MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 311 
For instance, if the constant a were unity, then the value of the sum- 
’ mation is 
T 1 il 
s=1( 2-7) 
which is equal closely to 7. On the other hand, if a were 0.1, the sum 
would be equal to 34r. 
11.7 Summary 
Theory which was developed for sampled-data systems has been 
adapted to other uses. Most important among these is the orderly 
approach to the setting up of a difference equation which approximates 
the differential equation of a system. If the system is linear, this differ- 
ence equation can be solved by means of a recursion formula. The 
interpretation of the process of numerical approximation by a physical 
sampled equivalent model assists considerably in setting up the numerical 
technique. For instance, the selection of quadrature interval is related 
to the choice of sampling interval in the physical model. Considerations 
of frequency response of the various elements and input functions affect 
the choice of sampling interval directly. The use of polygonal approxi- 
mations is equivalent to insertion of the triangle hold, which, though 
physically unrealizable, is useful for purposes of computation. 
The approach to the numerical solution of linear dynamical systems 
can take one of several forms. The first one replaces the continuous 
system with a sampled model containing samplers and data holds in 
carefully chosen places. This method is most closely related to the 
physical system itself. The next method approximates the Laplace 
transform of the desired variable, including all initial conditions. The 
use of z forms to approximate this transform makes possible its numerical 
inversion, but the relationship to the actual physical system is not so 
clear. Finally, for systems, particularly networks, where the initial 
conditions are associated with initial voltages across condensers or cur- 
rents in inductances, a technique which reduces the actual system to a set 
of interconnected integrators and summers is employed. Each of the 
integrators is then replaced by an approximate integration process with 
initial conditions introduced. This approach makes it relatively simple 
to apply initial conditions of the type mentioned. 
Linear systems with time-varying coefficients can also be solved by 
methods related to the theory of sampled-data systems. The setting up 
of a recursion formula in which the coefficients take on different values at 
each sampling instant is a relatively straightforward procedure. Finally, 
the evaluation of certain forms of infinite series, particularly those which 
