BEHAVIOR Ox SYSTEMS BETWEEN SAMPLING INSTANTS 203 
of that of the input sequence.*® As originally proposed, this method 
places a sampler whose period is one-half that of the input sampler at the 
output of the sampled system, as shown in Fig. 8.3a, in which it is seen 
that the input sampling interval is T while the output sampling interval 
is 77/2. Subsequent extensions of this theory”? have considered the case 
where the output sampler is operated at a period 7'/n, where n is any 
integer. This more general theory will be treated in the next chapter, 
and this discussion will consider only the case where n is taken as 2. 




——o 
Cle 

R(s) we C(s) 
(a) 
ere 
* i T/2 Cleo) 
Renae 
R(s) Rizo) Riz5) C(s) 
(0) 
Fia. 8.3. (a) Sampled-data system with double-rate output sampling. (b) Equivalent 
system. 
By operating the fictitious output sampler at twice the frequency of 
the input sampler, a value of the continuous output at the mid-point of 
the basic sampling interval is obtained. While this does not define the 
entire ripple, it does give an additional piece of information which enables 
the designer to estimate the true value of the ripple over the entire 
interval. It is seen that the input sampler closes at every other sampling 
instant of the output sampler. This suggests the use of a modified 
auxiliary variable z2, which is defined by 
Zo ene (8.8) 
It is seen that z2 is related to the variable z, as previously used, by 
45> ed (8.9) 
To obtain relationships between the double-rate sampled output and 
the input pulse sequence, reference is made to Fig. 8.3b. In this figure, 
the sampler operating at the basic interval T is preceded by another 
sampler operating at the interval 7/2. Since the second switch closes 
only at every other closure of the fictitious double-rate sampler, its intro- 
duction has not altered the input sequence to the continuous system in 
any way whatsoever. The z transform of the pulse sequence of the input 
applied to the continuous system is R(z). On the other hand, in view 
of (8.9), it is readily seen that this input sequence is also given by 
R(z) = Res") (8.10) 
