Signal.Detection.and.Estimation

oaktwig

Contents
Preface xv
Acknowledgments xvii
Chapter 1 Probability Concepts 1
1.1 Introduction 1
1.2 Sets and Probability 1
1.2.1 Basic Definitions 1
1.2.2 Venn Diagrams and Some Laws 3
1.2.3 Basic Notions of Probability 6
1.2.4 Some Methods of Counting 8
1.2.5 Properties, Conditional Probability, and Bayes’ Rule 12
1.3 Random Variables 17
1.3.1 Step and Impulse Functions 17
1.3.2 Discrete Random Variables 18
1.3.3 Continuous Random Variables 20
1.3.4 Mixed Random Variables 22
1.4 Moments 23
1.4.1 Expectations 23
1.4.2 Moment Generating Function and Characteristic Function 26
1.4.3 Upper Bounds on Probabilities and Law of Large
Numbers 29
1.5 Two- and Higher-Dimensional Random Variables 31
1.5.1 Conditional Distributions 33
1.5.2 Expectations and Correlations 41
1.5.3 Joint Characteristic Functions 44
1.6 Transformation of Random Variables 48
1.6.1 Functions of One Random Variable 49
1.6.2 Functions of Two Random Variables 52
1.6.3 Two Functions of Two Random Variables 59
1.7 Summary 65
Problems 65
Reference 73
Selected Bibliography 73
Chapter 2 Distributions 75
2.1 Introduction 75
2.2 Discrete Random Variables 75
2.2.1 The Bernoulli, Binomial, and Multinomial Distributions 75
2.2.2 The Geometric and Pascal Distributions 78
2.2.3 The Hypergeometric Distribution 82
2.2.4 The Poisson Distribution 85
2.3 Continuous Random Variables 88
2.3.1 The Uniform Distribution 88
2.3.2 The Normal Distribution 89
2.3.3 The Exponential and Laplace Distributions 96
2.3.4 The Gamma and Beta Distributions 98
2.3.5 The Chi-Square Distribution 101
2.3.6 The Rayleigh, Rice, and Maxwell Distributions 106
2.3.7 The Nakagami m-Distribution 115
2.3.8 The Student’s t- and F-Distributions 115
2.3.9 The Cauchy Distribution 120
2.4 Some Special Distributions 121
2.4.1 The Bivariate and Multivariate Gaussian Distributions 121
2.4.2 The Weibull Distribution 129
2.4.3 The Log-Normal Distribution 131
2.4.4 The K-Distribution 132
2.4.5 The Generalized Compound Distribution 135
2.5 Summary 136
Problems 137
Reference 139
Selected Bibliography 139
Chapter 3 Random Processes 141
3.1 Introduction and Definitions 141
3.2 Expectations 145
3.3 Properties of Correlation Functions 153
3.3.1 Autocorrelation Function 153
3.3.2 Cross-Correlation Function 153
3.3.3 Wide-Sense Stationary 154
3.4 Some Random Processes 156
3.4.1 A Single Pulse of Known Shape but Random Amplitude
and Arrival Time 156
3.4.2 Multiple Pulses 157
3.4.3 Periodic Random Processes 158
3.4.4 The Gaussian Process 161
3.4.5 The Poisson Process 163
3.4.6 The Bernoulli and Binomial Processes 166
3.4.7 The Random Walk and Wiener Processes 168
3.4.8 The Markov Process 172
3.5 Power Spectral Density 174
3.6 Linear Time-Invariant Systems 178
3.6.1 Stochastic Signals 179
3.6.2 Systems with Multiple Terminals 185
3.7 Ergodicity 186
3.7.1 Ergodicity in the Mean 186
3.7.2 Ergodicity in the Autocorrelation 187
3.7.3 Ergodicity of the First-Order Distribution 188
3.7.4 Ergodicity of Power Spectral Density 188
3.8 Sampling Theorem 189
3.9 Continuity, Differentiation, and Integration 194
3.9.1 Continuity 194
3.9.2 Differentiation 196
3.9.3 Integrals 199
3.10 Hilbert Transform and Analytic Signals 201
3.11 Thermal Noise 205
3.12 Summary 211
Problems 212
Selected Bibliography 221
Chapter 4 Discrete-Time Random Processes 223
4.1 Introduction 223
4.2 Matrix and Linear Algebra 224
4.2.1 Algebraic Matrix Operations 224
4.2.2 Matrices with Special Forms 232
4.2.3 Eigenvalues and Eigenvectors 236
4.3 Definitions 245
4.4 AR, MA, and armA Random Processes 253
4.4.1 AR Processes 254
4.4.2 MA Processes 262
4.4.3 ARMA Processes 264
4.5 Markov Chains 266
4.5.1 Discrete-Time Markov Chains 267
4.5.2 Continuous-Time Markov Chains 276
4.6 Summary 284
Problems 284
References 287
Selected Bibliography 288
Chapter 5 Statistical Decision Theory 289
5.1 Introduction 289
5.2 Bayes’ Criterion 291
5.2.1 Binary Hypothesis Testing 291
5.2.2 M-ary Hypothesis Testing 303
5.3 Minimax Criterion 313
5.4 Neyman-Pearson Criterion 317
5.5 Composite Hypothesis Testing 326
5.5.1 Θ Random Variable 327
5.5.2 θ Nonrandom and Unknown 329
5.6 Sequential Detection 332
5.7 Summary 337
Problems 338
Selected Bibliography 343
