MULTIRATE SAMPLED SYSTEMS 220. 
The open-loop system shown in Fig. 9.2 may also be analyzed by the 
technique illustrated in Fig. 9.6. In this case a mathematical or phantom 
sampling of the output is done by a switch operating at the rate 7'/n. 
The output of this switch, C(z,), can be readily calculated from single- 
rate-system theory based on a period of 7'/n sec. The actual output 
transform, C(z), can be represented symbolically as the z transform of 
R Clz,) 
(Zn) ‘A | ¢ | A en a C(z) 
n Th 
Fic. 9.6. Open-loop analysis of multirate system by phantom switch. 
Genor AIC): This operation is defined as follows. The output has 
the value c(k7T/n) at sampling instants separated by T’/n sec. Then 
transforms are defined as 
GG) = » é (: =| act (9.16) 
and C(z) 
c(IT) 2 (9.17) 
1=0 
These transforms are related and, since (9.17) contains only a portion of 
the samples of (9.16), it should be possible to derive C(z) from C(én). 
This is in fact the case. By the inversion theorem, 
C (: Ty nj =| Ose Ge. (9.18) 
The substitution of (9.18) in (9.17) with k = In gives 
» = i C(en)2n'™—! dzn |= 
l= 
1 fos Son] 
l1=0 
if 1 dzn 
Fos i OC rae el oe (9.19) 
eo @. 
C(z) 
The contour T on the z, plane must be so chosen that it encompasses all 
the poles of C(z,)/z, but excludes the poles contributed by the factor 
(1 — 2,21). The reason for this is that in the interchange of summation 
and integration in (9.19) requires that the infinite sum » (ize be 
1=0 
absolutely convergent. This is assured only if |z,”z—'| is less than unity. 
Thus the factor (1 — 2,27!) cannot be zero in the region over which (9.19) 
