Structural properties of the Wiener filter - Stability, smoothness properties, and FIR approximation behavior

Any Wiener filter can be interpreted as a cascade of a whitening and estimation filter. The whitening filter is determined due to the spectral factorization of the spectral density of the input signal. For the calculation of the estimation filter the spectral factorization as well as the so called plus-operator is needed. This correspondence investigates in detail the behavior of these two operations and studies the corresponding properties of both filters. Then the practical consequences for the overall Wiener Filter are discussed. It is shown that if the given spectral densities are smooth (Hölder continuous) functions, the resulting Wiener filter will always be stable and can be approximated arbitrarily well by a finite impulse response (FIR) filter. Moreover, the smoothness of the spectral densities characterizes how fast the FIR filter approximates the desired filter characteristic, and the correspondence gives a class of approximation polynomials which actually achieves the optimal approximation behavior. On the other hand, if the spectral densities are continuous, but not Hölder continuous, the resulting Wiener filter may not be stable.