96 SAMPLED-DATA CONTROL SYSTEMS 
transfer function G(z), the complex variable z is introduced. G(z)-is the 
pulse transfer function which upon inversion yields the pulse sequence 
whose sample values are g(k7’) contained in (5.37). Assuming that 
G(z) is the pulse transfer function of a stable system, it follows that the 
inequality 
Y WEN kl < @ (5.38) 
k=0 
is satisfied for the condition that |z-!| < 1 or, equivalently, that |z| > 1. 
In other words, the inequality given in (5.38) must be satisfied every- 
where outside the unit circle of the complex z plane. 
It is readily apparent that the following inequality also is true: 
i} 
Y geT ye" SY [g(bT)| le (5.39) 
k=0 
k=0 
since all the terms on the right-hand side of (5.39) are positive whereas 
those on the left-hand side may have mixed signs or be complex. Hence, 
a condition for satisfying (5.38) is that, for |z| > 1, 
[--} 
» g(kT)z-*§ < © (5.40) 
k=0 
This summation is recognized to be the pulse transfer function G(z) of the 
system, and the condition expressed in (5.40) is that G(z) be analytic 
everywhere outside the unit circle in the z plane. Thus, a stable system 
is characterized by a pulse transfer function satisfying this requirement. 
When use is made of the fact that the definition of z is e7*, it is seen that 
this requirement states that the pulse transfer function expressed in 
terms of the variable s be analytic in the right half of the s plane, a result 
which is not too surprising. 
Considering next a system described by a pulse transfer function con- 
taining poles which lie outside of the unit circle of the z plane, the condi- 
tion for instability can be found. For such a system there exists a z for 
which 
i} 
yok T)e* = (5.41) 
k=0 
for |z!| <1 or |z| > 1. Now the following sequence of inequalities is 
readily seen to be true: 
> WAT > Y ETI e+ >) o&T += > (6.42) 
k=0 k=0 k=0 
