Stochastic Methods and their Applications to Communications Stochastic Differential Equations Approach,9780470847411

Stochastic Methods and their Applications to Communications Stochastic Differential Equations Approach

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Edition: 1st
ISBN13: 9780470847411
ISBN10: 0470847417
Format: Hardcover
Pub. Date: 2004-09-03
Publisher(s): WILEY
List Price: $277.60

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Summary

Stochastic Methods & their Applications to Communications presents a valuable approach to the modelling, synthesis and numerical simulation of random processes with applications in communications and related fields. The authors provide a detailed account of random processes from an engineering point of view and illustrate the concepts with examples taken from the communications area. The discussions mainly focus on the analysis and synthesis of Markov models of random processes as applied to modelling such phenomena as interference and fading in communications. Encompassing both theory and practice, this original text provides a unified approach to the analysis and generation of continuous, impulsive and mixed random processes based on the Fokker-Planck equation for Markov processes. Presents the cumulated analysis of Markov processes Offers a SDE (Stochastic Differential Equations) approach to the generation of random processes with specified characteristics Includes the modelling of communication channels and interfer ences using SDE Features new results and techniques for the of solution of the generalized Fokker-Planck equationEssential reading for researchers, engineers, and graduate and upper year undergraduate students in the field of communications, signal processing, control, physics and other areas of science, this reference will have wide ranging appeal.

Author Biography

Professor Serguei L. Primak, The University of Western Ontario, Canada
Professor Primak is an Associate Professor at the University of the Western Ontario, Canada. His main areas of interest include modelling and performance evaluation of MIMO systems, Markov processes, non-Gaussian random processes and communications aspects of robotic assisted telesurgery. He has co-authored a book "Stochastic Methods and their Applications to Communications: Stochastic Differential Equations Approach", Wiley, 2004.

Professor Valeri Kontorovich, CINVESTAV-IPN, Mexico
Professor Kontorovich is a Professor at the CINVESTAV-IPN, Mexico. His main areas of interest include modelling and performance evaluation of MIMO systems, Markov processes, non-Gaussian random processes, fractal, electromagnetic compatibility and other related topics. Professor Kontorovich has co-authored a book "Stochastic Methods and their Applications to Communications: Stochastic Differential Equations Approach", Wiley, 2004, and has co-authored 4 other books (In Russian) and a large number of publications in the field of communications.

Vladimir Lyandres is the author of Stochastic Methods and their Applications to Communications: Stochastic Differential Equations Approach, published by Wiley.

