THE SAMPLING PROCESS 19 
ripple is a form of noise which is superimposed on the desired output and 
must be considered in the over-all design problem. 
2.3 The Impulse Sampling Approximation 
Except for the fact that it was periodic, the sampling or switching 
function p(t) used in the previous section was completely general. In 
practice, it is generally true that the elements comprising p(t) have a 
very small duration time y relative to the sampling interval T. In 
fact, in digital systems where the sample is in the form of a number whose 
magnitude represents the value of the function f(t) at a particular sam- 
pling instant, the switching-function elements have, in effect, an infinitely 
small duration time y. Because an extremely narrow pulse represents 
the physical situation accurately and also because of the resultant 
mathematical simplifications, a common form of switching-function 
element is an impulse, or Dirac, delta function, 6(é). This treatment 
of the sampling process !7*8-4° has been successfully applied to the analysis 
and synthesis of sampled-data svstems. 
The assumption of impulse sampling causes the expression for p(é) to 
become 
+ 
p(t) = ) 6(t — nT) (2.10) 
n=—o 
where 6(t — nT) represents an impulse of unit area at a time n7. In 
view of its extensive use in subsequent chapters, the impulse switching 
function is abbreviated to 
+0 
» (¢ — nT) & br(Z) (2.11) 
When the switching function is assumed to be an impulse train, the 
sampling operation may be thought of as impulse modulation and has 
been so referred to in the literature.® 
If the signal function is f(t), then the output of the impulse modulator 
may be expressed as 
FO = for (2.12) 
It is noted that since the impulse is infinitely narrow in the limit, the 
only significant value of f(t) is f(n7’), the value of the function at the 
instant of time when the impulse function appears. Thus, (2.12) may 
be rewritten 
FO = fT )ér(t) (2.13) 
The impulse sequence may now be interpreted as being a sequence of 
delta functions whose respective areas are equal to the magnitude of the 
