SAMPLED-DATA SYSTEMS WILL 
put which exists in the steady state and the transient terms which result 
from the poles of K(z) vanish as time approaches infinity, only the 
residues at the pole of the integrand introduced by the forcing function 
need be evaluated. Thus, the steady-state component of the output is 
given by 
Ce(nl) = K (eT) eineoT (5.64) 
An envelope which produces a pulse sequence such as that in (5.64) is a 
sinusoidal function c(t), which is given by 
c(t) = K (eT) eft (5.65) 
It is understood that this envelope is not unique since any number of 
other components may be included so long as they correspond to the 
value of the output at sampling instants. However, the envelope which 
has been selected is the simplest and lowest-frequency time function 
which fits these points. The frequency-response function is that complex 
function of frequency which relates the amplitude and phase of the output 
sinusoid in terms of the input sinusoid. In this case, referring to the 
frequency-response function as K(w), it is seen to be 
C(w) 
Ka(c) i — Rl) 

= "KO)-e (5.66) 
Since K(z) is periodic in wo, where wo is 27/7, it follows that the frequency- 
response function K(w) is also periodic in wo. 
EXAMPLE 
A pulsed network has a pulse transfer function G(z) given by 
1 
1 — e-4?z71 
cer 
It is desired to determine the frequency response G(w) of this pulsed 
network. Applying (5.66), there results 
if 
Oe) = a ote 
Typically, this frequency-response function plots as shown in Fig. 5.19. 
It is seen that the response is periodic in wo. 
The periodicity of the frequency-response function of sampled-data 
systems is disconcerting since it is not readily comparable to the common 
and familiar response functions found in continuous systems. A clearer 
understanding of the significance of these response functions is obtained 
by going back to the pole distributions of the pulse transfer functions in 
the z plane. If an over-all pulse transfer function K(z) is a ratio of poly- 
