# Thermal nuclear pairing within the self-consistent quasiparticle RPA

N. Dinh Dang1,2 and N. Quang Hung1,3 1 Theoretical Nuclear Physics Laboratory,
RIKEN Nishina Center for Accelerator-Based Science, 2-1 Hirosawa, Wako City,
351-0198 Saitama, Japan
2 Institute for Nuclear Science and Technique, Hanoi, Vietnam
3 Institute of Physics, Hanoi, Vietnam<EMAIL_ADDRESS>(N.D.D<EMAIL_ADDRESS>(N.Q.H.)

###### Abstract

The self-consistent quasiparticle RPA (SCQRPA) is constructed to study the
effects of fluctuations on pairing properties in nuclei at finite temperature
and $z$-projection $M$ of angular momentum. Particle-number projection (PNP)
is taken into account within the Lipkin-Nogami method. Several issues such as
the smoothing of superfluid-normal phase transition, thermally assisted
pairing in hot rotating nuclei, extraction of the nuclear pairing gap using an
improved odd-even mass difference are discussed. A novel approach of embedding
the PNP SCQRPA eigenvalues in the canonical and microcanonical ensembles is
proposed and applied to describe the recent empirical thermodynamic quantities
for iron, molybdenum, dysprosium, and ytterbium isotopes.

##  1 Introduction

Sharp phase transitions such as the superfluid-normal (SN) or shape ones are
prominent features of infinite systems such as metal superconductors, ultra-
cold gases, liquid helium, etc. They are well described by many-body theories
such as the BCS, RPA or quasiparticle RPA (QRPA). The situation changes in
finite small systems such as atomic nuclei, where strong quantal and thermal
fluctuations strongly or completely smooth out these sharp phase transitions.
It is well known that the conventional BCS, RPA or QRPA theories fail in a
number of cases in the description of the ground states as well as excited
states of these systems. The reason is that strong fluctuations invalidate the
assumptions, based on which the main equations of these theories have been
derived. Amongst these assumptions are the Cooper pairs, which violate the
particle-number conservation, and the closely related quasiboson-approximation
(QBA) used in the (Q)RPA, which violates the Pauli principle between the
fermion pairs. These assumptions cause the BCS and QRPA to break down at a
certain critical value $G_{c}$ of the pairing interaction parameter $G$, below
which the BCS theory only has a trivial solution with zero pairing gap
$\Delta=$ 0\. The same is true in the weak coupling region, where the
particle-particle RPA is valid but its solution also breaks down at $G\geq
G_{c}$. Meanwhile, the exact solution of the pairing problem exposes no
singularity at any $G$ [1]. Similarly, at finite temperature $T\neq$ 0, the
omission of quasiparticle-number fluctuations (QNF) within the BCS theory
leads to the collapse of the pairing gap at the critical temperature $T_{c}$,
corresponding to the temperature of the SN phase transition in infinite
systems. Meanwhile, the exact eigenvalues of the pairing problem embedded in
the canonical ensemble (CE) shows a smooth decreasing pairing energy with
increasing $T$ due to thermal fluctuations incorporated in the CE [2]. In
rotating nuclei, strong fluctuations also smear out the Mottelson-Valatin
effect, according to which the pairing gap, existing at zero angular momentum
$M=$ 0, would collapse at a certain critical angular momentum $M_{c}$. This
situation means that, in order to be reliable, the BCS, RPA, and/or QRPA
theories need to be corrected to include these effects of fluctuations when
applied to nuclei, in particular, the light ones. This is done within the
framework of the self-consistent QRPA (SCQRPA) presented in this work.

##  2 Formalism

We consider the pairing Hamiltonian $H=\sum_{k>0}\epsilon_{k}\hat{N}_{\pm
k}-G\sum_{kk^{\prime}}\hat{P}_{k}^{\dagger}\hat{P}_{k^{\prime}}~{}$, where
$\hat{N}_{\pm k}=a_{\pm k}^{\dagger}a_{\pm k}$ is the particle-number
operator, and
$\hat{P}_{k}=a_{k}^{\dagger}a_{-k}^{\dagger},\hat{P}_{j}=(\hat{P}_{j}^{\dagger})^{\dagger}$
are the pairing operators. The operators $a_{k}^{\dagger}$ and $a_{k}$ are
respectively the single-particle creation and destruction operators. This
Hamiltonian has been diagonalized exactly in [1]. The exact partition function
is constructed by embedding the exact eigenvalues into the CE as $Z_{\rm
Exact}(\beta)=\sum_{S}d_{S}\exp({-\beta\varepsilon_{S}^{\rm Exact}})$ , with
the degeneracy $d_{S}=2^{S}$, inverse temperature $\beta=1/T$, and
$S=0,2,...N$ being the total seniority of the system. Knowing the partition
function $Z$, one calculates the free energy $F$, entropy $S$, total energy
${\cal E}$, heat capacity $C$, and pairing gap $\Delta$ as $F=-T{\rm ln}Z(T)$,
$S=-{\partial F}/{\partial T}$, ${\cal E}=F+TS$, $C={\partial{\cal
E}}/{\partial T}$, and $\Delta=[-G({\cal
E}-2\sum_{k}\epsilon_{k}f_{k}+G\sum_{k}f_{k}^{2})]^{1/2}$, where $f_{k}$ is
the single-particle occupation number on the $k$th level obtained by averaging
the state-dependent occupation numbers $f_{k}^{(S)}$ within the CE [2].

