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
a_{n}q_{n}(t)
, for
n = 0, 1, 2,
and so on, where
q_{n}(t)
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
u_{1},u_{2},\cdots.
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
A_{\alpha} V(\alpha) e^{-u^{2}/2}
By taking the time average of
c(t) V(\alpha)
, where
c(t)
represents either the actual output or the desired output, we get the coefficients
A_{\alpha}
, which characterize the actual system or the system to be designed. Knowing
A_{\alpha}
, the synthesis procedure is obtained from the summation of
A_{\alpha} V(\alpha)
. By combining the output
c(t)
, obtained from the Ku-Wolf analysis, with the output
V(\alpha)
from the Laguerre network and Hermite function generator, we can get the-characterizing coefficients
A_{\alpha}
. It is suggested that the correlation of
a_n
, a set of random variables related to the random input, and
A_{\alpha}
, the characterizing coefficients, may shed light on a unified approach for the analysis and synthesis of nonlinear systems with random inputs.