Legacy Documentation

Time Series (2011)

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1.8 Spectral Analysis

We have so far studied stationary time series in terms of quantities that are functions of time. For example, the covariance function and correlation function are functions of the time lag. This approach is termed time series analysis in the time domain. Another approach is to analyze the time series in Fourier space or in the frequency domain. Although theoretically it provides a different representation of the same information, this approach can yield both powerful numerical methods of analysis and new insights. The techniques used in the frequency domain fall under the general rubric of spectral analysis and the fundamental tool is the Fourier transform. In this section we study time series in the frequency domain. First we introduce the concept of power spectrum and illustrate how to obtain the spectrum of a given ARMA process. Then we discuss how to get the estimated spectrum from time series data. Smoothing of spectra in both the time and frequency domains using various windows is also demonstrated.