Stochastic methods and their applications to communications: stochastic differential equations approach

439 Pages·2004·2.743 MB·English

by  Serguei Primak, Valeri Kontorovitch, Vladimir Lyandres

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
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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 Harmonic 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
viii CONTENTS
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 Harmonic Signal with a Constant Magnitude . . . . . . . . . . . . . . . 85
3.12.2 A Mixture of Harmonic Signals . . . . . . . . . . . . . . . . . . . . . . . . 86
3.12.3 Harmonic 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
CONTENTS ix
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 p ða Þ and pðIÞ . . . . . . 161
A n i
n
4.7.2.4 The Gram–Charlier Series for p(cid:2) ðxÞ and piðIÞ . . . . . . . . 163
R
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
x CONTENTS
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 Birth 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 Schro¨dinger 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
CONTENTS xi
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 (cid:2) ðxÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 305
0
6.3.3 High Intensity Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6.3.3.1 Zero Average Current Condition . . . . . . . . . . . . . . . . . . 310
6.3.3.2 Asymptotic Solution P ðxÞ. . . . . . . . . . . . . . . . . . . . . . 311
1
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 (cid:3) 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(1) 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 ð(cid:3);!Þ 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
xii CONTENTS
7.5 Synthesis of Compound Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.5.1 Compound (cid:3) 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 K-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 extraresourcewe haveset upa companionwebsite forour 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/
1
Introduction
1.1 PREFACE
A statistical approach to consideration of most problems related to information transmission
became dominant during the past three decades. This can be easily explained if one takes
into account that practically any real signal, propagation media, interference and even
information itself all have an intrinsically random nature.1 This is why D. Middleton, one of
the founders of modern communication theory, coined a term ‘‘statistical theory of
communications’’ [2]. The recent spectacular achievements in information technology are
based mainly on progress in three fundamental areas: communication and information
theory, signal processing and computer and related technologies. All these allow the
transmission of information at a rate close to that limited by the Shannon theorem [2].
In principle, the limitation on the speed of the transmission of information is defined by
the noise and interference which are inherently present in all communication systems. An
accurate description of such impairments is very important for a proper organization and
noise immunity of communication systems. The choice of a relevant interference model is a
crucial moment in design of communication systems and is an important step in their testing
and performance evaluation. The requirements, formulated to improve the performance of
systemsare often conflicting and hard to formulate in away convenient for optimization. On
one side, such models must accurately reflect the main features of the interference under
investigation. On the other hand, a maximally simple description is needed to be applicable
for the massive numerical simulation required to test modern communication system designs.
Historically, two simple processes, so-called White Gaussian Noise (WGN) and the
Poisson Point Process (PPP) have been widely used to obtain first rough estimates of a
system performance: the WGN model of the noise is a good approximation of an additive
wide band noise, caused by a variety of natural phenomena, while the PPP is a good model
for event modeling in a discrete communication channel. Unfortunately, in the majority of
realistic situations these basic models of the noise and errors are not adequate. In realistic
communication channels, a non-stationary non-Gaussian interference and noise are often
present. In addition, these processes are often band-limited, thus showing significant time
correlation, which cannot be represented by WGN and PPP. The following phenomena can
be considered as examples when the simplest models fail to provide an accurate description:
1Havingsaidthat,wewouldliketoacknowledgeanexponentiallyincreasingbodyofliteraturewhichdescribesthe
same phenomena using a rather different approach, based on the chaotic description of signals [1].
StochasticMethodsandTheirApplicationstoCommunications.
S.Primak,V.Kontorovich,V.Lyandres
#2004JohnWiley&Sons,Ltd ISBN:0-470-84741-7