DATA RECONSTRUCTION 45 
a number of outputs so computed that the output of the polynomial 
generator is corrected for the next interval. The nature of this correction 
is such that the polynomial for the interval in question passes through the 
previous number of samples equal to the order of the polynomial being 
generated. For lower-order holds, such as the first-order hold, this 
process is fairly evident. The full-velocity-correction system extrapo- 
lates a line which passes through the most recent data sample and the 
previous data sample. In the case of higher-order systems, the process is 
not as evident but will be clarified in subsequent discussions. 
r*(t) 
_{ orn+1)T 

e(n+1)T 
i rp(n+1)T 
(n—3)T (n—-1)T (n41)T 
(n—2)T nT 
Time 
Fic. 3.18. Porter-Stoneman polynomial extrapolation. 
Referring to Fig. 3.18, the objective of the extrapolator is to accept a 
sequence of pulses, r(n7’), and to fit a polynomial of appropriate order 
to these samples. This polynomial is called Q(t), will be assumed to pass 
through the point U, V, W, and X, and can be expressed as follows: 
Qt) = a+ a(t — nT) + a(t — nT)? + a3:(¢ — nT)? +--+ (8.24) 
where the various a’s are constants and the extrapolation holds for the 
interval from n7 to (n + 1)7. The dashed line in this interval is the 
extrapolated polynomial, which reaches a value r,(m + 1)T at the end of 
the pertinent sampling interval. The actual value of the sample at this 
instant is r(v + 1)7, and the difference between the actual and predicted 
samples at this instant is an error e(n + 1)7. If the predicted poly- 
nomial were exactly the same as the actual polynomial, this error would 
be zero and no alteration of the coefficients of (8.24) would be necessary. 
On the other hand, if there is an error e(n + 1)7, it shows that a correc- 
tion must be made on the various coefficients of (3.24) to cause the poly- 
nomial to fit the previous sample points during the next interval. 
Assuming that there is an error e(n + 1)T, a new polynomial P(t) 
must be generated to fit the function at the previous points now listed as 
V,W, X, Y, ete. This new polynomial is given by 
P(t) = bo + bit — (n + 1)T] + Bo[t — (n+ 1)TP? 
are Wal = (@ sede ae 2 (Gem) 
