30 SAMPLED-DATA CONTROL SYSTEMS 
extrapolator.”’ Figure 3.1 shows a sampled-data feedback control system 
in which the data-hold element is located immediately after the sampler. 
The signal is reconstructed from the sampled data into a continuous 
signal before being applied to the plant. In this context, the effective- 
ness of the data hold is measured by the difference between the recon- 
structed time function e,(t) and the actual error function e(t). In a 
later chapter, where digital compensation of sampled-data systems is 
considered, it will be shown that e,(t) is not necessarily an approximation 
of e(t) but rather of a modified error function so computed that it produces 
a desired compensating effect. In this chapter, only those aspects of 
the problem dealing with the reconstruction of the function without 
modification are considered. 
3.1 The Cardinal Data Hold 
In the previous chapter, it was shown that the signal is completely 
recoverable, provided that the Fourier transform spectrum terminates 
at a frequency equal to one-half the sampling frequency. An ideal 
filter which has unity transmission from zero frequency to one-half the 
sampling frequency and zero transmis- 
sion everywhere else is known as the 
cardinal data hold. Figure 3.2 shows 
the frequency response of such a filter 
which can be expressed by 
| Fjw)| 
F(jw) = 1.0 —W See (3.1) 
F(jw) = 0 —-W202W ; 
That such a filter is not physically 
realizable can be shown by inverting 
Fic. 3.2. Frequency response of cardi- vine Hommes transform eee 
a oltnicen! the impulsive response as follows: 
= ee cist df (3.2) 
Integrating this expression and simplifying, the impulsive response 
becomes 
-W=-7/T 0 +W=r/T 
Frequency, radians per sec. 
=o (3.3) 
Tv 

This impulsive response is plotted in Fig. 3.3, where it is seen that the 
filter is physically unrealizable because there is a finite response prior 
to the application of the exciting impulse at ¢ = 0. 
The significance of this result is that, even for those time functions 
which have a finite spectrum, it is not practical to expect a perfect 
