The Spectral Representation of a Stationary Process

Abstract

The spectral representation of a stationary process {X _t , t = 0, ± 1, ... } essentially decomposes {X _t } into a sum of sinusoidal components with uncorrelated random coefficients. In conjunction with this decomposition there is a corresponding decomposition into sinusoids of the autocovariance function of {X _t }. The spectral decomposition is thus an analogue for stationary stochastic processes of the more familiar Fourier representation of deterministic functions. The analysis of stationary processes by means of their spectral representations is often referred to as the “frequency domain” analysis of time series. It is equivalent to “time domain” analysis, based on the autocovariance function, but provides an alternative way of viewing the process which for some applications may be more illuminating. For example in the design of a structure subject to a randomly fluctuating load it is important to be aware of the presence in the loading force of a large harmonic with a particular frequency to ensure that the frequency in question is not a resonant frequency of the structure. The spectral point of view is particularly advantageous in the analysis of multivariate stationary processes (Chapter 11) and in the analysis of very large data sets, for which numerical calculations can be performed rapidly using the fast Fourier transform (Section 10.7).