1]Department of Physics, Davidson College, Davidson, NC, 28035, USA 2]Physics
Division, Argonne National Laboratory, Argonne, IL, 60439, USA 3]Department of
Physics, Florida State University, Tallahassee, FL, 32306, USA 4]Department of
Physics & Astronomy, Ursinus College, Collegeville, PA, 19426, USA

# Single-Neutron Adding on 34S

A. N. Kuchera<EMAIL_ADDRESS>C. R. Hoffman<EMAIL_ADDRESS>G. Ryan  I. B. D’Amato  O. M. Guarinello  P. S. Kielb  R. Aggarwal  S. Ajayi
A. L. Conley  I. Conroy  P. D. Cottle  J. C. Esparza  S. Genty  K. Hanselman
M. Heinze  D. Houlihan  B. Kelly  M. I. Khawaja  E. Lopez-Saavedra  G. W.
McCann  A. B. Morelock  L. A. Riley  A. Sandrik  V. Sitaraman  M. Spieker  E.
Temanson  C. Wibisono  I. Wiedenhöver  [  [  [  [

###### Abstract

Purpose: Single-neutron adding data was collected in order to determine the
distribution of the single-neutron strength of the $0f_{7/2}$, $1p_{3/2}$,
$1p_{1/2}$ and $0f_{5/2}$ orbitals outside of $Z=16,N=18$, 34S.

Methods: The 34S($d$,$p$)35S reaction has been measured at 8 MeV/u to
investigate cross sections to excited states in 35S. Outgoing proton yields
and momenta were analyzed by the Super-Enge Split-Pole Spectrograph in
conjunction with the CeBrA demonstrator located at the John D. Fox Laboratory
at Florida State University. Angular distributions were compared with
Distorted Wave Born Approximation calculations in order to extract single-
neutron spectroscopic overlaps.

Results: Spectroscopic overlaps and strengths were determined for states in
35S up through 6 MeV in excitation energy. Each orbital was observed to have
fragmented strength where a single level carried the majority. The single-
neutron centroids of the $0f_{7/2}$, $1p_{3/2}$, $1p_{1/2}$ and $0f_{5/2}$
orbitals were determined to be $2360^{+90}_{-40}$ keV, $3280^{+80}_{-50}$ keV,
$4780^{+60}_{-40}$ keV, and $\gtrsim 7500$ keV, respectively.

Conclusion: A previous discrepancy in the literature with respect to
distribution of the neutron $1p_{1/2}$ strength was resolved. The integration
of the normalized spectroscopic strengths, up to 5.1 MeV in excitation energy,
revealed fully-vacant occupancies for the $0f_{7/2}$, $1p_{3/2}$, and
$1p_{1/2}$ orbitals, as expected. The spacing in the single-neutron energies
highlighted a reduction in the traditional $N=28$ shell-gap, relative to both
the $1p$ spin-orbit energy difference ($N=32$) and the lower limit on the
$N=34$ shell spacing.

##  1 Introduction

The spectroscopic properties of nuclei consisting of protons occupying the
$1s-0d$ shells and neutrons occupying the $1s-0d$ or $0f-1p$ orbitals are
defined by intersecting behaviors. In general terms, there is a strong
interplay between the spacing of the evolving (spherical) single-particle
energies and the correlation energies provided by high-order, primarily
quadrupole-type, excitations. The latter manifest within a single-particle
framework through configurations containing various numbers of particle-hole
excitations from within the $1s-0d$ shells into the $0f-1p$ shells.

An observable impact of the aforementioned competition is the Island of
Inversion centered around 31Na which is characterized by a sharp transition
from near-spherical ground state shapes in the $N=20$ isotones 36S and 34Si to
deformed ground state shapes in 32Mg and 30Ne. The same scenario takes place
along the $N=28$ isotones moving from doubly-magic 48Ca down through the
deformed ground state of 42Si. Complementing the disappearance of these
traditional shell closures are the appearance of new shell gaps in nuclei such
as 24O ($N=16$) and 52,54Ca ($N=32,34$). The same underlying competition is
connected to other phenomena in region such as stark changes in the $Z=8$ to
$9$ neutron drip line locations, the presence of $p-$wave neutron-halo ground
states, the postulation of (proton) bubble densities, and modifications to the
spin-orbit force. Progress towards a quantitative description of the
spectroscopy in the region has been discussed in Refs. [1, 2, 3, 4, 5] and the
references therein.

Characterization of the single-neutron energy centroids of the $0f-1p$
orbitals determines their evolutionary trajectories across an isotopic chain
or along isotones. The pattern or summed-strengths of the single-neutron
overlap distributions provide complementary insight about the types and sizes
of the correlations present throughout these systems. Recent surveys of the
available single-neutron energy centroid data for the $0f-1p$ orbitals have
been reported in Refs. [6, 7, 8] across the Ne-Ca isotopes with $N=17-21$. The
surveyed data leveraged the recent influx of transfer reaction data collected
in inverse kinematics [9, 10, 7, 8], with the high-quality data available from
reactions on stable-targets. In the present work, the single-neutron strength
distributions and the single-neutron energy centroids were determined for the
corresponding $0f-1p$ orbitals outside of $N=18$ 34S where amongst other
aspects, previous experimental results have conflicted in their findings [11,
12].

We report on the single-neutron adding ($d$,$p$) reaction carried out at 8
MeV/$u$ using an enriched 34S target. The reaction populated states in 35S up
to an excitation energy, $E_{x}$, of $\sim 7.7$ MeV ($S_{n}=6.986$ MeV [13]).
The reaction probe and beam energy were ideal, in terms of the reaction
kinematics and the orbital angular momentum ($\ell$) matching, for
investigating overlaps with final states populated through neutron transfer
into the $1s0d$-$0f1p$ shells. The angular distributions from measured
relative differential cross sections, $d\sigma/d\Omega$, as a function of
center-of-mass angle ($\theta_{cm}$), were interpreted through a Distorted
Wave Born Approximation (DWBA) approach. Relative spectroscopic overlaps
(spectroscopic factors), C2S, were determined using known or assumed final-
state total angular momenta and parities, $J^{\pi}$. Note that here, the
isospin Clebsch-Gordon Coefficient, C $=1$ where neutron-adding takes place
onto an $N>Z$, isospin $T=1$, target. Single-neutron centroid energies,
$E_{cent}$, and single-neutron adding strengths,

$G_{+}=\frac{2J_{f}+1}{2J_{i}+1}\textrm{C}^{2}\textrm{S},$ (1)

where $J_{f,i}$ are the final and initial state spins ($J_{i}=0$ at present),
were further deduced from the $E_{x}$, $J^{\pi}$, $\ell$, and C2S information.

##  2 Background

The current status of the observed or extracted spectroscopic properties of
35S, first discovered in 1936 [14], may be found in the evaluated data files
of Ref. [15]. For states of interest to the present work, which lie below an
excitation energy of $\sim 5.1$ MeV, energies are known to better than 1 keV,
and all but one state have assigned $J^{\pi}$. Previous single-neutron adding
reactions on 34S fall into two classes. When $\gamma$-ray detection was used,
lifetime determinations of the 35S excited levels along with $J^{\pi}$
assignments were a primary focus. A number of other dedicated $(d$,$p$)
measurements were successful in determining energy levels, orbital angular
momenta, and single-neutron overlap information. Of those, Refs. [16, 17, 11,
12] are noted specifically due to discrepancies between results, for
completeness in terms of excitation energy coverage, or for their closeness in
reaction energy to the present work.

As the present work extracts the spectroscopic overlap between the 34S ground
state and excited levels in 35S, we note that 34S has been well studied to
date [15]. The ground state is known to consist of both protons and neutrons
confined to the $1s-0d$ orbitals. Of particular importance to the present work
is the amount of neutron $0f-1p$ shell contributions in the ground state.
Analyses from data collected in single-neutron removal reactions on the 34S
ground state, including both the ($p$,$d$) [18] and ($d$,$t$) [19] reactions,
show that less than a few percent of the total neutron removal cross section
contains $\ell=1$ or $3$ contributions. Hence, the ground state contains only
fractions ($\lesssim 0.1$) of $0f-1p$ neutron occupancies.

