72 SAMPLED-DATA CONTROL SYSTEMS 
The summations which have been factored out are recognized to be the 
generating functions for the output and input, respectively. Replacing 
them by C(z) and R(z), (4.52) becomes 
C(z) (l + byt + bee 2? + eee +t byz-*) 
= R(z)(ao + are“! + ane? + + + + + ame-™) (4.53) 
The pulse transfer function D(z) for the digital system is defined as 
relating the input generating function R(z) and output generating func- 
tion C(z) in the following manner: 
C(z) = D(z)R(2) (4.54) 
Using the result given in (4.53), the pulse transfer function for a system 
just described, 

C 
BOS 
_ 0 + aye! + aoe? + > + + + Anem™ 
A ee a ee (4.55) 
The pulse transfer function D(z) was obtained without recourse to the 
Laplace transform and the impulse approximation. The orders of the 
various 2’s serve to place the numbers in the proper position in the 
sequence and are treated in the same manner as the 2’s obtained through 
the Laplace transform and the definition of z as e7*. It is a fact that the 
algebraic manipulations are the same whether one deals with the pulsed 
filter or the digital system. For instance, if a number occurs at time mT, 
it has associated with it 2~”. If this number is to be delayed by one 
additional sample time, then it should have z~-”~! associated with it. 
It is readily apparent that multiplying z—” by 27, using the ordinary 
rules of algebra, produces the correct result. This is equally true when z 
is interpreted as e7*, because now e~”7* is multiplied by e-7* to yield 
(FAVES, 
The pulse transfer function for a digital system could have been derived 
in the same manner used for the pulsed filter. The only difference lies in 
the interpretation of the various steps. For instance, going back to 
(4.89), crx(mT) can be interpreted as the number resulting from the 
weighting of the nth input sample r(nT) by a weight g(m — n)T, where 
(m — n) is the “‘staleness”’ of the input sample. Carrying on the same 
manipulations of (4.40), (4.41), (4.42), and (4.43), one arrives at the 
convolution summation. The result (4.43) is now interpreted as the 
sequence in z, C(z), describing the output number sequence, and zg—” is the 
ordering variable which places each number in the correct sampling slot. 
Finally, (4.47) gives the definition of the pulse transfer function: 
0 
G(z) = > g(kT)z-# , (4.47), (4.56) 
k=0 
