The effect of instantaneous nonlinear devices on cross-correlation

Author

Leipnik, Roy

Volume

Issue

fYear

1958

fDate

6/1/1958 12:00:00 AM

Firstpage

Lastpage

Abstract

If $X_1(t), X_2(t)$ are two noises (stochastic processes), $f$ and $g$ are functions describing the action of two instantaneous nonlinear devices, we say that the $(m, n)$ cross-correlation property holds in case the cross-correlation of $f(X_1(t_1))$ with $g(X_2(t_2))$ is proportional to the cross-correlation of $X_1(t_2)$ with $X_2(t_2)$ , whenever $f$ and $g$ are polynomials of degrees not exceeding $m$ and $n$ , respectively. We take $m =\\infty$ or $n =\\infty$ to mean that $f$ or $g$ is any continuous function. The Barrett-Lampard expansion $^2$ of the second-order joint density of $X_1(t_1)$ and $X_2(t_2)$ is used to derive an expression for the cross-correlation of $f(X_1(t_1))$ and $g(X_2(t_2))$ . This expression yields necessary and sufficient conditions for the validity of the cross-correlation property in three cases: $X_1(t)$ and $X_2(t)$ stationary, $m, n$ unrestricted; $X_1(t)$ stationary, $m, n$ unrestricted; $X_1(t)$ stationary, $n = 1$ . Examples are constructed with the help of special orthonormal polynomials illustrating the necessity and sufficiency of the conditions.

Keywords

Correlation functions; Nonlinearities; Detectors; Gaussian processes; Information theory; Nonlinear distortion; Polynomials; Probability distribution; Random processes; Reactive power; Stochastic processes; Sufficient conditions; Terminology;

fLanguage

English

Journal_Title

Information Theory, IRE Transactions on

Publisher

ieee

ISSN

0096-1000

Type

jour

DOI

10.1109/TIT.1958.1057445

Filename

1057445

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=944327