Sorted spectral factorization of matrix polynomials in MIMO communications

Sorted spectral factorization of matrix polynomials is studied. Such type of factoring Hermitian matrix polynomials is the key step in calculating the optimum receive filter matrices in spatial/temporal decision-feedback equalization as well as the optimum transmit filter matrices in spatial/temporal Tomlinson-Harashima-type precoding schemes. Contrary to other approaches, we inherently consider asymptotic rather than finite-length results for transmission over MIMO channels with intersymbol interference. It is shown how the different types of factorizations can be transformed onto a prototype factorization task, which in turn can be solved by first performing an unsorted factorization and then determining the optimal processing order. An easy-to-use iterative algorithm for unsorted spectral factorization and the adjustment of the optimized order in DFE and precoding are explained. Numerical simulations cover the impact of sorted and unsorted spectral factorization on the performance of DFE and precoding schemes.