On a cross-correlation property for stationary random processes
Jerram L. Brown · IEEE Transactions on Information Theory · 1957
Given two stationary random processesx_1 (t)andx_2 (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 distributionp(x_1, x_2)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 ofp(x_1, x_2). The class of distributions satisfying this condition is a subclass of the set of distributions for which the cross-correlation property is valid.