# Ordinal UNLOC: Target Localization with Noisy and Incomplete Distance
Measures

Mahesh K. Banavar  Clarkson Center for Complex Systems Science ($C^{3}S^{2}$),
Clarkson University  Department of Electrical & Computer Engineering, Clarkson
University  Shandeepa Wickramasinghe  Clarkson Center for Complex Systems
Science ($C^{3}S^{2}$), Clarkson University  Department of Mathematics,
Clarkson University  Monalisa Achalla  Clarkson Center for Complex Systems
Science ($C^{3}S^{2}$), Clarkson University  Department of Electrical &
Computer Engineering, Clarkson University  Jie Sun  Clarkson Center for
Complex Systems Science ($C^{3}S^{2}$), Clarkson University  Department of
Mathematics, Clarkson University

###### Abstract

A main challenge in target localization arises from the lack of reliable
distance measures. This issue is especially pronounced in indoor settings due
to the presence of walls, floors, furniture, and other dynamically changing
conditions such as the movement of people and goods, varying temperature and
air flows. Here, we develop a new computational framework to estimate the
location of a target without the need for reliable distance measures. The
method, which we term Ordinal UNLOC, uses only ordinal data obtained from
comparing the signal strength from anchor pairs at known locations to the
target. Our estimation technique utilizes rank aggregation, function learning
as well as proximity-based unfolding optimization. As a result, it yields
accurate target localization for common transmission models with unknown
parameters and noisy observations that are reminiscent of practical settings.
Our results are validated by both numerical simulations and hardware
experiments.

###### Index Terms:

localization, ordinal data, rank aggregation, optimization, RSSI

##  I Introduction

A recent FCC document [1, 2] highlights the need for increasing the accuracy
of localization when users call from mobile devices from indoor environments.
While the FCC recommends accurate localization to within 3 meters within 30
seconds, in over 90 percent of the test calls the localization error is
greater than 100 meters [3]. With a majority of 911 calls now being made from
wireless devices [4], over 56 percent of which originate from indoor locations
[5], the need for accurate indoor localization is paramount. A poor
localization result can cause incorrect estimation of the room, the floor or
even the actual building from which the call originates. In practice, despite
being an important problem whose satisfactory solution can greatly improve
safety and user experience; and is critical in emergency situations such as
rescuing people from collapsed and/or burning buildings, accurate indoor
localization remains an open problem. In most buildings, including the
majority of schools, hospitals and shopping malls, there is generally no
existing infrastructure (such as an “indoor GPS”) to directly enable accurate
localization. While in most cases, it is in principle possible to set up such
an infrastructure with WiFi routers or Bluetooth beacons [6, 7, 8, 9, 10, 11,
12, 13], the indoor setting itself poses a challenge. This is because indoor
environments feature rich scattering and complex multipath due to the presence
of physical obstacles and barriers (e.g., walls, furniture, and people) and
other factors that make reliable measurements of distance unreliable and
accurate localization seemingly hopeless.

Here, we propose to tackle the target localization problem from a new
perspective. Instead of using the distance measured between anchors and the
target, we use comparative signal strengths from anchors pairs to the target.
Such ordinal data can typically be reliably obtained by comparing time-of-
arrival (TOA) or received signal strength indicators (RSSI) instead of
distances themselves, and do not require accurate account of unknown
environmental parameters or noise in the transmission models. Thus, the
crucial difference between the proposed approach and conventional localization
methods is that in our Ordinal UNLOC approach, accurate distance measures are
not required. Instead, all that is needed is to determine, between each given
pair of sensors, which one is closer to a given reference sensor. Based on
such ordinal data, we estimate the target location by a series of steps that
utilize techniques such as rank aggregation, function learning, and unfolding
optimization. We show that our approach generally produces accurate target
localization across several transmission models under unknown and changing
parameters as well as measurement noise, a scenario that is typical in
practical settings.

###  I-A Contributions of the Paper

The major contributions of this paper are as follows:

  * •

We introduce a new algorithm for localization called Ordinal UNLOC. This
algorithm combines rank aggregation, function learning, and unfolding
optimization to solve the localization problem.

  * •

We show that using only one-bit data (ordinal, pairwise comparisons), we can
accurately solve the localization problem.

  * •

We show through numerical simulations, that our method outperforms traditional
localization approaches in rich scattering environments.

  * •

Through hardware experiments, we show that Ordinal UNLOC outperforms
traditional localization.

  * •

We report that Ordinal UNLOC is only slightly worse off than fingerprinting-
based methods, but with significant lower overhead.

The rest of this paper is organized as follows. In Section III, the problem is
described. The Ordinal UNLOC algorithm introduced in this paper to solve the
localization problem is presented in Section IV. Simulation and hardware
experiments are discussed in Section V. Concluding remarks and future work are
presented in Section VI.

Figure 1: Sensor field (shaded in light gray) that contains anchors (sensors
at known locations, denoted by blue and white pins) and target nodes (sensors
at unknown locations, marked as green hexagonal prisms). Only comparative
pairwise distances are known, giving rise to ordinal data. Ordinal UNLOC is
used to estimate the location of target nodes based on ordinal data. The
method contains three main steps: rank aggregation (Section IV-A), function
learning (Section IV-B), and unfolding localization using approximate anchor-
to-target distances (Section IV-C).

##  II Related Work

Predominant research on indoor localization focuses on addressing the lack of
infrastructure, by utilizing wireless sensor networks (WSNs). A given area is
covered by placing a set of low-cost wireless sensors (called anchors). The
anchors, together with the target node that is to be localized, form a sensor
network via wireless communications (See Figure 1). Assuming that the distance
between the anchors and the target can be reasonably estimated, such
approximate anchor-to-target distances together with the known location of the
anchors are used to estimate the location of the target. Under this framework,
many methods have been developed for target localization based on noisy
distance measures, including multilateration and triangulation [14, 15, 16,
17, 18, 19, 20], and linear and nonlinear optimization [21, 22, 23, 24].
However, as noted earlier, these methods are not suitable for practical indoor
localization since the distances measured between the sensors, typically
inferred from proxies such as time and power signals, are generally unreliable
in indoor environments [25, 12, 26, 13, 8, 10, 27, 6, 28, 29, 30, 31, 32, 9].

In recent years, fingerprinting methods have become increasingly common for
localization with superior accuracy [9, 33, 29, 34, 35] where signal
distributions are mapped to specific regions in space. Though there are
performance advantages to using fingerprinting, it requires extensive mapping
of the environment, a priori. This presents a challenge in dynamic
environments that change frequently. It is also necessary that the training
database be sufficiently large to provide robustness to the algorithm. Several
methods to mitigate these issues have been developed including neural networks
[26], histogram interpolation to decrease the amount of sampling required
[34], and sensor fusion [8, 9, 13]. Methods to use the database for
localization include Bayesian estimators [9], and deep learning [13, 26, 36].

