Chemo-dynamical Evolution of Simulated Satellites for a Milky Way-like Galaxy

0000-0002-5661-033X]Yutaka Hirai
JSPS Research Fellow
Department of Physics and Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA
Astronomical Institute, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan
Joint Institute for Nuclear Astrophysics, Center for the Evolution of the Elements (JINA-CEE), USA

0000-0001-6196-5162]Evan N. Kirby
Department of Physics and Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA

0000-0002-9053-860X]Masashi Chiba
Astronomical Institute, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan

0000-0002-8758-8139]Kohei Hayashi
National Institute of Technology, Sendai College, 48 Nodayama, Medeshima-Shiote, Natori, Miyagi 981-1239, Japan
Astronomical Institute, Tohoku University, 6-3 Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8578, Japan
Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa, Chiba 277-8582, Japan

0000-0001-5261-4336]Borja Anguiano
Department of Physics and Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA

0000-0001-8226-4592]Takayuki R. Saitoh
Department of Planetology, Graduate School of Science, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe, Hyogo 657-8501, Japan

0000-0003-4656-0241]Miho N. Ishigaki
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa, Chiba 277-8582, Japan

0000-0003-4573-6233]Timothy C. Beers
Department of Physics and Astronomy, University of Notre Dame,
225 Nieuwland Science Hall, Notre Dame, IN 46556, USA
Joint Institute for Nuclear Astrophysics, Center for the Evolution of the Elements (JINA-CEE), USA

The chemical abundances of Milky Way's satellites reflect their star formation histories (SFHs), yet, due to the difficulty of determining the ages of old stars, the SFHs of most satellites are poorly measured. Ongoing and upcoming surveys will obtain around ten times more medium-resolution spectra for stars in satellites than are currently available. To correctly extract SFHs from large samples of chemical abundances, the relationship between chemical abundances and SFHs needs to be clarified. Here, we perform a high-resolution cosmological zoom-in simulation of a Milky Way-like galaxy with detailed models of star formation, supernova feedback, and metal diffusion. We quantify SFHs, metallicity distribution functions, and the $\alpha$-element (Mg, Ca, and Si) abundances in satellites of the host galaxy. We find that star formation in most simulated satellites is quenched before infalling to their host. Star formation episodes in simulated satellites are separated by a few hundred Myr owing to supernova feedback; each star formation event produces groups of stars with similar [$\alpha$/Fe] and [Fe/H]. We then perform a mock observation of the upcoming Subaru Prime Focus Spectrograph (PFS) observations. We find that Subaru PFS will be able to detect distinct groups of stars in [$\alpha$/Fe] vs. [Fe/H] space, produced by episodic star formation. This result means that episodic SFHs can be estimated from the chemical abundances of $\gtrsim$ 1,000 stars determined with medium-resolution spectroscopy.

§ INTRODUCTION

Satellite galaxies of the Milky Way (MW) are crucial for understanding galaxy formation <cit.>. Many satellites possess ancient stars; the histories of star formation and galaxy assembly are imprinted in the chemo-dynamical properties of such satellites. In our Galaxy, thanks to their relatively close distances from the Sun ($\approx$ 20–200 kpc), we can observe the chemo-dynamical properties of individual stars in the MW's satellites <cit.>.

Over 100 dwarf galaxies are identified within 3 Mpc <cit.>. Galaxies with stellar masses ($M_{*}$) less than $\approx10^9\,M_{\sun}$ are typically categorized as dwarf galaxies. Among them, gas-free dwarf galaxies with $M_{*} \gtrsim10^5\,M_{\sun}$ are called classical dwarf spheroidal galaxies (dSphs), while those with $M_{*} \lesssim10^5\,M_{\sun}$ are identified as ultrafaint dwarf galaxies (UFDs). Many of the dwarf galaxies in the Local Group are satellites of the MW or M31;  interactions with their more massive hosts could affect the chemo-dynamical properties of these satellites [Genina et al., 2019, Kvasova et al., 2024].

Satellites exhibit a wide variety of star formation histories (SFHs) and chemical abundances. The SFHs of Local Group dwarf galaxies can be derived by color-magnitude diagrams <cit.>. Weisz et al., 2014 comprehensively studied SFHs in the Local Group dwarf galaxies. They found that more massive systems tend to have more extended SFHs. They also showed that MW or M31 satellites have a shorter duration of star formation than those in the field populations.

Chemical abundances reflect the SFHs and nucleosynthesis pathways in satellites <cit.>. Kirby et al., 2011 analyzed metallicity distribution functions (MDFs) of the MW's satellites with Keck/DEIMOS with a chemical evolution model. They found that the MDFs of
more-luminous systems are well-fit with their Extra Gas Model, which assumes gas infall. However, their best-fit effective yields suggested that gas outflow also played an important role in the chemical evolution of less-luminous systems.

Thanks to the difference in the delay times between core-collapse supernovae (CCSNe) and type Ia supernovae (SNe Ia), the ratios of $\alpha$-elements (e.g., Mg, Ca, and Si) to Fe are often used as an indicator for the rate of chemical evolution. For example, Hill et al., 2019 reported high-resolution spectroscopy of 99 stars in the Sculptor dSph. They found that the decreasing trend of [$\alpha$/Fe][[X/Y] = =$\log(N_{\rm{X}}/N_{\rm{Y}})-\log(N_{\rm{X}}/N_{\rm{Y}})_{_\sun}$, where $N_{\rm{X}}$ and $N_{\rm{Y}}$ are the number densities of elements X and Y, respectively.] toward higher metallicity starts at [Fe/H] = $-1.8$. This metallicity is lower than the start of this trend in the MW, indicating that the chemical evolution of Sculptor dSph proceeded more slowly.

Numerical simulations have been performed to understand the SFHs and chemical evolution of dwarf galaxies <cit.>. Di Cintio et al., 2021 found that 25% of their simulated satellite dwarf galaxies exhibit an enhancement of star formation after infall to their host. In contrast, the star formation in satellites with little gas or small pericentric distances is quenched after infall due to ram pressure stripping. Escala et al., 2018 introduced the process of metal diffusion in cosmological zoom-in simulations of the Feedback in Realistic Environment (FIRE) project [Hopkins et al., 2014], and analyzed chemical abundances in their simulated dwarf galaxies. They found that the MDFs and intrinsic scatter in [$\alpha$/Fe] are similar in satellite and isolated dwarf galaxies, suggesting that internal chemical evolution plays a more important role than environmental effects.

Ongoing and upcoming surveys will significantly enlarge the number of stars in satellites of the MW with available spectroscopy <cit.>. For example, the Dark Energy Spectroscopic Instrument (DESI) Milky Way Survey will observe 7 million stars with magnitudes 16 $<r<$ 20 at Galactic latitudes $\abs{b}$ $>20^\circ$ [Cooper et al., 2023]. Their footprint includes 31 Local Group dwarf galaxies. This potentially could yield medium-resolution ($R \sim 5,000$) spectroscopy of the member stars in some of these galaxies from their centers to their outskirts.

The upcoming Subaru Prime Focus Spectrograph (PFS) will target 7 Local Group dwarf galaxies in their Galactic Archaeology survey [Takada et al., 2014]. Thanks to their wide field of view (1.25 square degrees) and massively multiplexed spectroscopic capability (2,394 fibers), they can obtain medium-resolution ($R \sim 5,000$) spectroscopy for stars with magnitudes $g\lesssim$ 23 in these galaxies. The Subaru PFS will yield radial velocities, [Fe/H], carbon, $\alpha$-elements, and nickel abundance measurements in each galaxy for $\approx$ 1,000 to 14,000 stars, more than ten times larger than the current numbers of stars with these measurements. Comparison with cosmological zoom-in simulations and these observations will greatly advance our understanding of the chemo-dynamical properties of dwarf galaxies.

This study aims to understand the relationship between star formation and chemical evolution in satellite galaxies. With our high-resolution cosmological zoom-in simulation of a MW-like galaxy, we examine SFHs, MDFs, and $\alpha$-element abundances in satellites with $M_{*}$ $\sim$ $10^5$–10$^7\,M_{\sun}$, corresponding to the mass ranges of satellite dSphs of the MW. We show how SFHs are reflected in MDFs and $\alpha$-element abundances using our simulation. We then evaluate the capability of upcoming surveys to reconstruct the SFHs from the chemical abundances of dwarf galaxies.

This paper is organized as follows. Section <ref> describes our code, the adopted initial conditions, and the procedures used for carrying out mock observations. In Section <ref>, we describe the chemo-dynamical properties of our simulated satellites. Section <ref> discusses how SFHs are reflected in chemical abundances, and how these can be observed in future surveys. Our conclusions are presented in Section <ref>.

§ METHODS

§.§ Code

We have computed the evolution of satellite galaxies in a cosmological zoom-in simulation of a MW-like galaxy performed by Hirai et al., 2022. In this simulation, we adopted the $N$-body/density-independent smoothed particle hydrodynamics code asura [Saitoh et al., 2008, Saitoh et al., 2009, Saitoh & Makino, 2013, Saitoh & Makino, 2016]. For cooling and heating calculations, we adopted cloudy ver. 13.05 [Ferland et al., 2013]. Gas particles probabilistically form stars if they are in a region with a number density of hydrogen atoms higher than 100 cm$^{-3}$, the temperature is lower than 1,000 K, and there are converging flows <cit.>.

Each star particle is treated as a simple stellar population (SSP) with the initial mass function (IMF) of Chabrier, 2003 from 0.1 $M_{\sun}$ to 100 $M_{\sun}$. Star particles with ages less than 10 Myr heat the surrounding gas to $10^4$ K [Fujii et al., 2021]. We implemented momentum-based supernova feedback following Hopkins et al., 2018. Metal diffusion was incorporated following Hirai & Saitoh, 2017. The cosmic ultra-violet (UV) heating was implemented following Haardt & Madau, 2012. The reionization is assumed to occur at a redshift ($z$) of 8.5. We also assumed the self-shielding model of Rahmati et al., 2013.

