** When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients.
** When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients.
For usage on computers, number theory and algebra, discrete arguments are often more appropriate, and are handled by the transforms :
Discrete-time Fourier transform: Equivalent to the Fourier transform of a "continuous" function that is constructed from the discrete input function by using the sample values to modulate a Dirac comb. When the sample values are derived by sampling a function on the real line, ƒ, the DTFT is equivalent to a periodic summation of the Fourier transform of ƒ. The DTFT output is always periodic. An alternative viewpoint is that the DTFT is a transform to a frequency domain that is bounded, the length of one cycle.
** When the input sequence is periodic, the DTFT output is also a Dirac comb function, modulated by the coefficients of a Fourier series which can be computed as a DFT of one cycle of the input sequence. The number of discrete values in one cycle of the DFT is the same as in one cycle of the input sequence.
** When the non-zero portion of the input sequence has finite duration, the DTFT is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one cycle of a periodic function and computing the DFT.
Generalized DFT, a generalization of the DFT and constant modulus transforms where phase functions might be of linear with integer and real valued slopes, or even non-linear phase bringing flexibilities for optimal designs of various metrics, e.g. auto- and cross-correlations.
Discrete-space Fourier transform is the generalization of the DTFT from 1D signals to 2D signals. It is called "discrete-space" rather than "discrete-time" because the most prevalent application is to imaging and image processing where the input function arguments are equally spaced samples of spatial coordinates. The DSFT output is periodic in both variables.
The use of all of these transforms is greatly facilitated by the existence of efficient algorithms based on a fast Fourier transform. The Nyquist-Shannon sampling theorem is critical for understanding the output of such discrete transforms.