In contrast to the work described above, we propose an algorithm that does not
require detailed distance measurements, or rely on pretrained, precollected,
and expensive methods such as with fingerprinting.

##  III System Model

A standard setup of a sensor field for indoor localization is shown in Figure
1, where $N$ sensors are classified depending on whether their locations are
known or unknown: $m$ sensors at known locations are called anchors and the
$n$ sensors at unknown locations are targets (note that $N=n+m$). We use
$\bm{y}_{i}=[y_{1i},\dots,y_{qi}]^{\top}\in\mathbb{R}^{q}$ and
$\bm{x}_{j}=[x_{1j},\dots,x_{qj}]^{\top}\in\mathbb{R}^{q}$ to represent the
coordinates of the $i$-th anchor ($i=1,\dots,m$) and those of the $j$-th
target ($j=1,\dots,n$) in a $q$-dimensional space, respectively. Typically
$q=2$ or $q=3$ but our analysis is valid for general $q$. We classify the
distances between sensor pairs into three types: anchor-to-anchor, anchor-to-
target, and target-to-target, which we represent using three matrices:
$\mathbf{D}^{Y}=[d^{Y}_{ij}]_{m\times m}$,
$\mathbf{D}^{X}=[d^{X}_{ij}]_{n\times n}$, and
$\mathbf{D}^{YX}=[d^{YX}_{ij}]_{m\times n}$, respectively, where
$d^{Y}_{ij}=\|\bm{y}_{i}-\bm{y}_{j}\|,~{}d^{X}_{ij}=\|\bm{x}_{i}-\bm{x}_{j}\|,~{}d^{YX}_{ij}=\|\bm{y}_{i}-\bm{x}_{j}\|.$
We also define an aggregated distance matrix $\mathbf{D}=[d_{ij}]_{N\times N}$
as

$\mathbf{D}=\begin{pmatrix}\mathbf{D}^{Y}&\mathbf{D}^{YX}\\\
\mathbf{D}^{XY}&\mathbf{D}^{X}\end{pmatrix},$ (1)

where $\mathbf{D}^{XY}$ is the transpose of $\mathbf{D}^{YX}$. When we refer
to the sensors without specifying the type, the first $m$ sensors are anchors
and the rest ($n$) are targets, so that sensor $i$ for $1\leq i\leq m$ refers
to the $i$-th anchor, whereas sensor $m+j$ refers to the $j$-th target node
($j=1,\dots,n$).

Since the anchor locations are known, the matrix $\mathbf{D}^{Y}$ can be
directly obtained. However, for many practical settings such as in indoor
environments, the rest of the block matrices $\mathbf{D}^{X}$,
$\mathbf{D}^{YX}$, and $\mathbf{D}^{XY}$ are generally unavailable [17, 18,
19, 37]. Here, we take a different approach. Instead of using unreliably
measured distances, we consider comparisons of distances (or their proximity
measures) which give rise to ordinal comparison data, which we describe as
follows. For each sensor triplet $(i,j,k)$, we compare the distance
measurements or proxies from sensor $i$ and from sensor $j$ to the reference
sensor $k$, respectively, to determine which sensor (between $i$ and $j$) is
closer to $k$. We denote the binary outcome of such a pairwise comparison as
$z^{(k)}_{ij}$, which generally depends on some unknown function $f$ of the
actual distances $d_{ik}$ and $d_{jk}$, represented as

$z^{(k)}_{ij}=f\left(d_{ik},d_{jk},\xi^{(k)}_{ij}\right),$ (2)

where $\xi^{(k)}_{ij}$ denotes noise in such a comparison. Due to skew
symmetry constraints ($i$ being closer to $k$ than $j$ is the same as $j$
being farther away from $k$ than $i$, and vice versa), we assume that
$\xi^{(k)}_{ij}=-\xi^{(k)}_{ji}$ and require that
$f(d,d^{\prime},\xi)=-f(d^{\prime},d,-\xi)$, which ensures that
$z^{(k)}_{ij}=-z^{(k)}_{ji}$ ($\forall i,j,k$). For the majority of the paper,
we focus on ordinal comparisons, defined by a “thresholding” function

$f(d,d^{\prime},\xi)=\operatorname{sgn}(d-d^{\prime}+\xi)\in\\{-1,1\\},$ (3)

where $\operatorname{sgn}(\cdot)$ is the signum function [38]. Note that this
is a special case of the logistic function
$f(d,d^{\prime},\xi)=1/\left[1+e^{-\beta(d-d^{\prime}+\xi)}\right]$ in the
limit of the parameter $\beta\rightarrow\infty$. Furthermore, in practice one
might only have measurements of physical quantities (“signals”) that serve as
proximity of distance, in which case

$f(d,d^{\prime},\xi)=\operatorname{sgn}(s(d)-s(d^{\prime})+\xi)\in\\{-1,1\\},$
(4)

where $s$ is a (monotonic) function of distance, such as power loss or time of
flight of signals between sensors.

The collection of all pairwise comparisons form $N$ square-shaped matrices:
$\\{\mathbf{Z}^{(k)}\\}$, where $\mathbf{Z}^{(k)}=[z^{(k)}_{ij}]_{N\times N}$
($k=1,\dots,N$), which we refer to as pairwise comparison matrices, can be
combined into single tensor $\mathbf{Z}$ (see Figure 2 for an example).

Figure 2: Visual illustration of the $\mathbb{Z}$ tensor as introduced in
Section III. Each small cube in the larger cube represents one entry in the
tensor, with a +1 represented by blue, -1 by red, and zero by gray. This
particular tensor was generated during one instance of the Monte-Carlo
simulation that resulted in Figure 3.

##  IV Ordinal UNLOC

Our localization algorithm, which we term Ordinal Unfolding-based Localization
(Ordinal UNLOC), consists of three main steps (see Fig. 1, right panel).
First, from the ordinal comparison data, we apply rank aggregation to obtain a
set of “dissimilarities”, which, for a given reference sensor, $k$, assigns a
score to each sensor that serves as a proxy to its distance to the $k$-th
sensor according to the ordinal data and rankings. Such sets of spatial
proximities (scores) are not unique, since any shifting and monotonic scaling
preserves the ranking of the scores. Consequently, the scores obtained in this
step cannot be directly used as approximate anchor-to-target distances for
localization.

Next, the estimated distance proximities among the anchors, together with the
known anchor-to-anchor distances, are used to fit functions that best
transform proximities into distances, and such functions are then applied to
obtain an estimation of anchor-to-target distances.

