# Can strangelets be detected in a large LAr neutrino detector?

Mihaela Pârvu  Ionel Lazanu 11footnotetext: Corresponding author.

###### Abstract

Predicted as possible bound states of up, down and strange quarks, strangelets
could be more energetically favourable and more stable than nuclear matter. In
this paper we explore the possibility of detecting such particles with the
future large liquid argon detectors developed for neutrino physics. Using
signals from ionization and scintillation, as well as measuring the range, we
suggest that a calorimetric TPC detector is able to put in evidence and to
discriminate between light strangelets and normal isotopes at intermediate
energies.

##  1 Introduction

Particles in the Standard Model (SM) have lifetimes spanning an enormous range
of magnitudes, from stable particle as electron, through the proton with mean
lifetime $>2.1\times 10^{29}$ years (or stable), up to the Z boson ($\approx
2\times 10^{-25}$ s). Models beyond the SM (BSM) typically predict new
particles with a variety of lifetimes that can generically have lifetimes long
compared to SM particles at the weak scale. The reasons include a) approximate
symmetries that stabilize the long-lived particle (LLP), b) small couplings
between the LLP and lighter states and c) suppressed phase space available for
decays [1].

If these particles are sufficiently long-lived, they can exist in the cosmos
either as Big Bang relics, are produced in the astrophysical processes, or as
secondary collision products and thus must be searches in accelerator-based
experiments at higher energies [1] or in non-collier experiments [2]. The
discovery of LLPs would address important questions in modern physics: such as
the origin and composition of dark matter and the unification of the
fundamental forces.

The new generation of neutrino experiments opens a window for searches of
physics BSM because: a) the combination of the high intensity proton beam
facilities and massive detectors for precision measurements of neutrino
oscillation parameters will help make BSM physics reachable even in low energy
regimes; b) large-mass detectors with highly precise tracking and energy
measurements, excellent timing resolution, and low energy thresholds will
enable the searches for BSM phenomena also from cosmogenic origin [3]. Thus,
experiments as DUNE or Hyper-Kamiokande have capabilities to make BSM physics
more reachable. In fact, the experiments have included this goal in their
scientific program [4, 5].

The early Universe started with equal matter and antimatter abundance in
thermal equilibrium, unless they have a very weak self-interaction cross
section. The particles would therefore be created when other particles
annihilate with each other but the rate at which this would occur is limited
by the mass of the particle and the temperature of the Universe. The species
in question would continue to follow the relic abundance set by this Boltzmann
suppression until it goes out of thermal equilibrium, at which point whatever
number of particles that are present per unit entropy is “frozen in”. At late
times, entropy density corresponds to photon density, but if other species
freeze out after the species in question, they will dump their entropy into
the photon bath and further reduce the relic abundance of the initial massive
particle [2]. A summary of possible types of particles are: LLPs as heavy
leptons and hadrons, WIMPs , Axion, Dark Atoms, Mirror Matter, Strange Quark
Matter, Magnetic monopoles (in different theories: Dirac, as dyson particle,
GUT monopole, extra dimensions, string theory), Q-balls, Black Holes.

Continuing the previous work [6], in the present paper we take LAr as an
example of active medium to explore its recording capabilities of signals
produced by strangelets.

##  2 Strangelets

###  2.1 About the structure and properties of the strangelets

The normal nuclear matter is composed by up and down quarks even if
measurements also show the presence of a strange component in the structure of
the proton, for example. Witten [7] and independently Farhi and Jaffe [8],
from theoretical considerations, suggested that at densities slightly above
nuclear matter density, a strange quark matter composed of up, down, and
strange quarks in roughly equal numbers could exist and it could be stable.
This is strange nuclear matter, dominated by strong interaction. In the frame
of several models existing in the literature, see for example [9, 10] it is
suggested that the Strange Quark Matter (SQM) could be energetically
favourable to nuclear matter, depending on the values of relevant parameters.

General features, common to all types of strangelets consist in: a) masses per
baryon below the mass per nucleon in nuclei (by assumption of stability), and
b) a very low charge-to-mass ratio. From observational point of view, it is
expected that high A/Z ratio compared to normal nuclei is a feature associated
with a possible strange structure. In environment they could be neutralised by
electrons and form unusual atoms or ions. In accord with the model developed
by Jes Madsen [11] for the strangelets with colour-flavour locked quark
matter, a $Z=0.3A^{2/3}$ dependence is predicted and for the strangelets
without pairing (“ordinary” strangelets), dependencies as $Z=0.1A$ for
$A\ll$700 and $Z=8A^{1/3}$ for $A\gg$700 are predicted, assuming a strange
quark mass of 150 MeV. An important problem is repesented by the mass range of
the existence of long-lived strangelets. In the absence of experimental
evidence, all the estimations are done in the frame of different models and,
furthermore, are model dependent. For example, Alford and co-workers, [12]
address the stability of electrically neutral bulk quark matter with respect
to fragmentation into a charge-separated strangelet and electron crust, and
the almost precisely equivalent question of whether large strangelets are
stable with respect to fission into smaller strangelets. The estimation of the
minimum value of $A$ for long-lived strangelets is model-dependent. A typical
minimum $A$ values are in the range $A_{min}\approx 10\div 600$, with up to
$Z\leq 70$ for ordinary strangelets, or $A\approx 680$ with $Z\approx 105$ in
the case of-colour–flavour locked strangelets. These predictions are all
consistent with the result $Z\approx 0.1A$ from the literature. If smaller
strangelets with mass number smaller than the minimum $A$ are created, they
will decay rapidly. If the ground state composition of strangelet consists of
equal numbers of quarks of the three types of quark flavours - which is the
most favourable state, thus it is neutral. Other, are expected to possess a
small positive electric charge can be neutralised by capturing electrons. In
accord with Madsen [22], the mass of a strangelet as a function of baryon
number, in the lowest energy ground state is given as:

$M(A)=874\mathrm{[MeV]}\times A+77\mathrm{[MeV]}\times
A^{2/3}+232\mathrm{[MeV]}\times A^{1/3},$ (2.1)

if the mass of strange quark is supposed to be 150 MeV and a bag constant
$B^{1/4}=145$ MeV. In this parameterization bulk quark matter is bound by 56
MeV/baryon and only strangelets with baryon number $A>23$ are stable relative
to ordinary nuclei, but this limit is proportional with both parameters of the
model - increasing the strange quark mass and/or the bag constant moves the
stability to higher $A$. However, these assumptions are not confirmed and
should be treated with precaution.