Chapter 6 Parameter Estimation 345
6.1 Introduction 345
6.2 Maximum Likelihood Estimation 346
6.3 Generalized Likelihood Ratio Test 348
6.4 Some Criteria for Good Estimators 353
6.5 Bayes’ Estimation 355
6.5.1 Minimum Mean-Square Error Estimate 357
6.5.2 Minimum Mean Absolute Value of Error Estimate 358
6.5.3 Maximum A Posteriori Estimate 359
6.6 Cramer-Rao Inequality 364
6.7 Multiple Parameter Estimation 371
6.7.1 θ Nonrandom 371
6.7.2 θ Random Vector 376
6.8 Best Linear Unbiased Estimator 378
6.8.1 One Parameter Linear Mean-Square Estimation 379
6.8.2 θ Random Vector 381
6.8.3 BLUE in White Gaussian Noise 383
6.9 Least-Square Estimation 388
6.10 Recursive Least-Square Estimator 391
6.11 Summary 393
Problems 394
References 398
Selected Bibliography 398
Chapter 7 Filtering 399
7.1 Introduction 399
7.2 Linear Transformation and Orthogonality Principle 400
7.3 Wiener Filters 409
7.3.1 The Optimum Unrealizable Filter 410
7.3.2 The Optimum Realizable Filter 416
7.4 Discrete Wiener Filters 424
7.4.1 Unrealizable Filter 425
7.4.2 Realizable Filter 426
7.5 Kalman Filter 436
7.5.1 Innovations 437
7.5.2 Prediction and Filtering 440
7.6 Summary 445
Problems 445
References 448
Selected Bibliography 448
Chapter 8 Representation of Signals 449
8.1 Introduction 449
8.2 Orthogonal Functions 449
8.2.1 Generalized Fourier Series 451
8.2.2 Gram-Schmidt Orthogonalization Procedure 455
8.2.3 Geometric Representation 458
8.2.4 Fourier Series 463
8.3 Linear Differential Operators and Integral Equations 466
8.3.1 Green’s Function 470
8.3.2 Integral Equations 471
8.3.3 Matrix Analogy 479
8.4 Representation of Random Processes 480
8.4.1 The Gaussian Process 483
8.4.2 Rational Power Spectral Densities 487
8.4.3 The Wiener Process 492
8.4.4 The White Noise Process 493
8.5 Summary 495
Problems 496
References 500
Selected Bibliography 500
Chapter 9 The General Gaussian Problem 503
9.1 Introduction 503
9.2 Binary Detection 503
9.3 Same Covariance 505
9.3.1 Diagonal Covariance Matrix 508
9.3.2 Nondiagonal Covariance Matrix 511
9.4 Same Mean 518
9.4.1 Uncorrelated Signal Components and Equal Variances 519
9.4.2 Uncorrelated Signal Components and Unequal
Variances 522
9.5 Same Mean and Symmetric Hypotheses 524
9.5.1 Uncorrelated Signal Components and Equal Variances 526
9.5.2 Uncorrelated Signal Components and Unequal
Variances 528
9.6 Summary 529
Problems 530
Reference 532
Selected Bibliography 532
Chapter 10 Detection and Parameter Estimation 533
10.1 Introduction 533
10.2 Binary Detection 534
10.2.1 Simple Binary Detection 534
10.2.2 General Binary Detection 543
10.3 M-ary Detection 556
10.3.1 Correlation Receiver 557
10.3.2 Matched Filter Receiver 567
10.4 Linear Estimation 572
10.4.1 ML Estimation 573
10.4.2 MAP Estimation 575
10.5 Nonlinear Estimation 576
10.5.1 ML Estimation 576
10.5.2 MAP Estimation 579
10.6 General Binary Detection with Unwanted Parameters 580
10.6.1 Signals with Random Phase 583
10.6.2 Signals with Random Phase and Amplitude 595
10.6.3 Signals with Random Parameters 598
10.7 Binary Detection in Colored Noise 606
10.7.1 Karhunen-Loève Expansion Approach 607
10.7.2 Whitening Approach 611
10.7.3 Detection Performance 615
10.8 Summary 617
Problems 618
Reference 626
Selected Bibliography 626
Chapter 11 Adaptive Thresholding CFAR Detection 627
11.1 Introduction 627
11.2 Radar Elementary Concepts 629
11.2.1 Range, Range Resolution, and Unambiguous Range 631
11.2.2 Doppler Shift 633
11.3 Principles of Adaptive CFAR Detection 634
11.3.1 Target Models 640
11.3.2 Review of Some CFAR Detectors 642
11.4 Adaptive Thresholding in Code Acquisition of Direct-
Sequence Spread Spectrum Signals 648
11.4.1 Pseudonoise or Direct Sequences 649
11.4.2 Direct-Sequence Spread Spectrum Modulation 652
11.4.3 Frequency-Hopped Spread Spectrum Modulation 655
11.4.4 Synchronization of Spread Spectrum Systems 655
11.4.5 Adaptive Thresholding with False Alarm Constraint 659
11.5 Summary 660
References 661
Chapter 12 Distributed CFAR Detection 665
12.1 Introduction 665
12.2 Distributed CA-CFAR Detection 666
12.3 Further Results 670
12.4 Summary 671
References 672
Appendix 675
About the Author 683
Index 685

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