New Algorithm for Spectral Factorization of Rational Matrix Functions with Applications to Paraunitary Filter Banks

Spectral factorization is the process by which a positive (scalar or matrix-valued) function S is expressed in the form S(t) = S+(t)S ∗ +(t), t ∈ T, where S+ can be analytically extended inside the unit circle T and S ∗ + is its Hermitian conjugate. There are multiple contexts in which this factorization naturally arises, e.g., linear prediction theory of stationary processes, optimal control, digital communications, etc. Spectral factorization is used to construct certain wavelets and multiwavelets as well. Therefore, many authors contributed to development different computational methods for spectral factorization. Unlike the scalar case, where an explicit formula exists for factorization, in general, there is no explicit expression for spectral factorization in the matrix case. The existing algorithms for approximate factorization are, therefore, more demanding in the matrix case.
The Janashia–Lagvilava algorithm [1, 2] is a relatively new method of matrix spectral factorization which proved to be effective [3, 4] and provides several generalizations. Nevertheless, the algorithm, as it was designed so far, was not able to factorize exactly even simple polynomial matrices. In the proposed work, we cast a new light on the capabilities of the method eliminating the above-mentioned flaw. In particular, we can factorize explicitly matrices whose rational entries in the lower-upper triangular factorization can be determined (indicating their poles inside T and the principle parts at these poles). This extension allows to construct rational paraunitary filter banks
with preassigned poles and zeros which are multidimensional lossless infinite impulse response filters and play an important role in linear time invariant systems.