  In this paper, we enhance a recent algorithm for approximate spectral
factorization of matrix functions, extending its capabilities to precisely
factorize rational matrices when an exact lower-upper triangular factorization
is available. This novel approach leverages a fundamental component of the
improved algorithm for the precise design of rational paraunitary filter banks,
allowing for the predetermined placement of zeros and poles. The introduced
algorithm not only advances the state-of-the-art in spectral factorization but
also opens new avenues for the tailored design of paraunitary filters with
specific spectral properties, offering significant potential for applications
in signal processing and beyond.
