This paper studies the statistical behavior of an affine combination of the outputs of two least mean-square (LMS) adaptive filters that simultaneously adapt using the same white Gaussian inputs. The purpose of the combination is to obtain an LMS adaptive filter with fast convergence and small steady-state mean-square deviation (MSD). The linear combination studied is a generalization of the convex combination, in which the combination factor ( ) is restricted to the interval (0,1). The viewpoint is taken that each of the two filters produces dependent estimates of the unknown channel. Thus, there exists a sequence of optimal affine combining coefficients which minimizes the mean-square error (MSE). First, the optimal unrealizable affine combiner is studied and provides the best possible performance for this class. Then two new schemes are proposed for practical applications. The mean-square performances are analyzed and validated by Monte Carlo simulations. With proper design, the two practical schemes yield an overall MSD that is usually less than the MSDs of either filter.

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This paper analyzes the statistical behavior of a sequential gradient search adaptive algorithm for identifying an unknown nonlinear system comprised of a discrete-time linear system H followed by a zero-memory nonlinearity g(.). The LMS algorithm tirst estimates H. The weights are then frozen. Recursions are derived for the mean and fluctuation behavior of LMS which agree with Monte Carlo simulations. When the nonlinearity is modelled by a scaled error function, the second part of the gradient scheme is shown to correctly learn the scale factor and the error function scale factor. Mean recursions for the scale factors show good agreement with Monte Carlo simulations.

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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