Gapped-Data APES - Spectral Analysis of Signals: The Missing Data Case - page 26

3.4.1 One-Dimensional Example
In this example, we consider the 1-D gapped-data spectral estimation. To imple-
ment GAPES, we choose K 2 N for the iteration steps and the final spectrum

is estimated on a finer grid with K

=

= 32. The initial filter length is chosen as

64 after the initialization step. We calculate the
corresponding WFFT spectrum via zero-padded FFT.
The true spectrum of the simulated signal is shown in Fig. 3.1(a), where we
have four spectral lines located at f 1

M 0 = 20, and we use M = N / 2 =

= 0 . 05 Hz, f 2 = 0 . 065 Hz, f 3 = 0 . 26 Hz, and

5. Besides
these spectral lines, Fig. 3.1(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 . 1 2 3 4

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

=

sequence has N 128 samples where the samples 23-46 and 76-100 are missing.

The data is corrupted by a zero-mean circularly symmetric complex white Gaussian
noise with variance

2
n

01.
In Fig. 3.1(b) the WFFT is applied to the data by filling in the gaps with
zeros. Note that the artifacts due to the missing data are quite severe in the spec-
trum. Figs. 3.1(c) and 3.1(d) show the moduli of the WFFT and APES spectra
of the complete data sequence, where the APES spectrum demonstrated superior
resolution compared to that of WFFT. Figs. 3.1(e) and 3.1(f ) illustrate the moduli
of the WFFT and APES spectra of the data sequence interpolated via GAPES.
Comparing Figs. 3.1(e) and 3.1(f ) with 3.1(c) and 3.1(d), we note that GAPES
can effectively fill in the gaps and estimate the spectrum.

σ = 0 .

3.4.2 Two-Dimensional Examples
GAPES applied to simulated data with line spectrum: In this example we con-
sider a data matrix of size 32 50 consisting of three noisy sinusoids, with fre-

quencies (1,0.8), (1,1.1), and (1.1,1.3) and amplitudes 1, 0.7, and 2, respectively,
embedded in white Gaussian noise with standard deviation 0.1. All samples in the
columns 10-20 and 30-40 are missing. The true spectrum is shown in Fig. 3.2(a)
and the missing-data pattern is shown in Fig. 3.2(b). In Fig. 3.2(c) we show the

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