Adaptive Filters
Theory and Applications
2. Edition May 2013
800 Pages, Hardcover
Wiley & Sons Ltd
ISBN:
978-1-119-97954-8
John Wiley & Sons
Short Description
This new edition of Adaptive Filters has been updated throughout to reflect the latest developments in this field, illustrating the much broader range of adaptive filters applications developed in recent years and clearly showing how the theory is modified for various applications. The book offers a thorough treatment of the theory of adaptive signal processing as well as in-depth study of applications such as OFDM, MIMO and smart antennas. Exercises and computer simulation problems are accompanied by MATLAB software on a related website. A highly useful reference for engineers, practitioners, and researchers.
This second edition of Adaptive Filters: Theory and Applications has been updated throughout to reflect the latest developments in this field; notably an increased coverage given to the practical applications of the theory to illustrate the much broader range of adaptive filters applications developed in recent years. The book offers an easy to understand approach to the theory and application of adaptive filters by clearly illustrating how the theory explained in the early chapters of the book is modified for the various applications discussed in detail in later chapters. This integrated approach makes the book a valuable resource for graduate students; and the inclusion of more advanced applications including antenna arrays and wireless communications makes it a suitable technical reference for engineers, practitioners and researchers.
Key features:
* Offers a thorough treatment of the theory of adaptive signal processing; incorporating new material on transform domain, frequency domain, subband adaptive filters, acoustic echo cancellation and active noise control.
* Provides an in-depth study of applications which now includes extensive coverage of OFDM, MIMO and smart antennas.
* Contains exercises and computer simulation problems at the end of each chapter.
* Includes a new companion website hosting MATLAB(r) simulation programs which complement the theoretical analyses, enabling the reader to gain an in-depth understanding of the behaviours and properties of the various adaptive algorithms.
1 Introduction 1
1.1 Linear Filters 1
1.2 Adaptive Filters 2
1.3 Adaptive Filter Structures 3
1.4 Adaptation Approaches 7
1.4.1 Approach Based on Wiener Filter Theory 7
1.4.2 Method of Least-Squares 8
1.5 Real and Complex Forms of Adaptive Filters 9
1.6 Applications 9
1.6.1 Modeling 10
1.6.2 InverseModeling 11
1.6.3 Linear Prediction 15
1.6.4 Interference Cancellation 20
2 Discrete-Time Signals and Systems 29
2.1 Sequences and z-Transform 29
2.2 Parseval's Relation 33
2.3 System Function 34
2.4 Stochastic Processes 36
2.4.1 Stochastic Averages 36
2.4.2 z-Transform Representations 38
2.4.3 The power spectral density 39
2.4.4 Response of Linear Systems to Stochastic Processes 41
2.4.5 Ergodicity and Time Averages 44
3 Wiener Filters 49
3.1 Mean-Squared Error Criterion 49
3.2 Wiener Filter - Transversal, Real-valued Case 51
3.3 Principle of Orthogonality 57
3.4 Normalized Performance Function 59
3.5 Extension to Complex-Valued Case 59
3.6 UnconstrainedWiener Filters 62
3.6.1 Performance Function 62
3.6.2 Optimum Transfer Function 65
3.6.3 Modeling 67
3.6.4 InverseModeling 70
3.6.5 Noise Cancellation 74
3.7 Summary and Discussion 80
4 Eigenanalysis and Performance Surface 91
4.1 Eigenvalues and Eigenvectors 91
