On Nonlinear Networks with Random Inputs

The paper attempts to combine the Wiener-Bose method for characterizing and synthesizing nonlinear systems with the Ku-Wolf method for analyzing nonlinear systems with random inputs. A simple partition theory is first presented. It is shown that a general nonlinear system can be partitioned into two portions: one linear portion with memory or storage, and one nonlinear portion which may also include linear elements. The partition method, the Taylor-Cauchy transform method, and the transform-ensemble method are developed, and illustrated by an example. It is shown that the output of a nonlinear system to a random input can be expressed as the summation of , for and so on, where depends upon the form of the functional representation of the modified forcing function or the actuating signal, and a. denotes a set of random variables which are related to the statistics of the random input. Wiener\´s theory of nonlinear systems is then reviewed. The Wiener-Bose method is outlined as follows. Let the output of a shot-noise generator be the standard probe for the study of non-linear systems. The standard random input is fed to a Laguerre network giving Laguerre coefficients The output of the over-all system is then expressed as Hermite function expansions of the Laguerre coefficients. By the ergodic hypothesis it is then possible to express the output as the summation of By taking the time average of , where represents either the actual output or the desired output, we get the coefficients , which characterize the actual system or the system to be designed. Knowing , the synthesis procedure is obtained from the summation of . By combining the output , obtained from the Ku-Wolf analysis, with the output from the Laguerre network and Hermite function generator, we can get the-characterizing coefficients . It is suggested that the correlation of , a set of random variables related to the random input, and , the characterizing coefficients, may sh- ed light on a unified approach for the analysis and synthesis of nonlinear systems with random inputs.