Digital Signal Processing Reference
In-Depth Information
The solution to (2.10) is readily obtained [33] as
S
−
1
(
ω
)
a
(
ω
)
ˆ
h
(
ω
)
=
)
.
(2.12)
)
S
−
1
(
a
H
(
ω
ω
)
a
(
ω
This is the forward-only APES filter, and the forward-only APES estimator in
(2.9) becomes
)
S
−
1
(
a
H
(
ω
ω
)
g
(
ω
)
α
ˆ
(
ω
)
=
)
.
(2.13)
)
S
−
1
(
a
H
(
ω
ω
)
a
(
ω
2.4 TWO-STEP FILTERING-BASED
INTERPRETATION
The APES spectral estimator has a two-step filtering interpretation: passing the
data
N
−
1
{
y
n
}
through a bank of FIR bandpass filters with varying center frequency
n
=
0
ω
, and then obtaining the spectrum estimate ˆ
α
(
ω
) for
ω
∈
[0
,
2
π
)from the filtered
data.
ω
, the corresponding
M
-tap FIR filter-bank is given by
(2.12). Hence the output obtained by passing
y
l
through the FIR filter
ˆ
h
(
For each frequency
ω
)canbe
written as
ˆ
h
H
(
)[
ˆ
h
H
(
)]
e
j
ω
l
ω
)
y
l
=
α
(
ω
ω
)
a
(
ω
+
w
l
(
ω
)
)
e
j
ω
l
=
α
(
ω
+
w
l
(
ω
)
,
(2.14)
ˆ
h
H
(
where
w
l
(
) denotes the residue term at the filter output and the
second equality follows from the identity
ω
)
=
ω
)
e
l
(
ω
ˆ
h
H
(
ω
)
a
(
ω
)
=
1
.
(2.15)
Thus, from the output of the FIR filter, we can obtain the LS estimate of
α
(
ω
)as
ˆ
h
H
(
α
ˆ
(
ω
)
=
ω
)
g
(
ω
)
.
(2.16)
2.5 FORWARD-BACKWARD AVERAGING
For ward-backward averaging has been widely used for enhanced performance in
many spectral analysis applications. In the previous section, we obtained the APES
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