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A Survey on Wireless Transmitter Localization Using Signal Strength Measurements

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Knowledge of deployed transmitters’ (Tx) locations in a wireless network improves many aspects of network management. Operators and building administrators are interested in locating unknown Txs for optimizing new Tx placement, detecting and removing unauthorized Txs, selecting the nearest Tx to offload traffic onto it, and constructing radio maps for indoor and outdoor navigation. This survey provides a comprehensive review of existing algorithms that estimate the location of a wireless Tx given a set of observations with the received signal strength indication. Algorithms that require the observations to be location-tagged are suitable for outdoor mapping or small-scale indoor mapping, while algorithms that allow most observations to be unlocated trade off some accuracy to enable large-scale crowdsourcing. This article presents empirical evaluation of the algorithms using numerical simulations and real-world Bluetooth Low Energy data.
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Review Article
A Survey on Wireless Transmitter Localization Using Signal
Strength Measurements
Henri Nurminen,1Marzieh Dashti,2and Robert Piché1
1Tampere University of Technology, Tampere, Finland
2NokiaBellLabs,Dublin,Ireland
Correspondence should be addressed to Henri Nurminen; henri.nurminen@tut.
Received  October ; Accepted  January ; Published  February 
Academic Editor: Sunwoo Kim
Copyright ©  Henri Nurminen et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Knowledge of deployed transmitters’ (Tx) locations in a wireless network improves many aspects of network management.
Operators and building administrators are interested in locating unknown Txs for optimizing new Tx placement, detecting and
removing unauthorized Txs, selecting the nearest Tx to ooad trac onto it, and constructing radio maps for indoor and outdoor
navigation. is survey provides a comprehensive review of existing algorithms that estimate the location of a wireless Tx given a
set of observations with the received signal strength indication. Algorithms that require the observations to be location-tagged are
suitable for outdoor mapping or small-scale indoor mapping, while algorithms that allow most observations to be unlocated trade
o some accuracy to enable large-scale crowdsourcing. is article presents empirical evaluation of the algorithms using numerical
simulations and real-world Bluetooth Low Energy data.
1. Introduction
Locating the wireless transmitters (Tx) in the network pro-
vides mobile network operators with important and relevant
information for a wide range of purposes, including nd-
ing rogue and nonfunctional access points (AP), planning
and operating the communication networks, and estimating
the radio frequency propagation properties of an area. Tx
location determination is also used when constructing radio
maps for localization services.
Every operator aims at providing good coverage so
that subscribers in most locations can access the network.
Competition between operators in providing the subscribers
with continuous and uninterrupted data usage prompt them
to nd unknown Txs that mainly belong to their competitors.
Based on the knowledge of the deployed Tx locations, oper-
ators decide optimal places for installing new infrastructure
within the area or steering the beam directions. An unknown
Tx can be a WLAN (wireless local area network) AP with
unlicensed spectrum or a femtocell AP whose spectrum is
licensed to the operator. ese Txs may be managed by
individuals or groups or the operator itself.
e operators ooad users from their G or G cells
to adjacent small cells or indoor femtocells when the trac
becomes heavy []. Knowing Txs’ locations and coverage
areas helps the operators to identify which cells are nearby.
Locating unknown Txs helps the administrators secure
the network when security loopholes are detected or when
there are intruders that breach the area managed by the
administrators []. Also when administrators update their
networkinfrastructurewithinanauthorizedarea,amapof
existing Tx locations helps to determine optimal locations for
new Txs.
Moreover, knowledge of Tx locations assists navigation
in environments where GNSS (Global Navigation Satellite
System) navigation is not feasible such as indoors. Indoor
navigation requires detailed knowledge of the network topol-
ogy of the building, and unmanaged Txs can also be used
provided that their locations are estimated. In many indoor
localization studies [–], it is assumed that Tx locations are
known a priori. is assumption is usually only valid for Txs
that belong to the owner of the infrastructure.
is survey provides the reader with a comprehensive
review on methods for locating wireless Txs using a set of
Hindawi
Wireless Communications and Mobile Computing
Volume 2017, Article ID 2569645, 12 pages
https://doi.org/10.1155/2017/2569645
Wireless Communications and Mobile Computing
Measurement © OpenStreetMap contributors
Tx
500 m
(a)
Measurement
Tx
10 m
(b)
F : An outdoor cellular base station (a) and an indoor BLE Tx
(b) with RSS measurement sets. Red color indicates a strong RSS.
measurements of the received signal strength (RSS). Most of
the presented methods can be applied to dierent types of
wireless networks, such as WLAN, Bluetooth Low Energy
(BLE), and cellular networks. Figure  shows examples of
an outdoor cellular base station and an indoor BLE Tx and
RSS measurement sets collected in the respective areas. RSS
information is available in reception reports of most wireless
networks’ receivers (Rx) without any special hardware or
soware modications [].
is article categorizes the methods based on two criteria:
measurement type and reference location requirement. e
measurement type can be the actual RSS from a Tx or just
the connectivity, that is, whether the Tx can be sensed or not.
Someofthereviewedmethodsrelyonlocatedmeasurements;
that is, they assume that every observation includes accurate
information about the location of the measurement. Some
methods assume that most of the observations are unlocated,
lacking the location information. e former are more accu-
rate but are costly to implement, while the latter are especially
suitableforcrowdsourcing.Weevaluatemethodsthatuse
located measurements through numerical simulations and
real-world BLE data.
e structure of this article is as follows: rstly, the
methods are presented in detail, connectivity-only methods
rst in Section , then RSS based methods that use mea-
surements with known locations in Section , and nally
RSS based methods that do not require all measurements to
be located in Section . Secondly, experimental results are
presented in Section , along with a table that summarizes the
basic practical properties of each method. Finally, Section 
presents the conclusions.
