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The Discrete-time Fourier Transform (DTFT) 65
8
Magnitude Part
6
|X|
4
2
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
frequency in units of π
1
Angle Part
radians/π
0.5
0
−0.5
−1
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
frequency in units of π
FIGURE 3.3 Plots in Example 3.5
MATLAB script:
>> n = 0:10; x = (0.9*exp(j*pi/3)).^n;
>> k = -200:200; w = (pi/100)*k;
>> X = x * (exp(-j*pi/100)) .^ (n’*k);
>> magX = abs(X); angX =angle(X);
>> subplot(2,1,1); plot(w/pi,magX);grid
>> xlabel(’frequency in units of pi’); ylabel(’|X|’)
>> title(’Magnitude Part’)
>> subplot(2,1,2); plot(w/pi,angX/pi);grid
>> xlabel(’frequency in units of pi’); ylabel(’radians/pi’)
>> title(’Angle Part’)
From the plots in Figure 3.3 we observe that X(e jω )isperiodic in ω but is not
conjugate-symmetric.
□
□ EXAMPLE 3.6 Let x(n) =(0.9) n , −10 ≤ n ≤ 10. Investigate the conjugate-symmetry property
of its discrete-time Fourier transform.
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