# Non-Linear Transformations of Stochastic Processes

## 1st Edition

**Editors:**P. I. Kuznetsov R. L. Stratonovich V. I. Tikhonov

**eBook ISBN:**9781483282688

**Imprint:**Pergamon

**Published Date:**1st January 1965

**Page Count:**514

## Description

Non-Linear Transformations of Stochastic Processes focuses on the approaches, methodologies, transformations, and computations involved in the non-linear transformations of stochastic processes.

The selection first underscores some problems of the theory of stochastic processes and the transmission of random functions through non-linear systems. Discussions focus on the transformation of moment functions for the general non-linear transformation; conversion formulas for correlation functions; transformation of moment functions for the simplest type of non-linear transformation; and normalization of the linear system of probability distribution laws. The text then ponders on quasi-moment functions in the theory of random processes and correlation functions in the theory of the Brownian motion generalization of the Fokker-Planck equation.

The manuscript elaborates on the correlation functions of random sequences of rectangular pulses; method of determining the envelope of quasi-harmonic fluctuations; and the problem of measuring electrical fluctuations with the aid of thermoelectric devices. The book then examines the effect of signal and noise on non-linear elements and the approximate method of calculating the correlation function of stochastic signals.

The selection is a dependable source of information for researchers interested in the non-linear transformations of stochastic processes.

## Table of Contents

Foreword to English Edition

Authors' Foreword

Chapter I Some Problems of the Theory of Stochastic Processes

1 The Transmission of Certain Random Functions Through Linear Systems

2 The Transmission of Random Functions Through Non-Linear Systems

3 Quasi-Moment Functions in the Theory of Random Processes

4 The Effect of Non-Normal Fluctuations on Linear Systems

5 Correlation Functions in the Theory of Brownian Motion Generalization of the Fokker-Planck Equation

6 A Note on the Mathematical Theory of Correlated Random Points

7 The Correlation Functions of Random Sequences of Rectangular Pulses

Chapter II The Effect of Noise on Certain Non-Linear Elements

8 A Method of Determining the Envelope of Quasi-Harmonic Fluctuations

9 The Effect of Electrical Fluctuations on a Detector (Envelope Method)

10 The Effect of Fluctuations on a Detector and Filter

11 A Note on the Problem of Measuring Electrical Fluctuations with the Aid of Thermoelectric Devices

12 The Response of Typical Non-Linear Elements to Normally Fluctuating Inputs

13 The Effect of Signal and Noise on Non-Linear Elements (The Direct Method)

14 Approximate Method of Calculating the Correlation Function of Stochastic Signals

15 The Effect of Fluctuations on the Operation of an Automatic Range Finder

Chapter III The Effect of Fluctuations on Oscillator Operation

16 The Effect of Electrical Fluctuations on a Valve Oscillator

17 The Effect of Small Fluctuations on Oscillator Operation

18 The Fluctuating Nature of the Establishment of Oscillation Amplitude in an Oscillator

19 The Effect of Slow Fluctuations on an Oscillator

20 The Effect of Noise on an Oscillator with Fixed Excitation

21 Oscillator Synchronization in the Presence of Noise

22 Establishment of the Synchronous Phase in a Self-Oscillatory Circuit in the Presence of Noise

23 Establishment of Amplitude in a Synchronized Oscillator in the Presence of Fluctuation Noise

24 The Effect of noise on the Operation of a Phase AFC Circuit

25 Operation of a Phase AFC Circuit in the Presence of Noise

26 Parametric Effect of a Random Force on Linear and Non-Linear Oscillatory Systems

27 The Simultaneous Parametric Effect of a Harmonic and Random Force on Oscillatory Systems

Chapter IV Random Function Excursions

28 On the Duration of Excursions of Random Functions

29 The Distribution of the Duration of Excursions of Normal Fluctuations

30 An Experimental Study of the Length Distribution Law of Fluctuation Excursions

31 Fluctuation Excursions and their Correlation

32 Probability Densities for the Duration of Fluctuation Excursions

33 The Effect of Differentiating and Integrating on the Average Number of Excursions

Chapter V Optimum Filtration

34 The Conditional Distribution of Correlated Random Points and the use of Correlation for Optimum Nitration of a Pulse Signal from Noise

35 A Contribution to the Theory of the Optimum Non-Linear Filtration of Random Functions

36 Conditional Markov Processes

37 Optimum Non-Linear Systems for Isolating a Signal with Constant Parameters from Noise

38 Application of the Theory of Markov Processes to the Optimum Nitration of Signals

39 Some Problems in Conditional Probability and Quasi-Moment Functions

Index

## Details

- No. of pages:
- 514

- Language:
- English

- Copyright:
- © Pergamon 1965

- Published:
- 1st January 1965

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483282688