The SCQRPA theory [3, 4] includes a set of BCS-based equations, corrected by
the effects of QNF, namely

$\Delta_{k}=\Delta+\delta\Delta_{k}~{},\hskip
5.69054pt\Delta=G\sum_{k^{\prime}}\langle{\cal D}_{k^{\prime}}\rangle
u_{k^{\prime}}v_{k^{\prime}}~{},\hskip
5.69054pt\delta\Delta_{k}=2G\frac{\delta{\cal N}_{k}^{2}}{\langle{\cal
D}_{k}\rangle}u_{k}v_{k}~{}.$ (1) $N=2\sum_{k}\bigg{[}v_{k}^{2}\langle{\cal
D}_{k}\rangle+\frac{1}{2}\big{(}1-\langle{\cal
D}_{k}\rangle\big{)}\bigg{]}~{}.$ (2)

where $u_{k}$ and $v_{k}$ are the Bogoliubov’s coefficients,

$u_{k}^{2}=\frac{1}{2}\bigg{(}1+\frac{\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\bigg{)}~{},\hskip
14.22636ptv_{k}^{2}=\frac{1}{2}\bigg{(}1-\frac{\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\bigg{)}~{},\hskip
5.69054ptE_{k}=\sqrt{(\epsilon^{\prime}_{k}-Gv_{k}^{2}-\lambda)^{2}+\Delta_{k}^{2}}~{},$
(3)

with the renormalized single-particle energies $\epsilon^{\prime}_{k}$

$\epsilon_{k}^{\prime}=\epsilon_{k}+\frac{G}{\langle{\cal
D}_{k}\rangle}\sum_{k^{\prime}}(u_{k^{\prime}}^{2}-v_{k^{\prime}}^{2})\bigg{(}\langle{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}\neq k}^{\dagger}\rangle+\langle{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}}\rangle\bigg{)}~{},$ (4)

$\langle{\cal D}_{k}\rangle=1-2n_{k}$, the quasiparticle-pair operators ${\cal
A}_{k}^{\dagger}=\alpha_{k}^{\dagger}\alpha_{-k}^{\dagger}$, ${\cal
A}_{k}=({\cal A}_{k}^{\dagger})^{\dagger}$, and $\delta{\cal N}_{k}^{2}\equiv
n_{k}(1-n_{k})$ is the QNF on $k$th level. To avoid level-dependent gaps
$\Delta_{k}$, the level-weighted gap
$\bar{\Delta}_{k}=\sum_{k}\Delta_{k}/\Omega$ ($\Omega$ is the number of
levels) is considered in the numerical results. Because of coupling to
collective vibrations beyond the quasiparticle mean field, the quasiparticle
occupation number $n_{k}$ is not given by a Fermi-Dirac distribution of free
fermions, but is found from the integral equation [4]

$n_{k}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\gamma_{k}(\omega)(e^{\beta\omega}+1)^{-1}}{[\omega-
E_{k}-M_{k}(\omega)]^{2}+\gamma_{k}^{2}(\omega)}d\omega~{},$ (5)

where the mass operator $M_{k}(\omega)$ and the quasiparticle damping
$\gamma_{k}(\omega)$ are functions of $n_{k}$, SCQRPA eigenvalues
$\omega_{\mu}$, SCQRPA ${\cal X}_{k}^{\mu}$ and ${\cal Y}_{k}^{\mu}$
amplitudes, SCQRPA phonon occupations numbers $\nu_{\mu}$, as well as $u_{k}$
and $v_{k}$. The SCQRPA submatrices $A$ and $B$ contain the screening factors
$\langle{\cal A}_{k}^{\dagger}{\cal A}^{\dagger}_{k^{\prime}}\rangle$ and
$\langle{\cal A}_{k}^{\dagger}{\cal A}_{k^{\prime}}\rangle$ so that the set of
SCQRPA equations should be solved self-consistently with Eqs. (1), (2) and (5)
to simultaneously determine $\bar{\Delta}$, chemical potential $\lambda$,
$n_{k}$, $\omega_{\mu}$, ${\cal X}_{k}^{\mu}$ and ${\cal Y}_{k}^{\mu}$. To
eliminate particle-number fluctuations inherent in the BCS theory, the Lipkin-
Nogami (LN) particle-number projection (PNP) [5] is applied on top of Eqs. (1)
and (2). The ensuing theory, called the LNSCQRPA theory, has also been
extended to include the finite $z$-projection $M$ of angular momentum
(noncollective rotation) [6]. The set of obtained equations is formally the
same except that now, depending on the single-particle spin projections
$\mp\gamma m_{k}$ with $\gamma$ being the angular velocity, one has two types
of quasiparticle occupation number, $n_{k}^{\pm}$, so that $\langle{\cal
D}_{k}\rangle=1-n_{k}-n_{-k}$. At $T=$ 0 and $M=$ 0 the SCQRPA theory reduces
to its zero temperature and non-rotating limit, where $\langle{\cal
D}_{k}\rangle=1/[1+2\sum_{\mu}({\cal Y}_{k}^{\mu})^{2}]$ [3].

##  3 Numerical results and discussions

Figure 1: Energies of the ground state (a) and first excited states for the
$N=\Omega=$ 10 as functions of $G$ at $T=M=$ 0\. $\omega_{ppRPA}={\cal
E}_{1}(N+2)-{\cal E}_{0}(N+2)$ with the ppRPA eigenvalues ${\cal E}_{i}$.

Shown in Fig. 1 are the energies of the ground state (a) and first excited
state (b) obtained at $T=M=$ 0 within several approximations as well as by
exactly diagonalizing the pairing Hamiltonian for the schematic model, which
consists of $\Omega$ doubly-folded equidistant levels with the single-particle
energies chosen as $\epsilon_{k}=k-(\Omega+1)/2$ MeV. The displayed results
are for the half-filled case with $N=\Omega=$ 10, and plotted as functions of
the pairing interaction parameter $G$. It is seen that the LNSCQRPA describes
rather well the exact energies of both the ground and first excited states
without any discontinuity in the region around $G_{c}$, where all other
approaches such as the RPA, QRPA, and SCQRPA collapse.

Figure 2: Level-weighted pairing gap $\bar{\Delta}$, total energy $E$, and
heat capacity $C$, as functions of $T$ for $N=\Omega=$ 10 (a - c) and 50 (d -
f) obtained within the FTBCS (dotted), FTBCS1 (thin solid), FTLN1 (thin
dashed), SCQRPA (thick solid), LNSCQRPA (thick dashed). The dash-dotted lines
for $N=$ 10 are the exact CE results.

The level-weighted gap, total energy, and heat capacity obtained for the
systems with $N=\Omega=$ 10 and 50 are shown as functions of temperature $T$
in Fig. 2. Beside the predictions by the SCQRPA, LNSCQRPA, as well as by their
corresponding limits, FTBCS1 and FTLN1, where coupling to QRPA is omitted
(i.e. $n_{k}$ is described by the Fermi-Dirac distribution for free fermions),
and the finite-temperature (FT) BCS results, the exact CE results are also
shown. This figure clearly demonstrates how QNF smooth out the sharp SN phase
transition in finite systems. The pairing gap never collapses, but decreases
monotonously with increasing $T$, whereas the spike at $T_{c}$ in the heat
capacity, which serves as a signature of sharp SN phase transition within the
FTBCS, becomes strongly depleted to a broad bump.