Finally, given the locations of anchors and estimated anchor-to-target
distances, we formulate a multidimensional unfolding optimization problem, the
solution of which provides an estimate of the location of the target.

In what follows, we describe each of these steps.

###  IV-A Rank aggregation from ordinal data

The first step in Ordinal UNLOC is to use a rank aggregation method to infer
spatial proximities from the given ordinal data. For each reference sensor
$k$, we seek a set of scores, denoted by
$\bm{\psi}^{(k)}=[\psi_{1k},\dots,\psi_{Nk}]^{\top}$, that serve as
proximities and ideally preserves the ordinal data as constraints, such that
$\psi_{ik}>\psi_{jk}$ if and only if $d_{ik}>d_{jk}$ (or in other words,
$z^{(k)}_{ij}=1$). Note that this is equivalent to requiring the ranking of
the entries in each $\bm{\psi}^{(k)}$ be identical (or as close as possible)
to the ranking of the entries of the $k$-th column of $\mathbf{D}$, which we
denote as $\bm{d}^{(k)}=[d_{1k},\dots,d_{Nk}]^{\top}.$

The problem of inferring the ranking or scoring of a set of items from their
pairwise comparisons is commonly known as “rank aggregation” [39, 40],
interpreting each pairwise comparison as assigning a local ranking between two
items, with the goal of obtaining an aggregated global ranking that preserves
these local rankings as much as possible. Several methods are available for
solving a rank aggregation problem, most of which compute a score for each
item based on the collection of ordinal data [41].

Here, we adopt a common, easy-to-implement, and effective method, referred to
as HodgeRank [42] or simply least squares (LS) ranking. The idea is to
formulate a linear least squares problem using the ordinal data as inputs, the
solution of which gives the estimated spatial proximities and approximately
preserves distance orderings from the comparison data. Consider an arbitrary
enumeration of the set of all ordered pairs of sensors, where the $\ell$-pair
is denoted by $(i_{\ell},j_{\ell})$, defining a set

$\displaystyle E$ $\displaystyle=$
$\displaystyle\\{(i_{\ell},j_{\ell}):\ell=1,\dots,M\\}$ (5) $\displaystyle=$
$\displaystyle\\{(i,j)\in\mathbb{N}^{2}:1\leq i<j\leq N\\},$

where $M=N(N-1)/2$. Thus, $E$ can be interpreted as the edge set of a complete
graph of $N$ nodes, and $\\{(i_{\ell},j_{\ell})\\}$ is an ordered list of all
the edges. From this, we define the corresponding incidence matrix,
$\mathbf{B}=[b_{\ell,q}]_{M\times N}$ [43], where

$b_{\ell,q}=\begin{cases}1,~{}\mbox{if $q=i_{\ell}$},\\\ -1,~{}\mbox{if
$q=j_{\ell}$},\\\ 0,~{}\mbox{otherwise}.\end{cases}$ (6)

For each sensor $k$, the ordinal data in matrix $\mathbf{Z}^{(k)}$ can be
effectively represented by a column vector

$\bm{z}^{(k)}=[z^{(k)}_{i_{1},j_{1}},\dots,z^{(k)}_{i_{M},j_{M}}]^{\top}\in\mathbb{R}^{M},$
(7)

following the same enumeration of pairs as in the incidence matrix. Then, the
LS ranking method solves a linear least squares problem to yield

$\bm{\psi}^{(k)}=\operatorname*{argmin}_{\bm{\psi}\in\mathbb{R}^{N},\bm{1}^{\top}\bm{\psi}=0}\|\mathbf{B}\bm{\psi}-\bm{z}^{(k)}\|.$
(8)

Such a solution can be computed in several ways, including for example, using
normal equations or singular value decomposition [44], to produce

$\bm{\psi}^{(k)}=(\mathbf{B}^{\top}\mathbf{B})^{\dagger}\mathbf{B}^{\top}\bm{z}^{(k)}+c\bm{1},$
(9)

where $(\cdot)^{\dagger}$ denotes the pseudoinverse of a matrix [44], the
matrix $\mathbf{B}^{\top}\mathbf{B}$ is also known as the graph Laplacian and
is generally noninvertible, and the constant $c$ is chosen such that the
condition $\bm{1}^{\top}\bm{\psi}^{(k)}=0$ is satisfied. Finally, we collect
the vectors $\bm{\psi}^{(k)}$ into a matrix of spatial proximities

$\bm{\Psi}=[\bm{\psi}^{(1)},\dots,\bm{\psi}^{(N)}].$ (10)

We partition this matrix in the same way as the aggregated distance matrix
$\mathbf{D}$, so that

$\bm{\Psi}_{N\times N}=\begin{pmatrix}\bm{\Psi}^{Y}_{m\times
m}&\bm{\Psi}^{YX}_{m\times n}\\\ \bm{\Psi}^{XY}_{n\times
m}&\bm{\Psi}^{X}_{n\times n}\end{pmatrix},$ (11)

where $\bm{\Psi}^{XY}$ is the transpose of $\bm{\Psi}^{YX}$.

###  IV-B Function Learning: estimating distances from spatial proximities

The second step of Ordinal UNLOC is to estimate the unknown anchor-to-target
distances from the matrix of spatial proximities, $\bm{\Psi}$, obtained by
rank aggregation in (11), together with the known anchor-to-anchor distances.
In particular, for each sensor $k$ that is an anchor (so $1\leq k\leq m$), we
learn a function $g_{k}$ that maps the estimated proximities with respect to
anchor $i$ into known distances $d_{ik}$, so that $d_{ik}\approx
g_{k}(\bm{\psi}^{(k)}_{i})$ ($i=1,\dots,m$). Since $d_{ik}$ is only known for
$1\leq i\leq m$ (i.e., distances between pairs of anchors), the function
$g_{k}$ needs to be inferred from the $k$-th column of the anchor-to-anchor
distance matrix $\mathbf{D}^{Y}$ and that of the dissimilarity matrix
$\bm{\Psi}^{Y}$, which we denote as $\bm{d}^{Y}_{k}$ and $\bm{\psi}^{Y}_{k}$,
respectively.

