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DFT Definition and Properties

Module by: Douglas L. Jones

Summary: 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 NN discrete-time samples to the same number of discrete frequency samples, and is defined as
Xk=∑n=0N-1xnⅇ-ⅈ2πnkN X k n N 1 0 x n 2 n k N (1)
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 NN discrete-frequency samples to the same number of discrete-time samples. The IDFT has a form very similar to the DFT,
xn=1N∑k=0N-1Xkⅇⅈ2πnkN x n 1 N k N 1 0 X k 2 n k N (2)
and can thus also be computed efficiently using FFTs.

DFT and IDFT properties

Periodicity

Due to the NN-sample periodicity of the complex exponential basis functions ⅇⅈ2πnkN 2 n k N in the DFT and IDFT, the resulting transforms are also periodic with NN samples.
Xk+N=Xk X k N X k xn=xn+N 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 NN samples.
xn-mmodNXkⅇ-ⅈ2πkmN x n m N X k 2 k m N The modulus operator pmodN p N means the remainder of pp when divided by NN. For example, 9mod5=4 9 5 4 and -1mod5=4 -1 5 4

Time Reversal

x-nmodN=xN-nmodNXN-kmodN=X-kmodN x n N x N n N X N k N X k N Note: time-reversal maps 00 0 0 , 1N-1 1 N 1 , 2N-2 2 N 2 , etc. as illustrated in the figure below.
image1.png image2.png
Subfigure 1.1: Original signal
Subfigure 1.2: Time-reversed
Figure 1: Illustration of circular time-reversal

Complex Conjugate

xn¯X-kmodN¯ x n X k N

Circular Convolution Property

Circular convolution is defined as xn*hn≐∑m=0N-1xmxn-mmodN ≐ 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: xn*hnXkHk x n h n X k H k

Multiplication Property

A similar property relates multiplication in time to circular convolution in frequency. xnhn1NXk*Hk x n h n 1 N X k H k

Parseval's Theorem

Parseval's theorem relates the energy of a length-NN discrete-time signal (or one period) to the energy of its DFT. ∑n=0N-1|xn|2=1N∑k=0N-1|Xk|2 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-valued data 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.
xn x n real  Xk=XN-kmodN¯ X k X N k N (This implies X0 X 0 , XN2 X N 2 are real-valued.)
Real part of X(k) is even image3.png
Subfigure 2.1: Even-symmetry in DFT sense
Imaginary part of X(k) is odd image4.png
Subfigure 2.2: Odd-symmetry in DFT sense
Figure 2: DFT symmetry of real-valued signal

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