Figure 3: Level-weighted pairing gaps $\bar{\Delta}$ for $N=\Omega=$ 10 as a
functions of temperature $T$ at various values of $M/M_{c}$ and angular
momentum $M$ at various values of $T/T_{c}$ within the FTBCS (a, c) and FTBCS1
(b, d) theories.

At finite angular momentum $M\neq$ 0, the FTBCS theory predicts the Mottelson-
Valatin effect, according to which, the zero-temperature pairing gap decreases
with increasing $M$ and collapses at $M=M_{c}$ because the angular momentum
blocks the levels close to the Fermi surface [Fig. 3 (a) and 3 (c)]. Thermal
effects relax the blocking, opening some levels around the Fermi surface for
pairing. This leads to the thermally assisted pairing gap (or pairing
reentrance), according to which at a certain $T=T_{1}$ the pairing gap becomes
finite even at $M>M_{c}$ [7, 8]. With increasing $T$ thermal effects again
break the pairs so that the gap disappears at $T=T_{2}>T_{1}$ [See Fig. 3 (a)
for $M/M_{c}\geq$ 1]. In finite systems, the QNF smooth out both the
Mottelson-Valatin transition and thermal assisted pairing, for instance, for
$N=\Omega=$ 10 with $G=$ 0.5 MeV at $T/T_{c}\geq$ 1, the gap only decreases
monotonously with increasing $M$ but never vanishes [See Fig. 3 (d) for
$M/M_{c}\geq$ 1], whereas at $M/M_{c}\geq$ 3, the pairing gap reappears at
$T>T_{1}$ but remains finite with further increasing $T$ [See Fig. 3 (b) for
$M/M_{c}\geq$ 3].

The odd-even mass difference contains the admixture with the contribution from
uncorrelated single-particle configurations, which increases with $T$.
Therefore, the simple extensions of this formula to obtain the three-point and
four-point gaps, in principle, do not hold at finite temperature. We propose
an improved odd-even mass difference formula at $T\neq$ 0, namely

$\widetilde{\Delta}^{(3)}(\beta,N)=\frac{G}{2}\bigg{[}(-1)^{N}+\sqrt{1-4\frac{S^{\prime}}{G}}\bigg{]}~{},\hskip
5.69054ptS^{\prime}=\frac{1}{2}\big{[}\langle{\cal
E}(N+1)\rangle_{\alpha}+\langle{\cal
E}(N-1)\rangle_{\alpha}\big{]}-\langle{\cal E}(N)\rangle_{\alpha}^{(0)}~{},$
(6)

where $\langle{\cal E}(N)\rangle_{\alpha}$ is the total energy of the system
with $N$ particles within the grand canonical ensemble (GCE) ($\alpha=$GC) or
CE ($\alpha=$C); $\langle{\cal E}\rangle_{\alpha}^{(0)}\equiv
2\sum_{k}\big{[}\epsilon_{k}-Gf_{k}^{(\alpha)}/2\big{]}f_{k}^{(\alpha)}$ with
$-G\sum_{k}[f_{k}^{(\alpha)}]^{2}$ coming from uncorrelated single-particle
configurations.

Figure 4: Pairing gaps extracted from the odd-even mass differences as
functions of $T$ for $N=$ 10 (a,c) and $N=$ 9 (b,d) ($\Omega=$ 10, $G=$ 0.9
MeV). The thin solid and thick solid lines denote the gap $\Delta^{(3)}(N)$
($\Delta^{(4)}(N)=[\Delta^{(3)}(N)+\Delta^{(3)}(N-1)]/2)$, and the improved
gap $\widetilde{\Delta}^{(i)}(\beta,N)$ ($i=$ 3, 4) from Eq. (6),
respectively. The dash-dotted lines are the canonical gaps $\Delta^{(i)}_{\rm
C}$.

Shown in Fig. 4 are the pairing gaps $\Delta^{(i)}(\beta,N)$ ($i=$ 3 and 4),
obtained for $N=$ 9 and 10 ($\Omega=$ 10) by using the simple extension of the
odd-even mass formula to $T\neq$ 0 as well as the modified gaps
$\widetilde{\Delta}^{(i)}(\beta,N)$ from Eq. (6), and the canonical gaps
$\Delta^{(i)}_{\rm C}$. It is seen in Fig. 4 that the naive extension of the
odd-even mass formula to $T\neq$ 0 fails to match the temperature-dependence
of the canonical gap $\Delta^{(i)}_{\rm C}$. The gap $\Delta^{(3)}(\beta,N=9)$
even turns negative at $T>$ 2.4 MeV, suggesting that such simple extension of
the odd-even mass difference to finite $T$ is invalid. The modified gap
$\widetilde{\Delta}^{(i)}(\beta)$ is found in much better agreement with the
canonical one, whereas the modified four-point gaps
$\widetilde{\Delta}^{(4)}(\beta)$ practically coincide with the canonical
gaps. The comparison in Fig. 4 suggests that formula (6) is a much better
candidate for the experimental gap at $T\neq$ 0, rather than the simple odd-
even mass difference.

Figure 5: CE heat capacities $C$ as functions of $T$ and MCE entropies $S$ as
functions of excitation energy $E^{*}$ for 56Fe, 94Mo, 162Dy, and 172Yb.
Experimental data are taken from [11].

In order to construct a feasible description for pairing within the CE, the
eigenvalues of the LNBCS and LNSCQRPA, obtained for each total seniority $S$
at $T=$ 0, are embedded into the CE by using the partition function
$Z_{\gamma}(\beta)=\sum_{S}d_{S}e^{-\beta\varepsilon_{S}^{\gamma}}$ ($\gamma=$
LNBCS, LNSCQRPA). The resulting approaches are called the CE-LNBCS and CE-
LNSCQRPA, respectively [10]. These solutions are also embedded into the
microcanonical ensemble (MCE) by using the Boltzmann’s definition for entropy,
$S({\cal E})={\rm ln}{W}({\cal E})$, where ${W}({\cal E})$ is the number of
accessible states within the energy interval (${\cal E},{\cal E}+\delta{\cal
E}$). The corresponding approaches are called the MCE-LNBCS and MCE-LNSCQRPA,
respectively [10].

