THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 957 
as follows: 
F(Z) = » res. Fe) a | (4.14) 
poles of 
F(p) 
This expression shows clearly that for all systems whose Laplace trans- 
form F(p) has a finite number of poles, the z transform is a finite poly- 
nomial in z or z~!. Also noted is that the order of the polynomial in z is 
no higher than the order of the polynomial in p. 
The result obtained in (4.14) is the same as that which is obtained by 
expanding F(p) into partial fractions and then applying the simple pro- 
cedure outlined in the example in Sec. 2.4, term by term. As a matter 
of fact, as a working method, expansion into partial fractions is often a 
more direct technique than the formal application of the residue formulas 
indicated by (4.14). Tables of z transforms are available,!4*13 and one 
is reproduced in Appendix I containing most of the common forms. 
Tables like this one list the Laplace transform F’(s) of the continuous 
function from which the pulse sequence is obtained, the time function 
f(nT) giving the value of the function at a sampling instant nT’, and the 
z transform F(z) which corresponds. 
EXAMPLE 
It is desired to determine the z transform F(z) corresponding to a 
Laplace transform F(s) given by 
20) = 
(s + a)(s + 6) 
It should be noted that a more formal statement of the problem would 
be to determine the z transform of the sampled sequence f*(t) obtained 
by sampling a continuous function f(f) whose Laplace transform is 
F(s). The shortened terminology given previously, however, is more 
convenient and more commonly used. Using the result of (4.14), F(z) 
becomes 

i i II 1 
—al—e 2! ay a—b1—e 27! 
The two terms may be combined over a common denominator, 


A@) = b 

1 e oT z-1 mee eel g-1 
F@) = a — b (1 — e°?z-)\(1 — e727!) 
4.3 Inversion of Z Transforms 
As in the case of continuous systems, the inversion of the z transform 
is an important operation which is often carried out in practical problems. 
The inversion theorem has been formalized** and may be derived by 