Although the analysis of the production mechanisms of strangelets is not
essential for our studies, it is useful to remember that there are several
scenarios proposed in the literature; cosmological origin, in energetic
nuclear collisions, collisions of strange stars for example. If the cosmologic
origin is considered, these particles are remnants of the cosmic QCD phase
transition [7, 13, 14, 15]. In high energetic nuclear collisions associated
the formation of quark-gluon plasma could be possible and thus strangelets
[16, 17, 18]. Collisions of strange stars and different other astrophysical
processes represent also a possibility to the formation of strangelets. See
for example the references [19, 20, 21].

###  2.2 Direct candidates for strangelets and exotic events interpreted as
signals of SQM

By consulting the literature, a small number of singular experimental events
interpreted as SQM can be identified.

  1. 1.

Events Saito [23]. In the analysis of background events from a counter-
technique experiment to study cosmic-ray nuclei above 8 GV/c rigidity two
events were observed with $Z\sim 14$, $A\sim 350$, and 450 amu, and which
cannot be accounted for by more conventional background. It is concluded that
as a confirmation of SQM at a flux level of $\sim 6\times
10^{-9}\mathrm{cm^{-2}}\mathrm{s^{-1}}\mathrm{sr^{-1}}$ and the theoretical
analysis was done by Kasuya and co-workers [24] considering an average value
$A=370$ amu and $Z=14$.

  2. 2.

Price’s event [25] with $Z=46$ and $A>1000$ is fully consistent with SQM
considering the ratio $Z/A$.

  3. 3.

An event with $Z=20$ and $A=460$ has been reported in [26].

  4. 4.

NA52 hypothetical event. One event with a mass 7.4 GeV/c2 and charge number
$Z=-1$, in the $p/Z=-100$GeV/c rigidity settings was observed [27].

  5. 5.

AMS-01 events. These events were identified by the Alpha Magnetic Spectrometer
(AMS-01) experiment, when the detector was flown on the space shuttle
Discovery during flight STS–91 (1998). First candidate has $Z/A=0.114\pm 0.01$
and a kinetic energy of 2.1 GeV, and has been assigned to 16He. The second
candidate was reconstructed as having a nuclear charge of $Z=8$ and a mass
$A=54^{+8}_{-6}$ which we will denote as 54O [28].

In literature are also mentioned a number of phenomena that are both puzzling
and extremely unusual or singular. i) Centauro events for which anomalous
composition of secondary particles with almost no neutral pions present
observed in emulsion chambers [29]. A re-analysis of this case done by Ohsawa
and co-workers [30] claimed an incorrect analysis of the experimental data.
ii) anomalous cosmic ray bursts from Cygnus X - 3 [31, 32]. iii) extraordinary
high luminosity gamma-ray bursts from the supernova remnant N49 in the Large
Magellanic Cloud [33]. Rybczynski and Zbigniew Włodarczyk –see for example
reference [34] where all previous calculations are cited, show that the
abundance of strangelets in the primary cosmic ray flux is about $2.4\times
10^{-5}$ at the same energy per particle. The analysis of the EAS data offers
a unique possibility to observe possible imprints of strangelets arriving from
outer space. These authors tried to explain the muon excess observed by the
Pierre Auger Observatory assuming the existence of strangelets. Results
suggest that the possible contribution of strangelets is negligible in the
limits of the precision of data.

Expected strangelet flux. Monreal [35] derived an equation to estimate the
abundance of strangelets in cosmic-ray propagation model. The author’s work
assumes a strangelet production rate of 10-10 solar masses ($M_{\odot}$) per
year in our Galaxy, with 10-5$M_{\odot}$ of strange matter released per
collision, and a collision every 30000 y. It includes the effects of
acceleration, interstellar propagation, and solar modulation using several
phenomenological models. If we assume that all of the released strange-star
material is converted into strangelets of charge $Z$ and and mass $A$ (GeV/c2
), the maximum flux $F$ is:

$F=2\times 10^{5}\mathrm{[m^{-2}y^{-1}sr^{-1}]}\times
A^{-0.467}Z^{-1.2}(R^{1.2})_{cutoff}.$ (2.2)

$R_{cutoff}$ is the either the geomagnetic cutoff or the solar-modulation
cutoff, whichever is greater. Similar estimations of the strangelet flux in
cosmic rays was done by Madsen in two successive papers [11, 22]. Madsen has
done distinct predictions:

-for “ordinary” strangelets:

$F_{total}\approx 2.5\times
10^{5}\mathrm{[m^{-2}y^{-1}sr^{-1}]}A^{-1.067}Z^{-0.6}\Phi^{-0.6}_{500}\Lambda.$
(2.3)

-for "CFL" strangelets:

$F_{total}\approx 2.8\times
10^{4}\mathrm{[m^{-2}y^{-1}sr^{-1}]}Z^{-2.2}\Phi^{-0.6}_{500}\Lambda.$ (2.4)

In Table 1 some estimations of the fluxes of strangelets is done, considering
“colour flavour locked quark matter and "ordinary” possibilities.

A | Colour flavour locked quark matter | Ordinary
---|---|---
| (Z,A) | Flux [m-2 yr-1 sr-1] | (Z,A) | Flux [m-2 yr-1 sr-1]
6 | (1,6) | 28000 | (1,6) | 36953
14 | (2,14) | 6094 | (2,14) | 9872
18 | (2,18) | 6094 | (2,18) | 7550
22 | (2,22) | 6094 | (2,22) | 6095
36 | (3,36) | 2497 | (4,36) | 2378
56 | (4,56) | 1326 | (6,56) | 1163
70 | (5,70) | 812 | (7,70) | 836
350 | (14,350) | 84 | (35,350) | 57
450 | (17,450) | 55 | (45,450) | 38
Table 1: Strangelets fluxes estimations.

The strangelets are influenced by the solar wind when entering the inner parts
of the Solar System. Here, $\Phi_{500}=\Phi/(500\mathrm{MeV})$. A good fit to
the solar modulation of the cosmic ray spectrum can be given in terms of a
potential model, where the charged particle climbs an electrostatic potential
of order $\Phi=500$ MeV (the value changes by a factor of less than 2 during
the 11 (22) year solar cycle). The second quantity, $\Lambda$, has the
explicit form:

$\Lambda=\left(\frac{\beta_{SN}}{0.005}\right)^{1.2}\left(\frac{0.5\mathrm{cm^{-3}}}{n}\right)\left(\frac{M}{10^{-10}M_{\odot}\mathrm{yr^{-1}}}\right)\left(\frac{1000\mathrm{kpc^{3}}}{V}\right)\left(\frac{930\mathrm{MeV}}{m_{0}c^{2}}\right).$
(2.5)

The minimal rigidity is assumed to be given by the speed of a typical
supernova shock wave, $\beta_{SN}\sim 0.005$. The strangelet spectrum after
acceleration in supernova shocks is assumed to be a standard power law in
rigidity with index 2.2 as derived from observations of ordinary cosmic rays.
$n$ \- denotes the average hydrogen number density per cubic centimeter
($n\sim 0.5\mathrm{cm^{-3}}$) when averaging over denser regions in the
galactic plane and dilute regions in the magnetic halo). For a total
production rate of $M=10^{-10}M_{\odot}\mathrm{yr}^{-1}$ of baryon number $A$
strangelets spread evenly in an effective galactic volume $V$. A typical value
for the effective galactic volume confining cosmic rays is $V=1000$ kpc3.

