SAMPLED-DATA SYSTEMS 87 
Substituting for C1(z) its equivalent from (5.1), the over-all relationship 
becomes 
Co(z) = Gi(z)G@2(z) R(z) (5.3) 
It is readily deduced that the over-all pulse transfer function is simply 
the product of the pulse transfer functions of the individual cascaded 
elements: 
G(z) = Gil(z)G2(z) (5.4) 
For emphasis, it will be repeated that the over-all pulse transfer function 
of two elements which are separated by a synchronous sampler is the 
product of the pulse transfer functions of the individual elements. 


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Fic. 5.1. Cascaded sampled elements. 
In contrast, two elements may be in cascade with a continuous connec- 
tion between them, as shown in Fig. 5.2. This case is of significance only 
when the two elements are themselves continuous elements capable of 
producing a continuous output between sampling instants. Digital ele- 
ments are inherently sampled since they can deliver only a number 
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Fic. 5.2. Cascaded continuous elements. 
sequence at their output and do not fall into the category under discus- 
sion. For the systems shown in Fig. 5.2, the excitation of the second 
element is by a continuous time function whose Laplace transform is 
Ci(s). The relation between this output and the input is 
Ci(s) = Gils) R*(s) (5.5) 
where R*(s) is the Laplace transform of the impulse sequence at the out- 
put of the sampler. The Laplace transform C2(s) of the final output 
time function is given by 
C2(s) = G(s) Cx(s) (5.6) 
and the relation between input and output Laplace transforms is obtained 
by combining (5.6) and (5.5): 
C2(s) = Gi(s)G2(s) R*(s) (5.7) 
This result could have been anticipated by observing that G1(s)G@2(s) is 
