A constructive algorithm for 2-D spectral factorization with rational spectral factors

Abstract

Spectral factorization of para-Hermitian polynomial matrices nonnegative on the imaginary axis is known to play a crucial role in signal and system theory. Such factorization, although always feasible in one-dimensional (1-D) case, can be carried out in two-dimensional (2-D) case in a somewhat modified form. In this modified form, even spectral factors of scalar polynomials must be rational matrices (with normal rank one), analytic in the Cartesian products of open right half planes. Despite this constrained form, such spectral factors are important in the context of 2-D systems. The feasibility of such spectral factorization is apparent in view of classical results from algebraic geometry, but a constructive proof of it has not been available in system theoretic literature. By a selection of elementary techniques borrowed from different sources, we give a constructive algorithm for the aforementioned spectral factorization of 2-D para-Hermitian scalar positive definite polynomials, and thus, indirectly provide a constructive proof of the corresponding result for para Hermitian polynomial matrices as well. An example illustrating the main aspect of the algorithm is included. Analog of the result for discrete time systems, and comparisons with known 2-D spectral factorization results of other types are also included.