74 
SAMPLED-DATA CONTROL SYSTEMS 
The pulse transfer function which expresses the process D(z) is thus 
Y(z) om 
Ne) =) diag 
Thus, if an input number sequence has a z transform X(z) and is multi- 
plied by D(z), the output sequence has the transform Y(z). Inversion 
of the latter will give the values of the integral at the various sampling 
times. 
More complex interpolations of x(t) in the interval under considera- 
tion result in more complex pulse transforms. For instance, using 
known numerical integration formulas, 
nT T 
il a(t) at = 5 [x(n — 1)T + 2(nT)] 
( eu 
n—1 
By a treatment similar to the simple case, this results in a recursion 
formula given by 
y(n) — y(n — YT = 5 [a(n — HT + a(n) 
Taking the z transform and evaluating the pulse transfer function for 
the process, 
Simpson’s 2 rule is stated in the more complex integration formula, in 
which a(t) is assumed to be fitted by a second-order polynomial in time. 
This formula states 
[v.20 sed 5 le(nT) Asin = TP oe 
Taking the z transform of both sides once again, there results the pulse 
transfer function for the process: 
=: + 4271+ 2? 
1-2? 
D(z) = 
These examples serve to illustrate the fact that a pulse transfer function 
can describe a purely numerical process in which there is no associated 
physical system. 
ac 
In applying these pulse transforms, care should be taken to take into 
count the error caused by the initial value of the number sequence 
resulting from the inversion of Y(z). Such an error arises when the 
integrand x(t) has an initial value x(0) other than zero. For instance, in 
