Stochastic Methods and Their Applications to Communications Stochastic Differential Equations Approach

Serguei Primak University of Western Ontario, Canada

Valeri Kontorovich Cinvestav-IPN, Mexico

Vladimir Lyandres Ben-Gurion University of the Negev, Israel

John Wiley & Sons, Ltd

Contents

1. Introduction 1 1.1 Preface 1 1.2 Digital Communication Systems 3

2. Random Variables and Their Description 7 2.1 Random Variables and Their Description 7

2.1.1 Definitions and Method of Description 7 2.1.1.1 Classification 7 2.1.1.2 Cumulative Distribution Function 8 2.1.1.3 Probability Density Function 9 2.1.1.4 The Characteristic Function and the

Log-Characteristic Function 10 2.1.1.5 Statistical Averages 11 2.1.1.6 Moments 12 2.1.1.7 Central Moments 12 2.1.1.8 Other Quantities 13 2.1.1.9 Moment and Cumulant Generating

Functions 14 2.1.1.10 Cumulants 15

2.2 Orthogonal Expansions of Probability Densities: Edgeworth and Laguerre Series 16 2.2.1 The Edgeworth Series 17 2.2.2 The Laguerre Series 20 2.2.3 Gram-Charlier Series 22

2.3 Transformation of Random Variables 23 2.3.1 Transformation of a Given PDF into an Arbitrary PDF 25 2.3.2 PDF of a Harmonie Signal with Random Phase 25

2.4 Random Vectors and Their Description 26 2.4.1 CDF, PDF and the Characteristic Function 26 2.4.2 Conditional PDF 28 2.4.3 Numerical Characteristics of a Random Vector 30

2.5 Gaussian Random Vectors 32 2.6 Transformation of Random Vectors 35

2.6.1 PDF of a Sum, Difference, Product and Ratio of Two Random Variables 37

2.6.2 Probability Density of the Magnitude and the Phase of a Complex Random Vector with Jointly Gaussian Components 39 2.6.2.1 Zero Mean Uncorrelated Gaussian Components

of Equal Variance 41

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2.6.2.2 Case of Uncorrelated Components with Equal Variances and Non-Zero Mean 41

2.6.3 PDF of the Maximum (Minimum) of two Random Variables 42 2.6.4 PDF of the Maximum (Minimum) of n Independent

Random Variables 44 2.7 Additional Properties of Cumulants 44

2.7.1 Moment and Cumulant Brackets 46 2.7.2 Properties of Cumulant Brackets 48 2.7.3 More on the Statistical Meaning of Cumulants 49

2.8 Cumulant Equations 49 2.8.1 Non-Linear Transformation of a Random Variable:

Cumulant Method 52 Appendix: Cumulant Brackets and Their Calculations 54

3. Random Processes 59 3.1 General Remarks 59 3.2 Probability Density Function (PDF) 60 3.3 The Characteristic Functions and Cumulative

Distribution Function 63 3.4 Moment Functions and Correlation Functions 64 3.5 Stationary and Non-Stationary Processes 70 3.6 Covariance Functions and Their Properties 71 3.7 Correlation Coefficient 74 3.8 Cumulant Functions 77 3.9 Ergodicity 77 3.10 Power Spectral Density (PSD) 80 3.11 Mutual PSD 82

3.11.1 PSD of a Sum of Two Stationary and Stationary Related Random Processes 83

3.11.2 PSD of a Product of Two Stationary Uncorrelated Processes . . . . 84 3.12 Covariance Function of a Periodic Random Process 85

3.12.1 Harmonie Signal with a Constant Magnitude 85 3.12.2 A Mixture of Harmonie Signals 86 3.12.3 Harmonie Signal with Random Magnitude and Phase 87

3.13 Frequently Used Covariance Functions 88 3.14 Normal (Gaussian) Random Processes 88 3.15 White Gaussian Noise (WGN) 95

4. Advanced Topics in Random Processes 99 4.1 Continuity, Differentiability and Integrability of a Random Process 99

4.1.1 Convergence and Continuity 99 4.1.2 Differentiability 100 4.1.3 Integrability 102

4.2 Elements of System Theory 103 4.2.1 General Remarks 103 4.2.2 Continuous SISO Systems 105 4.2.3 Discrete Linear Systems 107 4.2.4 MIMO Systems 109

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4.2.5 Description of Non-Linear Systems 110 4.3 Zero Memory Non-Linear Transformation of Random Processes 112

