80 SAMPLED-DATA CONTROL SYSTEMS 
The z transform which describes pulse sequences for negative as well as 
positive time is known as the two-sided z transform. 
Assuming now that r*(¢) represents a pulse sequence over positive and 
negative time, its representation in the time domain is 
+ 
r*(t) = > r(nT)5(t — nT) (4.68) 
n=—0 
This summation can be split into two summations, one ranging over all 
negative time and the other over all positive time, as follows: 
0 
+ 
r*() = > r(nT)5(t — nT) + > r(nT)8(t — nT) — r(0) (4.69) 
0 
n=-—e r= 
t-) 
It is necessary to subtract the central term r(0) because it appears twice in 
the summations, once as the last term of the first summation and once as 
the first term of the second summation. Taking the z transform of both 
summations, 
0 
+0 
R(z) = » r(nT)e-" + > r(nT)z-" — r(0)z (4.70) 
n=— n=0 
The first summation will be identified as R(z) and the second as R2(z). 
A change of index in R,(z) from n to m will be made in order to bring 
the form of R,(z) to the same as that of R2(z). The new index is defined 
as 
m=--—n (4.71) 
Making this change, 
0 
Ri(z) = > r(—mT) am (4.72) 
m=+o 
Reversing the limits in (4.72) has no effect since it indicates merely that 
all integral m must be included. Thus, 
+2 
Ri(z) = Y r(—mT)2™ (4.73) 
m=0 
The index now can be interpreted as indicating the position of the pulse 
measured from zero in the negative direction. The summation is valid 
only for positive values of the index m. Thus, the z transform of a two- 
sided pulse sequence can be expressed as 
R(z) = Ri(z) + Re(z) — r(O)z° (4.74) 
where F(z) describes the sequence for negative time and has the form 
