THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 85 
tems containing both digital elements and pulsed linear systems. These 
are common in the case of digitally compensated sampled-data control 
systems. In the optimization of sampled-data feedback control systems, 
it is often useful to use as a criterion the sum of the squares of the error 
pulses resulting from the application of some form of systematic input. 
It is possible to obtain this sum by direct utilization of the z transform of 
the pulse sequence in a contour integral. 
In some problems, the inputs and disturbances to a sampled-data sys- 
tem may be random time functions. To handle this type of input, which 
is assumed to extend over negative as well as positive time, a two-sided 
z transform can be used. Analogous in all respects to the two-sided 
Laplace transform for continuous systems, it forms the basis of the theory 
underlying sampled-data systems with random excitation. 
From all viewpoints, the z transformation is a powerful tool in the 
analysis and synthesis of sampled-data systems. It is not much more 
complex than the ordinary Laplace transform when used in similar prob- 
lems in continuous systems. In fact, there are many advantages, not 
the least of which are that inversion can be handled by means of calculat- 
ing machines and that many continuous problems are reduced to sampled 
models in order to take advantage of this fact. Tables of z transforms 
are available, thus further enhancing its value as a working tool. Sub- 
sequent chapters will make full use of the z transformation and its 
modifications. 
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