On Error-Saturation Nonlinearities in NLMS Adaptation

The effect of a saturation-type error nonlinearity in the weight update equation in normalized least mean-square (NLMS) adaptation is investigated for system identification for a white Gaussian data model. Nonlinear recursions are derived for the weight mean error and mean-square deviation (MSD) that include the effect of an error function (erf) saturation-type nonlinearity on the error sequence driving the algorithm. The nonlinear recursion for the MSD is solved numerically and shown in excellent agreement with Monte Carlo simulations, supporting the theoretical model assumptions. The theory is extended to tracking a Markov channel and accurately predicts the tracking behavior as well. The saturation behavior of the algorithm is easily studied by varying a single parameter in the error function, varying from a linear device to a hard limiter. For the white data case, the excess mean square-error (EMSE) is simply related to the MSD. The tradeoff between the extent of error saturation, steady-state EMSE, and algorithm convergence rate is studied using these results.