Table of Contents

1. Introduction 1(6)
1.1 Preface
1(2)
1.2 Digital Communication Systems
3(4)
2. Random Variables and Their Description 7(52)
2.1 Random Variables and Their Description
7(9)
2.1.1 Definitions and Method of Description
7(9)
2.1.1.1 Classification
7(1)
2.1.1.2 Cumulative Distribution Function
8(1)
2.1.1.3 Probability Density Function
9(1)
2.1.1.4 The Characteristic Function and the Log-Characteristic Function
10(1)
2.1.1.5 Statistical Averages
11(1)
2.1.1.6 Moments
12(1)
2.1.1.7 Central Moments
12(1)
2.1.1.8 Other Quantities
13(1)
2.1.1.9 Moment and Cumulant Generating Functions
14(1)
2.1.1.10 Cumulants
15(1)
2.2 Orthogonal Expansions of Probability Densities: Edgeworth and Laguerre Series
16(7)
2.2.1 The Edgeworth Series
17(3)
2.2.2 The Laguerre Series
20(2)
2.2.3 Gram-Charlier Series
22(1)
2.3 Transformation of Random Variables
23(3)
2.3.1 Transformation of a Given PDF into an Arbitrary PDF
25(1)
2.3.2 PDF of a Harmonic Signal with Random Phase
25(1)
2.4 Random Vectors and Their Description
26(6)
2.4.1 CDF, PDF and the Characteristic Function
26(2)
2.4.2 Conditional PDF
28(2)
2.4.3 Numerical Characteristics of a Random Vector
30(2)
2.5 Gaussian Random Vectors
32(3)
2.6 Transformation of Random Vectors
35(9)
2.6.1 PDF of a Sum, Difference, Product and Ratio of Two Random Variables
37(2)
2.6.2 Probability Density of the Magnitude and the Phase of a Complex Random Vector with Jointly Gaussian Components
39(3)
2.6.2.1 Zero Mean Uncorrelated Gaussian Components of Equal Variance
41(1)
2.6.2.2 Case of Uncorrelated Components with Equal Variances and Non-Zero Mean
41(1)
2.6.3 PDF of the Maximum (Minimum) of two Random Variables
42(2)
2.6.4 PDF of the Maximum (Minimum) of n Independent Random Variables
44(1)
2.7 Additional Properties of Cumulants
44(5)
2.7.1 Moment and Cumulant Brackets
46(2)
2.7.2 Properties of Cumulant Brackets
48(1)
2.7.3 More on the Statistical Meaning of Cumulants
49(1)
2.8 Cumulant Equations
49(5)
2.8.1 Non-Linear Transformation of a Random Variable: Cumulant Method
52(2)
Appendix: Cumulant Brackets and Their Calculations
54(5)
3. Random Processes 59(1)
3.1 General Remarks
59(1)
3.2 Probability Density Function (PDF)
60(3)
3.3 The Characteristic Functions and Cumulative Distribution Function
63(1)
3.4 Moment Functions and Correlation Functions
64(6)
3.5 Stationary and Non-Stationary Processes
70(1)
3.6 Covariance Functions and Their Properties
71(3)
3.7 Correlation Coefficient
74(3)
3.8 Cumulant Functions
77(1)
3.9 Ergodicity
77(3)
3.10 Power Spectral Density (PSD)
80(2)
3.11 Mutual PSD
82(3)
3.11.1 PSD of a Sum of Two Stationary and Stationary Related Random Processes
83(1)
3.11.2 PSD of a Product of Two Stationary Uncorrelated Processes
84(1)
3.12 Covariance Function of a Periodic Random Process
85(3)
3.12.1 Harmonic Signal with a Constant Magnitude
85(1)
3.12.2 A Mixture of Harmonic Signals
86(1)
3.12.3 Harmonic Signal with Random Magnitude and Phase
87(1)
3.13 Frequently Used Covariance Functions
88(1)
3.14 Normal (Gaussian) Random Processes
88(7)
3.15 White Gaussian Noise (WGN)
95(4)
4. Advanced Topics in Random Processes 99(90)
4.1 Continuity, Differentiability and Integrability of a Random Process
99(4)
4.1.1 Convergence and Continuity
99(1)
4.1.2 Differentiability
100(2)
4.1.3 Integrability
102(1)
4.2 Elements of System Theory
103(9)
4.2.1 General Remarks
103(2)
4.2.2 Continuous SISO Systems
105(2)
4.2.3 Discrete Linear Systems
107(2)
4.2.4 MIMO Systems
109(1)
4.2.5 Description of Non-Linear Systems
110(2)
4.3 Zero Memory Non-Linear Transformation of Random Processes
112(6)
4.3.1 Transformation of Moments and Cumulants
112(5)
4.3.1.1 Direct Method
115(1)
4.3.1.2 The Rice Method
116(1)
4.3.2 Cumulant Method
117(1)
