70 SAMPLED-DATA CONTROL SYSTEMS 
In the previous development, the pulse transfer function was derived 
for the case where a continuous element was included between the two 
samplers. Fully digital systems which perform arithmetic operations 
on number sequences to deliver processed number sequences at their out- 
put do not contain such an element. Nevertheless, if the operations 
which are performed are linear, it is possible to define a pulse transfer 
function which relates the input and output number sequences. Sche- 
matically, the digital system is illustrated as a block diagram in Fig. 4.8. 
c*(t) 
C(z) 
Fia. 4.8. Pulse transfer function of digital system. 

In this figure, the block D represents the arithmetic process being carried 
out. The input and output sampling switches indicate that the input 
to the digital system is a sequence of numbers and that the output is a 
sequence of processed numbers. In this simple representation, it is 
assumed that the switches are synchronous, that is, the output numbers 
are delivered simultaneously with the intake of a new number. If 
there is a significant computation delay, the output number will be 
synchronous but delayed by a fixed time from the input number. This 
can be taken into account by the insertion, after the output sampler, 
of a transportation lag or time delay equal to the computation time. 
The arithmetic computations carried out in the digital element may 
be of many different classes, but the usual form used in linear feedback 
control systems is a linear difference equation relating the input and out- 
put pulse sequences. Such an equation or recursion formula is written 
as follows: 
c(nT) + bicl(n — 1)T] + boc[(n — 2)T] + - + > + dicl(n — k)T] 
= aorel| 4 aun —) | 4 aor, — 2) Pe 
+ amr[(n — m)T] (4.49) 
where the various c’s are the output numbers at the instant corresponding 
to the argument, the various r’s the input numbers corresponding to the 
argument, and the b’s and a’s are constants. This linear relation can 
be interpreted as a formula through which the present output number 
c[nT] can be computed by taking weighted sums of a fixed group of input 
and output numbers. In the strict sense, this computation can be per- 
formed by storing k — 1 previous output numbers and m input numbers 
and adding them in the manner prescribed by (4.49). 
To obtain a pulse transfer function D(z) which relates the input and 
