# Asteroseismology of PG 1541$+$651 and BPM 31594 with TESS.

Alejandra D. Romero1, Gabriela Oliveira da Rosa1, S. O. Kepler1, Paul A.
Bradley2, Murat Uzundag3,4, Keaton J. Bell5,6, J. J. Hermes7, and G. R.
Lauffer1
1Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento
Gonçalves 9500, Brazil
2XCP-6, MS F-699 Los Alamos National Laboratory, Los Alamos, NM 87545
3Instituto de Física y Astronomía, Universidad de Valparaíso, Gran Bretaña
1111, Playa Ancha, Valparaíso 2360102, Chile
4European Southern Observatory, Alonso de Cordova 3107, Santiago, Chile
5DIRAC Institute, Department of Astronomy, University of Washington, Seattle,
WA-98195, USA
6NSF Astronomy and Astrophysics Postdoctoral Fellow
7Department of Astronomy & Institute for Astrophysical Research, Boston
University, 725 Commonwealth Ave., Boston, MA 02215, USA

E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the photometric data from TESS for two known ZZ Ceti stars, PG
1541+651 and BPM 31594. Before TESS, both objects only had observations from
short runs from ground-based facilities, with three and one period detected,
respectively. The TESS data allowed the detection of multiple periodicities,
12 for PG 1541$+$651, and six for BPM 31594, which enables us to perform a
detailed asteroseismological study. For both objects we found a representative
asteroseismic model with canonical stellar mass $\sim 0.61M_{\odot}$ and thick
hydrogen envelopes, thicker than $10^{-5.3}M_{*}$. The detection of triplets
in the Fourier transform also allowed us to estimate mean rotation periods,
being $\sim$ 22 h for PG 1541+651 and 11.6 h for BPM 31594, which is
consistent with range of values reported for other ZZ Ceti stars.

###### keywords:

surveys–(stars:) white dwarfs–stars: variables: general

††pubyear: 2018††pagerange: Asteroseismology of PG 1541$+$651 and BPM 31594
with TESS.–LABEL:lastpage

##  1 Introduction

White dwarf stars are the endpoint of the evolution for all stars with initial
masses up to 8.5-12 $M_{\odot}$ (Siess, 2007; Doherty et al., 2015; Lauffer et
al., 2018) when we consider single stellar evolution. This implies that more
than 97% of the stars in the Milky Way, including the Sun, will end their
lives as white dwarf stars. Therefore, the white dwarf population conveys an
important record of the evolution of all stellar populations in our Galaxy.
The most numerous class of white dwarf stars are the hydrogen atmosphere DAs,
comprising $\sim$ 87% of all white dwarfs known to date (Kepler et al., 2019;
Kepler et al., 2021a). In addition, variable DA white dwarf stars are also the
most numerous class of pulsating white dwarfs. They are known as ZZ Ceti stars
or DAVs, comprising $\sim$ 80% of all known pulsating white dwarfs (e.g.
Bognár et al., 2021). ZZ Ceti stars show periodic brightness variations due to
gravity driven g-mode pulsations, with periods from 70 to 3000 s and variation
amplitudes of 1–60 ppt (e.g. Fontaine & Brassard, 2008; Winget & Kepler, 2008;
Kepler & Romero, 2017; Córsico et al., 2019). The excitation mechanism is
related to the increase in opacity at the base of the hydrogen envelope, due
to partial ionization of hydrogen (Dolez & Vauclair, 1981; Winget et al.,
1982) and later the ability of the convection zone to react quickly to changes
in thermal structure (Brickhill, 1991; Goldreich & Wu, 1999). This occurs in a
narrow range of effective temperature, between $13\,500$ K and $10\,500$ K,
depending on the stellar mass. The instability strip for ZZ Cetis is pure,
meaning that all DA white dwarfs should show photometric variability within
these effective temperature range (Castanheira et al., 2007; Gianninas et al.,
2011).

As we move through the ZZ Ceti instability strip, there is a change in the
behaviour of the pulsation spectrum (Clemens, 1993; Mukadam et al., 2006). The
hot ZZ Cetis, near the blue edge of the instability strip, show stable
sinusoidal or sawtooth light curves, with a few short periods (<350 s) and
small variation amplitudes (1.5–20 mma). On the red edge, the cool ZZ Cetis
show non-sinusoidal light-curves and a collection of long periods (> 650 s),
with large amplitude variations (40–110 mma) that suffer from severe mode
interference, leading to the presence of linear combination frequencies and/or
harmonics in the Fourier transform. Around $11\,500$ K, the warm-like ZZ Cetis
show mixed characteristics from hot and cool members for short and long
periods, respectively.

Asteroseismology applied to white dwarf stars allows us to study the inner
structure and the evolutionary properties of these objects through the
observed period spectrum, in the same way as quakes on Earth allow for the
study of the inner core. The technique consists of a comparison between the
observed periods and the theoretical periods computed from representative
models, as inversion is not doable at present. Each pulsation mode propagates
in a specific region, providing information on that particular place inside
the star, where its amplitude has a maximum weight. In particular, the
thickness of the hydrogen and helium layers (Castanheira & Kepler, 2008, 2009;
Romero et al., 2012; Romero et al., 2013), the chemical composition of the
inner core and an estimate of the 12C$(\alpha,\gamma)$O16 reaction rate
(Metcalfe et al., 2002; De Gerónimo et al., 2017; Pepper et al., 2022;
Chidester et al., 2022), crystallization (Montgomery & Winget, 1999; Romero et
al., 2013), rotation velocity (Hermes et al., 2017b), and the properties of
the convective regions (Montgomery, 2007; Montgomery et al., 2020), can be
determined from the observed period spectrum through asteroseismology. In
addition, the determination of the rate of period change (Kepler et al., 2005)
can be used to study elementary particles such as neutrinos (Winget et al.,
2004) and axions (Córsico et al., 2012; Mukadam et al., 2013; Córsico et al.,
2016; Kepler et al., 2021b).

Since the discovery of the first ZZ Ceti star, HL Tau 76 (Landolt, 1968),
there are more than 400 ZZ Cetis reported to date (see for instance Bognar &
Sodor, 2016; Córsico et al., 2019; Romero et al., 2019b; Vincent et al., 2020;
Guidry et al., 2021; Romero et al., 2022). However, for most ZZ Cetis, only a
limited number of observed periods are known, usually detected as a result of
short runs from the discovery paper, and no follow-up observations were
performed.

The increase in the number of new ZZ Cetis and new detected periods was
boosted by recent space-based observations. For instance, the TESS satellite
(Ricker et al., 2014, 2015), launched in 2018 April 18, observed several ZZ
Ceti candidates and known pulsators (see for example Bognár et al., 2020;
Romero et al., 2022). This satellite observes between 150 and 300 white dwarf
stars every month. Each observation run last for a total of 27 days, in a 120
s-–cadence for selected objects, and a 20 s–cadence mode for a limited sample
of bright objects.

In particular, PG 1541+651 and BPM 31594 are two warm-like ZZ Cetis with three
and one periods, respectively, detected from ground-based discovery
observations (Vauclair et al., 2000; McGraw, 1976). The number of detected
modes considerably increased after they were observed by the TESS satellite,
allowing a more detailed asteroseismological analysis. In this work, we
present observations from TESS for PG 1541+651 and BPM 31594, along with a
detailed asteroseismological analysis. The paper is organized as follows: in
section 2 we present the results from previous observation runs for both
objects. We describe the data reduction process and results from TESS
observations in section 3. Section 4 is devoted to a detailed
asteroseismological analysis of the targets, and we summarized our findings in
section 5.

##  2 selected targets

Vauclair et al. (2000) reported the detection of photometric variability in PG
1541$+$651 (TIC 458484139), based on $\sim$9 h observations performed on the
2-m Pic du Midi and the 1.93-m Haute Provence Observatory telescopes. Three
periods were identified at 689, 564 and 757 s, with the 689 s period having
the largest amplitude111PG 1541+651 can also be found in the literature as PG
1541+650.. These periods show typical values for a ZZ Ceti in the middle of
the instability strip (Mukadam et al., 2006). The spectroscopic effective
temperature and surface gravity were determined by Gianninas et al. (2011),
$T_{\rm eff}=11\,560\pm 250$ K and $\log g=8.12\pm 0.038$, after applying the
3D convection correction (Tremblay et al., 2013), leading to a stellar mass of
$0.67\pm 0.03M_{\odot}$. Later on, Gentile Fusillo et al. (2021), using
photometry and parallax from Gaia eDR3, found $T_{\rm eff}=11\,607\pm 203$ K
and $\log g=8.035\pm 0.028$, with a significantly lower stellar mass of
$0.626\pm 0.018M_{\odot}$.

The pulsational variability of BPM 31594 (TIC 101014997) was first reported by
McGraw (1976) based on $\sim$13 h of observations at the Sutherland observing
station of the South African Astronomical Observatory, using the 0.76-m and
1.02-m telescopes. He found two periods at 617 s and 314 s, with an amplitude
of 0.18 mag. O’Donoghue et al. (1992), based on $\sim$300 h run at the 0.76-
1.02- and 1.9-m reflectors at the Sutherland observing station of the South
African Astronomical Observatory, also reported the detection of a period of
617 s, along with several harmonics.

Gianninas et al. (2011) obtained the atmospheric parameter for BPM 31594,
being $T_{\rm eff}=11\,500\pm 250$ K and $\log g=8.05\pm 0.038$ after applying
the 3D convection correction. With these values, the stellar mass is $0.63\pm
0.03M_{\odot}$. Later Fuchs (2017) found a slightly higher effective
temperature of $11\,786\pm 22$ K and a lower surface gravity $\log g=8.029\pm
0.006$, corresponding to a stellar mass of $0.615\pm 0.005M_{\odot}$, similar
to that found by Gianninas et al. (2011).

Note that, as is the case for many ZZ Ceti stars, no further follow-up
observations were published for these objects until the TESS satellite.

##  3 Data analysis

BPM 31594 was observed in Sectors 3 and 4 with 120 s–cadence, and in Sectors
30 and 31 with both 120 s and 20 s cadence. PG 1541$+$651 was observed in
Sectors 14 to 17 and 21 to 24 with 120 s cadence and in Sectors 41, 47, 48,
50, and 51 in 120 s and 20 s cadence.

