A performance bound for the LMS estimator

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 used to model this estimator obscure these dependencies; to simplify the analysis they restrict the estimator to the finite set of data sufficient to span the length of its filter. Thus the finite Wiener filter is 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 correspondence, we derive a bound on the LMS estimator which does not exclude the contributions from data outside its filter length. We give examples of this bound in cases where the LMS estimator outperforms the finite Wiener filter.