The CE heat capacities and MCE entropies for several nuclei are shown in Fig.
5 as functions of $T$ and excitation energy $E^{*}$, respectively. The single-
particle energies are calculated within the deformed Woods-Saxon potentials.
In order to have a consistent comparison with the recent experimental data in
[11], we carried out calculations by using the CE-LNBCS and CE-LNSCQRPA for
56Fe, where pairing is included within the complete $pf+g_{9/2}$ shell above
the 40Ca core. For Mo isotopes, pairing is included in the 22 degenerated
single-particle levels above the 48Ca core. For Dy and Yb the same is done on
top of the 132Sn core. It is clear from this figure that the CE-LNSCQRPA
results agree quite well with the experimental data [11], which are also
deducted from the CE. The MCE entropies, obtained within the MCE-LNBCS and
MCE-LNSCQRPA using ${\delta\cal E}=$1 MeV, are plotted versus the experimental
data. It is seen that the MCE-LNSCQRPA entropy not only offers the best fit to
the experimental data but also predicts the results up to higher $E^{*}>$ 10
MeV.

##  4 Conclusions

The proposed LNSCQRPA theory can describe without discontinuity the pairing
properties of hot noncollectively rotating nuclei at any values of pairing
interaction parameter $G$, temperature $T$, and angular momentum $M$. In the
limit of zero temperature and zero angular momentum, it offers the best fits
to the exact solutions in the weak coupling region with large particle numbers
for the energy of the first excited state, whereas the SCQRPA reproduces well
the exact one in the strong coupling region. In the limit of very large $G$
all the approximations predict nearly the same value as that of the exact one.
Under the effect of QNF, the paring gaps obtained at different values $M$ of
angular momentum decrease monotonously as $T$ increases, and do not collapse
even at hight $T$. The effect of thermally assisted pairing (pairing
reentrance) shows up in such a way that the pairing gap reappears at a given
$T_{1}>$ 0 and remains finite at $T>T_{1}$, in qualitative agreement with the
results of Ref. [9]. We suggest a novel formula to extract the pairing gap at
$T\neq$ 0 from the difference of total energies of even and odd systems, where
the contribution of uncorrelated single-particle motion is subtracted. Its
prediction is found in much better agreement with the canonical gap than the
simple extension of the odd-even mass formula to $T\neq$ 0\. Finally, we
embedded the solutions of the LNBCS and LNSCQRPA into the CE and MCE, and
found that the CE-LNSCQRPA predictions are in quite good agreements with the
exact results as well as the recent experimental data. The present approach
can also describe simultaneously and self-consistently the experimentally
extracted total energy, heat capacity, and entropy within both CE and MCE
treatments. It is simple and feasible for the application to larger finite
systems, where the exact matrix diagonalization and/or solving the Richardson
equation are impracticable to find all eigenvalues, whereas other methods,
such as the quantum Monte-Carlo calculations, are time consuming.

## References

## References

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  * [2] Hung N Q and Dang N D 2009 Phys. Rev. C 79 054328\.
  * [3] Hung N Q and Dang N D 2007 Phys. Rev. C 76 054302; 2008 Ibid. 77 029905(E).
  * [4] Dang N D and Hung N Q 2008 Phys. Rev. C 77 064315\.
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  * [8] Balian R, Flocard H and Vénéroni M 1999 Phys. Rep. 317 251\.
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  * [11] Melby E et al. 1999 Phys. Rev. Lett. 83 3150; Guttormsen M et al. 2000 Phys. Rev. C 62 024306; Schiller A et al. 2001 Phys. Rev. C 63 021306(R); Algin E et al. 2008 Phys. Rev. C 78 054321\.


# Power-Efficient Indoor Localization Using Adaptive Channel-aware Ultra-
wideband DL-TDOA

Sagnik Bhattacharya, Junyoung Choi, Joohyun Lee  Communications Standards
Research Team, Samsung Electronics, Seoul, South Korea
Email: {sagnik.b, juny.choi<EMAIL_ADDRESS>
(October 2023)

###### Abstract

Among the various Ultra-wideband (UWB) ranging methods, the absence of uplink
communication or centralized computation makes downlink time-difference-of-
arrival (DL-TDOA) localization the most suitable for large-scale industrial
deployments. However, temporary or permanent obstacles in the deployment
region often lead to non-line-of-sight (NLOS) channel path and signal outage
effects, which result in localization errors. Prior research has addressed
this problem by increasing the ranging frequency, which leads to a heavy
increase in the user device power consumption. It also does not contribute to
any increase in localization accuracy under line-of-sight (LOS) conditions. In
this paper, we propose and implement a novel low-power channel-aware dynamic
frequency DL-TDOA ranging algorithm. It comprises NLOS probability predictor
based on a convolutional neural network (CNN), a dynamic ranging frequency
control module, and an IMU sensor-based ranging filter. Based on the conducted
experiments, we show that the proposed algorithm achieves 50% higher accuracy
in NLOS conditions while having 46% lower power consumption in LOS conditions
compared to baseline methods from prior research.

###### Index Terms:

Ultra-wideband, localization, power-efficient, CNN, dynamic ranging frequency

##  I Introduction

Ultra-wideband (UWB) spectrum-based ranging offers high accuracy (e.g., 15 cm)
and it makes a topic of high research interest in recent years. The most
prominently used UWB-based ranging methods are double-sided two way ranging
(DS-TWR) and downlink-time-difference-of-arrival (DL-TDOA) [1]. In DL-TDOA
methods, there are pre-assigned clusters of UWB anchors. The anchors in a
cluster regularly exchange UWB messages containing timestamps and reply time
information. A user device in that area listens to those messages, extracts
the timestamp information, and triangulates its location coordinates in a
downlink manner, without requiring any uplink communication or centralized
computation. This makes the DL-TDOA based localization highly scalable with
increasing number of user devices, and thus the perfect candidate for large-
scale industrial deployments. However, the presence of temporary obstacles
(e.g. moving humans/cars) or permanent obstacles (e.g. walls/pillars in the
deployment scenario) often lead to signal blockage effects, hence resulting in
non-line-of-sight (NLOS) channel paths. This is detrimental to the
localization performance, and causes time synchronization errors and
localization errors. This problem is exacerbated in highly mobile or crowded
situations.

