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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 583
Scattering-Model-Based Methods for TOA Location
in NLOS Environments
Saleh Al-Jazzar, Member, IEEE, James Caffery, Jr., Member, IEEE, and Heung-Ryeol You, Member, IEEE
Abstract—In this paper, we address methods of mitigating one
of the major issues affecting wireless location accuracy in land
mobile terrestrial environments: nonline-of-sight (NLOS) propa-
gation. In order to improve location accuracy under such condi-
tions, we propose a novel methodology for NLOS environments
based on the use of scattering models to classify propagation
environments. The scattering models allow modeling of the NLOS
error so that the NLOS effect can be incorporated into a location
algorithm. Through the use of the scattering models, we develop
three novel location techniques based on the statistics of measured
ranges via moment matching, the expectation maximization al-
gorithm, and a Bayesian algorithm. Simulation results and dis-
cussion are given to illustrate the performance in typical NLOS
environments. The results show that the algorithms provide im-
provement over traditional location algorithms.
Index Terms—Geolocation, nonline-of-sight (NLOS) environ-
ments, scattering models.
I. INTRODUCTION
W
IRELESS location has received considerable attention
over the last few years due to the requirements set forth
by the FCC for Emergency-911 [1]. Further, with the emer-
gence of enhanced services for the third generation, wireless
communication systems has arisen the need for technologies to
enable and support those services, many of which require accu-
rate position location of mobile stations (MS) [2]. Traditional
approaches to position location assume a direct, or line of sight
(LOS), path that exists between the MS and each fixed station,
or base station (BS), used to perform radiolocation. Many
algorithms have been developed based on this assumption using
angles of arrival (or direction finding) [3], [4], times of arrival
(TOA) [5], [6], or time differences of arrival [7], [8] measured
at multiple BSs for network-based positioning or measured at
the MS for handset-based positioning. Unfortunately, in most
terrestrial wireless signal propagation environments, a LOS
propagation path does not exist to all of the BSs that participate
in locating a target MS, a condition which has become known
as nonline-of-sight (NLOS) propagation [9], [10]. For a TOA
Manuscript received December 21, 2004; revised March 1, 2006 and
March 9, 2006. This work was supported in part by Korea Telecom. Portions of
this paper were presented at the 55th IEEE Vehicular Technology Conference
(VTC), May 2002. The review of this paper was coordinated by Dr. R. Klukas.
S. Al-Jazzar is with the Department of Electrical and Computer Engineering,
Hashemite University, Zarqa 13115, Jordan (e-mail: saljazza@hu.edu.jo).
J. Caffery, Jr. is with the Department of Electrical and Computer Engineering
and Computer Science, University of Cincinnati, Cincinnati, OH 45221 USA
(e-mail: jcaffery@ececs.uc.edu).
H.-R. You is with the Korea Telecom, Seoul 137-792, Korea.
Digital Object Identifier 10.1109/TVT.2007.891491
location system, which is the method under consideration in
this paper, the range between the MS and each BS is measured.
However, NLOS propagation causes the measured ranges to be
considerably larger than the true, or LOS, distance between the
MS and each BS due to signal propagation around obstacles
such as buildings [9], [11]. Consequently, in NLOS conditions,
the traditional algorithms perform poorly [11], [12], leading
to the necessity of developing algorithms which are robust to
NLOS effects.
Several approaches for mitigating the error introduced due to
NLOS propagation have been addressed in literature [10], [11],
[13]–[19]. The method in [10] and [13] attempts to reconstruct
LOS TOA measurements from a series of both LOS and NLOS
TOA measurements made over time and assumes knowledge of
the NLOS standard deviation for identifying NLOS BSs. How-
ever, this approach requires occasional LOS links to each BS
and cannot be expected to be effective when signal propagation
over the measurement span is only through the NLOS paths.
Other approaches have been proposed to mitigate NLOS
effects when the objective is to locate MSs using single mea-
surements at a set of participating BSs. The algorithms in
[14]–[16] attempt to selectively remove or proportionately
weight NLOS corrupted measurements based on their deviation
from the majority-LOS BSs’ estimate using residual weighting
algorithms. Those with large deviation according to a specified
criterion are declared NLOS. The residual weighting method
is useful when the set of participating BSs is much larger than
the minimum of three required for 2-D location. Additionally,
the set of BSs must include both LOS and NLOS links to the
MS, with the best performance obtained when the number of
NLOS BSs is small. If the links between the MS and BSs are
all NLOS, the method cannot provide much improvement in
location accuracy.
Constrained optimization approaches have also been utilized
to exploit characteristics of the NLOS-propagation error. A
constrained nonlinear least-squares (LS) TOA algorithm was
presented in [11] which exploits the fact that the NLOS-
corrupted TOA measurements are larger than the true LOS
ranges. In the range scaling algorithm (RSA) in [17], which
operates with only three BSs, the NLOS error is mitigated
using TOA adjustment by scaling the NLOS-corrupted TOA
measurements using factors that are estimated from a con-
strained nonlinear optimization process. Another constrained
optimization procedure was presented in [18] that estimates the
NLOS error using a sequence of decreasing bound constraints
which attempts to correct the LS location estimate formed from
the NLOS-corrupted TOA measurements. The method was
shown to produce accurate location estimates in field trials, but
0018-9545/$25.00 © 2007 IEEE
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584 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Kim et al. do not describe how the bounds are formed, which is
critical to the success of the algorithm. Finally, the algorithm in
[19] employs a quadratic programming approach to compute a
maximum-likelihood (ML) estimate of the MS position using
constraints similar to those in [11]. Results showed that the
accuracy of this algorithm only improves if more than five BSs
are employed.
