Digital Signal Processing Reference
In-Depth Information
and
Q
2
(
ω
,ω
2
)
=
S
(
ω
,ω
2
)
+
[ ˆ
α
2
(
ω
,ω
2
)
a
(
ω
,ω
2
)
−
Z
(
ω
,ω
2
)]
1
1
1
1
1
2
)]
H
×
[ ˆ
α
2
(
ω
,ω
2
)
a
(
ω
,ω
2
)
−
Z
(
ω
,ω
,
(6.45)
1
1
1
where
S
(
ω
,ω
2
) and
Z
(
ω
,ω
2
)are defined as
1
1
L
1
−
1
L
2
−
1
L
1
−
1
L
2
−
1
1
L
1
L
2
1
L
1
L
2
z
l
1
,
l
2
z
l
1
,
l
2
S
(
ω
,ω
2
)
Γ
l
1
,
l
2
+
1
l
1
=
0
l
2
=
0
l
1
=
0
l
2
=
0
2
)
Z
H
(
−
Z
(
ω
,ω
ω
,ω
2
)
.
(6.46)
1
1
and
L
1
−
1
L
2
−
1
1
L
1
L
2
z
l
1
,
l
2
e
−
j
(
ω
1
l
1
+
ω
2
l
2
)
Z
(
ω
,ω
2
)
.
(6.47)
1
l
1
=
0
l
2
=
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
,ω
2
)
,
Q
(
ω
,ω
2
)
}
.
1
1
Step 1:
Use the most recent estimates of
in (6.39) and
(6.40) 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
,ω
2
)
,
Q
(
ω
,ω
2
)
}
1
1
{
α
(
ω
,ω
2
)
,
Q
(
ω
,ω
2
)
}
using (6.44) and (6.45).
1
1
Step 3:
Repeat steps 1 and 2 until practical convergence.
6.4 TWO-DIMENSIONAL MAPES VIA CM
Next we consider evaluating the spectrum on the
K
1
×
K
2
-point DFT grid. Instead
of dealing with each individual frequency (
ω
,ω
k
2
) separately, we consider the
k
1
following maximization problem:
K
1
−
1
K
2
−
1
L
1
−
1
L
2
−
1
1
L
1
L
2
max
−
ln
|
Q
(
ω
,ω
k
2
)
|−
k
1
µ
,
{
α
ω
k
1
,ω
k
2
)
,
ω
k
1
,ω
k
2
)
}
(
Q
(
k
1
=
0
k
2
=
0
l
1
=
0
l
2
=
0
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
)
e
j
(
ω
k
1
l
1
+
ω
k
2
l
2
)
k
2
)
y
l
1
,
l
2
−
α
Q
−
1
(
×
ω
,ω
(
ω
,ω
k
2
)
a
(
ω
,ω
,
k
1
k
1
k
1
(6.48)
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