We adopted the nucleosynthetic yields compiled in the Chemical Evolution Library <cit.>. CCSNe and SNe Ia are the dominant contributors to the evolution of the [$\alpha$/Fe] ratios. For CCSNe, we used the yields of Nomoto et al., 2013 with 13 $M_{\sun}$ to 40 $M_{\sun}$. Given the mass of the star particle, we integrated the IMF from the maximum stellar mass of the IMF to the lower stellar mass until the cumulative number of stars in the integration range became unity. This approach enabled the tracking of the contribution from CCSNe with different progenitor masses in sufficiently high-resolution simulations. When the stellar particle mass ($m_{*}$) was 4.5$\,\times\,10^3\,$$M_{\sun}$, the IMF for CCSNe (13–40 $M_{\sun}$) was divided into 100 bins. For SNe Ia, we assumed a delay-time distribution with a power-law index of $-1$, and a minimum delay time of 40 Myr, following Maoz et al., 2012. We also included the contribution of asymptotic giant branch (AGB) stars for stars with 1 to 8 $M_{\sun}$ [Karakas, 2010, Doherty et al., 2014]. We adopted the solar abundance of Asplund et al., 2009. Throughout this paper, we define [$\alpha$/Fe] as the weighted average of Mg, Si, and Ca ([$\alpha$/Fe] $=$ 0.35 [Mg/Fe] $+$ 0.25 [Si/Fe] $+$ 0.40 [Ca/Fe]) to mimic [$\alpha$/Fe] measured in Keck/DEIMOS [Kirby et al., 2008, Kirby et al., 2011].

§.§ Initial Conditions

A MW-like halo was selected from the cosmological simulation with a box size of (36 $h^{-1}$ Mpc)$^3$. We adopted cosmological parameters of $\Omega_{\rm{m}}\,=\,0.308$, $\Omega_{\Lambda}\,=\,0.692$, $\Omega_{\rm{b}}\,=\,0.0484$, and $H_{0}\,=\,67.8$ km s$^{-1}$$\>$Mpc$^{-1}$ [Planck Collaboration et al., 2016]. An initial condition for the zoom-in simulation was generated by music [Hahn & Abel, 2011]. We used the Amiga Halo Finder <cit.> to find the target halo. In this simulation, the initial masses of each particle in the finest region were $7.2\times10^4\,$$M_{\sun}$ for dark matter, $1.3\times10^4\,$$M_{\sun}$ for gas, and $4.5\times10^3\,$$M_{\sun}$ for stars. We set the gravitational softening length ($\epsilon_{\rm{g}}$) to 85 pc for dark matter and 82 pc for gas and stars. We performed the simulation from $z$ = 100 to 0.

In this simulation, we picked out satellites orbiting the central galaxy. We only considered those with a minimum of 10$^4$ dark matter and 10 star particles, and made sure that they were not false substructures introduced by the contamination from low-resolution particles. Table <ref> lists the simulated satellite galaxies selected for this study.

List of Simulated Satellite Galaxies at $z$ = 0.

Halo ID     $M_{\rm{halo}}$     $M_*$     $\langle {\rm [Fe/H]} \rangle$     $\sigma_{\rm{[Fe/H]}}$     $d$

($M_{\sun}$)   ($M_{\sun}$)      (dex)   (kpc)
9     $7.5\times10^9$     $7.5\times10^6$     $-1.95$     0.23     204.2

12     $4.7\times10^9$     $2.1\times10^7$      $-1.08$     0.58     148.8

36     $2.2\times10^9$     $1.1\times10^7$      $-1.43$     0.37     54.5

38     $2.5\times10^9$     $1.3\times10^5$      $-1.52$     0.52     198.7

40     $2.3\times10^9$     $3.9\times10^6$      $-1.52$     0.46     57.9

150     $5.9\times10^8$     $2.4\times10^5$      $-2.53$     0.24     190.7

151     $6.0\times10^8$     $7.7\times10^4$      $-2.89$     0.43     167.2

167     $5.2\times10^8$     $3.1\times10^4$      $-4.34$     0.14     206.6

199     $4.2\times10^8$     $2.8\times10^4$      $-3.42$     0.28     169.2

From left to right, the columns are the Halo ID, the total halo mass within the virial radius ($M_{\rm{halo}}$), the total stellar mass ($M_*$), the mean [Fe/H] ($\langle {\rm [Fe/H]} \rangle$), the dispersion of [Fe/H] ($\sigma_{\rm{[Fe/H]}}$), and the distance from the center of the central galaxy ($d$). $M_*$, $\langle {\rm [Fe/H]} \rangle$, and $\sigma_{\rm{[Fe/H]}}$ are computed within the half-mass radius.

§.§ Mock Observations

We performed mock observations for Subaru PFS (see Section <ref>)[Sanderson] et al., 2020 also discussed in detail mock observations of galaxy simulations.. For the mock observation, we computed the magnitudes of simulated stars. First, SSP particles were divided into individual stars. In this model, stars from 0.1 $M_{\sun}$ to 100 $M_{\sun}$ were probabilistically generated from SSP particles, following a Chabrier, 2003 IMF. Stars were generated until the total generated stellar mass exceeded the particle's mass. Then, the magnitudes of each star were computed using the isochrone table obtained from cmd 3.7[<http://stev.oapd.inaf.it/cgi-bin/cmd>] <cit.>.

We generated isochrones with ages from 4 Gyr to 13.8 Gyr and [M/H][[M/H] = $\log(Z/X)-\log(Z/X)_{\sun}$, where $X$ and $Z$ are the mass fractions of hydrogen and metals, respectively.] from $-2.0$ to 0.0 based on the PARSEC-COLIBRI stellar-evolutionary tracks [Bressan et al., 2012, Chen et al., 2014, Chen et al., 2015, Tang et al., 2014, Marigo et al., 2017, Pastorelli et al., 2019, Pastorelli et al., 2020]. With this tool, we computed apparent $V$-band magnitudes for stars in Halos 12 and 40. We assume Halos 12 and 40 are located at 147 kpc and 86 kpc from an observer to compare with the Fornax and Sculptor dSphs, respectively [McConnachie, 2012].

We then applied the Subaru PFS spectral synthesis pipeline <cit.> to compute observed uncertainties. The pipeline adopts synthetic spectra of K-giants and G-dwarfs for $-4.0 \leq$ [Fe/H] $\leq -0.5$. We calculated wavelength-dependent continuum signal-to-noise ratios with the Subaru PFS Exposure Time Calculator[<https://github.com/Subaru-PFS/spt_ExposureTimeCalculator>] using the simulated stars’ $V$-band magnitudes, assuming a three-hour exposure in the Subaru PFS median-resolution mode for K giants. Then, we computed uncertainties on [Fe/H] and [$\alpha$/Fe] by resampling the synthetic spectra hundreds of times from Gaussian-distributed per-pixel noise inversely proportional to the estimated signal-to-noise ratios.  The simulated chemical abundances of stars are varied within those estimated uncertainties.

§ RESULTS

§.§ Structures and Star Formation Histories

This paper mainly discusses the chemo-dynamical evolution of Halos 12, 40, and 150, listed in Table <ref>. The [$\alpha$/Fe] as a function of [Fe/H] for Halos 9 and 36 are shown in the Appendix. We select three these simulated dwarf galaxies based on their stellar mass (Halo 12: $2.1\times10^7M_{\sun}$, Halo 40: $3.9\times10^6M_{\sun}$, and Halo 150: $2.4\times10^5M_{\sun}$). These values are similar to those of the Fornax ($2.0\times10^7 M_{\sun}$), Sculptor ($2.3\times10^6 M_{\sun}$), and Draco ($2.9\times10^5 M_{\sun}$) dSphs [McConnachie, 2012]. Also, Halos 12, 40, and 150 currently contain no gas.

Figure <ref> shows the stellar mass distribution of Halos 12, 40, and 150 at $z$ = 0. The half-mass (light) radii of these galaxies are 1,334 pc (Halo 12), 874 pc (Halo 40), and 1,346 pc (Halo 150), respectively. The somewhat larger radii than the observed ones <cit.> are due to the spatial resolution of this simulation ($\epsilon_{\rm{g}}=85$ pc).

Stellar distribution of simulated satellite dwarf galaxies  for (a) Halo 12, (b) Halo 40, and (c) Halo 150. The color scale depicts each grid's log scale stellar-mass fraction. The stellar mass fraction is defined as $M_{*,\rm{grid}}/M_{*,\rm{tot}}$, where $M_{*,\rm{grid}}$ and $M_{*,\rm{tot}}$ are the stellar mass contained in each grid and total stellar mass of the galaxy, respectively. The grid size is 0.08 $\times$ 0.08 kpc$^2$. Most stars are spherically distributed at the center of their dark matter halo.

The simulated satellite dwarf galaxies exhibit various SFHs. Figure <ref> shows the cumulative SFHs of all satellites listed in Table <ref>. The SFHs of satellite galaxies are affected by SN feedback, cosmic reionization, and interactions with the host galaxy. This figure shows that more massive satellites tend to have extended SFHs, while less massive halos quench star formation earlier. Star formation in halos with $<10^9M_{\sun}$ (150, 151, 167, and 199) is quenched at $<$ 2 Gyr from the beginning of the simulation by cosmic reionization and SN feedback, while halos with $\geq\,10^9M_{\sun}$ form stars after the reionization epoch. Gas accreted before reionization in halos with $\geq\,10^9M_{\sun}$ self-shield the UV background, resulting in them surviving the reionization <cit.>. Hereafter, we focus on three satellites: Halos 12, 40, and 150.

Cumulative SFHs of simulated dwarf satellites, as listed in Table <ref>. Less massive halos (e.g., Halos 151, 167, and 199) tend to quench star formation earlier than more massive halos (e.g., Halos 9, 12, and 36). The cumulative SFH is defined as $M_{*}(t)/M_{*}(T)$, where $M_{*}(t)$ and $M_{*}(T)$ are the stellar masses at the cosmic time $t$ and the present ($z$ = 0), respectively.

The mass and the cosmic infall time also affect the SFHs. Figure <ref> shows the orbits (top panels), mass evolution (middle panels), and SFHs (bottom panels) of Halos (a) 12, (b) 40, and (c) 150. Halo 12 has the most recent infall time. The first pericentric passage (5 kpc) of this galaxy is 0.7 Gyr prior to the end of the simulation (Figure <ref> (a), top panel). Prior to pericentric passage, this galaxy experienced two star formation events separated by 2.9 Gyr (Figure <ref> (a), bottom panel). The first period of star formation starts at 0.1 Gyr and ends at 3.3 Gyr from the beginning of the simulation. During this period, stars are formed along with the accretion of material (Figure <ref> (a), middle panel). After SNe expel the gas away from the halo, the infall of the gas forms new stars. This interplay episodically forms stars for 3.2 Gyr.

