On the factorization of discrete-time rational spectral density matrices

For a discrete-time rational spectral density matrix
\Phi(z)
, the relationship between the factorizations
\Phi(z) = Z(z) + Z^{T} (z^{-1})
and
\Phi(z) = W(z)QW^{T}(z)
, in terms of minimal state space realizations of
Z(z)
and
W(z)
, 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
\Phi(z)
is discussed.