58 SAMPLED-DATA CONTROL SYSTEMS 
applying one of the Cauchy theorems of contour integration in the com- 
plex plane. It has been shown in (4.3) that the z transform of a sampled 
sequence f*(t) is given by 
F(z) = ‘s f(nT)e-" (4.15) 
n=0 
Expressed in open form, this summation is 
Fig) = fiO\e0 ah) ee tas ite lize ey 
+f(nT)z" +--+ (4.16) 
This expansion of F(z) about z = © is valid for positive time, that is, 
for positive n. The process of inversion requires that a relation be 
found which will give f(n7’) explicitly, Just as the inversion of F(s) 
requires a relation which gives f(¢). 
A step in arriving at the inversion theorem is to multiply the infinite 
sequence of (4.16) by z”~}, resulting in 
HG) ea — (Oat i Weatee eer et iit all) ecul 
+ f(nT)z1 + +--+ G17) 
The relation given in (4.17) is now in a form to which the Cauchy theorem 
in question may be applied. 
This particular Cauchy theorem states that if the integral J is defined 
by 
= 1 k 
— sil dz (4.18) 
and if Tis a closed contour which encloses the origin of the z plane, then 
I will have values given by 
0m ort 
T=1 =a 
baO~ dee sil (4.19) 
This theorem may be readily proved by taking I as a circle, as shown in 
Fig. 4.3, and then generalizing the result for any other contour which 
encloses the origin. If the function is regular in the region enclosed 
between the circle and the irregular contour, which 2* is, of course, 
then there is no contribution to the integral J beyond that made by the 
circular path. 
This theorem can now be applied to the expression for 2" -1F(z), as 
given in (4.17), by applying the results of (4.19) term by term. Thus, 
x} ih oF (2) de = fn) (4.20) 
