Discrete Fourier Transform
(DFT)
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Discrete Fourier Transform
• Definition - For a length-N sequence x[n],
defined for 0  n  N  1 only N samples of its
DTFT are required, which are obtained by
uniformly sampling X (e jw ) on the w-axis
between 0  w  2 at wk  2 k / N, 0  k  N  1
• From the definition of the DTFT we thus have
X [ k ]  X ( e jw )
0  k  N 1
N 1
w 2  k / N
  x[ n ] e  j 2  k / N ,
k 0
Discrete Fourier Transform (DFT)
Definition: The Discrete Fourier Transform (DFT) is defined by:
Where n = 0, 1, 2, …., N-1
The Inverse Discrete Fourier Transform (IDFT) is defined by:
where k = 0, 1, 2, …., N-1.
The Fast Fourier Transform (FFT) is a fast algorithm for evaluating
the DFT.
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Eeng 360 3
Examples
Tutorials
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