222 SAMPLED-DATA CONTROL SYSTEMS 
For large values of z,, where the series of (9.3) converges uniformly, sum- 
mation with respect to / and k may be interchanged, with the result 
Ce) — » r(kT) » g (2 _ er) gran (9.4) 
k 1=0 
=0 
In the second sum, it is always possible to find an integer 7 such that 
(1 — nk) = j, and the transform may then be written in terms of j as 
follows: 
EN ae 
cea — ») r(kT) > g (; =| Zin Ek) (9.5) 
0 j=0 
jp= 
The second sum in (9.5) is written from 7 = 0 rather than from 7 = —nk 
because the realizable impulse response g(t) is zero for negative values of 
the argument. The powers of z, can be separated out so that 
y (kT) (en) > 4 (; r) ac (9.6) 
k=0 j=0 
R(2n")G (Zn) (9.7) 
A word must be said about the notation in (9.7). As is evident from 
(9.6), the function R(z,”) is the z transform of the input R(s) (based on 
samples separated by 7’ sec) with the variable z replaced by 2,7. For 
example, if 
I 
Ce) 

it 
R(s) = — 
1 
then R(z) = ees 
and R(én") = >: 
As a matter of fact, it should not be surprising that z and z, are related. 
On a Laplace-transform basis it is already known that z = e*7, and it may 
be shown that z, = e*7/", so that in fact 2,” = z¢ and the operation indi- 
cated above is a simple change of variable. The transform G(z,) is the 
ordinary pulse transfer function of the linear system, based on a sample 
separation of 7T'/n sec. The variable z, identifies the period of the 
samples used in determining G(z,). 
The problem illustrated in Fig. 9.1 is the same as that which results 
from a search for ripple in the output of a single-rate system, and (9.7) 
expresses the solution to that problem. A more difficult case, from an 
analytical point of view, is shown in Fig. 9.2. In this case the continuous 