Prior research handles the problem by increasing the DL-TDOA ranging
frequency, i.e. the number of times DL-TDOA-based localization is conducted in
a given time. However, that leads to a heavy increase in user device power
consumption, provides very small performance improvement in NLOS conditions,
and no performance gain in LOS conditions. Another research deals with the
problem by introducing an auxiliary ranging process based on time-of-flight
(TOF) along with the primary DL-TDOA process [2]. While this improves the
localization accuracy to a higher degree, especially in NLOS conditions, the
process is not scalable to many user devices because the DS-TWR process
involves one-to-one communication between each anchor and each user device.

The ability of deep learning-based architectures to extract meaningful
patterns from given data, and use it to achieve near-optimal efficient
solutions to otherwise intractable problems in real-time, has led to its
widespread recent use in wireless optimization problems [3, 4]. Convolutional
neural networks (CNN) are extremely efficient in extracting practical spatial
or temporal features directly from raw data [3], and are thus extremely useful
in our wireless scenario.

In this paper, we propose a novel power-efficient channel-aware adaptive UWB
DL-TDOA ranging algorithm. The proposed algorithm comprises NLOS probability
predictor using a convolutional neural network (CNN) for the received UWB
signals, a dynamic DL-TDOA frequency control module, and an IMU sensor-based
ranging filter. We also adopt the localization predictor based on the
recurrent neural network (RNN) from [5] to further lower the required DL-TDOA
ranging frequency. The proposed algorithm is highly power-efficient, as well
as scalable, since it does not involve DS-TWR. We conduct localization
experiments on Samsung Galaxy S21 Ultra (GS21U) smartphones, connected to
Qorvo Decawave UWB chipset. We show that the proposed algorithm achieves 50.4%
higher localization accuracy, compared to baseline methods, in predominantly
NLOS conditions, while having 46.3% lower power consumption in LOS conditions.

Specifically, our novel contributions in this paper are the following:

  * •

We propose a novel CNN-based algorithm which takes as its input the received
signal channel impulse response (CIR) of the various anchors which are
currently participating in the DL-TDOA localization process, and predicts the
NLOS probability for each of those anchors.

  * •

We design a novel dynamic DL-TDOA ranging frequency module, which adjusts the
user device ranging frequency according to the average channel quality. In
addition, we also design an IMU sensor-based filter, which turns off the DL-
TDOA ranging module when the user is not moving. These two modules help reduce
power consumption.

  * •

To keep the localization accuracy unaltered during the times when the dynamic
ranging frequency is reduced, we use a recurrent neural network (RNN)-based
future localization predictor.

  * •

To verify the practical applicability of our proposed module architecture, we
deploy it on Samsung GS21U smartphones connected to Qorvo UWB chipsets.
Compared to prior work, we achieve $50.4\%$ higher localization accuracy in
NLOS conditions, while having $46.3\%$ lower power consumption in LOS
conditions.

##  II Related Work

There has been significant prior research on improving the performance of UWB
DL-TDOA-based localization [6, 7, 2, 8]. However, those works are primarily
concerned with either improving the localization accuracy [6, 7], or reducing
the power consumption [2]. The authors in [6] use a Kalman Filter-based
approach to improve the accuracy of time-of-arrival (TOA)-based localization.
Even though this is a low power consumption algorithm, the Kalman Filter-
predicted localization coordinates keep diverging from the ground truth values
in predominantly NLOS situations. This is because of the Kalman Filter’s
gaussian noise assumption, which is not true for the IMU sensor (accelerometer
and gyroscope) data, where the error has a bias and accumulates over time. The
authors in [2] use a combination of TDOA and time-of-flight (TOF)-based
measurements during localization. They achieve a $75\%$ user device-side power
consumption reduction compared to TOF-based localization, while maintaining
the same localization accuracy as baseline methods. The usage of TOF requires
the anchors to respond to individual user devices using response messages.
Hence, this algorithm is not scalable to many user devices. The authors in [4]
use deep learning on the UWB channel impulse response (CIR) to estimate the
probabilities of various factors which are representative of LOS/NLOS
conditions. While this method helps improve localization accuracy, it also
involves anchor-to-user device communication, which affects its scalability.

##  III Preliminaries of DL-TDOA localization

Figure 1: A DL-TDOA cluster comprising 4 anchors and a user device

Fig. 1 shows a DL-TDOA ranging cluster. The UWB anchors are fixed at pre-
defined locations and are assigned clusters [1]. We consider $M$ anchors,
which are all assigned the same cluster. These $M$ anchors periodically
exchange UWB messages containing timestamps. These messages are used by the
anchors to maintain time synchronization, which is crucial for successful DL-
TDOA localization. The user device, equipped with a UWB-enabled chipset,
“listens” to these messages and collects these timestamps. The user device
then calculates the TDOA values, and triangulates its location using least-
squares estimation. This eliminates the need of any uplink communication or
computation at the anchor. The presence of NLOS/multipath effects between the
user device and the anchor leads to inaccuracies in the measurements of reply
time, and hence the TDOA values. This adversely affects the least squares-
based estimation and leads to high localization errors.

##  IV Proposed Algorithm

Figure 2: The algorithmic flow for adaptive DL-TDOA localization.

(a) The CIR divided into signal and noise based on maxNoise

(b) The difference in the amplitude and the number of peaks

Figure 3: The example of effective CIR and difference between LOS/NLOS.

(a) LOS/NLOS classification model architecture with 4 CNN layers

(b) CNN layer architecture

Figure 4: CNN based model for LOS/NLOS classification using CIR input. Figure
5: The RNN model based LSTM for estimating the location.

Fig. 2 shows the proposed model for power-efficient channel-aware DL-TDOA-
based localization. We assume that there are $M$ anchors inside a cluster
involved in the DL-TDOA process. The UWB module in the user device collects
the UWB CIR and timestamps from $M$ anchors at the specified frequency. The
first step comprises a CNN-based NLOS probability predictor. It operates on
the received CIR from each anchor and predicts the NLOS probability for the
channel path between that anchor and the user device. If the average of those
NLOS probabilities, namely $p_{avg}$, is less than a pre-specified threshold
$p_{th}$ (=0.5) the situation is classified as predominantly LOS, and
predominantly NLOS otherwise.

