CHAPTER 4 
THE Z-TRANSFORM ANALYSIS OF LINEAR 
SAMPLED-DATA SYSTEMS 
It has been stated that linear difference equations relate the variables 
in linear sampled-data dynamical systems as do differential equations in 
linear continuous systems.‘® While classical methods can be employed 
to obtain the solution of difference equations, the use of transform 
methods results in considerable simplification and understanding of the 
problems associated with the analysis and synthesis of sampled-data 
systems. A form of transform calculus now known as the “‘z transforma- 
tion’’ has been available in one form or other for many years.**> Recent 
work has been directed toward the organization and unification of the 
theory. !7/1:5?,36,49,19,ete. Tt is now possible to apply z-transform techniques 
directly to the problem of sampled-data systems and, more particularly, 
feedback systems. The z transformation can be studied as a modifica- 
tion of the Laplace transformation or approached directly as the opera- 
tional calculus of number sequences. In some cases one approach is 
better than the other, but the resulting theory is the same and various 
theorems and rules can be derived readily. 
4.1 Introduction to the Z Transform 
It has been shown in Chap. 2 that the Laplace transform of an impulse 
sequence f*(t) has a Laplace transform F'*(s) which is given by the infinite 
summation 
F*(s) = > f(nT)e-"?s (4.1) 
n=0 
where f(nT) is the value of the function f(é) at sampling instants and 
for the condition that the impulse approximation is acceptable. It was 
also stated and demonstrated by example that infinite sequences like 
that of (4.1) could be expressed in closed form provided that the Laplace 
transform F’(s) of the original function f(¢) from which the sample sequence 
f*() was obtained is the ratio of polynomials in s. This will be proved 
rigorously in the next section. The important point here is that whether 
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