MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 291 
tinuous element, a number of locations are possible in the closed-loop case. 
For instance, the sampler could be placed in the error line, in the feedback 
line, or at some intermediate point in G(s) or H(s). In view of the fact 
that the sampler and data hold are most accurate for the lower-frequency 
components of a signal, it follows that the best location is always at that 
point where the frequency content of the signal is lowest. 
Assuming that the feedforward component of the system, G(s), is low- 
pass, it is evident that the most restricted bandwidth for the signal will be 
found at the output of this element. If H(s) is also low-pass, then the 
bandwidth at the output of this element is even more restricted. Figure 
11.66 shows the sampler and data-hold elements placed at the output of 
G(s), on the assumption that G(s) is low-pass and that H(s) is not. The 
accuracy of the computations based on this sampled model depends on 
the accuracy with which c;(t) is reproduced from a sequence of samples 
derived from c(t) by the triangle data hold. To compute the output 
c*(t), the z transform of the output C.(z) is found from the expression 
RG(z) 
CAO) = Te AO 
(11.8) 
Inversion of C,(z) by one of the standard methods will yield the desired 
output sequence c*(t) of the sampled model at sampling instants. To 
illustrate the technique, an example will be given. 
EXAMPLE 
The simple feedback system shown in Fig. 11.7a has a unit step r(¢) 
applied to its input. It is desired to compute the output c(t). As an 
arbitrary practical rule, the bandwidth of the system will be assumed 
to extend from zero frequency to an upper frequency at which the out- 
put is down 30 to 40 db below this level. It is assumed, therefore, that 
all components above this value are not significant. For the system 
used in the example, this frequency is 1 eps, at which frequency the 
response is down 32 db. The sampling frequency will be chosen at 
1 eps mainly for convenience, even though a higher frequency should 
have been chosen on the basis of using twice the frequency of the 
highest significant frequency component to be passed. 
The sampled model which will be used is shown in Fig. 11.7b, where 
it is seen that the triangle hold is placed in the feedback path to be con- 
sistent with the requirement that sampling take place at the point 
where the most restricted spectrum isfound. If the input r(¢) is a unit 
step function, then the elements to be substituted in (11.8) become 
1 
RO) Th ee aT 
