Digital Signal Processing Reference
In-Depth Information
•
In step 2, MAPES-CM uses the standard APES algorithm, which can
be efficiently implemented [34, 35] as discussed in Section 2.6. As for the
MAPES-EM spectral estimators
ˆ
α
ω
,ω
2
) and
ˆ
α
ω
,ω
2
), they have the
same structure as the APES estimator in (6.10), but with two differences.
The first one is using the conditional mean (
b
l
1
,
l
2
or
b
)ofthe missing
samples as their estimates. The second one is the additional terms (
Γ
l
1
,
l
2
and
Γ
l
1
,
l
2
) that involve the covariance matrices (
K
l
1
,
l
2
and
K
)ofthe missing data
in the
S
and
S
matrices. Because MAPES-EM uses different estimates for
the missing data at different frequencies, those techniques used to efficiently
implement APES cannot be applied here (see Section 2.6). As a result, no
efficient algorithms are available to calculate the corresponding spectral
estimate.
1
(
2
(
1
1
In summary, the MAPES-CM algorithm possesses a computational com-
plexity that is much lower than that of MAPES-EM.
6.6 NUMERICAL EXAMPLES
In this section, we present three numerical examples to illustrate the performance of
the MAPES algorithms for the 2-D missing-data spectral estimation problem. We
compare the MAPES algorithms with the WFFT and the GAPES [3]. A Taylor
window with order 5 and sidelobe level
35 dB is used to obtain the WFFT spectral
estimates. All the 2-D spectra are plotted with a dynamic range of 35 dB.
As in the 1-D case, we simply let the initial estimate of
−
2
)begiven
by the WFFT with the missing data samples set to zero. The initial estimate
of
Q
(
α
(
ω
,ω
1
2
) follows from the 2-D counterpart of (4.8), where again, the miss-
ing data samples are set to zero. For the initialization of GAPES, we consider
two cases. If the data missing pattern is arbitrary and no initial filter with a
proper size can be chosen, we use the same initial estimates of
ω
,ω
1
α
(
ω
,ω
2
) and
1
Q
(
2
) follows from
(3.29), where the missing samples are set to zero. If the data is gapped, as in examples
ω
,ω
2
)asfor MAPES, and the initial estimate of
H
(
ω
,ω
1
1
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