Most of these algorithms assume that there is at least one
LOS BS at which measurements are obtained [10], [13]–[16],
[18], or more than the minimum of three BSs (for 2-Ds) are
required for improved accuracy [14]–[16], [18], [19]. However,
two issues limit the application of these algorithms based on the
number of participating NLOS/LOS BSs. First, in macrocells,
the link between an MS and its serving BS is usually modeled
as NLOS, and similarly, the MS is generally considered NLOS
to the surrounding BSs since it is typically much farther from
them. For microcells, although the MS is typically modeled as
being LOS with its serving BS, one cannot expect the MS to be
LOS with the surrounding BSs. Thus, it is reasonable to expect
at most one LOS BS, the serving BS, in a terrestrial wireless
communication system. In the most general scenario, especially
for macrocells, all BSs can be expected to be the NLOS.
Second, the participation of a neighbor BS in positioning
is limited by its “hearability,” i.e., the ability to detect sig-
nals above a threshold level. This depends on factors such
as the density of BSs in the area of interest and system-
specific issues such as power control in code-division-multiple-
access (CDMA)-based systems. Therefore, while increasing the
number of BSs leads to better performance metrics in most
algorithms, this may not be realizable in practical systems. For
instance, the likelihood of finding three BSs with a received
signal-strength indicator stronger than −100 dB was only 35%
in rural areas, whereas it was approximately 84% in urban
areas [20]. It is clear, then, that TOA-based methods that require
more than the minimum number of BSs are not suitable for
many environments.
A further distinguishing characteristic of the existing NLOS-
mitigation algorithms is that they utilize a single TOA measure-
ment, usually assumed to be the TOA of the earliest arriving
multipath component in a fading environment. They do not
make use of the multipath signals from the channel to aid the
process of locating the MS in NLOS channels. In this paper, we
develop algorithms which make use of TOA measurements of
multipath components in order to provide improved accuracy
in the presence of NLOS error. The algorithms operate with
the minimum number of three nearest BSs and do not require
any to be LOS with the MS. Our approach is based on the use
of propagation scattering models which describe the multipath
scattering in the channel. Since scattering models are general
descriptions of multipath scattering in an environment, rather
than location-specific descriptions, we use the scattering mod-
els to classify environments based on their general scattering
characteristics as discussed in Section II. Thus, once an envi-
ronment has been classified, the scattering model associated
with that classification is used in the methods presented in
this paper. While many different scattering models exist in the
literature, we focus on two macrocell models for purposes of
illustrating the algorithm development.
We would also like to emphasize that the proposed algo-
rithms apply to any system based on ranging or TOA estima-
tion, and that we do not address the estimation of multipath
TOAs, but rather their use for location determination in an
NLOS environment. Methods for estimating multipath TOAs
can be found in [21]–[24].
The remainder of this paper is organized as follows.
Section II presents well-known scattering models and their
TOA probability distribution functions (pdfs). Section III dis-
cusses the problem formulation for the NLOS-mitigation algo-
rithms which are presented in Sections IV and V. Simulation
results are presented in Section VI which is followed by the
concluding remarks in Section VII.
II. S
CATTERING MODELS
A signal faces reflections/diffractions/scattering from differ-
ent physical scatterers as it propagates between the MS and BS
which results in the multipath fading phenomenon. Due to the
unique propagation conditions experienced by an MS at dif-
ferent locations in a cellular environment, an exact description
of the multipath profile, in particular, the multipath TOAs at
an MS, is not possible without extensive field measurements or
lengthy simulation using ray tracing. Due to the wide variability
from location to location, the scattering effects can be described
as being random in nature. Consequently, models for the dif-
ferent environments have been developed with the purpose of
describing environments “on average.” Such models include the
ring of scatterers (ROS) and disk of scatterers (DOS) models
for macrocellular environments and the elliptical model for mi-
crocellular environments [25]–[28]. Several variations on these
basic models have been presented in literature. For instance,
for the DOS formulation, the density of scatterers on the disk
has been assumed to be uniform [26], Gaussian [29], and
hyperbolic [30]. For the ROS formulation, the scatterers have
been assumed to be uniformly distributed around the ring [27].
All of these models are referred to as “single-bounce” models as
they are developed assuming that the signal undergoes a single
reflection during its propagation between the MS and BS. More
scattering models are given in [25].
The macro- and microcellular models are useful for classi-
fying the scattering characteristics of different environments.
In the case of macrocells, the ROS and DOS model classes
produce significantly different descriptions of the multipath
TOAs due to scattering, while within the DOS class of models,
the Gaussian and uniform distribution for scatterers do not pro-
duce drastically different TOA characteristics. Consequently,
by using the ring, disk, and elliptical models (regardless of the
underlying assumed distribution of scatterers in those models),
we are able to classify environments and utilize the model
descriptions to enhance wireless location under NLOS condi-
tions for those environments. In particular, due to the nature of
the models, the ROS model describes a sparse distribution of
scatterers around the MS, whereas the DOS model describes a
more dense distribution of scatterers, the density being deter-
mined by the underlying scatterer density assumption for the
disk (i.e., uniform, Gaussian, etc.). It is in this framework that
the algorithms in the next few sections are developed: A given
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AL-JAZZAR et al.: SCATTERING-MODEL-BASED METHODS FOR TOA LOCATION IN NLOS ENVIRONMENTS 585
Fig. 1. BS, MS, and scatterer geometry for the ROS scattering model.
environment is classified using a model according to its general
scattering characteristics, and that model is utilized to develop
algorithms that mitigate the NLOS-propagation effects.