We express $g_{k}$ using a basis expansion. Using the standard polynomial
basis, we have the representation

$g_{k}(\psi)=\sum_{l=0}^{\infty}c^{(k)}_{l}\psi^{l}.$ (12)

To preserve the ordering among the dissimilarities, we additionally require
that $g_{k}$ be a monotonic function. Under this additional constraint, we
here consider a truncated series to the first order so that $g_{k}$ becomes
just a linear function:

$g_{k}(\psi)=c^{(k)}_{0}+c^{(k)}_{1}\psi,$ (13)

where the coefficients $c^{(k)}_{0}$ and $c^{(k)}_{1}$ are to be determined
from the vectors $\bm{d}^{Y}_{k}$ and $\bm{\psi}^{Y}_{k}$. This can be done
via solving a linear regression problem, using standard least squares, for
example, producing

$\bm{c}^{(k)}=[c^{(k)}_{0},c^{(k)}_{1}]^{\top}=\operatorname*{argmin}_{\bm{c}\in\mathbb{R}^{2},c_{1}>0}\|[\bm{1},\bm{\psi}^{Y}_{k}]\bm{c}-\bm{d}^{Y}_{k}\|,$
(14)

for $k=1,\dots,m$ (every anchor).

From these coefficients, we compute an estimate of the distance from the
$k$-th anchor to the targets, using the formula

$\tilde{\bm{d}}^{XY}_{k}=[\tilde{d}^{XY}_{1k},\dots,\tilde{d}^{XY}_{nk}]^{\top}=c^{(k)}_{0}+c^{(k)}_{1}\bm{\psi}^{XY}_{k}.$
(15)

Repeating the procedure for all the anchors, $k=1,\dots,m$, produces a
preliminary estimate of the anchor-to-target distance matrix,
$\tilde{\mathbf{D}}^{XY}$ (or equivalently, $\tilde{\mathbf{D}}^{YX}$, by
transposing $\tilde{\mathbf{D}}^{XY}$).

Before moving on to the next step of localization, it is important to
recalibrate the estimated anchor-to-target distances, for the following
reason. Consider the $j$-th target node ($j\in\\{1,\dots,n\\}$), whose
distances to the anchors are preliminarily estimated to form the $j$-th column
of $\tilde{\mathbf{D}}^{YX}$, denoted by
$\tilde{\bm{d}}^{YX}_{j}=[\tilde{d}^{YX}_{1j},\dots,\tilde{d}^{YX}_{mj}].$ For
each anchor $k\in\\{1,\dots,m\\}$, the entry $\tilde{d}^{YX}_{kj}$ is obtained
from its corresponding function $g_{k}$, which generally differs from one
anchor to another. The error in fitting $g_{k}$ can cause the ordering of
$\tilde{d}^{YX}_{kj}$’s to differ from the ordering of the true distances
$d^{YX}_{kj}$, and even different from the estimated proximities
$\psi^{YX}_{kj}$. Therefore, for each target node $j$, to best ensure that the
estimated anchor-to-target distances preserve the ordering of estimated
proximities, we re-fit a linear function $g_{m+j}$ in the same form as (13),
but now with the goal of constructing a monotonic function that transforms the
anchor-to-target proximities to the preliminary estimate of anchor-to-target
distances, in place of the unknown distances, giving:

$\displaystyle\bm{c}^{(m+j)}$
$\displaystyle=[c^{(m+j)}_{0},c^{(m+j)}_{1}]^{\top}$
$\displaystyle=\operatorname*{argmin}_{\bm{c}\in\mathbb{R}^{2},c_{1}>0}\|[\bm{1},\bm{\psi}^{YX}_{j}]\bm{c}-\tilde{\bm{d}}^{YX}_{j}\|,$
(16)

for $j=1,\dots,n$, where
$\bm{\psi}^{YX}_{j}=[\psi^{YX}_{1j},\dots,\psi^{YX}_{mj}]$. Finally, we use
$g_{m+j}$ to recalibrate the estimated anchor-to-target distances, as

$\hat{\bm{d}}^{YX}_{j}=c^{(m+j)}_{0}+c^{(m+j)}_{1}\bm{\psi}^{YX}_{j},$ (17)

for $j=1,\dots,n$, thus forming the estimated anchor-to-target distance matrix
$\hat{\mathbf{D}}^{YX}$ and its transpose $\hat{\mathbf{D}}^{XY}$.

###  IV-C Unfolding localization from distance measures

For each target node $j$ (i.e., sensor $m+j$, where $1\leq j\leq n$), the
estimated anchor-to-target distances $\hat{\mathbf{D}}^{YX}$ together with the
locations of the anchors $\\{\bm{y}_{i}\\}$ can be used to infer the location
of the target $\bm{x}_{j}$. We achieve this by formulating an unfolding
optimization, with cost function [45]

$J(\bm{x},\mathbf{Y};\bm{\delta})=\sum_{i=1}^{m}\left(\|\bm{x}-\bm{y}_{i}\|^{2}-\delta_{i}\right)^{2},$
(18)

where the matrix $\mathbf{Y}=[\bm{y}_{1},\dots,\bm{y}_{m}]$,
$\bm{\delta}=[\delta_{1},\dots,\delta_{m}]^{\top}$, and the solution is termed
UNLOC [45]. Applying UNLOC to each column of the estimated distance matrix
$\hat{\mathbf{D}}^{YX}$ leads to the estimated location of all the targets.
That is, we compute, for each $j=1,\dots,n$,

$\hat{\bm{x}}_{j}=\operatorname*{argmin}_{\bm{x}}J(\bm{x},\mathbf{Y};\hat{\bm{d}}^{YX}_{j}).$
(19)

This can be solved in several ways, including those discussed in [21], a
global optimization routine [46], or generalized to include weights [45].

###  IV-D Computational Complexity

In this section, we discuss the computational complexity of Ordinal UNLOC with
reference to traditional localization methods. In terms of computational
complexity, the final step of Ordinal UNLOC is a traditional multilateral
algorithm such as the one discussed in [45]. Ordinal UNLOC is, therefore, more
complex than traditional ToA/RSS-based systems. However, while the traditional
methods require accurate distance or time estimates, our approach does not.
The cost in terms of computational complexity is therefore mitigated by the
fact that we can obtain accurate localization estimates even in rich
scattering environments.

##  V Experiments and Results

To validate Ordinal UNLOC and verify its effectiveness, we conduct a variety
of numerical simulations. In each such simulation, we place $m$ anchors as
well as a target uniformly at random in $[0,1]^{2}$ and obtain ordinal
comparison data from the threshold model (2)-(3) using Gaussian noise
$\mathcal{N}(0,\sigma^{2})$. Specifically, for $m=5$, we show an example of
the $\mathbf{Z}$ tensor in Fig. 2, for one realization of noise.

Figure 3: Dependence of localization error on the number of anchors. We plot
the root mean squared error (RMSE) of Ordinal UNLOC as a function of the
number of anchors $m$ and data collected at noise levels from $\sigma=0$ to
$\sigma=0.5$. The results are averaged over 5000 simulations and the locations
of the sensors are chosen randomly for every simulation run.

