Introduction to Adaptive Arrays

Front Matter......Page 1
Preface......Page 3
Table of Contents......Page 5
1. Introduction......Page 11
1.1 Motivation for Using Adaptive Arrays......Page 12
1.2 Historical Perspective......Page 13
1.3 Principal System Elements......Page 14
1.4 Adaptive Array Problem Statement......Page 15
1.5 Existing Technology......Page 17
1.5.1 Radar Technology......Page 18
1.5.1.1 Radiating Elements and Antenna Arrays......Page 19
1.5.1.3 Signal Processing......Page 22
1.5.2.1 Sonar Transducers and Hydrophones......Page 23
1.5.2.2 Sonar Arrays......Page 25
1.5.2.3 Beamformer......Page 28
1.6.2 Part 2......Page 29
1.7 Summary and Conclusions......Page 30
1.8 Problems......Page 31
1.9 References......Page 32
1. Introduction......Page 37
1.1 Motivation for Using Adaptive Arrays......Page 38
1.2 Historical Perspective......Page 39
1.3 Principal System Elements......Page 40
1.4 Adaptive Array Problem Statement......Page 41
1.5 Existing Technology......Page 43
1.5.1 Radar Technology......Page 44
1.5.1.1 Radiating Elements and Antenna Arrays......Page 45
1.5.1.3 Signal Processing......Page 48
1.5.2.1 Sonar Transducers and Hydrophones......Page 49
1.5.2.2 Sonar Arrays......Page 51
1.5.2.3 Beamformer......Page 54
1.6.2 Part 2......Page 55
1.7 Summary and Conclusions......Page 56
1.8 Problems......Page 57
1.9 References......Page 58
2. Adaptive Array Concept......Page 62
2.1.2 Signal Models......Page 63
2.1.3 Ideal Propagation Model......Page 65
2.2.1 Pair of Identical Sensors......Page 66
2.2.2 Linear Arrays......Page 70
2.2.3 Planar Arrays......Page 75
2.2.4 Conformal Arrays......Page 77
2.3.1 Enhanced Signal Reception by Adjustment of the Array Response......Page 78
2.3.2.1 Low Sidelobe Amplitude Tapers......Page 80
2.3.2.2 Thinned Arrays......Page 82
2.3.2.3 Null Synthesis Using the Unit Circle......Page 85
2.3.2.4 Null Synthesis Using Cancellation Beams......Page 87
2.4 Nulling Limitations due to Miscellaneous Array Effects......Page 91
2.5 Narrowband and Broadband Signal Processing Considerations......Page 95
2.6 Adaptive Array Performance Measure - Coverage Improvement Factor \0䌀䤀䘀......Page 99
2.7 Summary and Conclusions......Page 101
2.8 Problems......Page 102
2.9 References......Page 111
3. Optimum Array Processing: Steady-State Performance Limits and the Wiener Solution......Page 114
3.1 Mathematical Preliminaries......Page 115
3.1.1 Problem Formulation in Terms of Real Variables......Page 116
3.1.2 Correlation Matrices for Real Signals......Page 117
3.1.3 Revised Definitions Required for Complex Vector Quantities......Page 119
3.1.4 Correlation Matrices for Complex Signals......Page 120
3.2 Signal Descriptions for Conventional and Signal Aligned Arrays......Page 121
3.3.1 The MSE Performance Measure......Page 123
3.3.2 The SNR Performance Measure......Page 125
3.3.3 The ML Performance Measure......Page 132
3.3.4 The MV Performance Measure......Page 133
3.3.5 Factorization of the Optimum Solutions......Page 135
3.4 Optimum Array Processing for Broadband Applications......Page 136
3.4.1 Estimation of a Random Signal......Page 138
3.4.1.1 Gaussian Random Signal......Page 139
3.4.1.2 Non-Gaussian Random Signal......Page 140
3.4.1.3 Application of Random Signal Estimation Results to Optimum Array Processing......Page 141
3.4.2 Estimation of an Unknown, Nonrandom Signal......Page 142
3.4.2.2 Non-Gaussian Noise Case......Page 143
3.4.2.3 Application of Unknown, Nonrandom Signal Estimation Results to Optimum Array Processing......Page 144
3.4.3.2 Non-Gaussian Noise Case......Page 145
3.4.3.3 Application of Known Signal Detection Results to Optimum Array Processing......Page 147
3.4.4.1 Gaussian Case......Page 148
3.4.4.2 Gaussian Noise, Non-Gaussian Signal......Page 149
3.4.4.3 Application of Random Signal Detection Results to Optimum Array Processing......Page 150
