THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 67 
this approach, a continuous element having a transfer function G(s) 
receives a pulse sequence r*(¢) at its input. The output is sampled 
synchronously to produce the output sequence c*(t). Tf R(z) and C(z) 
are the z transforms of the input and output sequences, respectively, then 
G(z) is the pulse transfer function which relates them. 
In deriving G(z) for this pulsed continuous system, or pulsed filter, 
as it is sometimes called, the impulse approximation will be used. Thus, 
a train of impulses is applied to the input of the pulsed filter, and the 
r*(t) 
(m— n) — 
aus response 


Fic. 4.7. Contribution of r(n7’) to value of m’th output pulse by linear system whose 
transfer function is G(s). 
output is the sum of a sequence of impulse response functions of proper 
magnitude and spacing. A particular component of the output is the 
impulsive response resulting from the nth sample r(n7), as shown in 
Fig. 4.7. Here is shown the contribution to the output resulting from the 
application of an impulse whose area is r(n7') being sampled at the mth 
instant. This contribution to the total value of the output sample c(mT) 
is called c,(mT) and is given by 
Cri(mT) = r(nT)g(m — n)T (4.39) 
where g(m — n)T is the impulsive response of the system after an elapsed 
time of (m — n)T sec. The total value of the output at mT is the 
summation of all the contributions resulting from input pulses ranging 
fromt=Otot= mT. Thus, 
c(mT) = » r(nT)g(m — n)T (4.40) 
n=0 
It is noted here that the upper limit of the summation can be extended 
to infinity without effect since the impulsive response g(m — n)T is 
zero for all negative arguments. Hence, 
e(mT) = » r(nT)g(m — n)T (4.41) 
n=0 
