Digital Signal Processing Reference
In-Depth Information
3.7.
Sum of allpass filters: generalization.
Consider a filter bank with
M
filters
H
k
(
z
)givenby
M−
1
1
M
W
km
A
m
(
z
)
,
H
k
(
z
)=
m
=0
where
A
m
(
z
) are allpass (
|
A
m
(
e
jω
)
|
=1forall
ω
). Show that
H
k
(
z
)are
power complementary, that is,
M−
1
H
k
(
e
jω
)
|
2
=1
|
(P3
.
7)
k
=0
for all
ω.
Also show that
H
k
(
z
) are allpass complementary, that is,
M−
1
H
k
(
z
) = allpass
.
k
=0
3.8.
Biorthogonality
. Consider the transmultiplexer system of Fig. 3.26 and
assume
P
=
M
(no redundancy). Let
R
(
z
)and
E
(
z
) represent the
M
M
polyphase matrix of the synthesis and analysis bank, respectively, as in
Secs. 3.7.2-3.7.3.
×
1. Assuming that the channel
C
(
z
) = 1 and that there is no noise, show
that the necessary and su
cient condition for perfect symbol recovery
(
s
k
(
n
)=
s
k
(
n
) for all
k
)isgivenby
E
(
z
)
R
(
z
)=
I
.
2. Now consider the analysis/synthesis system shown in Fig. P3.8. Here
a single signal
s
(
n
) is first analyzed using the filter bank
{
H
k
(
z
)
}
,
andthisisfollowedbythesynthesisbank
. There are
M
filters in each bank, and the decimation ratios are equal to
M
as
well. This is called a maximally decimated analysis/synthesis bank
and finds application in data compression. Show that the necessary
and su
cient condition for perfect reconstruction (
{
F
k
(
z
)
}
s
(
n
)=
s
(
n
)) is
given by
R
(
z
)
E
(
z
)=
I
.
For square matrices, since
R
(
z
)
E
(
z
)=
I
is
satisfied if and only if
E
(
z
)
R
(
z
)=
I
,
we see that the condition for
perfect reconstruction in the transmultiplexer system is identical to
the corresponding condition in the analysis/synthesis system.
3. For the analysis/synthesis system shown in Fig. P3.8, show that the
condition for perfect reconstruction (
s
(
n
)=
s
(
n
)) can also be written
as
[
F
m
(
z
)
H
k
(
z
)]
↓M
=
δ
(
k
−
m
)
.
(P3
.
8)
Filter banks satisfying the condition (P3.8) are called biorthogonal fil-
ter banks. The biorthogonality condition (P3.8) is therefore equivalent
to perfect reconstruction for both the analysis/synthesis system and the
transmultiplexer system when
P
=
M.
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