We downloaded all light curves for BPM 31594 and PG 1541$+$651 from the
Mikulski Archive for Space Telescopes, which is hosted by the Space Telescope
Science Institute (STScI)222http://archive.stsci.edu/ in FITS format. The data
were processed based on the Pre-Search Data Conditioning pipeline (Jenkins et
al., 2016). We extracted times and fluxes (PDCSAP FLUX) from the FITS files.
The times are given in barycentric corrected dynamical Julian days (BJD –
2457000, corrected for leap seconds, see Eastman et al., 2010). For each
sector, the fluxes were converted into fractional variations from the mean,
that is, differential flux $\Delta I/I$, and transformed into amplitudes in
parts-per-thousand (ppt). The ppt unit corresponds to the milli-modulation
amplitude (mma) unit3331 mma= 1/1.086 mmag= 0.1% = 1 ppt; see, e.g., Bognar &
Sodor (2016).. The contamination level from other stars in the image is low,
with CROWDSAP444The CROWDSAP level indicates the ratio of the target flux to
the total flux. of 0.94 and 0.704 for BPM 31594 and PG 1541$+$651,
respectively. We sigma-clipped the data at 5$\sigma$ to remove the outliers
that appear above five times the median absolute deviation, that is, that
depart from the median by 5$\sigma$.

We computed the Fourier transforms (FTs) for all light curves and looked for
pulsations signatures above the detection limit, defined as 5$\left<A\right>$.
For pre-whitening, we employed our customized tool, in which, using a
nonlinear least-squares method, we simultaneously fit each pulsation frequency
in a waveform $A_{i}\sin(\omega_{1}t+\phi)$, with $\omega=2\pi/P$, and $P$ the
period. This iterative process was run, starting with the highest peak, until
no peak appeared above the detection limit. The uncertainties in the
frequencies are estimated as $1/T$, where $T$ is the duration of each block,
and vary from 0.11 $\mu$Hz to 0.43 $\mu$Hz.

###  3.1 PG 1541+651

The ZZ Ceti star PG 1541+651 was observed by TESS in Sectors 14–17, 21–24, 41,
47–48, and 50–51. For Sectors 41, 47, 48, 50, and 51 the observations were
taken with 20s–cadence, while for the previous sectors, the data correspond
only to 120s–cadence. For this object, we separate the data into five blocks,
corresponding to continuous observations runs, to avoid possible spurious
signals due to long gaps in the data. For each block, we obtained the Fourier
transform (FT) and identify the characteristic frequencies. Figure 1 shows the
FT for all concatenated sectors, Sectors 14–17, Sectors 21–24, Sector 41,
Sectors 47–48 and Sectors 50–51, from top to bottom. Note that the amplitude
scale is not the same for all the plots. The data corresponds to 120s–cadence
observations for all blocks, as there are no periods detected below the
Nyquist limit, corresponding to a period of 240 s.

The frequencies detected from each block are listed in Table 2, along with the
corresponding periods and amplitudes. As can be seen from this table, there
are only two frequencies that are present in all five blocks, but most of them
are detected in four of the five blocks. Other frequencies are present in some
blocks, but not detected in others. Note that not all the frequencies
correspond to independent pulsation modes, since there are several multiplets
in the period spectra (see section 3.1.1).

As can be seen from Figure 1, and also from the values listed in Table 1, the
amplitudes of the peaks in the FT vary from block to block. For the first
block (14-17) the dominant mode, that with the highest amplitude in the FT, is
at a frequency of 2480.5871 $\mu$Hz (403.1304 s). In the second block (21-24)
the amplitude of that mode decreases considerably, while the mode with a
frequency of $\sim$1847 $\mu$Hz ($\sim$ 541 s) shows the highest amplitude in
the FT. Finally, the amplitude for the mode with $\sim$1847 $\mu$Hz decreases
from the third to the fifth block, making the mode with a frequency of $\sim$
1458 $\mu$Hz ($\sim$685 s) the dominant mode from Sectors 41 to 51. This
change in amplitude, from a short to a longer period in a scale of a few years
is not likely to be related to convection (Montgomery et al., 2020).

In general, peaks corresponding to high-frequencies, higher than $\sim$1250
$\mu$Hz are narrow, with widths roughly matching the spectral window of the
observations. The peaks with frequencies below 1248 $\mu$Hz (periods longer
than $\sim$ 800 s) show a complex structure in the FT for all the blocks,
spreading their power over a broad band in the FT. This behaviour is in
agreement with the dichotomy found by Hermes et al. (2017a), based on 27 DAVs
observed with the Kepler satellite. These authors found that peaks
corresponding to periods longer than $\sim$800 s have substantially broader
mode widths than those with periods shorter than $\sim$800 s, and are most
likely representative of phase–unstable single modes, reminiscent of a damped
harmonic oscillator. Later, Montgomery et al. (2020) showed that the depth of
the convection zone changes during the pulsation cycle, causing the reflection
of the outgoing travelling wave to not be coherent. Since in most cases, modes
with periods longer than $\sim$800 s propagate all the way to the base of the
surface convection zone, these will be affected by the time-dependent position
of the outer turning point.

Figure 1: Fourier transform for the data for PG 1541+651. The top panel corresponds to the FT of all concatenated sectors. The data from second to sixth panels, from top to bottom, correspond to the blocks including Sectors 14–17, 21–24, 41, 47–48 and 50–51, with 120s–cadence. Note that the amplitude scale is different for each plot. Table 1: List of periods and amplitudes for PG 1541+651 for each block (top row). 14-17 |  | 21-24 |  | 41 |  | 47-48 |  | 50-51 |  | all |
---|---|---|---|---|---|---|---|---|---|---|---
$\Pi$ | Amp | $\Pi$ | Amp | $\Pi$ | Amp | $\Pi$ | Amp | $\Pi$ | Amp | $\Pi$ | Amp
s | ppt | s | ppt | s | ppt | s | ppt | s | ppt | s | ppt
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 302.5562 | 3.3489 | 302.5351 | 5.8632 | 302.5338 | 8.2818 | 302.5344 | 4.7417
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 375.7850 | 4.2753 | 375.1835 | 5.2731 | 375.5697 | 2.8014
401.0212 | 1.5348 | 401.0175 | 3.3980 | $\cdots$ | $\cdots$ | 400.9785 | 2.9746 | $\cdots$ | $\cdots$ | 400.9768 | 2.6916
403.1304 | 7.8849 | 403.1295 | 5.1636 | 403.1435 | 3.4118 | 403.1490 | 4.6771 | 403.1557 | 5.2909 | 403.1195 | 4.2203
541.6983 | 7.5621 | 541.5875 | 13.2365 | 541.3547 | 13.7774 | 541.3733 | 14.2022 | 541.2986 | 17.9046 | 541.2837 | 8.2155
545.4038 | 4.6067 | 545.4320 | 5.1746 | 545.5416 | 3.2471 | 545.5766 | 3.3115 | $\cdots$ | $\cdots$ | 545.5728 | 3.1326
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 568.6268 | 3.9720 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 621.8078 | 2.4711 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\cdots$ | 679.5963 | 3.0513 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\cdots$ | 685.1075 | 7.4502 | 685.8707 | 3.9720 | 685.7314 | 18.3264 | 685.8732 | 27.8657 | 685.9228 | 9.3018
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 730.4653 | 8.8848 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 730.4495 | 2.8126
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 781.2597 | 3.5981 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 801.2480 | 6.8186 | 800.5709 | 3.0613 | $\cdots$ | $\cdots$ | 801.0640 | 2.9695
806.5962 | 6.2639 | 808.3629 | 5.2433 | $\cdots$ | $\cdots$ | 810.6814 | 2.3961 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 815.4969 | 7.0925 | 814.9610 | 7.7355 | 814.4258 | 8.0823 | 814.9361 | 4.3494
$\cdots$ | $\cdots$ | 817.8851 | 3.0128 | $\cdots$ | $\cdots$ | 819.2914 | 2.7160 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$\cdots$ | $\cdots$ | 828.4669 | 3.8807 | 825.1054 | 4.6958 | 825.3679 | 2.9158 | $\cdots$ | $\cdots$ | 825.6349 | 2.5266
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 829.9834 | 5.2491 | 830.8310 | 10.9648 | 829.1089 | 12.0168 | 830.4214 | 4.3077
849.7289 | 6.6439 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
862.2140 | 4.8798 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
875.2320 | 3.8219 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$

####  3.1.1 Rotational splittings

White dwarf stars are considered slow rotators, with rotation periods that
range from hours to days (Kepler & Romero, 2017; Hermes et al., 2017a).
Rotation leads to a breaking of the degeneracy in pulsation frequencies,
causing a single $\ell$ mode to separate into the $2\ell+1$ components in the
azimuthal order $m$ (e.g. Unno et al., 1989).

For slow rotation we can consider that the frequency separation between the
central $m=0$ component and the $\pm m$ components are equal, and thus, a
rotation period can be estimated following the equation (Cowling & Newing,
1949; Ledoux, 1951):

$\frac{1}{P_{\rm rot}}=\frac{\Delta\nu_{k,\ell,m}}{m(1-C_{k\ell})}$ (1)

where $m$ is the azimuthal number and $C_{k\ell}$ is the rotational splitting
coefficient given by:

$C_{k,\ell}=\frac{\int_{0}^{R_{*}}\rho
r^{2}[2\xi_{r}\xi_{t}+\xi_{t}^{2}]dr}{\int_{0}^{R_{*}}\rho
r^{2}[\xi_{r}^{2}+\ell(\ell+1)\xi_{t}^{2}]dr}$ (2)

where $\rho$ is the density, $r$ is the radius and $\xi_{r}$ and $\xi_{t}$ are
the radial and horizontal displacement of the material (see Unno et al., 1989,
for details). In addition, the spherical degree and azimuthal order of the
modes can be identified from the rotationally split multiplets present in the
FT (e.g., Winget et al. 1991, 1994). In the asymptotic regime, for high radial
order modes, the value of the rotational splitting coefficient is
$C_{k\ell}\sim 1/\ell(\ell+1)$, being $\sim 0.5$ and $\sim 0.166$, for
$\ell=1$ and $\ell=2$ modes.

