Digital Signal Processing Reference
In-Depth Information
and
Q
2
(
S
(
)]
H
ω
)
=
ω
)
+
[ ˆ
α
2
(
ω
)
a
(
ω
)
−
Z
(
ω
)][ ˆ
α
2
(
ω
)
a
(
ω
)
−
Z
(
ω
,
(5.48)
where
S
(
ω
) and
Z
(
ω
)are defined as
−
−
L
1
L
1
1
L
1
L
S
(
z
l
z
l
)
Z
H
(
ω
)
Γ
l
+
−
Z
(
ω
ω
)
(5.49)
l
=
0
l
=
0
and
L
−
1
1
L
z
l
e
−
j
ω
l
Z
(
ω
)
.
(5.50)
l
=
0
The derivation of the MAPES-EM2 algorithm is thus complete, and a step-by-step
summary of this algorithm is as follows:
Step 0:
Obtain an initial estimate of
{
α
(
ω
)
,
Q
(
ω
)
}
.
Step 1:
Use the most recent estimates of
in (5.42) and (5.43) to
calculate
b
and
K
.Note that
b
canberegarded as the current estimate of the
missing sample vector.
Step 2:
Update the estimates of
{
α
(
ω
)
,
Q
(
ω
)
}
{
α
ω
,
ω
}
using (5.47) and (5.48).
Step 3:
Repeat steps 1 and 2 until practical convergence.
(
)
Q
(
)
5.5
ASPECTS OF INTEREST
5.5.1 Some Insights into the MAPES-EM Algorithms
Comparing
Q
1
(
Q
2
(
{
α
ˆ
1
(
ω
)
,
ω
)
}
in (5.26) and (5.27) [or
{
α
ˆ
2
(
ω
)
,
ω
)
}
in (5.47) and
Q
(
(5.48)] with
in (4.13) and (4.14), we can see that the EM algorithms
are doing some intuitively obvious things. In particular, the estimator of
{
α
ˆ
(
ω
)
,
ω
)
}
α
(
ω
)
b
l
estimates the missing data and then uses the estimate
{
}
(or
b
)asthough it were
correct. The estimator of
Q
(
) does the same thing, but it also adds an extra
term involving the conditional covariance
K
l
(or
K
), which can be regarded as a
generalized diagonal loading operation to make the spectral estimate robust against
estimation errors.
ω
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