Digital Signal Processing Reference
In-Depth Information
filter by using only forward data vectors. Here we show that forward-backward
averaging can be readily incorporated into the APES filter design by considering
both the forward and the backward data vectors.
Let the backward data subvectors (snapshots) be constructed as
[
y
N
−
l
−
1
y
N
−
l
−
2
···
y
N
−
l
−
M
]
T
y
l
=
,
l
=
0
,...,
L
−
1
.
(2.17)
We require that the outputs obtained by running the data through the filter both
forward and backward are as close as possible to a sinusoid with frequency
ω
. This
design objective can be written as
h
H
(
L
−
1
)
e
j
ω
l
h
H
(
)
e
j
ω
l
1
2
L
2
2
min
ω
)
y
l
−
α
(
ω
+
ω
)
y
l
−
β
(
ω
h
(
ω
)
,α
(
ω
)
,β
(
ω
)
l
=
0
h
H
(
s.t.
ω
)
a
(
ω
)
=
1
.
(2.18)
h
H
(
The minimization of (2.18) with respect to
α
(
ω
) and
β
(
ω
) gives ˆ
α
(
ω
)
=
ω
)
g
(
ω
)
ˆ
h
H
(
and
β
(
ω
)
=
ω
)
g
(
ω
), where
g
(
ω
)isthe normalized Fourier transform of
y
l
:
L
−
1
1
L
y
l
e
−
j
ω
l
g
(
ω
)
=
.
(2.19)
l
=
0
It follows that (2.18) leads to
)
S
fb
(
h
H
(
h
H
(
min
h
(
ω
ω
)
h
(
ω
)
s.t.
ω
)
a
(
ω
)
=
1
,
(2.20)
ω
)
where
)
g
H
(
)
g
H
(
g
(
ω
ω
)
+
g
(
ω
ω
)
S
fb
(
ˆ
R
fb
ω
)
−
(2.21)
2
with
L
−
1
1
L
ˆ
R
f
y
l
y
l
=
,
(2.22)
l
=
0
−
L
1
1
L
ˆ
R
b
y
l
y
l
=
,
(2.23)
l
=
0
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