The (J, J′)-spectral factorization of a general discrete-time system

For a discrete-time system whose transfer matrix function is completely general (arbitrary normal rank, strictly proper/imprope/polynomial, with poles and zeroes on the unit circle) we give existence conditions and constructive formulas for the (J, J´)-spectral factor. The computation of the spectral factor is done by employing generalized state-space realizations and numerically-sound algorithms based on a preliminary unitary projection which delineates a subsystem fulfilling all regularity assumptions. En route, we give formulas for the J-lossless conjugation of an arbitrary discrete-time system and the (J, J´)-lossless factorization of a marginally stable but otherwise general system.