Integral upper limits on the strangelet flux in cosmic rays was therefore set
for particles with charge 1 $\leq$ $Z$ $\leq$ 8 and mass 4 $\leq$ $A$ $\leq$
1.2 $\times$ 105 in terms of baryon number ($A$) was measured by PAMELA [36].
In figure 3 from the cited article are indicated these limits:
$1.5\times{10}^{3}\div 1.5\times{10}^{7}\left[\
\mathrm{m^{-2}y^{-1}{sr}^{-1}}\right]$, relative constant versus baryon
number, a different behaviour of the upper fluxes as predicted by Madsen.

###  2.3 Interactions of strangelets in matter

Because of the present hypotetical status of strangelets (not yet discovered),
it is very difficult to consider the correct and complete interactions
mechanisms in matter.

The strangelet is characterised by an electrical charge and thus their
interactions with electrons of atoms or molecules in the matter must exist.
For high energies of the strangelet ($v>2v_{Bohr}$), the relativistic Bethe
formula for the energy loss per distance travelled can be used. Here, Bohr
velocity refers to an electron in the innermost orbit of a hydrogen atom:
$v_{Bohr}=c/137=2.2\times 10^{8}$ cm/s. At low energies of the incident
particles the energy transferred to the recoil nuclei in accord with Lindhard
theory [37] is comparable with the electronic energy loss, thus they will
contribute to excitations and ionizations. As a result of electronic
interactions, ionizations or excitations are produced and thus scintillations
are the effects.

Of course, because the strangelets have quarks in their structure, strong
interactions must exist. Possible interactions are predicted in literature -
the strangelets has a high penetrability in atmosphere compared with the heavy
ions, but the effects are contradictory suggested: i) in an option, their mass
is reduced following collisions with atoms and molecules of the medium,
whereas ii) another approach posits that the mass and charge increase
following the accretion of nucleons. Wilk and Wodarczyk [38] interpret such a
penetrability of strangelets as indication of the existence of very heavy
lumps of SQM entering our atmosphere, which are then decreased in size during
their consecutive collisions with air nuclei (i.e., their original mass number
$A$ is reduced until $A=A_{crit}$) and finally decay be the evaporation of
neutrons. For strangelets the cross sections should be very small and hence a
geometric size much smaller than typical nuclear size. In the work of Paulucci
et al. [39] different possible mechanisms of interactions of strangelets with
ordinary matter are discussed: abrasion, fusion, fission, excitation and de-
excitation of the strangelets. It is shown that, although fusion may be
important for low-energy strangelets in the interstellar medium (thus
increasing the initial baryon number $A$), in the earth’s atmosphere the loss
of the baryon number should be the dominant process.

Alternatively, there exists a model based on the following hypothesis: in a
typical interaction between a strangelet and the nucleus of an atmospheric
atom, it is more probable for the strangelet to absorb neutrons from the
colliding nucleus, and as a result the mass of the strangelet increases in
every collision and it becomes more tightly bound [40, 41, 42].

Another aspect is that in the case of strangelet, as projectile with mass and
charge of certain sign, it can in principle capture particles of the opposite
sign and form an "atomic system". In the case of strangelets with intrinsic
negative charge, the probability to capture one or more protons is practically
improbable. For the strangelets with positive intrinsic charge, the formation
of one atomic system is possible. Thus, the determination of the average
equilibrium charge during penetration through matter is a difficult problem.
The problem was first discussed by Bohr [43, 44], extrapolated by Pierce and
Blann [45] and general treatment was done by Ziegler [46].

In this paper we are interested only in the interaction rate of strangelets in
the bulk volume of liquid noble gas used for their detection. The mechanisms
by which the strangelets interact in atmosphere are not relevant for the
present paper.

For the very small strangelets Gilson and Jaffe [9] show regularities as
"shell effects" and islands of metastability that are persistent for a large
intervals for the parameters of model. For the lower strangelets their results
are in disagreement with some predictions analysed in this paper. This region
of SQM is numerical investigated because it is important in the physics
searches Beyond the Standard Model in the next generation of the neutrino
experiments.

##  3 Interactions in a large LAr neutrino detector

For the next generation of neutrino experiments, liquid Argon (LAr) as a time
projection chamber technology (TPC) is currently being developed and is very
promising for neutrino detection. The DUNE detector will contain 4 modules of
10 kiloton of LAr each. For the existent prototypes, the two constructive
options consists in volumes of about 700 tonnes each. The detector will be
very promising for strangelets detection.

###  3.1 About scintillation yield in liquid argon and quenching phenomena

Doke and co-workers [47] developed a model for the scintillation mechanisms in
liquid argon. In accord with their suppositions, exist two distinct possible
ways for scintillations in argon. Liquid and gaseous Ar are transparent to
their own scintillation light.

Ideally, in liquid argon, the scintillation production process should be
associated only with the existence of the atomic excitation mechanism. In
fact, the scintillation from liquid argon has two decay time constants,
corresponding to the decay from the singlet state of an excited dimer and the
decay from its triplet state. Scintillation can also occur through a mechanism
that starts with an ionized atom.

  1. 1.

Ar∗:

Ar∗+Ar+Ar $\to$ Ar${}_{2}^{*}$+Ar

Ar${}_{2}^{*}$ $\to$ 2Ar + h$\nu$

  2. 2.

Ar+:

Ar++Ar $\to$ Ar${}_{2}^{+}$

Ar${}_{2}^{+}$ \+ e- $\to$ Ar∗∗ \+ Ar

Ar∗∗ $\to$ Ar∗ \+ heat

Ar∗ +Ar+Ar $\to$ Ar${}_{2}^{*}$ \+ Ar

Ar${}_{2}^{*}$ $\to$ 2Ar + h$\nu$

where h$\nu$ denotes the ultraviolet photon and the heat corresponds to a non-
radiative transition. In both processes, the excited dimer at the lowest
excited level should be de-excited to the dissociative ground state by
emitting a single UV photon (this hypothesis is not yet fully confirmed
experimentally), because the energy gap between the lowest excitation level
and the ground level is so large that there exists no decay channel such as
nonradiative transitions. The time constants of the singlet and triplet decays
have been measured in all phases and their lifetimes are 7.0$\pm$1.0 ns and
1600$\pm$100 ns respectively for liquid argon. In LAr, the spectrum is
dominated by an emission feature of 126.8 nm or 9.78 keV. Weak-emission
features in the wavelength range from 145 to 300 nm were observed.

