DATA RECONSTRUCTION 49 
It is characterized, however, by one important advantage. It is seen 
that the feedback system does not require a memory or storage device 
which holds past samples required to generate the higher-order differences 
required for extrapolation. The storage is implicit in the integrators, 
which also serve the purpose of generating the elements of the extrapo- 
lated polynomial. 
3.8 Exponential Extrapolation 
It was stated previously that in an extrapolation or data-reconstruction 
process, it was necessary to make an a priori assumption as to the form 
of the original function. In previous sections, it was assumed that the 
function either was actually a polynomial or could be adequately approxi- 
mated by one. Another viewpoint is that in which it is assumed that the 
original function is composed of exponentials in time or that it can be 
reconstructed by a set of exponentials. In view of the fact that a passive 
network has an impulsive response which is the sum of exponentials if its 
structure includes only lumped elements, it follows that the use of 
passive networks of this type in data reconstruction implies exponential 
extrapolation. 
There exists a large body of theory relating to the approximation and 
design of passive networks and filters. In their applications to sampled- 
data systems as data holds, however, only relative simple forms are used. 
This arises from the fact that the frequency range at which the networks 
are operated is so low that only a small number of components is usually 
tolerated and, further, that the ripple requirements are generally not too 
rigid. An approach to the selection of passive data-hold networks is that 
of approximating one of the polynomial extrapolators described previously 
by passive elements. Take as an example the zero-order hold whose 
transfer function is, from (3.14), given by 
Gi(s) = — he (3.33) 
This expression may be rearranged as follows: 
1 1 
Gr(s) = ie (1 = =) (3.34) 
The procedure is to expand the exponential term into an infinite series 
in s and simplifying the expression to the following: 
1+ Ts/2+7??/6+--- 
1+ Ts + Ts?/2 + T?s?/6 + -- - 
For an exact reproduction of the impulsive response of a zero-order hold, 
G(s) = 9) (3.35) 

