THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 71 
output number sequences, an approach is to introduce a “‘ generating 
function,” defined as follows: 
D(z) = » d(kT)2-# (4.50) 
k=0 
where d(kT) is the value of the number occurring at the kth sample 
and z—* is the ordering variable which establishes the position of the 
number in the sequence. The similarity to the z transform and the 
pulse transfer function is obvious; indeed, the generating function and the 
z transform are identical if z = e7. The generating function was 
originally introduced to handle weighting sequences found in mathe- 
matical statistics.2 The input and output sequences are described by 
generating functions R(z) and C(z), where the significance of z is also 
that of an ordering variable. 
Referring once again to (4.49), it is noted that the relation holds for all 
positive integer values of n. Thus, iterating the equation for all values 
of n and taking care to apply the correct ordering variable to each num- 
ber, there results the following equality: 
ve} eo [-<} 
» c(nT)z—" + by 2 c(nT)z"!4+ +--+ +b, » OG ae 
0 
n=0 =0 n= 
ao 
= Qo y AI a8 <> Oh » PINES qo 0 oe 
n=0 n=0 
(2) 
Wigs > r(nT)e-™—™ (4.51) 
n=0 
This complex relationship holds for any instant of time corresponding to a 
sampling instant. Since the time of the incidence of a particular number 
is contained in the exponent of z, then the equality at any sampling 
instant can be set up between numbers having the same power in z. 
If this is done for any particular value of n, it will be seen that the required 
relationship between numbers as expressed in (4.49) will be obtained. 
The lower index in the summations of (4.51) is zero, which indicates the 
fact that the values of the sequences for negative time are zero. 
Factoring out the common summations for each of the terms of (4.51), 
there results 
oO 
> c(nT)z-"(1 + bye? + bee? + +s + + dye-*) 
n=0 
) 
= » r(nT)2—"(ao + ayz7! + aoe? + > + + + amz7™) (4.52) 
n=0 
