MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 287 
gory, one of the more useful data-interpolation formulas is the one leading 
to a polygonal approximation of the continuous function. This approxi- 
mation is shown in Fig. 11.2, where it is seen that the reconstructed func- 
tion 7,(¢) is obtained by connecting the sample values r(nT’) by means of 
straight lines. The approximation is better than the one produced by 

Time 
Fig. 11.2. Polygonal approximation of continuous time function. 
either the zero-order or first-order data holds under the same conditions. 
The difficulty that would be experienced in attempting to construct 
physical devices embodying this form of data hold is that one would 
require physically unrealizable elements, as will be shown. For purposes 
of computation, however, this is of no consequence. 







ZA 
Time 
Fia. 11.3. Generating triangles for polygonal approximation. 
i 
AS 

In order to utilize the polygonal approximation in sampled-data models 
of continuous systems, it is necessary to derive the transfer function for 
the process. Referring to Fig. 11.3, it is seen that the polygonal approxi- 
mation can be obtained by means of a sequence of generating triangles so 
constructed that their altitudes are equal to the value of the ordinate at 
the sampling instant and their bases are twice the sampling interval. <A 
particular generating triangle at the third sampling instant is shown 
shaded in Fig. 11.3. Using the impulse approximation, a system must 
be found whose impulsive response is a generating triangle of the form 
shown here. 
