DATA RECONSTRUCTION 47 
(n + 1)T back to the point at (n — 2)T. Generalizing the result to an 
nth order polynomial, this form of correction will cause the newly gener- 
ated polynomial to fit the n + 1 most recent samples. 
The implementation of a scheme employing this principle is essentially 
that shown in Fig. 3.17, and the detailed technique for accomplishing 

Fic. 3.19. Generation of third-order polynomial. 
the result will now be considered. First, the polynomial generator is a 
cascaded number of integrators, as shown in Fig. 3.19 for the third-order 
case. To understand its operation, let it be assumed that the output 
of the last integrator, f(t), is a third-order polynomial given by 
ft) = Go + aut + gol? + gal? (3.30) 
where the various q’s are constants. The function f’(¢) appearing at the 
input of the last integrator is the derivative of (3.30) and is 
fO) = a + 2qat + 3qst? (3.31) 
and similarly, the inputs to the two preceding integrators, f’(¢) and 
jf (@), are 
f° O) = 22 + baat 
f''O = 64s (3.32) 
Thus, to generate a polynomial whose coefficients are the various q’s 
given in (3.30), it is necessary to apply steady signals whose values are 

Fig. 3.20. Porter-Stoneman-type polynomial data hold. 
63, 2q2, qi, and qo at the points in the integrator chain shown in Fig. 3.19. 
This simple concept can be used to generate the correction polynomials 
which are added to Q(t) to generate a sequence of corrected polynomials 
which fit the requisite number of past input samples. The block diagram 
of such a system is shown in Fig. 3.20, which contains the required number 
