Principles of Statistical Radiophysics 1

Elements of Random Process Theory

; Yurii A. Kravtsov ; Valeryan I. Tatarskii ; Alexander P. Repyev (Oversetter)

Principles of Statistical Radiophysics 1

Principles of Statistical Radiophysics is concerned with the theory of random functions (processes and fields) treated in close association with a number of ap plications in physics. Les mer
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Principles of Statistical Radiophysics 1

Principles of Statistical Radiophysics is concerned with the theory of random functions (processes and fields) treated in close association with a number of ap plications in physics. Primarily, the book deals with radiophysics in its broadest sense, i.e., viewed as a general theory of oscillations and waves of any physical l nature . This translation is based on the second (two-volume) Russian edition. It appears in four volumes: 1. Elements of Random Process Theory 2. Correlation Theory of Random Processes 3. Elements of Random Fields 4. Wave Propagation Through Random Media. The four volumes are, naturally, to a large extent conceptually interconnected (being linked, for instance, by cross-references); yet for the advanced reader each of them might be of interest on its own. This motivated the division of the Principles into four separate volumes. The text is designed for graduate and postgraduate students majoring in radiophysics, radio engineering, or other branches of physics and technology dealing with oscillations and waves (e.g., acoustics and optics). As a rule, early in their career these students face problems involving the use of random func tions. The book provides a sound basis from which to understand and solve problems at this level. In addition, it paves the way for a more profound study of the mathematical theory, should it be necessary2. The reader is assumed to be familiar with probability theory.

1. General Introduction.- 2. The Bernoulli Problem.- 2.1 The Physical Concept of Probability.- 2.2 Distribution Laws for Random Variables.- 2.3 The Binomial Distribution Law.- 2.4 Examples of Applications of the Binomial Law.- 2.5 Shot Effect. The Poisson Distribution.- 2.6 The De Moivre-Laplace Limit Theorem.- 2.7 Normal or Gaussian Distribution Law.- 2.8 Exercises.- 3. Random Pulses.- 3.1 Statement of the Problem.- 3.2 Characteristic Functions.- 3.3 Distribution Function for a Poisson Pulse Process.- 3.4 Covariance.- 3.5 Some Generalization of the Pulse Problem.- 3.6 Impulse Noise and the Central Limit Theorem.- 3.7 Exercises.- 4. Random Functions.- 4.1 General Definitions.- 4.2 Markov Processes.- 4.3 Stationary Processes.- 4.4 Moments of Random Functions.- 4.5 Correlation Theory.- 4.6 Probabilistic Convergence.- 4.7 Ergodicity of Random Processes.- 4.8 Exercises.- 5. Markov Processes.- 5.1 Preliminary Remarks.- 5.2 Smoluchowski Equation.- 5.3 Markov Process with Discrete States.- 5.4 Transition from Discrete Sequence to Processes with Continuous Sets of States Rayleigh Distribution.- 5.5 Some Generalizations of the Rayleigh Distribution.- 5.6 Continuous Markov Processes. The Einstein-Fokker-Planck Equation.- 5.7 Generalization to Multivariate Random Functions.- 5.8 Fluctuations in the Thomson Vacuum-Tube Oscillator.- 5.9 Fluctuations at Large Self-Oscillation Amplitudes.- 5.10 Rotational Brownian Motion. Random Refraction of a Ray.- 5.11 Stepwise Markov Processes. The Kolmogorov-Feller Equation.- 5.12 First Passage Problem.- 5.13 Exercises.- 6. Stochastic Differential Equations.- 6.1 Statement of the Problem.- 6.2 Random Functions with Independent Increments.- 6.3 Simple Example of a Stochastic Differential Equation.- 6.4 General Case of First-Order Equations and a Set of Such Equations for Gaussian Delta-Correlated Action.- 6.5 Stochastic Equation for Random Actions with Arbitrary Distribution Laws.- 6.6 Exercises.- References.

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