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Introduction to Statistical Signal Processing

Introduction to Statistical Signal Processing

Contents
Preface page ix
Acknowledgements xii
Glossary xiii
1 Introduction 1
2 Probability 10
2.1 Introduction 10
2.2 Spinning pointers and ipping coins 14
2.3 Probability spaces 22
2.4 Discrete probability spaces 44
2.5 Continuous probability spaces 54
2.6 Independence 68
2.7 Elementary conditional probability 70
2.8 Problems 73
3 Random variables, vectors, and processes 82
3.1 Introduction 82
3.2 Random variables 93
3.3 Distributions of random variables 102
3.4 Random vectors and random processes 112
3.5 Distributions of random vectors 115
3.6 Independent random variables 124
3.7 Conditional distributions 127
3.8 Statistical detection and classification 132
3.9 Additive noise 135
3.10 Binary detection in Gaussian noise 142
3.11 Statistical estimation 144
3.12 Characteristic functions 145
3.13 Gaussian random vectors 151
3.14 Simple random processes 152
3.15 Directly given random processes 156
3.16 Discrete time Markov processes 158
3.17 ⋆Nonelementary conditional probability 167
3.18 Problems 168
4 Expectation and averages 182
4.1 Averages 182
4.2 Expectation 185
4.3 Functions of random variables 188
4.4 Functions of several random variables 195
4.5 Properties of expectation 195
4.6 Examples 197
4.7 Conditional expectation 206
4.8 ⋆Jointly Gaussian vectors 209
4.9 Expectation as estimation 211
4.10 ⋆Implications for linear estimation 218
4.11 Correlation and linear estimation 221
4.12 Correlation and covariance functions 228
4.13 ⋆The central limit theorem 231
4.14 Sample averages 234
4.15 Convergence of random variables 236
4.16 Weak law of large numbers 243
4.17 ⋆Strong law of large numbers 245
4.18 Stationarity 249
4.19 Asymptotically uncorrelated processes 255
4.20 Problems 258
5 Second-order theory 275
5.1 Linear filtering of random processes 276
5.2 Linear systems I/O relations 278
5.3 Power spectral densities 284
5.4 Linearly filtered uncorrelated processes 286
5.5 Linear modulation 292
5.6 White noise 296
5.7 ⋆Time averages 299
5.8 ⋆Mean square calculus 303
5.9 ⋆Linear estimation and filtering 331
5.10 Problems 349
6 A menagerie of processes 363
6.1 Discrete time linear models 364
6.2 Sums of iid random variables 369
6.3 Independent stationary increment processes 370
6.4 ⋆Second-order moments of isi processes 373
6.5 Specification of continuous time isi processes 376
6.6 Moving-average and autoregressive processes 378
6.7 The discrete time Gauss–Markov process 380
6.8 Gaussian random processes 381
6.9 The Poisson counting process 382
6.10 Compound processes 385
6.11 Composite random processes 386
6.12 ⋆Exponential modulation 387
6.13 ⋆Thermal noise 392
6.14 Ergodicity 395
6.15 Random fields 398
6.16 Problems 400
Appendix A Preliminaries 411
A.1 Set theory 411
A.2 Examples of proofs 418
A.3 Mappings and functions 422
A.4 Linear algebra 423
A.5 Linear system fundamentals 427
A.6 Problems 431
Appendix B Sums and integrals 436
B.1 Summation 436
B.2 ⋆Double sums 439
B.3 Integration 441
B.4 ⋆The Lebesgue integral 443
Appendix C Common univariate distributions 446
Appendix D Supplementary reading 448
References 453
Index 457

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