Notes

Objectives
The fast Fourier transform (FFT) is a fast algorithm for calculating the Discrete Fourier Transform (DFT). The spectral components of the FFT are samples of the continuous DTFT of a finite length N-point signal. In certain cases it may be desireable to augment with zeros a signal (zero-pad) before taking its FFT.With Zero-padding, the DFT to give more closely spaced samples and approximates better the DTFT. While zero-padding does not increase the ability to perfectly resolve closely spaced frequencies, it does give a better idea of the true shape of the DTFT of the signal. Failure to zero-pad a signal may give misleading information about its spectrum. In order to better resolve frequencies in a signal, the length of the window should be increased. In other words, the signal should be examined over a longer period of time.

Another fundamental issue of the FFT is resolution and spectral leakage which is determined by the choice of the truncating window. Resolution is the ability to distinguish between two sinusoids that are closely spaced in frequency. Leakage is when components at one frequency affect the measurement ad into other frequencies and affect the components at those frequencies. If a signal contains 2 closely spaced sinusoids, spectral leakage may cause the spectral peak associated with each sinusoid to merge into one or a strong sinusoid may produce a measurement that will mask the measurement of a weaker sinusoid. This will cause the FFT to indicate that a single sinusoid is present instead of 2.