64 SAMPLED-DATA CONTROL SYSTEMS 
4.5 Delayed Z Transforms 
A very useful modification to the ordinary z transform was introduced 
by Barker! and was discussed in later papers on the subject.*° The 
modification treats the z transform of pulse sequences which are derived 
from time functions delayed by nonintegral multiples of the sampling 
frequency. In the case of sampled-data feedback control systems, these 
transforms find application in the analysis of systems having plants with 
transportation lags. Delayed z transforms are also useful in studying 
the behavior of a sampled function between sampling instants. The 

TT Pee AST ie: ke 

Time 
Fic. 4.5. Delayed pulse sequence used to derive G(z,A). 
delayed or modified z transform can be studied by referring to Fig. 4.5. 
Here is seen a continuous function f(t) which has been delayed by a time 
\T, where X is generally nonintegral. If d is integral, the result is trivial 
since the z transform of the resulting function is simply zF(z). 
An integer m is chosen such that it is the next highest integer after X. 
Thus, a number A can be defined so that 
m=r+A (4.32) 
where A is a positive number always less than unity. If F(s) is the 
Laplace transform of the function f(é), then the Laplace transform 
F(s,A) of the delayed function is 
GN eS) ee (4.33) 
which, from (4.32), can be written 
iM GIN)) = GRLLIE (GS) Geo (4.34) 
Since m is integral, it presents no problem in obtaining the z transform. 
The result is simply 
EGA iene (eA) (4.35) 
where F'(z,A) is defined as 
H@,A)i = ZF G)e*7| (4.36) 
