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Zeytinci et al. EURASIP Journal onWireless Communications and Networking 2013, 2013:178http://jwcn.eurasipjournals.com/content/2013/1/178

RESEARCH Open Access

Location estimation using RSS measurementswith unknown path loss exponentsMusa Bora Zeytinci, Veli Sari, Frederic Kerem Harmanciˆ, Emin Anarim and Mehmet Akar*

Abstract

The location of a mobile station (MS) in a cellular network can be estimated using received signal strength (RSS)measurements that are available from control channels of nearby base stations. Most of the recent RSS-based locationestimation methods that are available in the literature rely on the rather unrealistic assumption that signalpropagation characteristics are known and independent of time variations and the environment. In this paper, wepropose an RSS-based location estimation technique, so-called multiple path loss exponent algorithm (RSS-MPLE),which jointly estimates the propagation parameters and the MS position. The RSS-MPLE method incorporatesantenna radiation pattern information into the signal model and determines the maximum likelihood estimate ofunknown parameters by employing the Levenberg-Marquardt method. The accuracy of the proposed method isfurther examined by deriving the Cramer-Rao bound. The performance of the RSS-MPLE algorithm is evaluated forvarious scenarios via simulation results which confirm that the proposed scheme provides a practical positionestimator that is not only accurate but also robust against the variations in the signal propagation characteristics.

Keywords: Mobile positioning; Location estimation; Cramer-Rao bound; Received signal strength

1 IntroductionRecently, location estimation has been among the mostattractive research topics in the area of cellular commu-nications. With accurate position estimation, a varietyof applications and services such as emergency services,monitoring and tracking fraud protection, asset tracking,fleet management, mobile yellow pages, and even cellularsystem design and management can become feasible forcellular networks [1]. These potential applications of wire-less positioning have also been recognized by the IEEE,which set up a standardization group 802.15.4a for design-ing a new physical layer for low-data rate communicationscombined with positioning capabilities [2]. Furthermore,the Federal Communications Commission (FCC) in theUSA has required wireless providers to locate mobileusers within tens of meters for emergency 911 calls [3].The position of amobile station (MS) can be determined

using multiple radio signals transmitted or received bythe MS. Some location estimation methods like assisted

*Correspondence: [email protected]ˆDeceasedDepartment of Electrical and Electronics Engineering, Bogazici University,Bebek, Istanbul, Turkey

global positioning system (A-GPS) are based on signalstransmitted from satellites, while others rely on mea-surements of signals between MS and base stations (BS),the so-called network-based methods. Currently, the bestpositioning accuracy in cellular systems is provided by A-GPS at the expense of a significant increase in networkand handset complexity [4]. For instance, modificationson the handset such as an embedded GPS receiver anddeployment of locationmanagement units (LMU) into thenetwork are needed in order to operate A-GPS systems[5]. Compared to A-GPS, network-based methods are rel-atively less complex. Moreover, they can be used in manysituations where the A-GPS method cannot be applied,i.e., indoor positioning, but generally with a degradationin accuracy. So far, a wide variety of network-based posi-tioning techniques have been proposed which use mea-surements obtained within the cellular networks, such asreceived signal strength (RSS), time of arrival (TOA), timedifference of arrival (TDOA), and angle of arrival (AOA)methods [6-15].Positioning technology is often based on trilateration

in time-based methods like TOA and TDOA, in whichthe MS position is obtained as the intersection point of

© 2013 Zeytinci et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

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three circles constituted by distance estimates [16]. Assur-ance of proper operation of time-based methods requiresthe deployment of LMUs [17]. These network elementsperform timing measurements of all local transmitters,based on which the actual relative time difference foreach BS can be estimated. In a similar manner, AOA-based positioning methods require the implementation ofadaptive array antennas, by which the direction of signalarrival can be estimated. At the same time, the handsettypically requires software modifications to enable posi-tioning functionality with such a method. Consequently,these methods are not widespread in commercial systemsdue to their high deployment costs. Therefore, improvingthe accuracy of positioning systems based on the exist-ing cellular network infrastructure is desired, which is themain motivation of this study.The system model that is used for location estimation

in this paper is based on RSS measurements. One funda-mental MS function is to find the BSs with the strongestsignal strength for cell selection purposes. Thus, RSS-based methods can be implemented without any hard-ware enhancement in either the MS or the BS. Distanceinformation between MS and BS can be extracted fromRSS measurements using accurate information about thepropagation characteristics of the measurement chan-nel. However, in most of the RSS-based location esti-mation techniques in the literature, it is assumed thatthe propagation parameters of the measurement channelare accurately known a priori, either through a train-ing period or by assuming a perfect free-space channelcondition [18-24]. Such an approach results in degradedpositioning accuracy inmany practical application scenar-ios, including mobile tracking techniques in [25,26]. Morespecifically, both [25,26] study Monte Carlo-based mobil-ity tracking algorithms under the assumption that themobile moves in a 2D environment with input as accelera-tion. In [25], RSS data are used to predict mobile positionand velocity using particle filtering techniques, but themethod relies on known propagation parameters. In [26],the so-called interacting multiple model method has beenextended to predict the state of the mobile for indoorapplications by considering multiple fixed environmentparameters at the expense of an increase in the number ofrandom processes, whereas the technique herein relies oncontinuous parameter adaptation.In this paper, a method that aims to resolve major short-

comings of the existing RSS-based positioning techniquesis proposed. In particular, an RSS-based location esti-mation technique that jointly estimates the propagationparameters and the MS position is explained in Section 2.The Cramer-Rao bound (CRB) derivation and accuracyevaluation are given in Section 3 as a benchmark for per-formance comparison. In Section 4, simulation resultsunder various scenarios are presented to evaluate the

performance of the proposed algorithm. Finally, someconcluding remarks are given in Section 5.

