The effect of instantaneous nonlinear devices on cross-correlation
Roy B. Leipnik · IEEE Transactions on Information Theory · 1958
IfX_1(t), X_2(t)are two noises (stochastic processes),fandgare functions describing the action of two instantaneous nonlinear devices, we say that the(m, n)cross-correlation property holds in case the cross-correlation off(X_1(t_1))withg(X_2(t_2))is proportional to the cross-correlation ofX_1(t_2)withX_2(t_2), wheneverfandgare polynomials of degrees not exceedingmandn, respectively. We takem =\inftyorn =\inftyto mean thatforgis any continuous function. The Barrett-Lampard expansion^2of the second-order joint density ofX_1(t_1)andX_2(t_2)is used to derive an expression for the cross-correlation off(X_1(t_1))andg(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)andX_2(t)stationary,m, nunrestricted;X_1(t)stationary,m, nunrestricted;X_1(t)stationary,n = 1. Examples are constructed with the help of special orthonormal polynomials illustrating the necessity and sufficiency of the conditions.