60 SAMPLED-DATA CONTROL SYSTEMS 
There are many practical situations where the interest of the designer 
is only in the first few terms of the pulse sequence resulting from inver- 
sion. This is the case where transient response and overshoot are being 
studied in feedback control systems. For this purpose, an alternate 
approach to inversion can be employed. The z transforms encountered 
in practice are generally the ratio of polynomials in z or z~! as expressed 
by 
Qo Gil a @isae ot Aa Geom 
bo + byz-! + bez? +: + + bnew 
This transform can be expanded into a power series in z~! by the simple 
process of long division of the denominator into the numerator. Carry- 
ing out this numerical procedure, there will result 
F(2) = qo -b ie quer Ee (4.25) 
This power series in z~! is now identified with the z transform of an 
impulse sequence, where the power of z is the order or time at which 
the impulse exists and the various q’s are the areas of each impulse. The 
inversion process is now seen to be merely an arithmetic routine which 
can be carried out by means of a desk-calculator or similar methods. 
F(z) = (4.24) 


EXAMPLE 
It is desired to invert the z transform F(z) given by 
F() = 
oy =o eee 2 
Expanding into a power series in z~! by long division, 
1 + 1.22-! + 1.242-2 + 1.2482-3 + --: 
1 — 1.227! + 0.227? 
1.22-1 — 0.227 
1.22-1 — 1.442? + 0.242-8 
1.2422 — 0.24273 
1.242-2 — 1.4882—3 + 0.24824 
1.2482-8 . 



etc. 
The resultant power series shows that the first term of the sequence 
has a magnitude of 1, the second of 1.2, the third of 1.24, the fourth of 
1.248, etc. Thus, f*(é) is 
FC) = Oa 1-28¢— 1) DAS = 270) 
+ 1.24867 — 3T) + --> 
This method of inversion is useful in determining the first set of terms. 