The second star formation event begins when the accretion of a halo brings additional material to the halo at 6.2 Gyr. As with the first period of star formation, it is regulated by SN feedback. The star formation is quenched when feedback from CCSNe from the recent star formation ($t\lesssim$ 10 Myr ago) and SNe Ia from previous star formation ($t$ $\sim$ 1 Gyr ago) expel the gas from the galaxy at 9.5 Gyr.

Halo 40 has a shorter total duration of star formation, mainly due to the earlier infall time than that of Halo 12. Halo 40 crosses the main halo's virial radius ($R_{\rm{vir}}$) at 7.4 Gyr, while Halo 12 experiences its closest pericenter passage at 12.6 Gyr. Due to the early infall, repeated gas removal by ram pressure stripping prevents additional star formation in the later phase. The evolution of gas mass after the first infall is due to our analysis method. The increase in the tidal radius of the halo around the apocenter accretes more diffuse gas around the galaxy, resulting in the increase of the detected gas mass of this halo. Although gas mass evolution is shown here, these gas particles are not eligible to form stars.

Halo 40 experienced star formation in the first 2.8 Gyr. As shown in the bottom panel of Figure <ref> (b), there are five peaks of star formation, separated from 0.40 to 0.97 Gyr. The SFH in this halo is also mainly regulated by SN feedback. As shown in the bottom panel of Figure <ref> (b), stars are formed during cosmic reionization. After the star formation is quenched at 0.83 Gyr, an additional gas supply resumes star formation at 1.79 Gyr. Eventually, star formation is halted at 2.76 Gyr. This quenching is mainly caused by the heating by CCSNe from the recent star formation and SNe Ia from the previous star formation, due to their delay times. Since Halo 40 is located at a distance five times larger than the virial radius of the main halo at 2.76 Gyr, ram pressure stripping is unlikely to be the main cause responsible for the suppression of star formation.

Halo 150 has the shortest duration of star formation among the halos shown in Figure <ref>.  Figure <ref> (c) shows the cosmic time evolution of Halo 150. The top panel shows that this halo experienced at least two pericenter passages. Note that we cannot follow the mass evolution before 4.84 Gyr, because the progenitor halos are undetected by the halo finder. As shown in the bottom panel of Figure <ref> (c), the first episode of star formation lasts 0.47 Gyr, and is quenched by cosmic reionization. In this episode, 80% of its stars are formed. The second star formation event occurs at 1.66 Gyr, possibly because of the gas infall, but it is quenched quickly.

Cosmic time evolution of (a) Halo 12, (b) Halo 40, and (c) Halo 150. Top sub-panels: The orbital distance (blue) and the time evolution of the virial radius of the main halo (orange). Middle sub-panels: The dark matter (blue-solid) and gas (orange-dashed) mass evolution. Bottom sub-panels: SFHs. The grey line represents the epoch of reionization ($z = 8.5$). The light-grey shaded region in panel (c) means the halo finder cannot follow the mass evolution.

§.§ Chemical Abundances

The MDFs of stellar systems reflect their histories of star formation, gas infall, and gas outflow; Figure <ref> shows MDFs of Halos 12, 40, and 150. We also plot the observed MDFs of the Fornax, Sculptor, and Draco dSphs [Kirby et al., 2010]. It should be noted that the purpose of our study is not to reproduce the MDFs of the observed dSphs. Rather, we compare simulated and observed MDFs in Section <ref>.

The MDF of Halo 12 exhibits a bimodal distribution, reflecting two major star formation events (the bottom panel of Figure <ref> (a)). All stars with [Fe/H] $< -1.5$ are formed within 3.3 Gyr from the beginning of the simulation. These stars are mainly located in the outskirts of the galaxy. For stars with [Fe/H] $< -1.5$, 28.5% of them are within $r_{\rm{h}}$, while 71.5% of stars with [Fe/H] $\geq -1.5$ are within $r_{\rm{h}}$. As shown in the green-dashed line in Figure <ref> (a), the fraction of stars with [Fe/H] $< -1.5$ in the MDF is significantly decreased for stars within $r_{\rm{h}}$. Stars around [Fe/H] = $-1.2$ and [Fe/H] = $-0.8$ are associated with star formation events around 8.0 Gyr and 9.5 Gyr, respectively. As shown in the middle panel of Figure <ref> (a), these stars are formed from gas infall.

Figure <ref> (b) shows the MDF of Halo 40. The MDF is broadly distributed over
$-3.0\lesssim$ [Fe/H] $\lesssim-1.0$. Stars around [Fe/H] = $-$2.3, $-$1.8, and $-$1.3 reflect star formation at different cosmic times. For [Fe/H] $< -$2.5, all stars formed before 1.0 Gyr from the beginning of the simulation. For stars with $-$2.5 $<$ [Fe/H] $<$ $-$2.0, half of them are formed at $t\,<$ 1.0 Gyr, while others are formed at 1.7 $<$ $t$/(Gyr) $<$ 2.3, simultaneously with stars with $-$2.0 $<$ [Fe/H] $<$ $-$1.5. Stars with [Fe/H] $>-$1.5 have younger ages. All of these stars are formed after 2.2 Gyr. Although there is an overlap in the ages of each peak, the peaks in the MDFs indicate star formation at different cosmic times. In Figure <ref> (b), we also plot the MDF for stars within $r_{\rm{h}}$. Unlike for Halo 12, the MDFs are not largely affected by the spatial distribution of stars.

Figure <ref> (c) shows the MDF for Halo 150. As shown in Figure <ref> (c), Halo 150 exhibits two
star formation events. The second peak of star formation produces stars with $-$2.3 $<$ [Fe/H] $<$ $-$2.1, while the first star formation event mainly forms stars with [Fe/H] $< -$4.0. The number fraction of these ultra metal-poor (UMP) stars is 75.6% for all UMP stars and 64.9% for UMP stars within $r_{\rm{h}}$. These stars largely affect the median metallicity of this galaxy. The median metallicity is [Fe/H] = $-$4.36 for all stars, but [Fe/H] = $-$2.16 for stars with [Fe/H] $> -$4.0.

Simulated (blue-solid line) and observed (orange-dashed line) MDFs for (a) Halo 12 and Fornax, (b) Halo 40 and Sculptor, and (c) Halo 150 and Draco. The green-dashed line represents the MDFs for stars within $r_{\rm{h}}$. The simulated data do not include simulated observational errors.  Observed data are taken from Kirby et al., 2010.

Stars with different ages clearly differ in the [$\alpha$/Fe] vs. [Fe/H] space. Figure <ref> (a) shows [$\alpha$/Fe], as a function of [Fe/H], in Halo 12. This galaxy has two major star formation events (Figure <ref> (a)). The first event (13.7 Gyr to 10.5 Gyr ago) forms the decreasing trend of [$\alpha$/Fe] from [Fe/H] = $-2.5$ to [Fe/H] = $-1.0$. Also, there is roughly a $\sim$1 dex scatter in the [$\alpha$/Fe] ratios. The episodic star formation creates these features during the first major star formation event. The first star formation episode ($\geq$ 13 Gyr ago) forms the high-$\alpha$ ([$\alpha$/Fe] $>+$0.3) component. The interstellar medium (ISM)'s inhomogeneity results in a widely distributed metallicity ($-3.0<$ [Fe/H] $<-1.5$). The low-$\alpha$ ($-0.3<$ [$\alpha$/Fe] $<-0.1$) and very metal-poor ($-2.5<$ [Fe/H] $<-2.2$) component come from another dwarf galaxy accreted to Halo 12. The subsequent star formation episodes (12.0 Gyr to 10.5 Gyr ago) produce the decreasing trend of [$\alpha$/Fe] ratios due to the substantial contribution from SNe Ia.

In contrast, the second star formation event (7.6 Gyr to 4.3 Gyr ago) produces an increasing trend of the [$\alpha$/Fe] ratios for [Fe/H] $>-1.5$. This trend suggests that stars are preferentially formed from the ejecta of CCSNe. During the second major star formation event, stars are mainly produced at the galaxy's center. Young stars give rise to CCSNe mainly at the center, while SNe Ia occur in the more extended region due to their delay times; SNe Ia occur in the more distant places relative to the star-forming region. This difference in the spatial distribution results in the formation of stars reflecting the yields of CCSNe. Since Si also exhibits a similar behavior, AGB stars are unlikely to contribute to forming this trend.

Figure <ref> (b) shows [$\alpha$/Fe], as a function of [Fe/H], in Halo 40. From inspection, five peaks of star formation (Figure <ref> (b)) produce groups of stars with different [Fe/H] and [$\alpha$/Fe] ratios. The first peak of star formation (13.4 Gyr ago) produces stars with [Fe/H] $< -$2.3 and [$\alpha$/Fe] $>+$0.3. Since it is the earliest phase of the star formation, CCSNe are the dominant contributor to the enrichment, resulting in a flat trend of [$\alpha$/Fe] as a function of [Fe/H]. A few stars with [Fe/H] $>-$2.0 and [$\alpha$/Fe] $\approx$ 0.2 are formed from the ejecta of Population III CCSNe. The second peak of star formation (13.0 Gyr ago) forms stars with $-2.5<$ [Fe/H] $< -$2.0 and $+0.1<$ [$\alpha$/Fe] $<+$0.5. The contribution of SNe Ia from the stars produced in the first peak of star formation makes this second group of stars, with lower [$\alpha$/Fe] and higher [Fe/H] than the first group.

Subsequent star formation and the contributions of SNe Ia from the previous peaks of star formation produce groups of stars with lower [$\alpha$/Fe] and higher [Fe/H]. The third peak of star formation (12.0 Gyr ago) creates groups of stars with $-2.5<$ [Fe/H] $< -1.7$ and $-0.3<$ [$\alpha$/Fe] $<+0.2$. This group has the lowest [$\alpha$/Fe] ratios because of the contribution of SNe Ia from the previous two star formation peaks. The fourth peak of star formation (11.6 Gyr ago) produces stars with the same [Fe/H] range but higher [$\alpha$/Fe] ratios (0.0 $<$ [$\alpha$/Fe] $<+0.4$). This group of stars reflects the ejecta from CCSNe formed in the third peak of star formation. The final star formation event (11.0 Gyr ago) forms stars with $-1.5<$ [Fe/H] $< -1.0$ and $-0.2<$ [$\alpha$/Fe] $<+0.2$. Because of its short duration ($\sim$100 Myr), stars are mainly formed from the ejecta of CCSNe.

