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66 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
15
Magnitude Part
10
|X|
5
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 3.4 Plots in Example 3.6
Solution
Once again we will compute and plot X(e jω ) over two periods to study its
symmetry property.
MATLAB script:
>> n = -5:5; x = (-0.9).^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; axis([-2,2,0,15])
>> xlabel(’frequency in units of pi’); ylabel(’|X|’)
>> title(’Magnitude Part’)
>> subplot(2,1,2); plot(w/pi,angX/pi);grid; axis([-2,2,-1,1])
>> xlabel(’frequency in units of pi’); ylabel(’radians/pi’)
>> title(’Angle Part’)
From the plots in Figure 3.4 we observe that X(e jω )isnot only periodic in ω
but is also conjugate-symmetric. Therefore for real sequences we will plot their
Fourier transform magnitude and angle graphs from 0 to π.
□
3.1.3 SOME COMMON DTFT PAIRS
The discrete-time Fourier transforms of the basic sequences discussed in
Chapter 2 are very useful. The discrete-time Fourier transforms of some
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