Stochastic analysis of the LMS algorithm with a saturation nonlinearity following the adaptive filter output

This paper presents a statistical analysis of the least mean square (LMS) a lgorithm with a zero-memory scaled error function nonlinearity following th e adaptive filter output. This structure models saturation effects in activ e noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals. The problem is first defined as a no nlinear signal estimation problem and the mean-square error (MSE) performan ce surface is studied. Analytical expressions are obtained for the optimum weight vector and the minimum achievable MSE as functions of the saturation . These results are useful for adaptive algorithm design and evaluation. Th e LMS algorithm behavior with saturation is analyzed for Gaussian inputs an d slow adaptation. Deterministic nonlinear recursions are obtained for the time-varying mean weight and MSE behavior. Simplified results are derived f or white inputs and small step sizes. Monte Carlo simulations display excel lent agreement with the theoretical predictions, even for relatively large step sizes, The new analytical results accurately predict the effect of sat uration on the LMS adaptive filter behavior.