Multirate Digital Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 212

(a)

1 2 3 4 5 6 7 8 9 10111213141516171819
n

(b)

3 6 9 12151821242730333639424548515457
n

Fig. 5.4 Up-sampling process (L = 3)

5.2.2 Frequency-Domain Representation

As mentioned earlier, although down-sampling and up-sampling are linear oper-
ations, they are time variant. They therefore do not conform to standard LTI
transfer function relationships. It is nonetheless informative to examine the input
and output spectra of the down-sampling and up-sampling processes. Consider the
z-transform of the downsampler defined in ( 5.1 ), yields

Y ð z Þ¼ X

N 1

x ð Mn Þ z n : ð 5 : 3 Þ

n ¼ 0

One can not get a useful input-output relationship from ( 5.3 ) because of the way
the input signal is expressed. To solve this problem, one may introduce the
sampling sequence c M (n)as
c M ð n Þ¼ X

1

d ð n iM Þ¼ 1 ;

for n ¼ 0 ; M ; 2M ; ...

ð 5 : 4 Þ

0 ; otherwise

i ¼1

which can equivalently be expressed as (See Miscellaneous DSP Problems-D, Q1)

X

M 1

c M ð n Þ¼ 1
M

e j 2 M kn : ð 5 : 5 Þ

k ¼ 0