4.3.1 Transformation of Moments and Cumulants 112 4.3.1.1 Direct Method 115 4.3.1.2 The Rice Method 116

4.3.2 Cumulant Method 117 4.4 Cumulant Analysis of Non-Linear Transformation of Random Processes . . 118

4.4.1 Cumulants of the Marginal PDF 118 4.4.2 Cumulant Method of Analysis of Non-Gaussian

Random Processes 119 4.5 Linear Transformation of Random Processes 121

4.5.1 General Expression for Moment and Cumulant Functions at the Output of a Linear System 121 4.5.1.1 Transformation of Moment and Cumulant Functions . . . . 122 4.5.1.2 Linear Time-Invariant System Driven by a

Stationary Process 125 4.5.2 Analysis of Linear MIMO Systems 131 4.5.3 Cumulant Method of Analysis of

Linear Transformations 132 4.5.4 Normalization of the Output Process by a Linear System 137

4.6 Outages of Random Processes 140 4.6.1 General Considerations 140 4.6.2 Average Level Crossing Rate and the Average Duration

of the Upward Excursions 141 4.6.3 Level Crossing Rate of a Gaussian Random Process 145 4.6.4 Level Crossing Rate of the Nakagami Process 149 4.6.5 Concluding Remarks 152

4.7 Narrow Band Random Processes 152 4.7.1 Definition of the Envelope and Phase of

Narrow Band Processes 154 4.7.2 The Envelope and the Phase Characteristics 156

4.7.2.1 Blanc-Lapierre Transformation 156 4.7.2.2 Kluyver Equation 160 4.7.2.3 Relations Between Moments of pA„(an) andp,(/) 161 4.7.2.4 The Gram-Charlier Series for p^R(x) and £>,(/) 163

4.7.3 Gaussian Narrow Band Process 166 4.7.3.1 First Order Statistics 166 4.7.3.2 Correlation Function of the In-phase and Quadrature

Components 168 4.7.3.3 Second Order Statistics of the Envelope 169 4.7.3.4 Level Crossing Rate 172

4.7.4 Examples of Non-Gaussian Narrow Band Random Processes 173 4.7.4.1 K Distribution 173 4.7.4.2 Gamma Distribution 175 4.7.4.3 Log-Normal Distribution 175 4.7.4.4 A Narrow Band Process with Nakagami

Distributed Envelope 177

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4.8 Spherically Invariant Processes 181 4.8.1 Definitions 181 4.8.2 Properties 182

4.8.2.1 Joint PDF of a SIRV 182 4.8.2.2 Narrow Band SIRVs 183

4.8.3 Examples 184

5. Markov Processes and Their Description 189 5.1 Definitions 189

5.1.1 Markov Chains 190 5.1.2 Markov Sequences 203 5.1.3 A Discrete Markov Process 207 5.1.4 Continuous Markov Processes 212 5.1.5 Differential Form of the Kolmogorov-Chapman Equation 214

5.2 Some Important Markov Random Processes 217 5.2.1 One-Dimensional Random Walk 217

5.2.1.1 Unrestricted Random Walk 219 5.2.2 Markov Processes with Jumps 221

5.2.2.1 The Poisson Process 221 5.2.2.2 A Birfh Process 223 5.2.2.3 A Death Process 224 5.2.2.4 A Death and Birth Process 224

5.3 The Fokker-Planck Equation 227 5.3.1 Preliminary Remarks 227 5.3.2 Derivation of the Fokker-Planck Equation 227 5.3.3 Boundary Conditions 231 5.3.4 Discrete Model of a Continuous Homogeneous

Markov Process 234 5.3.5 On the Forward and Backward Kolmogorov Equations 235 5.3.6 Methods of Solution of the Fokker-Planck Equation 236

5.3.6.1 Method of Separation of Variables 236 5.3.6.2 The Laplace Transform Method 243 5.3.6.3 Transformation to the Schrödinger Equations 244

5.4 Stochastic Differential Equations 245 5.4.1 Stochastic Integrals 246

5.5 Temporal Symmetry of the Diffusion Markov Process 257 5.6 High Order Spectra of Markov Diffusion Processes 258 5.7 Vector Markov Processes 263

5.7.1 Definitions 263 5.7.1.1 A Gaussian Process with a Rational Spectrum 270

5.8 On Properties of Correlation Functions of One-Dimensional Markov Processes 271

6. Markov Processes with Random Structures 275 6.1 Introduction 275 6.2 Markov Processes with Random Structure and Their

Statistical Description 279

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6.2.1 Processes with Random Structure and Their Classification 279 6.2.2 Statistical Description of Markov Processes

with Random Structure 280 6.2.3 Generalized Fokker-Planck Equation for Random Processes with

