On a cross-correlation property for stationary random processes

Given two stationary random processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_1 (t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_2 (t)</tex> , the cross-correlation property of interest is the following: If one of the two processes is distorted by an instantaneous nonlinear device, then the cross correlation after the distortion is proportional to the cross-correlation function prior to the distortion. Using an expansion of the second-order joint probability distribution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x_1, x_2)</tex> introduced by Barrett and Lampard, a necessary and sufficient condition for the above cross-correlation property is given in terms of requirements on the expansion coefficients. In certain cases, the constant of proportionality involved in the cross-correlation property is equal to the "equivalent gain" of the nonlinear device as defined by Booton. A necessary and sufficient condition for these two constants to be identical is formulated in terms of the expansion coefficients of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x_1, x_2)</tex> . The class of distributions satisfying this condition is a subclass of the set of distributions for which the cross-correlation property is valid.