THE SAMPLING PROCESS 21 
- Without. attempting an exhaustive discussion of the problem, the 
applicability of the impulse approximation to the special case of feedback 
control systems will be discussed. The transfer functions of the elements 
of such systems are generally low-pass in frequency response. The essen- 
tial requirement in determining the validity of the impulse approxima- 
tion is that the impulse response of the feedforward (or any other transfer 
functions to which the pulses are applied) be acceptably equivalent to the 
pulse response. For low-pass systems this condition is usually satisfied 
if the pulse duration is small compared with the time constants of the 

nT nT+y¥ Time 

Fig. 2.8. Finite-width pulse. 
system. Although a pulse sequence would rarely be applied directly 
to the plant or continuous element of the control system without some 
form of data reconstruction, the impulse approximation can be assumed 
as being adequate if the condition mentioned above is met. On the 
other hand, in most practical systems the plant is preceded by a data- 
reconstruction component which is even less sensitive to pulse width 
than the plant, as will be seen later. 
In order to obtain a quantitative estimate of the error introduced by 
the impulse approximation, one can assume that the actual pulse is a 
flat-topped pulse whose amplitude is that of the sampled function f(t) 
at the sampling instant »7 and whose duration is y, as illustrated in 
Fig. 2.8. As seen in the figure, the pulse may be thought of as being the 
sum of a step function at the time n7 and a delayed negative step func- 
tion at a time (n7J’+ vy). Thus, 
p(nT) = f(nT)[u(t — nT) — ult — nT — y)] (2.17) 
where p(nT) is the pulse initiated at the sampling instant n7J and f(nT) 
is the value of f(t) at the same instant. The Laplace transform of this 
pulse, P,(s), is given by 
P,(s) = f(nT) (<7 as | | (2.18) 
