The correlation function of Gaussian noise passed through nonlinear devices

Abstract

This paper is concerned with the output autocorrelation function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{y}</tex> of Gaussian noise passed through a nonlinear device. An attempt is made to investigate in a systematic way the changes in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{y}</tex> when certain mathematical manipulations are performed on some given device whose correlation function is known. These manipulations are the "elementary combinations and transformations" used in the theory of Fourier integrals, such as addition, differentiation, integration, shifting, etc. To each of these, the corresponding law governing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{y}</tex> is established. The same laws are shown to hold for the envelope of signal plus noise for narrow-band noise with spectrum symmetric about signal frequency. Throughout the text and in the Appendix it is shown how the results can be used to establish unknown correlation function quickly with main emphasis on power-law devices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y = x^{m}</tex> with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> either an integer or half integer. Some interesting recurrence formulas are given. A second-order differential equation is derived which serves as an alternative means for calculating <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{y}</tex> .