ite SAMPLED-DATA CONTROL SYSTEMS 
nomials in z given by 

MOVe (2 — 201)(2 — Zoo) * + + (@ — Zon) 
(7 Zyl) (25a) ee (Ge Zpk) 
then its pole and zero distributions will be as shown in Fig. 5.20. If zis 
taken equal to e7, it traces a path in the z plane which is the unit circle. 
In that case, the denominator will consist of the complex numbers repre- 
sented by the sinors z — 2»; indicated in Fig. 5.20. It is readily seen that 
if any poles are near the unit circle, the magnitude of this sinor becomes 
very small, with the result that the magnitude of K(w) becomes very 
large at this frequency. Thus, if poles are very close to the unit circle, 
(5.67) 

Fia. 5.19. Frequency-response function for system used in example. 
they produce a pronounced frequency overshoot. The frequency at 
which this overshoot occurs depends on the angular location of the poles 
in question. For instance, a pole which is near the unit circle at an 
angle of 45° indicates a frequency overshoot at one-eighth of the sampling 
frequency. This condition is anal- 
Im ogous to the condition of dominance 
ae of poles in determining the time- 
domain response. A pole located 
near the unit circle will have a 
dominant effect on the transient 
response. 
Re Indirectly, the frequency re- 
sponse can indicate the location of 
poles of the over-all pulse transfer 
_ function, and all the properties asso- 
ciated with frequency response in 
continuous systems can be trans- 
ferred to sampled-data systems. 
The fact that the frequency re- 
sponse of a sampled-data system 
is periodic causes no concern since the repeated spectra are merely 
Fic. 5.20. Component sinors contribut- 
ing to frequency-response function. 
