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

6
SAMPLING AND
ANALOG-TO-DIGITAL
CONVERSION
As briefly discussed in Chapter 1, analog signals can be digitized through sampling and
quantization. This analog-to-digital (AID) conversion sets the foundation of modern
digital communication systems. In the AID converter, the sampling rate must be large
enough to permit the analog signal to be reconstructed from the samples with sufficient accuracy.
The sampling theorem, which is the basis for determining the proper (lossless) sampling
rate for a given signal, has played a huge role in signal processing, communication theory, and
AID circuit design.
6. 1 SAMPLING THEOREM
We first show that a signal g(t) whose spectrum is band-limited to B Hz, that is,
G(f) = 0
for Ill > B
can be reconstructed exactly (without any error) from its discrete time samples taken uniformly
at a rate of R samples per second. The condition is that R > 2B. In other words, the minimum
sampling frequency for perfect signal recovery is fs = 2B Hz.
To prove the sampling theorem, consider a signal g(t) (Fig. 6.la) whose spectrum is bandlimited
to B Hz (Fig. 6.lb).* For convenience, spectra are shown as functions off as well as
of w. Sampling g(t) at a rate offs Hz means that we takefs uniform samples per second. This
uniform sampling can be accomplished by multiplying g(t) by an impulse train 8T s
(t) of Fig.
6.lc, consisting of unit impulses repeating periodically every T s seconds, where T s = 1/fs.
This results in the sampled signal g(t) shown in Fig. 6.ld. The sampled signal consists of
impulses spaced every T s seconds (the sampling interval). The nth impulse, located at t = nT s ,
has a strength g(nT s ) which is the value of g(t) at t = nT s . Thus, the relationship between the
* The spectrum G(J) in Fig. 6.lb is shown as real, for convenience. Our arguments are valid for complex G(J).
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