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Properties of the Discrete Fourier Transform 171
50
Real{DFT[x(n)]}
20
Imag{DFT[x(n)]}
40
30
20
10
0
10
−10
0
0 5 10
k
−20
0 5 10
k
50
DFT[xec(n)]
20
DFT[xoc(n)]
40
30
20
10
0
10
−10
0
0 5 10
−20
0 5 10
k
k
FIGURE 5.15 Plots of DFT symmetry properties in Example 5.10b
b. MATLAB script:
>> X = dft(x,11); Xec = dft(xec,11); Xoc = dft(xoc,11);
>> subplot(2,2,1); stem(n,real(X)); axis([-0.5,10.5,-5,50])
>> title(’Real{DFT[x(n)]}’); xlabel(’k’);
>> subplot(2,2,2); stem(n,imag(X)); axis([-0.5,10.5,-20,20])
>> title(’Imag{DFT[x(n)]}’); xlabel(’k’);
>> subplot(2,2,3); stem(n,real(Xec)); axis([-0.5,10.5,-5,50])
>> title(’DFT[xec(n)]’); xlabel(’k’);
>> subplot(2,2,4); stem(n,imag(Xoc)); axis([-0.5,10.5,-20,20])
>> title(’DFT[xoc(n)]’); xlabel(’k’);
From the plots in Figure 5.15 we observe that the DFT of x ec(n) isthe same as
the real part of X(k) and that the DFT of x oc(n) isthe same as the imaginary
part of X(k).
□
A similar property for complex-valued sequences is explored in Problem
P5.18.
5. Circular shift of a sequence: If an N-point sequence is shifted in
either direction, then the result is no longer between 0 ≤ n ≤ N − 1.
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