The range of different strangelets as a function of their energies is
presented in Figure 1.

At DUNE, as well as ProtoDUNE, the spatial resolution of the detector is 0.47
cm [48] and this quantity can be used as a selection to exclude candidates
from the present investigation (dashed blue line in the figure).

Figure 1: Ranges of various Strangelets in LAr.

The maximum number of scintillation photons produced for the deposited energy
$E_{0}$ by an ionizing particle, $N_{ph}(max)$ is:

$N_{ph}(max)=N_{ion}+N_{ex}=\frac{E_{0}}{W}\left(1+\frac{N_{ex}}{N_{ion}}\right)=\frac{E_{0}}{W^{\prime}}.$
(3.1)

Considering the ionization energy $W=23.6$ eV and the mean excitation energy
$W^{\prime}=19.5$ eV, the total energy loss through interactions with the
atomic electrons can be written as:

$\begin{split}&23.6N_{ion}+19.5N_{ex}=E_{e},\\\
&\frac{N_{ex}}{N_{ion}}=0.21.\end{split}$ (3.2)

If the scintillation quenching and recombinations are not taken into account,
one can derive the number of ionizations and excitations for any incident
particle.

In accord with Doke et. al. [49], the estimated maximum number of photons
emitted from liquid argon per deposited energy of 1 MeV is 5.13 × 104.

In all materials that emit fluorescence signals, these are nonlinear with
respect to the energy deposited by ionizing particles, in particular for high
ionizing particles. This behaviour is attributed to quenching of the primary
excitation by the high ionization density along the particle track which leads
to a decrease of the scintillation efficiency. The Birks ”law” is defined as
[50]:

$\frac{dS}{dx}=\frac{AdE/dx}{A+kBdE/dx}\to\begin{cases}AdE/dx&\text{for low
value }dE/dx\\\ \frac{A}{kB}=constant&\text{for high value }dE/dx\end{cases}$
(3.3)

The scintillation quenching factor $f_{l}$ is defined by the reduction factor
of scintillation intensity. As effect, the initial $N_{ex}/N_{ion}$ ratio is
modified, as is indicated in Fig. 2, as energy dependence. $f_{l}$ is defined
as:

$f_{l}=\frac{1}{1+kB\frac{dE}{dx}},$ (3.4)

where $kB=7.4\times 10^{-4}$ MeV-1g cm-2. The number of photons as a result of
scintillation process after taking into account the quenching mechanism will
be:

$N^{\prime}_{ex}=f_{l}N_{ex}.$ (3.5)

Figure 2: Energy dependence of excitations vs. ionizations.

In a LAr detector operating as calorimeter, an electric field is applied to
collect the free electrons and ions. The effects were discussed by Burdin and
co-workers [51]. The charge collected is then scaled appropriately to deduce
the total amount of deposited energy. However, some electron–ion pairs may
recombine before the ionization electron can be recorded and thus this effect
reduces the recorded charge and, consequently, the deposited energy can be
underestimated. The recombination effect is proportional to the ionization
density, that is, the energy deposited per unit length. If $r$ is the
recombination probability, a modification of Birks’ formula was written [52]
in the form:

$\begin{split}&r=\frac{AdE/dx}{1+BdE/dx}+C,\\\ &C=1-A/B,\end{split}$ (3.6)

where $dE/dx$ is the LET expressed in (MeV cm2/g), and $A$, $B$ and $C$ are
empirical parameters that depend on the noble element, density and drift
electric field. For LAr, assumes:

$\begin{split}&A=0.05\times E_{drift}^{-0.85}\mathrm{[kV/cm]},\\\
&B=0.06\times E_{drift}^{-0.85}\mathrm{[kV/cm]},\\\ &C=1/6.\end{split}$ (3.7)

From the experimental point of view, the escape probability $(1-r)$ is
proportional with $Q$ \- the recorded charge (or "visible") and $Q_{0}>Q$ is
the charge produced initially.

The number of electrons produced by the ionization process, after
recombinations, will be:

$N^{\prime}_{ion}=(1-r)N_{ion}.$ (3.8)

Based on the experience of the ProtoDUNE-SP the minimum intensity of the
electric field (EF) required for the vertical drift (VD) is 300 V/cm; the goal
is to set an EF of 500 V/cm in order to achieve the requirements of the
horizontal drift (HD) and dual-phase (DP) far detectors. Experimentally the
electric field for drift was increased to test up to 650 V/cm [53]. The
survive probability versus strangelet’s energy is represented in Fig. 3. In
this case, two different values for the drift electric field are considered:
300 V/cm and 650 V/cm respectively.

Figure 3: Survival probability for two different values of the electric field
(300 V/cm continuous line, 650 V/cm dashed line).

The new excitation to ionization ratio considering both the scintillation
quenching and the electric field effect on the recombination processes can be
expressed as:

$\frac{{N^{\prime}}_{ex}}{{N^{\prime}}_{ion}}=\frac{N_{ex}}{N_{ion}}\frac{f_{l}}{(1-r)}$
(3.9)

Excitations to ionization ratio after considering both the scintillation
quenching and recombinations are indicated in Fig. 4 for two distinct values
of the drift electric field.

Figure 4: Excitations to ionization ratio after considering both the
scintillation quenching and recombinations inside two values of the electric
field.

Except the energy range 1-3 MeV, for the entire considered region, the
corrected ratio between the number of photons generated by the scintillation
mechanism and the number of ionization electrons can be used for the
discrimination between the incident projectiles with different values for
($Z$, $A$). The increase of the electric drift field considerably decreases
the $N_{ex}/N_{ion}$ ratio, especially for very light strangelets if their
energies are in the range of 0.1-10 GeV. In all the previous calculations of
$dE/dx$ and ranges the SRIM code was used [54].

Figure 5: The linear energy loss via ionizations for the considered
strangelets with incident energies up to 10 GeV.

When the incident strangelet (as an electrically charged particle) enters the
active medium of the detector, it transfers its energy and it is partitioned
between interactions with atomic electrons generating ionizations and
excitations and interactions with the nuclei of the medium. Due to the kinetic
energy received by the recoil nuclei, the same types of processes as the
primary projectile are generated.

The passage of atomic particles through a target involves scattering and
energy loss due to nuclear and electronic interactions. In the present
discussion the contributions from energy losses in nuclear interactions was
not considered because for all kinetic energies range of the projectile
considered in the present paper, the nuclear cross sections are
approximatively two order of magnitude below electronic cross sections.

An essential aspect to be discussed is the ability of such a detection system
to discriminate unambiguously between experimental signals from strangelets
and the radioactive background from intrinsic radioactive isotopes and from
cosmic background.

The detection principle consists in measuring the signal generated by the
ionization electrons and the photons coming from scintillations allowing a
three-dimensional reconstruction of the event.