2. Connectivity Based Methods with
Located Observations
Connectivity based Tx localization algorithms assume that
the closer one is to the Tx, the higher is the probability
of observing the Tx when listening with a Rx device. e
observations consist of tuples (p𝑖,ID𝑖),wherep𝑖is the
reference position of the th measurement and ID𝑖is the set of
Tx identiers observed at the th measurement. Connectivity
actually means that the RSS exceeds the receiver’s sensitivity
threshold []. us, connectivity based methods in fact rely
on a very coarsely quantized RSS.
e simplest connectivity based Tx localization algorithm
is the (unweighted) centroid algorithm that was proposed for
the localization of a wireless sensor networks nodes by Bulusu
et al. []. e centroid algorithm has also been proposed and
tested at least in [–]. e estimate of the location of the th
Txisthemeanofthemeasurementlocations
m𝑗=1
#;ID𝑖𝑁
𝑖=1 ID𝑖⋅p𝑖,()
where #is the number of elements in set ,∈means
that is an element of the set ,is the total number of
measurements, and ⋅is the indicator function
=
0, statement is false
1, statement is true.()
e centroid location is the solution of the optimization
problem
arg min
m
𝑁
𝑖=1 ∈ID𝑖p𝑖m2,()
because()canbeexpressedastheweightedleastsquares
problem
arg min
m
10
2d
0𝑁
p1
p2
...p𝑁
...m
2,()
where 𝑖=∈ID𝑖and is the identity matrix, and
() follows from () by the weighted linear least squares
formula. Typically only the measurements where the th Tx
has been observed are used in the estimation of m𝑗;thatis,
the information of not observing the th Tx in a measurement
is omitted. In this sense, the centroid also has a probabilistic
interpretation as the maximum likelihood solution to the
measurement model
P∈ID𝑖|p𝑖,m𝑗∝−(1/2)(p𝑖m𝑗)T𝑆−1(p𝑖m𝑗),()
Wireless Communications and Mobile Computing
x (m)
y (m)
Measurement
True Tx location
Centroid
Robust centroid
F : Robust centroid (]=4) can outperform the conventional
centroid algorithm when there are outlier measurements.
where the positive-denite matrix isaconstantthatdoes
not aect the solution and Pdenotes probability. Koski et al.
[] also estimate the coverage area parameter matrix for the
purpose of online mobile Rx positioning.
e algorithm of Pich´
e [] can be considered a robus-
tied version of the centroid algorithm. is work considers
the assumption that there are outlier measurements with, for
example, erroneous reference position in the observation set
by relying on the Student’s -distribution that gives a higher
probability for occasionally receiving the signal far from the
Tx:
PID𝑖|p𝑖,m𝑗
∝1+1]p𝑖m𝑗T−1 p𝑖m𝑗−(]+2)/2 ,()
where is a constant that does not aect the solution and ]
R+is a model parameter degrees of freedom, and the closer ]
is to zero, the more robust the algorithm is. Based on the Stu-
dent’s model [], use an EM (expectation–maximization)
algorithm to solve the maximum a posteriori values of the
centroid and the coverage area matrix. Figure  shows a
localization scenario where four out of the  measurements
are outliers. In this scenario the robust centroid’s Tx location
estimates are signicantly closer to the true location than
the conventional centroid algorithm’s. Notice that both [,
] mainly concentrate on online positioning and do not
explicitly assume that the Tx is actually located at m𝑗;the
estimate rather models the center point of Tx’s coverage area.
e connectivity based methods are based on two
assumptions: Tx’s antenna is omnidirectional and measure-
ments are collected uniformly in the whole reception area
[, , ]. As Bulusu et al. [] point out, the performance of
the centroid algorithm is highly dependent on the data. Some
studies report [, ] that the Tx position estimate will be
biased towards areas with the highest measurement densities.
A possible solution to this problem is to model the thorough-
ness of the data collection in each location, which would also
introduce information on where the Tx is not hearable; this
information has been used for mobile Rx localization [].
Another approach is gridding, clustering the observations
in a regular grid so that each grid point represents all the
measurements in its vicinity, which can partly mitigate the
problem of uneven measurement distribution. Algorithms
for detecting insucient data collection and automatically
proposing new measurement locations have been proposed
[].
e centroid method is straightforward to understand
andimplement.ebasiccentroidalgorithmiscomputation-
ally light and the robustied version is still computationally
feasible for most purposes even though it is a constant factor
heavier than the basic centroid. Furthermore, the centroid
algorithms have a small number of tunable conguration
parameters, which might be advantageous if there is little
prior information on the Tx locations and signal propagation
models, and there is no risk of overcondent RSS models.
One important property of a method is whether the Tx
position estimate can be updated when new observations
appear without needing to access all the old observations. For
all the presented connectivity based methods, updateability
can be achieved with a very low cost; only the point estimate
and the number of samples used need to be stored in the
database.
3. RSS Based Methods with
Located Observations
issectionreviewsmethodsthatestimatetheTxlocation
using observations that consist of tuples (p𝑖,ID𝑖,r𝑖),wherep𝑖
is the reference location, ID𝑖is the list of Tx identiers, and r𝑖
is the vector of corresponding RSSs.
