Stochastic analysis of the filtered-X LMS algorithm in systems with nonlinear secondary paths

This paper presents a statistical analysis of the filtered-X LMS algorithm behavior when the secondary path (output of the adaptive filter) includes a nonlinear element. This system is of special interest for active acoustic noise and vibration control, where a saturation nonlinearity models the nonlinear distortion introduced by the power amplifiers and transducers. Deterministic nonlinear recursions are derived for Gaussian inputs for the transient mean weight, mean square error, and cross-covariance matrix of the adaptive weight vector at different times. The cross-covariance results provide improved steady-state predictions (as compared with previous results) for moderate to large step sizes. Monte Carlo simulations show excellent agreement with the behavior predicted by the theoretical models. The analytical and simulation results show that a small nonlinearity can have a significant impact on the adaptive filter behavior