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The Discrete Fourier Transform (DFT) is a mapping between a finite set of discrete points in a (primal) domain (time, space) and the dual frequency domain. DFT requires an input sequence which is discrete, such as a sampling from an analogue audio signal.

It it is commonly defined by the following summation:

$ X_k = \sum\limits_{n=0}^{N-1} x_n e^{-\frac{2i\pi}{N}kn} $

for $ k =0,1,..N-1 $.

It is invertible; the inversion is known as the IDFT (Inverse Discrete Fourier Transform) and is defined by the following summation:

$ x_n = \frac{1}{N} \sum\limits_{k=0}^{N-1} X_k e^{\frac{2i\pi}{N}nk}$

for $ n =0,1,..N-1 $.

It should not be confused with the , which is applied to an infinite set of discrete points, or the lesser-known , which is applied to continuous signals.