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Preface xix
1 Introduction 1
1.1 Array Processing ......................... 2
1.2 Applications ............................ 6
1.2.1 Radar ........................... 6
1.2.2 Radio Astronomy ..................... 7
1.2.3 Sonar ........................... 8
1.2.4 Communications ..................... 9
1.2.5 Direction Finding ..................... 10
1.2.6 Seismology ........................ 10
1.2.7 Tomography ....................... 11
1.2.8 Array Processing Literature ............... 12
1.3 Organization of the Book .................... 12
1.4 Interactive Study ......................... 14
2 Arrays and Spatial Filters 17
2.1 Introduction ............................ 17
2.2 Frequency-wavenumber Response and Beam Patterns ..... 23 ......................
2.3 Uniform Linear Arrays 37
2.4 Uniformly Weighted Linear Arrays ............... 42
2.4.1 Beam Pattern Parameters ................ 46
2.5 Array Steering .......................... 51
..................
2.6 Array Performance Measures 59
2.6.1 Directivity ........................ 60
2.6.2 Array Gain vs. Spatially White Noise (A,) ...... 63
2.6.3 Sensitivity and the Tolerance Factor .......... 66
2.6.4 Summary ......................... 70
2.7 Linear Apertures ......................... 71
Viii Contents
2.7.1 Frequency-wavenumber Response ............ 71
2.7.2 Aperture Sampling ..................... 74
2.8 Non-isotropic Element Patterns ................ 75
2.9 Summary ............................. 78
2.10 Problems ............................. 79
Synthesis of Linear Arrays and Apertures 90
3.1 Spectral Weighting ........................ 95
3.2 Array Polynomials and the z-Transform ............ 109
3.2.1 z-Transform ........................ 109
3.2.2 Real Array Weights ................... 110
3.2.3 Properties of the Beam Pattern Near a Zero ...... 114
3.3 Pattern Sampling in Wavenumber Space ............ 118
3.3.1 Continuous Aperture ................... 118
3.3.2 Linear Arrays ....................... 120
3.3.3 Discrete Fourier Transform ............... 122
3.3.4 Norms ........................... 126
3.3.5 Summary ......................... 128
3.4 Minimum Beamwidth for Specified Sidelobe Level ....... 128
3.4.1 Introduction ....................... 128
3.4.2 Dolph-Chebychev Arrays ................ 130
3.4.3 Taylor Distribution .................... 143
3.4.4 Villeneuve fi Distribution ................ 147
3.5 Least Squares Error Pattern Synthesis ............. 149
3.6 Minimax Design ......................... 156
3.6.1 Alternation Theorem ...... Y. ........... 159
3.6.2 Parks-McClellan-Rabiner Algorithm .......... 160
3.6.3 Summary ......................... 163
3.7. Null Steering ........................... 165
3.7.1 Null Constraints ..................... 165
3.7.2 Least Squares Error Pattern Synthesis with Nulls ... 166 ,3.8 Asymmetric Beams ........................ 173
3.9 Spatially Non-uniform Linear Arrays .............. 178
3.9.1 Introduction ....................... 178
3.9.2 Minimum Redundancy Arrays ............. 179
39.3 Beam Pattern Design Algorithm ............ 183
3.10 Beamspace Processing ...................... 192
3.10.1 Full-dimension Beamspace ................ 192
3.10.2 Reduced-dimension Beamspace ............. 193
3.10.3 Multiple Beam Antennas ................ 200
Contents ix
3.10.4 Summary ......................... 200
3.11 Broadband Arrays ........................ 200
3.12 Summary ............................. 204
3.13 Problems ............................. 207
4 Planar Arrays and Apertures 231
4.1 Rectangular Arrays ........................ 233
4.1.1 Uniform Rectangular Arrays .............. 233
4.1.2 Array Manifold Vector .................. 249
4.1.3 Separable Spectral Weightings ............. 251
4.1.4 2-D z-Transforms ..................... 251
4.1.5 Least Squares Synthesis ................. 253 ....
4.1.6 Circularly Symmetric Weighting and Windows 259
4.1.7 Wavenumber Sampling and 2-D DFT ......... 260
4.1.8 Transformations from One Dimension to Two Dimen-sions ............................ 264
4.1.9 Null Steering ....................... 269
4.1.10 Related Topics ...................... 272
4.2 Circular Arrays .......................... 274
4.2.1 Continuous Circular Arrays (Ring Apertures) ..... 275
4.2.2 Circular Arrays ...................... 280
4.2.3 Phase Mode Excitation Beamformers .......... 284
........................