For purposes of illustration due to space limitations, only
the macrocellular models with uniform scatterer distributions
are considered. However, we note that the methods developed
can be directly applied to other scatterer distributions and to
the elliptical microcell models. Prior to the development of the
algorithms, the ROS and DOS models are described, and the
pdfs of the multipath TOAs are presented.
A. ROS Model
In the ROS model shown in Fig. 1, the scatterers are
uniformly distributed on a ring which is centered about the
MS with a radius R
r
. Thus, the angular distribution of the
transmitted (or, equivalently, arriving) signal from (at) the MS
is uniformly distributed on [0, 2π]. This is the classical model
[31], [32] used to describe macrocellular environments in which
the BS is situated high on a tower and is away from the local
scatterers, whereas the MS is near the ground and is surrounded
by the local scatterers.
For the TOA location approach, we are interested in the
distribution of the TOAs according to the model in Fig. 1.
Denoting the jth multipath TOA at the ith BS as l
ij
, and
assuming a single bounce, the pdf of the TOA of an arriving
multipath signal component is given by [25]
p
ROS
(l
ij
)=
l
ij
− R
r
R
r
R
i
Ω
ij
,R
i
≤ l
ij
≤ R
i
+2R
r
(1)
where
R
i
= x
j
− x
ms
=
(x
ms
− x
i
)
2
+(y
ms
− y
i
)
2
(2)
denotes the true distance between the MS located at x
ms
=
[x
ms
,y
ms
] and the jth BS is located at x
i
=[x
i
,y
i
], and
Ω
ij
=
1 −
(R
i
− l
ij
)(R
i
+ l
ij
− 2R
r
)
2R
r
R
i
+1
2
.
Fig. 2. ROS, DOS, and RDOS pdfs are corrupted with an additive white
Gaussian measurement noise of standard deviation of 50 m, and R
i
=1km
and a radius of the scatterers of 0.3 km.
Noting that the NLOS error associated with the jth multipath
TOA, η
ij
is l
ij
− R
i
, we can determine the pdf for the NLOS
component as
p
ROS
(η
ij
)=
η
ij
+ R
i
− R
r
R
r
R
i
Ψ
ij
, 0 ≤ η
ij
≤ 2R
r
(3)
where
Ψ
ij
=
1 −
η
ij
[2(R
r
− R
i
) − η
ij
]
2R
r
R
i
+1
2
.
An example of the pdf in (1), assuming that R
i
=1km with
R
r
=0.3 km, is shown in Fig. 2, where additional additive
Gaussian noise has been added to the multipath TOA mea-
surements. We note that since all of the scatterers are a fixed
distance, R
r
from the MS, the radius of the ring determines
the magnitude of the NLOS-propagation error η
ij
and also the
density of the scatterers in the area surrounding the MS. For
larger radii, the scatterers near the MS are quite sparse. We also
notice that the peaks of the pdf are near the LOS distance (1 km)
and near the maximum NLOS distance (1.6 km). Thus, there is
a high probability of large NLOS error and of near-zero NLOS
error with this model.
B. DOS Model
In the DOS model shown in Fig. 3, the scatterers are located
on a solid circular disk of fixed radius R
d
centered about the
MS. The distance to a scatterer from the MS r
DOS
is uniformly
distributed in the range [0,R
d
], and the angle θ is uniformly
distributed in the range [0, 2π]. Clearly, this model is similar to
the ROS model (in terms of the assumptions regarding the local
scatterers around the MS and BS) but with a different density
of scatterers around the MS.
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586 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Fig. 3. BS, MS, and scatterer geometry for the DOS scattering model.
Referring to Fig. 3 and following the method
1
in [25] with
a uniform distribution of scatterers, we find the pdf of the jth
multipath TOA at the ith BS to be
p
DOS
(l
ij
)=
N(l
ij
)
D(l
ij
)
(4)
where
N(l
ij
)=
−l
2
ij
+ R
2
i
(l
ij
− 2R
d
)
·
(R
i
+2R
d
− l
ij
)(R
i
− 2R
d
+ l
ij
)
− 2
2l
4
ij
− 3l
2
ij
R
2
i
+ R
4
i
arctan(Q
ij
)
with
Q
ij
=
(l
ij
+ R
i
) tan
1
2
arccos
l
2
ij
+R
2
i
−2l
ij
R
d
2R
i
(l
ij
−R
d
)
l
2
ij
− R
2
i
and
D(l
ij
)=4πR
2
d
(l
ij
− R
i
)
3/2
(l
ij
+ R
i
)
3/2
.
An example pdf using the same parameters as the ROS model
is included in Fig. 2. As with the ROS model, the size of the disk
R
d
determines the magnitude of the NLOS error. Unlike the
ROS model, the density of the scatterers about the MS remains
uniform despite the disk radius. Consequently, the DOS model
is appropriate for modeling environments where the scatterers
around the MS are much denser. The density of the scatterers
can be easily changed by considering a different distribution of
scatterers around the MS [29], [30], as indicated previously. Fi-
nally, we note that for the DOS model, the probability of gener-
ating multipath TOAs with large NLOS error is quite low, while
the multipath TOAs near the LOS are highly probable, as indi-
cated by the peak of the distribution near the LOS distance R
i
.
