THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 81 
given in (4.73) and R2(z) describes the sequence in positive time and has 
the usual form. 
For the condition that the pulse sequence r*(¢) is even, that is, 
r(nT) = r(—nT), then (4.74) becomes 
R(z) = R.(z-) + R2(z) — r(O)z? (4.75) 
An interpretation of (4.75) is that for those poles of Re(z) which lie on or 
inside the unit circle, the poles of R2(z—!) lie on or outszde the unit circle, 
as shown in Fig. 4.11. For instance, if one of the poles is at 0.5 as shown, 
the other pole lies at 2.0. The significance of this fact can be understood 
by considering the inversion theorem given by (4.20): 
oe) = i [ Ae! ab (4.76) 
where the path of integration I is ordinarily taken as a contour or circle 
about the origin of radius sufficient to enclose all the singularities of R(z). 
By evaluating the residues at the 
various poles of the integrand so zplane Im 
enclosed, the pulse sequence can be 
evaluated for all positive time, or, 
equivalently, positive n. 
This interpretation of the inver- 
sion theorem in the z domain is sim- 
ilar to that of the ordinary Laplace 
transform on the s plane. The z 
transform R(z) may contain poles 
inside and outside the unit circle, par- 
ticularly if it is a two-sided z trans- 
form. This is in many ways analo- 
gous to the Laplace transform containing poles on the left half or right half 
of thesplane. In the latter case, the choice as to whether poles on the right 
half plane represent divergent time functions which are nonzero for posi- 
tive time or convergent time functions which are nonzero in negative time 
depends on the choice of contour. In order to produce a time function 
which is nonzero for all positive time, the contour must be chosen such 
that it consist of the imaginary axis displaced by a constant c so that all 
poles lie to the left and a closure is made to encompass the entire left half 
of the s plane. To produce a time function which is nonzero for all nega- 
tive time, a displaced imaginary axis and enclosure of the right half of 
the s plane is made to enclose all poles contributing to the function. 
Thus, the choice of integration path determines whether positive or nega- 
tive time is considered. 
In the case of the z domain, the integration path is a finite closed con- 
Unit circle 

Fia. 4.11. Location of poles of R(z) and 
RGE=}). 
