62 SAMPLED-DATA CONTROL SYSTEMS 
of the inversion process would require that only one-half the value of 
(4.27) be used, though in fact, in a practical system, the full value would 
be obtained at a time of 0+ after application of the impulse. If the 
z transform is viewed in the light of a generating function, as was sug- 
gested previously, rather than as an outgrowth of a strict application of 
Laplace-transform theory, this difficulty would not be experienced. For 
fle 

(N—1)T NT 
Fra. 4.4. Truncated pulse sequences used for evaluation of final value. 
purposes of practical feedback control systems, the convention is adopted 
that the initial value of the pulse sequence is the full value of f(0) as 
given by (4.27) and that the factor of $ indicated by the formal mathe- 
matics is not representative of the physical system and therefore should 
be ignored. 
The final value of the pulse sequence f*(é) can also be obtained directly 
from its z transform, F(z). To derive the result, the pulse sequence 
f%(t) is formed by truncating the sequence f*(t) at the Nth sample, 
where N is some large number. The truncated sequence is shown in 
Fig. 4.4a, and it is evident that its z transform is 
N 
F(z) = » f(T) (4.28) 
n=0 
If the truncated function is now delayed by one sample time 7, then a 
function f*(t — T) is formed, and the sequence is plotted in Fig. 4.40. 
It is evident that the z transform of this delayed function is the same 
as that of the truncated function, except that an additional delay z~! is 
included as a factor. Thus, 
N-1 
Fy(@) =F y(2) = Y f(nTyere (4,29) 
n=0 izt 
