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This paper presents a statistical analysis of the least mean square (LMS) algorithm with a zero-memory scaled error function nonlinearity following the adaptive filter output. This structure models saturation effects in active noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals. The problem is first defined as a nonlinear signal estimation problem and the mean-square error (MSE) performance 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. The LMS algorithm behavior with saturation is analyzed for Gaussian inputs and slow adaptation. Deterministic nonlinear recursions are obtained for the time-varying mean weight and MSE behavior. Simplified results are derived for white inputs and small step sizes. Monte Carlo simulations display excellent agreement with the theoretical predictions, even for relatively large step sizes. The new analytical results accurately predict the effect of saturation on the LMS adaptive filter behavior. Index Terms-Adaptive filters, adaptive signal processing, least mean square methods, transient analysis. I. INTRODUCTION A DAPTIVE algorithms are applicable to system identification and modeling, noise and interference cancelling, equalization, signal detection and prediction [1]-[3]. Most adaptive system analyses assume nonlinear effects can be neglected and model both the unknown system and the adaptive path as linear with memory. Linearity simplifies the mathematical problem and often permits a detailed system analyses in many important practical circumstances. However, more sophisticated models must be used when nonlinear effects are significant to the system behavior (i.e., amplifier saturation).

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