THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 65 
The symbol Z, used here for the first time, indicates the z transform 
corresponding to the Laplace transform following it. 
By treating the delayed transform in this manner, a table of advanced 
transforms F'(z,A) can be used for all situations. By the definition of the 
z transform, the expression for F'(z,A) can be stated as 
F(z,A) = ) fin + A)Tz (4.37) 
n=0 
Many of the advanced z transforms can be evaluated directly from (4.37), 
as seen in the example. 
EXAMPLE 
It is desired to derive the expression for the advanced z transform 
for a time function e~* which has been advanced AT. In this case, 
f@ is 
f(t) — eg —a(tt+AT) (t + Nel) = 0 
The z transform corresponding to this function is 
JAB IN) = » GF Vere ge 
n=0 
_ which can, by factoring out e~747, be written 
F(z,A) <= e aAT » e anTz—n 
The summation is an infinite geometric progression, as in the case of 
ordinary z transforms, and can be expressed in closed form, resulting in 
eaAT 
F(z,A) oa 1 pe eal g-1 
This simple illustration shows how advanced z transforms can be 
obtained by going back to the time domain and expressing the sequences 
in closed form. 
Another approach to evaluating the advanced transform is to use the 
method of complex convolution as expressed in (4.14). Since the path 
of integration used to obtain the z transform encompasses the left half 
of the s plane, the added term e47* vanishes at infinity and causes no 
difficulty in the evaluation of the integral by the method of residues. 
Thus, by direct application of (4.14), the advanced z transform F(z,A) 
