DATA RECONSTRUCTION 33 
To be exact, an infinite number of terms of (3.9) must be taken. 
Adapting (3.9) to the case of the data hold used in sampled-data 
reconstruction, it is seen that the time ¢ is taken at a particular sampling 
instant nT and the prediction time 7 is the time measured from this 
instant. Thus, taking 
= poll (3.10) 
substitution in (3.9) results in the expression 
r(nT +7) =r(nT) + = {r(nT) — r[(n — 1)T]}7 
+ aa (r(nP) — 2rl(m — 17) + ohm — 27) FSO 4 --- Bay) 
where it is understood that the extrapolation is taken from the time nT 
onward. It is seen that the data which are needed to implement the 
extrapolation are the values of the function r(¢) at sampling instants nT, 
(n — 1)T, (n — 2)T, etc., only. These are the values of the function 
which are available in a sampled-data system, and for this reason the 
Gregory-Newton extrapolation formula is very useful. 
The polynomial in 7 which is generated by (3.11) is seen to contain 
elements that are recognized as the various back differences of the 
sampled function. For instance, the second term contained in the 
(3.11) is the first back difference; the third term, the second back differ- 
ence; etc. The expression may be written in the form 
(GR 0) = eam EE ee 
T+ (8.12) 
where the various V"r(n7') are the back differences of nth order. Several 
practical conclusions can be drawn from the extrapolation formula in 
the form given in (3.12). First, to generate an extrapolated function 
which is derived from a completely general r(¢) containing finite back 
differences of all orders, an infinite number of back samples must be 
taken into account. On the other hand, if the original r(t) had been an 
nth order polynomial in time containing only a finite number of terms, the 
extrapolation would be perfect with only n + 1 prior samples taken into 
account. This is in consequence of the fact that a finite polynomial 
contains only n finite back differences, where n is the order of the poly- 
nomial, all others being zero. If an extrapolator is used which contains 
an insufficient number of back differences, the extrapolated function 
will be in error, and the problem of the designer is to keep the magnitude 
of this error within acceptable bounds. 
Equation (3.12) suggests a method of classification for data-hold 
systems of this type depending on the number of terms contained therein. 