4.3 Circular Apertures 289
..................
4.3.1 Separable Weightings 290
4.3.2 Taylor Synthesis for Circular Apertures ........ 294
4.3.3 Sampling the Continuous Distribution ......... 298
4.3.4 Difference Beams ..................... 299
4.3.5 Summary ......................... 304
4.4 Hexagonal Arrays ......................... 305
4.4.1 Introduction ....................... 305
4.4.2 Beam Pattern Design .................. 307
4.4.3 Hexagonal Grid to Rectangular Grid Transformation . 314
4.4.4 Summary ......................... 316
4.5 Nonplanar Arrays ......................... 316
4.5.1 Cylindrical Arrays .................... 317
4.5.2 Spherical Arrays ..................... 320
4.6 Summary ............................. 321
4.7 Problems ............................. 322
X Contents
5 Characterization of Space-time Processes . 332
5.1 Introduction ............................ 332
5.2 Snapshot Models ......................... 333
5.2.1 Frequency-domain Snapshot Models .......... 334
5.2.2 Narrowband Time-domain Snapshot Models ...... 349
5.2.3 Summary ......................... 352
5.3 Space-time Random Processes .................. 353
5.3.1 Second-moment Characterization ............ 353
5.3.2 Gaussian Space-time Processes ............. 359
5.3.3 Plane Waves Propagating in Three Dimensions .... 361
5.3.4 1-D and 2-D Projections ................. 365
5.4 Arrays and Apertures ...................... 369
5.4.1 Arrays ........................... 369
5.4.2 Apertures ......................... 374
5.5 Orthogonal Expansions ..................... 375
5.5.1 Plane-wave Signals .................... 377
5.5.2 Spatially Spread Signals ................. 385
5.5.3 Frequency-spread Signals ................ 390
5.5.4 Closely Spaced Signals .................. 393
5.5.5 Beamspace Processors .................. 393
5.5.6 Subspaces for Spatially Spread Signals ......... 394
5.6 Parametric Wavenumber Models ................ 394
5.6.1 Rational Transfer Function Models ........... 395
5.6.2 Model Relationships ................... 407
5.6.3 Observation Noise ........ .......... 408
2 .
5.6.4 Summary ......................... 414
5.7 Summary ............................. 414
5.8 Problems ............................. 415
6 Optimum Waveform Estimation 428
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
6.2 Optimum Beamformers . . . . . . . . . . . . . . . . . . . . . 439
6.2.1 Minimum Variance Distortionless Response (MVDR) Beamformers . . . . . . . . . . . . . . . . . . . . . . . 440
6.2.2 Minimum Mean-Square Error (MMSE) Estimators . . 446
6.2.3 Maximum Signal-to-Noise Ratio (SNR) . . . . . . . . 449
6.2.4 Minimum Power Distortionless Response (MPDR) Beam-formers . . . . . . . . . . . . . . . . . . . . . . . . . . 451
6.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 452
6.3 Discrete Interference . . . . . . . . . . . . . . . . . . . . . . . 452 Contents Xi
6.3.1 Single Plane-wave Interfering Signal .......... 453
6 3.2 Multiple Plane-wave Interferers 465
63.3 Summary: Discrete Interference .......................... 471
64. Spatially Spread Interference .................. 473
6.4.1 Physical Noise Models .................. 473
6.4.2 ARMA Models ...................... 474
65. Multiple Plane-wave Signals ................... 477
6.5.1 MVDR Beamformer ................... 477
6.5.2 MMSE Processors .................... 485
66. Mismatched MVDR and MPDR Beamformers ......... 488
6.6.1 Introduction ....................... 488
6.6.2 DOA Mismatch ...................... 490
6.6.3 Array Perturbations ................... 501
6.6.4 Diagonal Loading ..................... 505
6.6.5 Summary ......................... 510
67. LCMV and LCMP Beamformers ................ 513
6.7.1 Typical Constraints ................... 514
6.7.2 Optimum LCMV and LCMP Beamformers ...... 526
6.7.3 Generalized Sidelobe Cancellers ............. 528
6.7.4 Performance of LCMV and LCMP Beamformers ... 532
6.7.5 Quiescent Pattern (QP) Constraints .......... 547
6.7.6 Covariance Augmentation ................ 554
6.7.7 Summary ......................... 555
68. Eigenvector Beamformers .................... 556
6.8.1 Principal-component (PC) Beamformers ........ 560
6.8.2 Cross-spectral Eigenspace Beamformers ........ 567
6.8.3 Dominant-mode Rejection Beamformers ........ 569
6.8.4 Summary ......................... 573
69. Beamspace Beamformers ..................... 575
6.9.1 Beamspace MPDR .................... 576
6.9.2 Beamspace LCMP .................... 583
6.9.3 Summary: Beamspace Optimum Processors ...... 585
6.10 Quadratically Constrained Beamformers . . . . . . . . . . . . 585
6.11 Soft-constraint Beamformers . . . . . . . . . . . . . . . . . . 593
6.12 Beamforming for Correlated Signal and Interferences . . . . . 599
6.12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 599
6.12.2 MPDR Beamformer: Correlated Signals and Interference600
6.12.3 MMSE Beamformer: Correlated Signals and Interference603
6.12.4 Spatial Smoothing and Forward-Backward Averaging 605
6.12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 620
Xii Contents
6.13 Broadband Bearnformers ..................... 621
6.13.1 Introduction ....................... 621
6.13.2 DFT Beamformers .................... 627
6.13.3 Finite impulse response (FIR) Beamformers ...... 647
6.13.4 Summary: Broadband Processing ............ 664
6.14 Summary ............................. 666
6.15 Problems ............................. 669
7 Adaptive Beamformers 710
7.1 Introduction ............................ 710
7.2 Estimation of Spatial Spectral Matrices ............ 712
7.2.1 Sample Spectral Matrices ................ 714
7.2.2 Asymptotic Behavior ................... 717
7.2.3 Forward-Backward Averaging .............. 718
7.2.4 Structured Spectral Matrix Estimation ......... 726
7.2.5 Parametric Spatial Spectral Matrix Estimation .... 726
7.2.6 Singular Value Decomposition .............. 727
7.2.7 Summary ......................... 727
7.3 Sample Matrix Inversion (SMI) ................. 728
7.3.1 SINRsmi Behavior: MVDR and MPDR ........ 731
7.3.2 LCMV and LCMP Beamformers ............ 739
7.3.3 Fixed Diagonal Loading ................. 739
7.3.4 Toeplitz Estimators ................... 751
7.3.5 Summary ......................... 751
7.4 Recursive Least Squares (RLS) ................. 752
7.4.1 Least Squares Formulation ............... 752
7.4.2 Recursive Implement ation ................ 756
7.4.3 Recursive Implementation of LSE Beamformer .... 763
7.4.4 Generalized Sidelobe Canceller ............. 766
7.4.5 Quadratically Constrained RLS ............. 768
7.4.6 Conjugate Symmetric Beamformers .......... 773
7.4.7 Summary ......................... 777
7.5 Efficient Recursive Implementation Algorithms ........ 778
7.5.1 Introduction ....................... 778
7.5.2 QR Decomposition (QRD) ............... 779
7.6 Gradient Algorithms ....................... 789
7.6.1 Introduction ....................... 789
7.6.2 Steepest Descent: MMSE Beamformers ........ 791
7.6.3 Steepest Decent: LCMP Beamformer ......... 799
7.6.4 Summary ......................... 805
Contents Xiii
7.7 LMS Algorithms ......................... 805
7.7.1 Derivation of the LMS Algorithms ........... 806
7.7.2 Performance of the LMS Algorithms .......... 813
7.7.3 LMS Algorithm Behavior ................ 817
7.7.4 Quadratic Constraints .................. 822
7.7.5 Summary: LMS algorithms ............... 826
7.8 Detection of Signal Subspace Dimension ............ 82'7
7.8.1 Detection Algorithms .................. 828
7.8.2 Eigenvector Detection Tests ............... 841
7.9 Eigenspace and DMR Beamformers ............... 845
7.9.1 Performance of SMI Eigenspace Beamformers ..... 846
7.9.2 Eigenspace and DMR Beamformers: Detection of Sub- space Dimension ...................... 850
7.9.3 Subspace tracking .................... 860
7.9.4 Summary ......................... 863
7.10 Beamspace Beamformers ..................... 864
7.10.1 Beamspace SMI ...................... 865
7.10.2 Beamspace RLS ..................... 869
7.10.3 Beamspace LMS ..................... 872
7.10.4 Summary: Adaptive Beamspace Processing ...... 873
7.11 Broadband Beamformers ..................... 874
7.11.1 SMI Implementation ................... 875
