# Applied Methods of the Theory of Random Functions, Volume 89

## 1st Edition

**Authors:**A. A. Sveshnikov

**Editors:**I. N. Sneddon M. Stark

**eBook ISBN:**9781483222639

**Imprint:**Pergamon

**Published Date:**1st January 1966

**Page Count:**332

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## Table of Contents

Publisher's Note

Chapter I. General Properties of Random Functions

§ 1. The Theory of Random Functions as a Branch of the Theory of Probability

§ 2. The Basic Notations and Formulae of the Theory of Probability

§ 3. Random Functions and Methods of Describing Them

§ 4. Typical Problems, Solved by Means of the Theory of Random Functions

§ 5. Properties of the Correlation Function

§ 6. The Differentiation and Integration of Random Functions

§ 7. The Action of a Linear Operator on a Random Function

§ 8. A System of Random Functions. The Cross-Correlation Function

§ 9. Problems on Overshoots: The Mean Number of Overshoots of a Random Function Above a Given Level, the Mean Duration of an Overshoot

Chapter II. The Spectral Theory of Stationary Random Functions

§ 10. The Spectral Representation of Stationary Random Functions

§ 11. Examples of the Calculation of the Spectral Density of a Stationary Random Process

§ 12. The Spectral Density of a Linear Combination of a Stationary Random Function and its Derivatives. The Stationary Solution of a Linear Differential Equation With Constant Coefficients

§ 13. Examples of Spectral Densities and Correlation Functions in More Complicated Cases

§ 14. Determination of the Correlation Function of the Solution of a Non-Homogeneous Differential Equation with Constant Coefficients When the Right-Hand Side is Non-Stationary

§ 15. Linear Differential Equations with Variable Coefficients

§ 16. The Probability Characteristics of the Solutions of a System of Linear Equations

§ 17. The Density Distribution of the Solution of a Linear Differential Equation

Chapter III. Determination of Optimal Dynamical Systems

§ 18. Statement of the Problem of the Determination of Optimal Systems

§ 19. The General Solution of the Problem of Determining (Given the Infinite Past of the Process) the Optimal Dynamical System Representing the Operations of Smoothing (Filtering), Extrapolation and Differentiation

§ 20. Formulae For the Determination of the Optimal Transfer Function of a Dynamical System Performing Filtration, Extrapolation and Differentiation in the Case of Rational Spectral Densities of Signal and Noise

§ 21. Practical Formulae for the Optimal Transfer Function of a Dynamical System with Delay

§ 22. Optimal Smoothing, Extrapolation and Differentiation for a Finite Observation Time

§ 23. Examples of the Determination of Optimal Dynamical Systems for a Finite Observation Time

Chapter IV. Experimental Methods for the Determination of Characteristics of Random Functions

§ 24. The General Principles of the Determination of Characteristics of Random Functions From Experimental Data

§ 25. The Principles of the Construction of Correlators — Devices for the Determination of the Correlation Functions of Stationary Random Processes

§ 26. Determination of the Estimated Value of the Spectral Density

Chapter V. The Method of Envelopes

§ 27. The Method of Envelopes and the Derivation of General Formulae

§ 28. Application of the Method of Envelopes in the Case of a Narrow-Band Spectrum

Chapter VI. Some Supplementary Problems of the Theory of Random Functions

§ 29. Random Sequences

§ 30. Random Functions of Many Variables

§ 31. Canonical Expansions of Random Functions

References

Index

Other Titles in this Series

## Description

International Series of Monographs in Pure and Applied Mathematics, Volume 89: Applied Methods of the Theory of Random Functions presents methods of random functions analysis with their applications in various branches of technology, such as in the theory of ships, automatic regulation and control, and radio engineering.

This book discusses the general properties of random functions, spectral theory of stationary random functions, and determination of optimal dynamical systems. The experimental methods for the determination of characteristics of random functions, method of envelopes, and some supplementary problems of the theory of random functions are also deliberated.

This publication is intended for engineers and scientists who use the methods of the theory of probability in various branches of technology.

## Details

- No. of pages:
- 332

- Language:
- English

- Copyright:
- © Pergamon 1966

- Published:
- 1st January 1966

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483222639