is based exclusively on orthogonal transformations and standard reliable procedures for solving Lyapunov and Riccati equations. The formula can be applied even for descriptor systems, or for factorizing polynomial matrices with respect to the unit disk. The main novelty is that we remove All the restrictive assumptions made in the literature and allow the system to have arbitrary normal rank, poles and zeros on the unit circle, or at infinity, while the resulting spectral factor is given in standard state-space form.