The factorization of discrete-process spectral matrices

A technique is presented for salving the discrete version of the multidimensional Wiener-Hopf equation by spectral factorization. This equation is derived to establish a need for spectral factorization and to determine the requirements of the factors of the spectral matrix. The method of factoring the spectral matrix of continuous systems, developed by Davis, is then extended to discrete systems. More specifically, a matrix
H(z)
must be found such that the matrix of the spectra of the input signals equals the product of
H(z^{-1})
and
H^{T}(z)
. A technique for finding this matrix is presented. The nonanticipatoriness as well as the stability of the elements of
H(z)
and
H^{-1}(z)
must be and is guaranteed. It is then shown that the solution to the discrete Wiener-Hopf equation is unique.