Random Structure and Distributed Transitions 281 6.2.4 Moment and Cumulant Equations of a Markov Process

with Random Structure 288 6.3 Approximate Solution of the Generalized Fokker-Planck Equations 295

6.3.1 Gram-Charlier Series Expansion 296 6.3.1.1 Eigenfunction Expansion 296 6.3.1.2 Small Intensity Approximation 297 6.3.1.3 Form of the Solution for Large Intensity 302

6.3.2 Solution by the Perturbation Method for the Case of Low Intensities of Switching 304 6.3.2.1 General Small Parameter Expansion of Eigenvalues

and Eigenfunctions 304 6.3.2.2 Perturbation of *0(JC) 305

6.3.3 High Intensity Solution 310 6.3.3.1 Zero Average Current Condition 310 6.3.3.2 Asymptotic Solution P<yo{x) 311 6.3.3.3 Case of a Finite Intensity v 314

6.4 Concluding Remarks 317

7. Synthesis of Stochastic Differential Equations 321 7.1 Introduction 321 7.2 Modeling of a Scalar Random Process Using a First Order SDE 322

7.2.1 General Synthesis Procedure for the First Order SDE 322 7.2.2 Synthesis of an SDE with PDF Defined on a Part of the

Real Axis 326 7.2.3 Synthesis of A Processes 329 7.2.4 Non-Diffusion Markov Models of Non-Gaussian Exponentially

Correlated Processes 334 7.2.4.1 Exponentially Correlated Markov Chain—DAR(l) and

Its Continuous Equivalent 335 7.2.4.2 A Mixed Process with Exponential Correlation 341

7.3 Modeling of a One-Dimensional Random Process on the Basis of a Vector SDE 347 7.3.1 Preliminary Comments 347 7.3.2 Synthesis Procedure of a (A,a>) Process 347 7.3.3 Synthesis of a Narrow Band Process Using a Second

Order SDE 351 7.3.3.1 Synthesis of a Narrow Band Random Process Using a

Duffing Type SDE 352 7.3.3.2 An SDE of the Van Der Pol Type 356

7.4 Synthesis of a One-Dimensional Process with a Gaussian Marginal PDF and Non-Exponential Correlation 361

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7.5 Synthesis of Compound Processes 364 7.5.1 Compound A Process 365 7.5.2 Synthesis of a Compound Process with a Symmetrical PDF 367

7.6 Synthesis of Impulse Processes 369 7.6.1 Constant Magnitude Excitation 370 7.6.2 Exponentially Distributed Excitation 371

7.7 Synthesis of an SDE with Random Structure 371

8. Applications 377 8.1 Continuous Communication Channels 377

8.1.1 A Mathematical Model of a Mobile Satellite Communication Channel 377

8.1.2 Modeling of a Single-Path Propagation 380 8.1.2.1 A Process with a Given PDF of the Envelope

and Given Correlation Interval 380 8.1.2.2 A Process with a Given Spectrum and

Sub-Rayleigh PDF 383 8.2 An Error Flow Simulator for Digital Communication Channels 388

8.2.1 Error Flow in Digital Communication Systems 389 8.2.2 A Model of Error Flow in a Digital Channel with Fading 389 8.2.3 SDE Model of a Buoyant Antenna-Satellite Link 391

8.2.3.1 Physical Model 391 8.2.3.2 Phenomenological Model 392 8.2.3.3 Numerical Simulation 395

8.3 A Simulator of Radar Sea Clutter with a Non-Rayleigh Envelope 397 8.3.1 Modeling and Simulation of the AT-Distributed Clutter 397 8.3.2 Modeling and Simulation of the Weibull Clutter 404

8.4 Markov Chain Models in Communications 408 8.4.1 Two-State Markov Chain—Gilbert Model 408 8.4.2 Wang-Moayeri Model 409 8.4.3 Independence of the Channel State Model on the

Actual Fading Distribution 418 8.4.4 A Rayleigh Channel with Diversity 418 8.4.5 Fading Channel Models 419 8.4.6 Higher Order Models 421

8.5 Markov Chain for Different Conditions of the Channel 422

Index 433

As an extra resource we have set up a companion Website for our book containing supple-mentary material devoted to the numerical Simulation of stochastic differential equations and description, modeling and Simulation of impulse random processes. Additional reference information is also available on the Website. Please go to the following URL and have a look: ftp://ftp.wiley.co.uk/pub/books/primak/