2 RSS-based location estimationmodelUsing RSS together with a path loss (PL) and shadow fad-ing model, a distance estimate between the BS and theMScan be obtained. The propagation model used through-out this paper is a modified version of the log normalmodel that is widely used in the literature [27-30]. ThePL exponent (PLE) is the key parameter in the log normalmodel. An accurate value of the PLE is required in orderto obtain an accurate estimate of theMS-BS distance fromthe corresponding RSS measurement.In most of the existing studies of RSS-based location

estimation techniques in the literature, the channel modelis assumed to be known a priori, that is, the path losscharacteristics of the coverage area are considered known,either by assuming that the environment is a perfect freespace or by extensive measurement and modeling prior tothe deployment of location estimation systems. However,the PLE parameter is environment dependent [31,32].Even in the same environment, propagation characteris-tics may change considerably over a long period of time,e.g., due to seasonal and/or weather changes [33]. In [32],it is experimentally demonstrated in an omnidirectionalantenna system that the PLE is strongly dependent on thebase antenna height and the terrain category. Extensionof the experimental study in [32] to directional antennasystems can be found in [34] where the authors showthat antenna beamwidth has an additional gain reduc-tion influence on the PLE. In other experimental studiesthat consider path loss modeling for directional antennasystems [35,36], the authors compute different PLEs fordifferent areas of the cell that consist of different terrains(but not different PLEs for different directional anten-nas of the same base station). There are also relatedstudies that utilize single PLE for directional antennasystems [37-39].The proposed algorithm in this paper estimates the MS

location using the available RSS measurements withoutany need for a training period or any need for the knowl-edge of the PLE value in the path loss model. The PLEvalues are determined and calibrated in real time for everymobile using the RSSmeasurements. By incorporating theantenna radiation patterns of the sectors in a BS into thesignal model, an additional improvement in position esti-mates is provided. In determining the position of a givenmobile user, it is assumed that the PLE is the same for allsectors of a given base station, whereas the PLE might bedifferent for another. This is a quite realistic assumptionin light of the discussion of the related literature in theprevious paragraph, since the mobile user is in the samelocation with respect to the base station and directionalantennas are at the same height.


2.1 Antenna modelPoint to multipoint and cellular communication systemscommonly use fan-beam antennas for sector coverage. Anapproximate formula describing the normalized azimuthradiation pattern of the fan-beam antenna using threeparameters is given by:

A(φ) = exp[−a3dB(

|φ|φHBW

)τ], (1)

where a3dB is a unitless parameter that determines the pat-tern value at half power bandwidth (HBW) |φ| = φHBW(in radians) [40], i.e.,

a3dB = − ln(1√2) = 0.3465 (2)

and τ is used to match the pattern to a second field value,ν, at |φ| = |φν | where 0 < ν < 1, i.e.,

τ = ln(− ln(ν)a3dB )

ln(|φν |φ3dB

)(3)

In this paper, τ is evaluated at |φν | = π/2, and thevalue of ν is taken to be equal to 0.006 that is the inverseof the front-to-side ratio of the normalized pattern andφHBW = π/3 is the recommended value according to ETSIEN 301 215-2 Class CS 2 requirement in a local multipointdistribution service point-to-multipoint application [40](see Figure 1 for the radiation pattern used in this paper).

Figure 1 Normalized radiation pattern for 3-sector BSconfiguration.

2.2 Path loss modelIn cellular networks, anMSmeasures received signal codepower on a common pilot channel (CPICH) in Univer-sal Mobile Telecommunications System (UMTS) and RSSon a broadcast control channel (BCCH) in GSM systemsto determine the received signal level. Signal strengthmeasurements are indexed with base station and sectoralidentifiers k and s, respectively, in (4) [18] consideringthat transmitting cells use different channels (CPICHs orBCCHs):

Rk,s(x, y,αk) = PGA2k,s(x, y)

β(dk,s(x,y)

d0

)αk 10−vk,s/10. (4)

In (4), Rk,s is the received signal strength from sector s ofbase station k at point (x, y) (in watts); P is the total trans-mitted power on each sector s (in watts), andAk,s(x, y) andG are the normalized radiation pattern and the antennagain in azimuth direction, respectively, that belongs tothe corresponding cell of sector s; d0 (in meters) is thefree-space reference distance; d(x, y) (in meters) is the dis-tance between MS and BS at point (x, y); vk,s accountsfor shadow fading; αk is the PLE that is specific for kthbase station; and Ak,s(φ) is represented as Ak,s(x, y) sinceit is possible to determine φ with the knowledge of MSand BS coordinates. Herein, it is assumed that P, G, andβ are assumed to be known a priori and β is calculatedby [32]:

β =(4πd0

λ

)2, (5)

where λ is the wavelength in meters and d0 is chosen to beequal to 1 m in a microcell environment [31]. Rewritingthe RSS in decibel (dB), we have:

μk,s(dB)(θ) =10 log10(PGβ

)+ 20 log10 Ak,s(x, y)

− 10αk log10dk,s (x, y)

d0Rk,s(dB)(θ) = μk,s(dB)(θ) − vk,s (6)

where θ = [x y α1 . . . αk

]T .In this paper, it is assumed that the channels used by dis-

tant BSs may have different propagation characteristics.More specifically, the channels used by the cells that areserved by the same BS are assumed to have the same PLE,and this PLE is not necessarily the same for other chan-nels. Hence, the value of Rk,s and μk,s depends on αk inaddition to the mobile position, (x, y).


