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n::::l 5 Sampling 330 5.1 T he S ampling T heorem 5 .1-3 ~" UY (a) The signal t
(b) The PAM signal (c) The P WM (PDM) signal 331 Some Applications o f t he Sampling Theorem T he s ampling theorem is very i mportant in signal analysis, processing, a nd
t ransmission because i t allows us t o replace a continuous-time signal by a discrete
sequence of numbers. Processing a continuous-time signal is therefore equivalent
t o processing a discrete sequence of numbers. Such processing leads us directly
into t he a rea of digital filtering. In t he field of communication, t he t ransmission
of a continuous-time message reduces t o t he t ransmission of a sequence of numbers
using pulse trains. T he c ontinuous-time signal f (t) is sampled, a nd sample values
are used t o modify certain parameters of a periodic pulse train. We may vary the
amplitudes (Fig. 5.9b), widths (Fig. 5.9c), o r positions (Fig. 5.9d) of t he pulses in
proportion t o t he sample values of t he signal f (t). Accordingly, we have p ulseamplitude m odulation (PAM), p ulse-width m odulation ( PWM), or p ulse
p osition m odulation ( PPM). T he most i mportant form of pulse modulation today
is p ulse c ode m odulation ( PCM), discussed below. In all these cases, instead of
t ransmitting f (t), we t ransmit t he corresponding pulse-modulated signal. At t he
receiver, we read t he i nformation of t he p ulse-modulated signal a nd r econstruct t he
analog signal f (t). (d) The PPM signal N t) ..... F ig. 5 .9 Pulse modulated signals.
Use of Eqs. (3.66) yields Co =~ ............. 2
and C n = n" sin (n4"); t hat is, v'2 1 Co = . ....................... . "-.,........ Cs = - 571" , . .. 4' ........... Consequently
- 1 f (t) = f (t)PT(t) = - f(t) 4 v'2
+ -71" f(t) cos 2071"t + -1" f(t) cos 4071"t + - 3 f (t) cos 6071"t + ...
71" F ig. 5.10 Time-division multiplexing of two signals. and
- F(w) 1
= - F(w) + ~V 2
71" + 1 2071") + F(w + 2071")J + -1
[F(w - ;n[F(w - 6071")
371" V 2 271" 4011') + F(w + 4071")J + F(w + 6071")J + ... In the present case F(w) = 0.211.(2~")' The spectrum F(w) is depicted in Fig. 5.8e.
Observe t hat the spectrum consists of F(w) repeating periodically at the interval of 2071"
r ad/s (10 Hz). Hence, there is no overlap between cycles, and F(w) can be recovered by
using an ideallowpass filter of bandwidth 5 Hz. An ideallowpass filter of unit gain (and
bandwidth 5 Hz) will allow the first term on the right-side of the above equation to pass
fully and suppress all the other terms. Hence, the output y(t) is
y (t) = 1
4f(t) • O ne a dvantage of using pulse modulation is t hat i t permits t he simultaneous
transmission of several signals on a time-sharing basis ( time-division m ultiplexing, o r T DM). Because a pulse-modulated signal occupies only a p art of t he channel time, we c an t ransmit several pulse-modulated signals on the same chann...
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