Formulation and implementation of the non-stationary evolutionary Wiener filtering

Abstract

In this paper we consider solutions to the non-stationary Wiener filtering problem using the evolutionary spectral theory. Two cases of interest result from the uncorrelation between the desired signal and the noise. One constrains the support of the generating kernels of the signals and the other imposes orthogonality on their innovation processes. The latter condition is more general and our solution coincides with the one presented previously by Abdrabbo and Priestley. For the first case, we develop a new solution that depends directly on the Wold–Cramer models of the desired and noisy processes. Implementation is achieved in both cases by estimating the kernels for the Wold–Cramer representations from the spectra using the evolutionary maximum entropy spectral estimation. The connections of the Wiener filter with the Wiener–Hopf equations and with the special case of stationary processes are discussed. Although the developed Wiener filter is non-recursive, an approximate recursive filter is obtained using a nonlinear Kalman system identification method. Examples illustrating the filtering are given.