Spectrum Analysis Tutorial, Part 2: Properties and Applications of the Discrete Fourier Transform
David A. Jaffe · Computer Music Journal · 1987
In part one of this tutorial (Jaffe 1987), we introduced the discrete Fourier transform (DFT). To review, the DFT takes a waveform as input and produces as output the spectrum of that waveform. One way to understand this process is to consider the samples of the waveform as a vector and to see the DFT as the projection of this vector onto a set of complex sinusoidal basis vectors. In this manner, the DFT produces a sequence of spectral components equally spaced in frequency, with a length equal to that of the original waveform. Each element of the spectrum is a coefficient of the projection given by the inner product of the waveform with one of the basis sinusoids. This coefficient can be represented in polar coordinates to give the amplitude and phase of the corresponding sinusoid. The equation for the DFT is: