MULTIRATE SAMPLED SYSTEMS 225 
operating on the input shifted T/3 sec to the left. This sequence is 
weighted by 0, 9(27/3), g[T + (27/3)], ..., which is the impulse 
response sequence shifted to the right by T/3 sec. Therefore, the first and 
the second contributions to the output are obtained by the circuit shown 
in Fig. 9.4b. Finally, the last contribution necessary to complete the 
output is given by the sequence r(27/3), r[T + (2T/3)], r[2T + (27T/3)], 
. , which is obtained from the inputs shifted to the left by 27/3 
through a transfer function e@7/**), This input sequence is weighted by 
the sequence 0, g(7'/3), g/[T + (T/3)], . . . , which is effected by passing 
the input sequence into G and delaying the response by 27/3 sec. The 
total equivalent circuit is shown in Fig. 9.4c, where the common transfer 
function G is combined in one box. This figure shows clearly that one 
sampling switch has been ‘‘decomposed”’ into three switches which oper- 
ate at one-third the rate of the equivalent switch. The method of switch 
decomposition is general and may be applied to the analysis of any multi- 
rate system. 
From a mathematical point of view, the switch decomposition is based 
on the rather obvious fact that a convergent sequence fy + fi + fe + f3 + 
- may be summed in parts such as (fo + fs +fe+ °° °) + (fi + 
jase jaar? & 9) =P Gia se iis aPaear °° 3h Os tin wanegall, 
» f(kE) = » Y Acin ee) Ge SO Oa) 
k=0 
m=0 7=0 
Applying (9.10) to the expression for the output samples, (9.9), 
c(i a Dy | Gin +m) 2 ar jer - 7 Gin + m)| (9.11) 
The z transform of the output is 
o 
co y UM 4 
5, Yr[ont mt] Solar M2) + corn 
m=0 j=0 1=0 
Making the substitution 7 — 7 = k in the last sum of (9.12), 
O@) = > > r (sr + “ Caml > g (er — “ pe (1185) 
m=0 j=0 k=0 
In terms of the Laplace transforms of r(¢) and g(t), it is possible to write 
