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CHAPTER 9 
MULTIRATE SAMPLED SYSTEMS 
The sampled-data feedback systems which have been discussed so far 
have sampling switches, all of which operate with the same period or 
sampling rate. ‘These systems are single-rate systems. It is the purpose 
of this chapter to extend the treatment of sampled-data control to a 
study of multirate systems. A multirate sampled-data system is defined 
as a system which has at least two samplers which operate at different 
rates. An open-loop example of multirate operation was given in the 
last chapter, where a double-rate switch was used to read the intersample 
ripple. In Chap. 6 a method was described for the design of continuous 
shaping networks for single-rate systems by an approximation which 
resulted in the introduction of double- or higher-rate sampling. The 
practical value of multirate sampling, however, goes beyond the mere 
convenience of such systems used as an analytical tool. It will be shown, 
for example, how multirate digital controllers can be designed to improve 
the response of systems which are receiving data sampled at a lower rate. 
Another practical possibility exists in that large-scale control systems 
may have a number of links which are, for reasons beyond those pertain- 
ing to the control function, sampled at different rates. In particular, 
systems which include remote data-transmission links are most likely to 
involve several different data rates in the same control loop. For all 
these reasons, it is important to understand the analysis methods and 
the synthesis possibilities inherent in multirate feedback systems. 
The introduction of different sampling rates in the control loop requires 
a redevelopment of the basic system analysis techniques which were 
described in the early chapters for single-rate systems. At the present 
time the theory of multirate systems, which is largely due to Kranc,”*??:30 
is able to handle, in a practical pencil-and-paper manner, only those sys- 
tems containing sampling rates whose ratios are the ratios of small 
numbers. While more complicated cases, including sampling at widely 
different rates, can, in theory, always be analyzed, the complicated nature 
of the resulting equations forces one to think in terms of simulation or 
automatic computation as aids in their analysis and design. As in the 
previous chapters, the emphasis in this chapter will be on the single-loop 
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