Discrete Fourier series
A Fourier series is a representation of a function in terms of a summation of an infinite number of harmonically-related sinusoids with different amplitudes and phases. The amplitude and phase of a sinusoid can be combined into a single complex number, called a Fourier coefficient. The Fourier series is a periodic function. So it cannot represent any arbitrary function. It can represent either:
- (a) a periodic function, or
- (b) a function that is defined only over a finite-length interval; the values produced by the Fourier series outside the finite interval are irrelevant.
When the function being represented, whether finite-length or periodic, is discrete, the Fourier series coefficients are periodic, and can therefore be described by a finite set of complex numbers. That set is called a discrete Fourier transform (DFT), which is subsequently an overloaded term, because we don't know whether its (periodic) inverse transform is valid over a finite or an infinite interval. The term discrete Fourier series (DFS) is intended for use in lieu of DFT when the original function is periodic, defined over an infinite interval. DFT would then unambiguously imply only a transform whose inverse is valid over a finite interval. But we must again note that a Fourier series is a time-domain representation, not a frequency domain transform. So DFS is a potentially confusing substitute for DFT. A more technically valid description would be DFS coefficients.