4.2 Properties of Eigenvalues and Eigenvectors 92
4.3 Performance Surface 104
5 Search Methods 121
5.1 Method of Steepest-Descent 122
5.2 Learning Curve 128
5.3 Effect of Eigenvalue-Spread 132
5.4 Newton's Method 133
5.5 An Alternative Interpretation of Newton's Algorithm 136
6 LMS Algorithm 141
6.1 Derivation of LMS Algorithm 141
6.2 Average Tap-Weight Behavior of the LMS Algorithm 143
6.3 MSE Behavior of the LMS Algorithm 146
6.3.1 Learning Curve 148
6.3.2 Weight-Error Correlation Matrix 150
6.3.3 Excess MSE and Misadjustment 153
6.3.4 Stability 155
6.3.5 The Effect of Initial Values of TapWeights on the Transient Behavior of the LMS Algorithm 156
6.4 Computer Simulations 157
6.4.1 System Modeling 158
6.4.2 Channel Equalization 160
6.4.3 Adaptive Line Enhancement 164
6.4.4 Beamforming 166
6.5 Simplified LMS Algorithms 168
6.6 Normalized LMS Algorithm 172
6.7 Affine Projection LMS Algorithm 175
6.8 Variable Step-Size LMS Algorithm 178
6.9 LMS Algorithm for Complex-Valued Signals 181
6.10 Beamforming (Revisited) 183
6.11 Linearly Constrained LMS Algorithm 186
6.11.1 Statement of the Problem and Its Optimal Solution 187
6.11.2 Update Equations 188
6.11.3 Extension to the Complex-Valued Case 189
7 Transform Domain Adaptive Filters 209
7.1 Overview of Transform Domain Adaptive Filters 209
7.2 Band-Partitioning Property of Orthogonal Transforms 212
7.3 Orthogonalization Property of Orthogonal Transforms 212
7.4 Transform Domain LMS Algorithm 215
7.5 Ideal LMS-Newton Algorithm and Its Relationship with TDLMS 217
7.6 Selection of the transform T 217
7.6.1 A Geometrical Interpretation 218
7.6.2 A Useful Performance Index 222
7.6.3 Improvement Factor and Comparisons 223
7.6.4 Filtering View 226
7.7 Transforms 230
7.8 Sliding Transforms 232
7.8.1 Frequency Sampling Filters 232
7.8.2 Recursive Realization of Sliding Transforms 233
7.8.3 Non-recursive Realization of Sliding Transforms 237
7.8.4 Comparison of Recursive and Non-recursive Sliding Transforms 240
7.9 Summary and Discussion 245
8 Block Implementation of Adaptive Filters 255
8.1 Block LMS Algorithm 256
8.2 Mathematical Background 259
8.2.1 Linear Convolution Using the Discrete Fourier Transform 259
8.2.2 Circular Matrices 260
8.2.3 WindowMatrices andMatrix Formulation of the Overlap-SaveMethod 263
8.3 The FBLMS Algorithm 264
8.3.1 Constrained and Unconstrained FBLMS Algorithms 265
8.3.2 Convergence Behavior of the FBLMS Algorithm 267
8.3.3 Step-Normalization 268
8.3.4 Summary of the FBLMS Algorithm 269
8.3.5 FBLMS Misadjustment Equations 271
8.3.6 Selection of the Block Length 271
8.4 The Partitioned FBLMS Algorithm 272
8.4.1 Analysis of the PFBLMS Algorithm 273
8.4.2 PFBLMS Algorithm withM > L 276
8.4.3 PFBLMS Misadjustment Equations 279
8.4.4 Computational Complexity and Memory Requirement 279
8.4.5 Modified Constrained PFBLMS Algorithm 280
8.5 Computer Simulations 281
9 Subband Adaptive Filters 299
9.1 DFT Filter Banks 300
9.1.1 Weighted Overlap-Add Method for Realization of DFT Analysis Filter Banks 301
9.1.2 Weighted Overlap-Add Method for Realization of DFT Synthesis Filter Banks 302
9.2 Complementary Filter Banks 304
9.3 Subband Adaptive Filter Structures 308
9.4 Selection of Analysis and Synthesis Filters 311
9.5 Computational Complexity 313
9.6 Decimation Factor and Aliasing 314
9.7 Low-Delay Analysis and Synthesis Filter Banks 316
9.7.1 Design Method 316
9.7.2 Filters Properties 318
9.8 A Design Procedure for Subband Adaptive Filters 319
9.9 An Example 322
9.10 Comparison with FBLMS Algorithm 323
10 IIR Adaptive Filters 329
10.1 Output Error Method 330
10.2 Equation Error Method 336
10.3 Case Study I: IIR Adaptive Line Enhancement 339
10.3.1 IIR ALE Filter,W(z) 340
10.3.2 Performance Functions 340
10.3.3 Simultaneous Adaptation of s and w 344
10.3.4 Robust Adaptation of w 344
10.3.5 Simulation Results 346
10.4 Case Study II: Equalizer Design for Magnetic Recording Channels 349
10.4.1 Channel Discretization 351
10.4.2 Design Steps 352
10.4.3 FIR Equalizer Design 352
10.4.4 Conversion from FIR to IIR Equalizer 355
10.4.5 Conversion from z-Domain to s-Domain 355
10.4.6 Numerical Results 356
10.5 Concluding Remarks 358
11 Lattice Filters 363
11.1 Forward Linear Prediction 363
11.2 Backward Linear Prediction 365
11.3 Relationship Between Forward and Backward Predictors 366
11.4 Prediction-Error Filters 367
11.5 Properties of Prediction Errors 367
11.6 Derivation of Lattice Structure 370
11.7 Lattice as an Orthogonalization Transform 375
11.8 Lattice Joint Process Estimator 377
11.9 System Functions 377
11.10Conversions 378
11.10.1 Conversion Between Lattice and Transversal Predictors 379
11.10.2 Levinson-Durbin Algorithm 380
11.10.3 Extension of Levinson-Durbin Algorithm 382
11.11All-Pole Lattice Structure 383
11.12Pole-Zero Lattice Structure 385
11.13Adaptive Lattice Filter 385
11.13.1Discussion and Simulations 387
11.14AutoregressiveModeling of Random Processes 391
11.15Adaptive Algorithms Based on AutoregressiveModeling 392
11.15.1Algorithms 393
11.15.2 Performance Analysis 398
11.15.3 Simulation Results and Discussion 402
12 Method of Least-Squares 419
12.1 Formulation of Least-Squares Estimation for a Linear Combiner 420
12.2 Principle of Orthogonality 421
12.3 Projection Operator 423
12.4 Standard Recursive Least-Squares Algorithm 424
12.4.1 RLS Recursions 424
12.4.2 Initialization of the RLS Algorithm 427
12.4.3 Summary of the Standard RLS Algorithm 427
12.5 Convergence Behavior of the RLS Algorithm 430
12.5.1 Average Tap-Weight Behavior of the RLS Algorithm 430
12.5.2 Weight-Error Correlation Matrix 431
12.5.3 Learning Curve 432
12.5.4 Excess MSE and Misadjustment 435
12.5.5 Initial Transient Behavior of the RLS Algorithm 435
13 Fast RLS Algorithms 443
13.1 Least-Squares Forward Prediction 443
13.2 Least-Squares Backward Prediction 445
13.3 Least-Squares Lattice 447
13.4 RLSL Algorithm 450
13.4.1 Notations and Preliminaries 450
13.4.2 Update Recursion for the Least-Squares Error Sums 453
13.4.3 Conversion Factor 454
13.4.4 Update Equation for Conversion Factor 455
13.4.5 Update Equation for Crosscorrelations 457
13.4.6 RLSL Algorithm Using A Posteriori Errors 459
13.4.7 RLSL algorithm with Error Feedback 461
13.5 FTRLS Algorithm 463
13.5.1 Derivation of the FTRLS Algorithm 464
13.5.2 Summary of the FTRLS Algorithm 467
13.5.3 Stabilized FTRLS Algorithm 467
14 Tracking 473
14.1 Formulation of the Tracking Problem 473
14.2 Generalized Formulation of LMS Algorithm 474
14.3 MSE Analysis of the Generalized LMS Algorithm 475
14.4 Optimum Step-Size Parameters 479
14.5 Comparisons of Conventional Algorithms 481
14.6 Comparisons Based on Optimum Step-Size Parameters 484
14.7 VSLMS: An algorithm with Optimum Tracking Behavior 486
14.7.1 Derivation of VSLMS Algorithm 487
14.7.2 Variations and Extensions 488
14.7.3 Normalization of the Parameter rho 489
14.7.4 Computer Simulations 490
14.8 RLS Algorithm with Variable Forgetting Factor 494
14.9 Summary 497
15 Echo Cancellation 501
15.1 The Problem Statement 501
15.2 Structures and Adaptive Algorithms 504
