Matrix spectral factorization with perturbed data

A necessary condition for the existence of spectral factorization is positive definiteness a.e. on the unit circle of a matrix function which is being factorized. Correspondingly, the existing methods of approximate computation of the spectral factor can be applied only in the case where the matrix function is positive definite. However, in many practical situations an empirically constructed matrix spectral densities may lose this property. In the present paper we consider possibilities of approximate spectral factorization of matrix functions by their known perturbation which might not be positive definite on the unit circle.