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The discrete Fourier transform (DFT) and its inverse (IDFT) are the primary numerical transforms relating time and frequency in digital signal processing. The DFT has a number of important properties relating time and frequency, including shift, circular convolution, multiplication, time-reversal and conjugation properties, as well as Parseval's theorem equating time and frequency energy.

Dft

The discrete Fourier transform (DFT) is the primary transform used for numerical computation in digital signal processing. It is very widely used for spectrum analysis , fast convolution , and many other applications. The DFT transforms N discrete-time samples to the same number of discrete frequency samples, and is defined as

X k n N 1 0 x n 2 n k N
The DFT is widely used in part because it can be computed very efficiently using fast Fourier transform (FFT) algorithms.

Idft

The inverse DFT (IDFT) transforms N discrete-frequency samples to the same number of discrete-time samples. The IDFT has a form very similar to the DFT,

x n 1 N k N 1 0 X k 2 n k N
and can thus also be computed efficiently using FFTs .

Dft and idft properties

Periodicity

Due to the N -sample periodicity of the complex exponential basis functions 2 n k N in the DFT and IDFT, the resulting transforms are also periodic with N samples.

X k N X k x n x n N

Circular shift

A shift in time corresponds to a phase shift that is linear in frequency. Because of the periodicity induced by the DFT and IDFT, the shift is circular , or modulo N samples.

x n m N X k 2 k m N The modulus operator p N means the remainder of p when divided by N . For example, 9 5 4 and -1 5 4

Time reversal

x n N x N n N X N k N X k N Note: time-reversal maps 0 0 , 1 N 1 , 2 N 2 , etc. as illustrated in the figure below.

Original signal
Time-reversed
Illustration of circular time-reversal

Complex conjugate

x n X k N

Circular convolution property

Circular convolution is defined as x n h n m N 1 0 x m x n m N

Circular convolution of two discrete-time signals corresponds to multiplication of their DFTs: x n h n X k H k

Multiplication property

A similar property relates multiplication in time to circular convolution in frequency. x n h n 1 N X k H k

Parseval's theorem

Parseval's theorem relates the energy of a length- N discrete-time signal (or one period) to the energy of its DFT. n N 1 0 x n 2 1 N k N 1 0 X k 2

Symmetry

The continuous-time Fourier transform , the DTFT , and DFT are all defined as transforms of complex-valueddata to complex-valued spectra. However, in practice signals are often real-valued.The DFT of a real-valued discrete-time signal has a special symmetry, in which the real part of the transform values are DFT even symmetric and the imaginary part is DFT odd symmetric , as illustrated in the equation and figure below.

x n real  X k X N k N (This implies X 0 , X N 2 are real-valued.)

Real part of x(k) is even

Even-symmetry in DFT sense

Imaginary part of x(k) is odd

Odd-symmetry in DFT sense
DFT symmetry of real-valued signal

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Source:  OpenStax, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
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