CHAPTER 2 
THE SAMPLING PROCESS 
The characterizing feature of a sampled-data system is the fact that 
data appear at one or more places in the system as a sequence of pulses or 
numbers. The process which converts continuous data into such a 
sequence is called the sampling process. In view of its importance, the 
process must be thoroughly understood and represented mathematically. 
Fortunately, there exists a considerable body of literature in the field of 
communications theory where this problem has been considered. Before 
introducing some of this theory, a few concepts of sampling will be 
considered qualitatively, with particular emphasis on their application 
to control systems. 
A schematic representation of the sampling process is given in Fig. 2.1, 
where it is seen that a continuous time function f(t) is observed by 
means of a switch which closes briefly every 
a T sec. The mechanical device implied by 
this representation is not binding and is 
t) +z used only for pictorialsimplicity. Thesam- 
eee ida deal pling of the time function f(t) may be 
Fic. 2.1. Schematic represen- carried out by electronic devices or may 
a i aaa Bg CS ag only implicit in that the data exist 
only as a sequence of samples to begin with. The closure of the switch 
is of very short duration compared with the time between closures. 
This means that the value of the function at the output of the switch 
is the instantaneous value of the function f(¢) at the particular instant of 
time when the switch closes. The output of the switch is a sequence of 
pulses or numbers whose values are f(71), f(T), . - . , f(T), where 
T1, Tz, ..., Tn are the instants of time when the switch has closed. 
It is clear that some information has been lost in the process, since the 
output sequence contains no data in the interval between sampling 
instants. 
While theory has been developed to handle the case where the intervals 
between sampling instants are not equal, the majority of practical situ- 
ations in control systems have equal or quasi-equal sampling intervals. 
In this case, the output of the sampling switch produces a sequence of 
numbers f(T), f(27), . . . , f(nT’), where T is the sampling interval and 
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