Table 2: Frequency, period, frequency separation and $\ell$ and $m$ identification for multiplets components found for PG 1541+651. The values in italic are computed from the observed frequencies. The values of the periods longer 800 sec are computed as a simple mean of the values for blocks 41, 47-48 and 50-51. Freq | $\Pi$ | $\Delta\mu$ | $\ell$ | $m$
---|---|---|---|---
$\mu$Hz | s |  $\mu$Hz |  |
2493.744 | 401.004 | 6.609 | 1 | +1
2487.135 | 402.069 | - | 1 | 0
2480.526 | 403.140 | 6.609 | 1 | -1
1846.965 | 541.429 | 6.845 | 1 | +1
1840.015 | 543.474 | - | 1 | 0
1833.065 | 545.534 | 7.036 | 1 | -1
1471.462 | 679.596 | 6.580 | 1 | +1
1464.882 | 682.649 | - | 1 | 0
1458.302 | 685.728 | 6.580 | 1 | -1
1233.530 | 810.681 | 6.467 | 1 | -1
1227.053 | 814.961 | - | 1 | 0
1220.567 | 819.291 | 6.496 | 1 | -1
1248.581 | 800.909 | 21.528 | ? | +
1227.053 | 814.961 | - | ? | 0
1204.857 | 829.974 | 22.196 | ? | -
1176.846 | 849.729 | 17.041 | ? | +
1159.805 | 862.214 | - | ? | 0
1142.554 | 875.232 | 17.251 | ? | -

By analysing the frequency spectrum for PG 1541+651 from the TESS data, we
identify seven possible multiplets. The components for each multiplet are
listed in Table 2, along with the corresponding harmonic degree ($\ell$) and
the azimuthal order ($m$). We identify four of them as triplets, with harmonic
degree $\ell=1$, with a frequency separation between 6 and 7 $\mu$Hz. The
regions of interest in the Fourier transform are shown in Figure 2 for the
modes with central components at $\sim$ 2487 $\mu$Hz (top), $\sim$1840 $\mu$Hz
(middle) and $\sim$1465 $\mu$Hz (bottom) for the data from Sectors 21 to 24
and 120 s-cadence.

The remaining two multiplets (last ones in Table 2) show a frequency
separation between the components that is much larger than the others, and
thus we do not assign a harmonic degree. The region of the FT corresponding to
the multiplet with a central component at a frequency of 1159.81 $\mu$Hz is
depicted in Figure 3. The frequency separation is $\sim$17 $\mu$Hz, which is
larger than two times the separation between the multiplets identified as
dipole modes.

Figure 2: Portions of the Fourier transform for PG 1541+651 for the identified
triplets with the shorter periods with central components at $\sim$ 402 s
(top), $\sim$ 543 s (middle) and $\sim$ 683 s (bottom) for the observation
from Sectors 21 to 24 and 120s–cadence. The red line correspond to the
5$\langle{\rm A}\rangle$ detection limit. For the bottom panel, we include the
data for Sector 41 with 20s–cadence (dashed line) for completeness. The
frequency separation between the central and prograde and retrograde
components is between 6 and 7 $\mu$ Hz for all triplets shown in this figure.
Figure 3: Fourier transform for PG 1541+651 for the region between 1140 and
1185 $\mu$Hz for PG 1541+651, for Sectors 14-17 and 120s–cadence. The red line
correspond to the 5$\langle{\rm A}\rangle$ detection limit.

As stated before, the region of the FT for frequencies around 1225 $\mu$Hz
shows a quite complex structure that changes from one block to the next. This
can be seen from Figure 4, where we show the region of interest for all
blocks. From the first block, corresponding to Sectors 14 to 17, we only
detect one peak at $\sim$806 s. As we move forward, other components seem to
gain energy and reach amplitudes above the detection limit. In particular,
components with higher frequencies (shorter periods) seem to lose energy as we
move to more recent sectors, with the component centred at $\sim$829 s (1206
$\mu$Hz) being the one with the highest amplitude for the data corresponding
to the 50-51 block. If we consider the combined peaks detected in the five
blocks, we can identify five periods, as indicated in the fourth panel in
Figure 4, corresponding to the data from Sectors 47 and 48. We identify these
collections of frequencies as two possible multiplets centred at $\sim$815 s
(1226 $\mu$Hz). The first multiplet is identified as a $\ell=1$ mode, with a
$\Delta\nu\sim$ 6.5$\,\mu$Hz (see Table 2). The second multiplet shows a
separation of $\sim$22$\,\mu$Hz.

Figure 4: Fourier transform for the region between 1190 and 1260$\,\mu$Hz for
PG 1541+651, for Sectors 14-17, 21-24, 41, 47-48 and 50-51 from top to bottom.
Their respective spectral windows are shown on the right side. The red line
correspond to the 5$\langle{\rm A}\rangle$ detection limit. The approximate
values and ranges for the corresponding periods are depicted in the plot. On
the right panel we depict the spectral window for each block.

###  3.2 BPM 31594

BPM 31594 was observed by TESS in Sectors 3 and 4 with 120 s cadence, and in
Sectors 30 and 31 with 20 s-cadence. Thus, we separate the data into two
blocks, corresponding to continuous observation runs. Figure 5 shows the FT
for the two blocks, corresponding to Sectors 30 and 31 (bottom panel) and
Sectors 3 and 4 (middle panel). The FT for the concatenated data is shown in
the top panel of this figure.

Figure 5: Fourier transform for the two blocks of data for BPM 31594. Top
panel corresponds to the FT of the concatenated data. The middle panel
corresponds to the data from Sectors 3 and 4 and 120s–cadence, while the
bottom panel shows the FT for the data from Sectors 30 and 31 with
120s–cadence. Note that the amplitude scale is different for each plot.

The detected frequencies, periods and amplitudes for each block are listed in
Table 3. From the 03-04 block, we detect 6 periods, with a dominant peak at
617.89 s. From block 30-31, we detect five additional periods, and confirm the
ones present in block 03-04. The mode with the highest amplitude is by far the
one with a frequency of 1618.4 $\mu$Hz ($f_{3}$), in agreement with the
results from McGraw (1976) and O’Donoghue et al. (1992). We also detected the
prograde and retrograde components for this frequency, shown in Figure 6, and
thus we identify this mode as a dipole $\ell=1$ mode. Among the detected
frequencies, we look for linear combinations and harmonics by computing
combinations of the detected frequencies, considering that their amplitude
cannot be larger than that of the parent modes (Buchler et al., 1995). We
identify two harmonics of the main period, and three linear combinations of
the main peak and $f_{2}$, identified as such in the last column of Table 3.
Note that the peak corresponding to the frequency 3166.7 $\mu$Hz reported by
(McGraw, 1976) is present in the TESS data for both blocks. On the other hand,
sub-harmonics of the main period, previously reported by O’Donoghue et al.
(1992), are not present in the TESS data.

Table 3: List of frequencies, periods and amplitudes detected for BPM 31594. | 03-04 |  |  | 30-31 |  | ID
---|---|---|---|---|---|---
Freq | $\Pi$ | Amp | Freq | $\Pi$ | Amp |
$\mu$Hz | s | ppt |  $\mu$Hz | s | ppt |
3166.8722 | 315.7670 | 4.0095 | 3166.6852 | 315.7876 | 2.3515 | $f_{1}$
2487.8989 | 401.9456 | 5.7188 | 2488.6026 | 401.8319 | 6.3873 | $f_{2}$
1631.8405 | 612.8050 | 10.9446 | 1630.7763 | 613.2049 | 7.9732 | $f_{3}^{+}$
1618.3989 | 617.8946 | 39.114 | 1618.3725 | 617.9047 | 32.7478 | $f_{3}$
1604.9801 | 623.0607 | 8.8420 | 1605.9453 | 622.6862 | 8.6200 | $f_{3}^{-}$
1548.4505 | 645.8069 | 8.4213 | 1548.3318 | 645.8564 | 3.6726 | $f_{4}$
1461.1146 | 684.8069 | 2.9050 | 1460.7312 | 684.5886 | 1.6364 | $f_{5}$
$\cdots$ | $\cdots$ | $\cdots$ | 877.2352 | 1139.9452 | 1.5551 | $f_{6}$
$\cdots$ | $\cdots$ | $\cdots$ | 5726.9279 | 174.6137 | 1.7852 | 2$f_{3}+f_{2}$
$\cdots$ | $\cdots$ | $\cdots$ | 4858.3346 | 205.8318 | 2.9610 | 3$f_{3}$
$\cdots$ | $\cdots$ | $\cdots$ | 4108.3563 | 243.4063 | 2.6466 | $f_{3}+f_{2}$
3236.7978 | 308.9473 | 6.2662 | 3236.7464 | 308.9522 | 7.1622 | 2$f_{3}$
$\cdots$ | $\cdots$ | $\cdots$ | 871.7971 | 1147.0558 | 1.5437 | $f_{2}-f_{3}$
Figure 6: Fourier transform for BPM 31495 for the region between 1600 and 1620
$\mu$Hz, for Sectors 30-31 at 20s–cadence. The red line correspond to the
5$\langle{\rm A}\rangle$ detection limit.

##  4 Asteroseismology

In this section, we present a detailed asteroseismological study of PG
1541+651 and BPM 31594. We employ a grid of DA white dwarf representative
models, product of fully evolutionary computations that consider the evolution
of the progenitor star from the main sequence to the cooling curve. The models
were computed using the LPCODE evolutionary code (see Althaus et al., 2010;
Renedo et al., 2010; Romero et al., 2013; Romero et al., 2015, for details).
The model grid covers the mass region where C/O–core white dwarfs are
expected, with stellar masses between 0.493 and 1.05 $M_{\odot}$. The inner
chemical profile is determined by the model evolution, thus the internal
structure, from the C/O core to the hydrogen envelope, are consistent. In
addition, models with different thickness of the hydrogen layers are included
(Bradley, 1996; Cowling & Newing, 1949), with hydrogen masses ranging from
$4\times 10^{-4}M_{*}$ to $\sim 10^{-10}M_{*}$, where the upper limit depends
on the stellar mass, and is set by residual nuclear burning at the base of the
hydrogen layer (Romero et al., 2012; Romero et al., 2019a). We do not consider
hydrogen envelopes thinner than $10^{-10}M_{*}$ since, by the time the white
dwarf reaches the instability strip, the outer convective zone will be deep
enough to mix the hydrogen into the much ($\sim 100x$) more massive helium
layer, turning the star into a DB white dwarf (Romero et al., 2013; Ourique et
al., 2020; Cunningham et al., 2020). Our current model grid has 616 white
dwarf cooling sequences, corresponding to 35 stellar masses, totalling $\sim
74\,000$ models. For each model within the instability strip, we computed
adiabatic non-radial pulsations using the adiabatic version of the LP-PUL
pulsation code (see Córsico & Althaus, 2006, for details). This model grid has
been largely used in the past to perform asteroseismological studies of DA
pulsating white dwarfs (Romero et al., 2012, 2022).

To search for the asteroseismological model for each target, we seek for the
theoretical model that minimizes a quality function, given by:

$\chi^{2}=\frac{1}{N-1}\sqrt{\sum_{i=1}^{N}\left(\Pi_{i}^{\rm
teo}-\Pi_{i}^{\rm obs}\right)^{2}}$ (3)

where $N$ is the number of observed period and $\Pi_{i}^{\rm teo}$ is the
theoretical period that better fits the observed period $\Pi_{i}^{\rm obs}$.
We also compute other quality functions, always obtaining similar results (see
e.g. Bradley, 1998; Castanheira & Kepler, 2008; Córsico et al., 2009). In our
fit, we consider the identification of the harmonic degree, when present, and
restrict our solutions using the stellar mass and effective temperature
obtained from Gaia photometry and parallax.

