Characteristics of Response to Nonstationary White Noise: Theory

Abstract

New methods of analysis are developed to study characteristics of the response of linear systems to nonstationary stochastic input. An envelope of the response based on the Hilbert transformation of the process is used, and derivations are made with respect to stationary response properties. The analysis begins with a derivation of the statistics of the response of general linear systems to modulated white‐noise input. Then, the important case of oscillator systems is examined and, by use of functional operator concepts, approximations are obtained for the second‐order statistics of the oscillator response. It is found that the original expressions for these statistics, involving lengthy double integrals, become reduced to simple, closed‐form expressions, which are stated in terms of the oscillator properties and one convolution integral of the modulation function. The results provide physically meaningful insight into the characteristics of nonstationary response and its envelope. The relationships between the nonstationary and stationary resonse characteristics are explored in detail.