Generalized spectral factorization problem for discrete time polynomial matrices via quadratic difference forms

We address spectral factorization problems for discrete time polynomial matrices. The main concept used in the paper is based on quadratic differential/difference forms and dissipativeness similarly to van der Geest and Trentelman (1997), Trentelman and Rapisarda (1999) and Kaneko and Fujii (2000) which treat the polynomial matrices with no zeros on the jω axis or the unit circle. Here, by using some inherent techniques in discrete time, we expand the spectral factorization algorithms for polynomial matrices with zeros on the unit circle via quadratic difference forms. Moreover, we show that this algorithm is also available to the singular polynomial matrices in discrete time