Cumulant series expansion of hybrid nonlinear moments of complex random variables

A general theorem for zero-memory nonlinear transformations of complex stochastic processes is presented. It is shown that, under general conditions, the cross covariance between a stochastic process and a distorted version of another process can be represented by a series of cumulants. The coefficients of this cumulant expansion are expressed by the expected values of the partial derivatives, appropriately defined, of the function describing the nonlinearity. The theorem includes as a particular case the invariance property (Bussgang's (1952) theorem) of Gaussian processes, while holding for any joint distribution of the processes. The expansion in cumulants constitutes an effective means of analysis for higher-order-moment-based estimation procedures involving non-Gaussian complex processes. >