e RSS is negatively correlated with the distance between
the Tx and Rx. Attenuation of the signal strength (path loss,
PL) is due to both free space propagation loss governed by
theFriisequationandlossesgeneratedbyvariousobstruc-
tions in the environment [, Ch. ]. Accurate modeling of
these obstructions is in most practical cases infeasible, so
simplifying probabilistic models are commonly used. e
conventional probabilistic model in both outdoor and indoor
environments is the log-normal shadowing model [, Ch. ]
()=(0) 10log10
(0) +𝜎,()
where ()is the RSS in dBm (dB referenced to milliwatt) at
distance from the Tx, (0) is a reference distance, typically
m, (0) ((0))is the RSS at the reference distance,
is the PL exponent parameter, and 𝜎∼(0,2)is a
normally distributed shadowing term with variance 2.e
environment dependent PL parameters and 2are usually
estimated for a certain environment or for a certain Tx based
on data, while the transmission power dependent parameter
(0) canbeassumedtobeknownbasedontheTxproperties
[, Ch. ]. A typical assumption is that the shadowing term
𝜎is a statistically independent random variable for each
measurement, while another possible approach is to assume
spatially correlated shadowing; see, for example, the Gaussian
process based algorithm []. For localization purposes, it is
usually adequate to use distances in -dimensional Cartesian
coordinates; in the localization of cellular base station tower,
for example, Tx’s altitude aects the distance in the proximity
Wireless Communications and Mobile Computing
of the Tx, but as most data typically comes from farther
distances, this eect can be neglected [].
e signal shadowing consists of the so called small-scale
and large-scale shadowing components, and typically the PL
models model average of the both statistically, as accurate
analysis of the multipath propagation patterns that cause
the small-scale fading is not feasible in large systems. See
further discussion in [, Ch. .]. Currently most wireless
communication networks transmit continuous waveforms,
and optimization for impulse signals is out of the scope of
thisarticle.ItshouldbenotedthatincaseofmostWLANs,
for example, the mapping from the reported RSS indicator
to the actual RSS in dBm is unknown. is problem can be
circumvented, for example, by using RSS ratios [] or RSS
histogram []. e Rx can have one or multiple antennas,
and in the latter case the Rx device can either report all the
measurements separately or combine them into a single RSS
measurement.
3.1. Closed Form Solutions. A commonly proposed closed
form solution for Tx’s position using RSS measurements is
the weighted centroid algorithm that was proposed for the
localization of the wireless sensor nodes by Blumenthal et
al. []. It has been proposed for WLAN Tx localization, for
example, by [, ]. In the weighted centroid approach, the
estimate of the th Tx’s location is
m𝑗=1
𝑁
𝑖=1 𝑖𝑗𝑁
𝑖=1𝑖𝑗 ⋅p𝑖,()
where is a weighting function that depends on the RSS
and 𝑖𝑗 is the RSS of the signal transmitted by the th Tx and
received at the location p𝑖.Usuallytheweightsarechosenso
that the stronger the RSS, the greater the weight. e standard
weighting methods are the distance based weighting []
𝑖𝑗=∈ID𝑖⋅10𝜆𝑟𝑖𝑗 ()
and the RSS based weighting []
𝑖𝑗=∈ID𝑖𝑖𝑗 −min 𝜆.()
In both the weighting methods, ∈R+is a free parameter
and min is the signal detection threshold, that is, the lowest
possible RSS. ese weighting methods are compared for
wireless node localization in [] where it is found that the
two methods have equal average performance. e weighted
centroid location is the solution of the optimization problem
arg min
m𝑗
𝑁
𝑖=1𝑖𝑗 p𝑖m𝑗2,()
because()canbeexpressedastheweightedleastsquares
problem () by setting 𝑖=(𝑖𝑗),and()followsfrom
() by the linear least squares formula. A corresponding
probabilistic interpretation is that the weighting function
value in the measurements where the th Tx is observed
x (m)
y (m)
80
70
60
50
40
30
RSS (dBm)
Measurement
True Tx location
Centroid
Weighted c.
F : When the Tx is not located in the middle of its coverage
area, RSS based algorithms such as the distance-weighted centroid
can outperform the unweighted centroid.
follows the exponential distribution whose scale parameter
is inversely proportional to the squared Tx–Rx distance. By
the change of variables formula for PDFs (probability density
functions) this gives
𝑖𝑗 |∈ID𝑖,p𝑖,m𝑗
=p𝑖m𝑗2
2
d
d𝑖𝑗 −(‖p𝑖m𝑗2/𝑠2)𝑤(𝑟𝑖𝑗),()
where ∈R+is a constant that does not aect the
solution. To obtain the objective function of () for the
maximum likelihood solution, the Rx–Tx distance p𝑖m𝑗
is only to appear in the exponent, so it is removed from
the normalization constant by modeling the probability of
the RSS exceeding the signal detection threshold min to be
inversely proportional to the squared Rx–Tx distance
PID𝑖|p𝑖,m𝑗∝p𝑖m𝑗−2 ()
for distances exceeding a limit.
e comments regarding the unweighted centroid are
applicable to the weighted version as well, although modeling
of the RSS somewhat reduces the sensitivity to uneven data
density. Figure  shows a simulated example where Tx’s actual
location is not in the middle of its coverage area. In such a case
the weighted centroid algorithm outperforms the unweighted
centroid due to the RSS measurement information.
Another RSS based closed form solution is proposed
by Koo and Cha []. Earlier similar formulas have been
proposed for wireless sensor networks in []. e same
formulas are used in [] for distance measurement based
Wireless Communications and Mobile Computing
wireless transmitter positioning without the estimation of
the signal propagation parameter. Instead of the log-normal
shadowing model () [], use a dierent PL model
𝑖𝑗 |p𝑖,m𝑗,𝑗,𝑗
=𝑖𝑗;𝑗−𝑗p𝑖m𝑗2,2
𝑗2, ()
where (x;𝜇)is the PDF of the (possibly multivariate)
normal distribution with mean 𝜇and covariance matrix Σ
evaluated at x,and𝑗and 𝑗are the parameters of this non-
logarithmic PL model that are not directly related to the PL
parameters (0) and in (). (e notation is simplied from
[].) us, the distribution of two conditionally independent
RSSs’ dierence is
𝑘𝑗 −𝑖𝑗 |p𝑘,p𝑖,m𝑗,𝑗,𝑗
=𝑘𝑗 −𝑖𝑗 ;𝑗p𝑖m𝑗2p𝑘m𝑗2,22
𝑗2
=1
𝑗𝑘𝑖;𝑗𝑘𝑖 m𝑗
1𝑗,22, ()
where 𝑘𝑖 =p𝑖2p𝑘2,
𝑗𝑘𝑖 =2p𝑖p𝑘T𝑘𝑗 −𝑖𝑗 . ()
Given the at prior for the Tx position m𝑗and the improper
prior (1/𝑗)|1/𝑗|−1,theposteriorof(m𝑗,1/𝑗)is thus
given by the standard linear least squares (LLS) formulas
m𝑗,1𝑗|1:𝑁,p1:𝑁
=m𝑗
1𝑗;T
𝑗𝑗−1 T
𝑗𝑗,22T
𝑗𝑗−1,()
where
𝑗=𝑗1,2
𝑗3,4
...
