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 Author Primak, Serguei Title Stochastic Methods and Their Applications to Communications : Stochastic Differential Equations Approach Imprint Hoboken : John Wiley & Sons, Incorporated, 2004 ©2005
 Edition 1st ed Descript 1 online resource (448 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Note Stochastic Methods and Their Applications to Communications -- Contents -- 1. Introduction -- 1.1 Preface -- 1.2 Digital Communication Systems -- 2. Random Variables and Their Description -- 2.1 Random Variables and Their Description -- 2.1.1 Definitions and Method of Description -- 2.1.1.1 Classification -- 2.1.1.2 Cumulative Distribution Function -- 2.1.1.3 Probability Density Function -- 2.1.1.4 The Characteristic Function and the Log-Characteristic Function -- 2.1.1.5 Statistical Averages -- 2.1.1.6 Moments -- 2.1.1.7 Central Moments -- 2.1.1.8 Other Quantities -- 2.1.1.9 Moment and Cumulant Generating Functions -- 2.1.1.10 Cumulants -- 2.2 Orthogonal Expansions of Probability Densities: Edgeworth and Laguerre Series -- 2.2.1 The Edgeworth Series -- 2.2.2 The Laguerre Series -- 2.2.3 Gram-Charlier Series -- 2.3 Transformation of Random Variables -- 2.3.1 Transformation of a Given PDF into an Arbitrary PDF -- 2.3.2 PDF of a Harmonic Signal with Random Phase -- 2.4 Random Vectors and Their Description -- 2.4.1 CDF, PDF and the Characteristic Function -- 2.4.2 Conditional PDF -- 2.4.3 Numerical Characteristics of a Random Vector -- 2.5 Gaussian Random Vectors -- 2.6 Transformation of Random Vectors -- 2.6.1 PDF of a Sum, Difference, Product and Ratio of Two Random Variables -- 2.6.2 Probability Density of the Magnitude and the Phase of a Complex Random Vector with Jointly Gaussian Components -- 2.6.2.1 Zero Mean Uncorrelated Gaussian Components of Equal Variance -- 2.6.2.2 Case of Uncorrelated Components with Equal Variances and Non-Zero Mean -- 2.6.3 PDF of the Maximum (Minimum) of two Random Variables -- 2.6.4 PDF of the Maximum (Minimum) of n Independent Random Variables -- 2.7 Additional Properties of Cumulants -- 2.7.1 Moment and Cumulant Brackets -- 2.7.2 Properties of Cumulant Brackets -- 2.7.3 More on the Statistical Meaning of Cumulants 2.8 Cumulant Equations -- 2.8.1 Non-Linear Transformation of a Random Variable: Cumulant Method -- Appendix: Cumulant Brackets and Their Calculations -- 3. Random Processes -- 3.1 General Remarks -- 3.2 Probability Density Function (PDF) -- 3.3 The Characteristic Functions and Cumulative Distribution Function -- 3.4 Moment Functions and Correlation Functions -- 3.5 Stationary and Non-Stationary Processes -- 3.6 Covariance Functions and Their Properties -- 3.7 Correlation Coefficient -- 3.8 Cumulant Functions -- 3.9 Ergodicity -- 3.10 Power Spectral Density (PSD) -- 3.11 Mutual PSD -- 3.11.1 PSD of a Sum of Two Stationary and Stationary Related Random Processes -- 3.11.2 PSD of a Product of Two Stationary Uncorrelated Processes -- 3.12 Covariance Function of a Periodic Random Process -- 3.12.1 Harmonic Signal with a Constant Magnitude -- 3.12.2 A Mixture of Harmonic Signals -- 3.12.3 Harmonic Signal with Random Magnitude and Phase -- 3.13 Frequently Used Covariance Functions -- 3.14 Normal (Gaussian) Random Processes -- 3.15 White Gaussian Noise (WGN) -- 4. Advanced Topics in Random Processes -- 4.1 Continuity, Differentiability and Integrability of a Random Process -- 4.1.1 Convergence and Continuity -- 4.1.2 Differentiability -- 4.1.3 Integrability -- 4.2 Elements of System Theory -- 4.2.1 General Remarks -- 4.2.2 Continuous SISO Systems -- 4.2.3 Discrete Linear Systems -- 4.2.4 MIMO Systems -- 4.2.5 Description of Non-Linear Systems -- 4.3 Zero Memory Non-Linear Transformation of Random Processes -- 4.3.1 Transformation of Moments and Cumulants -- 4.3.1.1 Direct Method -- 4.3.1.2 The Rice Method -- 4.3.2 Cumulant Method -- 4.4 Cumulant Analysis of Non-Linear Transformation of Random Processes -- 4.4.1 Cumulants of the Marginal PDF -- 4.4.2 Cumulant Method of Analysis of Non-Gaussian Random Processes -- 4.5 Linear Transformation of Random Processes 4.5.1 General Expression for Moment and Cumulant Functions at the Output of a Linear System -- 4.5.1.1 Transformation of Moment and Cumulant Functions -- 4.5.1.2 Linear Time-Invariant System Driven by a Stationary Process -- 4.5.2 Analysis of Linear MIMO Systems -- 4.5.3 Cumulant Method of Analysis of Linear Transformations -- 4.5.4 Normalization of the Output Process by a Linear System -- 4.6 Outages of Random Processes -- 4.6.1 General Considerations -- 4.6.2 Average Level Crossing Rate and the Average Duration of the Upward Excursions -- 4.6.3 Level Crossing Rate of a Gaussian Random Process -- 4.6.4 Level Crossing Rate of the Nakagami Process -- 4.6.5 Concluding Remarks -- 4.7 Narrow Band Random Processes -- 4.7.1 Definition of the Envelope and Phase of Narrow Band Processes -- 4.7.2 The Envelope and the Phase Characteristics -- 4.7.2.1 Blanc-Lapierre Transformation -- 4.7.2.2 Kluyver Equation -- 4.7.2.3 Relations Between Moments of p(A(n)) (a(n)) and p(i)(I) -- 4.7.2.4 The Gram-Charlier Series for p(xR) (x) and p(i)(I) -- 4.7.3 Gaussian Narrow Band Process -- 4.7.3.1 First Order Statistics -- 4.7.3.2 Correlation Function of the In-phase and Quadrature Components -- 4.7.3.3 Second Order Statistics of the Envelope -- 4.7.3.4 Level Crossing Rate -- 4.7.4 Examples of Non-Gaussian Narrow Band Random Processes -- 4.7.4.1 K Distribution -- 4.7.4.2 Gamma Distribution -- 4.7.4.3 Log-Normal Distribution -- 4.7.4.4 A Narrow Band Process with Nakagami Distributed Envelope -- 4.8 Spherically Invariant Processes -- 4.8.1 Definitions -- 4.8.2 Properties -- 4.8.2.1 Joint PDF of a SIRV -- 4.8.2.2 Narrow Band SIRVs -- 4.8.3 Examples -- 5. Markov Processes and Their Description -- 5.1 Definitions -- 5.1.1 Markov Chains -- 5.1.2 Markov Sequences -- 5.1.3 A Discrete Markov Process -- 5.1.4 Continuous Markov Processes 5.1.5 Differential Form of the Kolmogorov-Chapman Equation -- 5.2 Some Important Markov Random Processes -- 5.2.1 One-Dimensional Random Walk -- 5.2.1.1 Unrestricted Random Walk -- 5.2.2 Markov Processes with Jumps -- 5.2.2.1 The Poisson Process -- 5.2.2.2 A Birth Process -- 5.2.2.3 A Death Process -- 5.2.2.4 A Death and Birth Process -- 5.3 The Fokker-Planck Equation -- 5.3.1 Preliminary Remarks -- 5.3.2 Derivation of the Fokker-Planck Equation -- 5.3.3 Boundary Conditions -- 5.3.4 Discrete Model of a Continuous Homogeneous Markov Process -- 5.3.5 On the Forward and Backward Kolmogorov Equations -- 5.3.6 Methods of Solution of the Fokker-Planck Equation -- 5.3.6.1 Method of Separation of Variables -- 5.3.6.2 The Laplace Transform Method -- 5.3.6.3 Transformation to the Schrödinger Equations -- 5.4 Stochastic Differential Equations -- 5.4.1 Stochastic Integrals -- 5.5 Temporal Symmetry of the Diffusion Markov Process -- 5.6 High Order