Periodogram and Its Use in Frequency-Domain Analysis of Time Series

2 Periodogram and Its Use in Frequency-Domain Analysis of Time Series 

Since it has been shown that the Fourier transform contains the same information as the autocorrelation function, one may wonder what is the advantage of using it. Basically, the Fourier transform provides a different perspective on the same information allowing for different features to be more prominent. In the case of the Fourier transform, the periodicities are made clear, while in the (partial) autocorrelation plots, the different orders are emphasised. 

The most common way to use the Fourier transform is to construct a periodogram that shows all the identifiable frequencies and their amplitudes. When using the fast Fourier transform to obtain the periodogram, only half of the values are plotted, since the other half is a mirror image (about /= 0). The periodogram is constructed as follows  

Obtain the complex spectral density function, f( ), using any appropriate method (most often the fast Fourier method). Let n be the number of data points in the signal. 

Set the centre point to be C = [0.5/iJ + 1, where [ J is the floor or round down to the nearest integer function. 

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