ON WEIGHT UPDATE SATURATION NONLINEARITIES IN LMS ADAPTATION

The effect of a saturation-type nonlinearity in the weight update equation in LMS adaptation is investigated for a white Gaussian data model. Nonlinear difference equations are derived for the weight first and second moments that include the effect of a 1-e‒ax saturation-type nonlinearity on the update term driving the algorithm. A nonlinear difference equation for the mean norm is explicitly solved via a differential equation approximation and integration by quadratures. The steady-state second moment weight behavior is evaluated approximately for the nonlinearity. Using these results, the tradeoff between the extent of weight update saturation, steady-state excess mean-square error, and rate of algorithm convergence is studied. For the same steady-state misadjustment error, 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, 2) as the number of bits is increased further, the additional bits buy very little in convergence speed, asymptotically approaching the behavior of the linear model. Thus, by comparison to previous results, there is no significant difference in the behavior of digital implementations of the LMS algorithm whether roundoff occurs before or after multiplying the error by the data. © 1990 IEEE