Spectra of Markov Diffusion Processes -- 5.7 Vector Markov Processes -- 5.7.1 Definitions -- 5.7.1.1 A Gaussian Process with a Rational Spectrum -- 5.8 On Properties of Correlation Functions of One-Dimensional Markov Processes -- 6. Markov Processes with Random Structures -- 6.1 Introduction -- 6.2 Markov Processes with Random Structure and Their Statistical Description -- 6.2.1 Processes with Random Structure and Their Classification -- 6.2.2 Statistical Description of Markov Processes with Random Structure -- 6.2.3 Generalized Fokker-Planck Equation for Random Processes with Random Structure and Distributed Transitions -- 6.2.4 Moment and Cumulant Equations of a Markov Process with Random Structure -- 6.3 Approximate Solution of the Generalized Fokker-Planck Equations -- 6.3.1 Gram-Charlier Series Expansion -- 6.3.1.1 Eigenfunction Expansion -- 6.3.1.2 Small Intensity Approximation 6.3.1.3 Form of the Solution for Large Intensity -- 6.3.2 Solution by the Perturbation Method for the Case of Low Intensities of Switching -- 6.3.2.1 General Small Parameter Expansion of Eigenvalues and Eigenfunctions -- 6.3.2.2 Perturbation of Y(0)(x) -- 6.3.3 High Intensity Solution -- 6.3.3.1 Zero Average Current Condition -- 6.3.3.2 Asymptotic Solution P(x) -- 6.3.3.3 Case of a Finite Intensity v -- 6.4 Concluding Remarks -- 7. Synthesis of Stochastic Differential Equations -- 7.1 Introduction -- 7.2 Modeling of a Scalar Random Process Using a First Order SDE -- 7.2.1 General Synthesis Procedure for the First Order SDE -- 7.2.2 Synthesis of an SDE with PDF Defined on a Part of the Real Axis -- 7.2.3 Synthesis of l Processes -- 7.2.4 Non-Diffusion Markov Models of Non-Gaussian Exponentially Correlated Processes -- 7.2.4.1 Exponentially Correlated Markov Chain-DAR(1) and Its Continuous Equivalent -- 7.2.4.2 A Mixed Process with Exponential Correlation -- 7.3 Modeling of a One-Dimensional Random Process on the Basis of a Vector SDE -- 7.3.1 Preliminary Comments -- 7.3.2 Synthesis Procedure of a (l, w) Process -- 7.3.3 Synthesis of a Narrow Band Process Using a Second Order SDE -- 7.3.3.1 Synthesis of a Narrow Band Random Process Using a Duffing Type SDE -- 7.3.3.2 An SDE of the Van Der Pol Type -- 7.4 Synthesis of a One-Dimensional Process with a Gaussian Marginal PDF and Non-Exponential Correlation -- 7.5 Synthesis of Compound Processes -- 7.5.1 Compound L Process -- 7.5.2 Synthesis of a Compound Process with a Symmetrical PDF -- 7.6 Synthesis of Impulse Processes -- 7.6.1 Constant Magnitude Excitation -- 7.6.2 Exponentially Distributed Excitation -- 7.7 Synthesis of an SDE with Random Structure -- 8. Applications -- 8.1 Continuous Communication Channels -- 8.1.1 A Mathematical Model of a Mobile Satellite Communication Channel 8.1.2 Modeling of a Single-Path Propagation 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 equation Essential 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 Description based on publisher supplied metadata and other sources Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries Link Print version: Primak, Serguei Stochastic Methods and Their Applications to Communications : Stochastic Differential Equations Approach Hoboken : John Wiley & Sons, Incorporated,c2004 9780470847411 Subject Telecommunication -- Mathematics.;Stochastic differential equations Electronic books Alt Author Lyandres, Vladimir Kontorovich, Valeri
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