Hilbert Spectral Description and Simulation of Nonstationary Random Processes

Abstract

A new method is proposed for characterization and simulation of a nonstationary random process based on samples of the process. The theoretical background is that of the Hilbert Huang Transform (HHT). Samples of a random process X(t) can be decomposed into a summation of modal functions whose Hilbert transforms can be used to describe the amplitude and frequency changes with time. A Hilbert spectrum is then defined to describe the time-varying spectral content of the sample process. The method is also extended to characterization of vector processes. It can be conveniently applied to simulation of non-stationary random processes based on observed sample functions. The simulated processes approach Gaussian and will have the Hilbert spectra equal to the target spectra. Unlike current procedures such as those based on the evolutionary process, no assumptions of any functional forms for the spectra are necessary for parameter estimation, which are unknown a priori. Applications to spectral characterization and simulation of multivariate earthquake ground motions show that the Hilbert spectra give a clear description of the spectral content change with time. The simulated samples have the desired frequency and amplitude variation with time. The method has great potential for engineering applications when dealing with non-stationary, nonlinear random processes.