Design of Wiener filters using a cumulant based MSE criterion

This paper proposes a cumulant (higher-order statistics) based mean-square-error (MSE) criterion for the design of Wiener filters when both the given wide-sense stationary random signal x(n) and the desired signal d(n) are non-Gaussian and contaminated by Gaussian noise sources. It is theoretically shown that the designed Wiener filter associated with the proposed criterion is identical to the conventional correlation (second-order statistics) based Wiener filter as if both x(n) and d(n) were noise-free measurements. As the latter, the former can also be obtained by solving a cumulant-based Wiener-Hopf equation associated with a (cumulant-based) orthogonality principle. Then a generalized cumulant projection theorem is proposed which includes the projection of cumulants to correlations associated with the proposed cumulant-based MSE criterion and that associated with Delopoulos and Giannakis' cumulant-based MSE criterion as special cases. Moreover, the proposed cumulant-based MSE criterion and Delopoulos and Giannakis' cumulant-based MSE criterion are equivalent for cumulant order M = 3. Some simulation results for system identification and time delay estimation are then provided to demonstrate the good performance of the proposed cumulant-based Wiener filter. Finally, we draw some conclusions.