Stochastic Methods and their Applications to Communications Stochastic Differential Equations Approach

by Primak, Serguei; Kontorovich, Valeri; Lyandres, Vladimir

• ISBN13: 9780470847411
• ISBN10: 0470847417
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-09-03
• Publisher: WILEY

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