Adaptive Filters

Behrouz Farhang-Boroujeny, University of Utah, USA

Preface xvii Acknowledgments xxi 1 Introduction 1 1.1 Linear Filters 1 1.2 Adaptive Filters 2 1.3 Adaptive Filter Structures 3 1.4 Adaptation Approaches 7 1.5 Real and Complex Forms of Adaptive Filters 9 1.6 Applications 9 2 Discrete-Time Signals and Systems 28 2.1 Sequences and z-Transform 28 2.2 Parseval s Relation 32 2.3 System Function 33 2.4 Stochastic Processes 35 Problems 44 3 Wiener Filters 48 3.1 Mean-Squared Error Criterion 48 3.2 Wiener Filter Transversal, Real-Valued Case 50 3.3 Principle of Orthogonality 55 3.4 Normalized Performance Function 57 3.5 Extension to Complex-Valued Case 58 3.6 Unconstrained Wiener Filters 61 3.7 Summary and Discussion 79 Problems 80 4 Eigenanalysis and Performance Surface 90 4.1 Eigenvalues and Eigenvectors 90 4.2 Properties of Eigenvalues and Eigenvectors 91 4.3 Performance Surface 104 Problems 112 5 Search Methods 119 5.1 Method of Steepest Descent 120 5.2 Learning Curve 126 5.3 Effect of Eigenvalue Spread 130 5.4 Newton s Method 131 5.5 An Alternative Interpretation of Newton s Algorithm 133 Problems 135 6 LMS Algorithm 139 6.1 Derivation of LMS Algorithm 139 6.2 Average Tap-Weight Behavior of the LMS Algorithm 141 6.3 MSE Behavior of the LMS Algorithm 144 6.4 Computer Simulations 156 6.5 Simplified LMS Algorithms 167 6.6 Normalized LMS Algorithm 170 6.7 Affine Projection LMS Algorithm 173 6.8 Variable Step-Size LMS Algorithm 177 6.9 LMS Algorithm for Complex-Valued Signals 179 6.10 Beamforming (Revisited) 182 6.11 Linearly Constrained LMS Algorithm 186 Problems 190 Appendix 6A: Derivation of Eq. (6.39) 205 7 Transform Domain Adaptive Filters 207 7.1 Overview of Transform Domain Adaptive Filters 208 7.2 Band-Partitioning Property of Orthogonal Transforms 210 7.3 Orthogonalization Property of Orthogonal Transforms 211 7.4 Transform Domain LMS Algorithm 213 7.5 Ideal LMS-Newton Algorithm and Its Relationship with TDLMS 215 7.6 Selection of the Transform T 216 7.7 Transforms 229 7.8 Sliding Transforms 230 7.9 Summary and Discussion 242 Problems 243 8 Block Implementation of Adaptive Filters 251 8.1 Block LMS Algorithm 252 8.2 Mathematical Background 255 8.3 The FBLMS Algorithm 260 8.4 The Partitioned FBLMS Algorithm 267 8.5 Computer Simulations 276 Problems 279 Appendix 8A: Derivation of a Misadjustment Equation for the BLMS Algorithm 285 Appendix 8B: Derivation of Misadjustment Equations for the FBLMS Algorithms 288 9 Subband Adaptive Filters 294 9.1 DFT Filter Banks 295 9.2 Complementary Filter Banks 299 9.3 Subband Adaptive Filter Structures 303 9.4 Selection of Analysis and Synthesis Filters 304 9.5 Computational Complexity 307 9.6 Decimation Factor and Aliasing 308 9.7 Low-Delay Analysis and Synthesis Filter Banks 310 9.8 A Design Procedure for Subband Adaptive Filters 313 9.9 An Example 316 9.10 Comparison with FBLMS Algorithm 318 Problems 319 10 IIR Adaptive Filters 322 10.1 Output Error Method 323 10.2 Equation Error Method 327 10.3 Case Study I: IIR Adaptive Line Enhancement 332 10.4 Case Study II: Equalizer Design for Magnetic Recording Channels 343 10.5 Concluding Remarks 349 Problems 352 11 Lattice Filters 355 11.1 Forward Linear Prediction 355 11.2 Backward Linear