## Periodogram

The periodogram is another long-time standard method for finding periodicity. Based on Fourier’s insight that a signal can be reproduced by a series of sines and cosines, Fourier analysis is used in nearly every field of engineering and physical sciences, from MRI and image processing to X-ray crystallography and geophysics. In the periodogram, Fourier analysis is used to calculate the amplitude (strength) at specific periods: N/1, N/2, N/3, through N/(N/2)=2, where N is the number of equidistant samples in the time series. Each sequence begins with N, and continues to 2 (the Nyquist limit). Mathematically speaking, assessment ends just before 2, not at 2, as there must be more than 2 points per cycle, according to the Nyquist limit, to identify a cycle. Only these specific periods can be calculated with the periodogram, so N should be chosen with this in mind. See Table 3 for sample period calculations using differing series lengths. Unlike other CAT techniques that allow finer control of the periods calculated, the periods (or frequencies) that can be calculated with the periodogram depend directly on the number of data points analyzed.

Interpretation: Periodograms in Figures 15 and 16 use the same data as the correlograms in Figures 11 and 12. Vertical lines, called spectral lines are the amplitude of the sine (or cosine) curves at each frequency; they indicate the prominence of signals present in the data. Figure 15 is very clean with 1 strong spectral peak indicated at 24-hr, and 2 weaker ones at 12 and 8 hr. By contrast, Figure 16, based on only 2 days of data, is less definitive, indicating the same 3 spectral peaks as most prominent, but with less resolution between the periods identified.

In general, a non-sinusoidal waveform (a waveform not composed of one pure sine or cosine) will be represented in a periodogram by a spectral line at the fundamental frequency, with additional (usually smaller) spectral peaks at the additional harmonics that define the waveform. Very rarely a signal can be represented by harmonic terms in the absence of a fundamental period.

Interpretation: Periodograms in Figures 15 and 16 use the same data as the correlograms in Figures 11 and 12. Vertical lines, called spectral lines are the amplitude of the sine (or cosine) curves at each frequency; they indicate the prominence of signals present in the data. Figure 15 is very clean with 1 strong spectral peak indicated at 24-hr, and 2 weaker ones at 12 and 8 hr. By contrast, Figure 16, based on only 2 days of data, is less definitive, indicating the same 3 spectral peaks as most prominent, but with less resolution between the periods identified.

In general, a non-sinusoidal waveform (a waveform not composed of one pure sine or cosine) will be represented in a periodogram by a spectral line at the fundamental frequency, with additional (usually smaller) spectral peaks at the additional harmonics that define the waveform. Very rarely a signal can be represented by harmonic terms in the absence of a fundamental period.