Figure <ref> (c) shows [$\alpha$/Fe] as a function of [Fe/H] in Halo 150. Although stars are too few to discuss the trend of the [$\alpha$/Fe] ratios, stars formed at different times exhibit distinct differences in [$\alpha$/Fe] ratios. Stars formed in $\geq$ 13.4 Gyr ago show [$\alpha$/Fe] $>+0.2$, reflecting the yields of CCSNe. Different [$\alpha$/Fe] ratios originate from CCSNe with different progenitor masses. A clear separation of star formation events (1.24 Gyr, Figure <ref> (c)) yields stars formed in the second star formation peak with lower [$\alpha$/Fe] ratios owing to the contribution of SNe Ia.

The $\alpha$-element distributions for (a) Halo 12, (b) Halo 40 , and (c) Halo 150.
The color bars indicate the ages of the stars.  The simulated data do not include simulated observational errors.

The dispersion of the [$\alpha$/Fe] ratios reflects the degree of the ISM's inhomogeneity. We quantified the scatter for [$\alpha$/Fe] in $-3<$ [Fe/H] $< -0.5$  following Escala et al., 2018. These authors defined the intrinsic scatter as the standard deviation of the distance distribution between stars' [$\alpha$/Fe] ratios and the cubic spline fitting curve for the data. For Halos 12 and 40, the intrinsic scatter of [$\alpha$/Fe] is 0.18 dex  and 0.16 dex, respectively. These are similar to the estimated intrinsic scatter [Escala et al., 2018] of the Fornax (0.14 dex) and Sculptor dSphs (0.078  dex), meaning that the simulated and observed satellites have ISM inhomogeneity that gives rise to scatter $\lesssim 0.2$ dex for the [$\alpha$/Fe] ratios.

The radial metallicity distribution reflects spatial variations in star formation. Star formation in the inner region of Halos 12 and 40 lasts longer than that in the outer region. Figures <ref> (a) and (b) show radial [Fe/H] distributions in Halos 12 and 40, respectively. Both galaxies have a negative slope of [Fe/H], as a function of the distance from the center, reflecting the difference in the spatial distribution of the stars with different ages. The youngest stars in these galaxies are located within 3 kpc, while stars with ages of $>$ 13 Gyr have a more extended spatial distribution to 5 kpc.

The radial [$\alpha$/Fe] distribution exhibits positive slopes (Figures <ref> (c) and (d)). Because newer stars located in the center of the galaxies are more affected by SNe Ia, the average [$\alpha$/Fe] ratio near the galactic center is lower than in the outskirts. These radial [Fe/H] and [$\alpha$/Fe] gradients are caused by old and metal-poor populations in the outskirts. This result highlights the importance of measuring the chemical abundances of stars in the outer regions of dwarf satellites.

Radial [Fe/H] distributions for (a) Halo 12, (b) Halo 40, and [$\alpha$/Fe] distributions for (c) Halo 12, and (d) Halo 40, respectively. The color bars indicate the ages of the stars. The simulated data do not include simulated observational errors. The red line is the least squares linear fit for the data. The slope is shown in each panel.

The kinematics of stars also differ among stars with different metallicities. Figures <ref> (a) and (b) show the line-of-sight velocities ($v_{\rm{los}}$) as a function of [Fe/H]. We computed $v_{\rm{los}}$ assuming that Halos 12 and 40 are located in the equatorial coordinates of Fornax and Sculptor [Hayashi et al., 2020], respectively, i.e., we observed Halos 12 and 40 respectively located in the positions of Fornax and Sculptor dSphs from the position of the Sun in the Milky Way. The dispersion of $v_{\rm{los}}$ for [Fe/H] $\leq -1.5$ is 19.3 km s$^{-1}$ in Halo 12 and 19.2 km s$^{-1}$ in Halo 40. On the other hand, stars with [Fe/H] $> -1.5$ have smaller dispersion: 15.0 km s$^{-1}$ (Halo 12) and 16.8 km s$^{-1}$ (Halo 40). These results confirm the existence of kinematical distinct populations in satellites <cit.>.

Line-of-sight velocities ($v_{\rm{los}}$) as a function of [Fe/H] in (a) Halo 12 and (b) Halo 40. The simulated data do not include simulated observational errors. The orange-dashed line shows the standard deviation of $v_{\rm{los}}$ as a function of [Fe/H].

§ DISCUSSION

§.§ Chemo-dynamical Evolution of Satellites

Here, we discuss the chemo-dynamical evolution of the MW's satellites by comparing simulations and observations. The relationship between orbits and SFHs has been argued to explain the variety of observed SFHs seen in MW's satellites. Miyoshi & Chiba, 2020 computed the orbital motions of MW's satellites, including Fornax, Leo I, Sculptor, and Draco, with a time-varying gravitational potential based on the Gaia Data Release 2 [Gaia Collaboration et al., 2018] proper motions, and compared them with SFHs. They found that the infall times of classical dSphs coincide well with the peak of the star-formation regions (SFRs), while UFDs had already been quenched before the infall times.

Simulated satellites have some similarities to galaxies analyzed by Miyoshi & Chiba, 2020. Halo 12 is similar to the Fornax dSph in terms of its stellar mass and SFH. Both galaxies have intermediate age (4–8 Gyr) and old ($>$ 10 Gyr) stellar populations. The orbit of Halo 12 is similar to that of Leo I. Both Halo 12 and Leo I experienced one pericenter passage throughout their orbits. Stellar mass, orbits, and SFHs are similar between Halo 40 and the Sculptor dSph. These galaxies formed most stars prior to their infall. Halo 150 is similar to the Draco dSph regarding stellar mass, orbits, and SFHs. These galaxies also comprise old ($>$ 10 Gyr) stellar populations. These results suggest that star formation in intermediate-age  and old stars in these galaxies was regulated by SN feedback and gas inflow, as we have argued in Section <ref>.

The major difference between our simulation and the MW's satellites is the star formation after infall. Our simulation does not exhibit enhancement of  the SFR at the time of the infall, which has been observed by Miyoshi & Chiba, 2020. Di Cintio et al., 2021 <cit.> showed that galaxies should satisfy two conditions to enhance the star formation after infall: (1) galaxies must have cold gas with at least 10$^{-2}$ times the virial mass of the halo at the time of the infall and (2) the pericentric distance should be larger than 10 kpc. None of the galaxies analyzed in this study satisfy these conditions.

The strength and treatment of SN feedback highly affect the SFHs and gas outflow of simulated dwarf galaxies. Since
galaxy formation simulations cannot resolve the evolution of SN remnants, we need to rely on subgrid feedback models <cit.>. Revaz & Jablonka, 2012 performed isolated dwarf galaxy simulations with different strengths of SN feedback. Their simulations showed that the star formation lasted $< 1$ Gyr in their strongest feedback case, while stars were continuously formed over 14 Gyr if they adopted a level of feedback 100 times less than the strongest one <cit.>. Xu et al., 2022 suggested that the mass-loading factor (the ratio of outflow rate and star formation rate) in dwarf galaxies ($M_{*}\sim 10^4$–10$^7M_{\sun}$) observed in extremely metal-poor representatives explored by the Subaru survey project <cit.> were $\sim$10 to 100 times lower than those predicted in galaxy formation simulations. These results highlight the importance of studying the effects of feedback on the SFHs of dwarf galaxies.

MDFs reflect the SFHs and gas infall/outflow of dwarf galaxies. Kirby et al., 2011 showed that Fornax dSph has a narrow MDF with $\sigma$ = 0.36 dex. The Leo I dSph also exhibits a similar MDF. Their chemical evolution model suggested that these galaxies experienced gas infall to shape the narrow MDF. Halo 12 also exhibits a narrow MDF ($\sigma$ = 0.20 dex) for stars with [Fe/H] $>-1.5$. As described in Section <ref>, these stars are formed by gas infall. These results suggest that gas infall plays an important role in the chemical evolution of the Fornax and Leo I dSphs.

The Sculptor dSph has a broader MDF ($\sigma$ = 0.46 dex) than those of the Fornax and Leo I dSphs [Kirby et al., 2013]. Kirby et al., 2011 found that none of their chemical evolution models reproduce Sculptor's MDF. This problem is resolved if they alter the SFH of the chemical evolution model to a more appropriate choice of parameters for SNe Ia and the SFH [Kirby et al., 2011, Homma et al., 2015]. Homma et al., 2015 interpreted Sculptor's SFH derived by de Boer et al., 2012 with a chemical evolution model similar to that of Kirby et al., 2011. They found that dSphs with a larger fraction of stars formed in the early phase have a more elongated low-metallicity tail of the MDF.

Halo 40 in our simulation also exhibits a broad MDF ($\sigma$ = 0.46 dex) similar to Sculptor's MDF. This broad MDF is formed by episodic star formation (Figure <ref> (b)), rather than the continuous SFH assumed in the one-zone chemical evolution models [Kirby et al., 2011, Homma et al., 2015]. From inspection of Figure <ref> (b), there are at least three distinct peaks in Halo 40's MDF formed by episodic star formation. If this is the case, upcoming spectroscopic surveys of dSphs could confirm whether or not the Sculptor dSph has an episodic SFH (see Section <ref>).

§.§ Prospects for Future Surveys

Identifying whether the MW's satellites have episodic star formation is critical to understanding the effects of SN feedback on their chemo-dynamical evolution and the nature of dark matter <cit.>. Pontzen & Governato, 2012 showed that large-scale bulk motion of gas caused by episodic star formation transforms the cusped density profile of dark matter to a cored one <cit.>. The dependence of SFHs on dark matter profiles in observed satellites is not well understood <cit.>. We need additional indicators to identify episodic star formation. As we have found in Figure <ref>, the episodic star formation creates groups of stars with similar [$\alpha$/Fe] and [Fe/H]. We need to search for this feature with observations.

Upcoming wide-field spectroscopic surveys will be able to measure chemical abundances for a sufficiently large number of stars to detect signatures of episodic SFH from chemical abundances <cit.>. For example, Subaru PFS will measure Fe and $\alpha$-element abundances for 14,000 and 6,900 stars in Fornax and Sculptor, respectively. In this subsection, we discuss how the simulated [$\alpha$/Fe] vs. [Fe/H] distribution (Figure <ref>) can be observed by Subaru PFS.