𝑗(𝑁−1)𝑁
,
𝑗=1,2
3,4
...
(𝑁−1)𝑁
.
()
In this formula, each RSS measurement is used only once
to avoid correlations between RSS dierences. A strength of
this LLS method is the existence of closed form formulas; the
method thus has rather low and predictable computational
cost, and convergence is not an issue. Addition of prior
information on the Tx location is also straightforward.
However, if the actual RSS follows the log-normal shadowing
model (), the approximation () can be crude.
3.2. Iterative Methods. Maximizing the likelihood of the Tx
position and possibly some model parameters 𝜃using the
model () leads to the nonlinear least squares (LS) problem
m𝑗,𝜃=arg min
(m,𝜃)
𝑁
𝑖=1 𝑖(m,𝜃)2,()
where
𝑖(m,𝜃)=𝑖𝑗 −(0) (𝜃)+10(𝜃)log10 p𝑖m()
is the model function of one measurement.
is optimization problem can be solved using various
nonlinear LS methods that are typically iterative algorithms
[]. e general form of the nonlinear LS problem is
arg min
xf(x)2,()
where fis a known nonlinear function and ⋅is the Euclidean
norm. Many solution methods are based on dierentiation,
either on the rst order derivative (gradient, Jacobian) such
as the steepest descent and Gauss–Newton (GN) methods or
on the second-order derivative (Hessian matrix) such as the
Newton method []. To the authors’ knowledge, the second-
order information has not been used in problem () because
of the diculty of analytical dierentiation. Given an initial
point x0, a GN iteration is
x𝑘+1 =x𝑘−T
𝑘𝑘−1 T
𝑘fx𝑘, ()
where 𝑘is the Jacobian matrix of the function fevaluated in
x𝑘.
e GN method has been applied to problem () in
[, ], for example. In this case the model function fis the
function hwhose th element is
hm,(0),𝑖=𝑖𝑗 −(0) +10log10 p𝑖m,()
and the th row of its Jacobian matrix is
m,(0),𝑖,:
=10
ln10p𝑖mT
p𝑖m2−110log10 p𝑖m. ()
Wireless Communications and Mobile Computing
80
75
70
65
RSS (dBm)
Measurement
Tru e
Centroid
Weighted c.
GN
x (m)
y (m)
F : Centroid algorithms can suer from biased sampling
more than PL model based nonlinear methods such as the GN. e
colored dots are measurement locations in a -dimensional map. A
simulated example.
If the parameters (0) and are known for a certain environ-
ment, the corresponding columns can be le out from the
matrix (). e GN algorithm can sometimes diverge. A less
divergence-prone GN version is the Levenberg–Marquardt
(LM) algorithm used for Tx localization in []. Alternatively,
the divergence can be addressed by using an additional
line search algorithm that ensures decrease of the objective
function value as in [].
As pointed out in [], if the posterior covariance matrix
is approximated by the covariance matrix of the linearized
model. the estimate can be updated when new measurements
are obtained. Including a Gaussian prior distribution (𝜇)
for xkeeps the problem as a nonlinear LS problem
arg min
xx𝜇TΣ−1 x𝜇+h(x)T−1h(x),()
for which a GN iteration is []
x𝑘+1 =x𝑘−Σ−1 +T
𝑘−1𝑘−1
×Σ−1 x𝑘𝜇+T
𝑘−1hx𝑘, ()
where =2⋅is the measurement noise covariance matrix,
and 2is the shadowing variance in (). is iteration enables
approximative updating of the estimate without storing the
oldobservationsbyusingthecovariancematrixupdate−1+
T
𝑘−1𝑘)−1. Notice that if there is enough knowledge of the
PL parameters the Tx location estimate can be outside the
obser vation area as illustrated by Figure .
e GN converges to a local minimum, so the choice of
the initial point x0is important. Proposed choices of x0in Tx
localization are the location of the strongest observation [],
the centroid of all observations [], or the result of a grid-
type algorithm,which is discussed in Section .. A drawback
of GN and LM is that if there are several separate areas of
strong measurements, the computed Tx location estimate
depends strongly on the initial point so that dierent strong
areas are not compared.
Due to the assumption of normally distributed shadow-
ing,theGNalgorithmcanbesensitivetooutliermeasure-
ments, where the RSS diers signicantly from the value
predicted by the PL model. Outlier removal procedures for
tacklingthisissuehavebeenproposedatleastin[].
3.3. Monte Carlo and Grid Methods. is section discusses
methodsthatarebasedonexplicitevaluationoftheTx
locations PDF at several points of the location space. In
grid methods prespecied evaluation points are used, while
Monte Carlo (MC) algorithms are based on pseudorandom
evaluation points.
Importance sampling is a basic form of MC sampling.