2.3 Problem formulation and the positioning algorithmBased on the path loss model described above, the distri-bution of Rk,s(dB)(θ) is Gaussian, i.e., we have:

fRk,s(dB)

(Rk,s(dB); θ

)= 1σv

√2π

exp[−

(Rk,s(dB) − μk,s(dB) (θ)

)22σv2

],

(7)

where fRk,s(dB)

(Rk,s(dB); θ

)denotes the pdf for the determin-

istic unknown parameter θ [41]. Assuming Rk,s(dB) is inde-pendent and identically distributed (i.i.d.), the likelihoodfunction, L(θ) can be written as:

L (θ) =n∏

k=1

m∏s=1

fRk,s(dB)

(Rk,s(dB); θ

), (8)

where n is the number of BSs that have RSS measure-ments for the MS andm is the number of sectors for eachBS. Taking into account that Rk,s(dB) is an i.i.d. Gaussianrandom variable, L(θ) becomes:

L (θ) = 1σv

√2π

exp

⎡⎢⎢⎢⎣−

n∑k=1

m∑s=1


)22σv2

⎤⎥⎥⎥⎦.

(9)

In order to obtain the maximum likelihood (ML) esti-mate, θML that best approximates the available data, L(θ)

should be maximized over θ (see [41], pp. 177). This inturn can be done by computing:

θML := argminθ

n∑k=1

m∑s=1


)2, (10)

where argmin is the value of θ for which the given func-tion is minimized over the given data set. Note that adistinctive advantage of ML estimate is that it can alwaysbe found for a given data set [41].In this paper, we obtain this estimate using a recur-

sive solution of a nonlinear least squares problem. To thisend, define J (θ) to be the cost function (which is to beminimized so that L (θ) in (9) is minimized):

J (θ) =n∑

k=1

m∑s=1

(Rk,s(dB) − μk,s(dB)(θ)

)2 (11)

and

f (θ) = R − μ(θ), (12)

where R =[R1,1, R1,2, . . .Rn,m−1, Rn,m]T and μ(θ) =[μ1,1, μ1,2, . . . , μn,m−1, μn,m]T , J (θ) can be repre-sented in vector form as:

J (θ) = (R − μ(θ))T (R − μ(θ)) = f (θ)T f (θ) (13)

Thus, any θ∗ that satisfies (14) is a solution of (10) iff′ (θ∗) is nonsingular, i.e.,

0 = J ′ (θ∗) = 2f′(θ∗)T f

(θ∗) (14)

In this paper, the Levenberg-Marquardt (LM) methodwhich is a modified version of the Gauss-Newton methodis employed to solve the nonlinear least squares problem[42,43]. The proposed algorithm which is summarized asAlgorithm 1 on the next page employs a linear approxi-mation of the nonlinear equations to find a least squaresestimate of these equations iteratively. The vector μ(θ)

that consists of nonlinear functions is linearized using theTaylor series expansion in which the second-order termsare omitted:

μk,s(dB)(θ) ≈ μk,s(dB)(θ(0)) + J(0)

(θ − θ(0)

). (15)

First-order Taylor series expansion is obtained forμk,s(dB)(θ)where θ(0) is the initial estimate for θML and J(0)is the Jacobian matrix of μk,s(dB)(θ) at θ(0). Consequently,the LM step size� at each iteration is obtained by solving:(

JT J + hI)� = JT f, (16)

where h > 0, e.g., for n = 2 BSs andm = 3 sectors for eachBS, in case θ = [

x, y,α1,α2]are unknown parameters, the

Jacobian matrix J can be represented as:

J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂(C11(θ))∂x

∂(C11(θ))∂y

∂(10α1 log(d1(θ)))∂α1

0∂(C12(θ))

∂x∂(C12(θ))

∂y∂(10α1 log(d1(θ)))

∂α10

∂(C13(θ))∂x

∂(C13(θ))∂y

∂(10α1 log(d1(θ)))∂α1

0∂(C21(θ))

∂x∂(C21(θ))

∂y 0 ∂(10α2 log(d2(θ)))∂α2

∂(C22(θ))∂x

∂(C22(θ))∂y 0 ∂(10α2 log(d2(θ)))

∂α2∂(C23(θ))

∂x∂(C23(θ))

∂y 0 ∂(10α2 log(d2(θ)))∂α2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

(17)

where C (θ)ks = 20 log (Aks (θ)) − 10αk log (dk (θ)) anddk represents the distance between kth BS and the MS,that is, dk(θ) = √

(x − xk)2 + (y − yk)2. Herein, k indexrepresents the BS, and s represents the sector that belongsto BS k. Since each channel used by different BSs may havea distinct PLE, α parameters are indexed with BS ID, k.In Algorithm 1, kmax is themaximumnumber of allowed

iterations (kmax = 400 is used in the simulations), and ε1is used to detect how close the estimate is to the desiredvalue (e.g., ε1 = 10−15 ). Both parameters are chosenby the user. The damping parameter of the LM algo-rithm, h, is positive, which guarantees that � is a descentdirection. Note that for large values of h, we have � =JT (θ)f (θ)/h, which implies a short step in the descentdirection, which in turn is good if the current iterate isfar from the solution. On the other hand, if h is small,then � is approximately equal to what we have from theGauss-Newton iteration. Since the damping parameter


Algorithm 1 LM-based algorithm for joint estimation of propagation parameters and mobile position1: (Initialization)

(i) Let d1 be the initial radial distance estimate obtained using Cell ID + Timing advance (Cell ID + Round triptime 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 ns.If ns = 1, then let φ be the azimuth angle of the serving sectorelse if ns > 1, utilize the LM method (as in step 3 below) with θ =[φ,α1] and J(θ) as follows:

J (θ) =

⎡⎢⎢⎣

∂(20 log(A11(φ)))∂φ

−10α1 log(d1)∂α1

∂(20 log(A12(φ)))∂φ

−10α1 log(d1)∂α1

∂(20 log(A13(φ)))∂φ

−10α1 log(d1)∂α1

⎤⎥⎥⎦ , (18)

where φ is the angle between serving BS and MS in polar coordinates and A1s(φ) is the radiation pattern ofthe sth sector of the serving BS.

φ together with d1 provides 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 θ = [x, y,α1,α2, . . . ,αn

](this is what we call the RSS multipath loss exponent

(RSS-MPLE) algorithm in this paper); otherwise, let θ = [x, y,α1

](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).3: LM method is employed to solve the stated nonlinear least squares problem iteratively.

(i) Set k = 0, θ = θ(k), h = max diag JT (θ)J(θ).(ii) While (k ≤ kmax) and (‖JT (θ)f(θ)‖ > ε1)

Set k = k + 1 and solve � from(JT (θ)J(θ) + hI

)� = JT (θ)f(θ)

Compute θnew = θ − �

Compute the step size ρ = (F(θ) − F(θnew))/(0.5�T (h� − f (θ)))

If ρ > 0 (i.e., step acceptable), then set θ = θ(k) = θnew, andh = hmax{1/100, h/10}else set h = 2h

end

influences both the direction and the size of the step, itsupdate is controlled by the gain ratio ρ in the algorithm.A large positive value of ρ indicates a good approxima-tion which allows us to decrease h so that the LM step iscloser to the Gauss-Newton step, whereas a small or nega-tive ρ is a poor approximation which requires an increaseof damping by twofold in order to get closer to the steepestdirection and hence increase chances of faster conver-gence. By this choice of parameters similar to [44], we haveobserved linear to superlinear convergence in our prob-lem, although it is harder to make specific statements onthe convergence rate for the problem in hand. However,it is well known that the Levenberg-Marquardt methodhas a quadratic rate of convergence when Jacobian is anonsingular square matrix and if the parameter is chosensuitably at each step. The condition of the nonsingular-ity of Jacobian is too strong, and it is not valid in ourproblem either. Although the authors show in [42,43] thatthe method has quadratic convergence under appropriateassumptions and the choice of the damping parameter, the

results are valid only locally. In the next section, we derivethe CRB bounds for the proposed method.

3 The Cramer-Rao lower boundThe CRB for RSS estimation depends on the strengthof signal, Gaussian random variable and path loss expo-nent. In radio propagation channel studies, the randomvariable v in the path loss model (4) is considered a zero-mean Gaussian random variable, i.e., N(0, σ 2

v ), while itsstandard deviation σ 2

v depends on the characteristics ofa specific environment [45]. In the computation of theCramer-Rao bound, we let pk,s = 10 log10 PG/β − Rk,s,which is the observed path loss in decibel from 1 m tod meters. Thus, the system of nonlinear equations forlocation estimation can be rewritten as [46]:

pk,s = gk,s(θ) + v, 1 ≤ k ≤ n, and 1 ≤ s ≤ 3, (19)

where gk,s(θ) = 10αk log10 dk − 20 log10 Ak,s(φ) and theunknown vector parameter θ =[ x y α1 . . . αk]T , k is the


index identifying the base stations, while s is the indexrepresenting the antenna sector. For each base station k,there are three antenna sectors s = 1, 2, 3 defined. In thissetting, the path loss observation pk defined in (19) has aprobability density function:

f (pk,s; θ) = 1√2πσv

exp(

− (pk,s − gk,s(θ))2

2σ 2v

), (20)

which is parameterized by the unknown vector param-eter θ . If we assume that pk,s, 1 ≤ k ≤ n, 1 ≤ s ≤3 are statistically independent observations (which is areasonable assumption as the transmitted powers areindependent), the joint distribution of observation vectorp =[ p1,1, p1,2, p1,3, . . . , pn,1, pn,2, pn,3] is obtained as:

f (p; θ) =n∏

k=1

3∏s=1

f (pk,s; θ). (21)

The CRB on the covariance matrix of any unbiasedestimator θ is defined as:

cov(θ) − F−1 ≥ 0, (22)

where F = −E[�θ(�θ ln f (p; θ))T ] is the Fisher informa-tion matrix (FIM) [41].Given the joint distribution of the observation vector

p in (21), the equivalent log-likelihood function can bedefined as:

l(θ) = − 12σ 2

v

n∑k=1

3∑s=1

(pk,s − gk,s(θ))2, (23)

from which the Fisher information matrix can be derivedas:

Fij =[F]ij = −E[

∂2l(θ)

∂θi∂θj

](24)

=⎧⎨⎩

1σ 2v

∑nk=1

∑3s=1(

∂gk,s(θ)

∂θi)2 if i = j;

1σ 2v

∑nk=1

∑3s=1

∂gk,s(θ)

∂θi∂gk,s(θ)

∂θjif i = j.

Recall that gk,s(θ) is defined by gk,s(θ) = 10αk log10 dk −20 log10 Ak,s(x, y) and θ =[ x, y,α1,α2, . . . ,αk]T . Subse-quently, the gradients of gk,s(θ) can be computed as:

∂gk,s(θ)

∂x= −10αk

ln 10ukxdk

− 20ln 10

∂ lnAk,s(x, y)∂x

∂gk,s(θ)

∂y= −10αk

ln 10ukydk

− 20ln 10

∂ lnAk,s(x, y)∂y

∂gk,s(θ)

∂α1=

(10ln 10

ln d1)

δk1

∂gk,s(θ)

∂α2=

(10ln 10

ln d2)

δk2

...∂gk,s(θ)

∂αl=

(10ln 10

ln dk)

δkl (25)

where ukx = xk−xdk , uky = yk−y

dk , δkl = 1 if k = l, and δkl = 0,otherwise.The (k + 2) × (k + 2) Fisher information matrix can be

represented as follows:

Fk+2,k+2 =

⎡⎢⎢⎢⎣

F1,1 F1,2 · · · F1,k+2F2,1 F2,2 · · · F2,k+2...