###  4.1 PG 1541+651

The list of periods for PG 1541+651 considered for the asteroseismological fit
is listed in Table 4. The period values are taken from Table 1. For periods
that are detected in more than one block, we consider the mean value, weighted
by the amplitude in the FT. In case of multiplets, we consider the value of
the period of the central component ($m=0$) from Table 2. For PG 1541+651 we
identify 12 independent modes; four of them are identified positively as
$\ell=1$ dipoles and one is identified as a $\ell=2$ quadrupole, given the
number of detected components and the frequency separation between them (see
section 3.1.1).

Table 4: Periods list for PG 1541+651 used for asteroseismology. For the periods that appear as multiplets in the FT, we indicate the harmonic degree. Values for the modes marked in italic correspond to the central components of the multiplets shown in Figure 4. ID | Freq | $\Pi$ | $\ell$
---|---|---|---
|  $\mu$Hz | s |
$f_{1}$ | 3305.3641 | 302.5385 | -
$f_{2}$ | 2663.4520 | 375.7526 | -
$f_{3}$ | 2487.1350 | 402.0690 | 1
$f_{4}$ | 1840.0153 | 543.4738 | 1
$f_{5}$ | 1758.6227 | 568.6268 | -
$f_{6}$ | 1608.2140 | 621.8078 | -
$f_{7}$ | 1464.8820 | 682.6488 | 1
$f_{8}$ | 1368.9904 | 730.4653 | -
$f_{9}$ | 1279.9841 | 781.2597 | -
$f_{10}$ | 1227.0522 | 814.9613 | 1
$f_{11}$ | 1227.0522 | 814.9613 | 2
$f_{12}$ | 1159.8049 | 862.2140 | -

As can be seen from Table 1, most of the periods are not detected in all the
data blocks. In fact, only four modes are detected in four of the five the
blocks, whether it is the central components or at least one component of a
multiplet. These modes are the ones identified by $f_{3}$, $f_{4}$, $f_{7}$
and $f_{10}$, with periods of 402.0690, 543.4732, 682.6488 and 814.9609 s. For
the moment, we consider that the last period (814.96 s), corresponds to only
one mode with an unknown harmonic degree. We first perform an
asteroseismological fit using these four periods. The structural parameters of
two models with the lowest value of $\chi^{2}$ are listed in Table 5. Both
models are characterized by a stellar mass of 0.609 $M_{\odot}$, but the
hydrogen envelope mass is quite different. The model with the lowest value of
$\chi^{2}$ shows a thin hydrogen envelope of $4.7\times 10^{-9}M_{*}$ and an
effective temperature of $11\,800$ K, closer to the blue edge of the
instability strip for this stellar mass. The second model is characterized by
a hydrogen envelope almost three orders of magnitude thicker and an effective
temperature of $11\,290$ K, closer to the red edge of the instability strip,
which is in agreement with the period values larger than $\sim 400$ s (Mukadam
et al., 2006). The large difference in the hydrogen content of the two
seismological fits can be interpreted in terms of the core-envelope symmetry
(Montgomery et al., 2003) and the differences in the chemical structures
characterizing both models.

Table 5: Best fitting models for PG 1541+651 considering the four recurring periods. The stellar mass, hydrogen envelope and effective temperature are listed in columns 1, 2 and 3, respectively. The theoretical periods are listed in column 4, along with the harmonic degree and radial order. The value of the quality function is listed in the last column. M ($M_{\odot}$ | $\log(M_{H}/M_{*})$ |  $T_{\rm eff}$ [K] |  $\Pi$ [s] $(\ell,k)$ | $\chi^{2}$
---|---|---|---|---
0.609 |  $-$8.33 | 11800 | 397.5016 (1,5), 540.5612 (1,8), 686.4946 (1,11), 813.7084 (2,25) | 2.2535
0.609 |  $-$5.24 | 11290 | 400.3013 (1,6), 550.4957 (1,9), 678.4833 (1,12), 814.4608 (1,15) | 2.7890

Next, we perform an asteroseismological fit considering all 12 periods
detected for PG 1541+651, listed in Table 4. In this case, we consider that
the period of $\sim 815$ s corresponds to the central component of two
multiplets, that have harmonic degrees $\ell=1$ and $\ell=2$. The structural
parameters characterizing the best fit model are listed in Table 6, along with
the theoretical periods and the value of the quality function. Figure 7 shows
the inverse of $\chi^{2}$ as a function of the effective temperature and the
mass of the hydrogen envelope, for sequences characterized by a stellar mass
of the best fit model, being 0.609 M⊙. The effective temperature for the best
fit model (minimum of $\chi^{2}$) is in agreement with the determination from
Gentile Fusillo et al. (2021), within the uncertainties. The chemical profile
and the run of the Brunt–Väisälä and the Lamb frequencies for the best fit
model are shown in Figure 8.

Figure 7: Contour map with the value of the inverse of the quality function (gray scale) for PG 1541+651, considering models with stellar mass of 0.609 M⊙. The quality function is that given by equation 3, plotted as a function of the effective temperature and the logarithm of the hydrogen envelope mass.  Table 6: Best fitting model for PG 1541+651 considering all detected periods. The stellar mass, effective temperature and hydrogen envelope are listed in column 2. The theoretical periods, harmonic degree and radial order are listed in columns 3, 4 and 5, respectively. The value of the rotation kernel and the rotation period are listed in the columns 6 and 7, respectively, for the modes identified as multiplets. The value of the quality function is listed in the last row of column 2. Model |  |  $\Pi$ [s] | $\ell$ | $k$ | $C_{k\ell}$ | Prot [h]
---|---|---|---|---|---|---
M ($M_{\odot}$) | 0.609 | 305.6120 | 2 | 8 |  |
$T_{\rm eff}$ [K] | 11240 | 375.9419 | 2 | 11 |  |
$\log(M_{H}/M_{*})$ |  $-$5.35 | 402.9660 | 1 | 6 | 0.4877 | 21.53
$\chi^{2}$ | 1.5037 | 552.9017 | 1 | 9 | 0.4521 | 21.93
|  | 564.3766 | 2 | 18 |  |
|  | 616.4197 | 2 | 20 |  |
|  | 680.4286 | 1 | 12 | 0.4587 | 22.85
|  | 726.3037 | 2 | 24 |  |
|  | 782.4958 | 2 | 26 |  |
|  | 809.2072 | 2 | 27 | 0.1626 | 21.28*
|  | 821.7833 | 1 | 15 | 0.4946 | 21.66
|  | 867.1427 | 2 | 29 | 0.1642 | 27.08*
Figure 8: The square of the Brunt-Väisälä and the Lamb frequencies (top panel)
and the chemical profiles (bottom panel) for the best fit model for PG
1541+651 listed in Table 6, with a stellar mass of 0.609 $M_{\odot}$,
effective temperature of $11\,240$ K, and a hydrogen mass of $4.5\times
10^{-6}M_{*}$.

Figure 9 shows the run of the weight function $W$ for five of the six modes
identified as central components of multiplets. The vertical red lines
correspond to the position of the each chemical transition, being, from right
to left, the H–to–He transition, the base of the He buffer, and the point
where the carbon abundance is larger than the oxygen abundance in the C/O core
(see Fig. 8 for details). Note that the weight function is very sensitive to
the position of the H-to-He transition for all modes depicted in Figure 9.

Figure 9: Weight function $W$ corresponding to the modes with $(\ell,k)$ \-
(1,9), (1,12), (1,15), (2,27) and (2,29) of the asteroseismic model for PG
1541+651. Vertical dashed lines correspond to the position of the main
chemical transitions (see Fig. 8).

Finally, each theoretical mode has an associated value of the rotational
splitting coefficient $C_{k,\ell}$. The values for the $C_{k,\ell}$ for the
modes identified as multiplets are listed in column 6 of Table 6. Combining
this value with the observed frequency separation for each mode (third column
of Table 2), we can estimate the rotation period using equation 1. The
rotation periods are listed in the last column of Table 2. For the modes
identified as dipoles, with $\ell=1$, the values for the rotation period are
in agreement with each other, giving a value of $\sim$ 22 h. The rotation
period obtained for the mode with theoretical period 821.7833 s, identified as
$\ell=2$, shows a similar value, if we consider that the observed components
correspond to the azimuthal orders $m=\pm 2$. Thus, if we consider the five
multiplets with the shortest periods, we can estimate a rotation period of
$\sim$ 22 h.

The mode with a central component at 862.2140 s was also identified as a
multiplet. The frequency separation in this case is $\Delta\nu_{\ell,m}\sim
17\,\mu$Hz. The best fit model for PG 1541+651 fits this mode as a quadrupole
theoretical mode with $\ell=2$. The rotation period obtained for this mode
gives $\sim 13.5$ h if we consider the components to be $m=\pm 1$, or 27.1 h
if we consider the components to be $m=\pm 2$.

###  4.2 BPM 31594

For BPM 31594 we detected nine frequencies in the FT, with six modes
identified as independent modes. For our asteroseismological fit we consider
the period values from the 30-31 block, and fix the harmonic degree of the
main mode, with a period of 617.9 s, as $\ell=1$. The final list is shown in
Table 7.

Table 7: Periods list for BPM 31594 used for asteroseismology. For the period that appears as a multiplet in the FT, we indicate the harmonic degree. ID | Freq | $\Pi$ | $\ell$
---|---|---|---
|  $\mu$Hz | s |
$f_{1}$ | 3166.6852 | 315.7876 | -
$f_{2}$ | 2488.6026 | 401.8319 | -
$f_{3}$ | 1618.3725 | 617.9047 | 1
$f_{4}$ | 1548.3318 | 645.8564 | -
$f_{5}$ | 1460.7312 | 684.5886 | -
$f_{6}$ | 877.2352 | 1139.9452 | -

From our asteroseismological fit, we found a minimum in the quality function
for a stellar mass of 0.690 $M_{\odot}$. However, the luminosity of the model
leads to a distance of 37 pc, which is 7 pc closer than the distance
determined with Gaia DR3 parallax, of 44.3 pc (Gentile Fusillo et al., 2021).
We searched for a seismological solution that is compatible with the distance
determination. The structure parameters and the theoretical periods of this
model are listed in Table 8. Figure 10 depicts the inverse of $\chi^{2}$ as a
function of the effective temperature and the mass of the hydrogen envelope,
for sequences with a stellar mass of 0.632 M⊙, where a solution family can be
seen for the thickest envelope value. Figure 11 shows the run of the Brunt-
Väisälä and Lamb frequencies (top panel) and the chemical profile for the best
fit model (middle panel). In the bottom panel of Figure 11 we depict the
weight function $W$ for the theoretical mode with a period of 617.7 s. This is
the main observed mode, which is also identified as a triplet. Note that this
mode is quite sensitive to the H/He transition, and thus to the mass of the
hydrogen envelope.