In Figure 5 the linear energy loss, $dE/dx$, versus the ranges of light
strangelets is presented. At the same time, the value of the collected
electric charge can be unequivocally correlated with the range of event, this
representing an additional information in identification.

A distinct problem is the capability of these detectors to discriminate
between the signals of interest and the background. In accord with Farnese
[55], ICARUS Collaboration has estimated the background value to be around
$dE/dx\simeq 2.1$ MeV/cm. Thus, a clear discrimination between the considered
strangelets and the background can be done.

If we also consider the scintillation signals for the considered energy range
below 10 GeV, for strangelets with ($Z$, $A$) between (2,14) and (7,70), the
number of photons is $>10^{5}$ and can be completely discriminated from each
other. In LAr detectors, the intrinsic radioactive background is dominated by
the effect of the content of 39Ar in natural atmospheric argon. As a superior
limit, if the purification is not realized, the measured content of
radioactive 39Ar isotope will be $(8.0\pm 0.6)\times 10^{-16}$ g/g, or
$(1.01\pm 0.08)$ Bq/kg. The undesirable events induced by 39Ar are supposed to
be uniformly distributed in the active LAr volume, with an energy range up to
565 keV, the $\beta$ end-point energy, leading to very short tracks ($\sim 1$
mm) [56]. The emitted electrons are a source of noise for ionizations
(irrelevant in this case) and for scintillations. In order to reduce the level
of 39Ar, the use of underground argon is required [57]. In the DarkSide-50
Experiment the 39Ar activity was demonstrated to be 0.73 mBq/kg, that is 1400
lower than the activity of the atmospheric argon [58]. If we consider this
case, this component of the radioactivity introduces as a maximal value, a
noise around 200 photons, considerably below the signal generated by the
events of interest. In this estimation we supposed a tank of 10 kt of argon.
DUNE requires an average light yield of $>20$ photoelectrons/MeV with a
minimum of 0.5 photoelectron/MeV, corresponding to a photon detection
efficiency (PDE) of 2.6% and 1.3% [56], respectively.

##  4 Summary

In this paper we investigated the possibility that light strangelets, with
energies up to 10 GeV, can be directly observed as particles passing through
large volumes of LAr used as the active volume of a detector.

Light strangelets were investigated, this choice being justified because
experimentally there are a number of candidates as singular events reported in
experiments such as AMS-01, even though their existence is theoretically
controversial.

In this study the different experimental information obtained with a detector
operating as a calorimeter in the TPC mode were discussed. The use of signals
generated by electrons produced in the ionization process and scintillations
from atomic excitations, as well as the measurements of the linear energy loss
and the corresponding ranges, lead to the potential of such a detector to
discriminate between the signals of strangelets after ($Z$, $A$) at kinetic
energies up to 10 GeV - the limit considered in this study. It was showed that
lower electric drift field is beneficial for the proposed purpose.

##  5 Acknowledgment

This work was performed with the financial support of the contract no.
02/2020, PNCDI III 2015-2020, Programme 5, Module CERN-RO.

I.L. would like to thank UEFISCDI for partial support under project number PN-
III-P1-1.1-PD-2019-0178.

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# An Interpretable Low-complexity Model for Wireless Channel Estimation

Zainab R. Zaidi1, Tansu Alpcan1, Christopher Leckie2, and Sarah Erfani2
1Department of Electrical and Electronic Engineering, The University of
Melbourne, Australia
Email: $\\{$zainab.raziazaidi<EMAIL_ADDRESS>2 School of
Computing and Information Systems, The University of Melbourne, Australia
Email: $\\{$caleckie<EMAIL_ADDRESS>
###### Abstract

With the advent of machine learning, there has been renewed interest in the
problem of wireless channel estimation. This paper presents a novel low-
complexity wireless channel estimation scheme based on a tapped delay line
(TDL) model of wireless signal propagation, where a data-driven machine
learning approach is used to estimate the path delays and gains. Advantages of
this approach include low computation time and training data requirements, as
well as interpretability since the estimated model parameters and their
variance provide comprehensive representation of the dynamic wireless
multipath environment. We evaluate this model’s performance using Matlab’s
ray-tracing tool under static and dynamic conditions for increased realism
instead of the standard evaluation approaches using statistical channel
models. Our results show that our TDL-based model can accurately estimate the
path delays and associated gains for a broad-range of locations and operating
conditions. Root-mean-square estimation error remained less than $10^{-4}$, or
$-40$dB, for SNR $\geq 30$dB in all of our experiments.

The key motivation for the novel channel estimation model is to gain
environment awareness, i.e., detecting changes in path delays and gains
related to interesting objects and events in the field. The channel state with
multipath delays and gains is a detailed measure to sense the field than the
single-tap channel state indicator calculated in current OFDM systems.

###### Index Terms:

Channel estimation, tapped delay line model, ray-tracing, machine learning,
environment awareness

##  I Introduction

The extraordinary success of data-driven machine learning techniques in
estimating complex and non-linear system dynamics has led to the development
of novel wireless channel estimation (CE) and management approaches [1, 2, 3,
4, 5, 6, 7, 8, 9, 10]. However, machine learning (ML)-based channel estimation
models require substantial time and data for training, which makes them
unsuitable for real-time applications. Moreover, they are generally black-box,
or non-interpretable, and the complex patterns learned by the model cannot be
used to derive key insights about the radio environment. In this paper, we
propose a “gray-box” approach, where we use a mathematical model to represent
the channel and learn its parameters through a ML-inspired data-driven
approach. Our results show that this approach not only provides better channel
estimation than conventional methods, such as Least Squares, but the estimated
channel state is a more detailed measure to sense the field in an environment
awareness application. In our results section, we have also included a
preliminary case study of drone detection using our channel estimation model
with existing wireless signals.

The basic element of our approach is a Tapped-Delay Line (TDL) model of the
wireless channel. A TDL model represents the received signal as a
superposition of input signals traversing through line-of-sight or reflected
from various objects in the environment, each experiencing different delays
and attenuation. Due to movement of the mobile user or objects in the
environment, the number of these multipaths, their delays, and gains change
with time, which makes it extremely difficult to estimate and measure them
accurately.

ITU’s (International Telecommunication Union)
recommendations111https://www.itu.int/rec/R-REC-P.1407/en for parameterization
of multipath environments use power thresholds to determine the number of
multipath components. Not meant for real-time CE, ITU recommendations are for
characterization of different scenarios. Some efforts using ML have also been
proposed, e.g., [11] models the tap and gain estimation problem as multi-class
classification using deep neural network and requires training of the model by
a dictionary of channel models. This work improves the CE performance of a
previous method [12] which formulates the estimation of taps, for specific
transmitted and received signals in a Multi-Input Multi-Output (MIMO) system,
as an optimization problem.