.... . .

...Fk+2,1 Fk+2,2 · · · Fk+2,k+2

⎤⎥⎥⎥⎦ .

In the next section, we further define quantitative per-formance measures for location estimators based on CRB.

3.1 Accuracy measuresLet (x, y) be any unbiased location estimator. Then CRBin (22) provides a lower bound on the variance of theunbiased 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 mean-ingful performance measure of location estimators isbased on the geometric location estimation error ε =√

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

ε2rms = E[ (ε2]≥[F−1]11 +[F−1]22 , (27)

where εrms is defined as the root-MSE (RMSE) of locationestimators.Since we assume that the path loss exponent value for

each base station is independent of each other, the ele-ments of the Fisher information matrix that include theproduct of partial derivatives of distinct α values reduce tozero. Let Fx be the (k+1)×(k+1) special matrix obtainedby deleting the first row and column of F, i.e.,

Fx =

⎡⎢⎢⎢⎢⎢⎣

a f1 f2 f3 · · · fne1 d1 0 0 · · · 0e2 0 d2 0 · · · 0...

.... . . . . . . . .

...en 0 0 · · · 0 dn

⎤⎥⎥⎥⎥⎥⎦ , (28)

where its components that might be nonzero are shownwith parameters a, di, ei, and fi, i = 1, . . . , n. Similarly, letFy be the same type of special matrix obtained by deletingthe second row and column of F. Subsequently, (27) canbe rewritten as:

ε2rms = E[ ε2]≥[F−1]1,1 +[F−1]2,2 = det Fxdet(F)

+ det Fydet(F)

,

(29)


where det Fx (and similarly det Fy) can be computed as[20]:

det Fx =N∏

n=1dn.

(a −

N∑n=1

fnendn

)(30)

For the case of three base stations with unequal pathloss exponents, the FIM matrices are of size 5 × 5and the unknown vector parameter is given by θ =[ x, y,α1,α2,α3]T . After some algebraic manipulation,[ F−1]11 and [ F−1]22 can be expressed as:

[F−1]11 = (F22F33F44F55 − F223F44F55

− F224F33F55 − F2

25F33F44)/|F|[F−1]22 = (F11F33F44F55 − F2

13F44F55− F2

14F33F55 − F215F33F44)/|F|, (31)

where |F| is the determinant of the Fisher informationmatrix and εrms is defined as the RMSE of location esti-mators. The closed form expression of the CRB boundis not explicitly given here due to its complexity; instead,only the numerical solutions are presented. On the otherhand, each component of the FIM matrix is given in theAppendix.

4 Simulation resultsIn this section, simulation results for various scenariosare presented and discussed. We assume that all BSs inthe network have a three-sector configuration where eachsector belongs to a cell with identical coverage area andtransmit power. The considered network is composed ofthree BSs as shown in Figure 2. The cell radius is assumedto be 1 km for all cells. Sectoral antennas are modeled bythe antenna model described in Section 2. BSs are locatedat Cartesian coordinates [0, 0], [1500, 0], and [0, 1500]in meters. In order to evaluate the effect of the restric-tions mentioned in the GSM and UMTS specificationson the performance of the proposed algorithm, two casesfor the RSS measurements are considered separately. Inthe first case, exact measurements are used in the algo-rithm. In the second case, measurements are truncated ifthey are below or above the threshold values mentionedin the standards. In the simulations, it is assumed that BSlocated at coordinates [0, 0] is the serving cell and the MSdoes not change its serving cell (i.e., no handover occurs).At each realization, the MS point MS =[ x, y] is generatedrandomly with a uniform distribution inside the area inthe first quadrant of the Cartesian coordinates which isbounded by a circle centered at [0, 0] as in Figure 2.The proposed algorithm computes the ML estimate of

theMS position using the RSSmeasurements and concur-rently calibrates the PLE parameters of the channels occu-pied by different BSs. Recall that this algorithm is referredto as RSS-MPLE algorithm. In order to demonstrate the

improvement on the positioning accuracy provided by theRSS-MPLE algorithm, its performance is compared withthose of other algorithms, such as RSS with single PLEalgorithm (RSS-SPLE) [46], which finds and calibrates asingle PLE for all channels, and RSS with known PLE algo-rithm (RSS-KPLE) [18] in which PLE values are known asa priori. Furthermore, the Cramer-Rao bound has beenevaluated and compared with the RMSE results of theproposed algorithm.

4.1 Effect of truncated RSSmeasurementsRSS measurements below −110 dBm and above −48 dBmare truncated in GSM systems. In Figure 3, the effectof such truncation of RSS measurements in the perfor-mance of the proposed algorithm is investigated. Thecase in which all BSs have the same PLE value, i.e.,α1 = α2 = α3 = 3, is considered in this simulation, duringwhich RSS-MPLE and RSS-KPLE algorithms are operatedboth with truncated and original RSS measurements. Thetruncated measurements represent all RSS measurementsbelow −110 dBm and above −48 dBm. In Figure 3 (andthe subsequent figures in the paper), the following legendclarification is necessary to better interpret the results:

• ‘Truncated RSS are omitted’ means that RSSmeasurement are truncated and measurementsbelow −110 dBm and above −48 dBm are not usedin the simulations.

• ‘Truncated RSS are used’ means that RSSmeasurements being truncated and measurementsbelow −110 dBm and above −48 dBm are used in thesimulations.

−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

oo

rdin

ates

[m

]

Figure 2 Hypothetical network scheme used throughout thesimulations.MS positions are randomly distributed inside the circleon the first quadrant.


