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(I also applied an additional scale factor to the spectral results to compensate for the fact that fewer total samples were included in theaverage for the short samples. This caused the amplitude of the peak in the spectrum to be nominally the same in all five cases.)
As you can see in Figure 10 , there isn't much in the spectra to the right of about 50 spectral points. That is as it should be since the single sinusoid ineach time series was at the low end of the spectrum.
Figure 11 shows the same data as Figure 10 with only the first 50 frequency points plotted on the horizontal axis. The remaining 350 frequency points weresimply ignored. This provides a much better view of the structure of the peaks in the different spectra.
Figure 11. Spectra of five different sinusoids of different lengths. |
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I will begin the discussion with the bottom plot in Figure 11 , which is the computed spectrum for the single sinusoid having a length of 400 samples.
Ideally, since the time series was a single sinusoid, the spectrum should consist of a single non-zero value at the frequency of the sinusoid, (often referred to as a spectral line) and every other value in the spectrum should be zero.
However, because the computation of the spectrum involves the computation of average values resulting from the products of sinusoids, the ideal is not alwaysachieved. In order to achieve the ideal, it would be necessary to multiply and average over an infinite number of points. Anything short of that will result insome measurement error, as exhibited by the bottom plot in Figure 11 .
(The bottom plot in Figure 11 has a large peak in the center with every second point to the left and right of center having a zero value. Iwill explain this structure in more detail later.)
Moving from the bottom to the top in Figure 11 , each individual plot shows the result of shorter and shorter averaging windows. As a result, themeasurement error increases and the peak broadens for each successive plot going from the bottom to the top in Figure 11 . The plot at the top, with an averaging window of only 80 samples, exhibits the most measurement error and the broadestpeak.
(It should be noted, however, that even the spectra for the shorter averaging windows have some zero-valued points. Once you understand thereason for the zero-valued points, you can correlate the positions of those points to the length of the averaging windows in Figure 11 .)
Now I'm going to show you the detrimental impact of such spectral measurement errors. In particular, the failure of the average to converge on zero for shortaveraging windows limits the spectral resolution of the Fourier transform.
I will create five new time series, each consisting of the sum of two sinusoids with fairly closely-spaced frequencies. One sinusoid has 32 samplesper cycle as in Figures 10 and 11. The other sinusoid has 26 samples per cycle.
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