Digital Signal Processing Reference
In-Depth Information
and
Q
1
(
S
(
Z
(
ω
,ω
2
)
=
ω
,ω
2
)
+
[ ˆ
α
1
(
ω
,ω
2
)
a
(
ω
,ω
2
)
−
ω
,ω
2
)]
1
1
1
1
1
Z
(
2
)]
H
×
[ ˆ
α
1
(
ω
,ω
2
)
a
(
ω
,ω
2
)
−
ω
,ω
,
(6.27)
1
1
1
where
L
1
−
1
L
2
−
1
1
L
1
L
2
Z
(
z
l
1
,
l
2
e
−
j
(
ω
1
l
1
+
ω
2
l
2
)
ω
,ω
2
)
(6.28)
1
l
1
=
0
l
2
=
0
and
L
1
−
1
L
2
−
1
L
1
−
1
L
2
−
1
1
L
1
L
2
1
L
1
L
2
S
(
Z
(
ω
1
,ω
2
)
Z
H
(
Γ
l
1
,
l
2
+
z
l
1
,
l
2
z
l
1
,
l
2
−
ω
1
,ω
2
)
ω
1
,ω
2
)
.
l
1
=
0
l
2
=
0
l
1
=
0
l
2
=
0
(6.29)
This completes the derivation of the 2-D MAPES-EM1 algorithm, a step-by-step
summary of which is as follows:
Step 0:
Obtain an initial estimate of
{
α
(
ω
,ω
2
)
,
Q
(
ω
,ω
2
)
}
.
1
1
Step 1:
Use the most recent estimate of
in (6.20) and (6.21)
to calculate
b
l
1
,
l
2
and
K
l
1
,
l
2
,respectively. Note that
b
l
1
,
l
2
canberegarded as
the current estimate of the corresponding missing samples.
Step 2:
Update the estimate of
{
α
(
ω
,ω
2
)
,
Q
(
ω
,ω
2
)
}
1
1
using (6.26) and (6.27).
Step 3:
Repeat steps 1 and 2 until practical convergence.
{
α
(
ω
,ω
2
)
,
Q
(
ω
,ω
2
)
}
1
1
6.3.2 Two-Dimensional MAPES-EM2
MAPES-EM2 utilizes the EM algorithm by estimating the missing data simulta-
neously for all data snapshots. Let
y
=
vec[
Y
]
(6.30)
denote the vector of all the data samples. Recall that
γ
and
µ
denote the vectors
containing the available and missing elements of
y
,respectively, where
γ
has a size
of
g
×
1.
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