Digital Signal Processing Reference
In-Depth Information
and
Q
1
(
S
(
Z
(
Z
(
)]
H
ω
)
=
ω
)
+
[ ˆ
α
1
(
ω
)
a
(
ω
)
−
ω
)][ ˆ
α
1
(
ω
)
a
(
ω
)
−
ω
,
(5.27)
where
L
−
1
1
L
Z
(
z
l
e
−
j
ω
l
ω
)
(5.28)
l
=
0
and
L
−
1
L
−
1
1
L
1
L
S
(
Z
(
)
Z
H
(
Γ
l
z
l
z
l
ω
+
−
ω
ω
.
)
)
(5.29)
l
=
0
l
=
0
This completes the derivation of the MAPES-EM1 algorithm, a step-by-step
summary of which is as follows:
Step 0:
Obtain an initial estimate of
{
α
(
ω
)
,
Q
(
ω
)
}
.
Step 1:
Use the most recent estimate of
in (5.19) and (5.20) to
calculate
b
l
and
K
l
,respectively. Note that
b
l
canberegarded as the current
estimate of the corresponding missing samples.
Step 2:
Update the estimate of
{
α
(
ω
)
,
Q
(
ω
)
}
{
α
ω
,
ω
}
using (5.26) and (5.27).
Step 3:
Repeat steps 1 and 2 until practical convergence.
(
)
Q
(
)
0, which indicates that there is no available sample
in the current data snapshot
y
l
Note that when
g
l
=
S
g
(
l
) and
γ
l
do not exist and
S
m
(
l
)isan
M
M
identity matrix; hence, the above algorithm can still be applied by simply removing
any term that involves
,
×
S
g
(
l
)or
γ
l
in the above equations.
5.4 MAPES-EM2
Following the observation that the same missing data may enter in many snapshots,
we propose a second method to implement the EM algorithm by estimating the
missing data simultaneously for all data snapshots.
Recall that the available and missing data vectors are denoted as
γ
(
g
×
1
vector) and
µ
[(
N
−
g
)
×
1vector], respectively. Let
y
denote the
LM
×
1vector
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