Optimum multiple-input nonlinear system with Gaussian inputs

Methods are developed for optimizing a general, realizable, multi-input, single-output, non-linear system whose inputs are correlated, stationary, Gaussian random processes. The optimum system is nonlinear rather than linear if the desired output is not Gaussian. The output of the non-linear system is expressed as a sum of functional polynomials of the inputs. The only statistical information needed for this optimization is composed of the first-order autocorrelations and crosscorrelations among the inputs, and the higher-order crosscorrelations between the inputs and the desired output. This study is a generalization of Wiener\´s method for characterizing a single-input nonlinear system. A major problem which arises with the n-input system but not with the single-input system is the fact that a conventional variational procedure to optimize the n-input system would require the solution of an infinite set of simultaneous integral equations. The reason for this difference is the fact that all the terms in the expansion of the output of the single-input system are linearly independent while they are not for the -input system. A major result of this paper is the development of a novel indirect method of optimization that avoids the necessity of solving this infinite set of simultaneous integral equations. This indirect optimization procedure exploits an interesting relationship between optimum unrealizable systems and optimum realizable systems, and it makes use of the fact that optimum linear systems are imbedded in the optimum non-linear system.