##  3 Experiment

A 16-MeV deuteron beam (8 MeV/$u$) was accelerated by the 9-MV Super-FN tandem
Van-de-Graaff accelerator located at Florida State University’s John D. Fox
Superconducting Linear Accelerator Laboratory. Typical deuteron beam currents
were on the order of 12 electrical nano-Amperes. The beam impinged on a
carbon-backed sulfur target that was enriched in 34S. There was no known
information associated with the target thickness or the precise enrichment of
the 34S isotope. The presence of the more naturally-abundant isotope 32S was
estimated to be on the order of $\approx 1/10$ that of 34S based on the
reaction-yield information collected as discussed below. The experimental
techniques and layout discussed below are analogous to those described in
Refs. [20, 21, 22, 23, 24, 25] where additional details may be found.

The target was positioned upstream of the Super-Enge Split-Pole Spectrograph
(SE-SPS). The solid angle acceptance of the SE-SPS was fixed at 4.62 msr.
Continuous beam current integration took place with a Faraday Cup located
downstream of the target centered on the beam direction. The two dipole
magnets of the SE-SPS provide a focusing field for the charged reaction
products towards its focal plane detector suite where the protons were
spatially dispersed based on their momentum. Three magnetic field settings of
7.4 kilo-Gauss (kG), 7.8 kG, and 8.4 kG were chosen to ensure overlapping and
complete excitation energy coverage of 35S states through $E_{x}\approx 7.7$
MeV. In addition, the SE-SPS was positioned to collect data at six different
laboratory scattering angles,
$\theta_{lab}=10^{\circ},15^{\circ},20^{\circ},30^{\circ},35^{\circ}$, and
$50^{\circ}$, for two of the magnetic field settings (7.4 kG and 7.8 kG), and
a sub-set of those angles ($\theta_{lab}=15^{\circ},35^{\circ}$, and
$50^{\circ}$) for the third magnetic field setting (8.4 kG).

Figure 1: (a - c) Typical excitation energy spectra for the 34S($d$,$p$)35S
reaction at 8 MeV/$u$ and with the SE-SPS at $\theta_{lab}=20^{\circ}$. The
inset spectrum within (b) is for a sub-range of excitation energy and data
that were collected at $\theta_{lab}=30^{\circ}$. States identified as
belonging to 35S have been labeled by their excitation energy. The locations
of a few contaminant levels belonging to 33S, populated via the 32S($d$,$p$)
reaction, have been labeled by an asterisk (“*”) symbol. Other peaks in the
yield that go unlabeled throughout (a - c) are due to reactions on C and O
contaminants in the target.

The focal-plane detector suite consisted of a position-sensitive ionization
chamber and a plastic scintillator [23]. The identification of protons from
the ($d$,$p$) reactions was done by selecting the appropriate regions in the
$\Delta E-E$ spectra generated from the energy loss within the ion chamber
($\Delta E$) and remaining energy detected in the scintillator ($E$). Proton-
gated $Q$-value spectra were produced from the measured (dispersive) position
information in the ionization chamber. Furthermore, the $Q$-value spectra were
converted into calibrated $E_{x}$ spectra through a linear fit to the known
level information on 35S up to $E_{x}=5.1$ MeV [15]. Typical resolutions for
the $E_{x}$ spectra were on the order of $\sim 45$ keV FWHM. Example spectra
collected for 35S are shown in Fig. 1.

Figure 2: Excitation-energy gated coincident $\gamma$-ray spectra for protons
corresponding to (a) $E_{x}=1.571$ MeV in 35S and $E_{x}=3.221$ MeV in 33S,
and (b) $E_{x}=1.991$ MeV in 35S. The spectra comprise the integration of
events across all of the detectors of the CeBrA demonstrator. Protons were
accepted at $\theta_{lab}=37^{\circ}$ by the SE-SPS.

An additional set of data for the same 34S($d$,$p$) reaction at 8 MeV/$u$ was
taken with the CeBr3 Array (CeBrA) demonstrator surrounding the reaction
target of the SE-SPS [25]. In this measurement, the outgoing protons from the
$(d$,$p)$ reaction were detected in coincidence with de-exciting $\gamma$ rays
(Fig. 2). These data were primarily used in the present work to provide
information on some overlapping peaks produced from the 32S contaminants in
the target. The CeBrA demonstrator was comprised of two sizes of CeBr3
detectors, $2^{\prime\prime}\times 2^{\prime\prime}$ and
$3^{\prime\prime}\times 4^{\prime\prime}$. The SE-SPS was set to a fixed angle
of $\theta_{lab}=37^{\circ}$ and had a magnetic field setting of 7.8 kG.
Figures 2(a) and (b) display the coincident $\gamma$-ray energy spectra with
selected excitation energies in 35S. In particular, Fig. 2(b) highlights the
known 7/2${}^{-}\rightarrow 3/2^{+}$ 1991-keV transition in 35S which has
essentially a 100% branch from the 1.991-MeV state to the ground state [26].
No additional $\gamma$-ray transitions are present in the spectrum as this
state is well isolated from any anticipated contaminants in the SE-SPS focal
plane. A confirmation of the lifetime of the 1.991-MeV level was also made
from the present data as reported in Ref. [25].

Table 1: The single-neutron spectroscopic overlaps (C2S) and strengths ($G_{+}$) from the 34S($d$,$p$)35S reaction at 8 MeV/$u$. Uncertainties from both the fitting results as well as sensitivities to the DWBA optical model parameters ($5-10$% for $\ell>0$ and $10-15$% for $\ell=0$) were included. The C2S have been normalized through summed strengths as described in the text. The excitation energies ($E_{x}$) listed without uncertainties or without parenthesis on $J^{\pi}$, were adopted from previous work [15]. Only the upper limits on the strengths have been listed for states above $E_{x}=6$ MeV due to ambiguities in the DWBA fit results and unknown $J^{\pi}$ values.  $E_{x}$ [MeV] | $J_{f}$ | $\ell$ | C2S | $G_{+}$
---|---|---|---|---
0 | 3/2+ | 2 | 0.54(12) | 2.2(5)
1.571 | 1/2+ | 0 | 0.14(7)111Extracted from a multi-distribution fit to the data to account of the $\ell=1$ transfer to the 33S 3.22 MeV state. | 0.3(2)
1.991 | 7/2- | 3 | 0.87(9) | 7.0(7)
2.348 | 3/2- | 1 | 0.62(7) | 2.5(3)
2.718 | 5/2+ | 2 | 0.02(1) | 0.1(1)
2.939 | 3/2+ | 2 | 0.01(1)222Based on measured yields from only one or two SE-SPS angle settings. | $<0.1$
3.421 | 5/2+ | 2 | 0.02(1)222Based on measured yields from only one or two SE-SPS angle settings. | 0.1(1)
3.802 | 3/2- | 1 | 0.07(4)222Based on measured yields from only one or two SE-SPS angle settings. | 0.3(2)
4.189 | 1/2- | 1 | 0.17(2) | 0.3(1)
4.477333Assumed to be a doublet within the $E_{x}$ resolution at this energy. | 3/2+ | 2 | 0.07(2) | 0.3(1)
7/2- | 3 | 0.05(2) | 0.4(2)
4.903 | 1/2- | 1 | 0.78(8) | 1.6(2)
4.963 | 3/2- | 1 | 0.32(3) | 1.3(1)
5.058 | 7/2- | 3 | 0.08(1) | 0.6(1)
5.740(20) | (5/2-) | 3 | 0.03(1) | 0.2(1)
5.890(20) | (3/2+) | 2 | 0.11(2) | 0.4(1)
6.625(30) | – | (2) | – | $<0.3$
(3) | $<0.4$
6.915(30) | – | (1) | – | $<0.3$
(3) | $<0.4$
7.005(35) | – | (2) | – | $<0.3$
(3) | $<0.4$
7.150(35) | – | (0) | – | $<0.3$
(3) | $<0.1$
7.205(35) | – | (1) | – | $<0.1$
(2) | $<0.2$
7.495(40) | – | (1) | – | $<0.1$
(2) | $<0.1$
7.570(40) | – | (2) | – | $<0.1$
(3) | $<0.1$
7.640(40) | – | (2) | – | $<0.3$
(3) | $<0.4$

