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Sampling and Reconstruction in the z-Domain 151
□ EXAMPLE 5.5 Let x(n) =(0.7) n u(n). Sample its z-transform on the unit circle with N =5,
10, 20, 50 and study its effect in the time domain.
Solution
From Table 4.1 the z-transform of x(n) is
1
X(z) =
1 − 0.7z = z
, |z| > 0.7
−1 z − 0.7
We can now use MATLAB to implement the sampling operation
˜X(k) = X(z)| z=e j2πk/N , k =0, ±1, ±2,...
and the inverse DFS computation to determine the corresponding time-domain
sequence. The MATLAB script for N =5is as follows.
>> N = 5; k = 0:1:N-1; % sample index
>> wk = 2*pi*k/N; zk = exp(j*wk); % samples of z
>> Xk = (zk)./(zk-0.7); % DFS as samples of X(z)
>> xn = real(idfs(Xk,N)); % IDFS
>> xtilde = xn’* ones(1,8); xtilde = (xtilde(:))’; % Periodic sequence
>> subplot(2,2,1); stem(0:39,xtilde);axis([0,40,-0.1,1.5])
>> xlabel(’n’); ylabel(’xtilde(n)’); title(’N=5’)
The plots in Figure 5.3 clearly demonstrate the aliasing in the time domain,
especially for N =5and N = 10. For large values of N the tail end of x(n)
1.5
N=5
1.5
N=10
xtilde(n)
1
0.5
xtilde(n)
1
0.5
0
0 10 20 30 40
n
N=20
1.5
0
0 10 20 30 40
n
N=40
1.5
xtilde(n)
1
0.5
xtilde(n)
1
0.5
0
0 10 20 30 40
n
FIGURE 5.3 Plots in Example 5.5
0
0 10 20 30 40
n
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