One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 52

spectral estimates. Both MAPES algorithms perform quite well and their spectral
estimates are similar to the high-resolution APES spectrum in Fig. 5.1(b).
The MAPES-EM1 and MAPES-EM2 spectral estimates at different iter-
ations are plotted in Figs. 5.2(a) and 5.2(b), respectively. Both algorithms converge
quickly with MAPES-EM1 converging after 10 iterations while MAPES-EM2
after only 6.
The data restoration performance of MAPES-EM is shown in Fig. 5.3.
The missing samples are estimated using the averaging approach we introduced
previously. Figs. 5.3(a) and 5.3(b) display the real and imaginary parts of the inter-
polated data, respectively, obtained via MAPES-EM1. Figs. 5.3(c) and 5.3(d) show
the corresponding results for MAPES-EM2. The locations of the missing samples
are also indicated in Fig. 5.3. The missing samples estimated via the MAPES-
EM algorithms are quite accurate. More detailed results for MAPES-EM2 are
shown in Fig. 5.4. (Those for MAPES-EM1 are similar.) For a clear visualiza-
tion, only the estimates of the first three missing samples are shown in Fig. 5.4.
The real and imaginary parts of the estimated samples as a function of frequency
are plotted in Figs. 5.4(a) and 5.4(b), respectively. All estimates are close to the
corresponding true values, which are also indicated in Fig. 5.4. It is interesting to
note that larger variations occur at frequencies where strong signal components are
present.
The results displayed so far were for one randomly picked realization of the
data. Using 100 Monte Carlo simulations (varying the realizations of the noise, the
initial phases of the different spectral components, and the missing-data patterns),
we obtain the root mean-squared errors (RMSEs) of the magnitude and phase
estimates of the four spectral lines at their true frequency locations. These RMSEs
for WFFT, GAPES, and MAPES-EM are listed in Tables 5.1 and 5.2. Based on
this limited set of Monte Carlo simulations, we can see that the two MAPES-EM
algorithms perform similarly, and that they are much more accurate than WFFT
and GAPES. A similar behavior has been observed in several other numerical
experiments.