110 SAMPLED-DATA CONTROL SYSTEMS 
lies close to the unit circle, say, with a magnitude of 0.9, and all other 
poles lie well within the unit circle, say, with magnitudes of 0.2 or less, the 
former are dominant poles and determine the transient response of the 
system. 
5.7 Frequency Response of Sampled-data Systems 
While not as useful as in the analysis of continuous systems, frequency- 
response methods can be applied to sampled-data systems also. In fact, 
the pulse transfer locus which is a map of the unit circle may be regarded 
as a frequency locus since, by definition, z = e7*, and as z traces the unit 
circle on the z plane it is equivalent to 
Ris) _Rl2) C\s) Clz) tracing the imaginary axis on the s 
plane. Frequency-response concepts 
Fia. 5.18. System used to define fre- can be applied to the over-all response 
See response of sampled-data Ju)se transfer function K(z) of a sam- 
pled-data system. Referring to Fig. 
5.18, the block represents a sampled-data system whose internal con- 
figuration may be either open- or closed-cycle. The frequency response 
of this system is obtained by applying a sinusoidal signal at the input, 
shown as R(s), and determining the phase and amplitude of an equiv- 
alent sinusoidal envelope passing through the resulting sequence of 
samples at the output, shown as C(z). It is recognized that the actual 
output of the system before sampling, shown as C(s), is not sinusoidal but 
that it contains many frequency components at sampling frequency and 
related frequencies; however, it will be shown that the sampled output can 
describe an envelope which is sinusoidal. 
To determine analytic expressions for the frequency response, it will be 
assumed that the input is 
ii) ee (5.60) 
The z transform of r(¢) is obtained in the usual manner or from the tables, 
resulting in 
1 
— eleTz-1 
(5.61) 
If the pulse transfer function of the system is K(z), then the output 
sequence is given by the relationship 
Ci) = K(z¢) RE) (5.62) 
Inverting C(z) by means of the inversion integral, c(n7’) becomes 
1 at 
Since the frequency response represents only that component of the out- 
