One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 50

used as the initial spectral estimate for the MAPES-EM and GAPES algorithms.
The initial estimate of Q (

ω

) for MAPES-EM has been discussed before, and the

ω

initial estimate of h ( ) for GAPES is calculated from (2.12), where the missing

samples are set to zero. We stop the MAPES-EM and the GAPES algorithms
using the same stopping criterion in (5.51) with being selected as 10 − 3 and 10 − 2 ,

respectively. The reason we choose a larger for GAPES is that it converges rela-

tively slowly for the general missing-data problem and its spectral estimate would
not improve much if we used an

10 − 2 .All the adaptive filtering algorithms
considered (i.e. APES, GAPES, and MAPES-EM) use a filter length of M

<

= N / 2

for achieving high resolution.
The true spectrum of the simulated signal is shown in Fig. 5.1(a), where we
have four spectral lines located at f 1 = 0 . 05 Hz, f 2 = 0 . 065 Hz, f 3 = 0 . 26 Hz,

and .

Besides these spectral lines, Fig. 5.1(a) also shows a continuous spectral component
centered at 0.18 Hz with a width b

f 4 = 0 . 28 Hz with complex amplitudes α = α = α = 1 and α = 0 . 5 1 2 3 4

= 0 . 015 Hz and a constant modulus of 0.25.

The data sequence has N 128 samples among which 51 (40%) samples are miss-

ing; the locations of the missing samples are chosen arbitrarily. The data is corrupted
by a zero-mean circularly symmetric complex white Gaussian noise with variance
σ

=

2
n

.
In Fig. 5.1(b), the APES algorithm is applied to the complete data and the
resulting spectrum is shown. The APES spectrum will be used later as a reference
for comparison purposes. The WFFT spectrum for the incomplete data is shown
in Fig. 5.1(c), where the artifacts due to the missing data are readily observed.
As expected, the WFFT spectrum has poor resolution and high sidelobes and it
underestimates the true spectrum. Note that the WFFT spectrum will be used as
the initial estimate for the GAPES and MAPES algorithms. Fig. 5.1(d) shows the
GAPES spectrum. GAPES also underestimates the sinusoidal components and
gives some artifacts. Apparently, owing to the poor initial estimate of h (

= 0 . 01

k ) for the
incomplete data, GAPES converges to one of the local minima of the cost function
in (3.16). Figs. 5.1(e) and 5.1(f ) show the MAPES-EM1 and MAPES-EM2

ω