Kimetal.[]useamethodwheretheMCsamplesofthe
location of one Tx m(𝑘)
𝑗are generated from a prespecied
prior distribution and then given weights (𝑘)
𝑗based on the
training measurements and known PL parameters. Kim et al.
do not explain their weighting method, but the formula based
on the model () is
(𝑘)
𝑗=−(1/2𝜎2)∑𝑁
𝑖=1(𝑟𝑖𝑗 −𝑟(0)+10𝑛log10p𝑖m(𝑘)
𝑗‖)2.()
e MC estimate of the posterior mean is then the weighted
average of the samples
m𝑗=1
𝑁𝑝
𝑘=1 (𝑘)
𝑗
𝑁𝑝
𝑘=1(𝑘)
𝑗m(𝑘)
𝑗,()
where 𝑝is the number of MC samples. is method can
also be called a particle lter with the static state model
as in [], since the weights can be updated recursively. A
drawback is that importance sampling suers from sample
impoverishment in static state estimation [, Ch. .]: all
weight will over time concentrate to a few samples and
there will be little variability because of lack of dynamics.
is problem can to some extent be overcome by using
resampling techniques such as resample-move algorithm or
Markov Chain Monte Carlo techniques [, Ch. .].
For some models a solution to sample impoverishment
is Rao-Blackwellization [, Ch. .] that is proposed for
simultaneous mobile Rx and static Tx localization by Bruno
and Robertson []. ey do online Tx localization using a
Rao-Blackwellized particle lter (RBPF) so that the training
measurements’ locations are obtained by inertial positioning.
us, the distribution of the Rx locations is obtained by MC
sampling.edistributionofeachTxlocationisapproxi-
mated by a Gaussian mixture for each MC sample and each
value of the PL parameter (0). e PL exponent is assumed
to be known. is gives a recursive algorithm for joint
estimation of the Rx and Tx locations. is solution is suitable
for cases where the locations of the training measurements
are imprecise but form a time-series that can be ltered.
Han et al. [] propose a grid method, where a plane
=(p)isttedtothe-dimensionalpositionRSSspace
(p,:,𝑗)for each grid point. e direction of gradient of is
Wireless Communications and Mobile Computing
then considered as an estimate of Tx’s direction, and the Tx
location estimate is dened as the point that minimizes the
mean square error of the directions of the grid points. Han
et al. use a dense grid method for the minimization, but they
also suggest that more ecient optimization tools could be
used.
Some authors exploit the fact that the PL parameters
appear linearly in the measurement model given the Tx
location. us, the PL parameters can be tted analytically
to each point of a set of candidate Tx locations. Shrestha et
al.[]makealinearleastsquaretofthePLparameters
for every measurement assuming that the Tx is located in
the considered measurement location. e Tx estimate is
chosen to be the measurement location that minimizes the
meansquareerrorofthePLparametert.Dependenceofthe
measurement density can be reduced by using a regular grid
as the set of candidate points. is will make the algorithm
more exible but also increase the computational complexity.
Achtzehn et al. [] propose a genetic algorithm, but its
details are le unexplained. If the PL parameters (0),,and2
are assumed known, the Tx locations likelihood can simply
beevaluatedateachgridpoint[].Agridcanalsoassistthe
GN or LM algorithm so that each grid point gives an initial
point to the iterative algorithm [].
Grid algorithms can achieve arbitrary modeling accuracy,
but the computational complexity will increase rapidly along
with the state dimension and grid density. Furthermore,
optimal values of critical parameters such as grid density
and grid size may vary in dierent subregions in large-scale
systems.
4. Tx Localization with
Unlocated Observations
All methods presented this far rely on a set of measurements
with reference locations assumed known accurately or as
a probability distribution. However, this assumption is not
always realistic especially in indoor environments, where
accurate GNSS services are unavailable and manual entry
of reference locations is too laborious especially for data
collected by crowdsourcing. is section reviews algorithms
where Txs’ locations relative to other Txs is estimated using
unlocated observations and the undirected graph created by
connecting Txs that appear in a common observation. e
basic assumption is that the more frequently two Txs are
observedinthesamemeasurementlocation,thecloserto
each other they are probably located. It is also possible to
use the RSS: if two Txs’ signals are strong in a location,
the Txs are probably close to each other. e locations in
global coordinates, that is, the correct scaling and rotation of
the radiomap, are obtained by adding some measurements
with reference locations: it is assumed that when a Tx is
observed (with a high RSS) in a located measurement, the Tx
isprobablyclosetothismeasurementslocation.eprinciple
is illustrated in Figure .
Koo and Cha [] propose multidimensional scaling
(MDS). e RSSs in a measurement with more than one Tx
determine the dissimilarity between the observed Txs, and
Building
Unlocated meas.
Located meas.
Tx
Connected
F : Unlocated measurements connect several Txs and thus
give information on how close each Tx is to other Txs. Located
measurements give information on the observed Txs’ locations in
global coordinates. Combining both measurement types gives an
optimal estimate of the Tx location coordinates.
the MDS nds the -dimensional Tx locations whose mutual
distances best agree with the dissimilarity matrix. In [] the
dissimilarity of the mobile Rx and the Tx is dened to be a
certain decreasing function of the RSS, and the dissimilarity
of two Txs is the smallest sum of the Rx–Tx dissimilarities
observed in the same training measurement. e dissimilari-
ties of Txs that are not connected by a common measurement
are determined through the other dissimilarities by using a
graph construction. Since the dissimilarities are not simple
functions of distance and contain noise, the Tx localization
is a nonmetric MDS problem, for which iterative algorithms
exist []. If some reference locations are available, the
relative MDS location estimates are transformed to global
coordinates by an optimal scaling, rotation, and translation
given by Procrustes analysis []. A drawback of this algo-
rithm is that if two Txs are located close to each other but
the closest training measurement location is far from both,
the Koo–Cha dissimilarity will overestimate the distance
between the Txs, because the dissimilarity corresponds to
the distance via the closest training measurement location.
Furthermore, the most natural choice for the mapping from
the RSS to dissimilarity would be the exponential relation
derived from the log-normal shadowing model (), which is
dierentfromthechoiceof[].
