8 SAMPLED-DATA CONTROL SYSTEMS 
In the field of mathematics, Demoivre and Laplace? developed a 
form of transform calculus which could be applied to the solution of 
linear difference equations. ‘This approach was adapted to the solution 
of pulsed filters and sampled-data systems by Hurewicz,!7 who laid much 



Galvanometer 
Movement 
Chopper bar 


Bell crank 
Follower — 
cam 
Fic. 1.6. Sketch of chopper-bar galvanometer. 
of the basic groundwork for the transform method of analysis of sampled- 
data systems. Subsequent investigations! **47 further extended this 
initial work. The result of these efforts was the development and 
refinement of the so-called z transformation and its application to the 
analysis and synthesis of sampled-data systems. 
The z transformation is entirely analogous to the Laplace transforma- 
tion and its application to continuous systems. It turns out that, for 
systems having lumped constants, that is, those which are described by 
linear difference equations with constant coefficients, the z transformation 
gives expressions which are rational polynomial ratios in the variable z. 
This variable is complex and is related to the complex frequency s used 
in the Laplace transform by the relation z = e7*. In z-transform theory, 
such concepts as the transfer function, mapping theorems, combinatorial 
theorems, and inversion bear the same powerful relation to sampled-data 
systems as does the Laplace transformation to continuous systems. 
Without going into detail at this point, the general concept of the 
z transformation as applied to systems is shown in Fig. 1.7. Here the 
output number sequence of the system is related to the input number 
sequence by a linear difference equation. If the sampled output is c*(¢) 
and the input is r*(¢), and if the z transforms of these sequences are 
C(z) and R(z), respectively, a pulse transfer function G(z) can be found 
_which relates them in the following manner: 
C2) = G2) R(2) (1.2) 
