Digital Signal Processing Reference
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of APES [32]. In the complete-data case, these two approaches are equivalent in
the sense that from either of them we can derive the same full-data APES spectral
estimator. So at first, it might look counterintuitive that these two algorithms
(MAPES and GAPES) will perform differently for the missing-data problem (see
the numerical results in Section 5.7). We will now give a brief discussion about this
issue.
The difference between MAPES and GAPES concerns the way they esti-
mate
µ
when some data samples are missing. Although MAPES-EM estimates
each missing sample separately for each frequency
k
(and for each data snapshot
y
l
in MAPES-EM2) while GAPES estimates each missing sample by considering
all
K
frequencies together, the real difference between them concerns the different
criteria used in (3.16) and (5.3) for the estimation of
µ
: GAPES estimates the
missing sample
µ
based on a LS fitting of the filtered data,
h
H
(
ω
ω
k
)
y
l
.
On the other
hand, MAPES estimates the missing sub-sample
µ
directly from
based on an
ML fitting criterion. Because the LS formulation of APES focuses on the output
of the filter
h
(
{
y
l
}
ω
k
) (which is supposed to suppress any other frequency components
ω
ω
except
k
) when it tries to
estimate the missing data. This is why GAPES performs well in the gapped-data
case, since there a good estimate of
h
(
k
), the GAPES algorithm is sensitive to the errors in
h
(
k
)can be calculated during the initializa-
tion step. However, when the missing samples occur in an arbitrary pattern, the
performance of GAPES degrades. Yet the MAPES-EM does not suffer from such
a degradation.
ω
5.7 NUMERICAL EXAMPLES
In this section we present detailed results of a few numerical examples to demon-
strate the performance of the MAPES-EM algorithms for missing-data spec-
tral estimation. We compare MAPES-EM with WFFT and GAPES. A Taylor
window with order 5 and sidelobe level
−
35 dB is used for WFFT. We choose
32
N
to have a fine grid of discrete frequencies. We calculate the correspond-
ing WFFT spectrum via zero-padded FFT. The so-obtained WFFT spectrum is
K
=
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