In the first case, i.e., the case of a predominantly LOS situation, the
outputs from the CNN are sent to the dynamic frequency module, which updates
the frequency at which the user device conducts DL-TDOA-based localization.
This resulting updated frequency is sent to the UWB module. The IMU sensor-
based ranging filter also sends its output to the UWB module. These two inputs
together determine the next time instant when the UWB data is collected.
Finally, the DL-TDOA data collected from the UWB module is sent to the
localization module, which calculates the localization coordinates of the user
device using the least squares estimation [9].

In the second case, i.e., the predominantly NLOS situation, the NLOS
probabilities are sent to the healthy message selection module, which selects
a subset of the UWB messages collected from the $M$ anchors, which correspond
to LOS channel paths. If the total number of messages (each message from a
different anchor) in that subset is less than $3$, then the entire ranging
data for that round is discarded. If that is not the case, then the selected
DL-TDOA data in that subset are sent to the localization module. In addition
to the localization module, we adopt the RNN-based robust localization
predictor from [5]. This helps provide uninterrupted localization data in
cases where (1) the original collected UWB data is discarded as mentioned
above, or (2) the frequency of the UWB data collection is reduced by the
dynamic frequency module in a predominantly LOS condition.

The individual modules in the algorithmic flow are explained in detail as
follows:

###  IV-A Effective CIR (eCIR) Updater

The eCIR collection module is the first step in the proposed model, which
receives the received raw CIR data in the user device from the $M$ UWB
anchors. Fig. 3(a) shows a typical received CIR, which contains 1016 amplitude
values as a function of time. Here, FP_INDEX represents the first path index
or the beginning of the first path of the received CIR. All values prior to
FP_INDEX are irrelevant and hence removed. Here, maxNoise represents the noise
floor, hence all received values below that are essentially noise. Hence, we
see that all the essential and relevant CIR information is confined to $200$
values after the FP_INDEX, and hence we select only those $200$ values as the
effective CIR or the eCIR. The eCIR is sent to the NLOS probability predictor.

###  IV-B NLOS Probability Predictor

The NLOS probability predictor module, which is the first step of the proposed
algorithm, ingests the received eCIRs from the eCIR module. Each of the
received eCIRs are sent individually to the CNN-based model, which predicts
the corresponding NLOS signal probability. Fig. 4(a) shows the neural network
model architecture used for the NLOS probability prediction. The input layer
ingests the individual eCIRs. It is followed by 4 CNN layers, which are
efficient at meaningful feature extraction. The output from the last CNN layer
is sent to the flatten layer, which converts it into a 1-dimensional array.
This is taken as input by the dense layer, in which all neurons of the current
layer are connected with all neurons of the previous layer using weights. The
output of the dense layer is passed into the sigmoid activation layer, which
applies the sigmoid function and produces a number between 0 and 1. This
number is then passed through a low pass filter (LPF), which applies a
weighted mean of the current input and the previous stored output. Finally,
the output of the LPF represents the probability of the given input CIR
corresponding to a NLOS signal.

Fig. 4(b) shows the neural network layers constituting an individual CNN
layer. The four Conv1D layers each comprise a pre-defined number of 1
dimensional vectors, named filters. All of the filters have a pre-defined size
called kernel size. These filters help extract meaningful spatial/temporal
features from the input data. Each of these 1 dimensional filters is convolved
with the input to produce a 1 dimensional vector. These vectors are stacked
together to form a 2 dimensional matrix, which is then passed to the instance
normalization (IN) layer. The IN layer normalizes the incoming input along a
pre-specified axis. This, unlike the more traditional batch normalization, in
insensitive to training data statistics and hence helps prevent overfitting.
The rectified linear unit (ReLU) layer ingests the normalized output of the IN
layer, and applies the ReLU function on it elementwise. Next, the dropout
layer randomly masks the input from a pre-defined percentage of neurons at
runtime. This helps reduce overfitting caused by training data reliance. The
final layer, which is the maxpool layer, also has a pre-defined kernel size.
It applies the maxpool function, i.e., replacing a contiguous set of elements
with the maximum among them, along a specified axis. This helps reduce the
dimensionality of the data, which boosts efficiency and helps reduce
overfitting.

###  IV-C Dynamic Ranging Frequency Module

In the predominantly LOS situation, the dynamic frequency module updates the
DL-TDOA ranging frequency $f_{dyn}$. We calculate an intermediate value
$f^{{}^{\prime}}$ which is given by
$f^{{}^{\prime}}=\frac{f}{\lfloor\frac{p_{th}}{p_{avg}}\rfloor}$, where $f$ is
the default ranging frequency of $5$Hz. $f_{dyn}$ is given as

$\displaystyle f_{dyn}=\begin{cases}0.1Hz&\text{if }f^{{}^{\prime}}<0.1\\\
5Hz&\text{if }f^{{}^{\prime}}>5\\\ f^{{}^{\prime}}&\text{otherwise
}\end{cases}$ (1)

In predominantly LOS situations, i.e., when the average NLOS probability
$p_{avg}<p_{th}$, the dynamic frequency module reduces the ranging frequency
as per Eq. (1). The minimum and maximum allowed frequencies are $0.1$ Hz and
$5$ Hz, respectively. This leads to a reduction of power consumption.

###  IV-D Healthy Message Selector

In a predominantly NLOS situation, the received TDOA values, along with their
individual NLOS probabilities, are sent to the Healthy Message Selector
module. This module selects a subset of the TDOA values, for which the
corresponding average NLOS probability $p^{{}^{\prime}}_{avg}\leq p_{th}$. To
do this, it iteratively eliminates the TDOA from the set, for which the NLOS
probability is the highest, and then updates the new average NLOS probability
of the remaining TDOA values. If the number of TDOA values in that final
subset is $3$ or more, which is the minimum number required for 2D
localization, then the situation is determined to be “partially LOS”. In the
other case, i.e., if the number of remaining TDOA values in the final set is
$2$ or less, the situation is determined to be “fully NLOS”. In a partially
LOS situation, the subset of TDOA values is sent to the Least Squares
Estimation-based Localization module. In a fully NLOS situation, all the TDOA
values are discarded, and the RNN-based Localization Predictor module is used.

