1

Location Estimation Using RSS Measurements

with Unknown Path Loss Exponents
M. Bora Zeytinci, Veli Sari, F. Kerem Harmanc, Emin Anarm, Mehmet Akar
Abstract
The location of a mobile station (MS) in a cellular network can be estimated by using received signal
strength (RSS) measurements that are available from the control channels of nearby base stations. Most
of the recent RSS-based location estimation methods that are available in the literature rely on the rather
unrealistic assumption that the signal propagation characteristics are known and independent of time-
variations and the environment. In this paper, we propose an RSS-based location estimation technique,
so called multiple path loss exponent algorithm (RSS-MPLE), that jointly estimates the propagation
parameters and the MS position. The RSS-MPLE method incorporates the antenna radiation pattern
information into the signal model and determines the Maximum Likelihood estimate of the unknown
parameters by employing the Levenberg-Marquardt method. The accuracy of the proposed method is
further examined by deriving the Cramer-Rao bound. The performance of the RSS-MPLE algorithm is
evaluated for various scenarios via simulation results which conrm that the proposed scheme provides
a practical position estimator that is not only accurate but also robust against the variations in the signal
propagation characteristics.
Index Terms
Mobile positioning, Location Estimation
I. INTRODUCTION
Recently, location estimation has been among the most attractive research topics in the area of cellular
communications. With accurate position estimation, a variety of applications and services such as emer-
gency services, monitoring and tracking fraud protection, asset tracking, eet management, mobile yellow
The authors are with the Department of Electrical and Electronics Engineering, Bogazici University, Bebek,

Istanbul. Please ad-
dress all correspondence to Prof. Mehmet Akar. Tel: +90 212 3596854. Fax: +90 212 2872465 (email: mehmet.akar@boun.edu.tr).
October 13, 2012 DRAFT
2
pages, and even cellular system design and management can become feasible for cellular networks [1].
These potential applications of wireless positioning have also been recognized by the IEEE, which set up
a standardization group 802.15.4a for designing a new physical layer for low-data rate communications
combined with positioning capabilities [2]. Furthermore, the Federal Communications Commission (FCC)
in the US has required wireless providers to locate mobile users within tens of meters for emergency
911 calls [3].
The position of a mobile station (MS) can be determined by using multiple radio signals transmitted or
received by the MS. Some location estimation methods like assisted GPS (A-GPS) are based on signals
transmitted from satellites, while others rely on measurements of signals between MS and base stations
(BS), so called network based methods. Currently the best positioning accuracy in cellular systems is
provided by A-GPS at the expense of signicant increase in network and handset complexity [4]. For
instance, modications on the handset such as an embedded GPS receiver and deployment of location
management units (LMU) into the network are needed in order to operate A-GPS systems [5]. Compared
to A-GPS, network based methods are relatively less complex. Moreover, they can be used in many
situations where the A-GPS method cannot be applied, i.e., indoor positioning, but generally with a
degradation in the accuracy. So far, a wide variety of network based positioning techniques have been
proposed that use measurements obtained within the cellular networks, such as received signal strength
(RSS), time of arrival (TOA), time difference of arrival (TDOA) and angle of arrival (AOA) methods
[6], [7], [8], [9], [10], [11], [12], [13], [14], [15].
Positioning technology is often based on trilateration in time based methods like TOA and TDOA,
in which the MS position is obtained as the intersection point of three circles constituted by distance
estimates [16]. Assurance of proper operation of time based methods requires the deployment of LMUs
[17]. These network elements perform timing measurements of all local transmitters, based on which the
actual relative time difference for each BS can be estimated. In a similar manner, AOA based positioning
methods require the implementation of adaptive array antennas, by which the direction of the signal
arrival can be estimated. At the same time, the handset typically requires software modications to
enable positioning functionality with such a method. Consequently, these methods are not widespread in
commercial systems due to their high deployment costs. Therefore, improving the accuracy of positioning
systems based on the existing cellular network infrastructure is desired, which is the main motivation of
this study.
The system model that is used for location estimation in this paper is based on RSS measurements. One
fundamental MS function is to nd the BSs with the strongest signal strength for cell selection purposes.
October 13, 2012 DRAFT
3
Thus, RSS based methods can be implemented without any hardware enhancement in either the MS or
the BS. The distance information between MS and BS can be extracted from RSS measurements using
the accurate information about the propagation characteristics of the measurement channel. However, in
most of the RSS-based location estimation techniques in the literature, it is assumed that the propagation
parameters of the measurement channel are accurately known a priori, either through a training period,
or by assuming a perfect free space channel condition [18], [19], [20], [21], [22], [23]. Such an approach
results in degraded positioning accuracy in many practical application scenarios, including mobile tracking
techniques in [24], [25]. More specically, both [24], [25] study Monte Carlo based mobility tracking
algorithms under the assumption that the mobile moves in 2D environment with input as acceleration. In
[24], RSS data are used to predict mobile position and velocity by using particle ltering techniques but
the method relies on known propagation parameters. In [25], the so called Interacting Multiple Models
(IMM) method has been extended to predict the state of the mobile for indoor applications by considering
multiple xed environment parameters at the expense of an increase in the number of random processes,
whereas the technique herein relies on continuous parameter adaptation.
In this paper a method that aims to resolve major shortcomings of the existing RSS based positioning
techniques is proposed. In particular, an RSS based location estimation technique that jointly estimates the
propagation parameters and the MS position are explained in Section II. The Cramer-Rao Bound (CRB)
derivation and accuracy evaluation are given in Section III as a benchmark for performance comparison.
In Section IV, simulation results under various scenarios are presented to evaluate the performance of
the proposed algorithm. Finally, some concluding remarks are given in Section V.
II. RSS BASED LOCATION ESTIMATION MODEL
By using the received signal strength (RSS) together with a path loss (PL) and shadow fading model,
a distance estimate between the BS and the MS can be obtained. The propagation model used throughout
this paper is a modied version of the log normal model that is widely used in the literature [26], [27],
[28], [29]. The path loss exponent (PLE) is the key parameter in the log normal model. An accurate
value of the PLE is required in order to obtain an accurate estimate of the MS-BS distance from the
corresponding RSS measurement.
In most of the existing studies of the RSS-based location estimation techniques in the literature, the
channel model is assumed to be known a priori; that is, the pathloss characteristics of the coverage area
are considered known, either by assuming the environment is perfect free-space or by extensive measure-
ment and modelling prior to deployment of location estimation systems. However, the PLE parameter
October 13, 2012 DRAFT
4
is environment dependent [30], [31]. Even in the same environment, the propagation characteristics may
change considerably over a long period of time, e.g., due to seasonal and/or weather changes [32]. In
[31], it is experimentally demonstrated in an omnidirectional antenna system that the PLE is strongly
dependent on the base antenna height and the terrain category. Extension of the experimental study in
[31] to directional antenna systems can be found in [33] where the authors show that antenna beamwidth
has an additional gain reduction inuence on the PLE. In other experimental studies that consider path
loss modelling for directional antenna systems [34], [35], the authors compute different PLEs for different
areas of the cell that consist of different terrains (but not different PLEs for different directional antennas
of the same base station) . There are also related studies that utilize single PLE for directional antenna
systems [36], [37], [38].
The proposed algorithm in this paper estimates the MS location by using the available RSS measure-
ments without any need for a training period or any need for the knowledge of the PLE value in the
path loss model. The PLE values are determined and calibrated in real time for every mobile using the
RSS measurements. By incorporating the antenna radiation patterns of the sectors in a BS into the signal
model, an additional improvement in position estimates is provided. In determining the position of a
given mobile user, it is assumed that the PLE is the same for all sectors of a given base station whereas
the PLE might be different for another. This is a quite realistic assumption in light of the discussion of
the related literature in the previous paragraph, since the mobile user is in the same location with respect
to the base station, and directional antennas are at the same height.
A. Antenna Model
Point to multipoint and cellular communication systems commonly use fan-beam antennas for sector
coverage. An approximate formula describing the normalized azimuth radiation pattern of the fan-beam
antenna using three parameters is given by
A() = exp
_
a
3dB
(
||

