The Functional Wiener Filter

TL;DR

This paper introduces the functional Wiener filter (FWF), a closed-form solution in RKHS for Wiener filtering that leverages a new lagged RKHS based on correntropy, offering comparable performance to kernel adaptive filtering with less computation.

Contribution

The paper develops a novel functional Wiener filter in RKHS using a new lagged space, differing from traditional kernel adaptive filtering methods.

Findings

FWF achieves performance comparable to KAF in time series modeling.

FWF requires significantly less computational effort.

Two pre-imaging algorithms, FWFFP and FWFLM, are compared in performance.

Abstract

This paper presents a close form solution in Reproducing Kernel Hilbert Space (RKHS) for the famed Wiener filter, which we called the functional Wiener filter(FWF). Instead of using the Wiener-Hopf factorization theory, here we define a new lagged RKHS that embeds signal statistics based on the correntropy function. In essence, we extend Parzen$'$s work on the autocorrelation function RKHS to nonlinear functional spaces. The FWF derivation is also quite different from kernel adaptive filtering (KAF) algorithms, which utilize a search approach. The analytic FWF solution is derived in the Gaussian kernel RKHS with a constant computational complexity similar to the Wiener solution, and never composes nor employs the error as in conventional optimal modeling. Because of the lack of congruence between the Gaussian RKHS and the space of time series, we compare performance of two pre-imaging algorithms: a fixed-point optimization (FWFFP) that finds and approximate solution in the RKHS, and a local model implementation named FWFLM. The experimental results show that the FWF performance is on par with the KAF for time series modeling, and it requires far less computation.