MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 285 
replacement of the linear differential equation implied by G,(s) by the 
difference equation implied by G(z) with a quadrature interval of T. 
The accuracy of the output sequence obtained from the sampled model 
depends on the sophistication of the data-hold element in conjunction 
with the sampling interval T. The selection of this element is analogous 
to the selection of an interpolation technique in numerical methods for 
the integration of differential equations. In relating the choice of this 
element as well as the sampling interval, the use of a sampled model is 
generally clearer to the engineer and designer. For instance, if the input 
as expressed by r(t) contains significant spectral components up to some 
frequency F’, the sampling frequency should be high enough to pass them 
all, which means a sampling frequency more than 27. The combined 
data hold and G,(s) also enter into the choice of sampling frequency. If 
their combined frequency response effectively filters out high-frequency 
components of r(¢), the sampling frequency can be chosen with this in 
mind. Relating the numerical method to the physical problem in this 
manner has advantages. The application of the method can be shown 
by means of an example. 
EXAMPLE 
The response of a simple system whose transfer function G,(s) is 
1 
G,(s) = S25 Tl 
to a unit step function is to befound. The data hold which is selected 
is a zero-order hold whose transfer function is 
The sampling interval T is chosen to be 1| sec, resulting in a sampling 
frequency of 1 cps. This frequency is approximately six times the 
half-energy frequency of the system. 
If the input function is a unit step, it is evident from physical 
reasoning that the reconstructed function 7;(¢) is perfect, so that it is 
anticipated that the output obtained from the sampled model is perfect. 
The z transform of the input A(z) is 
1 
RG) ae 
The z transform of the process, G(z), is 
La Ge 
s(s + 1) 
0.632271 
1 — 0.368271 
G(z) =Z 
