D SAMPLED-DATA CONTROL SYSTEMS 
out as outputs should the necessity arise. Assuming for purposes of 
discussion that f(t) is a variable of interest, it is plotted in Fig. 1.1 asa 
continuous function of time. The plot or some analytic expression for 
f(t) will describe the function completely as a function of time. 
If, now, the value of f(¢) is read or sampled at equal intervals of time T 
so that the function is described by the sequence of numbers 
$0), (1), f2T), FBT), ... , fT), ... (1.1) 
it is seen that a limited description of the function f(é) has been given. 
For instance, the value of f(¢) at (1.57) is not available, so that a certain 

Fig. 1.1. The sampling operation. 
amount of information has been lost in the process of expressing f(t) 
as a number sequence given by (1.1). On the other hand, if the function 
is well-behaved, the intermediate values of f(t) can be interpolated 
between samples with acceptable accuracy. If the function is not well- 
behaved, it means that large and unpredictable variations in f(é) have 
occurred between sampling instants. The number sequence such as that 
of (1.1) then gives only a poor approximation of the variable. 
It is seen from this simple qualitative discussion that the sampling 
frequency must be related to the characteristics of the function being 
sampled, lest important information be lost in the sampling process. 
At the same time, if the sampling frequency is well chosen relative to the 
characteristics of the time function being sampled, only negligible 
information is lost in the sampling process. In the latter circumstance, 
the use of more samples would merely burden the system by carrying 
unessential information that could have been obtained by the simplest of 
interpolative processes. 
Considerations such as these suggest that continuous systems are 
capable of carrying and transmitting far more information than is 
required or justified by the dynamical-system characteristics. In the 
frequency domain, this is equivalent to stating that a capability of some 
components of the system to carry and transmit excessively large band- 
