ADAPTIVE FILTERING Edited by Lino García Morales Adaptive Filtering Edited by Lino García Morales Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Petra Zobic Technical Editor Teodora Smiljanic Cover Designer Jan Hyrat Image Copyright Designus, 2010. Used under license from Shutterstock.com First published July, 2011 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Adaptive Filtering, Edited by Lino García Morales p. cm. ISBN 978-953-307-158-9 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface IX Part 1 Fundamentals, Convergence, Performance 1 Chapter 1 Convergence Evaluation of a Random Step-Size NLMS Adaptive Algorithm in System Identification and Channel Equalization 3 Shihab Jimaa Chapter 2 Steady-State Performance Analyses of Adaptive Filters 19 Bin Lin and Rongxi He Chapter 3 The Ultra High Speed LMS Algorithm Implemented on Parallel Architecture Suitable for Multidimensional Adaptive Filtering 47 Marwan Jaber Chapter 4 An LMS Adaptive Filter Using Distributed Arithmetic - Algorithms and Architectures 89 Kyo Takahashi, Naoki Honma and Yoshitaka Tsunekawa Part 2 Complex Structures, Applications and Algorithms 107 Chapter 5 Adaptive Filtering Using Subband Processing: Application to Background Noise Cancellation 109 Ali O. Abid Noor, Salina Abdul Samad and Aini Hussain Chapter 6 Hirschman Optimal Transform (HOT) DFT Block LMS Algorithm 135 Osama Alkhouli, Victor DeBrunner and Joseph Havlicek Chapter 7 Real-Time Noise Cancelling Approach on Innovations-Based Whitening Application to Adaptive FIR RLS in Beamforming Structure 153 Jinsoo Jeong VI Contents Chapter 8 Adaptive Fuzzy Neural Filtering for Decision Feedback Equalization and Multi-Antenna Systems 169 Yao-Jen Chang and Chia-Lu Ho Chapter 9 A Stereo Acoustic Echo Canceller Using Cross-Channel Correlation 195 Shigenobu Minami Chapter 10 EEG-fMRI Fusion: Adaptations of the Kalman Filter for Solving a High-Dimensional Spatio-Temporal Inverse Problem 233 Thomas Deneux Chapter 11 Adaptive-FRESH Filtering 259 Omar A. Yeste Ojeda and Jesús Grajal Chapter 12 Transient Analysis of a Combination of Two Adaptive Filters 297 Tõnu Trump Chapter 13 Adaptive Harmonic IIR Notch Filters for Frequency Estimation and Tracking 313 Li Tan, Jean Jiang and Liangmo Wang Chapter 14 Echo Cancellation for Hands-Free Systems 333 Artur Ferreira and Paulo Marques Chapter 15 Adaptive Heterodyne Filters 359 Michael A. Soderstrand Preface A digital filter is a structure that transforms sequences of numbers to others from its input to its output (signals) and models thus the behavior of a real system. The model or transfer function is a simplified mathematical representation of the system. The structure of the filter consists of a few elements: delays, multipliers, adders and, less often, functions whose magnitude, combination and number determine its characteristics. An adaptive filter, however, is able to self-adjust the parameters of such elements in time (the coefficients of the multipliers for example) according to certain algorithm and thus the relationship between the input and output sequences to adapt itself to the changes of the complex system that represents. This update takes place, usually, by minimizing a cost function in an iterative scheme. Digital adaptive filters are, therefore, very popular in any implementation of signal processing where the system modelled and/or the input signals are time-variants; such as the echo cancellation, active noise control, blind channel equalization, etc., corresponding to problems of system identification, inverse modeling, prediction, interference cancellation, etc. Any design of an adaptive filter focuses its attention on some of its components: structure (transversal, recursive, lattice, systolic array, non-linear, transformed domain, etc.), cost function (mean square error, least squares), coefficient update algorithm (no memory, block, gradient, etc.); to get certain benefits: robustness, speed of convergence, misalignment, tracking capacity, computational complexity, delay, etc. This book is composed of 15 motivating chapters written by researchers and professionals that design, develop and analyze different combinations or variations of the components of the adaptive filter and apply them to different areas of knowledge. The first part of the book is devoted to the adaptive filtering fundamentals and evaluation of their performances while the second part presents structures and complex algorithms in specific applications. This information is very interesting not only for all those who work with technologies based on adaptive filtering but also for teachers and professionals X Preface interested in the digital signal processing in general and in how to deal with the complexity of real systems in particular: non-linear, time-variants, continuous, and unknown. Lino García Morales Audiovisual Engineering and Communication Department Polytechnic University of Madrid Spain [...]