MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 289 
for a sampling interval T of unity. The pulse transfer function of the 
combination is 
2. o@ =e 
ea oem 
which simplifies to 
_ 0.368 + 0.26427} 
Oe) = ae 
If the input is a unit ramp function as previously, the z transform of the 
output C(z) becomes 
BO) = 0.3682-!(1 + 0.722—1) 
1 — 2.372! + 1.7422 — 0137228 
Inverting C(z) by the method of long division, there results the sequence 
C(z) = 0.3682-! + 1.142-? + 2.052-* + 3.0024 + 4.0025 + -- - 
As expected, this sequence is exact because the polygonal approxima- 
tion reproduces the ramp function perfectly before it is applied to the 
G(z) 
T T 
1 
rt rt c 
a ares 
r,(t) cq it) 
Fic. 11.5. Integration process using polygonal approximation. 
continuous element. This demonstrates the effectiveness of relating 
the numerical method used to integrate the differential equation to 
the sampled-data model. 
In Sec. 4.6, it was shown how the pulse transfer function can be used to 
describe a numerical process and, by example, a number of well-known 
numerical integration formulas were so expressed. The physical interpre- 
tation of these numerical processes can further be clarified by considering 
them in terms of the preceding discussions. For instance, if the poly- 
gonal approximation is used, the process of integration can be repre- 
sented by the block diagram of Fig. 11.5, where B(s) represents a triangle 
data hold. The over-all pulse transfer function of the operation is given 
by 
eTs(1 od GONE 
GOl——Z Ts (11.6) 
which, by reference to the table in Appendix I, is found to be 
—1 
@@) = 4 ee (11.7) 

21—27} 
