68 SAMPLED-DATA CONTROL SYSTEMS 
This summation is the analogue of the convolution integral in continuous 
systems and is referred to as the convolution summation. 
Since the pulse transfer function is a relation between the z transforms 
of the input and output, it is recalled that C(z) is defined by 
ce) — y c(mT )e—™ (4.42) 
m=0 
Substituting for c(mT) from (4.41), C(z) becomes 
C(z) = » y r(nT)g(m — n)T2—™ (4.43) 
m=0n=0 
As an aid in simplifying this double summation, an auxiliary integer k 
is introduced, such that 
k=m—-—n (4.44) 
Eliminating the integer m by replacing it by its equivalent in (4.44), the 
summation of (4.43) can be rearranged as 
o ao 
Ce) = > d, maT gkT ere (4.45) 
k=—n n=0 
It is noted that for physically realizable systems, g(kT) has zero value 
for all negative values of k so that the lower limit of the first summation in 
k can have its lower limit replaced by zero. It is also noted that the 
various elements in (4.45) are functions either of k or n so that they may 
be separated to yield the following form: 
Ce g(kT)z-* AO Dae (4.46) 
i Di 
The second summation in (4.46) is recognized to be the z transform R(z) 
of the input pulse sequence r*(t). The first summation is defined as 
G(z), given by 
i] 
G(z) = YY g(kT)2-* (4.47) 
k=0 
Using this definition, the output z transform C(z) is given by 
C(z) = G(z) R(e2) (4.48) 
The relation between the output and input z transforms is given by 
(4.48), and G(z) is called the pulse transfer function. The definition of 
G(z) is contained in (4.47), where it is seen that it is the z transform of the 
sampled impulsive response g(¢). In other words, the pulse transfer 