HBW
)

_
(1)
where a
3dB
is a unitless parameter that determines the pattern value at half power bandwidth || =
HBW
(in radians) [46], i.e.,
a
3dB
= ln(
1

2
) = 0.3465 (2)
and is used to match the pattern to a second eld value, , at || = |

| where 0 < < 1, i.e.,

=
ln(
ln()
a3dB
)
ln(
||
3dB
)
(3)
October 13, 2012 DRAFT
5
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
Normalized Radiation Pattern for a 3 Sectored BS Configuration
Fig. 1. Normalized Radiation Pattern for 3-sector BS Conguration
In this paper, is evaluated at |

| = /2 and the value of is taken to be equal to 0.006 that is the

inverse of the front-to-side (F/S) ratio of the normalized pattern and
HBW
= /3 is the recommended
value according ETSI EN 301 215-2 Class CS 2 requirement in a Local Multipoint Distribution Service
(LMDS) point-to-multipoint application [46] (see Fig. 1 for the radiation pattern used in this paper).
B. Path Loss Model
In cellular networks, an MS measures received signal code power (RSCP) on common pilot channel
(CPICH) in UMTS and RSS on broadcast control channel (BCCH) in GSM systems to determine the
received signal level. Signal strength measurements are indexed with base station and sectoral identiers
k, s respectively in (4) [18] considering transmitting cells use different channels (CPICHs or BCCHs).
R
k,s
(x, y,
k
) =
PGA
2
k,s
(x, y)

_
dk,s(x,y)
d0
_
k
10
vk,s/10
(4)
In (4), R
k,s
is the received signal strength from the sector s of base station k at point (x, y) (in Watts); P
is the total transmitted power on each sector s (in Watts), A
k,s
(x, y) and G are the normalized radiation
pattern, and the antenna gain in azimuth direction respectively that belongs to the corresponding cell of
sector s; d
0
(in meters) is the free space reference distance; d(x, y) (in meters) is the distance between
MS and BS at point (x, y); v
k,s
accounts for shadow fading;
k
is the PLE that is specic for k
th
base
station; A
k,s
() is represented as A
k,s
(x, y) since it is possible to determine with the knowledge of
MS and BS coordinates. Herein, it is assumed that P,G and are assumed to be known a priori and
October 13, 2012 DRAFT
6
is calculated by [31]
=
_
4d
0

