140 SAMPLED-DATA CONTROL SYSTEMS 
network in Fig. 6.16 it is clear that 
(1 — 27)Z[P(s)/s] 
1+ (1 — 27)Z[Q(s)/s] 
is the pulse transfer function of the network. The expression (6.30) can 
be factored into two terms: 
N(z) = (6.30) 
Ni) = (2) D2) (6.31) 
where Dz) = A - es (6.32) 
aad Dere : (6.33) 
1+ (1 — 2-)Z[Q(s)/s] 
The factor D,(z) is basically the result of the series RC network, P(s), and 
D,(z) is the result of the feedback network, Q(s). The proof of the 
theorem that (6.29) can always be realized with the form of (6.30) 
depends upon the demonstration that D,(z) can be realized with essen- 
tially arbitrary zeros and D;(z) can be realized to provide arbitrary poles. 
If P(s) isa realizable RC transfer function, then all the poles of P(s) are 
simple and lie on the negative real axis of the s plane. A partial-fraction 
expansion of P(s)/s is thus of the form 
P(s) _ ki Pe eee 
eos > iat s;real, —-27 <8; <0 (6.34) 
a 
where the k; are arbitrary, except for a constant multiplying factor, and 
one of the s; is zero. The z transform of P(s)/s is therefore 
kz 
b, > —- (6.35) 
where a; = e~*7 and one of the a; is unity. Therefore, the form of D,(z) 
is, from (6.32) 
Den — ae 
— . ES a (6.36) 
The expression (6.36) has the following properties: 
1. The poles of D,(z) are in the range 0 < z; < 1. 
2. The zeros of D,(z) are arbitrary. 