C. Reverse DOS (RDOS) Model
Neither of the previous two models generates a pdf which
produces multipath TOAs having a high probability of large
NLOS error and low probability of small NLOS error. Conse-
quently, in order to provide a thorough examination of the effect
1
An alternative method for deriving this pdf is discussed in [27].
that NLOS propagation has on traditional location algorithms
and those developed in this paper, we use a heuristic model
which we term the RDOS model. In this case, the pdf generated
by the DOS model is reversed so that the peak of the pdf appears
at large TOA values, and there is a much smaller probability of
the TOAs near the true LOS value. This effectively models a
severe NLOS environment where large values of NLOS error
are predominate. Mathematically, we express the RDOS pdf in
terms of the DOS pdf by
p
RDOS
(l)=p
DOS
(2µ
DOS
− l) (5)
where µ
DOS
denotes the mean of the DOS pdf. An example
pdf of this model is shown in Fig. 2 where the RDOS pdf was
obtained by flipping the DOS pdf about its mean value.
III. P
ROBLEM FORMULATION FOR NLOS MITIGATION
USING SCATTERING MODELS
In order to make use of the scattering models and the sta-
tistical information they can generate, it is necessary to utilize
multiple measurements of TOAs at each BS that is used for
location. As multiple measurements of the first arriving TOA
(i.e., the TOA that is typically used for location purposes) may
not vary significantly from measurement to measurement, es-
pecially for a stationary MS, we utilize the multiple TOA mea-
surements that are available by measuring the TOAs of arriving
multipath signals at the BS. In this way, assuming the model is
stationary, the received multipath components must have been
received due to the reflections from scatterers at different dis-
tances from the BS, i.e., from different points on the scattering
model. Thus, the measurement of multipath TOAs at each BS
used for location enables us to estimate certain characteristics
of the scattering model and, consequently, determine the loca-
tion of the MS which is located at the center of the model.
The algorithms we develop are based on the assumption
that the multipath TOAs that occur due to reflections from the
scatterers surrounding the MS can be estimated. The multipath
channel can be modeled as
h(t)=
L
j=1
β
j
δ(t − τ
j
)
where β
j
and τ
j
are the propagation constant and the delay for
the jth path, respectively, and L is the number of multipaths
arriving at the jth BS from the MS. A key issue in this regard is
the ability to resolve the multipath components for estimation
purposes. We consider two multipath components with delays
τ
1
and τ
2
to be resolvable if their delay difference significantly
exceeds the inverse signal bandwidth: |τ
1
− τ
2
|W
−1
[33].
Further, we do not address methods to estimate multipath TOAs
as discussed in Section I but only use the multipath TOA
estimates for purposes of locating the MS.
With this in mind, the TOA measurement of the jth multipath
at the ith BS, separated by the true (LOS) distance of R
i
from
the MS, is represented as
l
ij
= R
i
+ η
ij
+ µ
ij
(6)
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AL-JAZZAR et al.: SCATTERING-MODEL-BASED METHODS FOR TOA LOCATION IN NLOS ENVIRONMENTS 587
where η
ij
is the NLOS component and µ
ij
is the measure-
ment noise which is considered as additive white Gaussian
noise (AWGN). For TOA systems, it is known that NLOS
propagation causes the signal to arrive from a path which is
longer than the true distance R
i
[9]–[11]; thus, η
ij
≥ 0 and
is typically much larger in magnitude than the Gaussian mea-
surement noise. A typical value for η
ij
has been measured in
the Global System for Mobile (GSM) communications system
which indicates that NLOS error can average between 500 and
700 m [9].
In the following sections, the measured multipath TOAs at
each BS l
ij
along with the macrocell models, which serve to
classify the scattering characteristics of the different environ-
ments, are used to estimate the location of the MS. The first
method is based on matching the moments of the measurements
with those of the model, and the remaining two methods make
use of parameter estimation techniques.
IV. J
OINT VARIANCE MAT C HI N G ALGORITHM (JVMA)
To determine the position of the MS when using the
scattering-model classification method, we must determine the
position of the model and its radius. For convenience and sim-
plicity of reference, we define the generic scatterer “radius” r as
r =
R
r
, ROS model
R
d
, DOS model
(7)
and is assumed to be fixed for all BS–MS pairs. This
assumption is reasonable since the environment surrounding
the MS is seen to be approximately the same for all BSs in an
NLOS situation, and the BSs are assumed to be positioned on a
higher level than its surroundings (i.e., macrocell). Thus, there
are three unknown parameters, x
ms
=[x
ms
,y
ms
] and r, which
can be determined using three independent equations.
From the pdfs in (1) and (4), the variance ˆσ
2
i
of the multipath
TOAs at BS
i
can be expressed mathematically in terms of R
i
and the scatterer radius r. The position of the scattering model,
or, equivalently, the position of the MS x
ms
, is directly related
to the LOS distances R
i
, according to (2).
Given this foundation, the approach for determining the MS
location is formed by generating expressions for ˆσ
2
i
, i =1, 2, 3,
from the pdfs in (1) and (4), which provides three equations in
the three unknowns x
ms
and r. These mathematical expressions
can then be equated to the variance σ
2
i
of the measured multi-
path TOAs at the three BSs. Solving the three equations for the
unknown position and radius produces the desired result. We
can then ignore r since only the location x
ms
is desired.
Using the pdfs developed in Section II, mathematical expres-
sions for the variance of the multipath TOAs ˆσ
2
i
can be found by
ˆσ
2
i
=
R
i
+2R
r
R
i
l
2
ij
p(l
ij
)dl
ij
− ˆµ
i
2
(8)
where ˆµ
i
is given by
ˆµ
i
=
R
i
+2R
d
R
i
l
ij
p(l
ij
)dl
ij
. (9)
In the following, the JVMA will be presented for the ROS and
DOS scattering models. The algorithm for RDOS is not shown
but is found in a manner identical to the DOS model, due to the
relationship between the models given in (5).