##  4 Methods and Results

###  4.1 Levels in 35S

The states in 35S residing below 5.1 MeV in excitation energy, known from
previous ($d$,$p$) measurements, have been identified in the calibrated
$E_{x}$ spectra of Fig. 1 and Table 1. No additional levels were observed up
through this $E_{x}$. Previously unknown states have been noted above
$E_{x}>5.1$ MeV in the figure and the table. The larger uncertainties on these
$E_{x}$ are due to the extrapolation of the linear energy calibration. Of
these additional states, a few have overlapping energies with known levels
[15], for instance from past $(d$,$p$) [11], neutron capture [27], or
($p$,3He) [28] measurements. However, no precise correspondences were
concluded. For some levels, yields may have been discarded for a specific SE-
SPS angle setting or two due to an overlapping contaminant line. Such
contaminants in the excitation energy spectra were identified through their
calculated kinematics and expected placement within the spectra, or by their
energy dependence as a function of SE-SPS angle setting. In a complementary
fashion, new levels suggested to belong to 35S were considered only if they
appeared at the same $E_{x}$ ($\sim$few keV) for at least three SE-SPS angle
settings.

The only state found to be directly impacted by a 33S contaminant level was
the $E_{x}=1.572$-MeV state. As was the case in previous work (for example see
Fig. 1 of Ref. [16]), it was similarly determined to be a doublet including
the 3/2- ($\ell=1$) state at 3.221 MeV in 33S. This determination was made
both from the kinematic calculations of the ($d$,$p$) reaction and the
supporting information provided by the $(d$,$p\gamma$) data. As viewed in Fig.
2(a), an expected 1/2${}^{+}\rightarrow 3/2^{+}$ 1572-keV transition is
observed in the $E_{x}=1.572$-MeV gated $\gamma$-ray spectrum. However, in
addition, an $\sim 840$-keV transition is also apparent. This transition
belongs to the 3/2${}^{-}\rightarrow 1/2^{+}\rightarrow 3/2^{+}$,
$2380\rightarrow 841$-keV, cascade in 33S. Due to low statistics and
$\gamma$-ray detection efficiencies at higher energies, the 2380-keV
transition, as well as the ground state 3221-keV transition, were not
observed.

Therefore, as discussed in more detail below, a multi-distribution fit was
applied to the angular distribution data for the 1.572-MeV state. Other states
in 33S, which did not interfere with 35S levels, were also confirmed in the
$E_{x}$ spectra and their C2S values have been extracted.

###  4.2 Angular distributions

The yields for identified states were collected at each SE-SPS angle setting
when no contaminant line(s) interfered, with the exception of the
$E_{x}=1.572$ MeV state. The differential cross sections, $d\sigma/d\Omega$,
were determined by normalizing to the integrated charge collected over the
duration of the data-collection period. The constant value of the SE-SPS solid
angle acceptance was applied, though not strictly needed, as an absolute scale
on the $d\sigma/d\Omega$ was not possible due to the unknown amount of 34S
isotope content in the target. When levels of interest overlapped across
multiple SE-SPS field settings, the $d\sigma/d\Omega$ from each setting were
combined in an un-weighted average. Each $\theta_{lab}$ scattering angle
setting of the SE-SPS was converted into $\theta_{cm}$ through the known
kinematics of the reaction.

Figure 3: The 34S($d$,$p$)35S differential cross sections, d$\sigma$/d$\Omega$
[arb. units] are given as a function of center-of-mass angle, $\theta_{cm}$,
by the black data points. Each angular distribution has been labeled by its
excitation energy in keV. Uncertainties on the data are represented by the
error bars when they are larger than the size of the symbol and they include
both the statistical and 15% systematic contributions (see text for additional
details). The calculated distributions based on the DWBA prescription
described in the text are represented by the lines and include four possible
$\ell$ transfer values (0 - dark-blue, dashed, 1 - yellow, solid, 2 - blue-
green, dot-dashed, and 3 - red, dotted). The resulting C2S from the fits of
the calculated to the experimental angular distributions are listed in Table
1.
Figure 4: Same as Fig. 3 for additional levels above $E_{x}=6$ MeV.

Experimental angular distributions are shown in Figs. 3 and 4 for 35S levels
in which yields could be measured in at least three SE-SPS angles. The
horizontal error bars on the data are smaller than the points size. The
vertical error bars on the $d\sigma/d\Omega$ values consist of uncertainties
associated with the yield determinations as well as a previously characterized
systematic uncertainty of 15% due to the charge integration [20, 21, 22, 23,
24]. The statistical contribution was only sizeable for the weaker states or
those above $E_{x}\approx 6$ MeV. Uncertainties due to non-uniformity in the
target thickness and changes in the deuteron beam spot parameters are expected
to be small and contained within the 15% uncertainty already applied.

###  4.3 Distorted Wave Analysis

Theoretical angular distributions representing the single-neutron transfer
cross sections as a function of $\theta_{cm}$ were calculated using the
Distorted Wave Born Approximation (DWBA) approach. The calculations were done
with the finite-range DWBA software package PTOLEMY [29]. The Argonne $v_{18}$
potential [30] was included to represent the deuteron bound-state wave
function. A Woods-Saxon potential was used for the final bound-state wave
function of the neutron with potential parameters of $R=r_{0}\textrm{A}^{1/3}$
[$r_{0}=1.25$ fm], $a=0.65$ fm, and $R^{C}=r^{C}_{0}\textrm{A}^{1/3}$
[$r^{C}_{0}=1.3$ fm], as well as spin-orbit parameters of $V_{so}=6.0$ MeV,
$R^{so}=r^{so}_{0}\textrm{A}^{1/3}$ [$r^{so}_{0}=1.1$ fm], and $a_{so}=0.65$
fm. The potential depth was varied to reproduce the binding energies of the
states in 35S below $E_{x}=6.5$ MeV. Above this energy, where only upper
limits on the C2S values were determined, a fixed binding energy corresponding
to $E_{x}=6.5$ MeV was used. This was due to the asymptotic behaviour of
low-$\ell$ orbitals in proximity to the one-neutron separation energy
($S_{n}=6.986$ MeV [13]). The choice of bound-state radius affected the
relative spectroscopic factors by less than 5%. The distorting optical
potentials of the incoming deuteron and outgoing proton were parameterized
through the global potentials defined in Refs. [31] and [32], respectively.
Other global optical potential parametrizations, such as those defined in
Refs. [33, 34, 35], produced relative C2S values within the quoted
uncertainties listed in Table 1.

The resulting fits of the calculated angular distributions to the measured
angular distributions are shown in Figs. 3 and 4. For the negative-parity 35S
states below $E_{x}<5.1$ MeV only the 4.477 MeV state(s) did not have a known
$J^{\pi}$ [15]. Each fit was consistent with the expected $\ell$ transfer
value for the known $J^{\pi}$ (Fig. 3). A multi-distribution fit was used for
the 4.477-MeV state due to conflicting information in the literature and the
possibility for a level doublet. A previous ($d$,$p$) measurement suggested
$J^{\pi}=7/2^{-}$ based on the angular distribution [11]. In contrast,
$(d$,$p\gamma$) data has shown multiple $\gamma$-ray branches to low-lying
states including the 1/2+ ground state [36, 37] which would not be favored
from a 7/2- level, and instead, suggests a $J^{\pi}\leq 5/2^{+}$ for the
level. A combined $\ell=2$ plus $\ell=3$ multi-distribution fit reproduced the
data and therefore a doublet has been adopted throughout this work.

