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The Discrete Fourier Transform 161
Magnitude of the DFT: N=16
4
3
|X(k)|
2
1
0
−1
0 2 4 6 8 10 12 14 16
k
200
Angle of the DFT: N=16
100
Degrees
0
−100
−200
0 2 4 6 8 10 12 14 16
k
FIGURE 5.7 The DFT plots of Example 5.7: N =16
1. Zero-padding is an operation in which more zeros are appended to the
original sequence. The resulting longer DFT provides closely spaced
samples of the discrete-time Fourier transform of the original sequence.
In MATLAB zero-padding is implemented using the zeros function.
2. In Example 5.6 all we needed to accurately plot the discrete-time
Fourier transform X(e jω )ofx(n) wasX 4 (k), the 4-point DFT. This
is because x(n) had only 4 nonzero samples, so we could have used the
interpolation formula (5.19) on X 4 (k) toobtain X(e jω ). However, in
practice, it is easier to obtain X 8 (k) and X 16 (k), and so on, to fill in
the values of X(e jω ) rather than using the interpolation formula. This
approach can be made even more efficient using fast Fourier transform
algorithms to compute the DFT.
3. The zero-padding gives us a high-density spectrum and provides a better
displayed version for plotting. But it does not give us a high-resolution
spectrum because no new information is added to the signal; only additional
zeros are added in the data.
4. To get a high-resolution spectrum, one has to obtain more data from
the experiment or observations (see Example 5.8 below). There are
also other advanced methods that use additional side information or
nonlinear techniques.
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