###  V-A Benchmark tests: effects of the number of anchors and noise

As the result of the Monte-Carlo simulations, as shown in Fig. 3, we found
that the root-mean-squared-error (RMSE) of localization decreases rapidly as
the number of anchors increases. Because ordinal comparisons inevitably
introduce information loss about the exact target location, a non-diminishing
localization error is expected, and is indeed observed in our simulations.
Both the quick improvement with increasing anchors and error saturation is
found across a range of noise levels, suggesting robustness of Ordinal UNLOC.

To further understand the non-perfectness of localization with ordinal
constraints, we compare the RMSE of localization with the rank correlation
between $\mathbf{D}$ and $\hat{\mathbf{D}}$, by plotting both versus
increasing noise levels. The results, shown in Fig. 4, validate our hypothesis
that increasing the number of anchors improves performance, and increasing
noise deteriorates performance. It can also be seen that as the rank
correlation between the distance matrices decreases, that is, as the estimates
of the cross-distances between the anchors and the targets reduce its quality,
the error in localization increases.

Figure 4: Dependence of localization error and rank correlation (Kendall
$\tau$) on the level of noise. Here we fix the number of anchors $m$ (either
$m=10$ or $m=20$), and use a single target node ($n=1$). We plot the RMSE of
localization and the rank correlation of $\mathbf{D}$ and $\hat{\mathbf{D}}$
as a function of the noise standard deviation $\sigma$.

###  V-B Localization under physical transmission models

Although we have shown the effectiveness of Ordinal UNLOC in simulations using
ordinal data, such ordinal data come directly from comparing distances under
white noise. However, effective noise in practice is expected to deviate from
this idealized scenario. For example, a sensor typically transmits and
receives signals that are distance-dependent. For a method to be potentially
useful, it must be able to deal with more realistic physical transmission
models, which is the focus of our next set of numerical experiments. We
consider two common models: received signal strength and time of arrival.

####  V-B1 Received Signal Strength (RSS)

Given a signal transmitted with power, $P_{T}$, the received power, $P_{R}$,
is given by [37]

$P_{R}=P_{T}\alpha d^{-G},$ (20)

where $\alpha$ is a constant that depends on the hardware and signal
characteristics and $G$ is the path-loss exponent, which equals two in free
space and is typically larger in indoor environments [47]. Given a sensor
field, we now assume that signals are transmitted with strength $P_{T}$ and
the received signal strength (RSS) is given according to (20). To mimic
realistic challenges of unknown environments, we conduct a simulation where
$G$ is drawn randomly in the interval $[a,b]$ with $a=2$ and $b=6$ [32].

We perform three numerical experiments. First, we run Ordinal UNLOC on the
RSSI values directly. This is compared with two other scenarios, where the
distance is estimated from the RSSI values first using (20), and then, UNLOC
[45] is applied in two ways. In the first scenario, a single estimate of the
path loss exponent, $G$, is available for the entire environment, as is
typically the case [32, 37]. In the second case, we assume Genie-aided
calibration using (20), where the path loss exponent for each link available
at all times. Performance for these three cases, where we plot localization
error (MSE) versus the number of anchors, is shown in Fig. 5. Clearly, having
all information in the Genie-aided case provides the best performance. In the
more realistic cases, Ordinal UNLOC outperforms traditional localization even
though some information about the path loss exponent is known to UNLOC, but no
information is available to Ordinal UNLOC except for the raw RSS values.

Figure 5: We compare UNLOC [45] with Ordinal UNLOC. Each transmit receive link
sees a different value of $G$ (case (iii)). For Ordinal UNLOC, we directly
compare the received power values. We then estimate the distances between
anchors and the target using (20). In one case, a fixed $G$, that is derived
using calibration is used. This is compared against the ideal “Genie-aided”
case, where instantaneous $G$ values of each link are available. While the
Genie-aided localization performs the best, Ordinal UNLOC directly with
received power outperforms UNLOC in a realistic setting.

####  V-B2 Time of Arrival (TOA)

In order to perform ranging using time of arrival, the signal propagation time
between the transmitter and the receiver needs to be measured. In indoor
environments, the time estimate is noisy, and may be modeled using a random
variable as

$\hat{\tau}\sim\mathcal{N}\left(\frac{d}{c},\sigma_{T}^{2}\right),$ (21)

where $d$ is the true distance between the transmitter and the receiver, $c$
is the speed of propagation, and $\sigma_{T}^{2}$ is the variance of the TOA
estimates on the channel [20, 48, 19]. We perform ordinal UNLOC on the TOA
measurements, where the time estimates for each transmit-receive pair are
drawn from the TOA model described in (21). We enlarge the sensor field to a
square of size 200 $\times$ 200 and use the normalized variance in order to
better show the effects of variations in the TOA measurements [20, 49]. We
plot the localization error vs the normalized variance $c\sigma_{T}^{2}$ for
different number of anchors in Figure 6.

Figure 6: Comparing Ordinal UNLOC and UNLOC for TOA-based localization;
$m=20$. For Ordinal UNLOC, the time-of-flight values are directly compared to
construct $\mathbf{Z}$. For UNLOC, first, the distance estimates between the
anchors and the targets are estimated from time-of-flight values using 21, and
the location is then estimated. As the uncertainty in time-of-flight values
increases, the UNLOC performance deteriorates much faster than Ordinal UNLOC.

###  V-C Hardware Experiments

Experiments were designed to validate our algorithm. Android devices were used
as transceivers and placed in a lab environment. Four Android devices were
designated as anchors and placed in fixed positions. One Android device was
designated as the target and placed at a location unknown to the system.
Bluetooth links were established between all the devices and received signal
strength indication (RSSI) data was collected. The Ordinal UNLOC algorithm was
applied to the collected data.

Figure 7: Hardware experiment to validate Ordinal UNLOC. Four anchors are
placed at the approximate corners of a rectangle and of many experiments
conducted, two targets placed within the anchor space are shown here. Here,
one of the targets is localized; the other target is individually localized in
Figure 8 and both targets are simultaneously localized in Figure 9. Ordinal
UNLOC is applied on both the actual cross-distances and the RSSI values. UNLOC
is applied on distance values derived from RSSI values. Ordinal UNLOC with no
calibration outperforms UNLOC with calibrated distance estimates. Averaging
all Ordinal UNLOC estimates with individual RSSI values provides an accurate
location estimate, validating the Ordinal UNLOC algorithm.

