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158 Chapter 5 THE DISCRETE FOURIER TRANSFORM
4
Magnitude of the DTFT
3
|X|
2
1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
frequency in π units
200
Angle of the DTFT
100
Degrees
0
−100
−200
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
frequency in π units
FIGURE 5.4 The DTFT plots in Example 5.6
angle part. Generally these angles should be ignored. The plot of DFT values
is shown in Figure 5.5. The plot of X(e jω )isalso shown as a dashed line for
comparison. From the plot in Figure 5.5 we observe that X 4 correctly gives
4 samples of X(e jω ), but it has only one nonzero sample. Is this surprising? By
looking at the 4-point x(n), which contains all 1’s, one must conclude that its
periodic extension is
˜x(n) =1, ∀n
which is a constant (or a DC) signal. This is what is predicted by the DFT
X 4(k), which has a nonzero sample at k =0(or ω =0)and has no values at
other frequencies.
□
□ EXAMPLE 5.7 How can we obtain other samples of the DTFT X(e jω )?
Solution
It is clear that we should sample at dense (or finer) frequencies; that is, we
should increase N. Suppose we take twice the number of points, or N = 8
instead of 4. This we can achieve by treating x(n) asan8-point sequence by
appending 4 zeros.
x(n) ={1, 1, 1, 1, 0, 0, 0, 0}
↑
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