A. JVMA for ROS Model
Applying (8) for the ROS model (r = R
r
) in (1) and using
the following series expansion approximation for arccos(x)
arccos(x) ≈
π
2
− x −
x
3
6
−
3x
5
40
we find the variance in terms of R
r
and R
i
to be
ˆσ
2
ROS,i
= R
2
i
+2R
i
R
r
+2R
2
r
− (ˆµ
ROS,i
)
2
−
2
π
317R
3
r
420R
i
+
R
5
r
35R
3
i
+
2R
7
r
1155R
i
(10)
where the mean ˆµ
ROS,i
can be found from (9) to be
ˆµ
ROS,i
= R
i
+ R
r
−
1
π
317R
2
r
420R
i
−
1
π
R
4
r
35R
3
i
−
1
π
2R
6
r
1155R
5
i
for i =1, 2, 3. Now, given a set of measured multipath TOAs
at each of the three BSs, we can equate (10), i =1, 2, 3,to
the calculated variances of the measured multipath TOAs. This
produces a set of three nonlinear equations which we iteratively
solve for ˆx
ms
, ˆy
ms
, and R
r
using the Newton–Raphson method
[34] with the error defined to be the square of the difference
between the variances from (10) and the measured variances.
B. JVMA for DOS Model
Unlike the ROS JVMA solution, there is no closed form
expression for ˆσ
2
DOS,i
for the DOS model. Thus, we incorporate
numerical techniques in the form of the Riemann sum for the
variance calculation. Thus
ˆσ
2
DOS,i
≈
R
i
+2R
d
l
ij
=R
i
l
ij
p
DOS
(l
ij
)l
ij
− (ˆµ
DOS,i
)
2
ˆµ
DOS,i
≈
R
i
+2R
d
l
ij
=R
i
l
ij
p
DOS
(l
ij
)l
ij
(11)
where l
ij
is the bin width used for the Riemann sum. Again,
we equate the three equations in (11) to the variances of the
measured multipath TOAs from each of the three BSs in order
to form the estimates of ˆx
ms
, ˆy
ms
, and R
d
. Due to the numerical
form of (11), we employ a numerical technique to determine the
solution to the set of equations. This is accomplished by finding
the minimum of the error surface that is defined by
Υ(ˆx
ms
, ˆy
ms
,R
d
)=
3
j=1
|ˆσ
i
− σ
i
|
2
which is evaluated numerically using the Nedler–Mead simplex
method [35].
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588 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
V. P ARAMETER ESTIMATION
The JVMA algorithm presented in the previous section es-
timates the MS coordinates together with the scattering-model
parameter r. In this section, we present two algorithms that first
estimate the true range between the MS and the BS then apply
the traditional location algorithms to estimate the MS location.
Since the multipath TOAs are produced from reflections of
the scatterers in the models, each multipath component will
have the same distribution, which is given by the specific model
used to classify an environment. In addition, we assume, due to
isotropic transmission of the signal from the MS and the ran-
dom locations of the scatterers, that the received multipath com-
ponents at each BS are independent. Consequently, the joint pdf
of the multipath TOAs at the ith BS will simply be the product
of the individual pdfs of the TOAs as given in (1) and (4) and
will continue to be functions of the parameters R
i
and r. In ad-
dition, the multipath TOA measurements are assumed to be cor-
rupted by zero-mean AWGN with variance σ
2
µ
according to (6).
From the pdfs for the models in Section II, we know that the
true LOS range between the MS and each BS is a parameter of
the pdfs and, consequently, can be estimated using parameter
estimation algorithms. Once the true range R
i
for each of the
three BSs has been estimated, then any traditional location al-
gorithm designed for LOS conditions, such as the Taylor series
algorithm [7] or the linear-line-of-positions (LLOP) method
in [36], can be applied to obtain an accurate estimate of the
MS position.
Several approaches can be used for estimating an unknown
parameter, or set of parameters, from a set of measurements
[37], [38]. In general, parameter estimation is used to estimate
a parameter θ of a pdf p(y; θ), where y is a random variable. The
estimate is determined from realizations of y, which we denote
as ˆy. Two of the algorithms that can be applied are the Bayesian
estimation algorithm [37] and the expectation maximization
(EM) algorithm [38]. The Bayesian and EM methods provide a
means of estimating θ. In the location estimation problem under
investigation, y represents the set of measured ranges L
i
=
[l
i1
,l
i2
,...,l
iN
m
] at the ith BS, and θ represents the parameters
[R
i
,r], where N
m
is the number of measured multipath TOAs
at each BS.
A. Bayesian Estimation
The Bayesian method is a parameter estimation technique
which is based on Bayes’ rule to calculate the pdf p(θ|y).
There are three criteria for which Bayesian estimation can be
applied [37]:
1) Minimum mean squared error (mmse)
ˆ
θ = E (θ|Y = y) .
2) Minimum mean absolute error
ˆ
θ = ω (p (θ|Y = y))
where ω(p(θ|Y = y)) is the median of the pdf.
3) Maximum a posterior
ˆ
θ = max (p (θ|y)) .
