IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 6, JUNE 2002 1327 Stochastic Analysis of the Filtered-X LMS Algorithm in Systems With Nonlinear Secondary Paths Márcio H. Costa, José Carlos M. Bermudez, Member, IEEE, and Neil J. Bershad, Fellow, IEEE Abstract—This paper presents a statistical analysis of the fil- tered-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 in- troduced by the power amplifiers and transducers. Deterministic nonlinear recursions are derived for Gaussian inputs for the tran- sient mean weight, mean square error, and cross-covariance matrix of the adaptive weight vector at different times. The cross-covari- ance results provide improved steady-state predictions (as com- pared with previous results) for moderate to large step sizes. Monte Carlo simulations show excellent agreement with the behavior pre- dicted by the theoretical models. The analytical and simulation re- sults show that a small nonlinearity can have a significant impact on the 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 identifi- cation and modeling, noise and interference cancelling, equalization, signal detection, and prediction [1], [2]. Most adaptive system analyses neglect nonlinear effects and model the unknown systems as linear with memory. In many im- portant practical circumstances, a linear model simplifies the mathematics and permits detailed system analysis. More sophisticated models must be used when nonlinear effects significantly impact actual system behavior [3]–[5]. Important nonlinear effects occur in active noise control (ANC) and active vibration control (AVC) systems, for example, [4]. ANC and AVC systems include acoustical/mechanical paths. Signal converters (A/D and D/A), power amplifiers, and transducers (speakers or actuators) can nonlinearly transform digital elec- trical signals into analog electrical or mechanical signals. This nonlinear effect is caused by overdriving the electronics or the Manuscript received August 13, 2001; revised February 26, 2002. This work was supported in part by the Brazilian Ministry of Education (CAPES) under Grant PICDT 0129/97-9 and by the Brazilian Ministry of Science and Tech- nology (CNPq) under Grant 352084/92-8. The associate editor coordinating the review of this paper and approving it for publication was Dr. Naofal Al-Dhahir. M. H. Costa is with the Grupo de Engenharia Biomédica, Escola de Engen- haria e Arquitetura, Universidade Católica de Pelotas, Pelotas, Brazil (e-mail: costa@atlas.ucpel.tche.br). J. C. M. Bermudez is with the Department of Electrical Engineering, Federal University of Santa Catarina, Florianópolis, Brazil (e-mail: bermudez@fast- lane.com.br). N. J. Bershad is with the Department of Electrical and Computer Engi- neering, University of California Irvine, Irvine, CA 96032 USA (e-mail: bershad@ece.uci.edu). Publisher Item Identifier S 1053-587X(02)04383-0. speakers/transducers in the secondary path 1 [6], [7]. Thus, the nonlinearities should be included in the mathematical model for accurate analysis. Bernhard et al. [8] briefly discussed such nonlinear effects but presented no analysis. Costa et al. [9], [10] recently studied the statistical behavior of the LMS algorithm for a memoryless nonlinear secondary path. Small nonlinearities were shown to significantly affect algorithm performance. These analytical re- sults provide important insights of the effect of a nonlinear sec- ondary path upon ANC and AVC system behavior. However, they do not provide information about secondary path impulse response effects. This paper provides a stochastic analysis of the FXLMS al- gorithm with a saturation nonlinearity as shown in Fig. 1. The function is a zero-memory saturation nonlinearity. 2 is the secondary path linear filter and is its estimate. Usually, is designed to duplicate . The mismatched case is analyzed here since perfect estimation cannot be achieved in practice. A degree of nonlinearity is defined that measures the impact of the nonlinearity on the achievable mean square error (MSE). De- terministic nonlinear recursions are derived for Gaussian inputs and slow adaptation for the transient mean weight, mean square error, and cross-covariance matrix of the adaptive weight vector at different times. The analytical and simulation results show that even a small nonlinearity can have a significant impact on the adaptive filter behavior. Contrary to the case studied in [10], the converged mean weight vector is not a scaled version of the primary path response. The steady-state solution depends on the nonlinearity , the secondary path impulse response , and the secondary path impulse response estimate . The cross-co- variance results provide improved steady-state predictions (as compared with previous results) for moderate to large step sizes. The models presented here generalize the linear case analyses in [11]–[13]. The models and the results for the linear case cor- respond to a degree of nonlinearity equal to zero. A wide variety of Monte Carlo simulations show excellent agreement with the theoretical predictions. II. ANALYSIS—FXLMS ALGORITHM TRANSIENT BEHAVIOR A. Problem Definition Fig. 1 shows a block diagram for the FXLMS algorithm [1] with a nonlinearity at the output of the adaptive filter. The 1 Secondary path is the usual term for the path leading from the adaptive filter output to the cancellation point [1]. 2 The modeling of nonlinear effects in amplifiers and tranducers is very com- plex, and there is no unique model for all situations [6]–[8]. Static nonlinearities have been used to model nonlinear effects in electronics and in transducers [3], [5]. 1053-587X/02$17.00 © 2002 IEEE