Figure 10: Contour map with the value of the inverse of the quality function (gray scale) for BPM 31594, considering models with stellar mass of 0.632 M⊙. The quality function is that given by equation 3, plotted as a function of the effective temperature and the logarithm of the hydrogen envelope mass.  Table 8: Best fitting model for BPM 31594 considering all detected periods. The stellar mass, effective temperature and hydrogen envelope are listed in column 2. The theoretical periods, harmonic degree and radial order are listed in columns 3, 4 and 5, respectively. The value of the quality function is listed in the last row of column 2. The value of the rotation kernel is listed in the last column. $\\#$ |  |  $\Pi$ [s] | $\ell$ | $k$ | $C_{k\ell}$
---|---|---|---|---|---
M ($M_{\odot}$) | 0.632 | 315.2605 | 2 | 10 |
$T_{\rm eff}$ [K] | 11560 | 406.3396 | 1 | 7 |
$\log(M_{H}/M_{*})$ |  $-$4.12 | 617.7196 | 1 | 12 | 0.4825
$\chi^{2}$ | 1.5500 | 644.1514 | 1 | 13 |
|  | 679.0426 | 1 | 14 |
|  | 1142.0466 | 2 | 44 |
Figure 11: The square of the Brunt-Väisälä and the Lamb frequencies (top
panel) and the chemical profiles (middle panel) for the best fit model for BPM
31594 listed in Table 8, with a stellar mass of 0.632 $M_{\odot}$, effective
temperature of $11\,560$ K, and a hydrogen mass of $7.6\times 10^{-5}M_{*}$.
Run of the weight function $W$ (bottom panel) for the main mode, with a period
of 617.7 s.

From the asteroseismological fit we also get the value of the rotational
splitting coefficient $C_{k\ell}$ (eq. 2) for all the theoretical periods. In
particular, for the main mode fitted with a theoretical period of 617.7 s the
value is $C_{1,12}=0.4825$. Following equation 1, with $\Delta\nu_{k,\ell,m}$
= 12.4$\,\mu$Hz, we obtain a rotation period of 11.57 h.

##  5 Conclusions

In this work we present a detailed astroseismological study of two warm–like
ZZ ceti stars, PG 1541+651 and BPM 31594, based on the photometric data
obtained by the TESS mission.

PG 1541+651 was observed in eight sectors with 120 s cadence, and also in five
sectors with both 120 and 20 s cadence, from Sectors 14 to 51, with 20
s–cadence data for five sectors. We found 12 modes, four of them identified as
triplets due to the presence of the rotational splitting components. From our
asteroseismological fit we found a representative model characterized by a
stellar mass of 0.609$M_{\odot}$, effective temperature of $11\,240$ K and a
mass of the hydrogen envelope of 4.5$\times 10^{-6}M_{*}$ (or 2.7$\times
10^{-6}M_{\odot}$). Due to the large number of observed modes we were able to
break the degeneracy in $M_{H}$ in our asteroseismological fit.

BPM 31594 was observed during the first year of the TESS mission, in Sectors 3
and 4, and during the third year in Sectors 30 and 31. We identify six
periods, being the dominant period as triplet. For this object we used the
restriction in stellar mass given by the distance and found a best fit model
characterized by $M_{*}=0.632M_{\odot}$, $T_{\rm eff}=11\,560$ K and a
$M_{H}=7.6\times 10^{-5}M_{*}$ (or 4.8$\times 10^{-5}M_{\odot}$), being the
thickest hydrogen envelope model for this stellar mass.

Rotation periods for both objects were obtained from the frequency separation
of the detected multiplets, being $\sim$ 22 h for PG 1541+651 and 11.57 h for
BPM 31594, in agreement with the values reported for other ZZ Ceti stars.

## Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, Conselho
Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq), and
Fundação de Amparo à Pesquisa do Rio Grande do Sul (FAPERGS) - Brasil. KJB is
supported by the National Science Foundation under Award AST-1903828.
Financial support from the National Science Centre under project No.
UMO-2017/26/E/ST9/00703 is acknowledged. J.J.H. acknowledges salary and travel
support through TESS Guest Investigator Programs 80NSSC19K0378 and
80NSSC20K0592. M.U. acknowledges financial support from CONICYT Doctorado
Nacional in the form of grant number No: 21190886 and ESO studentship program.
This paper includes data collected with the TESS mission, obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the TESS mission is provided by the NASA Explorer Program. This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of NASA’s Astrophysics Data System Bibliographic Services, and the SIMBAD
database, operated at CDS, Strasbourg, France.

## Data Availability

Data from TESS is available at the MAST archive
https://mast.stsci.edu/search/hst/ui/$#$/. Ground based data will be shared on
reasonable request to the corresponding author.

## References

  * Althaus et al. (2010) Althaus L. G., Córsico A. H., Bischoff-Kim A., Romero A. D., Renedo I., García-Berro E., Miller Bertolami M. M., 2010, ApJ, 717, 897
  * Bognar & Sodor (2016) Bognar Z., Sodor A., 2016, Information Bulletin on Variable Stars, 6184, 1
  * Bognár et al. (2020) Bognár Z., et al., 2020, A&A, 638, A82
  * Bognár et al. (2021) Bognár Z., Kalup C., Sódor Á., 2021, A&A, 651, A14
  * Bradley (1996) Bradley P. A., 1996, ApJ, 468, 350
  * Bradley (1998) Bradley P. A., 1998, ApJS, 116, 307
  * Brickhill (1991) Brickhill A. J., 1991, MNRAS, 251, 673
  * Buchler et al. (1995) Buchler J. R., Goupil M. J., Serre T., 1995, A&A, 296, 405
  * Castanheira & Kepler (2008) Castanheira B. G., Kepler S. O., 2008, MNRAS, 385, 430
  * Castanheira & Kepler (2009) Castanheira B. G., Kepler S. O., 2009, MNRAS, 396, 1709
  * Castanheira et al. (2007) Castanheira B. G., et al., 2007, A&A, 462, 989
  * Chidester et al. (2022) Chidester M. T., Farag E., Timmes F. X., 2022, ApJ, 935, 21
  * Clemens (1993) Clemens J. C., 1993, Baltic Astronomy, 2, 407
  * Córsico & Althaus (2006) Córsico A. H., Althaus L. G., 2006, A&A, 454, 863
  * Córsico et al. (2009) Córsico A. H., Althaus L. G., Miller Bertolami M. M., García-Berro E., 2009, A&A, 499, 257
  * Córsico et al. (2012) Córsico A. H., Althaus L. G., Miller Bertolami M. M., Romero A. D., García-Berro E., Isern J., Kepler S. O., 2012, MNRAS, 424, 2792
  * Córsico et al. (2016) Córsico A. H., et al., 2016, J. Cosmology Astropart. Phys., 2016, 036
  * Córsico et al. (2019) Córsico A. H., Althaus L. G., Miller Bertolami M. M., Kepler S. O., 2019, A&ARv, 27, 7
  * Cowling & Newing (1949) Cowling T. G., Newing R. A., 1949, ApJ, 109, 149
  * Cunningham et al. (2020) Cunningham T., Tremblay P.-E., Gentile Fusillo N. P., Hollands M., Cukanovaite E., 2020, MNRAS, 492, 3540
  * De Gerónimo et al. (2017) De Gerónimo F. C., Althaus L. G., Córsico A. H., Romero A. D., Kepler S. O., 2017, A&A, 599, A21
  * Doherty et al. (2015) Doherty C. L., Gil-Pons P., Siess L., Lattanzio J. C., Lau H. H. B., 2015, MNRAS, 446, 2599
  * Dolez & Vauclair (1981) Dolez N., Vauclair G., 1981, A&A, 102, 375
  * Eastman et al. (2010) Eastman J., Siverd R., Gaudi B. S., 2010, PASP, 122, 935
  * Fontaine & Brassard (2008) Fontaine G., Brassard P., 2008, PASP, 120, 1043
  * Fuchs (2017) Fuchs J. T., 2017, PhD thesis, University of North Carolina at Chapel Hill, United States
  * Gentile Fusillo et al. (2021) Gentile Fusillo N. P., et al., 2021, MNRAS, 508, 3877
  * Gianninas et al. (2011) Gianninas A., Bergeron P., Ruiz M. T., 2011, ApJ, 743, 138
  * Goldreich & Wu (1999) Goldreich P., Wu Y., 1999, ApJ, 511, 904
  * Guidry et al. (2021) Guidry J. A., et al., 2021, ApJ, 912, 125
  * Hermes et al. (2017a) Hermes J. J., et al., 2017a, ApJS, 232, 23
  * Hermes et al. (2017b) Hermes J. J., et al., 2017b, ApJ, 841, L2
  * Jenkins et al. (2016) Jenkins J. M., et al., 2016, in Chiozzi G., Guzman J. C., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 9913, Software and Cyberinfrastructure for Astronomy IV. p. 99133E, doi:10.1117/12.2233418
  * Kepler & Romero (2017) Kepler S. O., Romero A. D., 2017, in European Physical Journal Web of Conferences. p. 01011 (arXiv:1706.07020), doi:10.1051/epjconf/201715201011
  * Kepler et al. (2005) Kepler S. O., et al., 2005, ApJ, 634, 1311
  * Kepler et al. (2019) Kepler S. O., et al., 2019, MNRAS, 486, 2169
  * Kepler et al. (2021a) Kepler S. O., Koester D., Pelisoli I., Romero A. D., Ourique G., 2021a, MNRAS, 507, 4646
  * Kepler et al. (2021b) Kepler S. O., et al., 2021b, ApJ, 906, 7
  * Landolt (1968) Landolt A. U., 1968, ApJ, 153, 151
  * Lauffer et al. (2018) Lauffer G. R., Romero A. D., Kepler S. O., 2018, MNRAS, 480, 1547
  * Ledoux (1951) Ledoux P., 1951, ApJ, 114, 373
  * McGraw (1976) McGraw J. T., 1976, ApJ, 210, L35
  * Metcalfe et al. (2002) Metcalfe T. S., Salaris M., Winget D. E., 2002, ApJ, 573, 803
  * Montgomery (2007) Montgomery M. H., 2007, in Napiwotzki R., Burleigh M. R., eds, Astronomical Society of the Pacific Conference Series Vol. 372, 15th European Workshop on White Dwarfs. p. 635
  * Montgomery & Winget (1999) Montgomery M. H., Winget D. E., 1999, ApJ, 526, 976
  * Montgomery et al. (2003) Montgomery M. H., Metcalfe T. S., Winget D. E., 2003, MNRAS, 344, 657
  * Montgomery et al. (2020) Montgomery M. H., Hermes J. J., Winget D. E., Dunlap B. H., Bell K. J., 2020, ApJ, 890, 11
  * Mukadam et al. (2006) Mukadam A. S., Montgomery M. H., Winget D. E., Kepler S. O., Clemens J. C., 2006, ApJ, 640, 956
  * Mukadam et al. (2013) Mukadam A. S., et al., 2013, ApJ, 771, 17
  * O’Donoghue et al. (1992) O’Donoghue D., Warner B., Cropper M., 1992, MNRAS, 258, 415
  * Ourique et al. (2020) Ourique G., Kepler S. O., Romero A. D., Klippel T. S., Koester D., 2020, MNRAS, 492, 5003
  * Pepper et al. (2022) Pepper B. T., Istrate A. G., Romero A. D., Kepler S. O., 2022, MNRAS, 513, 1499
  * Renedo et al. (2010) Renedo I., Althaus L. G., Miller Bertolami M. M., Romero A. D., Córsico A. H., Rohrmann R. D., García-Berro E., 2010, ApJ, 717, 183
  * Ricker et al. (2014) Ricker G. R., et al., 2014, in Oschmann Jacobus M. J., Clampin M., Fazio G. G., MacEwen H. A., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 9143, Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave. p. 914320 (arXiv:1406.0151), doi:10.1117/12.2063489
  * Ricker et al. (2015) Ricker G. R., et al., 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Romero et al. (2012) Romero A. D., Córsico A. H., Althaus L. G., Kepler S. O., Castanheira B. G., Miller Bertolami M. M., 2012, MNRAS, 420, 1462
  * Romero et al. (2013) Romero A. D., Kepler S. O., Córsico A. H., Althaus L. G., Fraga L., 2013, ApJ, 779, 58
  * Romero et al. (2015) Romero A. D., Campos F., Kepler S. O., 2015, MNRAS, 450, 3708
  * Romero et al. (2019a) Romero A. D., Kepler S. O., Joyce S. R. G., Lauffer G. R., Córsico A. H., 2019a, MNRAS, 484, 2711
  * Romero et al. (2019b) Romero A. D., et al., 2019b, MNRAS, 490, 1803
  * Romero et al. (2022) Romero A. D., et al., 2022, MNRAS, 511, 1574
  * Siess (2007) Siess L., 2007, A&A, 476, 893
  * Tremblay et al. (2013) Tremblay P. E., Ludwig H. G., Steffen M., Freytag B., 2013, A&A, 559, A104
  * Unno et al. (1989) Unno W., Osaki Y., Ando H., Saio H., Shibahashi H., 1989, Nonradial oscillations of stars
  * Vauclair et al. (2000) Vauclair G., Dolez N., Fu J. N., Homeier D., Roques S., Chevreton M., Koester D., 2000, A&A, 355, 291
  * Vincent et al. (2020) Vincent O., Bergeron P., Lafrenière D., 2020, AJ, 160, 252
  * Winget & Kepler (2008) Winget D. E., Kepler S. O., 2008, ARA&A, 46, 157
  * Winget et al. (1982) Winget D. E., van Horn H. M., Tassoul M., Fontaine G., Hansen C. J., Carroll B. W., 1982, ApJ, 252, L65
  * Winget et al. (1991) Winget D. E., et al., 1991, ApJ, 378, 326
  * Winget et al. (1994) Winget D. E., et al., 1994, ApJ, 430, 839
  * Winget et al. (2004) Winget D. E., Sullivan D. J., Metcalfe T. S., Kawaler S. D., Montgomery M. H., 2004, ApJ, 602, L109


