A new prewhitening-based adaptive filter which converges to the Wiener-solution

A fundamental problem with the Least Mean Squares (LMS) and Normalized LMS (NLMS) algorithms is their slow convergence given a colored input signal. For white input signals, on the other hand, these simple and popular algorithms yield quite satisfactory performance. There are a stunning number of different algorithms that try to deal with this problem, all of which, in some fashion, use decorrelation techniques for whitening the input. In most cases this leads to increased complexity compared to the 2M multiplications/sample, M being the filter length, of the LMS. Of algorithms that keep the complexity as low as 2M + K for some small K, perhaps the simplest and most popular approach is what is known as prewhitening: the use of a short decorrelation FIR filter on the input signal. In this paper we show that an inherent problem with the prewhitening-based algorithms is that they will not converge to the exact Wiener-solution in the mean if the true system is slightly nonlinear or if it is linear but with an impulse response longer than M - that is, in the undermodeling case. A new decorrelation-filter based algorithm with equally low computational complexity, which does converge to the Wiener solution in the mean is then presented. Theoretical results and simulations are provided to back up these claims.