A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES
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1 A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES ROY M. HOWARD Department of Electrical Engineering & Computing Curtin University of Technology Perth, Australia WILEY
2 CONTENTS Preface xiii 1 A Signal Theoretic Introduction to Random Processes Introduction Motivation Book Overview 8 2 Background: Mathematics II 2.1 Introduction Set Theory Function Theory Measure Theory Measurable Functions Lebesgue Integration Convergence Lebesgue-Stieltjes Measure Lebesgue-Stieltjes Integration Miscellaneous Results Problems 62 3 Background: Signal Theory Introduction Signal Orthogonality Theory for Dirichlet Points 75
3 viii CONTENTS 3.4 Dirac Delta Fourier Theory Signal Power The Power Spectral Density The Autocorrelation Function Power Spectral Density-Autocorrelation Function Results for the Infinite Interval Convergence of Fourier Coefficients Cramer's Representation and Transform Problems Background: Probability and Random Variable Theory Introduction Basic Concepts: Experiments-Probability Theory The Random Variable Discrete and Continuous Random Variables Standard Random Variables Functions of a Random Variable Expectation Generation of Data Consistent with Defined PDF Vector Random Variables Pairs of Random Variables Covariance and Correlation Sums of Random Variables Jointly Gaussian Random Variables Stirling's Formula and Approximations to Binomial Problems Introduction to Random Processes Random Processes Definition of a Random Process Examples of Random Processes Experiments and Experimental Outcomes Prototypical Experiments Random Variables Defined by a Random Process Classification of Random Processes Classification: One-Dimensional RPs Sums of Random Processes Problems Prototypical Random Processes Introduction Bernoulli Random Processes Poisson Random Processes 246
4 CONTENTS ix 6.4 Clustered Random Processes Signalling Random Processes Jitter White Noise //Noise Birth^Death Random Processes Orthogonal Increment Random Processes Linear Filtering of Random Processes Summary of Random Processes Problems Characterizing Random Processes Introduction Time Evolution of PMF or PDF First-, Second-, and Higher-Order Characterization Autocorrelation and Power Spectral Density Correlation Notes on Average Power and Average Energy Classification: Stationarity vs Non-Stationarity Cramer's Representation State Space Characterization of Random Processes Time Series Characterization Problems PMF and PDF Evolution Introduction Probability Mass/Density Function Estimation Non/Semi-parametric PDF Estimation PMF/PDF Evolution: Signal Plus Noise PMF Evolution of a Random Walk PDF Evolution: Brownian Motion PDF Evolution: Signalling Random Process PDF Evolution: Generalized Shot Noise PDF Evolution: Switching in a CMOS Inverter PDF Evolution: General Case Problems The Autocorrelation Function Introduction Notation and Definitions Basic Results and Independence Information Sinusoid with Random Amplitude and Phase Random Telegraph Signal Generalized Shot Noise 424
5 X CONTENTS 9.7 Signalling Random Process-Fixed Pulse Case Generalized Signalling Random Process Autocorrelation: Jittered Random Processes Random Walk Problems Power Spectral Density Theory Introduction Power Spectral Density Theory Power Spectral Density of a Periodic Pulse Train PSD of a Signalling Random Process Digital to Analogue Conversion PSD of Shot Noise Random Processes White Noise //Noise PSD of a Jittered Binary Random Process PSD of a Jittered Pulse Train Problems Order Statistics Introduction Ordered Random Variable Theory Identical RVs With Uniform Distribution Uniform Distribution and Infinite Interval Problems Poisson Point Random Processes Introduction Characterizing Poisson Random Processes PMF: Number of Points in a Subset of an Interval Results From Order Statistics Alternative Characterization for Infinite Interval Modelling with Unordered or Ordered Times Zero Crossing Times of Random Telegraph Signal Point Processes: The General Case Problems Birth-Death Random Processes Introduction Defining and Characterizing Birth-Death Processes Constant Birth Rate, Zero Death Rate Process State Dependent Birth Rate - Zero Death Rate Constant Death Rate, Zero Birth Rate, Process 665
6 CONTENTS XI 13.6 Constant Birth and Constant Death Rate Process Problems The First Passage Time Introduction First Passage Time Approaches: Establishing the First Passage Time Maximum Level and the First Passage Time Solutions for the First Passage Time PDF Problems 695 Reference Material 709 References 717 Index 721
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