THE SAMPLING PROCESS 13 
T,2T, ...,nT are the sampling instants. ‘The sampled time function 
f(t) appearing at the output of the sampling switch, or sampler, as it is 
usually called, is referred to as f*(t), where the asterisk implies the 
sampled version of a function of time f(t). A graphical representation 
of f*(é) is shown in Fig. 2.2. In order to develop analytical approaches 
f*(t) 

SO yen Oa 



—-3T -2T -T 0 aed, She AT 
Time ——> 

Fic. 2.2. Sampled time function. 
to the sampled-data control-system problem, a mathematical description 
of this process must be obtained. 
2.1 Mathematical Description of Sampling Process 
A practical sampling operation cannot ignore the fact that the sampler 
remains closed for a finite, though short, length of time. For this 
condition, the ‘‘samples”’ are elements of finite duration y whose ampli- 
tudes during the closure interval follow the amplitude variations of the 
time function f(é) and are zero at all other times. This type of sampling 
operation is shown graphically in Fig. 2.3. 
The sampled function f*(£) appears here as a sequence of samples of 
finite duration y. The process may be thought of as being the result 
of multiplying a sampling function p(é) and a data-carrying function f(t), 
as shown in Fig. 2.3. This is a process of modulation, where a “‘carrier”’ 
p(t) is being modulated by a signal function f(t). Mathematically, the 
process may be represented by the expression 
FO = fOr) (2.1) 
In view of the fact that the sampling interval T is constant, thus making 
p(t) a periodic function, it is possible to expand p(t) into a Fourier series 
+0 
p=) Cee (2.2) 
k=—o0 
