Why a DFT is usually called an FFT in practice
Practical implementations of the DFT are usually based on one of the Cooley-Tukey ``Fast Fourier Transform'' (FFT) algorithms .8.1 For this reason, the matlab DFT function is called `fft', and the actual algorithm used depends primarily on the transform length .8.2 The fastest FFT algorithms generally occur when is a power of 2. In practical audio signal processing, we routinely zero-pad our FFT input buffers to the next power of 2 in length (thereby interpolating our spectra somewhat) in order to enjoy the power-of-2 speed advantage. Finer spectral sampling is a typically welcome side benefit of increasing to the next power of 2. Appendix A provides a short overview of some of the better known FFT algorithms, and some pointers to literature and online resources.
FFT of a Simple Sinusoid
Periodic Interpolation (Spectral Zero Padding)