15.2.1 Normalized LMS (NLMS) Algorithm 505
15.2.2 Affine Projection LMS (APLMS) Algorithm 508
15.2.3 Frequency Domain Block LMS Algorithm 509
15.2.4 Subband LMS Algorithm 511
15.2.5 LMS-Newton Algorithm 513
15.2.6 Numerical Results 514
15.3 Double-Talk Detection 522
15.3.1 Cohenerence Function 522
15.3.2 Double-Talk Detection Using the Coherence Function 523
15.3.3 Numerical Evaluation of the Coherence Function 523
15.3.4 Power Based Double-Talk Detectors 526
15.3.5 Numerical Results 528
15.4 Howling Suppression 530
15.4.1 Howling Suppression Through Notch Filtering 530
15.4.2 Howling Suppression by Spectral Shift 531
15.5 Stereophonic Acoustic Echo Cancellation 534
15.5.1 The Fundamental Problem 535
15.5.2 Reducing Coherence Between x1(n) and x2(n) 538
15.5.3 The LMS-Newton Algorithm for Stereophonic Systems 542
16 Active Noise Control 561
16.1 Broadband Feedforward Single-Channel ANC 563
16.1.1 System block diagram in the absence of the secondary path S1(z) 563
16.1.2 Filtered-X LMS algorithm 564
16.1.3 Convergence analysis 565
16.1.4 Adding the secondary path S1(z) 567
16.2 Narrowband Feedforward Single-Channel ANC 569
16.2.1 Waveform synthesis method 571
16.2.2 Adaptive notch filters 579
16.3 Feedback Single-Channel ANC 583
16.4 Multi-Channel ANC Systems 587
16.4.1 MIMO blocks/transfer functions 589
16.4.2 Derivation of the LMS algorithm for MIMO adaptive filters 590
17 Synchronization and Equalization in Data Transmission Systems 595
17.1 Continuous Time Channel Model 596
17.2 Discrete Time Channel Model and Equalizer Structures 601
17.2.1 Symbol-spaced equalizer 601
17.2.2 Fractionally-spaced equalizer 602
17.2.3 Decision feedback equalizer 604
17.3 Timing Recovery 604
17.3.1 Cost function 605
17.3.2 The optimum timing phase 607
17.3.3 Improving the cost function 610
17.3.4 Algorithms 611
17.3.5 Early-late gate timing recovery 611
17.3.6 Gradient-based algorithm 616
17.4 Equalizers Design and Performance Analysis 617
17.4.1 Wiener-Hopf equation for symbol-spaced equalizers 618
17.4.2 Numerical examples 624
17.5 Adaptation Algorithms 629
17.6 Cyclic Equalization 629
17.6.1 Symbol-spaced cyclic equalizer 630
17.6.2 Fractionally-spaced cyclic equalizer 636
17.6.3 Alignment of s(n) and x(n) 637
17.6.4 Carrier and timing phase acquisition and tracking 638
17.7 Joint Timing Recovery, Carrier Recovery
and Channel Equalization 640
17.8 Maximum Likelihood Detection 640
17.9 Soft Equalization 642
17.9.1 Soft MMSE equalizer 644
17.9.2 Statistical soft equalizer 646
17.9.3 Iterative channel estimation and data detection 652
17.10Single-InputMultiple-Output Equalization 653
17.11Frequency Domain Equalization 656
17.11.1 Packet structure 656
17.11.2 Frequency domain equalizer 657
17.11.3 Packet structure for fast tracking 659
17.11.4 Summary 660
17.12Blind Equalization 660
17.12.1 Examples of kurtosis 662
17.12.2 Cost function 662
17.12.3 Blind adaptation algorithm 665
18 Sensor Array Processing 669
18.1 Narrowband Sensor Arrays 670
18.1.1 Array topology and parameters 670
18.1.2 Signal subspace, noise subspace, and spectral factorization 673
18.1.3 Direction of arrival estimation 675
18.1.4 Beamforming methods 679
18.2 Broadband Sensor Arrays 689
18.2.1 Steering 689
18.2.2 Beamforming methods 690
18.3 Robust Beamforming 694
18.3.1 Soft-constraint minimization 697
18.3.2 Diagonal loading method 699
18.3.3 Methods based on sample matrix inversion 700
19 Code Division Multiple Access Systems 705
19.1 CDMA Signal Model 705
19.1.1 Chip-spaced users-synchronous model 706
