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The Discrete Fourier Transform 159
Magnitude of the DFT: N=4
4
3
|X(k)|
2
1
0
−1
200
0 0.5 1 1.5 2 2.5 3 3.5 4
k
Angle of the DFT: N=4
100
Degrees
0
−100
−200
0 0.5 1 1.5 2 2.5 3 3.5 4
k
FIGURE 5.5 The DFT plots of Example 5.6
This is a very important operation called a zero-padding operation. This operation
is necessary in practice to obtain a dense spectrum of signals as we shall
see. Let X 8 (k) bean8-point DFT, then
X 8 (k) =
7∑
n=0
x(n)W nk
8 ; k =0, 1,...,7; W 8 = e −jπ/4
In this case the frequency resolution is ω 1 =2π/8 =π/4.
MATLAB script:
>> x = [1,1,1,1, zeros(1,4)]; N = 8; X = dft(x,N);
>> magX = abs(X), phaX = angle(X)*180/pi
magX =
4.0000 2.6131 0.0000 1.0824 0.0000 1.0824 0.0000 2.6131
phaX =
0 -67.5000 -134.9810 -22.5000 -90.0000 22.5000 -44.9979 67.5000
Hence
X 8 (k) ={4, 2.6131e −j67.5◦ , 0, 1.0824e −j22.5◦ , 0, 1.0824e j22.5◦ ,
↑
0, 2.6131e j67.5◦ }
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