Signal Detection And Estimation (Artech House Radar Library)

711 Pages·2005·6.16 MB·English

by  Mourad Barkat

Signal Detection and Estimation
Second Edition
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Signal Detection and Estimation
Second Edition
Mourad Barkat
artechhouse.com
Library of Congress Cataloging-in-Publication Data
Barkat, Mourad.
Signal detection and estimation/Mourad Barkat.—2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 1-58053-070-2
1. Signal detection. 2. Stochastic processes. 3. Estimation theory. 4. Radar. I. Title.
TK5102.5.B338 2005
621.382'2—dc22 2005048031
British Library Cataloguing in Publication Data
Barkat, Mourad
Signal detection and estimation.—2nd ed.—(Artech House radar library)
1. Signal detection 2. Stochastic processes 3. Estimation theory
I. Title
621.3'822
ISBN-10: 1-58053-070-2
Cover design by Igor Valdman
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International Standard Book Number: 1-58053-070-2
10 9 8 7 6 5 4 3 2 1
To my wife and my children

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
vii
viii Signal Detection and Estimation
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
Contents ix
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