B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

6.1 Sampling Theorem 263
Figure 6.9
(al Non-bandlimited
signal
spectrum and its
sampled
spectrum G(f) .
(bl Equivalent
low-pass signal
spectrum G a (f)
constructed from
uniform samples
of g(t) at
sampling
rate 2B.
(a)
Aliasing
B
2B
Aliasing
Equivalent low-pass signal G a (f)
(b)
-B 0
B
This theoretical rate of communication assumes a noise-free channel. In practice, channel
noise is unavoidable, and consequently, this rate will cause some detection errors. In
Chapter 14, we shall present the Shannon capacity which determines the theoretical error-free
communication rate in the presence of noise.
6. 1 .4 Nonideal Practical Sampling Analysis
Thus far, we have mainly focused on ideal uniform sampling that can use an ideal impulse
sampling pulse train to precisely extract the signal value g(kT s ) at the precise instant of t =
kT s . In practice, no physical device can carry out such a task. Consequently, we need to
consider the more practical implementation of sampling. This analysis is important to the
better understanding of errors that typically occur during practical AID conversion and their
effects on signal reconstruction.
Practical samplers take each signal sample over a short time interval T p
around t = kT s .
In other words, every T s seconds, the sampling device takes a short snapshot of duration T p
from the signal g(t) being sampled. This is just like taking a sequence of still photographs
of a sprinter during an 100-meter Olympic race. Much like a regular camera that generates a
still picture by averaging the picture scene over the window T p
, the practical sampler would
generate a sample value at t = kT s by averaging the values of signal g(t) over the window T p
,
that is,
g1 (kT s ) = -
1 T p/2
1 g(kT s + t) dt
T p -T p/2
(6.1 9a)
Depending on the actual device, this averaging may be weighted by a device-dependent
averaging function q(t) such that
g1 (kT s ) = -
1 T p/2
1 q(t)g (kT s + t) dt
T p -T p/2
(6.19b)