19.1.2 Chip-spaced users-asynchronous model 708
19.1.3 Fractionally-spaced model 709
19.2 Linear Detectors 709
19.2.1 Conventional detector: the matched filter detector 710
19.2.2 Decorrelator detector 710
19.2.3 Minimum mean-squared error (optimal) detector 711
19.2.4 Minimum output energy (blind) detector 713
19.2.5 Soft detectors 716
19.3 Adaptation Methods 717
19.3.1 Conventional detector 717
19.3.2 Decorrelator detector 717
19.3.3 MMSE detector 717
19.3.4 MOE detector 718
19.3.5 Soft detectors 718
20 OFDM and MIMO Communications 721
20.1 OFDM Communication Systems 721
20.1.1 The principle of OFDM 721
20.1.2 Packet structure 724
20.1.3 Carrier acquisition 726
20.1.4 Timing acquisition 726
20.1.5 Channel estimation and frequency domain equalization 727
20.1.6 Estimation of Rhh and Rnunu 730
20.1.7 Carrier tracking methods 731
20.1.8 Channel tracking methods 740
20.2 MIMO Communication Systems 740
20.2.1 MIMO channel model 742
20.2.2 Transmission techniques for space-diversity gain 742
20.2.3 Transmission techniques and MIMO detectors for spacemultiplexing gain 747
20.2.4 Channel estimation methods 751
20.3 MIMO-OFDM 752
1.1 Linear Filters 1
1.2 Adaptive Filters 2
1.3 Adaptive Filter Structures 3
1.4 Adaptation Approaches 7
1.4.1 Approach Based on Wiener Filter Theory 7
1.4.2 Method of Least-Squares 8
1.5 Real and Complex Forms of Adaptive Filters 9
1.6 Applications 9
1.6.1 Modeling 10
1.6.2 InverseModeling 11
1.6.3 Linear Prediction 15
1.6.4 Interference Cancellation 20
2 Discrete-Time Signals and Systems 29
2.1 Sequences and z-Transform 29
2.2 Parseval's Relation 33
2.3 System Function 34
2.4 Stochastic Processes 36
2.4.1 Stochastic Averages 36
2.4.2 z-Transform Representations 38
2.4.3 The power spectral density 39
2.4.4 Response of Linear Systems to Stochastic Processes 41
2.4.5 Ergodicity and Time Averages 44
3 Wiener Filters 49
3.1 Mean-Squared Error Criterion 49
3.2 Wiener Filter - Transversal, Real-valued Case 51
3.3 Principle of Orthogonality 57
3.4 Normalized Performance Function 59
3.5 Extension to Complex-Valued Case 59
3.6 UnconstrainedWiener Filters 62
3.6.1 Performance Function 62
3.6.2 Optimum Transfer Function 65
3.6.3 Modeling 67
3.6.4 InverseModeling 70
3.6.5 Noise Cancellation 74
3.7 Summary and Discussion 80
4 Eigenanalysis and Performance Surface 91
4.1 Eigenvalues and Eigenvectors 91
4.2 Properties of Eigenvalues and Eigenvectors 92
4.3 Performance Surface 104
5 Search Methods 121
5.1 Method of Steepest-Descent 122
5.2 Learning Curve 128
5.3 Effect of Eigenvalue-Spread 132
5.4 Newton's Method 133
5.5 An Alternative Interpretation of Newton's Algorithm 136
6 LMS Algorithm 141
6.1 Derivation of LMS Algorithm 141
6.2 Average Tap-Weight Behavior of the LMS Algorithm 143
6.3 MSE Behavior of the LMS Algorithm 146
6.3.1 Learning Curve 148
6.3.2 Weight-Error Correlation Matrix 150
6.3.3 Excess MSE and Misadjustment 153
6.3.4 Stability 155
6.3.5 The Effect of Initial Values of TapWeights on the Transient Behavior of the LMS Algorithm 156
6.4 Computer Simulations 157
6.4.1 System Modeling 158
6.4.2 Channel Equalization 160
6.4.3 Adaptive Line Enhancement 164
6.4.4 Beamforming 166
6.5 Simplified LMS Algorithms 168
6.6 Normalized LMS Algorithm 172
6.7 Affine Projection LMS Algorithm 175
6.8 Variable Step-Size LMS Algorithm 178
6.9 LMS Algorithm for Complex-Valued Signals 181
6.10 Beamforming (Revisited) 183
6.11 Linearly Constrained LMS Algorithm 186
6.11.1 Statement of the Problem and Its Optimal Solution 187
