An Introduction to Statistical Signal

This book describes the essential tools and techniques of statistical signal processing. At every stage theoretical ideas are linked to specific applications in communications and signal processing using a range of carefully chosen examples. The book begins with a development of basic probability, random objects, expectation, and second order moment theory followed by a wide variety of examples of the most popular random process models and their basic uses and properties. Specific applications to the analysis of random signals and systems for communicating, estimating, detecting, modulating, and other processing of signals are interspersed throughout the book. Hundreds of homework problems are included and the book is ideal for graduate students of electrical engineering and applied mathematics. It is also a useful reference for researchers in signal processing and communications.

Preface

ix

Acknowledgements

xii

Glossary

xiii

1 Introduction

1

(9)

2 Probability

10

(72)

2.1 Introduction

10

(4)

2.2 Spinning pointers and flipping coins

14

(8)

2.3 Probability spaces

22

(22)

2.4 Discrete probability spaces

44

(10)

2.5 Continuous probability spaces

54

(14)

2.6 Independence

68

(2)

2.7 Elementary conditional probability

70

(3)

2.8 Problems

73

(9)

3 Random variables, vectors, and processes

82

(100)

3.1 Introduction

82

(11)

3.2 Random variables

93

(9)

3.3 Distributions of random variables

102

(10)

3.4 Random vectors and random processes

112

(3)

3.5 Distributions of random vectors

115

(9)

3.6 Independent random variables

124

(3)

3.7 Conditional distributions

127

(5)

3.8 Statistical detection and classification

132

(3)

3.9 Additive noise

135

(7)

3.10 Binary detection in Gaussian noise

142

(2)

3.11 Statistical estimation

144

(1)

3.12 Characteristic functions

145

(6)

3.13 Gaussian random vectors

151

(1)

3.14 Simple random processes

152

(4)

3.15 Directly given random processes

156

(2)

3.16 Discrete time Markov processes

158

(9)

3.17 *Nonelementary conditional probability

167

(1)

3.18 Problems

168

(14)

4 Expectation and averages

182

(93)

4.1 Averages

182

(3)

4.2 Expectation

185

(3)

4.3 Functions of random variables

188

(7)

4.4 Functions of several random variables

195

(1)

4.5 Properties of expectation

195

(2)

4.6 Examples

197

(9)

4.7 Conditional expectation

206

(3)

4.8 *Jointly Gaussian vectors

209

(2)

4.9 Expectation as estimation

211

(7)

4.10 *Implications for linear estimation

218

(3)

4.11 Correlation and linear estimation

221

(7)

4.12 Correlation and covariance functions

228

(3)

4.13 *The central limit theorem

231

(3)

4.14 Sample averages

234

(2)

4.15 Convergence of random variables

236

(7)

4.16 Weak law of large numbers

243

(2)

4.17 *Strong law of large numbers

245

(4)

4.18 Stationarity

249

(6)

4.19 Asymptotically uncorrelated processes

255

(3)

4.20 Problems

258

(17)

5 Second-order theory

275

(88)

5.1 Linear filtering of random processes

276

(2)

5.2 Linear systems I/O relations

278

(6)

5.3 Power spectral densities

284

(2)

5.4 Linearly filtered uncorrelated processes

286

(6)

5.5 Linear modulation

292

(4)

5.6 White noise

296

(3)

5.7 *Time averages

299

(4)

5.8 *Mean square calculus

303

(28)

5.9 *Linear estimation and filtering

331

(18)

5.10 Problems

349

(14)

6 A menagerie of processes

363

(48)

6.1 Discrete time linear models

364

(5)

6.2 Sums of iid random variables

369

(1)

6.3 Independent stationary increment processes

370

(3)

6.4 *Second-order moments of isi processes

373

(3)

6.5 Specification of continuous time isi processes

376

(2)

6.6 Moving-average and autoregressive processes

378

(2)

6.7 The discrete time Gauss-Markov process

380

(1)

6.8 Gaussian random processes

381

(1)

6.9 The Poisson counting process

382

(3)

6.10 Compound processes

385

(1)

6.11 Composite random processes

386

(1)

6.12 *Exponential modulation

387

(5)

6.13 *Thermal noise

392

(3)

6.14 Ergodicity

395

(3)

6.15 Random fields

398

(2)

6.16 Problems

400

(11)

Appendix A Preliminaries

411

(25)

A.1 Set theory

411

(7)

A.2 Examples of proofs

418

(4)

A.3 Mappings and functions

422

(1)

A.4 Linear algebra

423

(4)

A.5 Linear system fundamentals

427

(4)

A.6 Problems

431

(5)

Appendix B Sums and integrals

436

(10)

B.1 Summation

436

(3)

B.2 *Double sums

439

(2)

B.3 Integration

441

(2)

B.4 *The Lebesgue integral

443

(3)

Appendix C Common univariate distributions

446

(2)

Appendix D Supplementary reading

448

(5)

References

453

(4)

Index

457

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An Introduction to Statistical Signal Processing

Summary

Table of Contents

Supplemental Materials