40 SAMPLED-DATA CONTROL SYSTEMS 
3.5 Partial-velocity-correction First-order Hold 
A cursory examination of the function which is reconstructed by a 
first-order data hold as illustrated in Fig. 3.8 will show that under some 
circumstances, the amount of ripple content may be excessive. For 
instance, if the function from which the pulse train is derived contains 
sizable higher-order derivatives, the underlying assumption implied in a 
first-order data hold that all differences other than the first are zero can 
Full velocity correction 
Z _Partial velocity correction 
r(t) 
— Zero velocity correction 



r(t) 
Fic. 3.12. Partial-velocity-correction data hold. 
be seriously misleading. Heuristically, a means for reducing this error 
is that of extrapolating the function in any given sampling interval with 
only a fraction of, rather than the full, first difference. The process is 
illustrated in Fig. 3.12 in the first sampling interval from 0 to T sec. 
Here the extrapolated function is a straight line whose slope is only k 
times the first difference, where k is a number less than unity. 
It is difficult to arrive at a par- 
ticular value of k except for specific 
cases. ‘The frequency response can 
be used to estimate the value of k 
which will, on the average, result in 
extrapolations having least error. 
It is recalled that the frequency 
response of a full-velocity-correction 
zero-order hold showed an excessive 
transmission of components in the 
higher range of the passband from 0 
to 2r/T radians/sec. A criterion 
which can be applied to the choice of the velocity extrapolation constant k 
is that the frequency response in the passband be maximally flat. 
In order to set the value of k, it is necessary to obtain the transfer 
function of the partial-velocity data hold in a manner similar to that 
employed in the previous section. Figure 3.13 shows the impulsive 
response which will produce an extrapolation like that of Fig. 3.12. The 

Fic. 3.13. Impulsive response of partial- 
velocity-correction data hold. 