_
2
(5)
where is the wavelength in meters, and d
0
is chosen to be equal to 1 m in a microcell environment
[30]. Rewriting the RSS in dB, we have

k,s(dB)
() =10 log
10
_
PG

_
+ 20 log
10
A
k,s
(x, y)
10
k
log
10
d
k,s
(x, y)
d
0
R
k,s(dB)
() =
k,s(dB)
() v
k,s
(6)
where =
_
x y
1
. . .
k
_
T
In this paper, it is assumed that the channels used by distant BSs may have different propagation
characteristics. More specically, the channels used by the cells that are served by the same BS are
assumed to have the same PLE and this PLE is not necessarily the same for other channels. Hence, the
value of R
k,s
and
k,s
depends on
k
in addition to the mobile position, (x, y).
C. Problem Formulation and the Positioning Algorithm
Based on the path loss model described above, the distribution of R
k,s(dB)
() is Gaussian, i.e., we
have
f
Rk,s(dB)
_
R
k,s(dB)
;
_
=
1

2
exp
_

_
_
R
k,s(dB)

k,s(dB)
()
_
2
2
v
2
_

_ (7)
where f
Rk,s(dB)
_
R
k,s(dB)
;
_
denotes the pdf for the deterministic unknown parameter [43]. Assuming
R
k,s(dB)
is i.i.d., the likelihood function, L() can be written as
L() =
n

k=1
m

s=1
f
Rk,s(dB)
_
R
k,s(dB)
;
_
(8)
where n is the number of BSs that have RSS measurements for the MS and m is the number of sectors
for each BS. Taking into account that R
k,s(dB)
is an i.i.d. Gaussian random variable, L() becomes
L() =
1

2
exp
_

k=1
m

s=1
_
R
k,s(dB)

k,s(dB)
()
_
2
2
v
2
_

_
(9)
October 13, 2012 DRAFT
7
In order to obtain the maximum likelihood (ML) estimate,
ML
that best approximates the available data,
L() should be maximized over (see [43], pp. 177). This in turn can be done by computing

ML
:= arg min

k=1
m

s=1
_
R
k,s(dB)

k,s(dB)
()
_
2
(10)
where arg min is the value of for which the given function is minimized over the given data set. Note
that a distinctive advantage of ML estimate is that it can always be found for a given data set [43]. In
this paper, we obtain this estimate using a recursive solution of a nonlinear least squares problem. To this
end, dene J() to be the cost function (which is to be minimized so that L() in (9) is minimized)
J () =
n

k=1
m

s=1
_
R
k,s(dB)

k,s(dB)
()
_
2
(11)
and
f () = R() (12)
where R = [R
1,1
, R
1,2
, ...R
n,m1
, R
n,m
]
T
and () = [
1,1
,
1,2
, ...,
n,m1
,
n,m
]
T
, J () can be
represented in vector form as
J () = (R())
T
(R()) = f ()
T
f () (13)
Thus, any

that satises (14) is a solution of (10) if f

) is nonsingular, i.e.,
0 = J

) = 2f

)
T
f (

) (14)
In this paper, the Levenberg-Marquardt (LM) method which is a modied version of Gauss-Newton
method is employed to solve the nonlinear least squares problem. This algorithm employs a linear
approximation of the nonlinear equations to nd a least squares estimate of these equations iteratively.
The vector () that consists of nonlinear functions is linearized using Taylor series expansion in which
the second order terms are omitted:

k,s(dB)
()
k,s(dB)
(
(0)
) + J
(0)
_

(0)
_
(15)
First order Taylor series expansion is obtained for
k,s(dB)
() where
(0)
is the initial estimate for
ML
and J
(0)
is the Jacobian matrix of
k,s(dB)
() at
(0)
. Consequently, the LM step at each iteration is
obtained by solving
_
J
T
J + hI
_
= J
T
f
(16)
October 13, 2012 DRAFT
8
where h > 0, e.g., for n = 2 BSs and m = 3 sectors for each BS, in case = [x, y,
1
,
2
] are unknown
parameters, the Jacobian matrix J can be represented as
J =
_

_
(C11())
x
(C11())
y
(101 log(d1()))
1
0
(C12())
x
(C12())
y
(101 log(d1()))
1
0
(C13())
x
(C13())
y
(101 log(d1()))
1
0
(C21())
x
(C21())
y
0
(102 log(d2()))
2
(C22())
x
(C22())
y
0
(102 log(d2()))
2
(C23())
x
(C23())
y
0
(102 log(d2()))
2
_

_
(17)
where C ()
ks
= 20 log (A
ks
()) 10
k
log (d
k
()), and d
k
represents the distance between k
th
BS and
the MS, that is d
k
() =

(x x
k
)
2
+ (y y
k
)
2
. Herein, k index represents the BS and s represents
the sector that belongs to BS k. Since each channel used by different BSs may have a distinct PLE,
parameters are indexed with BS ID, k.
LM Based Algorithm for Joint Estimation of Propagation Parameters and Mobile Position
1) (Initialization)
(i) Let d
1
be the initial radial distance estimate obtained by using Cell ID + Timing Advance
(TA) (Cell ID + Round trip time in UMTS case). (Both of these measurements are available
from the serving cell)
(ii) Let the number of RSS measurements from different sectors of the same BS be n
s
.
If n
s
= 1, then let be the Azimuth angle of the serving sector
else if n
s
> 1, utilize the LM method (as in Step 3 below) with = [,
1
] and J() as
follows:
J () =
_

