By L. Morales
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Here, we only consider p 2 . Using (40) and Lemma 3, we can obtain the first-order and second-order partial derivatives of f e , e , expressed by p2 1 f e e p 1 e f e, e e p 1 p 2 e 2 p4 (41a) e in real-valued cases, and p p2 1 f e e , e e 2 p 2 p 4 2 . 1 f e e , e e e 2 p p 2 p4 2 f e, e e , e e e 4 (41b) 29 Steady-State Performance Analyses of Adaptive Filters in complex-valued cases, respectively. 1 becomes vp 2 b u 2 p 2 v , a u (44) and the steady-state performance for real LMP algorithms can be written as EMSE TEMSE opt u v2 p 2 a u vp 2 b u v 2 p2 .
The least mean squares (LMS) algorithm, which is the most popular adaptive filtering algorithm, has enjoyed enormous popularity due to its simplicity and robustness  . 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 , least-mean fourth (LMF) algorithm is proposed for applications in which the plant noise has a probability density function with short tail , 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 , .
2 Steady-State Performance Analyses of Adaptive Filters Bin Lin and Rongxi He College of Information Science and Technology, Dalian Maritime University, Dalian, China 1. Introduction 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  .
Adaptive Filtering by L. Morales