Raitoharju et al. [] propose several algorithms that use
unlocateddata.Basedontheirtests,theyrecommendaclosed
form solution called access point least squares (APLS). e
APLS is based on the model
m𝑖m𝑗∼0,2
1⋅,
m𝑘∼pk,2
2⋅, ()
where the Txs and are observed in the same measurement,
the Tx s located measurements’ mean location is p𝑘,and1
and 2are constants whose values do not aect the solution if
no prior distribution is used for the Tx locations. is results
in a linear Gaussian measurement model whose solution is
Wireless Communications and Mobile Computing
thestandardlinearleastsquaresformula.Raitoharjuetal.
[] also propose that the accuracy can be improved with the
cost of increased running time by applying a Gauss–Newton
method (), where the log-normal shadowing model with
xedPLparametervaluesisusedsothatbothTxlocations
andmobileRxlocationsareunknown.eGNalgorithm
is more accurate than the APLS due to modeling of RSS,
but multimodality of the posterior distribution can cause
convergence to nonglobal extrema [].
Chintalapudi et al. [] present a method that relies on
a genetic algorithm for nding initial points for iterative
optimization methods. In the rst phase, all initial points are
generated randomly; genetic algorithms are thus Monte Carlo
algorithms. e initial points are then treated in a manner
that depends on the objective function value (tness) of the
local maxima given by the iterative optimization method for
each initial point. e initial points with high tness are
retained, while the initial points with low tness are replaced
by generating new values, added random noise, or mixed
by random convex combinations. is cycle is iterated until
the solution stops improving. Chintalapudi et al. estimate
the mobile user location p𝑖,theTxlocationm𝑗,andthe
PL parameters (0) and jointly for each th measurement
and th Tx. Chintalapudi et al. use a tness function that
isbasedonthemeanabsoluteerror,butthestandardleast
squares approach of () can also be used for more standard
modeling and a wider range of optimization methods. e
genetic algorithm is capable of nding the global maximum
with a much higher probability than a single gradient descent
algorithm. e disadvantage is the increased computational
burden. Chintalapudi et al. discuss criteria to select a subset
of Txs and training data so that computational requirements
are somewhat reduced without losing accuracy signicantly.
5. Tests
5.1. Simulations. We implemented  Rx localization methods
with M. We simulated  Txs with  measure-
ments for each. We generated the measurement points from
bivariate normal distributions whose covariance matrices
were generated separately for each Tx from the Wishart
distribution with three degrees of freedom ((20m)2⋅,3).
Each measurement point was then assigned a RSS value
generated from the distribution
𝑖−70102log10 p𝑖m,62; ()
that is, the used PL parameters are (0) =−70,=2,
and =6, which are approximately in line with the values
(0) =−70.39,=1.32,and=5.85given in []. Each
Tx localization method that uses measurements with known
reference locations is then applied to each measurement set.
eparametervaluesusedinthetestswerethefollowing:
IntherobustcentroidalgorithmthenumberofEMiterations
was ve, and the number of degrees of freedom ]=4.
In the weighted centroid, we set min = −120dBm. We
optimized the parameter with a Monte Carlo simulation
0
1
2
3
Median error (m)
W.centroid-dist
W.centroid-RSS
10−2 10−1 100101102
𝜆
F : Optimization of the weighted centroid’s parameter
for distance based (black) and RSS based (grey) weighted centroid
algorithms. e values chosen for the further tests were . for
distance based and  for RSS based algorithm.
using . replications, and the median Tx positioning
error as a function of is shown in Figure . Based on
this, we set the parameter value to . for distance based
and  for RSS based weighted centroid. e GN iteration
wasterminatedwhenchangeintheTxlocationbetween
two successive iterations was less than  mm or aer 
iterations. e importance sampling used  Monte Carlo
samples. In the RSS gradient method the gradients were tted
for each point of the regular grid with -meter spacing so that
the grid squares that did not have any measurements were
removed. e window size of the gradient tting was chosen
according to the advice given in []: the window size was
increased until at least % of the grid points had at least
threemeasurementstotthegradient.egrid-point-wise
PL parameter tting method was based on a regular grid
using .-meter spacing and the square around the strongest
RSS measurement with side length  m.
e Tx localization error distributions are illustrated in
Figure . In these boxplots, the asterisks show the maximum
andminimumerrorofthemethod,andtheboxlevelsare%,
%, %, %, and % error quantiles. In the le subplot,
the measurement locations are generated from the bivariate
normal distributions. In the right subplot, the measurements
whose east coordinates are greater than those of the Tx are
removed; this test is done to study the robustness of the
methods to training data distributions that are not symmetric
with respect to the Tx location. Some of the algorithms can
be given prior information on the PL parameters. Note that
this kind of prior information is not always available in real-
worldscenarios.eredboxesinFigureshowtheerror
distributions when the PL parameters are given the prior
(0) −70,102, (a)
2,0.52. (b)
With the importance sampling method, estimation without
prior means using a prior with a large variance.
Wireless Communications and Mobile Computing
Centroid
Rob.centroid
W.centroid-dist
W.centroid-RSS
LLS
GN
Imp.sampling
RSS gradient
Grid-t
meas-t
0
5
10
15
20
25
30
Error (m)
(a)
Centroid
Rob.centroid
W.centroid-dist
W.centroid-RSS
LLS
GN
Imp.sampling
RSS gradient
Grid-t
Meas-t
0
5
10
15
20
25
30
Error (m)
(b)
F : Tx localization error distributions with simulated data. On the le, the measurements come from a point-symmetric bivariate
normal distribution, while on the right, the measurements east of the Tx are removed. e red boxes correspond to methods that use prior
information of the PL parameters.
