DATA RECONSTRUCTION 37 
organic form can be unstable when a sampled-data link is introduced in 
the closed loop. 
3.4 The First-order Data Hold 
The first-order data hold is a device which implements the first two 
terms of (3.11) or (8.12). Using the form of (8.11), the output of a first- 
order data-hold device is 
nie 5, Ss ata) 2) he ie 
T T (Gale) 
This expression shows that the extrapolated function in a particular 
sampling interval is linear and that the slope of the extrapolated function 
r(t), p(t} 
r(t) 

Fic. 3.8. Reconstruction of r(¢) by a first-order data hold. 
is equal to the first back difference. Figure 3.8 illustrates the recon- 
structed time function r,(¢) produced by such an extrapolation. The 
imperfections result from the fact that (3.17) omits the higher-order back 
differences, which, if nonzero in the original function, can produce con- 
siderable error. 
As in the case of the zero-order hold, it is desirable to obtain the 
transfer function of such a device by taking the Laplace transform of the 
impulsive response. It should be noted that the slope of the straight-line 
extrapolation is equal to the difference between the most recent sample 
and the previous sample divided by the sampling time interval. Thus, 
to obtain correctly the slope within an interval, the impulsive response 
of the hold in any given interval must contain the effect from the previous 
interval. Figure 3.9 shows the impulsive response of the first-order hold. 
In determining the correctness of this response, it is seen that at the 
onset of a pulse, the output rises immediately to the value of the pulse. 
A straight line is then generated whose slope is equal to the value of the 
sample at that instant divided by the sampling interval 7. It should 
be recalled that in its application, the first-order hold is actually sub- 