_
(20log(A11()))

101log(d1)
1
(20log(A12()))

101log(d1)
1
(20log(A13()))

101log(d1)
1
_

_
(18)
where is the angle between serving BS and MS in polar coordinates and A
1s
() is the
radiation pattern of the s
th
sector of the serving BS.
together with d
1
provide an initial estimate of the MS position in polar coordinates.
2) If the number of RSS measurements is greater than or equal to n+2 where at least one measurement
is available from each of the BSs, then let = [x, y,
1
,
2
, . . . ,
n
] (this is what we call the RSS
Multi Path Loss Exponent (RSS-MPLE) algorithm in this paper); otherwise let = [x, y,
1
] (in
this case, we can compute a single path loss exponent, and we refer to this algorithm as the RSS
Single Path Loss Exponent (RSS-SPLE) algorithm).
October 13, 2012 DRAFT
9
3) LM method is employed to solve the stated nonlinear least squares problem iteratively.
(i) Set k = 0, = (k), h = max diagJ
T
()J().
(ii) While (k k
max
) and (J
T
()f() >
1
)
Set k = k + 1 and solve from
_
J
T
()J() + hI
_
= J
T
()f()
Compute
new
=
Compute the step size = (F() F(
new
))/(0.5
T
(hf()))
If > 0 (i.e., step acceptable), then set = (k) =
new
, and h = hmax{1/100, h/10}
else set h = 2h
end
In the above algorithm, k
max
is the maximum number of allowed iterations (k
max
= 400 is used in the
simulations), and
1
is used to detect how close the estimate is to the desired value (e.g.,
1
= 10
15
).
Both parameters are chosen by the user. The damping parameter of the LM algorithm, h, is positive, which
guarantees that is a descent direction. Note that for large values of h, we have = J
T
()f()/h,
that implies a short step in the descent direction, which in turn is good if the current iterate is far from
the solution. On the other hand, if h is small, then is approximately equal to what we have from the
Gauss-Newton iteration. Since the damping parameter inuences both the direction and the size of the
step, its update is controlled by the gain ratio in the algorithm. A large positive value of indicates
a good approximation which allows us to decrease h so that LM step is closer to the Gauss-Newton
step; whereas a small or negative is a poor approximation which requires increase of the damping by
twofold in order to get closer to the steepest direction and hence increase chances of faster convergence.
By this choice of parameters similar to [39], we have observed linear to superlinear convergence in our
problem, although it is harder to make specic statements on the convergence rate for the problem in hand.
However, it is well known that the Levenberg-Marquardt method has a quadratic rate of convergence
when Jacobian is a nonsingular square matrix and if the parameter is chosen suitably at each step.
The condition of the nonsingularity of Jacobian is too strong, and it is not valid in our problem either.
Although the authors show in [40], [41] that the method has quadratic convergence under appropriate
assumptions and the choice of the damping parameter, the results are valid only locally.
In the next section we derive the CRB bounds for the proposed method.
October 13, 2012 DRAFT
10
III. THE CRAMER-RAO LOWER BOUND
The CRB for RSS estimation depends on the strength of signal, Gaussian random variable and path
loss exponent. In radio propagation channel studies, the random variable v in the pathloss model (4)
is considered a zero-mean Gaussian random variable, i.e., N(0,
2
v
), while its standard deviation
2
v
depends on the characteristics of a specic environment [42]. In the computation of the Cramer-Rao
bound, we let p
k,s
= 10 log
10
PG/ R
k,s
, which is observed pathloss in dB from 1 m to d. Thus,
the system of nonlinear equations for location estimation can be rewritten as [45]
p
k,s
= g
k,s
() + v, 1 k n, and 1 s 3 (19)
where g
k,s
() = 10
k
log
10
d
k
20 log
10
A
k,s
() and the unknown vector parameter = [x y
1
. . .
k
]
T
,
k is the index identifying the base stations while s is the index representing the antenna sector. For each
base station k, there are three antenna sectors s = 1, 2, 3 dened. In this setting, the pathloss observation
p
k
dened in (19), has probability density function
f(p
k,s
; ) =
1

2
v
exp
_

(p
k,s
g
k,s
())
2
2
2
v
_
(20)
which is parameterized by the unknown vector parameter . If we assume that p
k,s
, 1 k n, 1 s 3,
are statistically independent observations (which is a reasonable assumption as the transmitted powers
are independent) the joint distribution of observation vector p is obtained as
f(p; ) =
n

k=1
3

s=1
f(p
k,s
; ) (21)
The CRB on the covariance matrix of any unbiased estimator

is dened as
cov(

) F
1
0, (22)
where F = E[

ln f(p; ))
T
] is the Fisher information matrix [43].
Given the joint distribution of the observation vector p in (21), the equivalent log-likelihood function
can be dened as
l() =
1
2
2
v
n

k=1
3

s=1
(p
k,s
g
k,s
())
2
(23)
from which the Fisher Information matrix can be derived as
F
ij
= [F]
ij
= E
_

