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Sampling and Reconstruction of Analog Signals 87
x1(n)
1
0.8
0.6
0.4
0.2
Discrete Signal
Ts=0.2 msec
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
t in msec.
10
8
Discrete-time Fourier Transform
X1(w)
6
4
2
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
Frequency in π units
FIGURE 3.12 Plots in Example 3.19a
b. Here F s = 1000 < 4000. Hence there will be a considerable amount of aliasing.
This is evident from Figure 3.13, in which the shape of X(e jω )isdifferent
from that of X a(jΩ) and can be seen to be a result of adding overlapping
replicas of X a(jΩ).
□
3.4.3 RECONSTRUCTION
From the sampling theorem and the preceding examples, it is clear that if
we sample band-limited x a (t) above its Nyquist rate, then we can reconstruct
x a (t) from its samples x(n). This reconstruction can be thought of
as a 2-step process:
• First the samples are converted into a weighted impulse train.
∞∑
x(n)δ(t−nT s )=···+x(−1)δ(n+T s )+x(0)δ(t)+x(1)δ(n−T s )+···
n=−∞
• Then the impulse train is filtered through an ideal analog lowpass filter
band-limited to the [−F s /2,F s /2] band.
x(n) −→
Impulse train
conversion
−→
Ideal lowpass
filter
−→ x a (t)
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