Figure  shows that when the measurement data distri-
bution is point-symmetric, Gauss–Newton (GN) and grid-
point-wise t (grid-t) are the most accurate methods.
e importance sampling method is very close in accuracy
and it has exibility, for example, for extensions to non-
Gaussian models, but it requires a good prior distribution to
produce an ecient importance distribution. e accuracy of
the measurement point-wise t (meas-t) is limited by the
measurement point density and whether the measured area
covers the true Tx location. e gradient method performs
well with point-symmetric measurement sets, but suers
dramatically from removing the measurements of an area.
e reason for this can be that the method is based solely on
themeasurementgeometry;itdoesnotusethelogarithmic
shape of the propagation model. at is, in the west–east
direction there will mainly b e arrows pointing to east, and this
can deteriorate the accuracy in west–east direction. e linear
least square (LLS) method of [] suers from approximating
the logarithmic PL model with a linear one; the method seems
to t the linear PL model overweighting weak RSSs that are
the majority, and therefore the RSS peak location estimation
is biased.
e centroid algorithms that do not use RSSs perform
well in accuracy with point-symmetric data distributions.
e error is typically slightly higher than that of the GN,
but the overall performances can be regarded as competitive
considering the simplicity and computational ease of the
centroid methods. e centroid methods are robust against
deviations from the logarithmic PL model, but especially the
nonweighted centroids are sensitive to asymmetric data sets.
However, the weighted centroid still has accuracy slightly
lower but comparable with that of the GN. Robust centroid is
less accurate than the distance-weighted centroid, but slightly
more accurate than the nonweighted centroid due to non-
Gaussian coverage area.
Centroid
Rob.centroid
W.centroid-dist
W.centroid-RSS
LLS
GN
Imp.sampling
RSS gradient
Grid-t
Meas-t
0
20
40
60
80
100
120
Error (m)
F : Bimodal RSS distribution.
In some cases the distribution of RSS is not a function of
the distance only, but there can, for example, be several RSS
peaks, that is, areas governed by strong RSS measurements.
ese can be due to uneven terrain topology, reective build-
ing materials, or unmapped strong RSS areas, for example.
Figure  shows the Tx localization error distributions when
% of the training measurements are generated from a
normal distribution (𝜇,52⋅),where𝜇is a random point
close to the true Tx location. For each measurement point
we then generated the RSS 1fromthemodel()andthe
RSS 2fromthesamemodelusing𝜇as the Tx location. We
then set the actual RSS measurement to 0.71+0.32.
Figure  shows that the methods that perform best in the
unimodal RSS distribution’s case, that is, weighted centroid
and GN, have some large Tx localization errors with bimodal
RSS distribution. Weighted centroid and GN tend to choose
one RSS peak, the weighted centroid based on the strongest
measurements,andtheGNsolutionbasedontheinitialguess
 Wireless Communications and Mobile Computing
T : Tx localization RMSEs with real BLE data.
Algorithm RMSE w/o prior (m) RMSE with prior (m)
Centroid .
Robust centroid .
Weighted centroid (dist.) .
Weighted centroid (RSS) .
Linear least squares .
Gauss–Newton . .
Importance sampling . .
RSS gradient .
Point-wise t (regular grid) . .
Point-wise t (meas. pos.) . .
1
50
Floor
2
North (m)
0
East (m)
100
50
50 0
50
100
Measurement
Tx
F : Measurement and Tx locations in the test data set.
giventothealgorithm.Centroid,importancesampling,and
point-wise tting methods give more weight to the whole
RSS distribution and do not converge into nonglobal local
extrema.us,theweightedcentroidandGNhavemedian
accuracy close to the other methods, but they may require
some heuristics to cope with cases with multiple RSS peaks.
5.2. Real Bluetooth Low Energy Data. We in sta lled   Blu e-
tooth Low Energy (BLE) Txs in a building in the campus
of Tampere University of Technology. e ground truths of
thetheTxlocationsweremeasuredrelativetosomemap
objects using a measurement tape. Furthermore, we collected
measurements of the received BLE signal strengths using an
Android-run Samsung tablet device. e true location related
to each RSS measurement was obtained manually by clicking
an indoor map gure at each turn and interpolating between
the turns. Floor estimation was assumed perfect, so only
trainingdatacollectedinthetrueoorofeachTxwasused.
e locations of the Txs and the training measurements are
shown in Figure .
Figure  shows the Tx localization error distributions for
the real data test. e results mostly resemble those of the
simulation results with non-point-symmetric measurement
point distribution in Section .. e root-mean-s quare errors
(RMSE)ofthemethodsaregiveninTable.
Centroid
Rob.centroid
W.centroid-dist
W.centroid-RSS
LLS
GN
Imp.sampling
RSS gradient
Grid-t
Meas-t
0
5
10
15
20
25
30
Error (m)
F : Tx localization error distributions with real BLE data.
6. Concluding Remarks
is paper reviews and tests mathematical models and meth-
ods for wireless transmitter localization based on received
signal strength information. Empirical comparisons results
using simulated and real-world data are provided. e key
features of each presented method are summarized in Table .
Note that the column accuracy refers to how accurately the
methodcanbeadaptedtotheassumedsignalmodel,such
as the path loss model; the real-world localization error can
depend on the details of the scenario. Updateability means
that an algorithm for recursive updating without storing the
entire training database has been proposed.
e methods can be categorized based on what infor-
mation they use: RSS or only connectivity, with or without
known reference position. e methods that require refer-
ence positions are suitable for so called wardriving, that is,
outdoor network surveying where GNSS provides reference
positions, or for small-scale indoor mapping. e unlocated
methods trade o some accuracy to enable large-scale crowd-
sourcing even in GNSS-less environments. Computational
eciency and ease of updating the estimate without storing
large training databases are crucial in large-scale applications.
Wireless Communications and Mobile Computing 
T : Summary of the presented algorithms.
Algorithm
name Measurement Reference
location req.
Algorithm
type Accuracy Computat.
complexity
Implement.
complexity
Update-
ability
References &
Remarks
Centroid ID Accurate closed form Low Low Low Yes [, ]
Robust
centroid ID Accurate Iterative Low Low Medium Approx. [],robusttoerror
meas.