2
l()

j
_
(24)
=
_

_
1

2
v

n
k=1

3
s=1
(
gk,s()
i
)
2
if i = j;
1

2
v

n
k=1

3
s=1
gk,s()
i
gk,s()
j
if i = j.
October 13, 2012 DRAFT
11
Recall that g
k,s
() is dened by g
k,s
() = 10
k
log
10
d
k
20 log
10
A
k,s
(x, y) and = [x, y,
1
,
2
, ...,
k
]
T
.
Subsequently, the gradients of g
k,s
() can be computed as
g
k,s
()
x
=
10
k
ln 10
u
kx
d
k

20
ln 10
lnA
k,s
(x, y)
x
g
k,s
()
y
=
10
k
ln 10
u
ky
d
k

20
ln 10
ln A
k,s
(x, y)
y
g
k,s
()

1
=
_
10
ln 10
ln d
1
_

k1
g
k,s
()

2
=
_
10
ln 10
ln d
2
_

k2
.
.
.
g
k,s
()

l
=
_
10
ln 10
ln d
k
_

kl
(25)
where u
kx
=
xkx
dk
, u
ky
=
yky
dk
;
kl
= 1 if k = l, and
kl
= 0, otherwise.
The (k + 2) (k + 2) Fisher information matrix can be represented as follows:
F
k+2,k+2
=
_

_
F
1,1
F
1,2
F
1,k+2
F
2,1
F
2,2
F
2,k+2
.
.
.
.
.
.
.
.
.
.
.
.
F
k+2,1
F
k+2,2
F
k+2,k+2
_

_
In the next section, we further dene quantitative performance measures for location estimators based on
CRB.
A. Accuracy Measures
Let ( x, y) be any unbiased location estimator. Then CRB in (22) provides a lower bound on the variance
of the unbiased estimator ( x, y); that is,
E[( x x)
2
] [F
1
]
11
, E[( y y)
2
] [F
1
]
22
(26)
In location estimation applications a more meaningful performance measure of location estimators is
based on the geometric location estimation error =

( x x
k
)
2
+ ( y y
k
)
2
. The mean-squared error
(MSE) of any unbiased location estimator is lower bounded as (26) describes:

2
rms
= E[(
2
] [F
1
]
11
+ [F
1
]
22
(27)
where
rms
is dened as the root-MSE (RMSE) of location estimators.
October 13, 2012 DRAFT
12
Since we assume that the path loss exponent value for each base station is independent of each other,
the elements of the Fisher information matrix that include the product of partial derivatives of distinct
values reduce to zero. Let

F
x
be the (k + 1) (k + 1) special matrix obtained by deleting the rst row
and column of F, i.e.,

F
x
=
_

_
a f
1
f
2
f
3
f
n
e
1
d
1
0 0 0
e
2
0 d
2
0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
e
n
0 0 0 d
n
_

_
(28)
where its components that might be nonzero are shown with parameters a, d
i
, e
i
and f
i
, i = 1, . . . , n.
Similarly, let

F
y
be the same type of special matrix obtained by deleting the second row and column of
F. Subsequently, (27) can be rewritten as

2
rms
= E[
2
] [F
1
]
1,1
+ [F
1
]
2,2
=
det

F
x
det(F)
+
det

F
y
det(F)
(29)
where det

F
x
(and similarly det

F
y
) can be computed as [20]
det

F
x
=
N

n=1
d
n
.
_
a
N

n=1
f
n
e
n
d
n
_
(30)
In case the number of base stations are three and unequal pathloss exponent for each BS, the FIM
matrices are of size 5 5 and the unknown vector parameter is given by = [x, y,
1
,
2
,
3
]
T
. After
some algebraic manipulation, [F
1
]
11
and [F
1
]
22
can be expressed as
[F
1
]
11
=(F
22
F
33
F
44
F
55
F
2
23
F
44
F
55
F
2
24
F
33
F
55
F
2
25
F
33
F
44
)/|F|
[F
1
]
22
=(F
11
F
33
F
44
F
55
F
2
13
F
44
F
55
F
2
14
F
33
F
55
F
2
15
F
33
F
44
)/|F| (31)
where |F| is the determinant of the Fisher information matrix, and
rms
is dened as the root-MSE
(RMSE) of location estimators. The closed form expression of the CRB bound is not explicitly given
here due to its complexity; instead only the numerical solutions are presented. On the other hand, each
component of the FIM matrix is given in Appendix A.
IV. SIMULATION RESULTS
In this section, simulation results for various scenarios are presented and discussed. We assume that
all BSs in the network have three-sector conguration where each sector belongs to a cell with identical
October 13, 2012 DRAFT
13
500 250 0 250 500 750 1000 1250 1500 1750 2000
500
250
0
250
500
750
1000
1250
1500
1750
2000
X coordinates [m]
Y

c
o
o
r
d
i
n
a
t
e
s

[
m
]
Fig. 2. Hypothetical network scheme used throughout the simulations. MS positions are randomly distributed inside the circle
on the rst quadrant.
coverage area and transmit power. The considered network is composed of three BSs as shown in Fig. 2.
The cell radius is assumed to be 1 km for all cells. Sectoral antennas are modelled by the antenna model
described in Section II. BSs are located at Cartesian coordinates [0,0], [1500,0] and [0,1500] in meters.
In order to evaluate the effect of the restrictions mentioned in the GSM and UMTS specications on the
performance of the proposed algorithm, two cases for the RSS measurements are considered separately.
In the rst case, exact measurements are used in the algorithm. In the second case, measurements are
truncated if they are below or above the threshold values mentioned in the standards. In the simulations,
it is assumed that BS located at coordinates [0,0] is the serving cell and the MS does not change its
serving cell (i.e., no handover occurs). At each realization the MS point MS=[x,y] is generated randomly
with a uniform distribution inside the area in the rst quadrant of Cartesian coordinates which is bounded
by a circle centered at [0,0] as in Fig. 2.
October 13, 2012 DRAFT
14
0 1 2 3 4 5 6 7 8 9 10
0
100
200
300
400
500
600
Shadow Fading Standard Deviation [dB]
P
o
s
i
t
i
o
n
i
n
g