6.11.2 Update Equations 188
6.11.3 Extension to the Complex-Valued Case 189
7 Transform Domain Adaptive Filters 209
7.1 Overview of Transform Domain Adaptive Filters 209
7.2 Band-Partitioning Property of Orthogonal Transforms 212
7.3 Orthogonalization Property of Orthogonal Transforms 212
7.4 Transform Domain LMS Algorithm 215
7.5 Ideal LMS-Newton Algorithm and Its Relationship with TDLMS 217
7.6 Selection of the transform T 217
7.6.1 A Geometrical Interpretation 218
7.6.2 A Useful Performance Index 222
7.6.3 Improvement Factor and Comparisons 223
7.6.4 Filtering View 226
7.7 Transforms 230
7.8 Sliding Transforms 232
7.8.1 Frequency Sampling Filters 232
7.8.2 Recursive Realization of Sliding Transforms 233
7.8.3 Non-recursive Realization of Sliding Transforms 237
7.8.4 Comparison of Recursive and Non-recursive Sliding Transforms 240
7.9 Summary and Discussion 245
8 Block Implementation of Adaptive Filters 255
8.1 Block LMS Algorithm 256
8.2 Mathematical Background 259
8.2.1 Linear Convolution Using the Discrete Fourier Transform 259
8.2.2 Circular Matrices 260
8.2.3 WindowMatrices andMatrix Formulation of the Overlap-SaveMethod 263
8.3 The FBLMS Algorithm 264
8.3.1 Constrained and Unconstrained FBLMS Algorithms 265
8.3.2 Convergence Behavior of the FBLMS Algorithm 267
8.3.3 Step-Normalization 268
8.3.4 Summary of the FBLMS Algorithm 269
8.3.5 FBLMS Misadjustment Equations 271
8.3.6 Selection of the Block Length 271
8.4 The Partitioned FBLMS Algorithm 272
8.4.1 Analysis of the PFBLMS Algorithm 273
8.4.2 PFBLMS Algorithm withM > L 276
8.4.3 PFBLMS Misadjustment Equations 279
8.4.4 Computational Complexity and Memory Requirement 279
8.4.5 Modified Constrained PFBLMS Algorithm 280
8.5 Computer Simulations 281
9 Subband Adaptive Filters 299
9.1 DFT Filter Banks 300
9.1.1 Weighted Overlap-Add Method for Realization of DFT Analysis Filter Banks 301
9.1.2 Weighted Overlap-Add Method for Realization of DFT Synthesis Filter Banks 302
9.2 Complementary Filter Banks 304
9.3 Subband Adaptive Filter Structures 308
9.4 Selection of Analysis and Synthesis Filters 311
9.5 Computational Complexity 313
9.6 Decimation Factor and Aliasing 314
9.7 Low-Delay Analysis and Synthesis Filter Banks 316
9.7.1 Design Method 316
9.7.2 Filters Properties 318
9.8 A Design Procedure for Subband Adaptive Filters 319
9.9 An Example 322
9.10 Comparison with FBLMS Algorithm 323
10 IIR Adaptive Filters 329
10.1 Output Error Method 330
10.2 Equation Error Method 336
10.3 Case Study I: IIR Adaptive Line Enhancement 339
10.3.1 IIR ALE Filter,W(z) 340
10.3.2 Performance Functions 340
10.3.3 Simultaneous Adaptation of s and w 344
10.3.4 Robust Adaptation of w 344
10.3.5 Simulation Results 346
10.4 Case Study II: Equalizer Design for Magnetic Recording Channels 349
10.4.1 Channel Discretization 351
10.4.2 Design Steps 352
10.4.3 FIR Equalizer Design 352
10.4.4 Conversion from FIR to IIR Equalizer 355
10.4.5 Conversion from z-Domain to s-Domain 355
10.4.6 Numerical Results 356
10.5 Concluding Remarks 358
11 Lattice Filters 363
11.1 Forward Linear Prediction 363
11.2 Backward Linear Prediction 365
11.3 Relationship Between Forward and Backward Predictors 366
11.4 Prediction-Error Filters 367
11.5 Properties of Prediction Errors 367
11.6 Derivation of Lattice Structure 370
11.7 Lattice as an Orthogonalization Transform 375
11.8 Lattice Joint Process Estimator 377
11.9 System Functions 377
11.10Conversions 378
11.10.1 Conversion Between Lattice and Transversal Predictors 379
11.10.2 Levinson-Durbin Algorithm 380
