CHAPTER 11 
MISCELLANEOUS APPLICATIONS 
OF SAMPLED-DATA THEORY 
The theory which has been developed in previous chapters was applied 
specifically to linear dynamical systems containing one or more samplers 
operating periodically. Taken more abstractly, however, the z trans- 
formation bears the same relation to linear difference equations as does 
the Laplace transformation to linear differential equations. For this 
reason, sampled-data theory can serve to solve any problem which is 
represented exactly by a linear difference equation, as is the case in 
sampled-data systems, or approximately, as in other cases, where the 
difference equation is only an approximation to a differential equation. 
An advantage of the linear difference equation is that it can be integrated 
directly by use of desk calculators or digital computers. 
Linear continuous dynamical systems are described approximately by 
means of linear difference equations. The error which is incurred can 
be controlled by choice of quadrature interval and interpolation tech- 
nique employed. The former corresponds to the sampling interval and 
the latter to the form of data hold in a sampled-data system. For 
instance, a polygonal approximation, while physically unrealizable, serves 
as a good means of improving the computational accuracy, as contrasted 
to the use of a zero-order data-hold approximation. 
Methods of adapting sampled-data theory to the numerical solution of 
systems which can be approximated by difference equations will be con- 
sidered. There are a number of approaches to this problem, each having 
certain advantages and disadvantages. The use of a sampled-data model 
which bears topological similarity to the actual continuous system has the 
advantage of making possible a direct relation between the location of the 
samplers and the frequency content of the signal at those points. On the 
other hand, a direct approximation of the inverse Laplace transform of 
the variable of interest is convenient because it takes into account some 
initial conditions. In addition, this approach can be applied to time- 
varying systems. It will also be shown that sampled-data theory can 
also be used to evaluate certain infinite summations by application of the 
Poisson summation rule. Generally, this chapter will deal with some of 
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