3.4.5.2 Application of Unknown, Nonrandom Signal Detection Results to Optimum Array Processing......Page 151
3.4.6 Array Gain......Page 152
3.4.7 Criterion Invariant Array Processor......Page 153
3.5 Optimum Array Processing for Perturbed Propagation Conditions......Page 154
3.6 Polarization Sensitive Arrays......Page 157
3.7 Summary and Conclusions......Page 163
3.8 Problems......Page 164
3.9 References......Page 179
4. Gradient-Based Algorithms......Page 184
4.1.2 The Method of Steepest Descent......Page 185
4.1.3 Feedback Model of Steepest Descent......Page 187
4.2 The LMS Algorithm......Page 189
4.2.1 Convergence to the Wiener Solution......Page 191
4.2.2 Transient Response Characteristics for LMS Adaptation......Page 194
4.2.3 Weight Misadjustment during LMS Adaptation......Page 196
4.2.4 Practical Considerations for LMS Adaptation......Page 199
4.2.5 One-Mode and Two-Mode LMS Adaptation......Page 200
4.3 The Howells-Applebaum Adaptive Processor......Page 202
4.3.1 Adaptive Weight Equations......Page 204
4.3.2 Loop Noise Considerations......Page 210
4.3.3 Adaptive Array Behavior in Terms of Eigenvector Beams......Page 213
4.3.4.1 Assumptions and Initial Conditions......Page 217
4.3.4.2 Output Noise Power and SNR Degradation......Page 218
4.3.4.3 Eigenvalues and Eigenvectors of the Noise Covariance Matrix......Page 219
4.3.4.4 Performance Characteristics for Various Signal Conditions......Page 220
4.3.5 Hard Limiter Modification for N Adaptive Loops......Page 221
4.4 Introduction of Main Beam Constraints......Page 222
4.4.1 Pilot Signals......Page 223
4.4.2 Preadaption Spatial Filters......Page 225
4.4.3 Control Loop Spatial Filters......Page 227
4.5 Constraint for the Case of Known Desired Signal Power Level......Page 230
4.6 The DSD Algorithm......Page 232
4.6.1 Performance Penalty Incurred by Gradient Measurement......Page 234
4.6.2 Gradient Measurement Noise and Misadjustment in the Weight Vector......Page 235
4.7 The Accelerated Gradient Approach \0䄀䜀......Page 240
4.7.1 Algorithm Based on the Powell Accelerated Gradient Cycle......Page 243
4.8 Gradient Algorithm with Constraints......Page 244
4.8.1 Optimum Constrained Weight Vector Solution......Page 245
4.8.2 The Adaptive Algorithm......Page 249
4.8.3 Conditions Ensuring Convergence to the Optimum Solution......Page 250
4.8.4 A Useful Geometrical Interpretation......Page 252
4.9 Simulation Results......Page 255
4.10 Phase-Only Adaptive Nulling Using Steepest Descent......Page 258
4.11 Summary and Conclusions......Page 259
4.12 Problems......Page 261
4.13 References......Page 266
4. Gradient-Based Algorithms......Page 270
4.1.2 The Method of Steepest Descent......Page 271
4.1.3 Feedback Model of Steepest Descent......Page 273
4.2 The LMS Algorithm......Page 275
4.2.1 Convergence to the Wiener Solution......Page 277
4.2.2 Transient Response Characteristics for LMS Adaptation......Page 280
4.2.3 Weight Misadjustment during LMS Adaptation......Page 282
4.2.4 Practical Considerations for LMS Adaptation......Page 285
4.2.5 One-Mode and Two-Mode LMS Adaptation......Page 286
4.3 The Howells-Applebaum Adaptive Processor......Page 288
4.3.1 Adaptive Weight Equations......Page 290
4.3.2 Loop Noise Considerations......Page 296
4.3.3 Adaptive Array Behavior in Terms of Eigenvector Beams......Page 299
4.3.4.1 Assumptions and Initial Conditions......Page 303
4.3.4.2 Output Noise Power and SNR Degradation......Page 304
4.3.4.3 Eigenvalues and Eigenvectors of the Noise Covariance Matrix......Page 305
4.3.4.4 Performance Characteristics for Various Signal Conditions......Page 306
4.3.5 Hard Limiter Modification for N Adaptive Loops......Page 307
4.4 Introduction of Main Beam Constraints......Page 308
4.4.1 Pilot Signals......Page 309
4.4.2 Preadaption Spatial Filters......Page 311
4.4.3 Control Loop Spatial Filters......Page 313
4.5 Constraint for the Case of Known Desired Signal Power Level......Page 316