0 1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

Shadow Fading Standard Deviation [dB]

Pos

ition

ing

RM

SE

[m]

RSS−MPLE, Truncated RSS are omitted.RSS−SPLE, Truncated RSS are omitted.RSS−MPLE, Truncated RSS are used.RSS−KPLE, Truncated RSS are omitted.RSS−MPLE,RSS are not truncated.RSS−KPLE,RSS are not truncated.

Figure 3 Effect of RSS truncation on position RMSE.When PLE values for all BSs are 3 for σv between 0 to 10.

• ‘RSS are not truncated’ means that RSSmeasurements are not truncated.

From Figure 3, we note that the positioning RMSEobtained with RSS-MPLE algorithm does not exceed 20meven for σv = 10 if RSS measurements are not truncated.On the other hand, truncation of RSS measurements dra-matically degrades the performance of both RSS-MPLEand RSS-KPLE algorithms due to the decrease in thenumber of RSS measurements, i.e., the performance ofthe proposed algorithm is expected to improve with anincrease 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 whenincorporated in the RSS measurement set. Because ofthis, positioning accuracy of the RSS-MPLE algorithmseverely degrades when the truncated RSS measurementsare not omitted. Compared to the RSS-MPLE algorithm,

the RSS-KPLE algorithm performs better for all σv val-ues when truncated RSS measurements are used. This isan expected result since RSS-MPLE algorithm estimatesPLE values in addition to the coordinates with the samenumber of RSS measurements.

4.2 Effect of inaccurate knowledge of PLE valuesIn this subsection, the effect of inaccurate PLE values onpositioning accuracy is examined and depicted in Figure 4.Throughout the simulation, the PLE values α1 = 3.5, α2 =2.7, and α3 = 2.3 are used in the RSS-KPLE algorithm,whereas actual PLE values are α1 = 3.8, α2 = 3.0, andα3 = 2.6. Simulations are carried out with both truncatedand exact RSS measurements. As shown in Figure 4, theRSS-MPLE algorithm outperforms the rivals under mildshadow fading since the proposed algorithm is capable ofadapting PLE values in real time. On the other hand, thePLE inaccuracy has a drastic effect on positioning accu-racy of RSS-KLPLE. As σv increases, the adverse effect of

0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400


Pos

ition

ing

RM

SE

[m]

RSS−MPLE, Truncated RSS are omitted.RSS−SPLE, Truncated RSS are omitted.RSS−KPLE (PLE=3.8,3.0,2.5), Truncated RSS are omitted.RSS−MPLE,RSS are not truncated.RSS−KPLE (PLE=3.8,3.0,2.5),RSS are not truncated.

Figure 4 Effect of inaccurate PLEs on position RMSE. α1 = 3.5,α2 = 2.7, and α3 = 2.3 are used in RSS-KPLE, whereas actual PLEs areα1 = 3.8,α2 = 3.0, and α3 = 2.5.


0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400


Pos

ition

ing

RM

SE

[m]

RSS−MPLE, Truncated RSS are omitted.RSS−SPLE, Truncated RSS are omitted.RSS−KPLE, Truncated RSS are omitted.RSS−MPLE,RSS are not truncated.RSS−KPLE,RSS are not truncated.

Figure 5 Position RMSE in meters when PLE values for all BSs are 3 for σv between 0 to 10 dB.

the inaccurate PLE values diminishes since shadow fadingbecomes the main error source.

4.3 Effect of distinct PLE valuesThis subsection focuses on the performance analysis ofRSS-MPLE, RSS-SPLE, and RSS-KPLE algorithms underdifferent channel conditions. In the first scenario, it isassumed that all channels have the same PLE value. In thesecond scenario, PLE values differ for channels occupiedby different BSs.

4.3.1 Equal α1, α2, and α3Let us first consider the rather unrealistic case whereall BSs (i.e., all channels) are assumed to have the samePLE value which is equal to three (see Figures 5 and 6).In Figure 5, the RMSE for the positioning algorithms ofinterest is shown. Since PLE values of all channels areidentical, the RSS-SPLE algorithm is expected to outper-form the RSS-MPLE algorithm for the same number of

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Position Error [m

P(P

ositi

on E

rror

< x

)

RSS−MPLE, Truncated RSS are omitted.RSS−SPLE, Truncated RSS are omitted.RSS−KPLE, Truncated RSS are omitted.

[

Figure 6 CDF of positioning error for σv = 6 dB with PLE valuesof α1 = 3, α2 = 3, and α3 = 3.

RSS measurements, which is indeed the case as depictedin Figure 5.To evaluate the RSS-MPLE and RSS-SPLE algorithms

with respect to the FCC requirements, the positioningerror cumulative distribution function (CDF) is shown inFigure 6 for σv = 6 dB, which is a realistic value in amicro-cellular environment [31]. The RSS-SPLE and RSS-KPLEalgorithms satisfy the FCC requirements, which mandate67% CERP within 100 m and 95% CERP within 300 m.Although the RSS-MPLE algorithm does not satisfy theFCC requirements, this algorithm offers a solution forenvironments that possess distinct and variable PLEs.

4.3.2 Distinct α1, α2, and α3From Figure 5, it is seen that the RSS-SPLE algorithmhas good performance when all PLEs are equal. How-ever, if BSs have different α values, the RSS-MPLE algo-rithm is expected to outperform the rival algorithms sinceeach PLE is treated separately in the RSS-MPLE algo-rithm (see Figures 7 and 8). Moreover, as σv increases,the gap between the positioning RMSE of the RSS-MPLEand RSS-SPLE algorithms closes since the error varianceof the α estimates obtained with RSS-MPLE algorithmincreases. Such a scenario is simulated for BSs that havedifferent PLE values, i.e., for α1 = 3.5, α2 = 2.7, andα3 = 2.3.Figure 7 shows that the performance of the RSS-SPLE

algorithm deteriorates when PLE values of the BSs areunequal. The scenario considered in this simulation canbe experienced when a BS that is in the vicinity of the MSis in NLOS condition and other BSs are in LOS condi-tion with the MS. Compared to the RSS-SPLE algorithm,positioning accuracy of the RSS-MPLE algorithm doesnot change significantly under these conditions. On theother hand, position estimates obtained with RSS-SPLEalgorithm are erroneous due to the bias in the α esti-mate. Moreover, positioning accuracy of RSS-MPLE and


0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400


Pos

ition

ing

RM

SE

[m]

RSS−MPLE, Truncated RSS are omitted.RSS−SPLE, Truncated RSS are omitted.RSS−KPLE, Truncated RSS are omitted.RSS−MPLE,RSS are not truncated.RSS−KPLE,RSS are not truncated.