# Noise Learning Based Denoising Autoencoder

Woong-Hee Lee, Mustafa Ozger, Ursula Challita, and Ki Won Sung, This work was
supported by a Korea University Grant. This research was supported by the BK21
FOUR(Fostering Outstanding Universities for Research) funded by the Ministry
of Education(MOE, Korea) and National Research Foundation of Korea(NRF). This
work was partly funded by the European Union Horizon 2020 Research and
Innovation Programme under the EU/KR PriMO-5G project with grant agreement No
815191. (Corresponding author: Ki Won Sung.)W.-H. Lee is with the Department
of Control and Instrumentation Engineering, Korea University, Republic of
Korea (e-mail: woongheelee@korea.ac.kr).U. Challita is with Ericsson Research,
Stockholm, Sweden (e-mail: ursula.challita@ericsson.com).M. Ozger and K. W.
Sung are with the School of Electrical Engineering and Computer Science, KTH
Royal Institute of Technology, Stockholm, Sweden (e-mail:{ozger and
sungkw}@kth.se).

###### Abstract

This letter introduces a new denoiser that modifies the structure of denoising
autoencoder (DAE), namely noise learning based DAE (nlDAE). The proposed nlDAE
learns the noise of the input data. Then, the denoising is performed by
subtracting the regenerated noise from the noisy input. Hence, nlDAE is more
effective than DAE when the noise is simpler to regenerate than the original
data. To validate the performance of nlDAE, we provide three case studies:
signal restoration, symbol demodulation, and precise localization. Numerical
results suggest that nlDAE requires smaller latent space dimension and smaller
training dataset compared to DAE.

###### Index Terms:

machine learning, noise learning based denoising autoencoder, signal
restoration, symbol demodulation, precise localization.

##  I Introduction

Machine learning (ML) has recently received much attention as a key enabler
for future wireless communications [1, 2, 3]. While the major research effort
has been put to deep neural networks, there are enormous number of Internet of
Things (IoT) devices that are severely constrained on the computational power
and memory size. Therefore, the implementation of efficient ML algorithms is
an important challenge for IoT devices, as they are energy and memory limited.
Denoising autoencoder (DAE) is a promising technique to improve the
performance of IoT applications by denoising the observed data that consists
of the original data and the noise [4]. DAE is a neural network model for the
construction of the learned representations robust to an addition of noise to
the input samples [vincent2008extracting, 5]. The representative feature of
DAE is that the dimension of the latent space is smaller than the size of the
input vector. It means that the neural network model is capable of encoding
and decoding through a smaller dimension where the data can be represented.

The main contribution of this letter is to improve the efficiency and
performance of DAE with a modification of its structure. Consider a noisy
observation $Y$ which consists of the original data $X$ and the noise $N$,
i.e., $Y=X+N$. From the information theoretical perspective, DAE attempts to
minimize the expected reconstruction error by maximizing a lower bound on
mutual information $I(X;Y)$. In other words, $Y$ should capture the
information of $X$ as much as possible although $Y$ is a function of the noisy
input. Additionally, from the manifold learning perspective, DAE can be seen
as a way to find a manifold where $Y$ represents the data into a low
dimensional latent space corresponding to $X$. However, we often face the
problem that the stochastic feature of $X$ to be restored is too complex to
regenerate or represent. This is called the curse of dimensionality, i.e., the
dimension of latent space for $X$ is still too high in many cases.

What can we do if $N$ is simpler to regenerate than $X$? It will be more
effective to learn $N$ and subtract it from $Y$ instead of learning $X$
directly. In this light, we propose a new denoising framework, named as noise
learning based DAE (nlDAE). The main advantage of nlDAE is that it can
maximize the efficiency of the ML approach (e.g., the required dimension of
the latent space or size of training dataset) for capability-constrained
devices, e.g., IoT, where $N$ is typically easier to regenerate than $X$ owing
to their stochastic characteristics. To verify the advantage of nlDAE over the
conventional DAE, we provide three practical applications as case studies:
signal restoration, symbol demodulation, and precise localization.

The following notations will be used throughout this letter.

  * •

$\text{Ber},\text{Exp},\mathcal{U},\mathcal{N},\mathcal{CN}$: the Bernoulli,
exponential, uniform, normal, and complex normal distributions, respectively.

  * •

$\mathbf{x},\mathbf{n},\mathbf{y}\in\mathbb{R}^{P}$: the realization vectors
of random variables $X,N,Y$, respectively, whose dimensions are $P$.

  * •

$P^{\prime}(<P)$: the dimension of the latent space.

  * •

$\mathbf{W}\in\mathbb{R}^{P^{\prime}\times
P},\mathbf{W}^{\prime}\in\mathbb{R}^{P\times P^{\prime}}$: the weight matrices
for encoding and decoding, respectively.

  * •

$\mathbf{b}\in\mathbb{R}^{P^{\prime}},\mathbf{b}^{\prime}\in\mathbb{R}^{P}$:
the bias vectors for encoding and decoding, respectively.

  * •

$\mathcal{S}$: the sigmoid function, acting as an activation function for
neural networks, i.e., $\mathcal{S}(a)=\frac{1}{1+e^{-a}}$, and
$\mathcal{S}(\mathbf{a})=(\mathcal{S}(\mathbf{a}[1]),\cdots,\mathcal{S}(\mathbf{a}[P]))^{T}$
where $\mathbf{a}\in\mathbb{R}^{P}$ is an arbitrary input vector.

  * •

$f_{\theta}$: the encoding function where the parameter $\theta$ is
$\\{\mathbf{W},\mathbf{b}\\}$, i.e.,
$f_{\theta}(\mathbf{y})=\mathcal{S}(\mathbf{W}\mathbf{y}+\mathbf{b})$.

  * •

$g_{\theta^{{}^{\prime}}}$: the decoding function where the parameter
$\theta^{{}^{\prime}}$ is $\\{\mathbf{W}^{\prime},\mathbf{b}^{\prime}\\}$,
i.e.,
$g_{\theta^{{}^{\prime}}}(f_{\theta}(\mathbf{y}))=\mathcal{S}(\mathbf{W}^{\prime}f_{\theta}(\mathbf{y})+\mathbf{b}^{\prime})$.

  * •

$M$: the size of training dataset.

  * •

$L$: the size of test dataset.

##  II Method of nlDAE

(a) Training phase

(b) Test phase

Figure 1: An illustration of the concept of nlDAE. Figure 2: A simple example
of comparison between DAE and nlDAE: reconstruction error according to
$\sigma_{N}$.

In the traditional estimation problem of signal processing, $N$ is treated as
an obstacle to the reconstruction of $X$. Therefore, most of the studies have
focused on restoring $X$ as much as possible, which can be expressed as a
function of $X$ and $N$. Along with this philosophy, ML-based denoising
techniques, e.g., DAE, have also been developed in various signal processing
fields with the aim of maximizing the ability to restore $X$ from $Y$. Unlike
the conventional approaches, we hypothesize that, if $N$ has a simpler
statistical characteristic than $X$, it will be better to subtract from $Y$
after restoring $N$.