4.6 The DSD Algorithm......Page 318
4.6.1 Performance Penalty Incurred by Gradient Measurement......Page 320
4.6.2 Gradient Measurement Noise and Misadjustment in the Weight Vector......Page 321
4.7 The Accelerated Gradient Approach \0䄀䜀......Page 326
4.7.1 Algorithm Based on the Powell Accelerated Gradient Cycle......Page 329
4.8 Gradient Algorithm with Constraints......Page 330
4.8.1 Optimum Constrained Weight Vector Solution......Page 331
4.8.2 The Adaptive Algorithm......Page 335
4.8.3 Conditions Ensuring Convergence to the Optimum Solution......Page 336
4.8.4 A Useful Geometrical Interpretation......Page 338
4.9 Simulation Results......Page 341
4.10 Phase-Only Adaptive Nulling Using Steepest Descent......Page 344
4.11 Summary and Conclusions......Page 345
4.12 Problems......Page 347
4.13 References......Page 352
5.1 The Direct Matrix Inversion \0䐀䴀䤀 Approach......Page 355
5.1.1 Use of the Sample Covariance Matrix......Page 356
5.1.2 Use of the Sample Covariance Matrix and the Sample Cross-Correlation Vector......Page 360
5.2 Diagonal Loading of the Sample Covariance Matrix......Page 367
5.3.1.1 Givens Rotations......Page 369
5.3.1.2 Householder Transformations......Page 371
5.3.2 Cholesky Factorization......Page 372
5.4 Transient Response Comparisons......Page 373
5.6 Summary and Conclusions......Page 378
5.7 Problems......Page 379
5.8 References......Page 386
6.1 The Weighted Least Squares Error Processor......Page 388
6.2 Updated Covariance Matrix Inverse......Page 392
6.2.1 Recursive Algorithms Based on Factorization Methods......Page 395
6.3 Kalman Filter Methods for Adaptive Array Processing......Page 399
6.3.1 Development of a Kalman-Type Array Processor......Page 400
6.3.2 Speed of Convergence......Page 404
6.4 The Minimum Variance Processor......Page 406
6.5 Simulation Results......Page 410
6.7 Problems......Page 414
6.8 References......Page 416
7. Cascade Preprocessors......Page 418
7.1.1 Single-Stage Nolen Network......Page 419
7.1.2 Cascaded Networks......Page 422
7.1.3 Control of the Eigenvector Transformation Networks......Page 424
7.2.1 Problem Formulation......Page 426
7.2.2 Recognition of the Desired Signal......Page 428
7.3 Gram-Schmidt Orthogonalization Preprocessor......Page 430
7.3.1 Convergence of the Gram-Schmidt Cascade Preprocessor......Page 434
7.4 Simulation Results......Page 439
7.6 Problems......Page 443
7.7 References......Page 447
8. Random Search Algorithms......Page 449
8.1 Linear Random Search......Page 450
8.1.1 LRS Algorithm Stability......Page 451
8.2 Accelerated Random Search......Page 455
8.3 Guided Accelerated Random Search......Page 458
8.4 Genetic Algorithm......Page 460
8.5 Comparison of Random Search Algorithms......Page 467
8.6 Summary and Conclusions......Page 472
8.7 Problems......Page 473
8.8 References......Page 476
9. Adaptive Algorithm Performance Summary......Page 479
10. Compensation of Adaptive Arrays......Page 485
10.1.1 Error Analysis......Page 486
10.1.2 Quantization Errors......Page 487
10.2.1 Calibrated Source......Page 489
10.2.2 Signal Injection......Page 490
10.2.3 Near-Field Scan......Page 491
10.3 Broadband Signal Processing Considerations......Page 492
10.3.1 Distortionless Channel Transfer Functions......Page 493
10.3.2 Quadrature Hybrid and Tapped Delay Line Processing for a Least Mean Squares Array......Page 495
10.3.3 Performance Comparison of Four Array Processors......Page 500
10.3.4 Processor Transfer Functions......Page 505
10.4 Compensation for Mutual Coupling......Page 508
10.5.1 Two-Channel Interference Cancellation Model......Page 510
10.5.2 Signal Environment Assumptions......Page 512
10.5.3 Example: Results for Compensation of Multipath Effects......Page 514
10.5.4 Results for Compensation of Array Propagation Delay......Page 517
10.6 Analysis of Interchannel Mismatch Effects......Page 518