Among more general CE approaches, i.e., not using a TDL model, notable works
are [2, 4], which use Generative Adversarial Networks (GANs), and [7], which
incorporates variational autoencoders, to estimate the distribution of
received signals or the channel impulse response of time varying channels. A
large body of work in CE focuses on the problem of extrapolating channel
estimates for all operating frequencies and times in an OFDM MIMO system, when
the actual CE estimates are based on a handful of pilot symbols, mostly
calculated by classical single-tap methods, such as least squares. Compressed
sensing is one of the earlier solutions and most ML-based methods consider it
as a baseline, such as, Approximate Message Passing [1], deep image processing
and image restoration [3], attention based encoder-decoder [6, 9], unfolded
neural networks [8], and GANs [1, 5]. Noting the issue of computation time,
recent works suggest bi-directional RNN [10] and [13] recommends transfer
learning and federated/split/distributed learning to speed up the CE process.

Figure 1: A small example of a single transmitter and receiver pair with one
in-band interferer. We assume time-invariant multipaths in this example. (a)
scenario, (b) TDL-NN CE model, and (c) CE results and actual path gains using
Matlab simulations.

This paper focuses on the problem of CE using transmitted and received signals
leveraging the respective strengths of classical and ML techniques, which can
provide an approach for performing CE in real-time and can also facilitate
environment awareness where objects and events of interest in the field can be
detected by detecting changes in estimated channel states. In particular, our
contributions are as follows:

  * •

We propose a novel low-complexity data-driven channel estimation approach
based on a neural network with one hidden layer using the reference signals
already available in wireless communication systems.

  * •

We evaluate the CE model using Matlab’s ray-tracing tool instead of the
classical approach of employing statistical channel models, for increased
realism.

  * •

Our CE model shows accurate estimation (root-mean-square error is under
$10^{-4}$), or $-40$dB for different static and dynamic scenarios with
reasonable SNR ($\geq 30$dB).

  * •

We also show the potential of our CE model in finding outlier events in the
neighbourhood of the transmitter and receiver.

##  II System Model

The signal from a wireless transmitter can be represented as [14]:

$\displaystyle x(t)=\operatorname{Re}\\{s(t)e^{j2\pi f_{c}t}\\},$

where $s(t)$ is the baseband signal and $f_{c}$ is the carrier frequency. The
discrete time versions of the baseband and transmitted signal are
$\textbf{{s}}_{n}$ and $\textbf{{x}}_{n}$ respectively, both vectors of length
$K$. Considering that the signal will reach the receiver through line-of-sight
and various reflected paths, which experience different Doppler shifts, the
received signal $y(t)$ can be written as [14],

$\displaystyle
y(t)=\operatorname{Re}\left\\{\sum_{i=0}^{D(t)-1}{c_{i}(t)s(t-\tau_{i}(t))e^{j(2\pi
f_{c}(t-\tau_{i}(t))+{\phi}_{i}(t))}+w(t)}\right\\},$

where $c_{i}(t)$, $\tau_{i}(t)$, and ${\phi}_{i}(t)$ are the gain, delay, and
Doppler shift of the $i$th path, $i=0$ corresponds to the line-of-sight and
there are $D(t)$ distinct multipaths. $w(t)$ is the white Gaussian noise. In a
time varying channel model, the parameters
$\\{c_{i}(t),\tau_{i}(t),{\phi}_{i}(t)\\}~{}\forall{i}$, and $D(t)$ are random
processes. The channel impulse response can be written as

$\displaystyle
h(\theta_{h}(t))=\sum_{i=0}^{D(t)-1}{c_{i}(t)\delta(t-\tau_{i}(t))e^{j{\phi}_{i}(t)}},$
(1)

where $\theta_{h}(t)$, the parameters to describe channel impulse response are
defined as

$\displaystyle\theta_{h}(t)=\\{c_{i}(t),\tau_{i}(t),{\phi}_{i}(t)\\},~{}i=0,\cdots,D(t).$
(2)

As can be seen in the above expression, the total phase shift experienced by
the signal is $-2\pi f_{c}\tau_{i}(t)+{\phi}_{i}(t)$ due to multipaths and
Doppler effect. The discrete time received signal and received baseband signal
are denoted by $\textbf{{y}}_{n}$ and $\textbf{{z}}_{n}$ respectively, which
are also considered as $K$ length vectors.

###  II-A An Illustrative Example

Before formally presenting our CE approach, we illustrate the underlying
concept with the help of a small example as shown in Fig. 1. In Fig. 1a, there
is a single transmitter/receiver pair with one in-band interferer or
unintended signal leakage. For this case, CE is reduced to estimating
$\theta_{h}=\\{c_{i},\tau_{i}\\},~{}\forall i$. We assume we have $N$ samples
of transmitted signals, $\textbf{{x}}_{1,n}$ from the primary transmitter and
$\textbf{{x}}_{2,n}$ from the interferer, and the associated received signal
samples $\textbf{{y}}_{n}$, also given as

$\textbf{{y}}_{n}=\sum_{i=0}^{D_{1}}c_{1,i}\textbf{{x}}_{n-\tau_{1,i}}+\sum_{i=0}^{D_{2}}c_{2,i}\textbf{{x}}_{2,n-\tau_{2,i}}+\textbf{{w}}_{n},$

where $D_{1}$ and $D_{2}$ are the numbers of multipaths and $\textbf{{w}}_{n}$
is the white Gaussian noise.

Figure 1b shows our approach for estimating the path delays and gains. The
basic unit of our model, called TDL-NN, is a neural network (NN) with a single
input and single output and one hidden layer representing $L$ taps of constant
delays in multiples of $\tau$. The parameters $L$ and $\tau$ should cover all
possible distinct path delays. The output layer weights represent path gains
and are learned iteratively by minimizing the MSE (Mean Square Error) loss
function, i.e.,

$\frac{1}{N}\sum_{n=1}^{N}\|\textbf{{y}}_{n}-\hat{\textbf{{y}}}_{n}\|^{2}_{2},$

where
$\hat{\textbf{{y}}}_{n}=\sum_{m=1}^{2}\sum_{i=0}^{L}c_{m,i}\textbf{{x}}_{m,n-i\tau}$.

Using Matlab, we simulated the scenario of $N=10^{4}$ transmitted symbols,
with $20$ samples per symbol and each sample duration is $100$ns,
$\tau_{1,i}=\\{0,200,800\\}$ns, $\tau_{2,i}=\\{100,300,900\\}$ns. Both
transmitters use QPSK modulation, carrier frequency is 3MHz, and Gaussian
noise is added with SNR of 15dB. Figure 1c shows the actual tap gains and the
predicted gains from the PyTorch implementation of the model with delay
resolution $\tau=100$ns and $L=15$. It can be seen that our TDL neural network
accurately learned the path gains and delays. Our model uses a block size of
$10^{4}$, $5000$ epochs, and the Adam optimizer with learning rate of $0.01$.

