THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 61 
The process of long division can be organized into a numerical routine 
which is implemented either by manual desk-calculator methods or 
programmed on a digital computer. 
4.4 Initial and Final-value Theorems 
It is often desirable to ascertain the initial and final values of the 
pulse sequence which result from the inversion of a z transform. ‘These 
properties have been studied**!19 and can be summarized readily. The 
initial value of the pulse sequence can be obtained by noting that for 
physically realizable functions, F(z) can be expanded into a power series 
in 2! as has been done in previous sections. Thus, 
LQ) 2 UA ais UID aye 8 2 ae CLE ara 8 eG ZAG)) 
It is seen that by assigning a value of infinity to z, the only term having 
a value in (4.26) other than zero is the first term. Thus, the initial- 
value theorem is simply stated as 
(0) = lim F(z) (4.27) 
EXAMPLE 
The example used in Sec. 4.1 will be taken for an illustration of the 
initial-value theorem. It was found there that F(z) was given by 
1 e oT z-1 ae e eT z-1 
) a —b (1 — e°*?z-1)\(1 — e-@Fz-) 

The initial value f(0) obtained by substituting infinity for z becomes 
‘zero. Thus, the initial value of the pulse sequence f*(¢) resulting from 
the inversion of F(z) is zero. 
It has been pointed out** that for the class of z transforms for which 
the initial value is other than zero, a formal evaluation of the initial 
value of f*(¢) is only one-half that obtained from (4.27). This arises 
from the strict application of the complex-convolution integral of (4.10) 
used to obtain F(z). In this integral, it was implicit that the contribution 
to the integral resulting from integration along the infinite path used 
to close the contour was zero, in consequence of the fact that F(p) 
vanished for infinite arguments. For those functions whose impulsive 
response has an initial value, the order of p in the numerator and denom- 
inator polynomials is the same; hence for infinite values of p, F(p) does 
not vanish. 
It can be readily ascertained that the contribution to the value of F(z) 
of this infinite path leads to a result which produces only one-half the 
initial value of f*(t) as computed from (4.27). The formal application 
