278 SAMPLED-DATA CONTROL SYSTEMS 
the input delayed by one period. Between sampling periods, however, 
the continuous signal output is composed of a combination of a growing 
and a decaying exponential. The unstable plant or continuous portion 
of the filter is kept under control by the digital controller which pre- 
cedes it, but this control depends upon the exact cancellation of a pole 
outside the unit circle, which is not physically possible. A solution to 

ye 17,1 
c(t) 
e“1lr_ er 
Fia. 10.11. Block diagram of stabilized optimum filter. 
the problem may be obtained by realizing the filter as a closed-loop 
error-sampled system which has been adjusted to avoid the unstable 
cancellation, as discussed in Chap. 7. In the case of a filter such as is 
being discussed here, the system must be designed so as to maintain the 
transfer function given by (10.103). In this situation the necessary 
degree of freedom for avoiding the unstable cancellation is introduced 
by the addition of a feedback constant in the system. This additional 
constant is selected in such a manner that the digital controller in the 
closed loop is not required to cancel the unstable pole of the plant trans- 
fer function. A block diagram of the stabilized system is shown in 
Fig. 10.11. 
The example problem which has been worked out consists essentially of 
the design of a data hold or extrapolator for random data. It is instruc- 
tive to compare the operation of the optimum filter with that of a simpler 
and more common device. For example, how does the mean-square error 
of the optimum system compare with that of a clamp, or zero-order hold? 
This question can be answered simply by calculating the mean-square 
value of the difference between the output of the zero-order hold and the 
ideal output. As discussed in Chap. 6, the zero-order hold closely 
resembles a delay of half a sampling period, so that the error will be calcu- 
lated based on an ideal system of a half-period delay. A block diagram 
of the situation is shown in Fig. 10.12. For the configuration shown in 
Fig. 10.12, 
(e?(é)) = (fet) — ea()]?) 
(c?(t)) — 2(c(t)ca(t)) + (ca?(t)) 
= &..(0) — 28..,(0) + Besez(0) (10.105) 
where ca(t) equals r[é — (7/2)]. If r(t) is a stationary random signal, 
then the mean-square value of r(¢) delayed by half a sampling period, 
