THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 77 
through the block diagram it is seen that 
ei(nT) = r(nT) — dier(n — 1)T — boei(n — 2)T— --- (4.60) 
That this relation between samples is correct can be seen by taking all the 
r’s on one side and all the c,’s on the other side of the equation and then 
taking the z transform of both sides in the manner used on (4.49). Doing 
so, this results in 
R(z) = Ci(z)(1 + biz! + bez? + - - *) (4.61) 
which is the required relationship. 
The total output C(z) is now obtained by operating on the output 
Ci(z) by D(z). The significance of this operation is that ao times the 
present output sample is added to a; times the output sample one sample 
time previously, etc. These outputs are available in the system shown in 
Fig. 4.9 and by taking off weighted samples as shown in Fig. 4.10, the total 
ney C} es) 

Fic. 4.10. Implementation of system whose transfer function is D(z). 
output sequence C(z) is obtained. Thus, the output sample c(nT) at 
any particular sample time nT’ is obtained by the addition of weighted 
samples of the input r(nT) and intermediate output ci(nT). It is noted 
that a number of storage elements equal to the order of D(z) must be 
provided to hold the various numbers and that means for weighting these 
numbers by the various a’s and b’s must also be available. If a general- 
purpose digital computer is used to implement the operations in real time, 
the operations can be programmed into the computer. If a special-pur- 
pose computer is used, combinations of digital and analogue techniques 
can be used to best advantage. The implementation described in this 
section uses the minimum number of storage elements, although other 
methods employing more storage elements have been described in the 
literature.* | 