For the hardware experiment, anchors were placed approximately in the corners
of a rectangle of sides 4m $\times$ 5m. To avoid symmetry, that may play role
in the results, the anchors were not placed exactly in the corners, but were
moved away slightly. Within the convex hull of the four anchors, targets were
placed in 20 separate locations for evaluation. Since the data was collected
in a lab environment, due to multipath fading [50], some paths are more
reflective than others. Therefore, for each target location, from the over
10,000 measurements we collected, we reduced the number of localization
attempts to 100, by only selecting the paths with large line-of-sight power
using the Ricean $K$ factor estimator described in [51]. Once this was
accomplished, the reduced set of RSSI values was used to estimate the location
of each target 100 times, by using these RSSI values directly with Ordinal
UNLOC.

Two target nodes are shown below as examples. Using these two targets, we
consider three localization configurations. In the first two configurations,
the targets are localized individually (see Figure 7 and Figure 8). In the
third configuration, both targets are localized simultaneously (see Figure 9).
In each of the figures, the anchors are represented by green circles, and the
targets locations by the red diamonds.

Figure 8: Hardware experiment to validate Ordinal UNLOC. Four anchors are
placed at the approximate corners of a rectangle and of many experiments
conducted, two targets placed within the anchor space are shown here. Here,
one of the targets is localized; the other target is individually localized in
Figure 7 and both targets are simultaneously localized in Figure 9. Ordinal
UNLOC is applied on both the actual cross-distances and the RSSI values. UNLOC
is applied on distance values derived from RSSI values. Ordinal UNLOC with no
calibration outperforms UNLOC with calibrated distance estimates. Averaging
all Ordinal UNLOC estimates with individual RSSI values provides an accurate
location estimate, validating the Ordinal UNLOC algorithm.

Four scenarios were used for estimating the location of the target. In the
first idealized scenario, Ordinal UNLOC was used to estimate the location of
the target, when the ground-truth cross-distances between the target and the
anchors and the actual distances between the anchors is assumed to be known to
the algorithm. The second estimation scenario uses the RSSI values obtained
from the experiments directly. The procedure followed was as described above.

Figure 9: Hardware experiment to validate Ordinal UNLOC. Four anchors are
placed at the approximate corners of a rectangle and of many experiments
conducted, two targets placed within the anchor space are shown here. Here,
both of the targets are localized simultaneously; the targets are individually
localized in Figure 7 and Figure 8. Ordinal UNLOC is applied on both the
actual cross-distances and the RSSI values. Ordinal UNLOC with no calibration
outperforms UNLOC with calibrated distance estimates. Averaging all Ordinal
UNLOC estimates with individual RSSI values provides an accurate location
estimate, validating the Ordinal UNLOC algorithm.

The final scenario was conducted to provide insight into the effectiveness of
the Ordinal UNLOC algorithm in highly reflective environments. In this case,
we used (20) to estimate the distances between the target and the anchors,
with a fixed $G$ of 4, estimated using the techniques in [37]. With these
distance estimates, the UNLOC algorithm, described in Section IV-C and [45],
is used to estimate the target location. We compute estimates for all RSSI
values, and only show the best estimate.

Note that when multiple targets are simultaneously localized (see Figure 9),
we do not compare against the UNLOC algorithm, since UNLOC is designed to
localize one target at a time. Furthermore, to keep the figure from being
cluttered, we only show the averaged localization estimation when using RSSI
values with Ordinal UNLOC, and omit showing the individual estimates.

As can be seen from the results, the best performer is the case when the
actual distances are known. While this is not possible in a real localization
problem, it is used here to establish a performance benchmark. Using RSSI
values directly with ordinal UNLOC provides better performance in spite of not
needing to calibrate, and outperforms the (standard) UNLOC algorithm using
calibrated values for the operating environment. The location estimate
obtained by averaging all location estimates obtained by ordinal UNLOC on RSSI
values provides the best performance of all the realistic scenarios.

The Ordinal UNLOC algorithm was successful in estimating the location of the
target devices in three different configurations and four different scenarios,
in a rich scattering environment, outperforming conventional localization
methods. Recall that we had over 200,000 samples of data recorded for our
experiments, from which we calculated over 2000 location estimates from twenty
locations of targets. From this data, we estimate that the normalized error
when using Ordinal UNLOC is on the order of 0.02$m/m^{2}$. This is comparable
to, but a little worse than the normalized error expected from using carefully
calibrated fingerprinting methods (normally between 0.01$m/m^{2}$ and
0.001$m/m^{2}$) [35]. However, we should note here that fingerprinting methods
require significant precomputing and training, and are reliable only when the
environment does not change. This, in addition to our simulation results in
Sections V-A and V-B, validate the effectiveness of our approach.

###  V-D Latency

To implement Ordinal UNLOC successfully, we require one sample for good
localization. Based on our hardware experiments, we see that one in ten
samples provides an accurate estimate. That is, the latency of our approach is
on the order of 10 measurements per location estimate. In fingerprinting work
such as [9, 33, 29, 34, 52, 35, 53, 54, 36, 55, 56, 57, 58, 59], on the order
of 10-100 samples are needed for each estimate, showing that our work is
competitive when it comes to latency. Note that although there are advantages
to using fingerprinting methods for localization, these methods require
extensive mapping of the environment, a priori, leading to challenges in
dynamic environments, as well as maintaining a sufficiently large database to
provide robustness to the algorithm.

##  VI Conclusions and Future Work

In this paper, we present a new algorithm for performing localization with
limited and noisy measurement data between anchors and targets. Since
estimating distances from this limited data is typically unreliable, we
instead use the measurements to compute ordinal comparisons for the distances
between all the anchors and targets. We used the ordinal data and designed a
three-step “Ordinal UNLOC” algorithm for localization: (1) rank aggregation;
(2) function learning; and (3) unfolding localization. Through simulation
results, we demonstrate the effectiveness of our algorithm. Hardware
experiments, where Ordinal UNLOC is used to localize devices using Bluetooth
RSSI, were successful, suggesting that the algorithm can be used in real
settings, including rich scattering environments. Furthermore, unlike
traditional methods which require calibration depending on the operating
environment, Ordinal UNLOC can be directly applied to measured distance
proxies such as TOA or RSSI.

While we have demonstrated the effectiveness of the approach, we also note a
couple of limitations: the algorithm consists of a three stage process which
can become computationally expensive, and for the $\mathbb{Z}$ tensor to be
constructed, we require distance or distance proxy cross-information between
all nodes, anchors and targets.

Future work involves the investigation of the role of anchor locations on
localization. While ordinal data is one-bit data, another avenue of
exploration is the effect of increasing the number of bits available for
approximating incomplete and noisy measurements. We will investigate further,
the complexity and latency of Ordinal UNLOC and compare it with other methods
such as traditional multilateration and fingerprinting.