R
M
S
E

[
m
]
RSSMPLE, Truncated RSS are omitted.
RSSSPLE, Truncated RSS are omitted.
RSSMPLE, Truncated RSS are used.
RSSKPLE, Truncated RSS are omitted.
RSSMPLE,RSS are not truncated.
RSSKPLE,RSS are not truncated.
Fig. 3. Effect of RSS truncation on position RMSE when PLE values for all BSs are 3 for v between 0 10
The proposed algorithm computes the ML estimate of the MS position by using the RSS measurements
and concurrently calibrates the PLE parameters of the channels occupied by different BSs. Recall that
this algorithm is referred to as RSS with multiple PLE algorithm (RSS-MPLE). In order to demonstrate
the improvement on the positioning accuracy provided by the RSS-MPLE algorithm, its performance
is compared with those of other algorithms, such as RSS with single PLE algorithm (RSS-SPLE) [45],
which nds and calibrates a single PLE for all channels, and RSS with known PLE algorithm (RSS-
KPLE) [18] in which PLE values are known as a priori. Furthermore, the Cramer-Rao bound has been
evaluated and compared with the RMMSE results of the proposed algorithm.
A. Effect of Truncated RSS Measurements
RSS measurements below -110 dBm and above -48 dBm are truncated in GSM systems. In Fig. 3,
the effect of such truncation of RSS measurements in the performance of the proposed algorithm is
investigated. The case in which all BSs have the same PLE value, i.e.,
1
=
2
=
3
= 3 is considered
in this simulation, during which RSS-MPLE and RSS-KPLE algorithms are operated both with truncated
and original RSS measurements. Quantization of RSS measurements is not considered in this simulation
in order to examine the exclusive effect of truncation of RSS measurements on the positioning accuracy.
In addition, the effect of incorporating the truncated measurements into the algorithm is investigated.
October 13, 2012 DRAFT
15
0 1 2 3 4 5 6 7 8 9 10
0
50
100
150
200
250
300
350
400
Shadow Fading Standard Deviation [dB]
P
o
s
i
t
i
o
n
i
n
g

R
M
S
E

[
m
]
RSSMPLE, Truncated RSS are omitted.
RSSSPLE, Truncated RSS are omitted.
RSSKPLE (PLE=3.8,3.0,2.5), Truncated RSS are omitted.
RSSMPLE,RSS are not truncated.
RSSKPLE (PLE=3.8,3.0,2.5),RSS are not truncated.
Fig. 4. Effect of inaccurate PLEs on position RMSE. 1 = 3.5, 2 = 2.7, 3 = 2.3 are used in RSS-KPLE whereas actual
PLEs are 1 = 3.8, 2 = 3.0, 3 = 2.5.
Positioning RMSE obtained with RSS-MPLE algorithm does not exceed 20 m even for
v
= 10 if
RSS measurements are not truncated. On the other hand, truncation of RSS measurements dramatically
degrades the performance of both RSS-MPLE and RSS-KPLE algorithms due to the decrease in the
number of RSS measurements, i.e., the performance of the proposed algorithm is expected to improve
with an increase in the number of available measurements.
Since the truncated measurements represent all RSS measurements below -110 dBm and above -48
dBm, they introduce a large bias in the position estimate when incorporated in the RSS measurement set.
Because of this, positioning accuracy of the RSS-MPLE algorithm severely degrades when the truncated
RSS measurements are not omitted. Compared to the RSS-MPLE algorithm, the RSS-KPLE algorithm
performs better for all
v
values when truncated RSS measurements are used. This is an expected result
since RSS-MPLE algorithm estimates PLE values in addition to the coordinates with the same number
of RSS measurements.
B. Effect of Inaccurate Knowledge of PLE Values
In this subsection, the effect of inaccurate PLE values on positioning accuracy is examined and depicted
in Fig. 4. Throughout the simulation, the PLE values
1
= 3.5,
2
= 2.7,
3
= 2.3 are used in the RSS-
October 13, 2012 DRAFT
16
0 1 2 3 4 5 6 7 8 9 10
0
50
100
150
200
250
300
350
400
Shadow Fading Standard Deviation [dB]
P
o
s
i
t
i
o
n
i
n
g

