Digital Signal Processing Reference
In-Depth Information
and
,
TG
µ
(
ω
,ω
0
)
0
.
G
µ
(3.41)
TG
µ
(
ω
,ω
1
)
K
1
−
1
K
2
−
the criterion (3.38) can then be written as
α
G
µ
µ
G
γ
γ
2
min
µ
+
−
,
(3.42)
where
.
α
ˆ
(
ω
,ω
0
)
a
L
1
,
L
2
(
ω
,ω
0
)
0
0
.
α
(3.43)
α
ˆ
(
ω
,ω
1
)
a
L
1
,
L
2
(
ω
,ω
1
)
K
1
−
1
K
2
−
K
1
−
1
K
2
−
The closed-form solution of the quadratic problem (3.42) is easily obtained as
µ
=
G
H
G
µ
−
1
α
−
G
γ
γ
.
G
H
(3.44)
µ
µ
A step-by-step summary of 2-D GAPES is as follows:
Step 0:
Obtain an initial estimate of
{
α
(
ω
,ω
2
)
,
h
(
ω
,ω
2
)
}
.
1
1
Step 1:
Use the most recent estimate of
in (3.32) to estimate
µ
by minimizing the so-obtained cost function, whose solution is given by
(3.44).
Step 2:
Use the latest estimate of
µ
to fill in the missing data samples and estimate
{
α
{
α
(
ω
,ω
2
)
,
h
(
ω
,ω
2
)
}
1
1
K
1
−
1
,
K
2
−
1
0
by minimizing the cost function in (3.32)
based on the interpolated data. (This step is equivalent to applying 2-D
APES to the complete data.)
Step 3:
Repeat steps 1-2 until practical convergence.
(
ω
,ω
2
)
,
H
(
ω
,ω
2
)
}
1
1
k
1
=
0
,
k
2
=
3.4 NUMERICAL EXAMPLES
We now present several numerical examples to illustrate the performance of
GAPES for the spectral analysis of gapped data. We compare GAPES with
windowed FFT (WFFT). A Taylor window with order 5 and sidelobe level
−
35 dB
is used for WFFT.
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