This engaging introduction to random processes provides students with the critical tools needed to design and evaluate engineering systems that must operate reliably in uncertain environments. A brief review of probability theory and real analysis of deterministic functions sets the stage for understanding random processes, whilst the underlying measure theoretic notions are explained in an intuitive, straightforward style. Students will learn to manage the complexity of randomness through the use of simple classes of random processes, statistical means and correlations, asymptotic analysis, sampling, and effective algorithms. Key topics covered include: • Calculus of random processes in linear systems • Kalman and Wiener filtering • Hidden Markov models for statistical inference • The estimation maximization (EM) algorithm • An introduction to martingales and concentration inequalities. Understanding of the key concepts is reinforced through over 100 worked examples and 300 thoroughly tested homework problems (half of which are solved in detail at the end of the book).
Author(s): Bruce Hajek
Publisher: Cambridge University Press
Year: 2015
Language: English
Pages: 427
Hajek, Bruce. Random processes for engineers (CUP,2015)(ISBN 9781107100121)(600dpi)(427p) ......Page 4
Copyright v ......Page 5
Contents vii......Page 7
Preface xi ......Page 10
1.1 The axioms of probability theory 1 ......Page 13
1.2 Independence and conditional probability 5 ......Page 17
1.3 Random variables and their distribution 8 ......Page 20
1.4 Functions of a random variable 11 ......Page 23
1.5 Expectation of a random variable 16 ......Page 28
1.6 Frequently used distributions 20 ......Page 32
1.7 Failure rate functions 24 ......Page 36
1.8 Jointly distributed random variables 25 ......Page 37
1.10 Correlation and covariance 27 ......Page 39
1.11 Transformation of random vectors 29 ......Page 41
2.1 Four definitions of convergence of random variables 40 ......Page 52
2.2 Cauchy criteria for convergence of random variables 51 ......Page 63
2.3 Limit theorems for sums of independent random variables 55 ......Page 67
2.4 Convex functions and Jensen’s inequality 58 ......Page 70
2.5 Chemoff bound and large deviations theory 60 ......Page 72
3.1 Basic definitions and properties 73 ......Page 85
3.2 The orthogonality principle for minimum mean square error estimation 75 ......Page 87
3.3.1 Conditional expectation as a projection 79 ......Page 91
3.3.2 Linear estimators 81 ......Page 93
3.3.3 Comparison of the estimators 82 ......Page 94
3.4 Joint Gaussian distribution and Gaussian random vectors 84 ......Page 96
3.5 Linear innovations sequences 90 ......Page 102
3.6 Discrete-time Kalman filtering 91 ......Page 103
4.1 Definition of a random process 103 ......Page 115
4.2 Random walks and gambler’s ruin 106 ......Page 118
4.3 Processes with independent increments and martingales 108 ......Page 120
4.4 Brownian motion 110 ......Page 122
4.5 Counting processes and the Poisson process 111 ......Page 123
4.6 Stationarity 115 ......Page 127
4.8 Conditional independence and Markov processes 118 ......Page 130
4.9 Discrete-state Markov processes 122 ......Page 134
4.10 Space-time structure of discrete-state Markov processes 129 ......Page 141
5.1 A bit of estimation theory 143 ......Page 155
5.2 The expectation-maximization (EM) algorithm 148 ......Page 160
5.3 Hidden Markov models 152 ......Page 164
5.3.1 Posterior state probabilities and the forward-backward algorithm 153 ......Page 165
5.3.2 Most likely state sequence - Viterbi algorithm 157 ......Page 169
5.3.3 The Baum-Welch algorithm, or EM algorithm for HMM 158 ......Page 170
5.4 Notes 160 ......Page 172
6.1 Examples with finite state space 167 ......Page 179
6.2 Classification and convergence of discrete-time Markov processes 169 ......Page 181
6.3 Classification and convergence of continuous-time Markov processes 172 ......Page 184
6.4 Classification of birth-death processes 175 ......Page 187
6.5 Time averages vs. statistical averages 177 ......Page 189
6.6 Queueing systems, M/M/1 queue and Little’s law 178 ......Page 190
6.7 Mean arrival rate, distributions seen by arrivals, and PASTA 181 ......Page 193
6.8 More examples of queueing systems modeled as Markov birth-death processes 184 ......Page 196
6.9.1 Stability criteria for discrete-time processes 186 ......Page 198
6.9.2 Stability criteria for continuous-time processes 193 ......Page 205
7.1 Continuity of random processes 206 ......Page 218
7.2 Mean square differentiation of random processes 212 ......Page 224
7.3 Integration of random processes 216 ......Page 228
7.4 Ergodicity 223 ......Page 235
7.5 Complexification, Part I 229 ......Page 241
7.6 The Karhunen-Loeve expansion 231 ......Page 243
7.7 Periodic WSS random processes 238 ......Page 250
8.1 Basic definitions 248 ......Page 260
8.2 Fourier transforms, transfer functions* and power spectral densities 252 ......Page 264
8.3 Discrete-time processes in linear systems 258 ......Page 270
8.4 Baseband random processes 260 ......Page 272
8.5 Narrowband random processes 263 ......Page 275
8.6 Complexification, Part II 269 ......Page 281
9.1 Return of the orthogonality principle 280 ......Page 292
9.2 The causal Wiener filtering problem 283 ......Page 295
9.3 Causal functions and spectral factorization 284 ......Page 296
9.4 Solution of the causal Wiener filtering problem for rational power spectral densities 288 ......Page 300
9.5 Discrete-time Wiener filtering 292 ......Page 304
10.1 Conditional expectation revisited 304 ......Page 316
10.2 Martingales with respect to filtrations 310 ......Page 322
10.3 Azuma-Hoeffding inequality 313 ......Page 325
10.4 Stopping times and the optional sampling theorem 316 ......Page 328
10.5 Notes 321 ......Page 333
11.1 Some notation 325 ......Page 337
11.2 Convergence of sequences of numbers 326 ......Page 338
11.3 Continuity of functions 330 ......Page 342
11.4 Derivatives of functions 331 ......Page 343
11.5.1 Riemann integration 333 ......Page 345
11.5.3 Riemann-Stieltjes integration 335 ......Page 347
11.6 On convergence of the mean 336 ......Page 348
11.7 Matrices 339 ......Page 351
12 Solutions to even numbered problems 344 ......Page 356
References 411 ......Page 423
Index 412 ......Page 424
cover......Page 1