R
M
S
E

[
m
]
RSSMPLE, Truncated RSS are omitted.
RSSSPLE, Truncated RSS are omitted.
RSSKPLE, Truncated RSS are omitted.
RSSMPLE,RSS are not truncated.
RSSKPLE,RSS are not truncated.
Fig. 5. Position RMSE in meters when PLE values for all BSs are 3 for v between 0 10 dB
KPLE algorithm whereas actual PLE values are
1
= 3.8,
2
= 3.0,
3
= 2.6. Simulations are carried
out with both truncated and exact RSS measurements. As shown in Fig. 4, the RSS-MPLE algorithm
outperforms the rivals under mild shadow fading since the proposed algorithm is capable of adapting PLE
values in real time. On the other hand, the PLE inaccuracy has a drastic effect on positioning accuracy of
RSS-KLPLE. As
v
increases, the adverse effect of the inaccurate PLE values diminishes since shadow
fading becomes the main error source.
C. Effect of Distinct PLE Values
This subsection focuses on the performance analysis of RSS-MPLE, RSS-SPLE and RSS-KPLE
algorithms under different channel conditions. In the rst scenario, it is assumed that all channels have
the same PLE value. In the second scenario, PLE values differ for channels occupied by different BSs.
1) Equal
1
,
2
,
3
(see Figs. 56): Let us rst consider the rather unrealistic case where all BSs
(i.e., all channels) are assumed to have the same PLE value which is equal to three. In Fig. 5, the RMSE
for the positioning algorithms of interest is shown. Since PLE values of all channels are identical, the
RSS-SPLE algorithm is expected to outperform the RSS-MPLE algorithm for the same number of RSS
measurements, which is indeed the case as depicted in Fig. 5.
To evaluate the RSS-MPLE & RSS-SPLE algorithms with respect to the FCC requirements, the
October 13, 2012 DRAFT
17
0 100 200 300 400 500 600
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Position Error [m]
P
(
P
o
s
i
t
i
o
n

E
r
r
o
r

<

x

)
RSSMPLE, Truncated RSS are omitted.
RSSSPLE, Truncated RSS are omitted.
RSSKPLE, Truncated RSS are omitted.
Fig. 6. CDF of the positioning error for v = 6 dB with PLE values of 1 = 3, 2 = 3, 3 = 3.
positioning error CDF is shown in Fig. 6 for
v
= 6dB, which is a realistic value in a microcellular
environment [30]. The RSS-SPLE and RSS-KPLE algorithms satisfy the FCC requirements, which
mandate 67% CERP within 100m and 95% CERP within 300m. Although the RSS-MPLE algorithm
does not satisfy the FCC requirements, this algorithm offers a solution for environments that possess
distinct and variable PLEs.
2) Distinct
1
,
2
,
3
(see Figs. 78): From Fig. 5 it is seen that the RSS-SPLE algorithm has good
performance when all PLEs are equal. However, if BSs have different values, the RSS-MPLE algorithm
is expected to outperform the rival algorithms since each PLE is treated separately in the RSS-MPLE
algorithm. Moreover, as
v
increases, the gap between the positioning RMSE of the RSS-MPLE and RSS-
SPLE algorithms closes since the error variance of the estimates obtained with RSS-MPLE algorithm
increases. Such a scenario is simulated for BSs that have different PLE values, i.e., for
1
= 3.5,
2
= 2.7
and
3
= 2.3.
Fig. 7 shows that the performance of the RSS-SPLE algorithm deteriorates when PLE values of the
BSs are unequal. The scenario considered in this simulation can be experienced when a BS that is in the
October 13, 2012 DRAFT
18
0 1 2 3 4 5 6 7 8 9 10
0
50
100
150
200
250
300
350
400
Shadow Fading Standard Deviation [dB]
P
o
s
i
t
i
o
n
i
n
g

R
M
S
E

[
m
]
RSSMPLE, Truncated RSS are omitted.
RSSSPLE, Truncated RSS are omitted.
RSSKPLE, Truncated RSS are omitted.
RSSMPLE,RSS are not truncated.
RSSKPLE,RSS are not truncated.
Fig. 7. Position RMSE with PLE values : 1 = 3.5, 2 = 2.7, 3 = 2.3
vicinity of the MS is in NLOS condition and other BSs are in LOS condition with the MS. Compared to the
RSS-SPLE algorithm, positioning accuracy of the RSS-MPLE algorithm does not change signicantly
under these conditions. On the other hand, position estimates obtained with RSS-SPLE algorithm are
erroneous due to the bias in the estimate. Moreover, positioning accuracy of RSS-MPLE and RSS-KPLE
algorithms are close even when PLE values vary. Thus, RSS-MPLE algorithm satises the requirements
for a position estimate with low error variance, independent from unknown propagation parameters, i.e.,

v
and values. The positioning error CDF shown in Fig. 8 indicates that the accuracy of mobile
positioning can substantially be improved by employing the RSS-MPLE algorithm.
D. CRB Performance Evaluation
In this subsection, we evaluate the CRB bound for equal PLE case and depict the positioning RMSE
for the proposed RSS-MPLE algorithm in Fig. 9. From Fig. 9, it is noted that the RMSE error is relatively
high in the middle area while it shows better results at boundary areas. Actually, this is expected due
to the pattern contribution in location estimation since more signal measurements are available for such
cases. Fig. 10 depicts the CRB results for the same scenario of proposed algorithm RSS-MPLE. From
October 13, 2012 DRAFT
19
0 100 200 300 400 500 600
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Position Error [m]
P
(
P
o
s
i
t
i
o
n

