Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 289

x ( n ) x ( n ) y ( n )

-

v ( n ) y ( n )

-

z

M 1

z

M

M

(a)

y ( n )

x ( n ) y ( n )

L

v ( n )

-

x ( n )

z -

L

z

L 1

(b)

Solution: In both cases H(z) = z -1 , that is a time delay. In Fig. a , the right-hand
side can be expressed as

y ð n þ 1 Þ¼ x ð Mn Þ
y ð n Þ¼ x ð M ð n 1 ÞÞ¼ x ð Mn M Þ

while the left-hand side is

v ð n Þ¼ x ð n M Þ
y ð n Þ¼ v ð Mn ÞÞ¼ x ð Mn M Þ

that is, both sides are of Fig. a are equivalent. The up-sampler shown in Fig. b
can be verified similarly. In both cases the right-hand side implementation is more
computationally efficient since after the down-sampler or before the up-sampler
the filter H(z) will operate at lower sampling rate. On the other hand, the left-hand
side implementation is wasteful as it processes a lot of zero terms.