For purposes of demonstrating the Bayesian parameter esti-
mation approach to the location problem under consideration,
we employ only the MMSE criterion in the algorithm that
follows. Thus, according to the first criterion above, the location
problem reduces to one of finding the pdf p(γ|L
i
), where γ =
[R
i
,r] is the vector of unknown parameters to be estimated, and
L
i
=[l
i1
,...,l
iN
m
] is the vector containing the N
m
measured
multipath TOAs at the ith BS. From Baye’s rule, we know
p(γ|L
i
)=
p (L
i
|γ) p(γ)
p(L
i
)
but p(γ) is unknown, so an alternative method of applying
the Bayesian mmse approach [37] is followed which estimates
the unknown parameters iteratively according to the following
manner.
1) Assume r.
2) Estimate
ˆ
R
i
from p(R
i
|L
i
,r).
3) Estimate ˆr by matching the variance of the measured
multipath TOAs in L
i
with the estimated variance.
4) Repeat steps 2) and 3) until the convergence criteria is
satisfied.
With regard to the third step, for each BS, we match the variance
of the measured TOAs σ
i
with the calculated variance given R
i
and L
i
, ˆσ
i
(r|L
i
,R
i
), i.e.,
3
i=1
|ˆσ
i
(r|L
i
,R
i
) − σ
i
|
2
=0 (12)
which is chosen since it has a single solution at ˆσ
i
= σ
i
for
all i. This cost function is minimized iteratively using the
Nedler–Mead algorithm [35].
Finally, we note that this procedure estimates both the true
LOS ranges R
i
and the model parameter r for each BS. How-
ever, since we are only interested in R
i
for location purposes,
we discard the results for r for each BS.
B. EM Estimation
A common method for performing parameter estimation
via measured data is the ML method. In this approach, the
parameters R
i
and r are estimated to maximize the pdf of the
observed data p(L
i
|R
i
,r) [39]. The maximization of the pdf
p(L
i
|R
i
,r) is analytically complicated because the pdf is non-
linear. Consequently, an alternative method for performing the
maximization is through the use of the iterative EM algorithm
[39]. The EM algorithm is primarily used for maximizing the
pdf of the observed data from incomplete measurement data.
The observed data might be incomplete because of the limited
available information in either the time or spatial domain [39].
If the set of the complete data is denoted as X and the set of
incomplete data is Y, then
Y = F (X )
where F (·) is a noninvertible transformation. This relation
indicates that the complete data are only observed through a
noninvertible transformation, where this transformation gener-
ally causes a reduction in the available data [39].
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AL-JAZZAR et al.: SCATTERING-MODEL-BASED METHODS FOR TOA LOCATION IN NLOS ENVIRONMENTS 589
Denoting the unknown parameters as θ =[R
i
,r],theEM
algorithm performs the maximization of p(L
i
|θ) via the fol-
lowing iterative steps.
1) Initialize the parameter θ with an initial guess θ
0
.
2) (E-step) Estimate the likelihood function: Q(θ, θ
n−1
)=
E[ln p(X , Y|θ)|X ,θ
n−1
].
3) (M-step) Compute the ML estimate of θ using
Q(θ, θ
n−1
), and set θ
n
= arg max Q(θ, θ
n−1
).
4) Set n = n +1.
5) Repeat steps 2) through 4) until a convergence criterion
is satisfied.
In cases where the likelihood function is analytically in-
tractable, making the second step difficult to complete, the
EM function can still be applied. Rather than completing the
E- and M-steps as shown above, the E-step can be computed
by minimizing (12) to estimate the radius r. Then, the M-step
is performed for the pdf of R
i
, p(R
i
|L
i
,r). Unfortunately, this
pdf is complex due to the forms of the pdfs derived from the
scattering models of Section II, and there is no straightforward
method to perform this maximization. The approach which we
apply is the Nedler–Mead numerical method [35] to solve for
the parameter that will maximize this pdf.
VI. S
IMULATION RESULTS
Computer simulation was performed to assess the perfor-
mance of the algorithms developed for mitigating NLOS-
propagation effects and to compare their performance to
traditional location algorithms. We assume that the BSs are
located at (0, 0), (8.66, 0), and (4.33, 7.5) km. The MS is
randomly placed in the geographic area covered by the BSs. For
the DOS and RDOS cases, the value of l
ij
in the Riemann
sums is chosen to be 1 m. For the Bayesian and ML-EM
algorithms, to estimate R
i
and r, we compute the variance
which requires a complicated integration [see (12) and (8)]
which is replaced by numerical integration using Riemann
summation, the precision for which is 1 m for R
i
and 10 m
for r. The Gaussian measurement noise of the multipath TOA
measurements in (6) is zero mean with variance σ
2
µ
=50m
2
.
To evaluate the performance for the algorithms using both
the DOS and ROS model classifications, we first examined
the performance of the JVMA, Bayesian, and EM algorithms
when the simulated scattering environment matched the model
assumption, i.e., ROS algorithms were applied to a simulated
ROS environment, while DOS algorithms were applied to a
DOS scattering environment. Fig. 4 shows the performance
of the three proposed algorithms when applied to a simulated
ROS environment for different numbers of available multipath
TOA measurements (N
m
=4 and N
m
=8) versus the ROS
radius R
r
. Also, shown for comparison purposes are the LOS-
based LLOP method [11] which is applied to the first (i.e.,
smallest) multipath TOA measured at each BS and the RSA
[17] which is an NLOS-mitigation algorithm. Increasing the
radius R
r
has the effect of increasing the size of the NLOS error,
thereby indicating a more harsh environment, and we see that
the location error for all algorithms increases as R
r
is increased.