Figure <ref> shows Subaru PFS mock observations of [$\alpha$/Fe] vs. [Fe/H] for Halos 12 and 40. Procedures for the mock observations are described in Section <ref>. Typical observational uncertainties added to the simulated data are $\sigma\approx$ 0.13 dex and 0.14 dex for the [$\alpha$/Fe] and [Fe/H] ratios, respectively. Compared to Figure <ref>, the scatter in the [$\alpha$/Fe] ratios have been increased. Nevertheless, we can still identify groups of stars having similar [$\alpha$/Fe] and [Fe/H] associated with episodic star formation.

Subaru PFS mock observations (black dots) of [$\alpha$/Fe] vs. [Fe/H] for Halos 12 (top panel) and 40 (bottom panel). Red symbols are the abundances for Fornax  (top panel) and Sculptor (bottom panel) observed with Keck/DEIMOS [Kirby et al., 2011].

The top panel of Figure <ref> compares mock observed abundances of Halo 12 and the Fornax dSph. With Keck/DEIMOS, Kirby et al., 2011 found scatter in [$\alpha$/Fe] ratios and a lack of correlation with [Fe/H] in Fornax. Their results suggested that such scatter could arise from bursty star formation or inhomogeneity of the ISM. Mock observed [$\alpha$/Fe] ratios in Halo 12 also exhibit scatter for stars with [Fe/H] $>-1.5$. Due to the observed uncertainties, detailed structures of [$\alpha$/Fe] ratios seen in Figure <ref> (a) cannot be observed, and these structures are observed as scatter. As we have argued in Section <ref>, the scatter of [$\alpha$/Fe] ratios likely come from the enhanced contribution of CCSNe, due to bursty star formation and inhomogeneous chemical abundances in the ISM. This result is consistent with the suggestion by Kirby et al., 2011.

Stars with [Fe/H] $< -1.5$ in Figure <ref> (top) highlight the importance of observing the Fornax dSph with a wide-field multiplexed spectrograph. In Figure <ref> (a), we have shown that most stars with [Fe/H] $< -1.5$ are located outside of $r_{\rm{h}}$. Even after applying observed uncertainties, we can still see the decreasing trend of [$\alpha$/Fe] as a function of [Fe/H] and scatter associated with the peaks of episodic star formation. Since the current sample [Kirby et al., 2011] is limited to the center of the Fornax dSph ($\lesssim$ 400 pc), we cannot constrain the chemical evolution in the outskirts of this galaxy. We will be able to investigate the most metal-poor tail of the MDF and [$\alpha$/Fe] ratios by obtaining spectroscopy out to the tidal radius <cit.> of the Fornax dSph.

There are limitations on the ability of medium-resolution spectroscopy to identify dwarf galaxies accreted to the Fornax dSph with [$\alpha$/Fe] ratios. In Figure <ref>, we find a low-$\alpha$ ($-0.3<$ [$\alpha$/Fe] $<-0.1$) and very metal-poor ($-2.5<$ [Fe/H] $<-2.2$) component, which is from an accreted dwarf galaxy. However, the distinction of this component is unclear, due to the observed uncertainties in Figure <ref> (top). This result suggests that measuring velocity distribution (Figure <ref>) and high-resolution spectroscopy for chemical abundances of stars on the outskirts is necessary to distinguish accreted components. For example, most stars with [Fe/H] $\leq-2.5$ in Halo 12 come from accreted dwarf galaxies. Their line-of-sight velocity dispersion is 22.3 km s$^{-1}$, while that of stars with [Fe/H] $>-2.5$ shows 16.7 km s$^{-1}$ (Figure <ref>). These differences in velocity dispersion could be measured in future surveys.

The bottom panel of Figure <ref> compares mock observed [$\alpha$/Fe], as a function of
[Fe/H], in Halo 40 and the Sculptor dSph. In the mock observation, the groups of stars with similar [$\alpha$/Fe] and [Fe/H] formed in episodic star formation. For [Fe/H] $< -2.0$, these groups are typically separated with 0.5 and 0.4 dex in [Fe/H] and [$\alpha$/Fe], respectively. However, the number of stars (375) observed in Keck/DEIMOS [Kirby et al., 2011] is insufficient to identify such groups of stars. With Subaru PFS, we expect to measure [$\alpha$/Fe] and [Fe/H] for 6,900 stars in the Sculptor dSph. As shown in this mock observation, in the planned survey we will confirm whether there is episodic star formation occurring every few hundred Myr by identifying chemical clumps.

In this subsection, we have shown that [$\alpha$/Fe] vs. [Fe/H] measured by medium-resolution spectroscopy for $\gtrsim$ 1,000 stars can confirm signatures of episodic star formation in the Fornax and Sculptor dSphs. Thanks to our high-resolution cosmological zoom-in simulation, we can discuss the detailed chemo-dynamical structures of satellite galaxies with $\gtrsim$ 10$^6$ $M_{\sun}$. However, due to the resolution limit, we cannot constrain the SFHs and chemical abundances of galaxies with $\lesssim$ 10$^5$ $M_{\sun}$. The SFHs of poorly resolved galaxies tend to be more bursty, because there are too many synchronized SNe from a star particle <cit.>. Hopkins et al., 2018 showed that simulated galaxies should have $> 100$ star particles to result in a convergence of SFHs. This result means that simulations of MW-like galaxies with a mass resolution of $\sim10\,M_{\sun}$ is required to resolve SFHs of the smallest satellites ($\lesssim10^3\,M_{\sun}$). Such simulations could be achieved by resolving the computational scaling issue using deep learning [Hirashima et al., 2023]. We expect that a comparison with upcoming wide-field spectroscopic surveys and high-resolution cosmological simulations will improve our capability to reconstruct the chemo-dynamical evolution of satellites from chemical abundances.

§ CONCLUSIONS

In this study we performed a high-resolution cosmological zoom-in simulation of a MW-like galaxy. With this simulation, we find that the SFHs, MDFs, and [$\alpha$/Fe] ratios of three simulated satellite galaxies are similar to the MW's satellites (Fornax, Sculptor, and Draco dSphs). We also performed a mock observation of medium-resolution spectra using the Subaru PFS spectral synthesis pipeline.

In our simulation, we find that star formation in most simulated satellites is quenched before their infall to the host (Figure <ref>). Star formation episodes in simulated satellites are separated by a few hundred Myr. Such episodic star formation is regulated by SN feedback. For the Fornax-like galaxy (Halo 12), gas infall induces additional star formation at $\approx$ 6–10 Gyr from the beginning of the simulation.

Simulated MDFs reflect SFHs and gas infall/outflow. The narrow MDF for [Fe/H] $>-1.5$ in Halo 12 is formed in the additional star formation due to gas infall (Figure <ref> (a)). This feature is similar to the Fornax dSph. In contrast, the Sculptor-mass galaxy (Halo 40) exhibits a broad MDF (Figure <ref> (b)). This MDF has at least three distinct peaks formed by episodic star formation.

The [$\alpha$/Fe] ratios, as a function of [Fe/H], reflect the ages of stars (Figure <ref>). The oldest stars have [$\alpha$/Fe] $\gtrsim +0.4$. Subsequent enrichment by SNe Ia decreases the [$\alpha$/Fe] ratios. Stars with similar ages formed in episodic star formation events
comprise groups with similar [$\alpha$/Fe] and [Fe/H]. The bursty star formation and inhomogeneity of the ISM form scattered [$\alpha$/Fe] ratios at [Fe/H] $> -1.5$.

Our mock observations find that the groups of stars with similar [$\alpha$/Fe] and [Fe/H] formed by episodic star formation can be identified by upcoming multiplexed medium-resolution spectra surveys (Figure <ref>). We can test whether satellites have episodic star formation with [$\alpha$/Fe] ratios measured with medium-resolution spectra for $\gtrsim$ 1,000 stars. We also find that metal-poor stellar populations can be found in the outskirts of the galaxy. These results indicate that comparison with upcoming spectroscopic surveys and high-resolution cosmological simulations will greatly improve our understanding of the chemo-dynamical evolution of satellite galaxies.

We are grateful to an anonymous referee for carefully reviewing the manuscript. This work was supported in part by JSPS KAKENHI Grant Numbers JP22KJ0157, JP21H04499, JP21K03614, JP22H01259, JP24K00669, JP20H01895, JP21K13909, JP23H04009, JP22K03688, JP24K07095, JP20H05855, MEXT as “Program for Promoting Researches on the Supercomputer Fugaku" (Structure and Evolution of the Universe Unraveled by Fusion of Simulation and AI; Grant Number JPMXP1020230406), JICFuS, grants PHY 14-30152; Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-CEE), and OISE-1927130: The International Research Network for Nuclear Astrophysics (IReNA), awarded by the US National Science Foundation.  E.N.K. acknowledges support from NSF CAREER grant AST-2233781. Numerical computations and analysis were carried out on Cray XC50 and computers at the Center for Computational Astrophysics, the National Astronomical Observatory of Japan and the  Yukawa Institute Computer Facility. This research made use of NASA's Astrophysics Data System.

AHF [Gill et al., 2004, Knollmann & Knebe, 2009],
astropy [Astropy Collaboration et al., 2013, Astropy Collaboration et al., 2018],
CELib [Saitoh, 2017],
Cloudy [Ferland et al., 2013],
MUSIC [Hahn & Abel, 2011]

Figure <ref> shows [$\alpha$/Fe], as a function of [Fe/H], for Halos 9 and 36, which have sufficient data to plot. Halo 9 has two major star formation events in 2 Gyr. The first burst forms stars with [Fe/H] $\lesssim -2.0$, and the second burst produces stars with [Fe/H] $\gtrsim -2.0$. As a result of Fe enrichment by SNe Ia, the [$\alpha$/Fe] ratios decrease toward higher metallicity. The inhomogeneity of the spatial metallicity distribution of the ISM due to CCSNe produces scatter of the [$\alpha$/Fe] ratios. On the other hand, Halo 36 has a more extended SFH (Figure <ref>). The first star formation event creates stars with constant [$\alpha$/Fe] $\approx$ +0.3 due to the contribution from CCSNe. The two subsequent star formation events are influnced by SNe Ia. There is a decreasing trend of [$\alpha$/Fe] ratios toward higher [Fe/H]. Similar to Halos 12 and 40, both galaxies have stars with different [$\alpha$/Fe] and [Fe/H], depending on their ages.

Same as Figure <ref>, but for Halos 9 (left) and 36 (right).
The color bars indicate the ages of the stars.