11.10.3 Extension of Levinson-Durbin Algorithm 382
11.11All-Pole Lattice Structure 383
11.12Pole-Zero Lattice Structure 385
11.13Adaptive Lattice Filter 385
11.13.1Discussion and Simulations 387
11.14AutoregressiveModeling of Random Processes 391
11.15Adaptive Algorithms Based on AutoregressiveModeling 392
11.15.1Algorithms 393
11.15.2 Performance Analysis 398
11.15.3 Simulation Results and Discussion 402
12 Method of Least-Squares 419
12.1 Formulation of Least-Squares Estimation for a Linear Combiner 420
12.2 Principle of Orthogonality 421
12.3 Projection Operator 423
12.4 Standard Recursive Least-Squares Algorithm 424
12.4.1 RLS Recursions 424
12.4.2 Initialization of the RLS Algorithm 427
12.4.3 Summary of the Standard RLS Algorithm 427
12.5 Convergence Behavior of the RLS Algorithm 430
12.5.1 Average Tap-Weight Behavior of the RLS Algorithm 430
12.5.2 Weight-Error Correlation Matrix 431
12.5.3 Learning Curve 432
12.5.4 Excess MSE and Misadjustment 435
12.5.5 Initial Transient Behavior of the RLS Algorithm 435
13 Fast RLS Algorithms 443
13.1 Least-Squares Forward Prediction 443
13.2 Least-Squares Backward Prediction 445
13.3 Least-Squares Lattice 447
13.4 RLSL Algorithm 450
13.4.1 Notations and Preliminaries 450
13.4.2 Update Recursion for the Least-Squares Error Sums 453
13.4.3 Conversion Factor 454
13.4.4 Update Equation for Conversion Factor 455
13.4.5 Update Equation for Crosscorrelations 457
13.4.6 RLSL Algorithm Using A Posteriori Errors 459
13.4.7 RLSL algorithm with Error Feedback 461
13.5 FTRLS Algorithm 463
13.5.1 Derivation of the FTRLS Algorithm 464
13.5.2 Summary of the FTRLS Algorithm 467
13.5.3 Stabilized FTRLS Algorithm 467
14 Tracking 473
14.1 Formulation of the Tracking Problem 473
14.2 Generalized Formulation of LMS Algorithm 474
14.3 MSE Analysis of the Generalized LMS Algorithm 475
14.4 Optimum Step-Size Parameters 479
14.5 Comparisons of Conventional Algorithms 481
14.6 Comparisons Based on Optimum Step-Size Parameters 484
14.7 VSLMS: An algorithm with Optimum Tracking Behavior 486
14.7.1 Derivation of VSLMS Algorithm 487
14.7.2 Variations and Extensions 488
14.7.3 Normalization of the Parameter rho 489
14.7.4 Computer Simulations 490
14.8 RLS Algorithm with Variable Forgetting Factor 494
14.9 Summary 497
15 Echo Cancellation 501
15.1 The Problem Statement 501
15.2 Structures and Adaptive Algorithms 504
15.2.1 Normalized LMS (NLMS) Algorithm 505
15.2.2 Affine Projection LMS (APLMS) Algorithm 508
15.2.3 Frequency Domain Block LMS Algorithm 509
15.2.4 Subband LMS Algorithm 511
15.2.5 LMS-Newton Algorithm 513
15.2.6 Numerical Results 514
15.3 Double-Talk Detection 522
15.3.1 Cohenerence Function 522
15.3.2 Double-Talk Detection Using the Coherence Function 523
15.3.3 Numerical Evaluation of the Coherence Function 523
15.3.4 Power Based Double-Talk Detectors 526
15.3.5 Numerical Results 528
15.4 Howling Suppression 530
15.4.1 Howling Suppression Through Notch Filtering 530
15.4.2 Howling Suppression by Spectral Shift 531
15.5 Stereophonic Acoustic Echo Cancellation 534
15.5.1 The Fundamental Problem 535
15.5.2 Reducing Coherence Between x1(n) and x2(n) 538
15.5.3 The LMS-Newton Algorithm for Stereophonic Systems 542
16 Active Noise Control 561
16.1 Broadband Feedforward Single-Channel ANC 563
16.1.1 System block diagram in the absence of the secondary path S1(z) 563
16.1.2 Filtered-X LMS algorithm 564
16.1.3 Convergence analysis 565
16.1.4 Adding the secondary path S1(z) 567
16.2 Narrowband Feedforward Single-Channel ANC 569
