MULTIRATE SAMPLED SYSTEMS 229 
tion. The latter form could have been derived by evaluating the 
residues at the three poles contributed by (1 — 23827). 
In addition to the analysis of systems as illustrated by the example 
above, the relation (9.19) can be used to show general properties of a 
z transform in terms of the z, transform. For example, if C(z,) has a pole 
at z, = a, then C(z) has a pole atz = a”. The function C(z,”), formed 
by replacing z by z,” in C(z), has n poles of magnitude a located at equal 
angles around a circle of radius a, as illustrated in Fig. 9.7. This last 
result can be obtained from the fact that if C(z) has a pole at z = a”, then 
C(z,”) has poles where 
2,” = a” (9.22) 
Since e??"" = 1, (9.22) is equivalent to 
Cae Nae UO, Me Paral 8G 
and therefore en ae 11) eo a (9.23) 
which is the desired result. In addition to the added insight given by 
Fig. 9.7 and the relation (9.23) into the characteristics of multirate trans- 
forms, these results will be very useful in design problems to be described 
later. 
9.2 Analysis of Closed-loop Multirate Systems 
The analysis of closed-loop multirate systems is most readily carried 
out by the use of the switch-decomposition method described in the last 
section to reduce the problem to asin- 
gle-rate system. Such a reduction is 
possible only if all sampling rates in 
the system are integral multiples of 
a single basic rate. For example, 
the rates 2/T, 3/T, and 4/T are all 
multiples of the basic rate1/T. The 
complexity of the analysis problem 
is directly related to the number of 
parallel switches which result from 
the decomposition process. For the 
three rates listed above, it would be | 

D Pole of Cizg) 
necessary to perform three decom- Fic. 9.7. Sketch of corresponding poles 
positions, leading to two, three, and of C’(Zn) and Cn"). 
four parallel switches, respectively. If the rate 3/7 were missing, then 
decomposition could be accomplished with respect to a basic rate of 2/7’ 
and the problem would be greatly simplified. The method of analysis is 
straightforward and best illustrated by example. The techniques used in 
