Digital Signal Processing Reference
In-Depth Information
and
ˆ
R
f
ˆ
R
b
+
ˆ
R
fb
=
.
(2.24)
2
Note that here we use
ˆ
R
f
instead of
ˆ
R
to emphasize on the fact that it is estimated
from the forward-only approach. The solution of (2.20) is given by
S
−
1
fb
(
ω
)
a
(
ω
)
ˆ
h
fb
(
ω
)
=
)
.
(2.25)
)
S
−
1
a
H
(
ω
ω
ω
fb
(
)
a
(
Because of the following readily verified relationship
J y
L
−
l
−
1
,
y
l
=
(2.26)
we have
J g
∗
(
)
e
−
j
ω
(
L
−
1)
g
(
ω
)
=
ω
,
(2.27)
ˆ
R
b
J
ˆ
R
f
=
J
,
(2.28)
and
)
g
H
(
)
g
H
(
)]
T
J
g
(
ω
ω
)
=
J
[
g
(
ω
ω
,
(2.29)
where
J
denotes the exchange matrix whose antidiagonal elements are ones and the
remaining elements are zeros. So
S
fb
(
ω
)can be conveniently calculated as
S
f
(
J S
f
(
ω
)
+
ω
)
J
S
fb
(
ω
)
=
,
(2.30)
2
where
S
f
(
ˆ
R
f
)
g
H
(
ω
)
−
g
(
ω
ω
)
.
(2.31)
ˆ
h
fb
(
Given the forward-backward APES filter
ω
), the forward-backward
spectral estimator can be written as
)
S
−
1
a
H
(
ω
fb
(
ω
)
g
(
ω
)
ˆ
α
fb
(
ω
)
=
)
.
(2.32)
)
S
−
1
a
H
(
ω
fb
(
ω
)
a
(
ω
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