On the factorization of discrete-time rational spectral density matrices

Abstract

For a discrete-time rational spectral density matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(z)</tex> , the relationship between the factorizations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(z) = Z(z) + Z^{T} (z^{-1})</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(z) = W(z)QW^{T}(z)</tex> , in terms of minimal state space realizations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z(z)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(z)</tex> , are derived in a straightforward way, without resorting to the use of the bilinear transformation. The obvious application of this result to performing a spectral factorizafion of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Phi(z)</tex> is discussed.