###  IV-E RNN-based Localization Predictor

Fig. 5 shows the RNN model architecture used for future localization
coordinates prediction. The input comprises the UWB TDOA values and the IMU
sensor (accelerometer and gyroscope) data as a continuous incoming stream. The
sequence generator stacks the UWB TDOA and IMU values for the past $N_{in}$
timesteps and generates sequences of dimension $(N_{in},(M-1)+3+3)$. Here, $M$
is the number of anchors currently participating in the DL-TDOA ranging
session. Hence, we have $M-1$ TDOA values. The IMU sensor, i.e. the
accelerometer and gyroscope readings along the X, Y, and Z axes contribute
another $6(=3+3)$ values. The generated sequence is sent as input to the RNN.
The RNN comprises a long short-term memory (LSTM) encoder-decoder structure.
LSTM cells are extremely efficient in extracting practical temporal
information from given time series data. The encoder architecture comprises a
set of LSTM blocks, which ingest the input sequence and generate compressed
embeddings. The decoder again consists of another set of LSTM blocks which
ingest the generated embeddings and provide an output of dimensions
$(N_{out},2)$. This is followed by a dense layer. Finally, the output from the
dense layer is passed through a low pass filter (LPF) to smoothen out random
fluctuations in localization coordinates. The output of the LPF, which is of
dimension $(N_{out},2)$ provides the predicted localization coordinates for
the next $N_{out}$ timesteps.

It is to be noted that, as a result of the RNN-based localization prediction,
we always have the predicted localization coordinates for $N_{out}$ future
timesteps available to us. During the DL-TDOA localization process, there are
cases of signal reception failure, which lead to non-availability of collected
TDOA values. In such cases, the predicted localization coordinates at that
timestep are used to calculate the predicted TDOA values. For example, suppose
the pre-provided fixed locations of the $M$ anchors in the current cluster are
given by $\\{(x_{i},y_{i})~{}|~{}i\in\mathbb{Z},1\leq i\leq M\\}$, where the
first anchor ($i=1$) is the initiator. If the RNN-predicted localization
coordinates of the user device for a particular future timestep are
$(x_{ud},y_{ud})$, then the predicted TDOA value corresponding to the $j^{th}$
responder, $\alpha_{j}$ is given by:

$\alpha_{j}=\frac{1}{c}(|\sqrt{(x_{j}-x_{ud})^{2}+(y_{j}-y_{ud})^{2}}-\\\
\sqrt{(x_{1}-x_{ud})^{2}+(y_{1}-y_{ud})^{2}}|)~{}\forall j\in\mathbb{Z},2\leq
j\leq M$ (2)

where $c$ is the speed of light in vacuum. These predicted (or “augmented”)
TDOA values can be used as input to the RNN at the corresponding future
timestep, if the Qorvo chipset fails to obtain external UWB signal at that
time. Hence, we always obtain localization predictions using the RNN, even
during signal reception failures. This perennial supply of augmented TDOA
values, along with the IMU sensor data, makes the localizer robust to
multipath effects and penetration losses.

##  V Baseline Methods

We compare the performance of our proposed power-efficient high-accuracy UWB
DL-TDOA localization algorithm to several baseline methods from prior
research. Baseline 1 includes the traditional least squares estimation-based
DL-TDOA localization [9]. This method has a static ranging frequency of
$200~{}ms$. It also lacks any predictive power, i.e., it fails to compute
localization coordinates in the absence of UWB signals from the minimum number
of anchors that is necessary for triangulation, i.e. $3$. Our baseline 2 is
from [2], which uses a combination of UWB TOF and DL-TDOA values for
localization. Baseline 3 is adopted from [4]. The algorithm used in this paper
infers several LOS/NLOS-related channel conditions using deep learning on the
received CIR in two-way-ranging (TWR), and then accordingly adjusts the Kalman
filter for localization prediction and estimation.

##  VI Experimental Results

Figure 6: The real-world experiment setting, showing the user holding the device, and the anchor installed on the ceiling. TABLE I: UWB and IMU sensor configuration parameters Parameter | Value
---|---
UWB Ranging Block Duration |  $200$ms
UWB Ranging | DL-TDOA
UWB Mode | Multicast
No. of Anchors | $7$
IMU Sensor Frequency |  $16$Hz
TABLE II: Model Parameters CNN hyperparameter | Value
---|---
Conv layer {1,2,3,4} kernel size | {9,13,17,17}
Conv layer {1,2,3,4} no. of filters | {64,64,128,256}
Maxpool kernel size | 2
Dropout rate | 0.1
Dense layer no. of neurons | 64
Optimizer | Adam
Loss function | Binary cross-entropy
Batch size | 50
Maximum no. of training epochs | 100
RNN hyperparameter | Value
$N_{in}$ | 16
LSTM layer {1,2,3,4} no. of units | {32,32,32,64}
Dense layer no. of neurons | 64
Optimizer | Adam
Loss function | Mean-squared error
Batch size | 10
Maximum no. of training epochs | 50

###  VI-A Simulation Setup

Fig. 6 shows the simulation setup for the performance evaluation of the
proposed algorithm. The lab environment is rectangular of dimensions
$9~{}m\times 6~{}m$, and contains $7$ Qorvo anchors placed on the ceiling at a
height of $2.7~{}m$ as shown in Fig. 6. The user device is Samsung GS21, which
is connected to a Qorvo chipset using a USB port. Squares marked on the floor
surface with pre-provided location coordinates provide the ground truth
localization data. The user carries the Samsung S20U device, attached to the
Qorvo chipset by cable, in a handheld manner. The Android app running on the
device collects the TDOA, CIR, and IMU sensor data.

###  VI-B Model Implementation

User device: We implement the application on the commercial smartphone (i.e.,
SM-N986B) running on Android 11 to show the feasibility. Both the CNN-based
NLOS probability predictor and RNN-based localization predictor are
implemented using TensorFlow 2.10.0 and converted to TensorFlow-Lite for
mobile deployment. Other modules including IMU are implemented using Android
Studio.

UWB: We implement the UWB module on the user device based on the example code
from the Qorvo DWM1001 [10]. The DL-TDOA operation in Qorvo board is
implemented by following the user manual using SEGGER Embedded Studio for ARM
5.32a [11, 12]. The UWB ranging and IMU sensor parameters used for configuring
the user device and the installed anchors are shown in Table I.

