Digital Signal Processing Reference
In-Depth Information
Inserting (4.8) into (4.7) yields the following concentrated cost function (with
changed sign)
)
e
j
ω
l
H
,
−
=
)
=
L
1
y
l
)
e
j
ω
l
y
l
1
L
Q
G
(
ω
−
α
(
ω
)
a
(
ω
−
α
(
ω
)
a
(
ω
(4.9)
α
l
=
0
ˆ
R
which is to be minimized with respect to
α
(
ω
)
.
By using the notation
g
(
ω
)
,
,
and
S
(
) defined in (2.6), (2.7), and (2.11), respectively, the cost function
G
in (4.9)
becomes
ω
=
)
ˆ
R
2
a
(
)
a
H
(
H
(
)
a
H
(
)
g
H
(
G
+|
α
(
ω
)
|
ω
ω
)
−
g
(
ω
)
α
ω
ω
)
−
α
(
ω
)
a
(
ω
ω
=
)]
H
ˆ
R
)
g
H
(
−
g
(
ω
ω
)
+
[
α
(
ω
)
a
(
ω
)
−
g
(
ω
)][
α
(
ω
)
a
(
ω
)
−
g
(
ω
(4.10)
=
)
I
)]
H
,
S
(
S
−
1
(
ω
+
ω
)[
α
(
ω
)
a
(
ω
)
−
g
(
ω
)][
α
(
ω
)
a
(
ω
)
−
g
(
ω
(4.11)
S
(
where
ω
)can be recognized as an estimate of
Q
(
ω
). Making use of the identity
I
AB
=
I
BA
,weget
+
+
)
1
)]
.
S
(
)]
H
S
−
1
(
G
=
ω
+
[
α
(
ω
)
a
(
ω
)
−
g
(
ω
ω
)[
α
(
ω
)
a
(
ω
)
−
g
(
ω
(4.12)
Minimizing
G
with respect to
α
(
ω
)yields
)
S
−
1
(
a
H
(
ω
ω
)
g
(
ω
)
ˆ
α
(
ω
)
=
)
.
(4.13)
)
S
−
1
(
a
H
(
ω
ω
)
a
(
ω
ω
Making use of the calculation in (4.10), we get the estimate of
Q
(
)as
Q
(
S
(
)]
H
ω
)
=
ω
)
+
[ ˆ
α
(
ω
)
a
(
ω
)
−
g
(
ω
)][ ˆ
α
(
ω
)
a
(
ω
)
−
g
(
ω
.
(4.14)
Q
(
In the APES algorithm, ˆ
α
(
ω
)isthe sought spectral estimate and
ω
)isthe
estimate of the nuisance matrix parameter
Q
(
ω
).
4.3 REMARKS ON THE ML FITTING CRITERION
The phrase ML fitting criterion used above can be commented as follows. In some
estimation problems, using the exact ML method is computationally prohibitive or
even impossible. In such problems one can make a number of simplifying assump-
tions and derive the corresponding ML criterion. The estimates that minimize the
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