## References

  * [1] “Federal communications commission, wireless e911 location accuracy requirements,” 2014.
  * [2] “Federal communications commission, Fourth Report and Order in PS Docket No. 07-114,” 2015.
  * [3] Working Group 3, “E9-1-1 location accuracy: Indoor localization test bed report,” _The Communications Security, Reliability and Interoperability Council_ , March 2013.
  * [4] “Consumer reports magazine, “for 911, is a cell phone as safe as a landline?”,” Available Online at http://www.consumerreports.org/cro/magazine-archive/2011/january/electronics/best-cell-phones/911-from-cell-phone/index.htm (2016/07/15).
  * [5] C. Carle, “Indoor location: The mobile revolution starts now” directions magazine,” Available Online at http://www.directionsmag.com/articles/indoor-location-the-mobile-revolution-starts-now/334122 (2016/07/15).
  * [6] P. Tragas, A. Kalis, C. Papadias, F. Ellinger, R. Eickhoff, T. Ussmuller, R. Mosshammer, M. Huemer, A. Dabek, D. Doumenis, and A. Kounoudes, “Resolution: Reconfigurable systems for mobile local communication and positioning,” in _2007 16th IST Mobile and Wireless Communications Summit_ , July 2007, pp. 1–5.
  * [7] A. Matic, A. Popleteev, V. Osmani, and O. Mayora-Ibarra, “An indoor positioning system based on a WiFi router and FM beacons,” _In Proc. 18th Telecommunications forum TELFOR 2010_ , 2010.
  * [8] K. Chintalapudi, A. P. Iyer, and V. N. Padmanabhan, “Indoor localization without the pain,” _Proceedings of the sixteenth annual international conference on Mobile computing and networking - MobiCom_ , 2010.
  * [9] R. Faragher and R. Harle, “Location fingerprinting with bluetooth low energy beacons,” _IEEE Journal on Selected Areas in Communications_ , vol. 33, no. 11, pp. 2418 – 2428, November 2015.
  * [10] M. E. Rida, F. Liu, Y. Jadi, A. A. A. Algawhari, and A. Askourih, “Indoor location position based on bluetooth signal strength,” _2nd International Conference on Information Science and Control Engineering_ , 2015\.
  * [11] R. Diwate and D. Rewadkar, “Indoor tracking using Wi-Fi routers on a smartphone,” _International Journal of Science and Research (IJSR)_ , vol. 4, no. 7, pp. 2319–2322, 7 2015.
  * [12] S. Chai, R. An, and Z. Du, “An indoor positioning algorithm using bluetooth low energy RSSI,” in _International Conference on Advanced Material Science and Environmental Engineering (AMSEE 2016), Thailand_ , June 2016.
  * [13] P. Kriz, F. Maly, and T. Kozel, “Improving indoor localization using bluetooth low energy beacons,” _Mobile Information Systems_ , vol. 2016, pp. 1–11, 2016.
  * [14] N. Patwari, A. O. Hero, M. Perkins, N. S. Correal, and R. J. O’dea, “Relative location estimation in wireless sensor networks,” _IEEE Transactions on signal processing_ , vol. 51, no. 8, pp. 2137–2148, 2003.
  * [15] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, R. L. Moses, and N. S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” _IEEE Signal processing magazine_ , vol. 22, pp. 54–69, 2005\.
  * [16] H. Wymeersch, J. Lien, and M. Z. Win, “Cooperative localization in wireless networks,” _Proceedings of the IEEE_ , vol. 97, pp. 427–450, February 2009\.
  * [17] M. Willerton, M. Banavar, X. Zhang, A. Manikas, C. Tepedelenlioglu, A. Spanias, T. Thornton, E. Yeatman, and A. Constantinides, “Sequential wireless sensor network discovery using wide aperture array signal processing,” in _Signal Processing Conference (EUSIPCO), 2012 Proceedings of the 20th European_. IEEE, 2012.
  * [18] X. Zhang, M. K. Banavar, M. Willerton, A. Manikas, C. Tepedelenlioglu, A. Spanias, T. Thornton, E. Yeatman, and A. G. Constantinides, “Performance comparison of localization techniques for sequential wsn discovery,” in _Sensor Signal Processing for Defence (SSPD 2012)_ , Sept 2012, pp. 1–5.
  * [19] X. Zhang, C. Tepedelenlioğlu, M. Banavar, and A. Spanias, “Crlb for the localization error in the presence of fading,” in _Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on_. IEEE, 2013.
  * [20] X. Zhang, C. Tepedelenlioglu, M. Banavar, and A. Spanias, _Node Localization in Wireless Sensor Networks_. Morgan and Claypool, 2016. [Online]. Available: http://ieeexplore.ieee.org.ezproxy1.lib.asu.edu/xpl/articleDetails.jsp?arnumber=7809451
  * [21] A. Beck, P. Stoica, and J. Li, “Exact and approximate solutions of source localization problems,” _IEEE Transactions on Signal Processing_ , vol. 56, no. 5, pp. 1770–1778, May 2008.
  * [22] F. K. W. Chan and H. C. So, “Efficient weighted multidimensional scaling for wireless sensor network localization,” _IEEE Transactions on Signal Processing_ , vol. 57, no. 11, pp. 4548–4553, Nov 2009.
  * [23] G. Wang and K. Yang, “Efficient semidefinite relaxation for energy-based source localization in sensor networks,” in _2009 IEEE International Conference on Acoustics, Speech and Signal Processing_ , April 2009, pp. 2257–2260.
  * [24] M. R. Gholami, H. Wymeersch, E. G. Ström, and M. Rydström, “Wireless network positioning as a convex feasibility problem,” _EURASIP Journal on Wireless Communications and Networking_ , vol. 2011, no. 1, p. 161, Nov 2011\.
  * [25] A. Ozer and E. John, “Improving the accuracy of bluetooth low energy indoor positioning system using kalman filtering,” _International Conference on Computational Science and Computational Intelligence (CSCI)_ , 2016.
  * [26] M. Altini, D. Brunelli, E. Farella, and L. Benini, “Bluetooth indoor localization with multiple neural networks,” _IEEE 5th International Symposium on Wireless Pervasive Computing_ , 2010.
  * [27] F. Palumbo, P. Barsocchi, S. Chessa, and J. C. Augusto, “A stigmergic approach to indoor localization using bluetooth low energy beacons,” _12th IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS)_ , 2015.