4.4 Cumulant Analysis of Non-Linear Transformation of Random Processes
118(3)
4.4.1 Cumulants of the Marginal PDF
118(1)
4.4.2 Cumulant Method of Analysis of Non-Gaussian Random Processes
119(2)
4.5 Linear Transformation of Random Processes
121(19)
4.5.1 General Expression for Moment and Cumulant Functions at the Output of a Linear System
121(10)
4.5.1.1 Transformation of Moment and Cumulant Functions
122(3)
4.5.1.2 Linear Time-Invariant System Driven by a Stationary Process
125(6)
4.5.2 Analysis of Linear MIMO Systems
131(1)
4.5.3 Cumulant Method of Analysis of Linear Transformations
132(5)
4.5.4 Normalization of the Output Process by a Linear System
137(3)
4.6 Outages of Random Processes
140(12)
4.6.1 General Considerations
140(1)
4.6.2 Average Level Crossing Rate and the Average Duration of the Upward Excursions
141(4)
4.6.3 Level Crossing Rate of a Gaussian Random Process
145(4)
4.6.4 Level Crossing Rate of the Nakagami Process
149(3)
4.6.5 Concluding Remarks
152(1)
4.7 Narrow Band Random Processes
152(29)
4.7.1 Definition of the Envelope and Phase of Narrow Band Processes
154(2)
4.7.2 The Envelope and the Phase Characteristics
156(10)
4.7.2.1 Blanc-Lapierre Transformation
156(4)
4.7.2.2 Kluyver Equation
160(1)
4.7.2.3 Relations Between Moments of ρΑ&eta(αη) and ρi(Iota)
161(2)
4.7.2.4 The Gram-Charlier Series for ρξR(χ) and ρi(Iota)
163(3)
4.7.3 Gaussian Narrow Band Process
166(7)
4.7.3.1 First Order Statistics
166(2)
4.7.3.2 Correlation Function of the In-phase and Quadrature Components
168(1)
4.7.3.3 Second Order Statistics of the Envelope
169(3)
4.7.3.4 Level Crossing Rate
172(1)
4.7.4 Examples of Non-Gaussian Narrow Band Random Processes
173(8)
4.7.4.1 Kappa Distribution
173(2)
4.7.4.2 Gamma Distribution
175(1)
4.7.4.3 Log-Normal Distribution
175(2)
4.7.4.4 A Narrow Band Process with Nakagami Distributed Envelope
177(4)
4.8 Spherically Invariant Processes
181(8)
4.8.1 Definitions
181(1)
4.8.2 Properties
182(2)
4.8.2.1 Joint PDF of a SIRV
182(1)
4.8.2.2 Narrow Band SIRVs
183(1)
4.8.3 Examples
184(5)
5. Markov Processes and Their Description 189(86)
5.1 Definitions
189(28)
5.1.1 Markov Chains
190(13)
5.1.2 Markov Sequences
203(4)
5.1.3 A Discrete Markov Process
207(5)
5.1.4 Continuous Markov Processes
212(2)
5.1.5 Differential Form of the Kolmogorov-Chapman Equation
214(3)
5.2 Some Important Markov Random Processes
217(10)
5.2.1 One-Dimensional Random Walk
217(4)
5.2.1.1 Unrestricted Random Walk
219(2)
5.2.2 Markov Processes with Jumps
221(6)
5.2.2.1 The Poisson Process
221(2)
5.2.2.2 A Birth Process
223(1)
5.2.2.3 A Death Process
224(1)
5.2.2.4 A Death and Birth Process
224(3)
5.3 The Fokker-Planck Equation
227(18)
5.3.1 Preliminary Remarks
227(1)
5.3.2 Derivation of the Fokker-Planck Equation
227(4)
5.3.3 Boundary Conditions
231(3)
5.3.4 Discrete Model of a Continuous Homogeneous Markov Process
234(1)
5.3.5 On the Forward and Backward Kolmogorov Equations
235(1)
5.3.6 Methods of Solution of the Fokker-Planck Equation
236(9)
5.3.6.1 Method of Separation of Variables
236(7)
5.3.6.2 The Laplace Transform Method
243(1)
5.3.6.3 Transformation to the Schrodinger Equations
244(1)
5.4 Stochastic Differential Equations
245(12)
5.4.1 Stochastic Integrals
246(11)
5.5 Temporal Symmetry of the Diffusion Markov Process
257(1)
5.6 High Order Spectra of Markov Diffusion Processes
258(5)
5.7 Vector Markov Processes
263(8)
5.7.1 Definitions
263(16)
5.7.1.1 A Gaussian Process with a Rational Spectrum
270(1)
5.8 On Properties of Correlation Functions of One-Dimensional Markov Processes
271(4)
6. Markov Processes with Random Structures 275(46)
6.1 Introduction
275(4)
6.2 Markov Processes with Random Structure and Their Statistical Description
279(16)
6.2.1 Processes with Random Structure and Their Classification
