Digital Signal Processing Reference
In-Depth Information
b
l
from
µ
l
in (5.14) as follows:
subtracting the conditional mean
S
g
(
l
)
γ
l
)
e
j
ω
l
S
m
(
l
)
µ
l
+
−
α
(
ω
)
a
(
ω
=
S
m
(
l
)(
µ
l
b
l
)
+
S
g
(
l
)
γ
l
)
e
j
ω
l
.
S
m
(
l
)
b
l
−
+
−
α
(
ω
)
a
(
ω
(5.21)
The cross-terms that result from the expansion of the quadratic term in (5.14)
vanish when we take the conditional expectation. Therefore the expectation step
yields
E
1
)
Q
i
−
1
(
i
−
1
(
γ
l
,
µ
l
}|
α
γ
l
}
,
L
ln
p
(
{
(
ω
)
,
Q
(
ω
))
|{
α
ˆ
ω
)
,
ω
tr
Q
−
1
(
S
m
(
l
)
K
l
S
m
(
l
)
l
=
0
L
−
1
)
1
L
=−
M
ln
π
−
ln
|
Q
(
ω
)
|−
ω
)
e
j
ω
l
H
+
S
g
(
l
)
γ
l
+
)
e
j
ω
l
S
g
(
l
)
γ
l
+
S
m
(
l
)
b
l
S
m
(
l
)
b
l
−
α
(
ω
)
a
(
ω
−
α
(
ω
)
a
(
ω
.
(5.22)
Maximization:
The maximization part of the EM algorithm produces up-
dated estimates for
α
(
ω
) and
Q
(
ω
). The normalized expected surrogate log-
likelihood (5.22) can berewritten as
tr
Q
−
1
(
,
Γ
l
l
=
0
L
−
1
+
z
l
)
e
j
ω
l
z
l
)
e
j
ω
l
)
1
L
H
−
M
ln
π
−
ln
|
Q
(
ω
)
|−
ω
−
α
(
ω
)
a
(
ω
−
α
(
ω
)
a
(
ω
(5.23)
where we have defined
Γ
l
S
m
(
l
)
K
l
S
m
(
l
)
(5.24)
and
S
g
(
l
)
γ
l
+
S
m
(
l
)
b
l
z
l
.
(5.25)
According to the derivation in Chapter 4, maximizing (5.23) with respect to
α
(
ω
)
and
Q
(
ω
) gives
)
S
−
1
(
)
Z
(
a
H
(
ω
ω
ω
)
α
ˆ
1
(
ω
)
=
(5.26)
)
S
−
1
(
a
H
(
ω
ω
)
a
(
ω
)
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