Prediction 357 11.3 Relationship Between Forward and Backward Predictors 359 11.4 Prediction-Error Filters 359 11.5 Properties of Prediction Errors 360 11.6 Derivation of Lattice Structure 362 11.7 Lattice as an Orthogonalization Transform 367 11.8 Lattice Joint Process Estimator 369 11.9 System Functions 370 11.10 Conversions 370 11.11 All-Pole Lattice Structure 376 11.12 Pole-Zero Lattice Structure 376 11.13 Adaptive Lattice Filter 378 11.14 Autoregressive Modeling of Random Processes 383 11.15 Adaptive Algorithms Based on Autoregressive Modeling 385 Problems 400 Appendix 11A: Evaluation of E[ua(n)xT(n)K(n)x(n)uTa (n)] 407 Appendix 11B: Evaluation of the parameter 408 12 Method of Least-Squares 410 12.1 Formulation of Least-Squares Estimation for a Linear Combiner 411 12.2 Principle of Orthogonality 412 12.3 Projection Operator 415 12.4 Standard Recursive Least-Squares Algorithm 416 12.5 Convergence Behavior of the RLS Algorithm 421 Problems 430 13 Fast RLS Algorithms 433 13.1 Least-Squares Forward Prediction 434 13.2 Least-Squares Backward Prediction 435 13.3 Least-Squares Lattice 437 13.4 RLSL Algorithm 440 13.5 FTRLS Algorithm 453 Problems 460 14 Tracking 463 14.1 Formulation of the Tracking Problem 463 14.2 Generalized Formulation of LMS Algorithm 464 14.3 MSE Analysis of the Generalized LMS Algorithm 465 14.4 Optimum Step-Size Parameters 469 14.5 Comparisons of Conventional Algorithms 471 14.6 Comparisons Based on Optimum Step-Size Parameters 475 14.7 VSLMS: An Algorithm with Optimum Tracking Behavior 477 14.8 RLS Algorithm with Variable Forgetting Factor 485 14.9 Summary 486 Problems 488 15 Echo Cancellation 492 15.1 The Problem Statement 492 15.2 Structures and Adaptive Algorithms 495 15.3 Double-Talk Detection 512 15.4 Howling Suppression 521 15.5 Stereophonic Acoustic Echo Cancellation 524 Appendix 15A: Multitaper method 542 Appendix 15B: Derivation of the Two-Channel Levinson Durbin Algorithm 549 16 Active Noise Control 551 16.1 Broadband Feedforward Single-Channel ANC 553 16.2 Narrowband Feedforward Single-Channel ANC 559 16.3 Feedback Single-Channel ANC 573 16.4 Multichannel ANC Systems 577 Appendix 16A: Derivation of Eq. (16.46) 582 Appendix 16B: Derivation of Eq. (16.53) 583 17 Synchronization and Equalization in Data Transmission Systems 584 17.1 Continuous Time Channel Model 585 17.2 Discrete Time Channel Model and Equalizer Structures 589 17.3 Timing Recovery 593 17.4 Equalizers Design and Performance Analysis 606 17.5 Adaptation Algorithms 617 17.6 Cyclic Equalization 618 17.7 Joint Timing Recovery, Carrier Recovery, and Channel Equalization 628 17.8 Maximum Likelihood Detection 629 17.9 Soft Equalization 631 17.10 Single-Input Multiple-Output Equalization 643 17.11 Frequency Domain Equalization 645 17.12 Blind Equalization 649 Problems 654 18 Sensor Array Processing 659 18.1 Narrowband Sensor Arrays 660 18.2 Broadband Sensor Arrays 678 18.3 Robust Beamforming 683 Problems 692 19 Code Division Multiple Access Systems 695 19.1 CDMA Signal Model 695 19.2 Linear Detectors 699 19.3 Adaptation Methods 707 Problems 709 20 OFDM and MIMO Communications 711 20.1 OFDM Communication Systems 711 20.2 MIMO Communication Systems 730 20.3 MIMO OFDM 743 Problems 743 References 746 Index 761