10.6.1 Example: Effects of Amplitude Mismatching......Page 521
10.6.2 Results for Compensation of Selected Amplitude Mismatch Model......Page 523
10.6.3 Example: Effects of Phase Mismatching......Page 525
10.6.4 Results for Compensation of Selected Phase Mismatch Model......Page 526
10.7 Summary and Conclusions......Page 527
10.8 Problems......Page 528
10.9 References......Page 531
10. Compensation of Adaptive Arrays......Page 533
10.1.1 Error Analysis......Page 534
10.1.2 Quantization Errors......Page 535
10.2.1 Calibrated Source......Page 537
10.2.2 Signal Injection......Page 538
10.2.3 Near-Field Scan......Page 539
10.3 Broadband Signal Processing Considerations......Page 540
10.3.1 Distortionless Channel Transfer Functions......Page 541
10.3.2 Quadrature Hybrid and Tapped Delay Line Processing for a Least Mean Squares Array......Page 543
10.3.3 Performance Comparison of Four Array Processors......Page 548
10.3.4 Processor Transfer Functions......Page 553
10.4 Compensation for Mutual Coupling......Page 556
10.5.1 Two-Channel Interference Cancellation Model......Page 558
10.5.2 Signal Environment Assumptions......Page 560
10.5.3 Example: Results for Compensation of Multipath Effects......Page 562
10.5.4 Results for Compensation of Array Propagation Delay......Page 565
10.6 Analysis of Interchannel Mismatch Effects......Page 566
10.6.1 Example: Effects of Amplitude Mismatching......Page 569
10.6.2 Results for Compensation of Selected Amplitude Mismatch Model......Page 571
10.6.3 Example: Effects of Phase Mismatching......Page 573
10.6.4 Results for Compensation of Selected Phase Mismatch Model......Page 574
10.7 Summary and Conclusions......Page 575
10.8 Problems......Page 576
10.9 References......Page 579
11. Direction of Arrival Estimation and Related Topics......Page 580
11.1 Periodogram......Page 581
11.3 Music Algorithm......Page 582
11.4 The Maximum Entropy Method......Page 585
11.4.1 Partially Known Autocorrelation Function......Page 586
11.4.2 Unknown Autocorrelation Function......Page 589
11.4.3 Extension to Multichannel Spectral Estimation......Page 592
11.5 Comparing DOA Algorithm Spectra......Page 595
11.6 Sequential Implementation of Bayes Optimal Array Processor......Page 596
11.6.1 Estimate and Plug Array Processor......Page 597
11.6.2 Sequential Optimal Array Processor......Page 598
11.7 Parameter Estimation via Subspace Fitting......Page 600
11.8 The Maximum Likelihood Estimate......Page 602
11.8.1 Mathematical Preliminaries......Page 603
11.8.2 Maximum Likelihood Estimation of the Direction of Arrival for Stochastic Signals......Page 605
11.8.3 Maximum Likelihood Estimation of the Direction of Arrival for Deterministic Signals......Page 607
11.9 Cramer-Rao Lower Bound on AOA Estimation Error......Page 608
11.10 Fisher Information Matrix and CR Bound for General Cases......Page 610
11.12 Problems......Page 611
11.13 References......Page 618
12.1 Beam Switching......Page 621
12.2 Space-Time Adaptive Processing......Page 623
12.3 MIMO......Page 631
12.4 Reconfigurable Antennas and Arrays......Page 637
12.5 Performance Characteristics of Large Sonar Arrays......Page 642
12.6 Adaptive Processing for Monopulse Tracking Antennas......Page 644
12.7 Partially Adaptive Arrays......Page 646
12.7.1 Adaptive Subarray Beamforming......Page 652
12.7.2 Element-Level Adaptivity......Page 654
12.7.3 Eigenspace Beamformers......Page 657
12.7.4 The J Subbeam Approach......Page 659
12.7.5 The Subarray Design Approach......Page 660
12.9 Problems......Page 661
12.10 References......Page 662
Appendix A: Frequency Response Characteristics of Tapped-Delay Lines......Page 665
A.1 References......Page 668
A......Page 669
B......Page 670
C......Page 671
D......Page 673
E......Page 674
G......Page 675
J......Page 676
M......Page 677
P......Page 679
Q......Page 681
R......Page 682
S......Page 683
T......Page 685
Z......Page 686