On a cross-correlation property for stationary random processes

Given two stationary random processes x₁ (t) and x₂ (t), 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 p (x₁, x₂) 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 p (x₁, x₂). The class of distributions satisfying this condition is a subclass of the set of distributions for which the cross-correlation property is valid.