7.11.2 LMS Implementation ................... 878
7.11.3 GSC: Multichannel Lattice Filters ........... 884
7.11.4 Summary ......................... 885
7.12 Summary ............................. 885
7.13 Problems ............................. 887
8 Parameter Estimation I: Maximum Likelihood 917
81. Introduction. ........................... 917
82. Maximum Likelihood and Maximum a posteriori Estimators . 920
8.2.1 Maximum Likelihood (ML) Estimator ......... 922
8.2.2 Maximum a posteriori (MAP) Estimator ....... 924
8.2.3 Cramer-Rao Bounds ................... 925
83. Parameter Estimation Model .................. 933
8.3.1 Multiple Plane Waves .................. 933
8.3.2 Model Perturbations ................... 936
8.3.3 Parametric Spatially Spread Signals .......... 938
8.3.4 Summary ......................... 938
84. Cramer-Rao Bounds ....................... 938
8.4.1
8.4.2
8.4.3
8.4.4
8.4.5
8.4.6
8.5 Maximum
8.5.1
8.5.2
8.5.3
8.5.4
8.5.5
8.5.6
Contents
Gaussian Model: Unknown Signal Spectrum ...... 939 Gaussian Model: Uncorrelated Signals with Unknown Power ........................... 958 Gaussian Model: Known Signal Spectrum ....... 967 Nonrandom (Conditional) Signal Model ........ 971 Known Signal Waveforms ................ 978 Summary ......................... 980 Likelihood Estimation ................ 984
Maximum Conditional Weighted Asymptotic Wideband Summary
Likelihood Estimation ............ 984 Maximum Likelihood Estimators ..... 1004 Subspace Fitting ............... 1009 Performance ................. 1014 Signals .................... 1015 ......................... 1018
8.6 Computational Algorithms ................... 1018
8.6.1 Optimization
8.6.2 Alternating
8.6.3 Expectation
8.6.4 Summary
Techniques ................ 1018 Maximization Algorithms ......... 1025 Maximization Algorithm ......... 1031 ......................... 1037
8.7 Polynomial Parameterization .................. 1037
8.7.1 Polynomial Parameterization .............. 1038
8.7.2 Iterative Quadratic Maximum Likelihood (IQML) . . 1039
8.7.3
8.7.4
8.8 Detection
8.9 Spatially
8.9.1
8.9.2
8.9.3
Polynomial WSF (MODE) ............... 1045 Summary ......................... 1053 of Number of Signals ................. 1054 Spread Signals ..................... 1055 Parameterized S(&+) .................. 1055 Spatial ARMA Process ................. 1062
Summary .........................
8.10 Beamspace algorithms ......................
8.10.1 Introduction .......................
8.10.2 Beamspace Matrices ...................
8.10.3 Beamspace Cramer-Rao Bound
8.10.4 Beamspace Maximum Likelihood
8.10.5 Summary .........................
8.11 Sensitivity, Robustness, and Calibration
1062 1062 1062 1065 ............. 1073 ............ 1081 1088 ............ 1088
8.11.1 Model Perturbations .................. 1089
8.11.2 Cram&-Rao Bounds ................... 1090
8.11.3 Sensitivity of ML Estimators .............. 1098
8.11.4 MAP Joint Estimation .................. 1099
Contents xv
8.11.5 Self-Calibration Algorithms ............... 1101
8.11.6 Summary ......................... 1102
8.12 Summary ............................. 1102
8.12.1 Major Results ....................... 1102
8.12.2 Related Topics ...................... 1105
8.12.3 Algorithm complexity .................. 1108
8.13 Problems ............................. 1109
9 Parameter Estimation II 1139
9.1 Introduction ........................... 1139
......................
9.2 Quadratic Algorithms 1140
9.2.1 Introduction ....................... 1140
..................
9.2.2 Beamscan Algorithms 1142 ...............
9.2.3 MVDR (Capon) Algorithm 1144
9.2.4 Root Versions of Quadratic Algorithms ........ 1147
9.2.5 Performance of MVDR Algorithms ........... 1148
9.2.6 Summary ......................... 1149
9.3 Subspace Algorithms ....................... 1155
9.3.1 Introduction ....................... 1155
9.3.2 MUSIC .......................... 1158
9.3.3 Minimum-Norm Algorithm ............... 1163
9.3.4 ESPRIT .......................... 1170
9.3.5 Algorithm Comparison .................. 1189
9.3.6 Summary ......................... 1190
.........................
