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The Properties of the DTFT 73
2
Real part of X
10
Imaginary part of X
1
5
Re(X)
0
Im(X)
0
–1
–5
–2
–1 –0.5 0 0.5 1
frequency in π units
2
Transform of even part
–10
–1 –0.5 0 0.5 1
frequency in π units
10
Transform of odd part
1
5
XE
0
XO
0
–1
–5
–2
–1 –0.5 0 0.5 1
frequency in π units
–10
–1 –0.5 0 0.5 1
frequency in π units
FIGURE 3.6 Plots in Example 3.12
>> error2 = max(abs(XO-j*XI)) % Difference
error2 =
1.8033e-019
% graphical verification
>> subplot(2,2,1); plot(w/pi,XR); grid; axis([-1,1,-2,2])
>> xlabel(’frequency in pi units’); ylabel(’Re(X)’);
>> title(’Real part of X’)
>> subplot(2,2,2); plot(w/pi,XI); grid; axis([-1,1,-10,10])
>> xlabel(’frequency in pi units’); ylabel(’Im(X)’);
>> title(’Imaginary part of X’)
>> subplot(2,2,3); plot(w/pi,real(XE)); grid; axis([-1,1,-2,2])
>> xlabel(’frequency in pi units’); ylabel(’XE’);
>> title(’Transform of even part’)
>> subplot(2,2,4); plot(w/pi,imag(XO)); grid; axis([-1,1,-10,10])
>> xlabel(’frequency in pi units’); ylabel(’XO’);
>> title(’Transform of odd part’)
From the plots in Figure 3.6 we observe that the real part of X(e jω ) [or the
imaginary part of X(e jω )] is equal to the discrete-time Fourier transform of
x e(n) [or x o(n)].
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