The figure indicates that the proposed algorithms performed
better than the LLOP algorithm, as expected, since the ROS
Fig. 4. Mean estimation error for ROS with four and eight multipath TOAs
measured at each BS.
Fig. 5. Mean estimation error for DOS with four and eight multipath TOAs
measured at each BS.
model has a significant probability of large NLOS error, and the
new algorithms incorporate information regarding the NLOS
characteristics. Also, the EM and Bayesian algorithms outper-
formed the RSA NLOS-mitigation algorithm, and of the three
proposed algorithms, the EM performed best. The figures also
indicate that the performance is enhanced as N
m
is increased,
since the TOA variance and pdf information are more reliable
when more multipath measurements are available.
Similar to Fig. 4, the results for the DOS-based algorithms
in a simulated DOS scattering environment are shown in Fig. 5
for N
m
=4and N
m
=8multipath TOA measurements at each
BS versus the size of the scattering disk R
d
. Again, we find
that the performances of all algorithms degrade as the disk
radius is increased but not as significantly as with the ROS-
based algorithms in the previous figure, due to the large peak
in the DOS TOA pdf near the LOS TOA. In this scenario,
the LLOP algorithm performs well (as would most traditional
LOS-based algorithms) for the same reason. In other words,
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590 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Fig. 6. Mean estimation error for RDOS with four and eight multipath TOAs
measured at each BS.
the randomly generated multipath TOAs are more likely to be
near the LOS value, and consequently, LOS-based algorithms
may perform well. From the figure, we find that the Bayesian
and JVMA algorithms performed very closely and better than
the other algorithms. The superior performance of the Bayesian
algorithm is due to the fact that the TOA pdfs have a large
peak near the LOS TOA, causing E[R
i
|L
i
,r] to be close to
thetrueLOSR
i
. Likewise, the narrow highly peaked pdf
causes variance computations for JVMA to be more accurate
for smaller N
m
.
For the heuristic model RDOS, which is used to evaluate per-
formance in an environment where there is a high probability of
large NLOS error, results are shown in Fig. 6 for N
m
=4and
N
m
=8multipath TOA measurements versus the radius of the
model. In this scenario, the EM and RSA algorithms show a
significantly better performance than the other algorithms. The
EM algorithm performed better than the Bayesian algorithm
due to the fact that the pdfs have a peak far away from the
LOS, and E[R
i
|L
i
,r] is expected to be greater than the true
LOS range R
i
. The JVMA algorithm performs better than the
Bayesian and LLOP algorithms.
According to the results of the simulations, the JVMA and
Bayesian algorithm gave a better performance than the EM
algorithm for the DOS environment (when there is a high
probability of LOS), while the EM algorithm performed better
in the ROS and RDOS environments where the probability
of large NLOS error is high. The JVMA algorithm is usually
performed in between the EM and Bayesian algorithms for the
scattering environments considered.
The location algorithms developed in Sections IV and V were
based on classifying a propagation environment and utilizing
the appropriate model-based algorithm for that environment.
While the results in Figs. 4–6 assumed that the classification
perfectly matched the environment, in reality, the classification
will not be exact. Instead, the model used to classify an en-
vironment will be the closest match to that environment. In
these cases, there will be some mismatch between the model
Fig. 7. Mean estimation error for the ROS-environment mismatch with eight
multipath TOAs measured at each BS.
Fig. 8. Mean estimation error for the DOS-environment mismatch with eight
multipath TOAs measured at each BS.
chosen and the actual scattering characteristics. In order to
evaluate the performance of the algorithms when the model
is mismatched from the actual environment, we consider two
simulation scenarios. The first considers the scenario where the
classification model for the environment is chosen incorrectly,
e.g., a DOS model-based algorithm used for a ROS environ-
ment or vice versa. The second scenario examines the more
likely case where an environment is classified properly, but
there exist some variations between the ideal models used for
classification and the actual environment.
For the first model mismatch scenario, the results in Figs. 7
and 8 show the robustness of the algorithms when they are ap-
plied to environments different from what they were designed.
Fig. 7 shows the DOS algorithm when used for a ROS environ-
ment. The EM and Bayesian algorithms for DOS were the most
robust when applied to the ROS environment. Fig. 8 shows the
ROS algorithm when it is used for a DOS environment. The
Bayesian algorithm proved to be the most robust.
For the second mismatch scenario, the locations of scatterers
in the ROS and DOS models were distorted by adding Gaussian
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AL-JAZZAR et al.: SCATTERING-MODEL-BASED METHODS FOR TOA LOCATION IN NLOS ENVIRONMENTS 591
Fig. 9. Mean estimation error for the ROS environment with the scatterers
radii that are corrupted with a Gaussian noise of standard deviation 0.2 R
r
with
eight multipath TOAs measured at each BS.
Fig. 10. Mean estimation error for the DOS with the scatterers radii that
are corrupted with a Gaussian noise of standard deviation 0.2 R
d
with eight
multipath TOAs measured at each BS.
noise to the scatterer positions. The Gaussian noise was zero
mean with a standard deviation of 0.2r, where r is the radius of
the disk or ring, depending on the model used. This formulation
randomizes the model in proportion to the radius. The results
for the proposed ROS- and DOS-based algorithms, and the
LLOP and RSA algorithms, are shown in Figs. 9 and 10 for the
distorted ROS- and DOS-simulated environments, respectively.