We first look into the mechanism of DAE to build neural networks. Recall that
DAE attempts to regenerate the original data $\mathbf{x}$ from the noisy
observation $\mathbf{y}$ via training the neural network. Thus, the parameters
of a DAE model can be optimized by minimizing the average reconstruction error
in the training phase as follows:

$\theta^{*},\theta^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\mathbf{x}^{(i)},g_{\theta^{{}^{\prime}}}(f_{\theta}(\mathbf{y}^{(i)}))\big{)},$
(1)

where $\mathcal{L}$ is a loss function such as squared error between two
inputs. Then, the $j$-th regenerated data $\tilde{\mathbf{x}}^{(j)}$ from
${\mathbf{y}}^{(j)}$ in the test phase can be obtained as follows for all
$j\in\\{1,\cdots,L\\}$:

$\tilde{\mathbf{x}}^{(j)}=g_{\theta^{{}^{\prime}*}}(f_{\theta^{*}}(\mathbf{y}^{(j)})).$
(2)

It is noteworthy that, if there are two different neural networks which
attempt to regenerate the original data and the noise from the noisy input,
the linear summation of these two regenerated data would be different from the
input. This means that either $\mathbf{x}$ or $\mathbf{n}$ is more effectively
regenerated from $\mathbf{y}$. Therefore, we can hypothesize that learning
$N$, instead of $X$, from $Y$ can be beneficial in some cases even if the
objective is still to reconstruct $X$. This constitutes the fundamental idea
of nlDAE.

The training and test phases of nlDAE are depicted in Fig. 1. The parameters
of nlDAE model can be optimized as follows for all $i\in\\{1,\cdots,M\\}$:

$\theta_{nl}^{*},{\theta}_{nl}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\mathbf{n}^{(i)},g_{\theta^{\prime}}(f_{\theta}(\mathbf{y}^{(i)}))\big{)}.$
(3)

Notice that the only difference from (1) is that $\mathbf{x}^{(i)}$ is
replaced by $\mathbf{n}^{(i)}$. Let $\tilde{\mathbf{x}}_{nl}^{(j)}$ denote the
$j$-th regenerated data based on nlDAE, which can be represented as follows
for all $j\in\\{1,\cdots,L\\}$:

$\tilde{\mathbf{x}}_{nl}^{(j)}=\mathbf{y}^{(j)}-g_{\theta_{nl}^{{}^{\prime}*}}(f_{\theta_{nl}^{*}}(\mathbf{y}^{(j)})).$
(4)

To provide the readers with insights into nlDAE, we examine two simple
examples where the standard deviation of $X$ is fixed as 1, i.e.,
$\sigma_{X}=1$, and that of $N$ varies. $Y=X+N$ is comprised as follows:

  * •

Example 1: $X\sim\mathcal{U}(0,2\sqrt{3})$ and
$N\sim\mathcal{N}(0,\sigma_{N})$.

  * •

Example 2: $X\sim\text{Exp}(1)$ and $N\sim\mathcal{N}(0,\sigma_{N})$.

Fig. 2 describes the performance comparison between DAE and nlDAE in terms of
mean squared error (MSE) for the two examples111Throughout this letter, the
squared error and the scaled conjugate gradient are applied as the loss
function and the optimization method, respectively.. Here, we set $P=12$,
$P^{\prime}=9$, $M=10000$, and $L=5000$. It is observed that nlDAE is superior
to DAE when $\sigma_{N}$ is smaller than $\sigma_{X}$ in Fig. 2. The gap
between nlDAE and DAE widens with lower $\sigma_{X}$. This implies that the
standard deviation is an important factor when we select the denoiser between
DAE and nlDAE.

These examples show the consideration of whether $X$ or $N$ is easier to be
regenerated, which is highly related to differential entropy of each random
variable, $H(X)$ and $H(N)$ [marsh2013introduction]. The differential entropy
is normally an increasing function over the standard deviation of the
corresponding random variable, e.g., $H(N)=\log(\sigma_{N}\sqrt{2\pi e})$.
Naturally, it is efficient to reconstruct a random variable with a small
amount of information, and the standard deviation can be a good indicator.

##  III Case Studies

To validate the advantage of nlDAE over the conventional DAE in practical
problems, we provide three applications for IoT devices in the following
subsections. We assume that the noise follows Bernoulli and normal
distributions, respectively, in the first two cases, which are the most common
noise modeling. The third case deals with noise that follows a distribution
expressed as a mixture of various random variables. For all the studied use
cases, we select the DAE as the conventional denoiser as a baseline for
performance comparison. We present the case studies in the first three
subsections. Then, we discuss the experimental results in Sec. III-D.

###  III-A Case Study I: Signal Restoration

In this use case, the objective is to recover the original signal from the
noisy signal which is modeled by the corruptions over samples.

####  III-A1 Model

The sampled signal of randomly superposed sinusoids, e.g., the recorded
acoustic wave, is the summation of samples of $k$ damped sinusoidal waves
which can be represented as follows:

$\mathbf{x}=\Big{\\{}\sum_{l=1}^{k}V_{l}e^{-\gamma_{l}n\Delta t}\cos(2\pi
f_{l}n\Delta t)\Big{\\}}_{n=0}^{P-1},$ (5)

where $V_{l}$, $\gamma_{l}$, and $f_{l}$ are the peak amplitude, the damping
factor, and the frequency of the $l$-th signal, respectively. Here, the time
interval for sampling, $\Delta t$, is set to satisfy the Nyquist theorem,
i.e., $\frac{1}{2\Delta t}>\max\\{f_{1},\cdots,f_{k}\\}$. To consider the
corruption of $\mathbf{x}$, let us assume that the probability of corruption
for each sample follows the Bernoulli distribution $\text{Ber}(p_{cor})$,
which indicates the corruption with the probability $p_{cor}$. In addition,
let $\mathbf{b}\in\\{0,1\\}^{P}$ denote the realization of
$\text{Ber}(p_{cor})$ over $P$ samples. Naturally, the corrupted signal,
$\mathbf{y}\in\mathbb{R}^{P}$, can be represented as follows:

$\mathbf{y}=\mathbf{x}+C\mathbf{b},$ (6)

where $C$ is a constant representing the sample corruption.

####  III-A2 Application of nlDAE

Based on (6), the denoised signal $\tilde{\mathbf{x}}_{nl}^{(j)}$ can be
represented by

$\tilde{\mathbf{x}}_{nl}^{(j)}=\mathbf{x}^{(j)}+C\mathbf{b}^{(j)}-g_{\theta_{nl}^{{}^{\prime}*}}(f_{\theta_{nl}^{*}}(\mathbf{x}^{(j)}+C\mathbf{b}^{(j)}),$
(7)

where

$\theta_{nl}^{*},{\theta}_{nl}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}C\mathbf{b}^{(i)},g_{\theta^{\prime}}(f_{\theta}(\mathbf{x}^{(i)}+C\mathbf{b}^{(i)}))\big{)}.$

####  III-A3 Experimental Parameters

We evaluate the performance of the proposed nlDAE in terms of the MSE of
restoration. For the experiment, the magnitude of noise $C$ is set to 1 for
simplicity. In addition, $V_{l}$, $\gamma_{l}$, and $f_{l}$ follow
$\mathcal{N}(0,1)$, $\mathcal{U}(0,10^{3})$, and $\mathcal{U}(0,10\text{
kHz})$, respectively, for all $l$. The sampling time interval $\Delta t$ is
set to $0.5\times 10^{-4}$ second, and the number of samples $P$ is $12$. We
set $P^{\prime}=9$, $p_{cor}=0.9$, and $M=10000$ unless otherwise specified.

###  III-B Case Study II: Symbol Demodulation

Here, the objective is to improve the symbol demodulation quality through
denoising the received signal that consists of channel, symbols, and additive
noise.

####  III-B1 Model

Consider an orthogonal frequency-division multiplexing (OFDM) system with $P$
subcarriers where the subcarrier spacing is expressed by $\Delta f$. Let
$\mathbf{d}\in\mathbb{C}^{P}$ be a sequence in frequency domain.
$\mathbf{d}[n]$ is the $n$-th element of $\mathbf{d}$ and denotes the symbol
transmitted over the $n$-th subcarrier. In addition, let $K$ denote the pilot
spacing for channel estimation. Furthermore, the channel impulse response
(CIR) can be modeled by the sum of Dirac-delta functions as follows:

$h(t,\tau)=\sum_{l=0}^{L_{p}-1}\alpha_{l}\delta(t-\tau_{l}),$ (8)

where $\alpha_{l}$, $\tau_{l}$, and $L_{p}$ are the complex channel gain, the
excess delay of $l$-th path, and the number of multipaths, respectively. Let
$\mathbf{x}\in\mathbb{C}^{P}$ denote the discrete signal obtained by $P$-point
fast Fourier transform (FFT) after the sampling of the signal experiencing the
channel at the receiver, which can be represented as follows:

${}\mathbf{x}=\mathbf{d}\odot\mathbf{h}=\\{\mathbf{d}[n]\sum_{l=0}^{L_{p}-1}\alpha_{l}e^{-j2\pi
n\Delta f\tau_{l}}\\}_{n=0}^{P-1},$ (9)

where $\odot$ denotes the operator of the Hadamard product. Here,
$\mathbf{h}\in\mathbb{C}^{P}$ is the channel frequency response (CFR), which
is the $P$-point FFT of $h(t,\tau)$. In addition, let
$\mathbf{n}\in\mathbb{C}^{P}$ denote the realization of the random variable
$N\sim\mathcal{CN}(0,\sigma_{N})$. Finally,
$\mathbf{y}(=\mathbf{d}\odot\mathbf{h}+\mathbf{n})$ is the noisy observed
signal.

Our goal is to minimize the symbol error rate (SER) over $\mathbf{d}$ by
maximizing the quality of denoising $\mathbf{y}$. We assume the method of
channel estimation is fixed as the cubic interpolation [6] to focus on the
performance of denoising the received signal.

(a)

(b)

(c)

(d)

Figure 3: Case study I (signal restoration): MSE according to (a) the
dimension of latent space; (b) the size of training dataset; (c) $p_{cor}$;
and (d) the depth of neural networks.

(a)

(b)

(c)

(d)

Figure 4: Case study II (symbol demodulation): SER according to (a) the
dimension of latent space; (b) the size of training dataset; (c) SNR; and (d)
the depth of neural networks.

(a)

(b)

(c)

(d)

Figure 5: Case study III (precise localization): Localization error according
to (a) the dimension of latent space; (b) the size of training dataset; (c)
$p_{NLoS}$; and (d) the depth of neural networks.

