50 SAMPLED-DATA CONTROL SYSTEMS 
an infinite number of terms of (3.35) would be required. However, an 
approximation to the result may be obtained by taking only a finite 
number of terms in the expansion of e7*. For instance, if only the first 
two terms of the expansion are taken, G(s) becomes 
IL 
Gils) Teor 
(8.36) 
A simple resistance-capacitance network can be used to obtain this 
impulsive response. As matter of fact, in many applications, particularly 
those where the sampling frequency is high, simple RC networks of this 
type are used as data holds. From the frequency-response point of view, 
the network is low pass, with a half-power frequency of 2r/T radians/sec. 
This should be compared with the response of a zero-order data hold, as 
shown in Fig. 3.7. 
By taking additional terms in the expansion of e7’, a better approxima- 
tion to the zero-order data hold may be obtained. Taking an additional 
term, there results the approximation 
1G teetligy 2 
EAC) ts + Ts + T?s?/2 
(3.37) 

This transfer function can be realized by passive elements, but it can be 
readily verified that the poles of the transfer function are complex and 
that the physical implementation of the network requires inductive, 
capacitative, and resistive elements. While this is no serious theoretical 
obstacle, the need for passive elements of impractical dimensions often 
results in feedback control systems having sampling periods measured in 
seconds and minutes. For this reason, the use of passive networks of 
any but the simplest form, such as that given in (3.36), are not common. 
3.9 Summary 
The data hold may be regarded as an element which reconstructs the 
continuous function from which a sequence of data samples is obtained. 
Except for certain cases, this reconstruction is approximate at best, and 
the difference between the output of the data hold and the actual function 
from which the sequence of samples was derived is known as ripple. 
Since data samples are available at each sampling instant, the output 
of the data hold for a particular sampling interval is readjusted as each 
data sample is received. In view of the requirement of physical realiz- 
ability, only past data samples can be used in estimating the output 
of the data hold. From the viewpoint of the frequency domain, data 
holds are low-pass filters which pass the signal spectrum and reject the 
spurious side spectra which result from the sampling process. 
