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tabulated transforms of specific functions

Since the DTFT is also a continuous Fourier transform (of a comb function), the Fourier series also applies to it.  Thus, when  is periodic, with period N,   is another Dirac comb function, modulated by a Fourier series.   And the integral formula for the series simplifies to:

for all integer values of k.

Since the DTFT is periodic, so is .  And it has the same period (N) as the input function.  This transform is also called DFT, particularly when only one period of the output sequence is computed from one period of the input sequence.

When  is not periodic, but its non-zero portion has finite duration (N),   is continuous and finite-valued.  But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of  that was analyzed.  The same discrete set is obtained by tréating N as if it is the period of a periodic function and computing the Fourier series / DFT.

The inverse transform of  does not produce the finite-length sequence,   (It takes the inverse of , to do that.)  The inverse DFT can only reproduce the entire time-domain if the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain, discrete-time functions.   An alternative viewpoint is that the periodicity is the time-domain consequence of approximating the continuous-domain function, , with the discrete subset, .   N can be larger than the actual non-zero portion of .   The larger it is, the better the approximation (also known as zero-padding).

The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.

See Discrete Fourier transform for much more information, including: