ENEE630 ADSP
RECITATION 4 w/ solution
October 1/2, 2009
1.
Find the efficient structure for fractional decimation in Fig. R4.1, with
H
(
z
) =
10
X
n
=0
nz

n
.
Please first apply the Type 1 polyphase decomposition. Illustrate your implementation.
Figure R4.1:
Solution:
The intermediate block diagram is shown in the left figure, and the final result is
on the right.
The polyphase components are as follows,
E
0
(
z
) = 3
z

1
+ 6
z

2
+ 9
z

3
, E
1
(
z
) =
1 + 4
z

1
+ 7
z

2
+ 10
z

3
, E
2
(
z
) = 2 + 5
z

1
+ 8
z

2
.
E
00
(
z
) = 3 + 9
z

1
, E
01
(
z
) = 6
z

1
, E
10
(
z
) =
4 + 10
z

1
, E
11
(
z
) = 1 + 7
z

1
, E
20
(
z
) = 5
z

1
, E
21
(
z
) = 2 + 8
z

1
.
Note:
For this problem, the solution is not unique.
For instance, in recitation we used the trick
z

2
=
z
4
z

6
when dealing with the third branch. However, since
z

2
can directly interchange the
order with the upsampler, the following intermediate block diagram is also correct.
2.
Prove that the block diagram shown in Fig. R4.2(a) is a perfect reconstruction system. Based
on this, answer the following questions (No knowledge of QMF bank is needed though.
We will
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 Spring '10
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 Following, Christopher Nolan, Fig

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