####  III-B2 Application of nlDAE

To consider the complex-valued data, we separate it into real and imaginary
parts. $\Re$ and $\Im$ denote the operators capturing real and imaginary parts
of an input, respectively. Thus, $\tilde{\mathbf{x}}_{nl}^{(j)}$ is the
regenerated $\mathbf{d}^{(j)}\odot\mathbf{h}^{(j)}$ by denoising
$\mathbf{y}^{(j)}$, which can be represented by

$\begin{split}\tilde{\mathbf{x}}_{nl}^{(j)}&=\Re(\mathbf{y}^{(j)})-g_{\theta_{nl,R}^{{}^{\prime}*}}(f_{\theta_{nl,R}^{*}}(\Re(\mathbf{y}^{(j)})))\\\
&+i(\Im(\mathbf{y}^{(j)})-g_{\theta_{nl,I}^{{}^{\prime}*}}(f_{\theta_{nl,I}^{*}}(\Im(\mathbf{y}^{(j)})))),\end{split}$
(10)

where

$\begin{split}\theta_{nl,{R}}^{*},{\theta}_{nl,{R}}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\Re(\mathbf{n}^{(i)}),g_{\theta^{\prime}}(f_{\theta}(\Re(\mathbf{y}^{(i)})))\big{)},\\\
\theta_{nl,{I}}^{*},{\theta}_{nl,{I}}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\Im(\mathbf{n}^{(i)}),g_{\theta^{\prime}}(f_{\theta}(\Im(\mathbf{y}^{(i)})))\big{)}.\end{split}$

Finally, the receiver estimates $\mathbf{h}$ with the predetermined pilot
symbols, i.e., $\mathbf{d}[nK+1]$ where $n=0,1,\cdots$, and demodulates
$\mathbf{d}$ based on the estimate of $\mathbf{h}$ and the regenerated
$\tilde{\mathbf{x}}_{nl}$.

####  III-B3 Experimental Parameters

The performance of the proposed nlDAE is evaluated when $L=5000$. For the
simulation parameters, we set $4$ QAM, $P=12$, $\Delta f=15$ kHz, $L_{p}=4$,
and $K=3$. We further assume that $\alpha\sim\mathcal{CN}(0,1)$ and
$\tau\sim\mathcal{U}(0,10^{-6})$. Furthermore, $P^{\prime}=9$, SNR$=5$ dB, and
$M=10000$ unless otherwise specified. We also provide the result of non-ML
(i.e., only cubic interpolation).

###  III-C Case Study III: Precise Localization

The objective of this case study is to improve the localization quality
through denoising the measured distance which is represented by the quantized
value of the mixture of the true distance and error factors.

####  III-C1 Model

Consider a 2-D localization where $P$ reference nodes and a single target node
are randomly distributed. We estimate the position of the target node with the
knowledge of the locations of $P$ reference nodes. Let
$\mathbf{x}\in\mathbb{R}^{P}$ denote the vector of true distances from $P$
reference nodes to the target node when $X$ denotes the distance between two
random points in a 2-D space. We consider three types of random variables for
the noise added to the true distance as follows:

  * •

$N_{N}$: ranging error dependent on signal quality.

  * •

$N_{U}$: ranging error due to clock asynchronization.

  * •

$N_{B}$: non line-of-sight (NLoS) event.

We assume that $N_{N}$, $N_{U}$, $N_{B}$ follow the normal, uniform, and
Bernoulli distributions, respectively. Hence, we can define the random
variable for the noise $N$ as follows:

$N=N_{N}+N_{U}+R_{NLoS}N_{B},$ (11)

where $R_{NLoS}$ is the distance bias in the event of NLoS. Note that $N$ does
not follow any known probability distribution because it is a convolution of
three different distributions. Besides, we assume that the distance is
measured by time of arrival (ToA). Thus, we define the quantization function
$\mathcal{Q}_{B}$ to represent the measured distance with the resolution of
$B$, e.g., $\mathcal{Q}_{10}(23)=20$. In addition, the localization method
based on multi-dimensional scaling (MDS) is utilized to estimate the position
of the target node [7].

####  III-C2 Application of nlDAE

In this case study, we consider the discrete values quantized by the function
$\mathcal{Q}_{B}$. Here, $\tilde{\mathbf{x}}_{nl}^{(j)}$ can be represented as
follows:

$\tilde{\mathbf{x}}_{nl}^{(j)}={\mathcal{Q}_{B}(\mathbf{y}}^{(j)})-g_{\theta_{nl,R}^{{}^{\prime}*}}(f_{\theta_{nl,R}^{*}}({\mathcal{Q}_{B}(\mathbf{y}}^{(j)}))),$
(12)

where

$\theta_{nl,{R}}^{*},{\theta}_{nl,{R}}^{{}^{\prime}*}=\operatorname*{arg\,min}_{\theta,\theta^{{}^{\prime}}}\frac{1}{M}\sum_{i=1}^{M}\mathcal{L}\big{(}\mathcal{Q}_{B}(\mathbf{n}^{(i)}),g_{\theta^{\prime}}(f_{\theta}(\mathcal{Q}_{B}(\mathbf{y}^{(i)})))\big{)}.$

Thus, $\tilde{\mathbf{x}}_{nl}$ is utilized for the estimation of the target
node position in nlDAE-assisted MDS-based localization.

####  III-C3 Experimental Parameters

The performance of the proposed nlDAE is evaluated via $L=5000$. In this
simulation, $12$ reference nodes and one target node are uniformly distributed
in a $100\times 100$ square. We assume that
$N_{N}\sim\mathcal{N}(0,10),N_{U}\sim\mathcal{U}(0,20),N_{B}\sim\text{Ber}(0.2)$,
and $R_{NLoS}=50$. The distance resolution $B$ is set to $10$ for the
quantization function $\mathcal{Q}_{B}$. Note that $P^{\prime}=9$,
$p_{NLoS}=0.2$, and $M=10000$ unless otherwise specified. We also provide the
result of non-ML (i.e., only MDS based localization).

###  III-D Analysis of Experimental Results

Fig. 3(a), Fig. 4(a), and Fig. 5(a) show the performance of the three case
studies with respect to $P^{\prime}$, respectively. nlDAE outperforms non-ML
and DAE for all ranges of $P^{\prime}$. Particularly with small values of
$P^{\prime}$, nlDAE continues to perform well, whereas DAE loses its merit.
This means that nlDAE provides a good denoising performance even with an
extremely small dimension of latent space if the training dataset is
sufficient.

The impact of the size of training dataset is depicted in Fig. 3(b), Fig.
4(b), and Fig. 5(b). nlDAE starts to outperform non-ML with $M$ less than 100.
Conversely, DAE requires about an order higher $M$ to perform better than non-
ML. Furthermore, nlDAE converges faster than DAE, thus requiring less training
data than DAE.

In Fig. 3(c), Fig. 4(c), and Fig. 5(c), the impact of a noise-related
parameter for each case study is illustrated. When the noise occurs according
to a Bernoulli distribution in Fig. 3(c), the performance of ML algorithms
(both nlDAE and DAE) exhibits a concave behavior. This is because the variance
of $\text{Ber}(p)$ is given by $p(1-p)$. Similar phenomenon is observed in
Fig. 5(c) because the Bernoulli event of NLoS constitutes a part of
localization noise. As for non-ML, the performance worsens as the probability
of noise occurrence increases in both cases. Fig. 4(c) shows that the SER
performance of nlDAE improves rapidly as the SNR increases. In all
experiments, nlDAE gives superior performance than other schemes.

Thus far, the experiments have been conducted with a single hidden layer. Fig.
3(d), Fig. 4(d), and Fig. 5(d) show the effect of the depth of the neural
network. The performance of nlDAE is almost invariant, which suggests that
nlDAE is not sensitive to the number of hidden layers. On the other hand, the
performance of DAE worsens quickly as the depth increases owing to overfitting
in two cases.

In summary, nlDAE outperforms DAE over the whole experiments. nlDAE is
observed to be more efficient for the underlying use cases than DAE because it
requires smaller latent space and less training data. Furthermore, nlDAE is
more robust to the change of the parameters related to the design of the
neural network, e.g., the network depth.

##  IV Conclusion and Future Work

We introduced a new denoiser framework based on the neural network, namely
nlDAE. This is a modification of DAE in that it learns the noise instead of
the original data. The fundamental idea of nlDAE is that learning noise can
provide a better performance depending on the stochastic characteristics
(e.g., standard deviation) of the original data and noise. We applied the
proposed mechanism to the practical problems for IoT devices such as signal
restoration, symbol demodulation, and precise localization. The numerical
results support that nlDAE is more efficient than DAE in terms of the required
dimension of the latent space and the size of training dataset, thus rendering
it more suitable for capability-constrained conditions. Applicability of nlDAE
to other domains, e.g., image inpainting, remains as a future work.
Furthermore, information theoretical criteria of decision making for the
selection between or a combination of DAE and nlDAE is an interesting further
research.

## References

  * [1] U. Challita, H. Ryden, and H. Tullberg, “When machine learning meets wireless cellular networks: Deployment, challenges, and applications,” _IEEE Communications Magazine_ , vol. 58, no. 6, pp. 12–18, 2020.
  * [2] M. Chen, U. Challita, W. Saad, C. Yin, and M. Debbah, “Artificial neural networks-based machine learning for wireless networks: A tutorial,” _IEEE Communications Surveys & Tutorials_, vol. 21, no. 4, pp. 3039–3071, 2019.
  * [3] A. Azari, M. Ozger, and C. Cavdar, “Risk-aware resource allocation for URLLC: Challenges and strategies with machine learning,” _IEEE Communications Magazine_ , vol. 57, no. 3, pp. 42–48, 2019.
  * [4] Y. Sun, M. Peng, Y. Zhou, Y. Huang, and S. Mao, “Application of machine learning in wireless networks: Key techniques and open issues,” _IEEE Communications Surveys & Tutorials_, vol. 21, no. 4, pp. 3072–3108, 2019.
  * [5] Y. Bengio, L. Yao, G. Alain, and P. Vincent, “Generalized denoising auto-encoders as generative models,” _Advances in neural information processing systems_ , pp. 899–907, 2013.
  * [6] S. Coleri, M. Ergen, A. Puri, and A. Bahai, “Channel estimation techniques based on pilot arrangement in OFDM systems,” _IEEE Transactions on broadcasting_ , vol. 48, no. 3, pp. 223–229, 2002.
  * [7] I. Dokmanic, R. Parhizkar, J. Ranieri, and M. Vetterli, “Euclidean distance matrices: Essential theory, algorithms, and applications,” _IEEE Signal Processing Magazine_ , vol. 32, no. 6, pp. 12–30, 2015.