The figures show that the model mismatch in this case did
not cause significant increase in the mean location error for
the proposed algorithms. Of the three proposed algorithms, the
EM algorithm had the worst performance degradation in the
distorted models. For the distorted ROS environment in Fig. 9,
the Bayesian algorithm outperformed the other proposed algo-
rithms, while in Fig. 10, for the distorted DOS environment,
the JVMA performed the best. We also note that the RSA
algorithm, which is indifferent to the type of simulated scatter
Fig. 11. RMS error for the estimation of the LOS TOA for the Bayesian and
EM algorithms with N
m
=8for the different scattering-model radii of the
ROS and DOS models.
environment, showed a better performance than the proposed
algorithms for the distorted ROS-simulated environment and
performed nearly as well as JVMA for the DOS environ-
ment. However, we found that the algorithm often had trouble
converging for both the undistorted and distorted scattering
models used.
Since the parameter estimation algorithms, i.e., the Bayesian
estimation and EM algorithms of Section V, form estimates
of the LOS TOAs for each BS, simulations were performed
to assess the accuracy of the LOS estimates. Fig. 11 shows
the root-mean-square (rms) error for the Bayesian and EM
estimators versus the scattering-model radius for both the DOS
and ROS models. Consistent with Fig. 5, the EM algorithm
performs poorly for the DOS model but better than the Bayesian
algorithm for ROS. As expected, the LOS estimation accuracy
decreases as the NLOS error increases, i.e., the scattering radius
increases.
Figs. 12 and 13 show the performance of the proposed
algorithms versus the number of measured multipaths N
m
for
the ROS and DOS environments, respectively, for the differ-
ent scattering radii (R
r
or R
d
) and TOA measurement noise
power σ
2
µ
. In Fig. 12, we notice that for small R
r
and σ
2
µ
,
the rms error can be kept under 100 m for N
m
as small as
two. When σ
2
µ
is increased, N
m
=6 is required to keep the
RMS error less than 100 m. N
m
=4 produces an rms error
less than 120 m for the Bayesian and EM algorithms, while
JVMA produces an error of 100 m. For large NLOS values
(i.e., large R
r
), a larger number of multipath TOAs (N
m
≥ 8)
are required to produce an rms error less than 100 m. The trends
for the DOS scattering environment in Fig. 13 are slightly better
(except for the EM algorithm) in requiring about two fewer
multipath TOAs to obtain similar rms error as the ROS model
for the same scattering radius and the TOA-measurement-error
power. Overall, the size of the NLOS error (via larger scattering
radii) has a much greater impact on the choice of N
m
than the
TOA measurement error. Clearly, the more resolvable multipath
TOAs that can be measured, the better the performance that will
be achieved.
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592 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Fig. 12. RMS estimation error for the ROS environment versus the number
of measured multipaths N
m
for the different ROS radii R
r
and measurement
noise power σ
2
µ
.
Fig. 13. RMS estimation error for the DOS environment versus the number
of measured multipaths N
m
for the different DOS radii R
d
and measurement
noise power σ
2
µ
(m
2
).
VII. CONCLUSION
In this paper, we have presented new algorithms for im-
proving wireless location accuracy in NLOS environments. The
algorithms were based on scattering models which were used to
classify different types of fading environments and utilize the
multipath measurements rather than only the earliest arriving
multipath component. The algorithms showed significant im-
provement over a traditional LOS-based algorithm and another
NLOS-mitigation algorithm (RSA), which utilizes only three
BSs for location. The amount of improvement depended on the
type of environment, indicating that the proposed algorithms
are each preferable in different scattering environments. The
algorithms were also shown to be robust when the environments
are incorrectly classified and when the classification model does
not match perfectly with the actual channel conditions, which
is a more likely scenario.
Finally, although two specific scattering models or classifi-
cations were considered in this paper that were specific to the
macrocells, the approaches used for developing the algorithms
can be applied to any model for any environment. Thus, the
algorithms can be extended to other macrocell models as well
as micro- and picocell models.
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Saleh Al-Jazzar (S’01–M’04) received the B.Sc.
degree in electrical engineering from University of
Jordan, Amman, Jordan, in 1997, the M.Sc. degree
in RF engineering from University of Leeds, Leeds,
U.K., in 1998, and the Ph.D. degree from University
of Cincinnati, Cincinnati, OH, in 2004.
Currently, he is with the Department of Electrical
and Computer Engineering, Hashemite University,
Zarqa, Jordan. His research interests are mainly in
wireless communication, with emphasis on wireless
location, channel estimation, and tracking, multiuser
code-division-multiple-access (CDMA) system, and channel equalization.
James Caffery, Jr. (S’90–M’99) received the Bach-
elor of Science degree in electrical engineering
(summa cum laude) from Bradley University, Peoria,
IL, in 1992 and the M.S. and Ph.D. degrees in elec-
trical engineering from the Georgia Institute of Tech-
nology, Atlanta, GA, in 1993 and 1998, respectively.
In 1999, he joined the Department of Electrical
and Computer Engineering and Computer Science,
University of Cincinnati, Cincinnati, OH, where he
is currently an Assistant Professor. His research in-
terests include CDMA systems, multiuser detection
and parameter estimation, and wireless location systems.
Heung-Ryeol You (M’03) received B.S., M.S. and
Ph.D. degrees in electronic engineering from Yonsei
University, Seoul, Korea.
In 1987, he joined Korea Telecom Authority
(now, KT) and conducted various projects related
to microwave transmission, personal communication
service (PCS), and IMT-2000 system development.
Since 1999, he has been involved in the development
of wireless position location technologies of CDMA
mobile networks and the portable Internet network.
Currently, he is a Director of the Mobile Broadband
Service Division of Service Development Laboratory, Korea Telecom, Seoul.
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