MULTIRATE SAMPLED SYSTEMS Depa 
error-sampled configuration, which displays the essential characteristics 
of the problem in the simplest possible way. 
9.1 Analysis of Open-loop Multirate Systems 
As in the ease of single-rate sampled systems, the important concept in 
dealing with linear multirate systems is the pulse transfer function. Asa 
first example, consider the system shown in Fig. 9.1, where an input F is 
sampled with the uniform period T 
and applied to the continuous sys- ees ee) Gis) 
temG. The output of G is sampled z 7 
at an increased rate with period T7/n Fic. 9.1. Multirate system with slow 
toformasequenceofoutputsamples ™Ptt arog! Haste Olu SEIN: 
whose transform (which will be defined shortly) is designated C(z,). The 
analysis problem requires that the input transform F(z) be related to this 
output transform. The necessary relation can be established directly from 
the convolution summation developed earlier for single-rate systems. By 
adding up the impulse responses of the linear system G, the continuous 
output c(t) is calculated to be 
Clzn) 
AO) = » r(kT)g(t — kT) (9.1) 
k=0 
The samples which appear at the output of the “‘fast’’ switch are the 
values of (9.1) at the times ¢ = /T’/n, or 
(UBD) = > r(kT)g C a KT) (9.2) 
k=0 
If the transform of this output is to be of use in later calculations, it 
must obviously include all the samples in (9.2); that is, the output trans- 
form must be defined on samples separated by 7'/n sec rather than the 
T sec which separate samples of the input r*(é). To distinguish the 
transform variables according to the separation between successive 
samples which they represent, the variable z, will be used in the pulse 
transform of samples separated by 7'/n sec and the variable z retained for 
sequences separated by 7 sec. In all cases, the definitions will be made 
clear by the example problem. For the case illustrated in Fig. 9.1 and 
described by Eq. (9.2), the z, transform of the output is defined as 
oo 
G@,) = y CU cna 
= > r(kT)g (Z — er) a (9 3) 
1=0 k=0 
