Two-Dimensional MAPES via Expectation Maximization and Cyclic Maximization - Spectral Analysis of Signals: The Missing Data Case - page 79

example. Fig. 6.1(c) shows the GAPES spectrum, which also underestimates the
sinusoidal components and gives some artifacts due to the poor initial estimate of
H (

ω ,ω

2 ). The MAPES-CM spectrum is plotted in Fig. 6.1(d). Figs. 6.1(e) and
6.1(f ) show the MAPES-EM1 and MAPES-EM2 spectra, respectively. All the
MAPES algorithms perform quite well and their missing-data spectral estimates
are similar to the high-resolution complete-data APES spectrum in Fig. 6.1(a).
The MAPES and GAPES spectral estimates at different iterations are plot-
ted in Figs. 6.2(a)-6.2(d). Among the MAPES algorithms, MAPES-CM is the
fastest to converge, after three iterations. MAPES-EM converges more slowly, with
MAPES-EM1 converging after 11 and MAPES-EM2 after 9 iterations. Because
of the relatively poor initial filter-bank H (

1

2 ) used in this example, the GAPES
algorithm performs relatively poorly and converges relatively slowly, after ten it-
erations. In the following examples where the gapped-data initialization step in
Chapter 3 can be applied, GAPES usually converges faster, within a few iterations.
Forillustration purposes, in Figure 6.2 we only plot the first four iterations of each
method and note that the convergence behavior of each algorithm does not change
significantly in the remaining iterations.

ω ,ω

1

6.6.2 Performance Study
In this example, we illustrate the performance of the MAPES algorithms for 2-D
spectral estimation. We consider a 16 16 data matrix consisting of three 2-D

sinusoids (signals 1, 2, and 3) at normalized frequencies (4/16, 5/16), (6/16, 5/16),
and (10/16, 9/16) and with complex amplitudes equal to 1, 0.7, and 2, respectively,
embedded in zero-mean circularly symmetric complex Gaussian white noise with
standard deviation 0.1. All the samples in rows 4, 8, 11, 14, and in columns 3, 6,
7, 11, 12, 14 are missing, which amounts to over 50% of the total number of data
samples. The true amplitude spectrum is plotted in Fig. 6.3(a) with the estimated
amplitude values given next to each sinusoid. Each spectrum is obtained on a
64

×

64 grid. The WFFT spectrum of the complete data is shown in Fig. 6.3(b)
along with the estimated magnitudes of the sinusoids at their true locations. The

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