Spectral Analysis

Abstract

This chapter can be omitted without any loss of continuity. The reader with no background in Fourier or complex analysis should go straight to Chapter 5. The spectral representation of a stationary time series {X _t } 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 processes of the more familiar Fourier representation of deterministic functions. The analysis of stationary processes by means of their spectral representation is often referred to as the “frequency domain analysis” of time series or “spectral analysis.” 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 sinusoidal component with a particular frequency to ensure that this is not a resonant frequency of the structure. The spectral point of view is also particularly useful in the analysis of multivariate stationary processes and in the analysis of linear filters. In Section 4.1 we introduce the spectral density of a stationary process {X _t }, which specifies the frequency decomposition of the autocovariance function, and the closely related spectral representation (or frequency decomposition) of the process {X _t } itself. Section 4.2 deals with the periodogram, a sample-based function from which we obtain estimators of the spectral density. In Section 4.3 we discuss time-invariant linear filters from a spectral point of view and in Section 4.4 we use the results to derive the spectral density of an arbitrary ARMA process.