For 35S states above $E_{x}>5.1$ MeV which could not be associated with any
previously known levels or known $J^{\pi}$, angular distributions considering
$0\leq\ell\leq 3$ neutron transfers were independently fit to the angular
distributions. No multi-$\ell$ fit combinations were considered. Only the
5.740 MeV and 5.890 MeV levels produced distinct results (Fig. 3). For states
above $E_{x}>6$ MeV, the fits which produced reasonable $\chi^{2}$ values have
been displayed in Fig. 4.

In the case of the 1.571-MeV state in 35S, due to the discussed overlap with
the 33S 3.221-MeV state populated from the $\ell=1$ ($1p_{3/2}$) neutron
transfer on the 32S in the target, a multi-parameter fit was performed on the
measured angular distribution. The DWBA angular distributions for the 33S
$\ell=1$ component was calculated for the proper kinematics of the
32S($d$,$p$)33S reaction at the same beam energy.

##  5 Discussion

The 34S($d$,$p$)35S spectroscopic overlaps, C2S, determined from fits of the
DWBA angular distributions to the experimental angular distributions (Figs. 3
and 4), have been listed in Table 1 for the states having established $\ell$
values ($E_{x}\lesssim 6$ MeV). In the cases of the 5.740(20) MeV and
5.890(20) MeV levels, the $J^{\pi}$ assignments are purely speculative. The
uncertainties on the C2S values incorporate variations in the adopted DWBA
parameters and were on the order of 5%-10% for $\ell>0$ transfer and
$\sim$10%-15% for any $\ell=0$ distributions. Further uncertainties from the
choice and outcome of the angular distribution fits were included and were on
the order of $\sim 5-10$% for the key states of interest below $E_{x}<5.1$ MeV
(excluding the multi-distribution fits).

For the $\ell=1$, $3/2^{-}$ state at $E_{x}=3.802$ MeV, only the yield from a
single angle was measured. In this case, the C2S was extracted by first
normalizing the 3.802 MeV yield to that of the $\ell=1$ 4.963-MeV level at the
same angle. An additional correction was then applied based on the ratio of
the DWBA cross sections for the two different $E_{x}$ values at the
corresponding angle. The uncertainty for the C2S value was dominated by the
statistical uncertainty of the 4.963-MeV data point. The same procedure was
applied for the more weakly-populated $\ell=2$, 5/2+ levels at 2.939 and 3.421
MeV, where yields for only two angles were measured. In those cases, the
2.718-MeV state provided the normalization and the final value was the average
of the two points, though large uncertainties persisted due to statistics.

The C2S values listed in Table 1 have been normalized through a procedure
based on the well-known Macfarlane and French sum rules [29]. In the present
case, the normalization value was determined from the $\ell=3$ strengths
summed up through $E_{x}=5.1$ MeV and divided by the expected $0f_{7/2}$
vacancy, i.e. $\sum G_{+}(\ell=3)/(2j+1)$, where $j=7/2$. The measured
strength over this excitation energy range was assumed to belong solely to the
$\nu 0f_{7/2}$ orbital (and not the higher-lying $\nu 0f_{5/2}$ orbital). The
$\ell=3$ strength from the multi-distribution fit done on the 4.477 MeV-state
was included, even though it only contributed $\sim$5% the total strength. The
assumption made of a fully unoccupied $0f_{7/2}$ orbital, hence, not requiring
the single-neutron removal strength ($G_{-}$), was supported by the less than
a percent relative C2S values obtained through various neutron-removal
reactions on 34S [18, 19]. The normalization value determined from the
$\ell=3$ strength was commonly applied to all of the C2S values, independent
of $\ell$. It can also be noted, that for this specific reaction, the
procedure outlined above is equivalent to normalizing the sum of the
corresponding $\ell=3$, C2S values to equal 1.

As noted above, states in 33S were populated in the 32S($d$,$p$) reaction due
to the presence of 32S in the target. C2S values were extracted for three of
these states as checks of the analysis procedures as well as to estimate of
the amount of relative 34S to 32S present in the target. The three levels,
including the 7/2- 2.935-MeV ($\ell=3$) level, the 3/2- 3.221-MeV ($\ell=1$)
level, and the 1/2- 5.711-MeV ($\ell=1$) level, appeared at $E_{x}\approx
1.29$ MeV, 1.57 MeV, and 4.06 MeV in the 35S $E_{x}$ spectra, respectively
(denoted by asterisks in Fig. 1). The relative C2S values for the 2.935-MeV
7/2- state [C2S = 0.08(1)], the 3.221-MeV 3/2- state [C2S = 0.06(2)], and the
5.711-MeV 1/2- state [C2S = 0.07(1)] are similar to their reported values of
$\sim 0.5$, $\sim 0.9$, and $\sim 0.5$, respectively [15].

The ratio of the extracted 33S C2S values to corresponding fragments of
similar strength in 35S, i.e. those with C2S $\gtrsim$ 0.6, sets the
contributions of the 33S states at $<10$% of those of 35S. This sets the
34S/32S target material ratio on the order of $\approx 10$. It also confirms
that non-identified strength from any other 33S levels would contribute
negligibly to the 35S yields.

Figure 5: (a) Integrated experimental single-neutron strength ($\sum G_{+}$)
as a function of excitation energy ($E_{x}$) in 35S. The horizontal lines
indicate the total vacancy expected for each orbital. Note that the integrated
$0f_{7/2}$, $\ell=3$ strength was used to determine the normalization factor
of the C2S, i.e. $\sum G_{+}=8$ was ensured for this orbital. Arrows indicate
the single-neutron energy centroid energies for each corresponding orbital.
(b) The distribution of the extracted single-neutron spectroscopic strength
(C2S) in 35S up through $E_{x}\approx 6$ MeV (Table 1). Uncertainties are
shown when larger than the data points and each orbital is distinguished by
color and marker type.

The validity of a commonly applied normalization was demonstrated through an
inspection of the normalized integrated strength for the neutron $1p_{3/2}$
and $1p_{1/2}$ orbitals. Over the same excitation energy range, $E_{x}\leq
5.1$ MeV, $\sum G_{+}$ produced values of 4.1(6) and 1.9(3), for the
$1p_{3/2}$ and $1p_{3/2}$ orbitals, respectively. These are in agreement with
the vacancy expectations of $4$ and $2$. The evolution of these summed
strengths as a function of $E_{x}$ are presented in Fig. 5(a). Congruently,
the normalized summed C2S values for each the $0f_{7/2}$, $1p_{3/2}$ and
$1p_{1/2}$ neutron orbitals are $=1.00(12)$, 1.01(14), and 0.95(10),
respectively.

While the choice of an upper limit of summed strength at $E_{x}=5.1$ MeV is
somewhat arbitrary, it is clear from the integrated strengths displayed in
Fig. 5(a) that similar amounts of strengths are being equally missed across
the three orbitals. Conservatively, an overall uncertainty of 10-20% should be
considered on the relative vacancies for variations in the missed strength. If
all possible strength from the levels between 5.1 - 8 MeV were considered, an
additional amount of normalized strength of $\lesssim 2$ may be present. This
is about 15% of the total vacancy ($8+4+2=14$) without inclusion of the
$0f_{5/2}$, and about 10% of a full vacancy of $20$ if this upper orbital is
included. This additional uncertainty has not been explicitly applied to the
errors in Table 1 or Fig. 5(a).

The distribution of the normalized C2S values is shown as a function of
$E_{x}$ in Fig. 5(b). There is a fragment carrying the majority of the
strength for each orbital with values of $\sim$90%, $\sim$60%, and $\sim$80%,
for the $0f_{7/2}$, $1p_{3/2}$ and $1p_{1/2}$ orbitals, respectively. Other
fragments, ranging from $\sim$5 - 30% of the integrated strength comprise the
remaining strength below $E_{x}=5.1$ MeV. A similarly dominant fragment
corresponding to strength in the $\nu 0f_{5/2}$ orbital was not observed
through our $E_{x}\sim$ 7.7 MeV sensitivity range.