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textcite Textcite

# Practical Channel Splicing using OFDM Waveforms for Joint Communication and
Sensing in the IoT

Sigrid Dimce1, Anatolij Zubow1, Alireza Bayesteh2,
Giuseppe Caire3, and Falko Dressler1 {dimce, zubow<EMAIL_ADDRESS>berlin.de<EMAIL_ADDRESS>
<EMAIL_ADDRESS>1 Telecommunication Networks, TU Berlin, Germany  2 Huawei,
Canada  3 Communications and Information Theory, TU Berlin, Germany

###### Abstract

Channel splicing is a rather new and very promising concept. It allows to
realize a wideband channel sounder by combining multiple narrow-band
measurements. Among others, channel splicing is a sparse sensing techniques
suggested for use in joint communication and sensing (JCAS), channel
measurements and prediction using cheap hardware that cannot measure wideband
channels directly such as in the internet of things (IoT). This work validates
the practicality of a channel splicing technique by integrating it into an
OFDM-based IEEE 802.11ac system, which we consider representative for many IoT
solutions. Our system allows computing both the channel impulse response (CIR)
and the channel frequency response (CFR). In this paper, we concentrate on the
impact of the number of sub-bands in our study and show that even using only
50% of the overall spectrum leads to very accurate CIR measures. We validate
the system in simulation and confirm the results in an experimental in-door
scenario using software defined radios.

###### Index Terms:

Joint communication and sensing, JCAS, channel sounder, channel splicing,
internet of things, IoT

##  I Introduction

JCAS is becoming more important in different application domains, both in 6G
as well as for the IoT [1, 2, 3, 4]. This also triggered an ongoing discussion
of appropriate waveforms, orthogonal time frequency space (OTFS) being
considered a possible compromise [5]. However, most existing communication
systems are based on orthogonal frequency division multiplexing (OFDM), so,
integrating sensing here is very important [6].

Channel sounding is the core functionality required for JCAS, but also for
channel estimation and channel prediction techniques. For example, commodity
WiFi devices have been used for purposes other than wireless communication
such as indoor localization and channel sounding due to the relatively low
cost [7, 8]. Channel sounding relies on the channel state information (CSI) of
the communication link. In case of a single-antenna device, the CSI is
equivalent to the CFR, which provides channel information in the frequency
domain. Through Fourier transformation, the channel is characterized in the
delay domain, represented by the CIR. The CIR describes the multipath channel
over the delay domain. However, the accuracy of the estimated CIR is limited
by the channel bandwidth (BW) supported by the system. Precise measurement of
multipath components is only possible using wideband signals. In the delay
domain, the resolution is equal to $1/\mathit{BW}$, which, multiplied with the
speed of light, gives the necessary difference between the distance of two
distinct paths. For instance, a bandwidth of $20\text{\,}\mathrm{MHz}\text{/}$
implies that the distance traveled by the signal from two distinct paths
should have at least a difference of $15\text{\,}\mathrm{m}\text{/}$, so that
the paths are distinguishable at the receiver [8].

Figure 1: Illustration of the channel splicing concept

However, in general, wideband sounding is quite complex and energy inefficient
for existing communication systems. Also, many IoT systems can only process
narrow-band subchannels. A possible solution to this issue is given by multi-
band sensing or spectrum splicing [9]. Conceptually, channel splicing means
measuring multiple narrow-band sub-bands/subchannels and then combining the
results to obtain, ideally, the same results as generated by a single wideband
measurement. The concept is depicted in Figure 1.

Initially, spectrum splicing was developed targeting indoor localization
applications [8, 10, 11], and later extended to other applications such as
human sensing [12]. This technique allows a single WiFi device to transmit
packets and extract CSI in the multiple frequency bands tens of megahertz
wide. The collected information through multiple bands is combined mimicking a
wideband channel. Channel splicing exploits the sparse nature of the CIR and
applies sparse recovery method on the collected data to obtain CIR with high
resolution. The delay associated with the shortest path enables estimating the
signal time of flight (ToF), which is used for localization. As we will
demonstrate in this paper, splicing can also be performed on a subset of the
sub-bands with only small reduction in accuracy. Despite its advantages,
splicing faces several challenges due to transceiver impairments.

A few algorithms have been developed aiming to enhance the splicing
performance and resolve the impairments challenges [8, 7, 10, 9, 11, 13]. Our
work builds upon the theoretical concepts of spectrum splicing presented in
[8]. We first implement a communication system in Matlab based on the IEEE
802.11ac standard. We perform channel estimation based on the least square
(LS) estimation technique both in time and frequency domain. We extended the
system by spectrum splicing allowing to combine CFR measurements from multiple
bands. The channel sounder utilizes software defined radio (SDR) components,
we used two USRP N310 in the lab, for transmitting/receiving the signal over
the air. We validate the practical use of splicing technique in simulations
and indoor experiments by splitting a wide channel bandwidth into narrower
bands, collect the CFR over these bands, perform splicing on the collected
data and compare the estimated CIR with the wide channel. Our results build
the basis for OFDM-based JCAS solutions and for low-cost IoT channel sounding.

Our main contributions can be summarized as follows:

  * •

We implement a practical OFDM communication system based on IEEE 802.11ac and
the LS estimation technique on the receiver to compute both CIR and CFR.

  * •

We extend the system by incorporating the spectrum splicing to investigate the
impact of the number of sub-bands, and the accuracy of the sensing approach.

  * •

We validate the splicing technique in a controlled simulation environment as
well as in a set of indoor experiments focusing on the numbers of multipath
component.

##  II Related Work

Channel sounding is a crucial technique for generating knowledge to
characterize the wireless channel in a certain frequency band. The principle
of sounding is to transmit a known baseband signal up-converted at the
frequency of interest, which is post-processed on the receiver side for
extracting metrics (CIR, CFR) that provide channel information. Several
sounding techniques were developed over the years with the sole purpose to
accurately characterize the propagation channel. The two most common
techniques are the spread spectrum sliding correlator [14] and the OFDM-based
system [15]. Both techniques have demonstrated undeniable success at lower
frequencies, as well as at millimiter-wave (mmWave) frequency band [16, 17].
The OFDM-based system consists of multiple subcarriers, where each subcarrier
sees the channel as flat, hence, allowing it to overcome the main drawback of
wideband transmission in terms of frequency selectivity. Nevertheless, the
system requires a large peak-to-average-power ratio (PAPR) as well as tight
receiver synchronization. On the other hand, the spread spectrum sliding
correlator “spreads” the carrier signal over a large bandwidth by mixing it
with a binary PN sequence. The received signal is mixed with a slower
identical version of the PN sequence, which makes the system less vulnerable
towards interference, but more complex in terms of hardware and software
implementation compared to the OFDM system. Thus, choosing one of the
aforementioned sounding techniques is application specific and requires to
compromise between robustness and complexity.

Propagation information is provided in the delay and frequency domain,
respectively, CIR and CFR, which are computed by the channel estimation
techniques. Among the existing ones, the LS estimation is the most common
method characterized by low computational complexity. Yet, in a few
application scenarios, this technique yields inferior performance [18].
Another well-known estimation method is the minimum mean square error (MMSE)
[15], which minimizes the channel estimation error. However, MMSE leads to
high computational complexity and requires prior channel statistic
information, which sometimes is not available. Beside these traditional
techniques, new models based on deep learning are being developed [18],
targeting performance improvement, and using LS estimation for training their
model.

Reducing system costs by using existing hardware infrastructure for
applications such as channel sounding is becoming more crucial. In this
aspect, spectrum splicing methods are attracting high attention in the recent
years. Most of the existing works target indoor localization applications [7,
13, 10, 8], and a few others e.g., human sensing [12]. The developed methods
aim estimating the multipath component (MPC) delays precisely considering the
present of hardware distortions. For instance, Chronos [7] is an indoor
positioning system, which estimates sub-nanosecond ToF using compressed
sensing sparse recovery methods on the collected CFRs over multiple bands. The
algorithm addresses the phase offset issue as a result of hopping between
frequency bands. Splicer [11] is a software-based system that splices multiple
CSIs measurements and achieves single-AP localization. The authors propose
several techniques for hardware impairment corrections. The work in [13]
presents a two-stage global estimation scheme, which aims to improve
estimation accuracy firstly by achieving initial delay estimation based on a
coarse signal model and then a global delay estimation. Other authors [10] use
the shift-invariance structure in the multiband CSI and propose a weighted
gridless subspace fitting algorithm for the delay estimation. The fundamental
limits and optimization of multi-band splicing in terms of the time delays are
analyzed and presented in [9]. The statistical resolution limit is derived for
the delay resolution and an algorithm is proposed to solve the parameters
optimization problem. A grid-based multi-band splicing technique is presented
in [8]. The method is characterized by low-complexity and can easily scale to
large-dimensional problems. Besides localization, other mechanisms, such as
WiRIM [12], utilize spectrum splicing for human sensing applications.

We integrate the theoretical concepts proposed in [8] into an OFDM-based JCAS
system. We validated our practical implementation of channel splicing both in
simulation as well as an experimental indoor scenario.

##  III Spectrum Splicing Architecture

Our channel sounder is based on the spectrum splicing technique presented in
[8]. The developed method exploits the sparse nature of the CIR and utilizes a
grid-based compressed sensing technique to estimate the path delays and
amplitude. On a high abstraction level, the idea is to use subchannels to
reconstruct the CIR with high resolution as shown in Figure 1.

###  III-A Spectrum Splicing Concept

The system is based on OFDM and packets are transmitted over M frequency
bands, where each band is composed of N subcarriers. The subcarriers are
indexed according to the integer set
$\mathcal{N}=\\{-\frac{N-1}{2},...,\frac{N-1}{2}\\}$, where $N$ is an odd
integer. The presented technique performs the estimation based on the pilot
signals, which in the receiver side can be written as

$y[m,n]=H[m,n]S_{m,n}+z[m,n],\quad m\in[M],n\in\mathcal{N}$ (1)

where $S_{m,n}=1$ is the pilot symbol transmitted over the n-th subcarrier of
band $m$ ($f_{m,n}$), which without loss of generality is assumed to be 1;
$H[m,n]$ denotes the CFR in the same subcarrier, and the
$z[m,n]\sim\mathcal{CN}(0,\,1/\acs{SNR})$ is the additive white Gaussian noise
(AWGN) channel.

