SC505 STOCHASTIC PROCESSES Class Notes c Prof. D. Castanon~ & Prof. W. Clem Karl Dept. of Electrical and Computer Engineering Boston University College of Engineering 8 St. Mary's Street Boston, MA 02215 Fall 2004 2 Contents 1 Introduction to Probability 11 1.1 Axioms of Probability . 11 1.2 Conditional Probability and Independence of Events . 13 1.3 Random Variables . 13 1.4 Characterization of Random Variables . 14 1.5 Important Random Variables . 19 1.5.1 Discrete-valued random variables . 19 1.5.2 Continuous-valued random variables . 21 1.6 Pairs of Random Variables . 24 1.7 Conditional Probabilities, Densities, and Expectations . 27 1.8 Random Vectors . 28 1.9 Properties of the Covariance Matrix . 31 1.10 Gaussian Random Vectors . 33 1.11 Inequalities for Random Variables . 35 1.11.1 Markov inequality . 35 1.11.2 Chebyshev inequality . 36 1.11.3 Chernoff Inequality . 36 1.11.4 Jensen's Inequality . 37 1.11.5 Moment Inequalities . 37 2 Sequences of Random Variables 39 2.1 Convergence Concepts for Random Sequences . 39 2.2 The Central Limit Theorem and the Law of Large Numbers . 43 2.3 Advanced Topics in Convergence . 45 2.4 Martingale Sequences . 48 2.5 Extensions of the Law of Large Numbers and the Central Limit Theorem . 50 2.6 Spaces of Random Variables . 52 3 Stochastic Processes and their Characterization 55 3.1 Introduction . 55 3.2 Complete Characterization of Stochastic Processes . 56 3.3 First and Second-Order Moments of Stochastic Processes . 56 3.4 Special Classes of Stochastic Processes . 57 3.5 Properties of Stochastic Processes . 59 3.6 Examples of Random Processes . 61 3.6.1 The Random Walk . 61 3.6.2 The Poisson Process . 62 3.6.3 Digital Modulation: Phase-Shift Keying . 65 3.6.4 The Random Telegraph Process . 66 3.6.5 The Wiener Process and Brownian Motion . 67 3.7 Moment Functions of Vector Processes . 68 3.8 Moments of Wide-sense Stationary Processes . 69 4 CONTENTS 3.9 Power Spectral Density of Wide-Sense Stationary Processes . 71 4 Mean-Square Calculus for Stochastic Processes 75 4.1 Continuity of Stochastic Processes . 75 4.2 Mean-Square Differentiation . 77 4.3 Mean-Square Integration . 79 4.4 Integration and Differentiation of Gaussian Stochastic Processes . 83 4.5 Generalized Mean-Square Calculus . 83 4.6 Ergodicity of Stationary Random Processes . 86 5 Linear Systems and Stochastic Processes 93 5.1 Introduction . 93 5.2 Review of Continuous-time Linear Systems . 93 5.3 Review of Discrete-time Linear Systems . 96 5.4 Extensions to Multivariable Systems . 98 5.5 Second-order Statistics for Vector-Valued Wide-Sense Stationary Processes . 98 5.6 Continuous-time Linear Systems with Random Inputs . 99 6 Sampling of Stochastic Processes 105 6.1 The Sampling Theorem . 105 7 Model Identification for Discrete-Time Processes 111 7.1 Autoregressive Models . 111 7.2 Moving Average Models . 113 7.3 Autoregressive Moving Average (ARMA) Models . 115 7.4 Dealing with non-zero mean processes . 116 8 Detection Theory 117 8.1 Bayesian Binary Hypothesis Testing . 118 8.1.1 Bayes Risk Approach and the Likelihood Ratio Test . 119 8.1.2 Special Cases . 121 8.1.3 Examples . 123 8.2 Performance and the Receiver Operating Characteristic . 125 8.2.1 Properties of the ROC . 128 8.2.2 Detection Based on Discrete-Valued Random Variables . 131 8.3 Other Threshold Strategies . 135 8.3.1 Minimax Hypothesis Testing . 136 8.3.2 Neyman-Pearson Hypothesis Testing . 137 8.4 M-ary Hypothesis Testing . 139 8.4.1 Special Cases . 140 8.4.2 Examples . 141 8.4.3 M-Ary Performance Calculations . 144 8.5 Gaussian Examples . 146 9 Series Expansions and Detection of Stochastic Processes 149 9.1 Deterministic Functions . 149 9.2 Series Expansion of Stochastic Processes . 150 9.3 Detection of Known Signals in Additive White Noise . 154 9.4 Detection of Unknown Signals in White Noise . 156 9.5 Detection of Known Signals in Colored Noise . 157 CONTENTS 5 10 Estimation of Parameters 159 10.1 Introduction . 159 10.2 General Bayesian Estimation . 160 10.2.1 General Bayes Decision Rule . 160 10.2.2 General Bayes Decision Rule Performance . 161 10.3 Bayes Least Square Estimation . 162 10.4 Bayes Maximum A Posteriori (MAP) Estimation . 167 10.5 Bayes Linear Least Square (LLSE) Estimation . 174 10.6 Nonrandom Parameter Estimation . 181 10.6.1 Cramer-Rao Bound . 182 10.6.2 Maximum-Likelihood Estimation . 185 10.6.3 Comparison to MAP estimation . 187 11 LLSE Estimation of Stochastic Processes and Wiener Filtering 189 11.1 Introduction . 189 11.2 Historical Context . 190 11.3 LLSE Problem Solution: The Wiener-Hopf Equation . 191 11.4 Wiener Filtering . 192 11.4.1 Noncausal Wiener Filtering (Wiener Smoothing) . 193 11.4.2 Causal Wiener Filtering . ..