66 SAMPLED-DATA CONTROL SYSTEMS 
becomes 
F(z,A) = > res. [Fisyerr oa (4.38) 
poles of 
F(s) 
For practical convenience, various transforms of this type are listed in 
Appendix II, where they can be readily associated with the corresponding 
Laplace transforms. 
EXAMPLE 
The same transform evaluated in the preceding example will be used 
to illustrate the application of (4.38). In this case, the Laplace 
transform F'(s) of the time function j(¢) is 
if! 
34a 
The transform has a pole at —a, and the residue at this pole results in 
F(s,A) = 
eATs 
e—aAT 
i = eel g— aL 
AGA) e— 
This result is the same as that obtained in the previous example. 
4.6 The Convolution Summation and Pulse Transfer Function 
One of the concepts of great value in the analysis of linear systems 
is that of the transfer function which relates the output and input 
of the system. For continuous systems whose performance is described 
G(z) 
eee Gis) Ss 
R(s) @ Cis) C(z) 
Fic. 4.6. Pulsed linear system showing definition of pulse transfer function. 

by a set of linear differential equations with constant coefficients, the 
transfer function is the Laplace transform of the impulsive response. 
In linear sampled-data systems, there exists an analogous transfer function 
known as the pulse transfer function,! or pulsed transfer function.* In 
subsequent discussions, the former nomenclature will be used. The 
pulse transfer function relates the output and input pulse sequences of a 
linear sampled system. 
Just as in the derivation of the continuous transfer function in which a 
convolution integral is employed, the derivation of the pulse transfer 
function uses a convolution summation. As an aid in deriving the 
required relationships, a system diagram shown in Fig. 4.6 is used. In 