Assuming that the propagation environment is comprised of K scatters, the CIR
over the delay domain is given as

$h(\tau)=\sum_{k=1}^{K}c_{k}\delta(\tau-\tau_{k}),$ (2)

where $\delta(\cdot)$ stands for Dirac’s delta function,
$\tau_{k}\in[0,1/f_{s})$ is the delay associated with each path $k$ ($f_{s}$
\- subcarrier spacing), and $c_{k}\in\mathbb{C}$ is corresponding amplitude.
The parameter of gain and delay are independent of the frequency band. On the
other hand, the CFR samples are computed from the CIR via the Fourier
transformation, and can be written as

$H[m,n]=\mathscr{F}\\{h(\tau)\\}\mid_{f=f_{m,n}}=\sum_{k=1}^{K}c_{k}e^{-j2\pi
f_{m,n}\tau_{k}},$ (3)

for $m\in[M],n\in\mathcal{N}$. Furthermore, during the transmission, the pilot
samples are affected by several distortions caused by the hardware devices,
which lead to the phase term

$\psi[m,n]=-2\pi(\delta_{m}nf_{s}+\phi_{m}),\quad m\in[M],n\in\mathcal{N}$ (4)

where $\delta_{m}\in[0,1/f_{s})$ stands for the timing offset due to the
packet detection delay (PDD) and the sampling frequency offset (SFO), and
$\phi_{m}\in[0,1)$ represents the phase offset due to the carrier frequency
offset (CFO) between transmitter and receiver, and the phase offset due to
switching channel bands. The two parameters, $\delta_{m},\phi_{m}$ differ from
one band to the other such that, $\psi[m,n]$ is a linear function in each
band, of the subcarrier index with different slope and constant term. As a
result, the received pilot samples, including the distortion component are
represented in the following form

$\displaystyle y[m,n]$ $\displaystyle=e^{j\psi[m,n]}H[m,n]+z[m,n]$ (5)
$\displaystyle=e^{j\psi[m,n]}\sum_{k=1}^{K}c_{k}e^{-j2\pi
f_{m,n}\tau_{k}}+z[m,n]$
$\displaystyle=\sum_{k=1}^{K}c_{k}e^{-j2\pi(f_{m,0}\tau_{k}+\phi_{m})}e^{-j2\pi
nf_{s}(\delta_{m}+\tau_{k})}+z[m,n]$

where for each band the subcarriers are assumed to be equispaced with a space
equal to $f_{s}$ and $f_{m,n}=f_{m,0}+nf_{s},n\in\mathcal{N}$, and $f_{m,0}$
being the carrier frequency of band $m$.

The proposed spectrum splicing technique aims to estimate the CIR based on
noisy and distorted pilot samples, using the following steps:

  1. 1.

For each band $m\in[M]$ estimate and remove the distortion parameters
$\delta_{m},\phi_{m}$. The estimation is performed using the sparse recovery
technique of atomic norm denoising (AND).

  2. 2.

Splice the clean pilot data to obtain a high-resolution estimated CIR using
the orthogonal maching pursuit (OMP) sparse recovery technique.

  3. 3.

Resolve ambiguities using a hand-shaking procedure between the two
communication nodes.

In this work, we focus on the second step of multi-band splicing and perform
the estimation based on the very high throughput long training field (VHT-LTF)
of 802.11ac frame. On the receiver side, time and CFO is estimated and
corrected.

The multi-band splicing technique consists of merging the measurements
conducted over several bands, increasing the resolution of the estimated CIR
by expanding the measurement bandwidth. For instance, the resolution over the
delay domain obtained from the measurements over a single band is given as
$(\Delta\tau)_{1}=1/Nf_{s}$, whereas over $M$ frequencies band it increases to
$(\Delta\tau)_{1}=1/MNf_{s}$. Considering that the CIR is sparse, the authors
in [8] used compressed sensing, and more specifically the OMP sparse recovery
method to recover the CIR. Firstly, a vector is defined containing the
subcarriers per band f($m$) = $[f_{m,-(N-1)/2},...,f_{m,(N-1)/2}]^{T}$ and for
all the bands
$\textbf{f}=[\textbf{f}(1)^{T},...,\textbf{f}(M)^{T}]^{T}\in\mathbb{R}^{MN}$.
Similarly, a vector is defined containing the clean CFR samples per band
$\tilde{\textbf{y}}(m)=[\tilde{y}[m,-(N-1)/2],...,\tilde{y}[m,(N-1)/2]]^{T}$
and for all the bands
$\tilde{\textbf{y}}=[\tilde{\textbf{y}}(1)^{T},...,\tilde{\textbf{y}}(M)^{T}]^{T}\in\mathbb{C}^{MN}$.
Furthermore, the elements of the $\tilde{\textbf{y}}$ can be written as

$[\tilde{\textbf{y}}]_{i}=\mathscr{F}\\{h_{0}(\tau)\\}\mid_{[\textbf{f}]_{i}}+[\tilde{\textbf{z}}_{i}]\,\quad
i=1,...,MN$ (6)

with $[\tilde{\textbf{z}}_{i}]$ representing the AWGN plus the error due to
the phase-distortion removal procedure. In order to apply the OMP method, a
uniform grid of size $G$ is defined over the delay domain as
$\mathfrak{G}=\\{0,1/G,...,G-1/G\\}/f_{s}$, and a dictionary D as
$\textbf{D}=[\textbf{d}(0),...,\textbf{d}(G-1)]\in\mathbb{C}^{MN\times G}$,
where $G>\\!\\!>MN$ and each column $\textbf{d}(i)$ given as

$\textbf{d}(i)=\dfrac{1}{\sqrt{MN}}[e^{-j2\pi[\textbf{f}]_{1}(\dfrac{i}{G})/f_{s}},...,e^{-j2\pi[\textbf{f}]_{MN}(\dfrac{i}{G})/f_{s}}]^{T}\in\mathbb{C}^{MN},$
(7)

where $i=0,1,...,G-1$. For values of $G=2MN$, or $G=3MN$ the grid
$\mathfrak{G}$ is considered dense and the vector in (6) can be approximated
to

$\tilde{\textbf{y}}\approx\textbf{D}\textbf{h}_{0}+\tilde{\textbf{z}}$ (8)

where $\textbf{h}_{0}\in\mathbb{C}^{G}$ is a discrete approximation for
$h_{0}$, and is estimated using the OMP sparse recovery method and the given
$\tilde{\textbf{y}}$ samples. The OMP method is a greedy iterative algorithm,
that selects a column of the dictionary D, at each iteration, such that it has
the highest correlation with the current residual and it repeats until a
convergence condition is met [19]. For each selected column, the non-zero
coefficients are computed using the least-square method, such that they
approximate the measurement vector $\tilde{\textbf{y}}$. The algorithm stops
once the number of the selected dictionary columns reaches the sparsity order
of $\textbf{h}_{0}$, which is given as input.

###  III-B Practical Channel Splicing for OFDM Systems

In the following, we describe the system architecture and the estimation
technique used for generating the CIR and CFR. The SDR-based channel sounder
builds upon USRP-components to perform the over-the-air communication, and
Matlab for the software implementation.

The OFDM transmitter is implemented according to the 802.11ac standard. This
standard support the signal bandwidths of
$204080160\text{\,}\mathrm{MHz}\text{/}$. Despite the different signal
bandwidths, the OFDM subcarrier spacing is kept fixed to
$312.5\text{\,}\mathrm{kHz}\text{/}$, whereas the number of the subcarriers
changes accordingly. Firstly, the transmitter converts the payload message
into a sequence of bits, which later are binary phase shift keying (BPSK) and
OFDM modulated. For over-the-air experiments, the generated baseband signal is
forwarded to the USRP SDR, which upconverts it to the carrier frequency of
interest. The connection between Matlab and the USRP is realized through the
recently released Wireless Testbench
toolbox111https://de.mathworks.com/products/wireless-testbench.html, which
supports high-speed data transmission.

We also use preamble detector functionality provided by the toolbox, allowing
capturing only the signal of interest for offline analysis. Once packets in
the air are detected by the preamble detector, the signal is captured,
downconverted, and the raw data is stored into a binary file for further post-
processing on the host computer. During post-processing, the time and carrier
frequency offset are estimated using the frame preamble and compensated per
each received packet. The channel is estimated based on the VHT-LTF sequence
in the preamble using the LS estimation technique, and the obtained CIR and
CFR are stored. Finally, the signal is decoded and demodulated, and the
transmitted payload message is recovered. The collected CFR samples over
multiple narrow frequency bands are used as input to the spectrum splicing
technique for estimating the wideband channel.

Channel estimation is performed using LS estimation time or frequency domain
approach [15] on the VHT-LTF. The time-domain approach computes the CIR as

$\hat{h}=X^{\dagger}y$ (9)

where y are the received samples, $\hat{h}$ is the CIR and $X^{\dagger}$ is
the Moore-Penrose (pseudo) inverse of the Toeplitz matrix $X$. Likewise, the
frequency-domain approach acquires the CFR as

$\hat{H}=Y./X$ (10)

where Y are the received samples, $\hat{H}$ is the CFR and the X are the
transmitted samples. Both CIR and CFR can be computed from each other using
the fast Fourier transformation (FFT) and inverse FFT (IFFT).

In the following, we describe the validation of our implementation in
simulation as well as results from first over-the-air experiments in an indoor
scenario.

##  IV Performance Evaluation

Channel splicing aims to estimate a wideband channel by combining multiple
narrow-band channel measurements. This work validates the splicing technique
presented in [8] in a controlled simulated environment and a real-world
scenario, exploiting the flexibility of the developed tool to switch between
theory and practice. Currently, the validation focuses only on the accuracy of
the estimated delay, ignoring the amplitude, which will be analyzed in future
work.

###  IV-A Simulations

Splicing is validated in simulations at the frequency band
$5\text{\,}\mathrm{GHz}\text{/}$ by splitting the wide
$160\text{\,}\mathrm{MHz}\text{/}$ bandwidth into narrower sub-bands,
specifically $2\times$80\text{\,}\mathrm{MHz}\text{/}$$,
$4\times$40\text{\,}\mathrm{MHz}\text{/}$$, and
$8\times$20\text{\,}\mathrm{MHz}\text{/}$$ sub-bands. Simulations are run at
the central frequencies listed in Table I.