... -10 -12 -14 -16 -18 -20 0 200 400 600 800 10 00 12 00 Number of iterations 14 00 16 00 18 00 Fig 10 MSE performance of the NLMS algorithm for various step-sizes (mu) 2000 16 Adaptive Filtering Comparison Performance at mu=0.5 -4 Fixed mu Random mu -6 -8 -10 MSE (dB) -12 -14 -16 -18 -20 -22 -24 0 200 400 600 800 10 00 12 00 Number of Iterations 14 00 16 00 18 00 2000 Fig 11 MSE performances of fixed and random step-size... MSE(dB) -10 -12 -14 -16 -18 -20 0 200 400 600 800 10 00 12 00 Number of Iterations 14 00 16 00 18 00 Fig 12 MSE performances for fixed and random step-size NLMS algorithms 2000 Convergence Evaluation of a Random Step-Size NLMS Adaptive Algorithm in System Identification and Channel Equalization 17 Comparison performance at mu =1. 3 -6 Fixed mu Random mu -8 MSE (dB) -10 -12 -14 -16 -18 0 200 400 600 800 10 00 12 00... Equalization 15 MSE performance of the NLMS for various step-sizes 0 mu=0. 01 mu=0.05 mu=0 .1 mu=0.3 mu=0.5 -5 MSE (dB) -10 -15 -20 -25 -30 -35 0 200 400 600 800 10 00 12 00 Number of iterations 14 00 16 00 18 00 2000 Fig 9 MSE performance of the NLMS algorithm for various step-sizes (mu) MSE performance of the NLMS for various step-sizes -6 mu=0.7 mu=0.9 mu =1. 1 mu =1. 3 mu =1. 5 -8 MSE (dB) -10 -12 -14 -16 -18 -20... [11 ] 2 ξ (n) = E[ ε ( n) ] (11 ) Where E denotes expectation Substituting (10 ) into (11 ) yields: (n)e(n) e(n) 2 − 2 μ E Re u 2 u(n) u(n) 2 ξ (n + 1) − ξ (n) = μ 2 E (12 ) where u (n) is the undisturbed error signal defined by: u (n) = ε H (n) u( n) The bounded range of the normalized step-size parameter can be found from (12 ) as: (13 ) 8 Adaptive Filtering. .. u(n) y(n) Adaptive Filter e(n) Adaptive Algorithm Fig 3 The block diagram of the adaptive channel equalization 10 Adaptive Filtering In a transversal adaptive filter, the input vector Un and the weight vector Wn at the time of nth iteration are defined as follows: U n = [un , un − 1 , , un − M + 1 ]T (22) Wn = [ w0 , w1 , , wM 1 ]T (23) Where un is the filter input and wi, (i=0, 1, …, M -1) is the... follows: H1(z) = 1. 0 + 0.5z -1 ( 31) H2(z) = 0.5 + z -1 (32) The digital signal transmitted through the channel was bipolar BPSK with values of ± 1 and the channel was corrupted by AWGN with SNR = 30dB The order of the filter was 12 Adaptive Filtering set to 12 for both minimum phase, H1(z), and non-minimum phase, H2(z), channels The step size for the NLMS algorithm was chosen from (0. 01) to (0 .1) and the... various step-sizes over CH -1 Convergence Evaluation of a Random Step-Size NLMS Adaptive Algorithm in System Identification and Channel Equalization Fig 6 MSE performances of the two algorithms for SNR =15 dB over CH -1 Fig 7 MSE performances of the two algorithms for SNR =10 dB over CH -1 13 14 Adaptive Filtering Fig 8 MSE performances of the two algorithms for SNR=5 dB over CH -1 4.3.2 Adaptive system identification... limited between 1 and -1 The general equation for a sum of weighted time delayed Telephone channel impulse responses can be written as H(z) = h0+h1z -1+ h2z-2+….hnz-n (19 ) Two types of channels are considered here, the minimum phase (CH -1) and the nonminimum phase (CH-2) channels, which are given, respectively, below: H(z) = 1. 0 + 0.5z -1 (20) H(z) = 0.5 + z -1 ( 21) The discrete time model for the adaptive channel... convergence [5] 3 .1. 1 Adaptive filter A general form of the adaptive filter is shown in Figure 1, where an input signal u(n) produces an output signal y(n), then the output signal y(n) is subtracted from the desired response d(n) to produce an error signal e(n) The input signal u(n) and error signal e(n) are combined together into an adaptive weight-control mechanism The weight controller 6 Adaptive Filtering. .. and control applications [1] Adaptive filters are self learn As the signal into the filter continues, the adaptive filter coefficients adjust themselves to achieve the desired result, such as identifying an unknown filter or cancelling noise in the input signal Figure 1 represents the adaptive filter, comprising the adaptive filter and the adaptive weight control mechanism An adaptive Finite Impulse . yields: 2 (1) () ()() () nn unen un μ εε += − (10 ) To study the stability performance of adaptive filters, the mean-square deviation may be identified [11 ]. 2 () [()]nE n ξε = (11 ) Where. 4 .1 Adaptive channel equalization Adaptive channel equalization in digital communication systems is perhaps the most heavily exploited area of application for adaptive filtering algorithms. Adaptive. Correlation 19 5 Shigenobu Minami Chapter 10 EEG-fMRI Fusion: Adaptations of the Kalman Filter for Solving a High-Dimensional Spatio-Temporal Inverse Problem 233 Thomas Deneux Chapter 11 Adaptive- FRESH