Due to discrepancies between previous ($d$,$p$) measurements, it is worth
pointing out that the present results are in agreement with the results of
Ref. [11], specifically those of potential set 2 listed in their Table 2. The
present work shows disagreement with Ref. [12] where a larger fragment of
strength was extracted for the $\ell=1$, $E_{x}=4.963$-MeV state. It is
possible that due to the large number of isotopes in their target, the
additional strength did not belong to the 34S($d$,$p$) reaction.

Single-neutron centroid energies, $E_{cent}$, for the $0f_{7/2}$
($E_{cent}=2360^{+90}_{-40}$ keV), $1p_{3/2}$ ($E_{cent}=3280^{+80}_{-50}$
keV), and $1p_{1/2}$ ($E_{cent}=4780^{+60}_{-40}$ keV) orbitals were
determined from their ($2J+1$)C2S-weighted excitation energy distributions.
The $E_{cent}$ values are indicated in Fig. 5(a) by arrows for each
corresponding orbital. Only the levels below $E_{x}<5.1$ MeV determined the
$E_{cent}$, however, their associated errors reflect possible contributions
from higher-lying strength (Table 1). The lack of sizeable $\ell=3$ strength
which could be associated with the neutron $0f_{5/2}$ orbital, places a lower
limit on its $E_{cent}$ at $\gtrsim 7.5$ MeV.

The $E_{cent}$ are not designated as the effective single-particle energies
considering the energy-weighted summation did not include the single-neutron
removal strengths [29], albeit they have been found to be small [18, 19]. It
should be emphasized, that due to the observed fragmentation of the single-
neutron strength, if one were to incorrectly interpret _only_ the major
fragments as the centroid energies for the $0f_{7/2}$, $1p_{3/2}$ and
$1p_{1/2}$ orbitals, the inappropriate $E_{cent}$ energies would be different
by $\sim 350$ keV, $\sim 900$ keV, and $\sim 120$ keV, respectively.

The energy spacing between the orbitals is an indication of the spherical
neutron shell gaps outside of $N=18$, 34S. The neutron $1p_{3/2}-0f_{7/2}$
centroid energy difference is 920${}^{+120}_{-140}$ keV and is the spacing for
the traditional $N=28$ shell gap. Similarly, the
$1p_{1/2}-1p_{3/2}=1500^{+110}_{-120}$ keV energy difference reflects the
neutron $1p$ spin-orbit partner splitting. Its size is also that of the $N=32$
sub-shell gap, assuming a normal ordering and filling of the $1p_{3/2}$
orbital. Relative to the upper most $\nu 1p_{1/2}$ centroid, the estimated
$0f_{5/2}-1p_{1/2}$ energy spacing reflecting the $N=34$ gap is $\gtrsim 2.5$
MeV. This lower limit is larger than both the traditional $N=28$ gap and the
($N=32$) spin-orbit partner energy difference in this system. For
completeness, we note that the limit on $0f$ spin-orbit energy spacing is
$0f_{5/2}-0f_{7/2}\gtrsim 5000$ keV.

To make a comparison, we explore the corresponding $N=21$ single-neutron
energy gaps outside of $N=20$, 36S and based on the C2S data extracted in Ref.
[38]. They return energy separations of $\approx 1200$ keV for $N=28$,
$\approx 1500$ keV for $N=32$, and $\gtrsim 2700$ keV for $N=34$, similar to
those extracted in the present work for orbitals outside of 34S. Although
34,36S are stable isotopes they each have a reduced $N=28$ traditional shell-
gap spacing relative to their corresponding $N=32$ and $34$ gaps. A reduced
$N=28$ energy gap is known to persist in the $Z=16$, $N\approx 28$ neutron-
rich isotopes being one of the key components leading to the lower-lying
intruder configurations in 44S (and 42Si) [1, 2, 3, 4, 5].

The energy ordering of the orbitals outside of 34S and 36S follows those found
in the Ca isotopes. In both 41Ca and 49C [15], the neutron $0f_{5/2}$ centroid
resides at sizeable energy ($>1$ MeV) above the neutron $1p_{1/2}$ energy,
thus generating an $N=34$ sub-shell gap. Similarly, data collected on a number
of neutron-rich $N=17,19$ and $21$ isotopes, including those in the present
work, demonstrate the persistence of this sub-shell spacing down through at
least $Z=14$ [10, 8]. The $1p_{1/2}\rightarrow 0f_{5/2}$ ordering is in
opposition to the level-energy sequence determined for the lowest $1/2^{-}$
and $5/2^{-}$ states in $N\approx Z$ 57Ni [39], which implies reversed or
near-degenerate $0f_{5/2}-1p_{1/2}$ neutron orbital energies (Fig. 5 of Ref.
[40]). Recent work has also revisited the available data which determines the
evolution of these orbitals as a function of proton $0f_{7/2}$ occupancy
($Z=20-28$) in the $N=29$ isotones [20, 21, 22]. Further discussion points on
the underlying mechanisms which produce the energy shifts are contained within
Refs. [1, 2, 3, 4, 5].

Considering the single-neutron centroid energies relative to the 35S one-
neutron separation energy, $S_{n}-E_{cent}$, where $S_{n}=6.986$ MeV [13],
provides some perspective on influences from weak-confinement or continuum-
couplings on their relative energies. The $S_{n}-E_{cent}$ values for the
three orbitals of interest are roughly, $4630$ keV ($0f_{7/2}$), $3710$ keV
($1p_{3/2}$), $2210$ keV ($1p_{1/2}$), and $\lesssim 0$ keV ($0f_{5/2}$). The
two $1p$ orbitals have the additional confinement of $\sim$1 MeV beyond the
particle threshold due to their $\ell=1$ orbital angular momentum barriers.
The $\ell=3$ orbitals even more so, with barriers of $\gtrsim 3$ MeV. While
the $\ell=1$ orbitals would be most susceptible to confinement effects, in 35S
they appear reasonably-well bound. Similarly, though the $0f_{5/2}$ neutron
orbital centroid is likely above the $S_{n}$, it would still be confined by a
few MeV.

##  6 Summary and Conclusions

The single-neutron overlaps, strengths, and centroid energies were determined
from the cross sections measured in the 34S($d$,$p$) reaction at 8 MeV/$u$ for
the $0f_{7/2}$, $1p_{3/2}$, $1p_{1/2}$ and $0f_{5/2}$ neutron orbitals. The
experiment took place at the Florida State University John D. Fox Laboratory
and utilized the Super-Enge Split-Pole Spectrograph to analyze the outgoing
reaction protons. In a complementary measurement, the CeBrA demonstrator [25]
provided $\gamma$-ray detection around the target location for the same
reaction. In total, over 20 previously established or candidate states in 35S
were observed up through $E_{x}\approx 7.7$ MeV.

Primarily focused on the extracted C2S below $E_{x}<5.1$ MeV, a consistent
picture of the vacancies for the $0f_{7/2}$, $1p_{3/2}$ and $1p_{1/2}$
orbitals emerged. Each showed fragmentation of the their strength across two
or more individual states. The single-neutron energy centroids showed a
reduced energy spacing at $N=28$, consistent with the situation in the more
neutron-rich $Z=16$ isotopes. The neutron $1p_{3/2}$-$1p_{1/2}$ energy
spacing, established with the proper $1p_{1/2}$ strengths, defined the spin-
orbit energy difference at $Z=16$, $N=19$. A lower limit on the $0f_{5/2}$
centroid energy placed it above the $1p_{1/2}$ orbital. This was in agreement
with the energy ordering in other nearby isotopes including those of the
neutron-rich Ca isotopes.

Acknowledgements

This work was supported by the National Science Foundation (NSF), United
States, under Grant No. PHY-2012522 (FSU) and is based upon work supported by
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics,
under Contract No. DE-AC02-06CH11357 (Argonne). Targets were provided by the
Center for Accelerator Target Science (CATS) at Argonne National Laboratory.