Table I: Simulation measurement center frequencies Bandwidth | Center frequency
---|---
$80\text{\,}\mathrm{MHz}\text{/}$ | $4.965.04\text{\,}$\mathrm{missing}\mathrm{revert_as}\mathrm{=}{presentation}\mathrm{]}{\text{/}\mathrm{GHz}}{\mathrm{GHz}\text{/}}$$
$40\text{\,}\mathrm{MHz}\text{/}$ | $4.944.985.025.06\text{\,}$\mathrm{missing}\mathrm{revert_as}\mathrm{=}{presentation}\mathrm{]}{\text{/}\mathrm{GHz}}{\mathrm{GHz}\text{/}}$$
$20\text{\,}\mathrm{MHz}\text{/}$ | $4.934.954.974.995.015.035.055.07\text{\,}$\mathrm{missing}\mathrm{revert_as}\mathrm{=}{presentation}\mathrm{]}{\text{/}\mathrm{GHz}}{\mathrm{GHz}\text{/}}$$

The generated CFRs are combined into an array and used as input to the
spectrum splicing technique. The estimated CIR is compared towards the full
$160\text{\,}\mathrm{MHz}\text{/}$ CIR obtained utilizing the LS method. In
the simulated environment, we up-/down-convert to the carrier frequency of
$5\text{\,}\mathrm{GHz}\text{/}$ in software and make use of a the Matlab
Rayleigh channel. The channel model allows defining the number of the MPCs and
the time delay for each path. We investigate the algorithm performance in two
scenarios, and with different number of multipath components and sub-bands
width.

####  IV-A1 Different Frequency Resolution

The first scenario consists of $\textit{K}=2$ MPCs, with delays and average
powers set to $\\{0,18.75\\}$\mathrm{ns}\text{/}$$ and
$\\{0,-2\\}$\mathrm{dB}\text{/}$$. The splicing technique is applied over the
collected CFR samples for different sub-bands width at the corresponding
center frequencies. The results are presented in Figure 2 and depicts the
estimated two paths (indicated by the markers) along with the computed
$160\text{\,}\mathrm{MHz}\text{/}$ CIR. Our technique correctly estimates the
two $160\text{\,}\mathrm{MHz}\text{/}$ peaks despite the sub-channel width. In
the considered scenario, the CFR samples are collected over all the sub-bands.
However, we further investigate the algorithm performance, looking at only
half of the sub-bands. Similarly, the results are shown in Figure 3.

Figure 2: Estimated peaks using different sub-bands width compared towards the
$160\text{\,}\mathrm{MHz}\text{/}$ CIR. Figure 3: Estimated peaks using 50% of
the sub-bands, for different sub-bands width, compared towards
$160\text{\,}\mathrm{MHz}\text{/}$ CIR.

As can be seen, the results for the $2\times$40\text{\,}\mathrm{MHz}\text{/}$$
and the $1\times$80\text{\,}\mathrm{MHz}\text{/}$$ show that we can correctly
estimate the two strongest paths observed by the
$160\text{\,}\mathrm{MHz}\text{/}$ CIR, despite the missing samples. The
results for the $4\times$20\text{\,}\mathrm{MHz}\text{/}$$ experiment, on the
other hand, indicate that we fail estimating correctly the second path, with a
difference up to 2 samples (1 sample = 1/160MHz =
$6.25\text{\,}\mathrm{ns}\text{/}$).

####  IV-A2 Multiple Paths

Figure 4: Estimated MPCs using spectrum splicing based on
$40\text{\,}\mathrm{MHz}\text{/}$ sub-bands for the scenario with 4 paths.

In the second scenario, we consider $\textit{K}=4$ MPCs, with delays and
average powers set to $\\{0,18.75,200,218.75\\}$\mathrm{ns}\text{/}$$ and
$\\{0,0,-2,0\\}$\mathrm{dB}\text{/}$$. Splicing is applied on CFRs samples
collected over all and $50\%$ of the $40\text{\,}\mathrm{MHz}\text{/}$ sub-
bands, with the purpose to estimate the $160\text{\,}\mathrm{MHz}\text{/}$
CIR. The results are presented in Figure 4 and illustrate that splicing
applied over all the sub-bands correctly estimate the $\textit{K}=4$ peaks
delay, shown by the markers. Yet, the algorithm fails to estimate the third
peak when only $50\%$ of the sub-bands are used.

To summarize, the simulation results show that, channel splicing correctly
estimates the paths delay which are distinguishable only by a wide channel, in
case it is applied over the CFR samples collected over all the narrower sub-
bands. The algorithm performance in some cases deteriorate when $50\%$ of the
sub-bands are utilized. As future work we aim to improve the algorithm
performance when only $50\%$ of the sub-bands are utilized which would speed
up the estimation of wider channels in the range of, e.g.,
$1\text{\,}\mathrm{GHz}\text{/}$.

###  IV-B Indoor Lab-Experiments

Table II: Experimental measurement center frequencies Bandwidth | Center frequency
---|---
$40\text{\,}\mathrm{MHz}\text{/}$ | $2.432.47\text{\,}$\mathrm{missing}\mathrm{revert_as}\mathrm{=}{presentation}\mathrm{]}{\text{/}\mathrm{GHz}}{\mathrm{GHz}\text{/}}$$
$20\text{\,}\mathrm{MHz}\text{/}$ | $2.422.442.462.48\text{\,}$\mathrm{missing}\mathrm{revert_as}\mathrm{=}{presentation}\mathrm{]}{\text{/}\mathrm{GHz}}{\mathrm{GHz}\text{/}}$$
Figure 5: Estimated peaks from measurements conducted over all frequency bands
and compared towards the $80\text{\,}\mathrm{MHz}\text{/}$ CIR. Figure 6:
Estimated peaks from measurements conducted over 50% of the sub-bands and
compared towards the $80\text{\,}\mathrm{MHz}\text{/}$ CIR.

To verify the accuracy of the spectrum splicing technique, measurements are
conducted in real-world scenarios. Two USRP N310, correspondingly the
transmitter and receiver, are connected through 1 Gbps Ethernet to a host
computer, in which two instances of Matlab are set up. The experiments are
performed indoor, at $2.4\text{\,}\mathrm{GHz}\text{/}$ using signal bandwidth
of $80\text{\,}\mathrm{MHz}\text{/}$ and a communication distance of
$4\text{\,}\mathrm{m}\text{/}$. Next, the wide bandwidth is divided into
$2\times$40\text{\,}\mathrm{MHz}\text{/}$$,
$4\times$20\text{\,}\mathrm{MHz}\text{/}$$ sub-bands and raw data is collected
at the center frequencies listed in Table II.

To estimate the wide bandwidth CIR, spectrum splicing is applied on the
collected CFR samples over all and half of the sub-bands. The results are
presented in Figures 5 and 6, respectively. The plots depict the estimated two
strongest paths for different sub-channel width (illustrated by the markers),
compared towards the $80\text{\,}\mathrm{MHz}\text{/}$ CIR. In all cases the
method correctly compute the strongest $80\text{\,}\mathrm{MHz}\text{/}$ paths
in terms of delay.

Figure 7: Computed ecdf for 2 paths estimated from splicing applied over
$20\text{\,}\mathrm{MHz}\text{/}$ sub-bands over 100 packets.

However, differently from simulations, the amplitude of the CIR slightly
fluctuates over the collected packets at each band as a consequence of the
dynamic environment, even though the transmitter and the receiver are static.
This slight fluctuation impacts the OMP algorithm on delay estimation.

Therefore, to study the algorithm performance, we collected 100 packets and
computed the ecdf for the estimated two strongest paths for the 100 packets,
over all and 50% $20\text{\,}\mathrm{MHz}\text{/}$ sub-bands. The ground truth
are the two $80\text{\,}\mathrm{MHz}\text{/}$ peaks delay, respectively,
$0\text{\,}\mathrm{ns}\text{/}$ and $12.5\text{\,}\mathrm{ns}\text{/}$. The
results are given in Figure 7 and show that for such narrow bands, the method
sometimes overestimate the delays. Considering that the resolution of the wide
band is
$1/$80\text{\,}\mathrm{MHz}\text{/}$=$12.5\text{\,}\mathrm{ns}\text{/}$$, the
first path is sometimes overestimated up to 2 samples, whereas the second path
up two 3 samples. As a future work, we aim to optimize the method, such that
the impact of the amplitude variation is ignored.

##  V Conclusion

Following the key concepts of joint communication and sensing, we realized an
OFDM-based channel sounder using spectrum splicing. Spectrum splicing allows
to use measurements of multiple narrow-band subchannels to obtain precise
wideband measurements. In particular, we validated the low-complexity grid-
based spectrum splicing algorithm in simulations and real-world-scenario –
making it feasible for internet of things (IoT) solutions. The algorithm is
integrated into an IEEE 802.11ac based communication system, where CIR and CFR
are estimated using the LS estimation technique. For indoor lab experiments,
we implemented the system using an USRP-based SDR. We were particularly
interested in the ability to obtain wideband channel properties using only a
subset of the narrow-band subchannels. Our system allows a very good
estimation of the multipath components in terms of time delay both in
simulation as well as in indoor experiments.

We see this work as a first step towards measurements of even wider band
channels in the mmWave bands. In future work, we plan further experiments also
covering longer distances and more dynamic scenarios. In addition, we aim to
analyze the algorithm performance in computing the amplitude for the estimated
multi-path components.

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  *[JCAS]: joint communication and sensing
  *[IoT]: internet of things
  *[CIR]: channel impulse response
  *[CFR]: channel frequency response
  *[OTFS]: orthogonal time frequency space
  *[OFDM]: orthogonal frequency division multiplexing
  *[CSI]: channel state information
  *[BW]: channel bandwidth
  *[ToF]: time of flight
  *[LS]: least square
  *[SDR]: software defined radio
  *[mmWave]: millimiter-wave
  *[PAPR]: peak-to-average-power ratio
  *[MMSE]: minimum mean square error
  *[MPC]: multipath component
  *[CFRs]: channel frequency response
  *[CSIs]: channel state information
  *[AWGN]: additive white Gaussian noise
  *[PDD]: packet detection delay
  *[SFO]: sampling frequency offset
  *[CFO]: carrier frequency offset
  *[AND]: atomic norm denoising
  *[OMP]: orthogonal maching pursuit
  *[VHT-LTF]: very high throughput long training field
  *[BPSK]: binary phase shift keying
  *[FFT]: fast Fourier transformation
  *[IFFT]: inverse FFT
  *[MPCs]: multipath component