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86 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
1
Analog Signal
xa(t)
0.8
0.6
0.4
0.2
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
t in msec.
2
Continuous-time Fourier Transform
Xa(jW)*1000
1.5
1
0.5
0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Frequency in KHz
FIGURE 3.11 Plots in Example 3.18
MATLAB script:
% Analog Signal
>> Dt = 0.00005; t = -0.005:Dt:0.005; xa = exp(-1000*abs(t));
% Discrete-time Signal
>> Ts = 0.0002; n = -25:1:25; x = exp(-1000*abs(n*Ts));
% Discrete-time Fourier transform
>> K = 500; k = 0:1:K; w = pi*k/K;
>> X = x * exp(-j*n’*w); X = real(X);
>> w = [-fliplr(w), w(2:K+1)]; X = [fliplr(X), X(2:K+1)];
>> subplot(2,1,1);plot(t*1000,xa);
>> xlabel(’t in msec.’); ylabel(’x1(n)’)
>> title(’Discrete Signal’); hold on
>> stem(n*Ts*1000,x); gtext(’Ts=0.2 msec’); hold off
>> subplot(2,1,2);plot(w/pi,X);
>> xlabel(’Frequency in pi units’); ylabel(’X1(w)’)
>> title(’Discrete-time Fourier Transform’)
In the top plot in Figure 3.12, we have superimposed the discrete signal x 1(n)
over x a(t) toemphasize the sampling. The plot of X 2(e jω ) shows that it is a
scaled version (scaled by F s = 5000) of X a(jΩ). Clearly there is no aliasing.
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