###  VI-C Model Training

For training the CNN-based NLOS probability predictor model, we collect
volunteer data in both LOS and NLOS conditions. This CIR data for training is
collected at a default frequency of $5$ Hz, and we collect a total of $4000$
LOS samples and $4000$ NLOS samples. To prevent the overfitting scenario
associated with training data shortage, this collected dataset is combined
with publicly available open datasets.

For the RNN-based localization predictor, we collect training data and train
the model as in [5]. The UWB DL-TDOA data (along with the CIR data) is
collected by the Qorvo module at a frequency of $5$Hz. We can choose the value
of $N_{in}$, which is the number of timesteps for which we send the TDOA data
to the RNN. We experiment with various values of $N_{in}$ and obtain the best
localization performance for $N_{in}=16$. The IMU sensor data is also sent as
input to the RNN. We collect new IMU sensor readings every $62.5$ ms. Assuming
that the average human walking speed is 2 steps/second, and that we want to
send the IMU data corresponding to the latest 2 steps into the RNN, we send 1
second’s worth of IMU data. We want to keep the timestep dimension of the UWB
DL-TDOA and IMU data to be the same, i.e. $16$. Hence we should send $16$ IMU
data samples, covering the data for $1~{}s$. Hence, the required IMU data
interval is $1/16=62.5~{}ms$. The various parameters used for the proposed
model implementation are showed in Table II.

Figure 7: Change in UWB DL-TDOA frequency, as dictated by the dynamic ranging
frequency module, with change in average NLOS probability.

###  VI-D Performance Evaluation

Fig. 7 shows the change in DL-TDOA ranging frequency with the change in the
NLOS probability as predicted by the algorithm. The X-axis values increase
from $0$, indicating one corner of the rectangular lab, i.e., $(9~{}m,6~{}m)$
to $\sqrt{(9/2)^{2}+(6/2)^{2}}=5.41~{}m$, indicating the center of the
rectangle. As the user device moves along a straight line from one corner of
the cluster toward the center, the scenario changes from a predominantly NLOS
situation to a predominantly LOS situation. As a result, the dynamic ranging
frequency keeps decreasing, leading to lower power consumption. The piece-wise
nature of the graph is because of the floor function in the definition of
$f^{{}^{\prime}}$ in Eq. (1).

TABLE III: Average power consumption and average localization error achieved by proposed and baseline algorithms |  Average Current (mA) |  Localization Error (cm)
---|---|---
LOS Scenario |  |
Least Squares |  30.8 |  3.5
TDOA+TOF |  45.7 |  2.8
NLOS Prob+KF |  36.4 |  3.3
Proposed Algorithm |  19.7 |  3.3
NLOS Scenario |  |
Least Squares |  30.8 |  17.6
TDOA+TOF |  45.7 |  12.8
NLOS Prob+KF |  41.3 |  11.5
Proposed Algorithm |  28.2 |  6.7

Table III shows the average localization error and the average user device
current consumption per UWB ranging block, obtained during UWB DL-TDOA
localization, using the proposed and baseline algorithms. As explained in
Section V, baselines 1 (Least Squares), 2 (TDOA+TOF) and 3 (NLOS Prob+KF)
refer to the least-squares estimation, the combination of TDOA and TOF
methods, and the deep learning-based adaptive kalman filter methods
respectively. Table III is divided into two parts, corresponding to the
experiment scenario being predominantly LOS or predominantly NLOS. The
predominantly LOS situation is the ideal lab environment in which the user
walks around with the smartphone in a handheld manner. The predominantly NLOS
situation is created by partially blocking all of the anchors in the lab. From
Table III, we see that in predominantly LOS conditions, the localization error
achieved by all the algorithms are quite similar, with Baseline 2 being
slightly better. This is because it uses a combination of TDOA and TOF values,
leveraging the higher accuracy of TOF-based localization. As for the average
current consumption, Baseline 2 performs worst. This is because of the
message-heavy two-way-ranging involved in the TOF phase, where there is one-
to-one communication between the user device and each anchor, unlike the DL-
TDOA process. Baseline 3 uses deep learning-based NLOS classification and
accordingly updates the Kalman Filter (KF) parameters for localization
prediction. Although it uses DL-TDOA localization, it is computationally
heavy, and thus has a higher current consumption than the traditional least-
squares estimation-based DL-TDOA localization, i.e. Baseline 1. Finally, the
proposed algorithm has the lowest current consumption of $19.7$ mA. This is
attributed to the dynamic ranging frequency operation, which reduces the DL-
TDOA ranging frequency during conditions of low NLOS probability, thus saving
power. This decrease in ranging frequency does not, however, affect its
localization accuracy, because of the augmented TDOA values supplied by the
RNN-based localization predictor. Overall, we achieve $46.3\%$ lower power
consumption compared to baseline methods.

For the predominantly NLOS situation, the gap between the average current
consumption of the proposed algorithm and Baseline 1 decreases to $2.6$ mA.
This is because, in a predominantly NLOS situation, the dynamic ranging
frequency algorithm does not reduce the ranging frequency as in the LOS
situation. This is required to maintain sufficient localization accuracy in a
NLOS scenario. Baseline 2 remains unaffected in terms of current consumption.
Baseline 3, which uses a NLOS classifier-based kalman filter-adaptation, has
to make repeated changes to the noise and covariance matrices of the filter,
and thus has a higher current consumption in the NLOS case. The CNN-based NLOS
probability prediction, the high ranging frequency, and the RNN-based
localization prediction using augmented TDOA, all lead to the superior
localization accuracy of the proposed algorithm. All three baseline algorithms
are heavily affected by the NLOS-based localization inaccuracies. Baseline 3,
with its deep learning-based NLOS classification and adaptive kalman
filtering, is able to perform slightly better than baseline 2, which uses a
combination of TDOA and TOF. Baseline 1, the least-squares-based estimation,
is worst affected by the NLOS and multipath effects and has no interpolation
or prediction capabilities to estimate future localization coordinates
beforehand. Overall, we achieve $50.4\%$ lower localization error compared to
baseline methods.

##  VII Conclusions

In this paper, we propose a novel channel-aware low-power UWB DL-TDOA
localization algorithm. The proposed algorithm is scalable, because it does
not involve one-to-one anchor-user device communications. It is highly
accurate, as well as power-efficient by virtue of its dynamic ranging
frequency setting. We obtain $50\%$ higher localization accuracy compared to
baseline methods in predominantly NLOS conditions, and $46\%$ lower power
consumption in LOS conditions.

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