Digital Signal Processing Reference
In-Depth Information
where the objective function is the summation over the 2-D frequency grid of all
the frequency-dependent complete-data likelihood functions in (6.8) (within an
additive constant). We solve the above optimization problem via a CM approach.
First, assuming that the previous estimate
i
−
1
θ
i
−
1
(
formed from
{
α
ˆ
ω
,ω
k
2
),
k
1
Q
i
−
1
(
is available, we maximize (6.48) with respect to
µ
. This step can
be reformulated as
ω
,ω
k
2
)
}
k
1
−
K
2
−
1
K
1
1
y
k
2
)
D
i
−
1
(
k
2
)
−
1
H
i
−
1
(
min
µ
−
α
ˆ
ω
,ω
k
2
)
ρ
(
ω
,ω
ω
,ω
k
1
k
1
k
1
k
1
=
0
k
2
=
0
×
y
k
2
)
,
i
−
1
(
ˆ
−
α
ω
,ω
k
2
)
ρ
(
ω
,ω
(6.49)
k
1
k
1
D
i
−
1
(
where
y
,
ρ
(
ω
,ω
k
2
), and
ω
,ω
2
) have been defined previously. Recalling
k
1
1
that
y
S
m
µ
,wecan easily solve the optimization problem in (6.49) as its
objective function is quadratic in
µ
:
=
S
g
γ
+
K
1
−
1
k
2
)
−
1
S
m
K
1
−
1
K
2
−
1
K
2
−
1
S
m
D
i
−
1
(
S
m
D
i
−
1
(
k
2
)
−
1
µ
=
ω
,ω
ω
,ω
k
1
k
1
k
1
=
0
k
2
=
0
k
1
=
0
k
2
=
0
i
−
1
(
×
[
ˆ
α
ω
,ω
k
2
)
ρ
(
ω
,ω
−
S
g
γ
]
.
k
2
)
(6.50)
k
1
k
1
A necessary condition for the inverse in (6.50) to exist is that
L
1
L
2
M
1
M
2
>
N
1
N
2
−
g
, which is always satisfied.
Once an estimate
µ
has become available, we reestimate
{
α
(
ω
,ω
k
2
)
}
and
k
1
{
by maximizing (6.48) with
µ
replaced by
µ
. This can be done by
maximizing each frequency term separately:
Q
(
ω
,ω
k
2
)
}
k
1
L
1
−
l
1
=
0
L
2
−
l
2
=
0
1
1
1
L
1
L
2
max
k
2
)
−
ln
|
Q
(
ω
,ω
k
2
)
|−
k
1
α
(
ω
k
1
,ω
k
2
)
,
Q
(
ω
,ω
k
1
y
l
1
,
l
2
−
α
k
2
)
e
j
(
ω
k
1
l
1
+
ω
k
2
l
2
)
H
(
ω
,ω
k
2
)
a
(
ω
,ω
k
1
k
1
k
2
)
y
l
1
,
l
2
−
α
k
2
)
e
j
(
ω
k
1
l
1
+
ω
k
2
l
2
)
,
(6.51)
Q
−
1
(
×
ω
,ω
(
ω
,ω
k
2
)
a
(
ω
,ω
k
1
k
1
k
1
which reduces to the 2-D APES problem.
Acyclic maximization of (6.48) can be implemented by the alternating max-
imization with respect to
µ
and, respectively,
α
(
ω
,ω
k
2
) and
Q
(
ω
,ω
k
2
). A step-
k
1
k
1
by-step summary of 2-D MAPES-CM is as follows:
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