Figure 7 Position RMSE with PLE values. α1 = 3.5, α2 = 2.7, and α3 = 2.3.

RSS-KPLE algorithms are close even when PLE valuesvary. Thus, RSS-MPLE algorithm satisfies the require-ments for a position estimate with low error variance,independent from unknown propagation parameters, i.e.,σv and α values. The positioning error CDF shown inFigure 8 indicates that the accuracy of mobile positioningcan substantially be improved by employing the RSS-MPLE algorithm.

4.4 CRB performance evaluationIn this subsection, we evaluate the CRB bound for theequal PLE case and depict the positioning RMSE forthe proposed RSS-MPLE algorithm in Figure 9. FromFigure 9, it is noted that the RMSE is relatively high inthe middle area, while it shows better results at bound-ary areas. Actually, this is expected due to the pattern

contribution in location estimation since more signalmeasurements are available for such cases. Figure 10depicts the CRB results for the same scenario of the pro-posed algorithm RSS-MPLE. From Figures 9 and 10, itcan be concluded that the performance degrades up to450 m in RMSE and 400 m in CRB. Moreover, it is clearlyseen that the radiation pattern has a significant effect onlocation estimation.

5 ConclusionsIn this paper, a practical positioning method that canbe implemented in mobile networks with simple modi-fications in the existing infrastructure is presented. Theproposed method, the so-called RSS-MPLE, is based onRSS measurements and jointly estimates the MS positionand the propagation parameters, namely the PLE value of

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Position Error [m]

P(P

ositi

on E

rror

< x

)

RSS−MPLE, Truncated RSS are omitted.RSS−SPLE, Truncated RSS are omitted.RSS−KPLE, Truncated RSS are omitted.

Figure 8 CDF of positioning error for σv = 6 dB with PLE values of α1 = 3.5, α2 = 2.7, and α3 = 2.3.


x(m)

y(m

)

RMSE

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

50

100

150

200

250

300

350

400

450

Figure 9 Positioning RMSE in meters while α = 3 for all BSs.

x(m)

y(m

)

CRLB

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

50

100

150

200

250

300

350

400

Figure 10 Cramer-Rao bound in meters when α = 3 for all BSs.


the measurement channel. The RSS-MPLE method doesnot need a training period to estimate the PLE valueof the channel. The most significant feature of the pro-posed RSS-MPLE algorithm is its ability to separatelycalibrate the PLE value of each channel occupied by dif-ferent BSs. Moreover, the RSS-MPLE algorithm incor-porates the antenna radiation pattern information thatprovides additional improvement in positioning accuracy.Via extensive simulations, the performance of the pro-posed method has been compared with those of theexisting algorithms in terms of positioning RMSE, bias,availability, and CERP under different environmental con-ditions by changing PLE and SNR values. Simulationresults indicate that the RSS-MPLE algorithm is robustagainst variations in the PLE values under differentenvironment conditions.

6 Appendix6.1 Cramer-Rao bound for the three BS caseThe components of the Fisher information matrix aregiven as below:

F11 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂x)2

= 1σ 2v

n∑k=1

3∑s=1

(−10αkln 10

ukxdk

− 20ln 10


)2

F12 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂x)(

∂gk,s(θ)

∂y)

= 1σ 2v

n∑k=1

3∑s=1

(−10αkln 10

ukxdk

− 20ln 10


)

(−10αkln 10

ukydk

− 20ln 10


)

F13 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂x)(

∂gk,s(θ)

∂α1)

= 1σ 2v

3∑s=1

(−10α1ln 10

u1xd1

− 20ln 10

∂ lnA1,s(x, y)∂x

)(10ln 10

ln d1),

F14 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂x)(

∂gk,s(θ)

∂α2)

= 1σ 2v

3∑s=1

(−10αkln 10

u2xd2

− 20ln 10


)(10ln 10

ln d2),

F15 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂x)(

∂gk,s(θ)

∂α3)

= 1σ 2v

3∑s=1

(−10α3ln 10

u3xd3

− 20ln 10


)(10ln 10

ln d3),

F21 = F12

F22 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂y)2

= 1σ 2v

n∑k=1

3∑s=1

(−10αkln 10

ukydk

− 20ln 10


)2

F23 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂y)(

∂gk,s(θ)

∂αk)

= 1σ 2v

3∑s=1

(−10α1ln 10

u1yd1

− 20ln 10

∂ lnA1,s(x, y)∂y

)(10ln 10

ln d1),

F24 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂y)(

∂gk,s(θ)

∂α2)

= 1σ 2v

3∑s=1

(−10α2ln 10

u2yd2

− 20ln 10


)(10ln 10

ln d2),

F25 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂y)(

∂gk,s(θ)

∂α3)

= 1σ 2v

3∑s=1

(−10α3ln 10

u3yd3

− 20ln 10


)(10ln 10

ln d3),

F31 = F13F32 = F23

F33 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂α1)(

∂gk,s(θ)

∂α1)

= 1σ 2v

3∑s=1

(10ln 10

ln d1)2

F34 = 0, F35 = 0

F41 = F14, F42 = F24

F44 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂y)(

∂gk,s(θ)

∂α2)

= 1σ 2v

3∑s=1

(10ln 10

ln d2)2

F43 = 0, F45 = 0

F51 = F15, F52 = F25F53 = F35 = 0, F54 = F45 = 0

F55 = 1σ 2v

n∑k=1

3∑s=1

(∂gk,s(θ)

∂α3)(

∂gk,s(θ)

∂α3)

= 1σ 2v

3∑s=1

(10ln 10

ln d3)2

Competing interestsThe authors declare that they have no competing interests.


