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The Properties of the DTFT 71
60
Magnitude of X
1
Angle of X
|X|
40
20
radiands/pi
0.5
0
−0.5
0
−1 −0.5 0 0.5 1
frequency in π units
60
Magnitude of Y
−1
−1 −0.5 0 0.5 1
frequency in π units
1
Angle of Y
|Y|
40
20
radians/pi
0.5
0
−0.5
0
−1 −0.5 0 0.5 1
frequency in π units
−1
−1 −0.5 0 0.5 1
frequency in π units
FIGURE 3.5 Plots in Example 3.9
>> xlabel(’frequency in pi units’); ylabel(’|Y|’)
>> title(’Magnitude of Y’)
>> subplot(2,2,4); plot(w/pi,angle(Y)/pi); grid; axis([-1,1,-1,1])
>> xlabel(’frequency in pi units’); ylabel(’radians/pi’)
>> title(’Angle of Y’)
From the plots in Figure 3.5, we observe that X(e jω )isindeed shifted by π/4
in both magnitude and angle.
□
□ EXAMPLE 3.10 To verify the conjugation property (3.8), let x(n) beacomplex-valued random
sequence over −5 ≤ n ≤ 10 with real and imaginary parts uniformly distributed
between [0, 1]. The MATLAB verification is as follows.
>> n = -5:10; x = rand(1,length(n)) + j*rand(1,length(n));
>> k = -100:100; w = (pi/100)*k; % frequency between -pi and +pi
>> X = x * (exp(-j*pi/100)).^(n’*k); % DTFT of x
% conjugation property
>> y = conj(x); % signal conjugation
>> Y = y * (exp(-j*pi/100)).^(n’*k); % DTFT of y
% verification
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