Weig ht ed
centroid ID, RSS Accurate Closed form Low Low Low Yes []
Linear least
squares ID, RSS Accurate Closed form Low Low Low Yes []
GN/LM ID, RSS Accurate Iterative Medium Medium Medium Approx. [–], local
minima
Importance
sampling ID, RSS Accurate Monte Carlo High Medium Low Yes [], sample
impoverishment
RSS gradient ID, RSS Accurate Grid, iterative Medium High High No []
grid-point-
wise
t
ID, RSS Accurate
Grid, closed
form or
iterative
High High Medium No [] (meas. pos.)
[] (regular grid)
RBPF ID, RSS Time-series,
distribution Monte Carlo High High High Yes []
MDS ID, RSS No Iterative Medium Medium High No [], various
methods
APLS ID No Closed form Low Low Low Yes []
GN ID, RSS No Iterative Medium Medium Medium Approx. []
Genetic
algorithm ID, RSS No MC, iterative High High High No []
An example of such a system is ubiquitous indoor position-
ing, which requires ecient initialization, improving, and
updating of large-scale radio maps that contain not only -
dimensional locations but also oor information.
Competing Interests
e authors declare that there is no conict of interests
regarding the publication of this paper.
Acknowledgments
e authors are grateful to Simo Ali-L¨
oytty, Jukka Talvitie,
Lauri Wirola, and Jari Syrj¨
arinne for enlightening conver-
sations. Henri Nurminen receives funding from Tampere
University of Technology Graduate School, the Foundation
of Nokia Corporation, and Tekniikan edist¨
amiss¨
a¨
ati¨
o.
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... The most important factor of proper planning was the access point placement optimization devices, where it has a significant impact on the coverage optimization, operation, and management of the networks more than the development of characteristics behavior of wireless networks 6 . It is worth mentioning that the location of the AP device in Wireless Local Area Network (WLAN) has improved a wide range of purposes such as locating the non-functional AP devices and estimating the propagation characteristics of the wireless signal 7 . ...
... These challenges can be summarized in the serious effects of different building materials including the wall thickness on signal penetration inside the building. Moreover, the effects of various sources of interference caused by improper AP and channel configuration 7 . Additionally, the Multipath Propagation (MP) describes the process of receiving several copies of the original signal distributed over varying time delays and attenuation caused by the reflection, diffraction and diffusion of these signals 8 . ...
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Optimizing the Access Point (AP) deployment has a great role in wireless applications due to the need for providing an efficient communication with low deployment costs. Quality of Service (QoS), is a major significant parameter and objective to be considered along with AP placement as well the overall deployment cost. This study proposes and investigates a multi-level optimization algorithm called Wireless Optimization Algorithm for Indoor Placement (WOAIP) based on Binary Particle Swarm Optimization (BPSO). WOAIP aims to obtain the optimum AP multi-floor placement with effective coverage that makes it more capable of supporting QoS and cost-effectiveness. Five pairs (coverage, AP deployment) of weights, signal thresholds and received signal strength (RSS) measurements simulated using Wireless InSite (WI) software were considered in the test case study by comparing the results collected from WI with the present wireless simulated physical AP deployment of the targeted building - Computer Science Department at University of Baghdad. The performance evaluation of WOAIP shows an increase in terms of AP placement and optimization distinguished in order to increase the wireless coverage ratio to 92.93% compared to 58.5% of present AP coverage (or 24.5% coverage enhancement on average).
... However, its accuracy is lower and require many numbers of access points (AP's) devices to improve its accuracy [2]. On the other hand, ranged-based method could achieve better accuracy where several methods have been presented to measure the time of arrival (ToA), angle-ofarrival (AoA), time difference of arrival (TDoA) and received signal strength (RSS) to obtain the distance or angles, which in turn would be used to calculate the location coordination's [7][8][9][10]. Several types of these measurements require extra hardware and cost. ...
... It can be deduced that some Rx's were closer to the actual coordination as the Rx in 2nd floor ranging between (1-6) with error ranging between (0.002-0.2) m. However, points of Rx (7)(8)(9)(10)(11) were different in large manner due to the utilization of direction antenna and its allocation. The error ranging in this case were up to 10 m. ...
... Localization-based indoor scenario suffers from bunch of challenges as compared to outdoor localization due to the complexity of indoor environment [6], the observed effect of different materials inside the building on the dissemination of signal and the need for higher accuracy at a lower cost and no extra hardware support [7].In contrast, Indoor localization market had been suffered from rapidly increasing lately and the growth is in a persistent basis [8,9]. The locating of the Radio wave (RW) transmitter in a wireless network (WN) has a significance over an extensive range of purposes [1], such as the locating of the non-functional Access point (AP), operating and managing the WN and estimating the propagation characteristics of the RW [10]. ...
... RSS was one of the most utilized measurements in recent localization systems and wireless coverage estimation methods [10], because its works together with the powered noise to define the value of the Signal to Noise Ratio (SNR) for the signals. Based on Shannon low [24], SNR represents the valuable capacity of the communication system. ...
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... The materials used to construct this scene using WI Package are mainly dependent on the characteristics of the electrical analog parameters. Therefore, the International Telecommunication Union (ITU) proposes that each material has conductivity and permittivity, and this conductivity and permittivity depend on the type of material as well as on the frequency used in the simulation environment [22,23,24,25]. The conductivity (σ) and permittivity (ε) values of the frequencies that have been used in the work environment are listed in Table 2. Table 2. Conductivity and permittivity values of various materials used in building the simulated scene model. ...
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... This work used link layer data such as the Received Signal Strength Indicator (RSSI) to estimate the location of users within an environment. It is noteworthy that most of the existing effective techniques are based on RSSI [25]. However, the RSSI information is provided by only a few protocols within the link-layer of the receiver. ...
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