This wide applicability stems from many useful properties of the transforms:
See Fourier series for more information, including the historical development.
The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:
The Fourier series coefficients (and inverse transform), are defined by:
Applications of the DTFT are not limited to sampled functions. See Discrete-time Fourier transform for more information on this and other topics, including:
The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.Fourier transforms on arbitrary locally compact abelian topological groups