Two-Dimensional MAPES via Expectation Maximization and Cyclic Maximization - Spectral Analysis of Signals: The Missing Data Case - page 77

discussed in Sections 6.6.2 and 6.6.3, we follow the initialization step considered in
Chapter 3.
We stop the iteration of all iterative algorithms whenever the relative change
in the total power of the 2-D spectra corresponding to the current ( ˆ

i (

α ω ,ω

k 2 )) k 1

i − 1 (

and previous ( ˆ α ω ,ω

k 2 )) estimates is smaller than a preselected threshold (e.g., k 1

10 − 2 ):
K 1 − 1
k 1

=

0 K 2 − 1

− K 1 − 1

0 K 2 − 1

i ( 2 i − 1 ( 2

| α ˆ ω ,ω

k 2 )

| | α ˆ ω ,ω

k 2 )

|

k 1 k 1

= k 2 = 0 k 1 = k 2 = 0

≤ .

K 1 − 1
k 1 0

K 2 − 1

| α ˆ

i − 1 (

ω ,ω k 2 ) |

2

k 1

= k 2 = 0

(6.55)

6.6.1 Convergence Speed
In our first example, we study the convergence properties of the MAPES algo-
rithms. We use a 1-D example for simplicity. (Note that a similar 1-D example was
considered in Chapter 5 but without the MAPES-CM algorithm, which has been
introduced in this chapter.)
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,

= . α = α = α = α = .

and 5.

Besides these spectral lines, Fig. 5.1(a) also shows a continuous spectral component
centered at 0.18 Hz with a width of 0.015 Hz and a constant modulus of 0.25. The
data sequence has N 1

f 4 0 28 Hz with complex amplitudes 1 and 0 1 2 3 4

1) samples out of which 51 (40%) samples are
missing; the locations of the missing samples are chosen arbitrarily. The data are
corrupted by a zero-mean circularly symmetric complex white Gaussian noise with
standard deviation 0.1.
In Fig. 6.1(a), 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. 6.1(b). 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 initial estimate for both the GAPES and MAPES algorithms in this

= 128 ( N 2 =