Digital Signal Processing Reference
In-Depth Information
Step 0:
Obtain an initial estimate of
{
α
(
ω
,ω
k
2
)
,
Q
(
ω
,ω
k
2
)
}
.
k
1
k
1
{
α
ω
,ω
,
ω
,ω
}
Step 1:
Use the most recent estimates of
(
k
2
)
Q
(
k
2
)
in (6.50) to
k
1
k
1
estimate the missing samples.
Step 2:
Update the estimates of
using 2-D APES applied
to the data matrices with the missing sample estimates from step 1 [see (6.10)
and (6.11)].
Step 3:
Repeat steps 1 and 2 until practical convergence.
{
α
(
ω
,ω
2
)
,
Q
(
ω
,ω
k
2
)
}
1
k
1
6.5 MAPES-EM VERSUS MAPES-CM
Consider evaluating the spectrum for all three MAPES algorithms on the same
DFT grid. Since all three algorithms iterate step 1 and step 2 until practical conver-
gence, we can compare their computational complexity separately for each step.
•
In step 1, MAPES-CM estimates the missing samples via (6.50), which
canberewritten in a simplified form as
=
S
m
D
s
S
m
−
1
S
m
D
υ
−
S
m
D
s
S
g
γ
,
µ
(6.52)
where
K
1
−
1
K
2
−
1
[
D
i
−
1
(
k
2
)]
−
1
D
s
ω
,ω
(6.53)
k
1
k
1
=
0
k
2
=
0
and
K
1
−
1
K
2
−
1
[
D
i
−
1
(
k
2
)]
−
1
ˆ
−
i
1
(
D
υ
ω
,ω
α
ω
,ω
k
2
)
ρ
(
ω
,ω
k
2
)
.
(6.54)
k
1
k
1
k
1
k
1
=
0
k
2
=
0
D
i
−
1
(
When computing
D
s
and
D
υ
, the fact that
k
2
)isblock diagonal
canbeexploited to reduce the computational complexity. Comparing (6.52)
with (6.39) and (6.40) [or (6.20) and (6.21)], which have to be evaluated
for each frequency (
ω
,ω
k
1
l
2
) for (6.20) and
(6.21)], we note that the computational complexity of MAPES-CM is
much lower.
ω
,ω
k
2
) [and for each snapshot (
l
1
,
k
1
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