Response of multidimensional nonlinearities to inputs which are separable processes

The property of separability of a process with respect to n random processes is defined and examples of such a process are given. The equivalent gains of a multidimensional nonlinearity (m.d.n.l.), obtained by Kazakov, are shown to constitute the best linear model of the m.d.n.l. in the Wiener sense if, and only if, each of the inputs is separable with respect to all the inputs. An extended version of Price´s theorem is stated and proved, and its use in the determination of the output autocorrelation function for an m.d.n.l. is demonstrated. A series representation of the output autocorrelation function for an m.d.n.l. is also derived. The concepts of d.c. incremental gain and modified nonlinearity are extended to the multidimensional case, and their use in the analysis of closed-loop systems containing an m.d.n.l. is indicated. Finally, the applicability of these results to the general m.d.n.l. with n inputs and r outputs is demonstrated.