9.4 Linear Prediction 1194
.....................
......................
9.5 Asymptotic Performance 1195
9.5.1 Error Behavior 1195 ..........
9.5.2 Resolution of MUSIC and Min-Norm 1203 ..........
9.5.3 Small Error Behavior of Algorithms 1211
9.5.4 Summary ......................... 1233
................
9.6 Correlated and Coherent Signals 1233
9.6.1 Introduction ....................... 1233
.........
9.6.2 Forward-Backward Spatial Smoothing 1235
9.6.3 Summary. ........................ 1241
9.7 Beamspace Algorithms ...................... 1243
9.7.1 Beamspace MUSIC .................... 1243
9.7.2 Beamspace Unitary ESPRIT .............. 1247
9.7.3 Beamspace Summary .................. 1251
9.8 Sensitivity and Robustness ................... 1251
...........................
9.9 Planar Arrays 1255
xvi Contents
9.9.1 Standard Rectangular Arrays .............. 1255
9.9.2 Hexagonal Arrays .................... 1272
9.9.3 Summary: Planar Arrays ................ 1279
9.10 Summary ............................. 1279
9.10.1 Major Results ....................... 1279
9.10.2 Related Topics ...................... 1282
9.10.3 Discussion ......................... 1285
9.11 Problems ............................. 1285
10 Detection and Other Topics 1318
10.1 Optimum Detection ....................... 1318
10.1.1 Classic Binary Detection ................. 1319
10.1.2 Matched Subspace Detector ............... 1320
10.1.3 Spatially Spread Gaussian Signal Processes ...... 1321
10.1.4 Adaptive Detection .................... 1323
10.2 Related Topics .......................... 1327
10.3 Epilogue .............................. 1329
10.4 Problems ............................. 1329
A Matrix Operations 1340
A.1 Introduction ............................ 1340
A.2 Basic Definitions and Properties ................ 1341
.....................
A.2.1 Basic Definitions 1341 ......................
A.2.2 Matrix Inverses 1347
A.2.3 Quadratic Forms ..................... 1348
...................
.....................
A.2.4 Partitioned Matrices 1349
A.2.5 Matrix products 1351 ....................
A.2.6 Matrix Inequalities 1356
A.3 Special Vectors and Matrices .................. 1356
A.3.1 Elementary Vectors and Matrices ............ 1356
A.3.2 The vet(A) matrix .................... 1358
A.3.3 Diagonal Matrices .................... 1359
A.3.4 Exchange Matrix and Conjugate Symmetric Vectors . 1361
A.3.5 Persymmetric and Centrohermitian Matrices ..... 1362
A.3.6 Toeplitz and Hankel Matrices .............. 1364
A.3.7 Circulant Matrices .................... 1365
A.3.8 Triangular Matrices ................... 1366
............
A.39 Unitary and Orthogonal Matrices 1367 ..................
A.3.10 Vandermonde Matrices 1368
A.3.11 Projection Matrices ................... 1369
Contents xvii
....................
A.3.12 Generalized Inverse 1370
A.4 Eigensystems ........................... 1372
A.4.1 Eigendecomposition ................... 1372
.....................
A.4.2 Special Matrices 1376
A.5 Singular Value Decomposition .................. 1381
A.6 QR Decomposition ........................ 1387
.......................
A.6.1 Introduction 1387
A.6.2 QR Decomposition .................... 1388
A.6.3 Givens Rotation ..................... 1390
A.6.4 Householder Transformation ............... 1394
A.7 Derivative Operations ...................... 1397
A.7.1 Derivative of Scalar with Respect to Vector ...... 1397
A.7.2 Derivative of Scalar with Respect to Matrix ...... 1399
A.7.3 Derivatives with Respect to Parameter ......... 1401
A.7.4 Complex Gradients .................... 1402
B Array Processing Literature 1407
B.1 Journals .............................. 1407
B.2 Books ............................... 1408
B.3 Duality .............................. 1409
C Notation 1414
C.l Conventions ............................ 1414
C.2 Acronyms ............................. 1415
......................
C.3 Mathematical Symbols 1418
C.4 Symbols .............................. 1419 |
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发表于 2008-3-4 23:17:20
