In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally-spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values. It can be said to convert the sampled function from its original domain (often time or position along a line) to the frequency domain.
The input samples are complex numbers (in practice, usually real numbers), and the output coefficients are complex too. The frequencies of the output sinusoids are integer multiples of a fundamental frequency, whose corresponding period is the length of the sampling interval. The combination of sinusoids obtained through the DFT is therefore periodic with that same period. The DFT differs from the discrete-time Fourier transform (DTFT) in that its input and output sequences are both finite; it is therefore said to be the Fourier analysis of finite-domain (or periodic) discrete-time functions.
DFT AND ITS PROPERTIES
The sequence of N complex numbers x0, ..., xN−1 is transformed into another sequence of N complex numbers according to the DFT formula:
* It completely describes the discrete-time Fourier transform (DTFT) of an N-periodic sequence, which comprises only discrete frequency components. (Discrete-time Fourier transform#Periodic data) * It can also provide uniformly spaced samples of the continuous DTFT of a finite length sequence. (Sampling the DTFT) * It is the cross correlation of the input sequence, xn, and a complex sinusoid at frequency k/N. Thus it acts like a matched filter for that frequency.
f the expression that defines the DFT is evaluated for all integers k instead of just for , then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N. The periodicity can be shown directly from the definition:...