Bounding the performance of the LMS estimator for cases where performance exceeds that of the finite Wiener filter

The least-mean-square (LMS) estimator is a nonlinear estimator with information dependencies spanning the entire set of data fed into it. The traditional analysis techniques which are used to model this estimator obscure this, restricting the estimator to the finite set of data sufficient to span the length of its filter. The finite Wiener filter is thus often considered a bound on the performance of the LMS estimator. Several papers have reported the performance of the LMS filter exceeding that of the finite Wiener filter. In this paper, we will demonstrate a bound on the LMS estimator, which does not exclude the contributions from data outside its filter length, and which demonstrates the ability of the LMS estimator to outperform the finite Wiener filter in certain cases.