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# Indoor Millimeter Wave Localization using Multiple Self-Supervised Tiny
Neural Networks

Anish Shastri, Andres Garcia-Saavedra, Paolo Casari This work received support
from the European Commission’s Horizon 2020 Framework Programme under the
Marie Skłodowska-Curie Action MINTS (GA no. 861222). A. Shastri and P. Casari
are with the Department of Information Engineering and Computer Science,
University of Trento, Italy. E-mails: {anish.shastri<EMAIL_ADDRESS>Garcia-Saavedra is with NEC Laboratories Europe, Heidelberg, Germany. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We consider the localization of a mobile millimeter-wave client in a large
indoor environment using multilayer perceptron neural networks (NNs). Instead
of training and deploying a single deep model, we proceed by choosing among
multiple tiny NNs trained in a self-supervised manner. The main challenge then
becomes to determine and switch to the best NN among the available ones, as an
incorrect NN will fail to localize the client. In order to upkeep the
localization accuracy, we propose two switching schemes: one based on a Kalman
filter, and one based on the statistical distribution of the training data. We
analyze the proposed schemes via simulations, showing that our approach
outperforms both geometric localization schemes and the use of a single NN.

##  I Introduction

Millimeter wave (mmWave) technologies support not just multi-Gbit/s data rates
for next-generation applications such as augmented reality (AR) and 8-K video
streaming, but also high-accuracy location systems [1]. The use of large
antenna arrays results in highly directional beams, leading to quasi-optical
propagation. Because mmWave communications are short-ranged due to atmospheric
attenuation (especially around the 60 GHz band), and are easily blocked by
obstacles, it is common to consider dense deployments of mmWave access points
(APs) [2]. In such scenarios, location information becomes an extremely useful
tool to optimize the performance of mmWave networks [3].

Existing localization algorithms employ the geometric properties of mmWave
signals such as the angle of arrival (AoA), angle of departure (AoD), and time
of flight (ToF) to localize a mmWave device. However, they require the
knowledge of the indoor area, e.g., the locations of the APs and corresponding
anchors, geometry of the room, device orientation, etc [1]. In practice,
distributing and maintaining this information is not always feasible. Machine
learning and deep learning techniques have also been considered for high-
accuracy indoor localization [4]. However, they rely on collecting large
training datasets, which is often burdensome, and the resulting models are
often computationally complex for resource-constrained devices. In our
previous conference paper [5], we proposed a lightweight shallow NN model to
map angle difference-of-arrival (ADoA) measurements to client location
coordinates. These shallow NNs relieve the training data collection process by
exploiting the location labels obtained from a bootstrapping localization
algorithm (here, we refer to bootstrapping as the data annotation technique
that automatically labels the training data). However, large environments or
indoor spaces with irregular geometries require bigger models, which end up
performing well only in those spaces for which they have been trained. Hence,
a single model can rarely generalize to generic environments.

In this paper, we advocate that using multiple tiny NN models to cover
different sections (enclosed overlapping/non-overlapping sub-area) of an
indoor space is better than training a single model. These NNs, one for a
given section of the indoor space, are trained using location labels obtained
from a geometric bootstrapping localization algorithm. Training multiple NNs
offers two key advantages: ($i$) the models require less training data,
thereby reducing both the training complexity of the NN and that of the
bootstrapping localization algorithm; ($ii$) the resulting localization scheme
is device-centric and can be easily scaled up to multiple clients. In this
work, we obtain the training labels in a self-supervised manner by resorting
to JADE [6] as the bootstrapping algorithm [5]. This is because JADE requires
zero knowledge of the indoor environment, and employs ADoA measurements as
input (same as our tiny NNs). In order to select the best NN to localize the
mobile client, we propose two schemes: one based on a Kalman filter (KF), that
exploits the track information of the client, and another based on out-of-
distribution detection (ODD), that exploits the statistical distribution of
the training labels to select the best NN for localization.

Figure 1: The workflow of our proposed approaches to select the best NN model
for accurate indoor localization.

The outline of the paper is as follows: Section II presents a brief review of
the existing localization schemes; Section III elaborates on our proposed
localization schemes; Section IV presents the results of our simulation
campaign; finally, we draw our conclusions in Section V.

##  II Related work

Geometry-based schemes exploit the angle and range information to localize
mmWave devices [7], e.g., ADoAs are used to triangulate a client’s location in
[8]. Blanco _et al._ exploit AoA and ToF measurements from 60-GHz 802.11ad-
based and sub-6 GHz routers to trilaterate the client location [9]. The
authors of [10] process the channel impulse response (CIR) measurements from
an field-programmable gate array (FPGA)-based 802.11ad implementation in order
to estimate ToF and AoDs, and compute the client locations. They then apply a
Kalman filter to smooth out the estimated trajectory.

Deep learning methods have also been explored to localize a mmWave device. For
example, received signal strength indicator (RSSI) and signal-to-noise ratio
(SNR) beam indices from IEEE 802.11ad-based mmWave routers help learn a multi-
task model for location and orientation estimation [4], [11]. The authors of
[12] designed a dual-decoder neural dynamic learning framework to reconstruct
the intermittent beam-training measurements from these routers sequentially,
and thus estimate a client’s trajectory. Vashisht _et al._ used SNR
fingerprints to train a multilayer perceptron regression model and filter
these coarse estimates using a Kalman filter [13].

##  III Proposed localization scheme

###  III-A Problem statement and main idea

The main objective of this work is to employ multiple self-supervised tiny NNs
to localize a client moving in a large indoor environment in a distributed
fashion. A key aspect then is to decide when to switch to the right NN model,
given that prior information about the ground truth locations and the map of
the indoor environment is not available.

Fig. 1 illustrates the workflow of our proposed localization scheme, which
consists of two phases. In the offline training phase, NN $m$ (corresponding
to section $m$ of a large indoor space) is trained using the ADoA values
$\tilde{\boldsymbol{\phi}}$ computed from the AoA measurements ${\bf\Phi}_{m}$
(obtained by processing channel state information (CSI) at the client as in
[9, 14]). These are fed to the bootstrapping localization algorithm to obtain
training labels $\chi_{m}$ and their corresponding mean-covariance matrix pair
($\boldsymbol{\mu}_{m},{\bf\Sigma}_{m})$. These labels are used to train NN
$m$. Note that the input to both our NN model and the bootstrapping
localization algorithm are the same, i.e, $\tilde{\boldsymbol{\phi}}$.

For the online training phase, we propose two schemes. The first scheme
exploits the location estimates from the NN $m$ to track the evolution of the
client’s state. We exploit the state innovation ${\bf y}_{t}$ and the
normalized innovation squared (NIS) metric $\beta$ of the predicted state,
which measures how accurately the Kalman filter predicted the measurements, to
choose the best NN. The second scheme exploits the statistical parameters of
the training labels to compute the distance between the NN estimates from $m$
different training label distributions. The idea is to compute the distance
between the NNs’ estimates and the distribution of the labels (in this work,
we resort to the Mahalanobis distance). As the wrongly estimated location will
be far from the distribution corresponding to the true label, we refer to this
scheme as _out-of-distribution detection_.

###  III-B Kalman filter (KF)-based decision scheme

The KF-based scheme involves two stages: the trajectory tracking phase and the
decision phase.

Kalman tracking phase. In this phase, we track the evolution of the client
trajectory, as estimated by our trained NNs, using a Kalman filter [15]. Let
the state of the client at time $t$ be
$\mathbf{s}_{t}=[x_{t},\dot{x}_{t},y_{t},\dot{y}_{t}]^{T}$, where $x$ and $y$
are the 2-D coordinates of the client, and $\dot{x}$ and $\dot{y}$ are the $x$
and $y$ component of it’s velocity. The state $\mathbf{s}$ evolves in time
following a constant-velocity (CV) model. The evolution of the state
$\mathbf{s}$ at $t$ is given as
$\mathbf{s}_{t}=\mathbf{F}_{t}\mathbf{s}_{t-1}+\mathbf{w}_{t}$, where
$\mathbf{F}_{t}=\boldsymbol{I}_{2}\,\scriptsize{\otimes}\left[\begin{array}[]{cc}1&\Delta
t\\\ 0&1\end{array}\right]$ is the state transition matrix that transforms the
state of the client at time step $t-1$ to $t$, and
$\mathbf{w}_{t}\sim\mathcal{N}(0,\mathbf{Q}_{t})$ represents the zero-mean
Gaussian distributed process noise with covariance matrix $\mathbf{Q}_{t}$.
Here, $\Delta t=1$ s, $\otimes$ is the Kronecker product, and
$\boldsymbol{I}_{2}$ is the $2\times 2$ identity matrix. The predicted state
representing the 2D location of the client is
$\mathbf{H}\mathbf{s}_{t}+\mathbf{r}_{t}$, where $\mathbf{H}$ is the
observation matrix given by diag$(1,0,1,0)$ and
$\mathbf{r}_{t}\sim\mathcal{N}(0,\mathbf{V}_{t})$ is the zero mean Gaussian
observation noise, with the observation noise covariance matrix
$\mathbf{V}_{t}$.