  * [28] M. R. Gholami, R. M. Vaghefi, and E. G. Ström, “Rss-based sensor localization in the presence of unknown channel parameters,” _IEEE Transactions on Signal Processing_ , vol. 61, no. 15, pp. 3752–3759, Aug 2013.
  * [29] H. Jin, C. Xu, and K. Lyons, “Corona: Positioning adjacent device with asymmetric bluetooth low energy RSSI distributions,” _Proceedings of the 28th Annual ACM Symposium on User Interface Software and Technology - UIST_ , 2015.
  * [30] F. Benedetto, G. Giunta, and E. Guzzon, “A novel “single-path” vs. “few-path” test based on higher order statistics to possibly start-up coherent combining,” _Journal of Applied Research and Technology_ , vol. 13, no. 1, pp. 7–19, 2015.
  * [31] Y. Wang, Q. Yang, G. Zhang, and P. Zhang, “Indoor positioning system using euclidean distance correction algorithm with bluetooth low energy beacon,” _2016 International Conference on Internet of Things and Applications (IOTA)_ , 2016.
  * [32] A. Juri, T. Arslan, and F. Wang, “Obstruction-aware bluetooth low energy indoor positioning,” _Proceedings of the 29th International Technical Meeting of The Satellite Division of the Institute of Navigation (ION GNSS+ 2016)_ , 2016.
  * [33] Y. Zhuang, J. Yang, Y. Li, L. Qi, and N. El-Sheimy, “Smartphone-based indoor localization with bluetooth low energy beacons,” _Sensors_ , vol. 16, no. 5, p. 596, Apr 2016.
  * [34] F. Daniş and A. Cemgil, “Model-based localization and tracking using bluetooth low-energy beacons,” _Sensors_ , vol. 17, no. 11, p. 2484, Oct 2017\.
  * [35] S. Khandker, J. Torres-Sospedra, and T. Ristaniemi, “Analysis of received signal strength quantization in fingerprinting localization,” _Sensors_ , vol. 20, no. 11, p. 3203, Jun 2020.
  * [36] T. Koike-Akino, P. Wang, M. Pajovic, H. Sun, and P. V. Orlik, “Fingerprinting-based indoor localization with commercial mmwave wifi: A deep learning approach,” _IEEE Access_ , vol. 8, pp. 84 879–84 892, 2020\.
  * [37] M. K. Banavar and K. Mack, “Localization using wireless signals,” US Patent US10 117 051B2, 2017.
  * [38] J. Spanier and K. Oldham, _An Atlas of Functions_. Hemisphere Publishing Corporation, 1987.
  * [39] X. Chen, P. N. Bennett, K. Collins-Thompson, and E. Horvitz, “Pairwise ranking aggregation in a crowdsourced setting,” in _Proceedings of the sixth ACM international conference on Web search and data mining_. ACM, 2013, pp. 193–202.
  * [40] A. Ammar and D. Shah, “Efficient rank aggregation using partial data,” in _ACM SIGMETRICS Performance Evaluation Review_ , vol. 40. ACM, 2012, pp. 355–366.
  * [41] S. Negahban, S. Oh, and D. Shah, “Rank centrality: Ranking from pairwise comparisons,” _Operations Research_ , vol. 65, no. 1, pp. 266–287, 2016\.
  * [42] X. Jiang, L.-H. Lim, Y. Yao, and Y. Ye, “Statistical ranking and combinatorial hodge theory,” _Mathematical Programming_ , vol. 127, no. 1, pp. 203–244, 2011.
  * [43] J. A. Bondy and U. S. R. Murty, _Graph theory with applications_. Macmillan, 1976.
  * [44] G. Golub and C. V. Loan, _Matrix computations_. JHU Press, 2012.
  * [45] J. Sun, T. Yang, K. Mack, and M. Banavar, “UNLOC: optimal unfolding localization from noisy distance data,” _Sampling Theory in Signal and Image Processing (STSIP)_ , vol. 17, no. 2, p. 183–195, 2018.
  * [46] T. Cox and M. Cox, _Multidimensional scaling_. CRC press, 2000.
  * [47] C. A. Balanis, _Antenna Theory: Analysis and Design_. Wiley-Interscience, 2005.
  * [48] X. Zhang, C. Tepedelenlioglu, M. K. Banavar, and A. Spanias, “Distributed location detection in wireless sensor networks,” in _Signals, Systems and Computers, 2013 Asilomar Conference on_. IEEE, 2013.
  * [49] X. Zhang, C. Tepedelenlioğlu, M. K. Banavar, A. Spanias, and G. Muniraju, “Location estimation and detection in wireless sensor networks in the presence of fading,” _Physical Communication_ , vol. 32, pp. 62 – 74, 2019\.
  * [50] A. Goldsmith, _Wireless Communications_. Cambridge University Press, 2005.
  * [51] C. Tepedelenlioglu, A. Abdi, and G. B. Giannakis, “The ricean k factor: estimation and performance analysis,” _IEEE Transactions on Wireless Communications_ , vol. 2, no. 4, pp. 799–810, 2003.
  * [52] A. Moreira, I. Silva, F. Meneses, M. J. Nicolau, C. Pendao, and J. Torres-Sospedra, “Multiple simultaneous wi-fi measurements in fingerprinting indoor positioning,” in _2017 International Conference on Indoor Positioning and Indoor Navigation (IPIN)_. IEEE, 2017, pp. 1–8.
  * [53] P. Chen, J. Shang, and F. Gu, “Learning rssi feature via ranking model for wi-fi fingerprinting localization,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 2, pp. 1695–1705, 2019.
  * [54] Z. Chen, H. Zou, J. Yang, H. Jiang, and L. Xie, “Wifi fingerprinting indoor localization using local feature-based deep lstm,” _IEEE Systems Journal_ , 2019.
  * [55] J. Yoo and S. Park, “Fingerprint variation detection by unlabeled data for indoor localization,” _Pervasive and Mobile Computing_ , vol. 67, p. 101219, 2020.
  * [56] L. Zheng, B.-j. Hu, J. Qiu, and M. Cui, “A deep learning based self-calibration time-reversal fingerprinting localization approach on wi-fi platform,” _IEEE Internet of Things Journal_ , 2020.
  * [57] J. He and H. C. So, “A hybrid tdoa-fingerprinting-based localization system for lte network,” _IEEE Sensors Journal_ , 2020.
  * [58] W. Y. Al-Rashdan and A. Tahat, “A comparative performance evaluation of machine learning algorithms for fingerprinting based localization in dm-mimo wireless systems relying on big data techniques,” _IEEE Access_ , vol. 8, pp. 109 522–109 534, 2020.
  * [59] X. Zhu, W. Qu, T. Qiu, L. Zhao, M. Atiquzzaman, and D. O. Wu, “Indoor intelligent fingerprint-based localization: Principles, approaches and challenges,” _IEEE Communications Surveys & Tutorials_, vol. 22, no. 4, pp. 2634–2657, 2020.