For the example presented in the first chapter, several small discrepancies are noted between the constituent sinusoids listed in Table 1.1 and the peak spectral values shown in the Fourier transform periodogram in Fig. 1.4. For example, the spectral peak in Fig. 1.3 is located at 12.00 Hz, as the input given in Table 1.1 would indicate. However, the peak at 12.00 Hz has an amplitude of 5.5703 V/root-Hz, rather than 5.50 V like the input signal. There is also the matter of what appears to be missing phase information in the frequency-domain portrayal. Some portions of the discrepancies in amplitude between the Fourier transform output and the original signal can be explained by the presence of the random (uniform) noise component that was added. However, some aspects of the discrepancies are also due to intrinsic features of using Fourier transforms to determine the spectral content of time-series signals. For example, the main spectral peak in Fig. 1.4 occurs at 12.00 Hz, but the occurrence of a Fourier basis component at exactly 12.00 Hz is partly fortuitous. The Fourier transform periodogram is limited to displaying spectral information at certain discrete frequencies. If 12.00 were not an integer multiple of the time span of data collection, then there would not be a Fourier basis component at exactly 12.00 Hz.
Preliminary Background for the Fourier Analysis of Mathematical Functions