82 SAMPLED-DATA CONTROL SYSTEMS 
tour, which is usually taken as the unit circle. It has been shown in 
Sec. 4.3 that, for functions having poles outside the unit circle, the pulse 
sequences in positive time are divergent and, conversely, for functions 
having poles inside the unit circle, the pulse sequences in positive time are 
convergent. ‘The pulse sequences for all poles contained inside the unit 
circle (or, more generally, inside the contour of integration) are zero for 
negative time. On the other hand, the pulse sequences for all poles con- 
tained outside the unit circle are zero for all positive time and convergent 
for negative time. Just as in the case of the Laplace transform, the selec- 
tion of the condition of nonzero value for positive or negative real time 
depends on the selection of the path of integration and the manner in 
which it divides the poles of R(z) as being within or without the contour. 
Since the main interest is in pulse sequences which are bounded in both 
negative and positive time, the unit circle is a natural choice for the 
integration path, and the inversion formula for a two-sided z transform 
containing poles inside and outside the unit circle becomes 
r(nT) = » res. R(z)e""2 on SO (4.77) 
all poles 
outside 
unit circle 
and ri) — » res. R(z)2z"—} n2z0O (4.78) 
all poles 
inside 
unit circle 
The two-sided z transform can thus be used to describe pulse sequences 
both in the positive and negative time domains, subject to the restrictions 
and conventions outlined in this section. 
EXAMPLE 
A two-sided z transform R(z) is given by 
1 1 
Up ogee La 
It is assumed that qg is a number less than unity, so that the first term in 
R(z) contains a pole outside the unit circle and the second term contains 
a pole inside the unit circle. Thus, the first term represents the pulse 
sequence for negative time and the second term for positive time. 
Applying the inversion procedure outlined previously and as expressed 
by (4.77) and (4.78), the positive and negative time pulse sequences 
can be found. First R(z)z"~! is brought to the form 
GAS ar) 
G2 (2a) 
where the pole inside the unit circle is at q and the pole outside the unit 
circle is at q7t. 

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