MULTIRATE SAMPLED SYSTEMS 223 
output is given by the usual expression of superposition 
AG) 5 ; ( =) ; ( a =) (9.8) 
k=0 
The output of the second sampler is a periodic impulse train modulated 
by c(é) with a period of T sec. The corresponding sequence of output 
values is given by 
CUia— ) r (: =) g (ur —k =| (9.9) 
An attempt to take the z transform of (9.9) by multiplying by z~ and 
summing on / leads to difficulty because it is not possible to make a direct 
change of variable to separate out 
that part of the transform which Rs} Rie.) Ce) 
depends on the input from that part or — Gis) ani 
of the transform which depends on n 
Fig. 9.2. Multirate system with fast 
the system. The source of the difi- . 
input and slow output sampling. 
culty may be readily seen from a 
graph (shown in Fig. 9.3) of the component signals from g(t) which make 
up the output c(t). In this graph n = 3 and all input samples are taken 
to be the same value to avoid confusion as far as possible. It is noticed 
A: Output at sampling instants 

Time 
Fic. 9.3. Signal components of output of system of Fig. 9.2 with triple-rate input. 
from Fig. 9.3 that there are in fact three distinct sequences of weights given 
by the system weighting function g(t) to the input samples. The first 
weighting sequence is marked with crosses and applies to input samples 
which occur at 0, 7, 27, .... This sequence is g(0), g(7), g(27), 
The second weighting sequence, distinct from the first, is marked 