The Kalman filter performs two steps: _prediction_ and _model update_. The
_prediction_ step estimates the current _a priori_ state of the client
$\mathbf{s}_{t|t-1}$ based on the previous _a posteriori_ estimate
$\mathbf{s}_{t-1|t-1}$, i.e.,
$\mathbf{s}_{t|t-1}=\mathbf{F}_{t}\mathbf{s}_{t-1|t-1}+\mathbf{w}_{t}$, and
also computes the _a priori_ state covariance matrix
$\mathbf{P}_{t|t-1}=\mathbf{F}_{t}\mathbf{P}_{t-1|t-1}\mathbf{F}^{T}_{t}$. The
_model update_ equations correct the existing state predictions and the
covariance matrix using the measurement vector $\hat{\mathbf{x}}_{t}$ and the
updated Kalman gain. The prediction error $\hat{\mathbf{y}}_{t}$ is the
innovation of the Kalman filter and is given as

$\hat{\mathbf{y}}_{t}=\hat{\mathbf{x}}_{t}-\mathbf{H}\mathbf{s}_{t|t-1}.$ (1)

This is used to correct the predicted state of the client as
$\mathbf{s}_{t|t}=\mathbf{s}_{t|t-1}+\mathbf{K}_{t}\hat{\mathbf{y}}_{t}$,
where $\mathbf{K}_{t}$ is the Kalman gain.

We exploit the innovation along with the innovation covariance
$\mathbf{G}_{t}=\mathbf{H}\mathbf{P}_{t|t-1}\mathbf{H}^{T}+\mathbf{r}_{t}$, to
compute

$\beta_{t}=\mathbf{y}^{T}_{t}\mathbf{G}^{-1}_{t}\mathbf{y}_{t}.$ (2)

We also compute the Euclidean distance $\delta_{t}$ between the predicted
state $\mathbf{H}\mathbf{s}_{t|t-1}$ and the current measurement
$\hat{\mathbf{x}}_{t}$, given as
$\delta_{t}=||\mathbf{H}\mathbf{s}_{t|t-1}-\hat{\mathbf{x}}_{t}||_{2}$. We use
$\hat{\mathbf{y}}_{t}$ and $\beta_{t}$ to implement the decision scheme to
switch among the NNs.

KF-based decision scheme. The measurements used by the Kalman filter are the
location estimates obtained from NN $m$, i.e., the NN model trained in section
$m$ of the indoor space. NN $m$ will be able to estimate the client location
accurately, as long as the client is moving within region $m$. Thus, a Kalman
filter will be able to predict the location of the client with a low
$\delta_{t}$. At any time step $k$, as the client moves into another region,
NN $m$ will be unable to estimate the location of the client accurately. This
is because the client will not be able to listen to all the multipath
components from the APs of the previous region. However, the Kalman filter
would predict the location estimate based on the model learned up to time step
$k-1$, thus the Kalman predicted state would be closer to the location of the
client. As the current NN will be unable to accurately localize the client,
this would result in a large $\delta_{k}$ and an even larger $\beta_{k}$. When
$\beta_{k}$ exceeds a user-defined threshold $\eta$, we feed the corresponding
ADoA measurements $\tilde{\boldsymbol{\phi}}_{k}$ to the $M$ trained NN models
to compute the location estimate $\hat{\mathbf{x}}_{k,m}$, where
$m=1,\cdots,M$. We select the NN $m^{*}$ that minimizes the Euclidean distance
between $\hat{\mathbf{x}}_{k,m}$ and the Kalman-predicted location
$\mathbf{H}_{k}\mathbf{s}_{k|k-1}$:

$m^{*}=\operatorname*{arg\,min}_{m}\delta_{k,m},$ (3)

where
$\delta_{k,m}=||\mathbf{H}_{k}\mathbf{s}_{k|k-1}-\hat{\mathbf{x}}_{k,m}||_{2}$,
with the best estimate of the client being
$\hat{\mathbf{x}}^{*}_{k}=\hat{\mathbf{x}}_{k,m^{*}}$. The subsequent set of
measurements for the Kalman filter would be the locations estimated by NN
$m^{*}$. We repeat this procedure to switch to the right NN whenever the
$\beta$ metric exceeds a threshold $\eta$. In our evaluation, we set $\eta=2$.
This aligns with the theoretical result that the expectation of $\beta$ should
equal the number of degrees of freedom of the Kalman filter (i.e., the 2
coordinates of the client in our case) [16].

###  III-C Out-of-distribution detection (ODD) switching scheme

In this approach, we exploit the statistical properties of the location labels
used to train the NNs of each section of the indoor space. Specifically, we
compute the mean and the covariance of the training labels distribution
$\chi_{m}$ corresponding to region $m$ obtained from the offline training
phase. Each distribution $\chi_{m}$ is characterized by its mean-covariance
pair $(\boldsymbol{\mu}_{m},\boldsymbol{\Sigma}_{m})$. In the online testing
phase, at any time instance $k$, if the distance between the current location
estimate $\hat{\bf x}_{k}$ and the previous estimate $\hat{\bf x}_{k-1}$
exceeds a user-defined threshold $\zeta$, i.e., $||\hat{\bf x}_{k}-\hat{\bf
x}_{k-1}||_{2}>\zeta$, where $\zeta$ is the distance threshold (1 m in our
case), the ADoA values $\tilde{\boldsymbol{\phi}}_{k}$ at test location $k$
are fed to $M$ NNs. This results in $M$ location estimates $\hat{\bf
x}_{k,m}$. We then compute the Mahalanobis distance $\rho_{k,m}$ of $\hat{\bf
x}_{k,m}$ from each of the distributions $\chi_{m}$ as

${\rho}_{k,m}=\sqrt{(\hat{\bf{x}}_{k,m}-\boldsymbol{\mu}_{m})^{T}\boldsymbol{\Sigma}^{-1}_{m}({\bf\hat{x}}_{k,m}-\boldsymbol{\mu}_{m})}\,.$
(4)

The Mahalanobis distance measures the distance of a point from a distribution
of points, where, the smaller the distance, the closer the point to the
distribution. We finally choose the best NN as
$m^{*}=\operatorname*{arg\,min}_{m}\,\rho_{k,m}$, and its corresponding
estimate $\hat{\bf x}_{k,m^{*}}$ as the best location estimate. Whenever the
location estimated by NN $m^{*}$ is far from the previous estimate, i.e., the
Euclidean distance between the current estimate and the previous estimate
exceeds the distance threshold $\zeta$ (chosen after exhaustive search), we
repeat the above procedure and switch to another NN model.

###  III-D Tiny neural network architecture

We resort to a 4-layer tiny NN model with
$(N_{i},\,N_{h_{1}},\,N_{h_{2}},\,N_{h_{3}},\,2)$ neurons in each layer. Here,
$N_{i}$ is greater than or equal to the number of ADoAs from visible anchors
[5], $N_{h_{1}}=\lceil{\kappa N_{i}\rceil}$, $N_{h_{2}}=N_{h_{1}}$,
$N_{h_{3}}=\lceil{N_{h_{2}}/2\rceil}$, and $\lceil\cdot\rceil$ represents the
ceiling function. The NN has two output neurons, corresponding to the 2D
coordinates of the client. The NN learns a non-linear regression function
$\mathcal{F}(\tilde{\boldsymbol{\phi}}_{t})$ between the ADoAs
$\tilde{\boldsymbol{\phi}}_{t}$ and the client location $\hat{\bf x}_{t}$. The
regression problem is framed to minimize the mean-square error (MSE) between
the self-supervised training labels and the locations estimates obtained from
the NN. The NN employs the rectified linear unit (ReLU) activation function
and Adam optimizer. The hyperparameters used for tuning the NN to minimize the
loss function are the factor $\kappa$, dropout rate $p$, learning rate $r$,
and the training batch size $b$.