E
r
r
o
r

<

x

)
RSSMPLE, Truncated RSS are omitted.
RSSSPLE, Truncated RSS are omitted.
RSSKPLE, Truncated RSS are omitted.
Fig. 8. CDF of positioning error for v = 6 dB with PLE values of 1 = 3.5, 2 = 2.7, 3 = 2.3.
Figs. 910, it can be concluded that the performance degrades up to 450 m in RMSE error; and 400 m in
CRB. Moreover, it is clearly seen that the radiation pattern has signicant effect on location estimation.
V. CONCLUSION
In this paper, a practical positioning method that can be implemented in mobile networks with simple
modications in the existing infrastructure is presented. The proposed method, so called RSS-MPLE,
is based on RSS measurements and jointly estimates the MS position and the propagation parameters,
namely the PLE value of the measurement channel. The RSS-MPLE method does not need a training
period to estimate the PLE value of the channel. The most signicant feature of the proposed RSS-
MPLE algorithm is its ability to separately calibrate the PLE value of each channel occupied by different
BSs. Moreover, the RSS-MPLE algorithm incorporates the antenna radiation pattern information that
provides additional improvement in positioning accuracy. Via extensive simulations, the performance of
the proposed method has been compared with those of the existing algorithms in terms of positioning
RMSE, bias, availability and CERP under different environmental conditions by changing PLE and SNR
October 13, 2012 DRAFT
20
x(m)

y
(
m
)
RMSE
0 100 200 300 400 500 600 700 800 900 1000
0
100
200
300
400
500
600
700
800
900
1000
50
100
150
200
250
300
350
400
450
Fig. 9. Positioning RMSE in meters while = 3 for all BSs
values. Simulation results indicate that the RSS-MPLE algorithm is robust against variations in the PLE
values under different environment conditions.
APPENDIX
CRAMER-RAO BOUND FOR THE THREE BS CASE
The components of the Fisher information matrix are given as below:
F
11
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
x
)
2
=
1

2
v
n

k=1
3

s=1
(
10
k
ln 10
u
kx
d
k

20
ln 10
ln A
k,s
(x, y)
x
)
2
F
12
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
x
)(
g
k,s
()
y
)
=
1

2
v
n

k=1
3

s=1
(
10
k
ln 10
u
kx
d
k

20
ln 10
ln A
k,s
(x, y)
x
)
(
10
k
ln 10
u
ky
d
k

20
ln 10
ln A
k,s
(x, y)
y
)
October 13, 2012 DRAFT
21
x(m)

y
(
m
)
CRLB
0 100 200 300 400 500 600 700 800 900 1000
0
100
200
300
400
500
600
700
800
900
1000
50
100
150
200
250
300
350
400
Fig. 10. Cramer-Rao bound in meters when = 3 for all BSs
F
13
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
x
)(
g
k,s
()

1
)
=
1

2
v
3

s=1
(
10
1
ln 10
u
1x
d
1

20
ln 10
ln A
1,s
(x, y)
x
)(
10
ln 10
ln d
1
),
F
14
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
x
)(
g
k,s
()

2
)
=
1

2
v
3

s=1
(
10
k
ln 10
u
2x
d
2

20
ln 10
ln A
2,s
(x, y)
x
)(
10
ln 10
ln d
2
),
F
15
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
x
)(
g
k,s
()

3
)
=
1

2
v
3

s=1
(
10
3
ln 10
u
3x
d
3

20
ln 10
ln A
3,s
(x, y)
x
)(
10
ln 10
ln d
3
),
October 13, 2012 DRAFT
22
F
21
= F
12
F
22
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
y
)
2
=
1

2
v
n

k=1
3

s=1
(
10
k
ln 10
u
ky
d
k

20
ln 10
ln A
k,s
(x, y)
y
)
2
F
23
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
y
)(
g
k,s
()

k
)
=
1

2
v
3

s=1
(
10
1
ln 10
u
1y
d
1

20
ln 10
ln A
1,s
(x, y)
y
)(
10
ln 10
ln d
1
),
F
24
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
y
)(
g
k,s
()

2
)
=
1

2
v
3

s=1
(
10
2
ln 10
u
2y
d
2

20
ln 10
ln A
2,s
(x, y)
y
)(
10
ln 10
ln d
2
),
F
25
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
y
)(
g
k,s
()

3
)
=
1

2
v
3

s=1
(
10
3
ln 10
u
3y
d
3

20
ln 10
ln A
3,s
(x, y)
y
)(
10
ln 10
ln d
3
),
F
31
= F
13
F
32
= F
23
F
33
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()

1
)(
g
k,s
()

1
)
=
1

2
v
3

s=1
(
10
ln 10
ln d
1
)
2
F
34
= 0, F
35
= 0
F
41
= F
14
, F
42
= F
24
F
44
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()
y
)(
g
k,s
()

2
)
=
1

2
v
3

s=1
(
10
ln 10
ln d
2
)
2
F
43
= 0, F
45
= 0
October 13, 2012 DRAFT
23
F
51
= F
15
, F
52
= F
25
F
53
= F
35
= 0, F
54
= F
45
= 0
F
55
=
1

2
v
n

k=1
3

s=1
(
g
k,s
()

3
)(
g
k,s
()

3
)
=
1

2
v
3

s=1
(
10
ln 10
ln d
3
)
2
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