##  III TDL-NN Channel Estimation Approach

Figure 2: TDL-NN channel estimation for a scenario with multiple transmitters
and one receiver.

In this section, we formally present our TDL neural network approach for
channel estimation. All wireless communication systems use pilots or reference
signals, which are known sequences of symbols transmitted over the wireless
channel and the respective received signals are used for channel state
estimation. Figure 2 shows $\textbf{{x}}_{m,n}$ transmitted reference signals
from one primary transmitter and $M-1$ multiple interferers. The received
reference signal is $\textbf{{y}}_{n}$. Our CE model is trained using
$\textbf{{x}}_{m,n}$ and $\textbf{{y}}_{n}$ for $m={1,~{}\cdots,~{}M}$ and
$n={1,~{}\cdots,~{}N}$, where $N$ is the available reference symbols for
training.

The model shifts each input or transmitted signal, $\textbf{{x}}_{m,n}$, from
$0\times\tau$ to $L\times\tau$, where $\tau$ is the tap delay resolution in
the hidden layer. It then computes the weights of the output layers, or
coefficients $c_{m,i}$ for all taps and also for all $M$ sub-models, by
minimizing the MSE loss function as shown in the Fig. 2. Our implementation in
PyTorch uses $N$ as the block-size and a learning rate of $0.01$. The
computation complexity of our model grows as $\mathcal{O}(N\times L\times e)$,
where $e$ is the number of epochs. As an option, we also implemented weight
pruning in our model to push the smaller weights to 0. Note that our model
uses the time shift idea as in conventional time delay neural networks [15],
which were proposed for time series analysis, but otherwise it is a customized
model to suit our application of channel estimation.

Path delays and gains change with time as mobile users or objects in the
environment move. Weather conditions, such as, rain, wind, etc., also impact
path gain. For a dynamic environment, the multipath delays and gains should be
updated as new pilot symbols are received. Figure 2 shows one snapshot of our
approach, which estimates $c_{m,i}(n)$ for each path tap $i$ at time $n$ by
updating the model weights using pilot transmitted and received signals
$\textbf{{x}}_{m,n}$ and $\textbf{{y}}_{n}$. Note that we are not explicitly
estimating Doppler phase shift $\phi_{i}(t)$ associated with the path $i$ from
the model, instead tracking the variations in path delays and gains.

Moreover, we remark that our TDL neural network is a realization of a typical
Steepest Gradient Descent (SGD) algorithm iteratively computing coefficients
of a linear predictor when the objective function is to minimize mean square
error between actual observations and predictions. However, using SGD in a
neural network setting with parallelization of computations is much more
effective and an order of magnitude faster than the conventional iterative
approach.

Furthermore, in systems using OFDM modulation, such as 4/5G or LTE, reference
signals or pilots are embedded in subframes along with the payload. Channel
estimation requires these pilots to be extracted first and then used to
estimate the channel. In this case, baseband signals can be used for CE
instead of transmitted or received signals with a carrier.

##  IV Experimental Results

###  IV-A Ray-tracing Simulations

In order to validate our TDL-based CE approach, we used Matlab’s ray-tracing
tool to generate received signal data. To the best of our knowledge, this is
the first CE paper which uses ray-tracing data for validation. Prior
approaches have used statistical channel models, such as Rayleigh or Rician
models, to generate the data [2, 4]. Use of ray-tracing not only allows us to
assess the performance of our CE method by comparing it with the ground truth,
it also facilitates testing the model efficacy under a wide-range of practical
scenarios.

####  IV-A1 Static Multipath Scenario

(a)

(b)

(c)

(d)

Figure 3: Static multipath scenario, (a) scene with one transmitter and
receiver, (b) actual and estimated path delays and gains, taps 5 and 17 have 2
different rays and the estimation shows the combined effect, (c) scene of (a)
with two interferers, (d) estimation error versus SNR for scenarios of one
transmitter with no interferer (1 tx), one interferer (2 txs), and 2
interferers (3 txs). Figure 4: Estimation error versus SNR of TDL-NN channel
estimation model in different urban, suburban, and rural scenarios.

We placed a transmitter and receiver in the Melbourne Central Business
District scenario as shown in Fig. 3a. The transmitter is configured to
transmit QPSK modulated signal with $900$MHz carrier frequency and $50$dBm
power. It is placed at a height of 10m on the roof of a tall building and uses
a directional antenna with azimuth angle of 30°. The receiver antenna’s angle
is selected as 120°, and it is at a height of 1m from the ground as shown in
Fig. 3a.

The maximum number of reflections in Matlab’s ray-tracing tool can be set as
10 (R2023b). Here, the tool finds 8 paths to the receiver. White Gaussian
noise is added to the faded signal with SNR of $30$dB. The signal is amplified
by $20$dB on reception. $10,000$ symbols, with $100$ samples per symbol, are
generated and transmitted through the multipath channel. Each sample duration
is $20$ns corresponding to a sampling frequency of $5\times 10^{7}$
samples/second. Unless otherwise stated, all the experiments in this paper
uses the same configurations.

As shown in Fig. 3b, our TDL-NN model accurately finds channel taps and gains.
Taps 5 and 17 show discrepancies, but the estimated gain is the collective
impact of two rays with the same path delay. With weight pruning, smaller tap
coefficients can be pushed to 0. However, for an unknown or time varying
channel, this may not be advisable.

We also calculated the RMS (Root Mean Square) estimation error for different
SNR conditions, as shown in Fig. 3d. Here, each point is averaged over 10
different estimation runs with $10,000$ training samples. In a high noise
environment, the estimation error becomes excessive. We added interferers in
our example as shown in Fig. 3c and estimated path delays and gains for all
transmitted signals. As expected, the presence of multiple transmitters
worsens the estimation error, however, the estimation accuracy of TDL-NN model
remains high, for reasonable SNR, despite the presence of in-band
interference.

(a)

(b)

Figure 5: Changes in signal multipaths as the mobile user moves around a city
block, (a) scene with changing rays when the receiver is at different
locations, and (b) actual and estimated path delays and gains.

Figure 4 shows the estimation error vs. SNR plots for two urban scenarios
(Melbourne Central Business District, Victoria, Australia and Manhattan, NY,
USA), two suburban scenarios (Mount Waverley, Victoria, Australia and
Chapelford, UK), and two rural areas (Wodonga, Victoria, Australia and Golden,
CO, USA). The urban scenarios are more challenging due to the number of
multipaths, whereas suburban areas have comparatively smaller number of
reflections, but line-of-sight may be obstructed. Also, path attenuation may
be excessive due to larger distances between users and base-stations in a
sparse coverage situation. Rural areas on the other hand have mostly just the
line-of-sight. Our TDL based CE approach performs exceptionally well for all
conditions for reasonable SNR conditions as shown in Fig. 4.