##  IV Simulation Results

###  IV-A Simulation environment

Figure 2: U-shaped room used for simulations. The colored boxes represent the
three sections of the room and the grey line represents the test trajectory.

We validate the performance of our proposed schemes via a simulation campaign.
We simulate human motion trajectories in an irregular U-shaped room as shown
in Fig. 2, comprising two vertical rectangular sections (S-1 and S-3) of size
5 m $\times$ 18 m and 6 m $\times$ 18 m, connected by a 20 m wide horizontal
section (S-2). We deploy seven mmWave APs to cover the indoor space. We
collect AoAs from all the APs and their corresponding virtual anchors (VAs)
(mirror images of the physical APs with respect to each wall of the room) at
each client location along a trajectory using a ray tracer. To simulate
realistic noisy measurements, we perturb the AoAs with zero-mean Gaussian
noise of standard deviation $\sigma=5^{\circ}$.

We train each model with $\approx 800$ locations within each boxed section
(S-1, S-2, S-3) in Fig. 2. The trained NNs have the following architecture:
NN-1 and NN-2 have $(53,48,48,24,2)$ neurons in each layer with $\kappa=0.9$,
$p=0.1$, $r=0.003$, $b=50\%$, and $\kappa=0.9$, $p=0.05$, $r=0.004$, $b=50\%$
respectively, while NN-3 has $(53,43,43,22,2)$ neurons with $\kappa=0.8$,
$p=0.1$, $r=0.003$, $b=50\%$. We test the trained NNs on a trajectory
comprising 388 client locations across the three sections of the room (grey
line).

###  IV-B Analysis of the switching schemes

Figure 3: $\beta$ and localization error obtained when using NN-1.

We first analyze the performance of the KF-based switching scheme. Fig. 3
illustrates the metric $\beta$ computed while using NN-1 to track the client
along the test trajectory. The figure also presents the localization error
when using NN-1. Initially, when NN-1 estimates the client locations in S-1,
$\beta$ is close to zero. As the client crosses the border of S-1 and S-2
(around location 140), the errors of NN-1’s estimates increase. These
estimates significantly deviate from the client’s state as predicted by the
Kalman filter, resulting in a large $\delta_{t}$ and hence an even larger
$\beta$ value, leading to the appearance of peaks. Conversely, with every
subsequent wrong estimates by NN-1, the Kalman filter keeps predicting the
client’s location to be around the wrong estimates, resulting again in low
values of $\beta$. This trend can be observed when the client moves in S-2 and
S-3.

We now analyze the ODD scheme. Fig. 4 illustrates the Mahalanobis distance
between the location estimates obtained from different NNs and the
distribution of the training data $\chi_{m}$ corresponding to each section of
the room, as a function of the client location. The locations estimated by the
NNs and lying within the distribution of the training data consistently yield
$\rho\leq 2$. We also observe that a large set of locations in the common
sections of the room, i.e., C-1 (client locations from 90 to 140) and C-2
(from 235 to 270), were localized accurately by the NNs sharing the overlapped
regions, as they compute the ADoAs from the same set of visible anchors. Thus,
these estimates have similar $\rho$ values, as they are part of two $\chi$s.
In such situations, we can exploit the Euclidean distance between the current
location estimate and the previous location estimate to choose the right NN.

While the values of $\rho$ for NN-1 and NN-3 are low in their respective
regions and easily distinguishable thanks to ADoA measurements from a disjoint
set of APs, the $\rho$ values for location estimates from NN-2 are low for the
entire trajectory. This is because a vast majority of the locations in S-2 can
compute the ADoA values from multipath components (MPCs) arriving from all the
APs. Thus, NN-2 estimates the client locations closer to all the three
$\chi$s. However, the localization error (yellow dashed line) is still large,
especially for the locations in the non-overlapping regions of S-1 and S-3.
This implies that NN-2’s location estimates, even though highly erroneous, lie
within the other two distributions. Thus, we remark that the ODD technique is
more appropriate in environments where each section is illuminated by a
disjoint set of APs.

Figure 4: Mahalanobis distance between the estimated client locations and
$\chi_{m}$ for each NN. The dashed line represents the localization error of
the test trajectory using NN-2. Figure 5: Localization error using the three
NNs and estimated switching points using both the KF-based (triangles) and
ODD-based (squares) switching schemes. The shaded sections represent the
overlap regions C-1 and C-2.

The switching points obtained from the KF-based (triangles) and ODD-based
(squares) switching schemes are depicted in Fig. 5. The dotted and shaded
sections are the borders and the common areas C-1 and C-2, between the S-1 and
S-2, and S-2 and S-3 respectively. The ideal switching points are within the
boundaries of C-1 and C-2 (see Fig. 2), from location index 90 to 140 (C-1)
and from 235 to 270 (C-2). We observe that both schemes decide to switch NN
models within the ideal switching area. However, unlike the ODD scheme, the
KF-based scheme tends to be more robust, as it uses the motion model of the
client to predict its future location.

Figure 6: Location-wise estimation error after reconstructing the trajectory
upon switching among the 3 NNs using the KF-based switching scheme.

Fig. 6 illustrates the localization errors at each location along the test
track, where we consider the NN switching points obtained from the KF-based
scheme, and concatenate the location estimates of each NN. Here, blue hues
represent errors $\leq 1$ m whereas green to red hues correspond to higher
errors. We observe sub-meter errors at most locations, and slightly larger
errors near the bottom left and right corners, and at the switching points,
where the NN being used cannot accurately localize the client. Fig. 7 presents
the cumulative distribution function (CDF) of the localization errors of the
reconstructed trajectory and of the trajectory estimated using the JADE
algorithm, a single self-supervised NN, and the geometric ADoA algorithm [8].
We observe that JADE and our self-supervised NNs outperform the ADoA scheme
[5]. While JADE achieves sub-meter localization errors in 80% of the cases
(mean error of 0.74 m), a single NN performs better with sub-meter errors in
about 87% of the cases (mean error of 0.63 m). Likewise, choosing the right NN
via the KF\- or ODD-based schemes achieves sub-meter errors in about 90% of
the cases. We note that running JADE on the entire trajectory results in large
errors, especially when moving closer to the bottom left and right corners of
the room. This is because noisy ADoA values largely offset the estimates of
the associated anchors. Moreover, using a single NN instead of multiple NNs
results in large training and bootstrapping algorithm complexity, and a single
NN typically needs additional hidden layers to accurately localize a client in
complex-shaped environments. Thus, training a different, smaller NNs (each for
a different portion of the environment) and employing our proposed switching
schemes can help accurately localize a client in large or irregular rooms.

##  V Conclusions

In this letter, we proposed to employ multiple self-supervised tiny NN models
to accurately localize a client as it moves across a large indoor environment.
We presented two schemes based on innovation of Kalman filters and statistical
distribution of training labels, to dynamically choose the best NN model to
accurately localize a client. Results show that the trajectory reconstructed
upon switching to the right NNs using the proposed schemes achieves $\leq$1 m
localization error in 90% of the cases. We can conclude that both schemes can
aid to dynamically switch to the best NN. However, the KF-based scheme is more
robust, owing to its ability to track client’s motion, thus predicting its
trajectory using NN estimates. In contrast, the ODD scheme could be a feasible
choice in large distributed environments, where different sections of the
indoor space are illuminated by a disjoint set of APs.

Figure 7: CDF of the trajectory reconstructed using KF-based switching scheme
against JADE.

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