The Stability and Continuity Behavior of the Spectral Factorization in the Wiener Algebra With Applications in Wiener Filtering

This paper investigates the stability and the continuity behavior of the spectral factorization and of the Wiener filter in the bounded-input bounded-output (BIBO) stability norm. It shows that if the minimum of the given spectra becomes not smaller than half its norm, there exist uniform upper bounds on the stability norm of the spectral factor and Wiener filter. If on the other hand, the minimum becomes smaller than a quarter of its norm, no such upper bounds can exist. In the second part, it is shown that every BIBO-stable spectral density is a continuity point of the spectral factorization. From this, it is derived that the Wiener filter always depends continuously on the data in the BIBO-norm. These results are compared with energy stable systems. It turns out that every continuous spectrum is a discontinuous point for the spectral factorization in the energy norm. It follows that the Wiener filter depends not continuous on the data in this norm.