279(1)
6.2.2 Statistical Description of Markov Processes with Random Structure
280(1)
6.2.3 Generalized Fokker-Planck Equation for Random Processes with Random Structure and Distributed Transitions
281(7)
6.2.4 Moment and Cumulant Equations of a Markov Process with Random Structure
288(7)
6.3 Approximate Solution of the Generalized Fokker-Planck Equations
295(22)
6.3.1 Gram-Charlier Series Expansion
296(8)
6.3.1.1 Eigenfunction Expansion
296(1)
6.3.1.2 Small Intensity Approximation
297(5)
6.3.1.3 Form of the Solution for Large Intensity
302(2)
6.3.2 Solution by the Perturbation Method for the Case of Low Intensities of Switching
304(6)
6.3.2.1 General Small Parameter Expansion of Eigenvalues and Eigenfunctions
304(1)
6.3.2.2 Perturbation of Ψ0(&chi)
305(5)
6.3.3 High Intensity Solution
310(12)
6.3.3.1 Zero Average Current Condition
310(1)
6.3.3.2 Asymptotic Solution ρinfinity(χ)
311(3)
6.3.3.3 Case of a Finite Intensity ν
314(3)
6.4 Concluding Remarks
317(4)
7. Synthesis of Stochastic Differential Equations 321(56)
7.1 Introduction
321(1)
7.2 Modeling of a Scalar Random Process Using a First Order SDE
322(25)
7.2.1 General Synthesis Procedure for the First Order SDE
322(4)
7.2.2 Synthesis of an SDE with PDF Defined on a Part of the Real Axis
326(3)
7.2.3 Synthesis of λ Processes
329(5)
7.2.4 Non-Diffusion Markov Models of Non-Gaussian Exponentially Correlated Processes
334(13)
7.2.4.1 Exponentially Correlated Markov Chain-DAR(1) and Its Continuous Equivalent
335(6)
7.2.4.2 A Mixed Process with Exponential Correlation
341(6)
7.3 Modeling of a One-Dimensional Random Process on the Basis of a Vector SDE
347(14)
7.3.1 Preliminary Comments
347(1)
7.3.2 Synthesis Procedure of a (λ ω) Process
347(4)
7.3.3 Synthesis of a Narrow Band Process Using a Second Order SDE
351(13)
7.3.3.1 Synthesis of a Narrow Band Random Process Using a Duffing Type SDE
352(4)
7.3.3.2 An SDE of the Van Der Pol Type
356(5)
7.4 Synthesis of a One-Dimensional Process with a Gaussian Marginal PDF and Non-Exponential Correlation
361(3)
7.5 Synthesis of Compound Processes
364(5)
7.5.1 Compound A Process
365(2)
7.5.2 Synthesis of a Compound Process with a Symmetrical PDF
367(2)
7.6 Synthesis of Impulse Processes
369(2)
7.6.1 Constant Magnitude Excitation
370(1)
7.6.2 Exponentially Distributed Excitation
371(1)
7.7 Synthesis of an SDE with Random Structure
371(6)
8. Applications 377(56)
8.1 Continuous Communication Channels
377(11)
8.1.1 A Mathematical Model of a Mobile Satellite Communication Channel
377(3)
8.1.2 Modeling of a Single-Path Propagation
380(8)
8.1.2.1 A Process with a Given PDF of the Envelope and Given Correlation Interval
380(3)
8.1.2.2 A Process with a Given Spectrum and Sub-Rayleigh PDF
383(5)
8.2 An Error Flow Simulator for Digital Communication Channels
388(9)
8.2.1 Error Flow in Digital Communication Systems
389(1)
8.2.2 A Model of Error Flow in a Digital Channel with Fading
389(2)
8.2.3 SDE Model of a Buoyant Antenna-Satellite Link
391(6)
8.2.3.1 Physical Model
391(1)
8.2.3.2 Phenomenological Model
392(3)
8.2.3.3 Numerical Simulation
395(2)
8.3 A Simulator of Radar Sea Clutter with a Non-Rayleigh Envelope
397(11)
8.3.1 Modeling and Simulation of the Kappa-Distributed Clutter
397(7)
8.3.2 Modeling and Simulation of the Weibull Clutter
404(4)
8.4 Markov Chain Models in Communications
408(14)
8.4.1 Two-State Markov Chain-Gilbert Model
408(1)
8.4.2 Wang-Moayeri Model
409(9)
8.4.3 Independence of the Channel State Model on the Actual Fading Distribution
418(1)
8.4.4 A Rayleigh Channel with Diversity
418(1)
8.4.5 Fading Channel Models
419(2)
8.4.6 Higher Order Models
421(1)
8.5 Markov Chain for Different Conditions of the Channel
422(11)
Index 433

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