Steady-State Performance Analyses of Adaptive Filters

Abstract

Adaptive filters have become a vital part of many modern communication and control systems, which can be used in system identification, adaptive equalization, echo cancellation, beamforming, and so on [l]. The least mean squares (LMS) algorithm, which is the most popular adaptive filtering algorithm, has enjoyed enormous popularity due to its simplicity and robustness [2] [3]. Over the years several variants of LMS have been proposed to overcome some limitations of LMS algorithm by modifying the error estimation function from linearity to nonlinearity. Sign-error LMS algorithm is presented by its computational simplicity [4], least-mean fourth (LMF) algorithm is proposed for applications in which the plant noise has a probability density function with short tail [5], and the LMMN algorithm achieves a better steady state performance than the LMS algorithm and better stability properties than the LMF algorithm by adjusting its mixing parameter [6], [7]. The performance of an adaptive filter is generally measured in terms of its transient behavior and its steady-state behavior. There have been numerous works in the literature on the performance of adaptive filters with many creationary results and approaches [3]-[20]. In most of these literatures, the steady-state performance is often obtained as a limiting case of the transient behavior [13]-[16]. However, most adaptive filters are inherently nonlinear and time-variant systems. The nonlinearities in the update equations tend to lead to difficulties in the study of their steady-state performance as a limiting case of their transient performance [12]. In addition, transient analyses tend to require some more simplifying assumptions, which at times can be restrictive. Using the energy conservation relation during two successive iteration update , N. R. Yousef and A. H. Sayed re-derived the steady-state performance for a large class of adaptive filters [11],[12], such as sign-error LMS algorithm, LMS algorithm, LMMN algorithm, and so on, which bypassed the difficulties encountered in obtaining steady-state results as the limiting case of a transient analysis. However, it is generally observed that most works for analyzing the steady-state performance study individual algorithms separately. This is because different adaptive schemes have different nonlinear update equations, and the particularities of each case tend to require different arguments and assumptions. Some authors try to investigate the steadystate performance from a general view to fit more adaptive filtering algorithms, although that is a challenge task. Based on Taylor series expansion (TSE), S. C. Douglas and T. H. Meng obtained a general expression for the steady-state MSE for adaptive filters with error