Influence of bandwidth on some nonlinear transformations of a Gaussian random process

Predictions of the response probability density function for bandlimited white noise excitation of nonlinear systems often employ the so-called Fokker-Planck technique. While these predictions are exact for white noise, there exist no general criteria for assessing their accuracy in physical situations. This paper endeavors to shed some light on the problem and perhaps point the way toward better understanding of the inherent difficulties involved. First, a brief discussion of more typical response functions for a first-order nonlinear system is presented on the basis of a very narrow-band low-pass excitation and on a very wide-band low-pass excitation spectrum. Such limiting cases generally exhibit substantially different behavior. Next, some experimental results for the same system driven by variable-bandwidth low-pass Gaussian noise is presented. These results indicate substantial deviation from the Fokker-Planck predictions when the excitation bandwidth is comparable with an equivalent system bandwidth. Moreover, it was found that the data at high response levels were bounded by the Fokker-Planck predictions on one hand and by a prediction based on a pointwise transformation of an extremely low-pass excitation process on the other. Bandwidths ranging between three and fifteen times the system's equivalent bandwidth are required for validity of the Fokker-Planck approximation depending on the nature of the non-linearity and the response level in question.