####  IV-A2 Mobile Receiver Scenario

In this experiment, we picked an open-access GPS track of a walker around a
block of streets, representing a receiver, as shown in Fig. 5a. Here a
transmitter is placed 10m above the roof of a nearby building. Other
configurations are the same as the static scenario. The colours of the GPS
track shows the signal strength at that location, Figure 5a shows the receiver
at multiple locations during the walk and changing multipath conditions.

This track has 405 points, sampling walker’s location every second. We
generated $1000$ transmitted symbols for each location, each symbol is sampled
205 times at the rate of $5\times 10^{7}$ samples/second. The estimation
results are shown in Fig. 5b where it can be seen that the TDL-NN model is
tracking the changing multipaths at every step accurately. The discrepancy in
actual and estimated path gains is due to multiple paths with the same delay,
and estimated gains are showing the cumulative impact. The lower left-hand
image in Fig. 5b shows some sporadic path delays observed during the walk by
lighter pixels. The right-hand image of estimated tap gains shows that our
model was able to catch them as well.

We also repeated the experiment with SNR of $30$dB and $20$dB and also for a
GPS trace, containing 400 points, of a bike ride in NY, USA. The RMS
estimation error for both scenarios and for different SNR conditions are shown
in Table I, where it can be seen that TDL-NN model effectively estimates the
dynamic channels.

TABLE I: RMS Estimation error versus SNR Scenario | SNR = 40dB | SNR = 30dB | SNR = 20dB
---|---|---|---
Walk, Melbourne | $3.3\times 10^{-6}$ | $3.3\times 10^{-5}$ | $3.2\times 10^{-4}$
Bike-ride, NY | $2.7\times 10^{-6}$ | $2.7\times 10^{-5}$ | $2.8\times 10^{-4}$

###  IV-B Comparison with other Channel Estimation Methods

Figure 6: Performance comparison in terms of BER, 1- channel estimated from
TDL-NN with MLSE equaliser, 2,3- linear equalizer with LMS and RLS algorithm
for channel estimation, 4- perfect channel with MLSE equalizer, 5- CGAN
channel model with AE to model Tx/Rx [4].

As discussed above, our TDL-NN model is a novel approach and is different from
classical channel estimation. In order to compare our results with prior art,
we calculate the BER (Bit Error Rate) after equalization and demodulation
using the channel estimates provided by our TDL-NN model. Using BER, we can
also compare our model with the CE methods, not necessarily estimating path
delays and gains.

We used Matlab’s MLSE equalizer with our channel estimates to equalize the
received signal, and BER is calculated for $90,000$ received signal samples
for different SNR conditions for a scenario with 3 distinct multipaths. MLSE
equalizer employs the Viterbi algorithm for decoding, which can also perform
poorly in high noise conditions. In order to provide a fair comparison to TDL-
NN, we also plot BER vs. SNR for a perfect channel using ray-tracing ground
truth in Fig. 6. TDL-NN closely tracks MLSE equalization with the perfect
channel and the errors in case of TDL-NN is largely due to MLSE equalization.

TDL-NN is also compared with Matlab’s linear equalizer using LMS (Lsast Mean
Square) and RLS (Recursive Least Square) algorithms and performs very well
compared to both as shown in the Fig. 6. However, we observe smaller BER with
CGAN [4]. The approach in [4] models the end to end communication system with
an autoencoder (transmitter with 5 and receiver with 8 hidden layers models)
with CGAN (generator with 4 and discriminator with 6 hidden layers) modelling
the channel effects and BER shows the overall performance, not just that of
the CGAN. Moreover, the channel is trained using $5\times 10^{6}$ data samples
with SNR = 35dB for all cases in [4] and the results are shown here after 32
iterations which took approximately 4 days on the same GPU, where TDL-NN is
trained with $10,000$ data samples and $5,000$ training epochs in less than
$1/2$ hour. We remark that for more complex channels with non-linear effects,
GANs CE might be better than linear TDL-NN, but its computation time and
training requirements make it unsuitable for practical scenarios. In contrast,
our model can be refined for real-time applications. Moreover, TDL-NN
estimates multipath delays and gains and provides visibility into the channel,
which is especially useful for environment sensing and awareness.

###  IV-C Example Application: Drone Detection

In this section, we elaborate further on the interpretability of our TDL-NN
model using a preliminary case study of detecting an authorised drone in an
urban scenario. Our aim here is to show the potential of our TDL-NN approach,
and detection of interesting events using estimated channel state is still
work-in-progress. We created a scene using Matlab’s ray-tracing tool where
four mobile users on an urban street are transmitting to four base-stations,
mounted on the roofs, while a drone fly over the street at an height of 10m as
shown in Fig. 7. The communication parameters are set as previous examples.
During it’s flight, drone impacts some of the signal paths, which results in
changes in channel state, both in delay and gains. For this preliminary
example, we assume no traffic on the road. The objective is to detect the
instances where a change happens. We used existing change/novelty detection
algorithms for this purpose which can be trained on normal data and anomalies
can be detected as outliers. For this experiment, we assume to have training
data from the four base-stations without the interference of drone. The
results of drone detection are presented in Table II. This is an on-going
research project and we are building novel detection models based on the
correlation between successively estimated channel states to detect sporadic
events. Nevertheless, this example show the promising potential of our
approach for environment awareness application.

Figure 7: Scene with four mobile users transmitting each to a base-station, which are continuously monitoring channel states. A drone flying on the street impacts the signal multipaths as shown in insets. TABLE II: Drone Detection Results Model | Precision | Recall | F1 score
---|---|---|---
k-means | 0.6667 | 0.5 | 0.5717
PCA with k-means | 0.6667 | 0.6667 | 0.667
OCSVM | 0.1667 | 0.6667 | 0.2667
LOF | 0.2 | 1 | 0.3333
CBLOF | 0.2143 | 1 | 0.3529
DeepSVDD | 0.1875 | 1 | 0.3158

##  V Conclusion

In this paper, we present a practical low-complexity channel estimation
approach based on a tapped delay line model. In our approach, the tap delays
and gains are estimated through a neural network with one hidden layer. Our
extensive experiments, using Matlab’s ray-tracing tool for generation of
transmitted and received signal samples, show that this simple model can
estimate the time-varying channel with reasonable accuracy in the presence of
white noise and also in-band interference. This approach also enables
developing a novel environmental awareness application where interesting
objects and events can be detected through the changes in the channel state.

Although Matlab ray-tracing is an excellent resource for this research, we
understand that our experiments are limited by how accurately ray-tracing is
performed by the tool. The next step in this research is to apply the CE model
in a real environment.

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