16.2.1 Waveform synthesis method 571
16.2.2 Adaptive notch filters 579
16.3 Feedback Single-Channel ANC 583
16.4 Multi-Channel ANC Systems 587
16.4.1 MIMO blocks/transfer functions 589
16.4.2 Derivation of the LMS algorithm for MIMO adaptive filters 590
17 Synchronization and Equalization in Data Transmission Systems 595
17.1 Continuous Time Channel Model 596
17.2 Discrete Time Channel Model and Equalizer Structures 601
17.2.1 Symbol-spaced equalizer 601
17.2.2 Fractionally-spaced equalizer 602
17.2.3 Decision feedback equalizer 604
17.3 Timing Recovery 604
17.3.1 Cost function 605
17.3.2 The optimum timing phase 607
17.3.3 Improving the cost function 610
17.3.4 Algorithms 611
17.3.5 Early-late gate timing recovery 611
17.3.6 Gradient-based algorithm 616
17.4 Equalizers Design and Performance Analysis 617
17.4.1 Wiener-Hopf equation for symbol-spaced equalizers 618
17.4.2 Numerical examples 624
17.5 Adaptation Algorithms 629
17.6 Cyclic Equalization 629
17.6.1 Symbol-spaced cyclic equalizer 630
17.6.2 Fractionally-spaced cyclic equalizer 636
17.6.3 Alignment of s(n) and x(n) 637
17.6.4 Carrier and timing phase acquisition and tracking 638
17.7 Joint Timing Recovery, Carrier Recovery
and Channel Equalization 640
17.8 Maximum Likelihood Detection 640
17.9 Soft Equalization 642
17.9.1 Soft MMSE equalizer 644
17.9.2 Statistical soft equalizer 646
17.9.3 Iterative channel estimation and data detection 652
17.10Single-InputMultiple-Output Equalization 653
17.11Frequency Domain Equalization 656
17.11.1 Packet structure 656
17.11.2 Frequency domain equalizer 657
17.11.3 Packet structure for fast tracking 659
17.11.4 Summary 660
17.12Blind Equalization 660
17.12.1 Examples of kurtosis 662
17.12.2 Cost function 662
17.12.3 Blind adaptation algorithm 665
18 Sensor Array Processing 669
18.1 Narrowband Sensor Arrays 670
18.1.1 Array topology and parameters 670
18.1.2 Signal subspace, noise subspace, and spectral factorization 673
18.1.3 Direction of arrival estimation 675
18.1.4 Beamforming methods 679
18.2 Broadband Sensor Arrays 689
18.2.1 Steering 689
18.2.2 Beamforming methods 690
18.3 Robust Beamforming 694
18.3.1 Soft-constraint minimization 697
18.3.2 Diagonal loading method 699
18.3.3 Methods based on sample matrix inversion 700
19 Code Division Multiple Access Systems 705
19.1 CDMA Signal Model 705
19.1.1 Chip-spaced users-synchronous model 706
19.1.2 Chip-spaced users-asynchronous model 708
19.1.3 Fractionally-spaced model 709
19.2 Linear Detectors 709
19.2.1 Conventional detector: the matched filter detector 710
19.2.2 Decorrelator detector 710
19.2.3 Minimum mean-squared error (optimal) detector 711
19.2.4 Minimum output energy (blind) detector 713
19.2.5 Soft detectors 716
19.3 Adaptation Methods 717
19.3.1 Conventional detector 717
19.3.2 Decorrelator detector 717
19.3.3 MMSE detector 717
19.3.4 MOE detector 718
19.3.5 Soft detectors 718
20 OFDM and MIMO Communications 721
20.1 OFDM Communication Systems 721
20.1.1 The principle of OFDM 721
20.1.2 Packet structure 724
20.1.3 Carrier acquisition 726
20.1.4 Timing acquisition 726
20.1.5 Channel estimation and frequency domain equalization 727
20.1.6 Estimation of Rhh and Rnunu 730
20.1.7 Carrier tracking methods 731
20.1.8 Channel tracking methods 740
20.2 MIMO Communication Systems 740
20.2.1 MIMO channel model 742
20.2.2 Transmission techniques for space-diversity gain 742
20.2.3 Transmission techniques and MIMO detectors for spacemultiplexing gain 747
20.2.4 Channel estimation methods 751
20.3 MIMO-OFDM 752
