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Digital Signal Processing Handbook P19

Characterizing the Performance of Adaptive Filters 19.3 Analytical Models, Assumptions, and Definitions System Identification Model for the Desired Response Signal • Statistical Models for the Input Signal • The Independence Assumptions • Useful Definitions 19.4 Analysis of the LMS Adaptive Filter Mean Analysis • Mean-Square Analysis 19.5 Performance Issues Basic Criteria for Performance • Identifying Stationary Systems • Tracking Time-Varying Systems Normalized Step Sizes • Adaptive and Matrix Step Sizes • Other Time-Varying Step Size Methods 19.6 Selecting Time-Varying Step Sizes Scott C. Douglas University of Utah Markus Rupp Bell Laboratories Lucent Technologies 19.7 Other Analyses of the LMS Adaptive Filter 19.8 Analysis of Other Adaptive Filters 19.9. | Scott C. Douglas et. Al. Convergence Issues in the LMS Adaptive Filter. 2000 CRC Press LLC. http www.engnetbase.com . Convergence Issues in the LMS Adaptive Filter Scott C. Douglas University of Utah Markus Rupp Bell Laboratories Lucent Technologies 19.1 Introduction 19.2 Characterizing the Performance of Adaptive Filters 19.3 Analytical Models Assumptions and Definitions System Identification Model for the Desired Response Signal Statistical Models for the Input Signal The Independence Assumptions Useful Definitions 19.4 Analysis of the LMS Adaptive Filter Mean Analysis Mean-Square Analysis 19.5 Performance Issues Basic Criteria for Performance Identifying Stationary Systems Tracking Time-Varying Systems 19.6 Selecting Time-Varying Step Sizes Normalized Step Sizes Adaptive and Matrix Step Sizes Other Time-Varying Step Size Methods 19.7 Other Analyses of the LMS Adaptive Filter 19.8 Analysis of Other Adaptive Filters 19.9 Conclusions References 19.1 Introduction In adaptive filtering the least-mean-square LMS adaptive filter 1 is the most popular and widely used adaptive system appearing in numerous commercial and scientific applications. The LMS adaptive filter is described by the equations W n C 1 W n C j n e n X n e n d n WT n X n 19.1 19.2 where W n wo n wi n WL i n T is the coefficient vector X n x n x n 1 x n L C 1 T is the input signal vector d n is the desired signal e n is the error signal and p. n is the step size. There are three main reasons why the LMS adaptive filter is so popular. First it is relatively easy to implement in software and hardware due to its computational simplicity and efficient use of memory. Second it performs robustly in the presence of numerical errors caused by finite-precision arithmetic. Third its behavior has been analytically characterized to the point where a user can easily set up the system to obtain adequate performance with only limited knowledge about the input and desired response signals. 1999 by CRC Press LLC Our .