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ENEE 630 Fall’2009 Homework #4
Problem 1
Suppose the Mchannel maximally decimated QMF bank is aliasfree, and let T(z)
be the distortion function. Suppose we deﬁne a new ﬁlter bank in which the analysis
and synthesis ﬁlters are interchanged, that is
F
k
(
z
) are the analysis ﬁlters and
H
k
(
z
) are
the synthesis ﬁlters. Show that the resulting system is free from aliasing and has the
same distortion function T(z). So we can swap each
F
k
(
z
) with the corresponding
H
k
(
z
)
without changing the input/output properties.
(Hint: Use AC matrix formulation cleverly.)
Problem 2
Consider Fig. P2 with
T
=
W
*
(a uniform DFT analysis bank). Suppose
R
k
(
z
)
are chosen as in Eq (1.1) below, so that the product
R
k
(
z
)
E
k
(
z
) is independent of k. This
ensures that aliasing has been cancelled.
R
k
(
z
) =
Y
l
6
=
k
E
l
(
z
)
(1
.
1)
Figure : P2
a)
First as a review, verify that the uniformshift relations
H
k
(
z
) =
H
0
(
zW
k
) and
F
k
(
z
) =
W

k
F
0
(
zW
k
) hold, where
W
=
e

j
2
π
M
.
b)
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 Spring '10
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