On error-saturation nonlinearities in LMS adaptation

The effect of a saturation-type error nonlinearity in the weight update equation in least-mean-squares (LMS) adaptation is investigated for a white Gaussian data model. Nonlinear difference equations are derived for the eight first and second moments, which include the effect of an error function (erf) saturation-type nonlinearity on the error sequence driving the algorithm. A nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures. The steady-state second-moment weight behavior is evaluated exactly for the erf nonlinearity. Using the above results, the tradeoff between the extent of error saturation, steady-state excess mean-square error, and rate of algorithm convergence is studied. The tradeoff shows that (1) starting with a sign detector, the convergence rate is increased by nearly a factor of two for each additional bit, and (2) as the number of bits is increased further, the additional bit by very little in convergence speed, asymptotically approaching the behavior of the linear algorithm