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8.2 Discrete Fourier Transform
The discrete Fourier transform (DFT) is a frequency domain representation of finite-extent sequences. The DFT is a decomposition of a finite-extent sequence by a family of complex exponential sequences. The DFT of a discrete-time sequence of length N is itself a sequence of length
The DFT plays a central role in the implementation of many signal and image processing algorithms. Particularly important has been the discovery of a fast algorithm, the fast Fourier transform (FFT) [
], that reduces the number of complex multiplications needed to compute an
-point DFT by a factor of
discrete Fourier transform of
inverse discrete Fourier transform of
Fourier basis sequences
The 1D discrete Fourier transform (1D DFT) is a unique decomposition of a discrete-time, finite-length signal onto a finite family (basis) of discrete, periodic, complex exponential sequences of the form
, where k,n,N
(integers). The ratio
is the fundamental frequency. The symmetric DFT formulation is given by the following pair of equations:
X[k] is known as the
-point DFT of the sequence x[n]. Equations (8.2.1) and (8.2.2) are the analysis and synthesis equations, respectively.
This loads the
We conclude this section with a couple of examples of computing the DFT. We first calculate the DFT of the following sequence:
Here is the sequence x[n] of length 32 and its DFT.
The 2D discrete Fourier transform (2D DFT) is a straightforward extension of the 1D formulation. The DFT X[k
] of a 2D sequence
] , with
, is defined as
for 0≤ n
Next we obtain the DFT of a 2D sequence defined by the following formula:
Here is the N×N matrix representation of this sequence (N=8).
Here is the 32-point DFT of the sequence x[n
Here we plot the magnitude and phase of the 2D example sequence.
Finally, we compute the 2D DFT of one of the example images.
Here is the original image.
Here we display the magnitude (left) and phase (right) spectra of the image.
In visualizing the frequency response of a sequence, it is common to use a centered DFT domain.
The energy compaction property of the DFT is clearly visible. Most of the signal's energy is concentrated at the four corners of the DFT magnitude matrix, which are the low-frequency zones of the transform domain. Note that a large majority of the Fourier coefficients have relatively small values and therefore contribute little to the signal's total energy.
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