CHAPTER 8 
BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 
The z transformation describes the behavior of the sampled-data sys- 
tem at sampling instants only, unless a modified or delayed form of the 
transformation is employed. Despite this apparent limitation, the 
simplicity, clarity, and ready manipulation of the standard z transforma- 
tion makes it a most valuable tool. On the other hand, where critical 
evaluation of the performance of sampled-data filters or control systems 
is to be made, the behavior of the system between sampling instants is an 
important factor. For instance, some systems which are designed to 
have finite settling time at sampling instants do not necessarily settle to 
an equilibrium condition between sampling instants. There will be 
over- and undershoots in the output of such systems, even though the 
response at sampling instants is perfect after the finite settling time has 
passed. A comprehensive design procedure should include behavior 
during the sampling intervals, as well as at sampling instants. 
The behavior of the system between sampling instants has been 
referred to as “‘ripple.”’ Under certain conditions, this ripple may take 
the form of ‘hidden oscillations,’ !?24 whose amplitude might increase 
without limit with time despite the fact that the response at sampling 
instants is perfect. While this phenomenon is of academic interest, it 
can be readily avoided in practice and is therefore only of secondary 
importance. Various techniques are available for the study of the per- 
formance of sampled-data systems between sampling instants. These 
will be discussed in this chapter. While there are advantages and dis- 
advantages in the application of the various methods, it is generally true 
that each method should be thoroughly understood in order that it can 
be applied to optimum advantage in each circumstance. 
8.1 Approximation of Ripple Using Infinite Summation 
If a linear continuous system is subjected to a sampled input, the out- 
put at sampling instants is fully described by the z transform. On the 
other hand, if the continuous output is desired, the Laplace transform of 
the continuous output must be obtained. Referring to Fig. 8.1, the 
transfer function of the continuous system is G(s). The input to the 
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