AcknowledgementsThe authors would like to thank the referees for their constructive commentswhich improved the exposition of the paper.The authors dedicate this article in memory of one of its co-authors, FredericKerem Harmanci, who passed away between the completion of his article andits publication.

Received: 13 October 2012 Accepted: 16 June 2013Published: 26 June 2013

References1. JJ Caffery,Wireless Location in CDMA Cellular Radio Systems. (Springer,

Heidelberg, 2000)2. IEEE P802.15.4a/D4, Part 15.4: Wireless Medium Access Control(MAC) and

Physical Layer (PHY) Specifications for Low-Rate wireless personal areanetworks (LR-WPANs): Amendment of IEEE Std 802.15.4, July 2006.(IEEE, Piscataway, 2006)

3. FC Commission, Revision of the Commission’s Rules to Ensure Compatibilitywith Enhanced 911 Emergency Calling Systems, FCC 94-102.(FC Commission, Washington DC, 1996)

4. J Borkowski, J Niemela, J Lempiäinen, Cellular location techniquessupporting A-GPS positioning. IEEE Vehicular Technol. Conf.1, 429–433 (2005)

5. F Gustafsson, F Gunnarsson, Mobile positioning using wireless networks:Possibilities and fundamental limitations based on available wirelessnetwork measurements. IEEE Signal Process. Mag. 22(4), 41–53 (2005)

6. K Cheung, H So, W-K Ma, Y Chan, Least squares algorithms fortime-of-arrival-based mobile location. IEEE Trans. Signal Process.52(4), 1121–1130 (2004)

7. BJ Dil, PJM Havinga, in Australasian Telecommunication Networks andApplications Conference, ATNAC 2010 (Auckland, 31 October –3 November 2010)

8. B Lakmali, D Dias, Database correlation for GSM location in outdoor & indoorenvironments. The 4th International Conference on Information andAutomation for Sustainability, Colombo, 12-14 December 2008.(IEEE, Piscataway, 2008), pp. 42–47

9. S Ahonen, H Laitinen, Database correlation method for UMTD location.IEEE Vehicular Technol. Conf. 4, 2696–2700 (2003)

10. M Spirito, A Mattioli, Preliminary experimental results of a GSM mobilephones positioning system based on timing advance. IEEE VehicularTechnol. Conf. 4, 2072–2076 (1999)

11. M Spirito, On the accuracy of cellular mobile station location estimation.IEEE Trans. Vehicular Technol. 50(3), 674–685 (2001)

12. M Khalaf-Allah, Nonparametric Bayesian filtering for location estimation,position tracking and global localization of mobile terminals in outdoorwireless environments. EURASIP J Adv. Signal Process. 2008, 1–14 (2008)

13. J Borkowski, J Lempiäinen, Pilot correlation positioningmethod for urbanUMTS networks. Proceedings of the 11th EuropeanWireless Conference,Nicosia, 10–13 April 2005. vol. 2. (IEEE, Piscataway, 2005), pp. 465–469

14. W Pan, J Wu, Z Jiang, Y Wang, X You,Mobile position tracking byTDOA-Doppler hybrid estimation inmobile cellular system, IEEE InternationalConference on Communications, Glasgow, 24–28 June 2007. (IEEE,Piscataway, 2007), pp. 4670–4673

15. O Türkyilmaz, F Alagöz, G Gür, T Tugcu, Environment-aware locationestimation in cellular networks. EURASIP J. Adv. Signal Process.2008, 276456 (2008)

16. A Sayed, A Tarighat, N Khajehnouri, Network-based wireless location:challenges faced in developing techniques for accurate wireless locationinformation. IEEE Signal Process. Mag. 22(4), 24–40 (2005)

17. J Borkowski, J Lempiäinen, Practical network-based techniques for mobilepositioning in UMTS. EURASIP J. Appl. Signal Process. 2006, 012930(2006)

18. K Budka, D Calin, B Chen, D Jeske, A Bayesian method to improve mobilegeolocation accuracy. IEEE Vehicular Technol. Conf. 2, 1021–1025 (2002)

19. KMK Chu, KRPH Leung, Ng JK-Y, CH Li, Locatingmobile stations withstatistical directional propagationmodel in AINA ’04. Proceedings of the 18thInternational Conference on Advanced Information Networking andApplications, Washington, 29 - 31 March 2004.(IEEE, Piscataway, 2004), pp. 230–235

20. Y Qi, H Kobayashi, Cramer-Rao Lower bound for geolocation innon-line-of-sight environment. IEEE Int. Conf. Acoustics, Speech, SignalProcess. (ICASSP). 3, 2473–2476 (2002)

21. R Yamamoto, H Matsutani, H Matsuki, T Oono, H Ohtsuka, Positionlocation technologies using signal strength in cellular systems. IEEEVehicular Technol. Conf. 4, 2570–2574 (2001)

22. M Aso, T Saikawa, T Hattori, Maximum likelihood location estimationusing signal strength and the mobile station velocity in cellular systems.Vehicular Technol. Conf. 2, 742–746 (2003)

23. J Zhou, J-Y Ng, A data fusion approach tomobile location estimation basedon ellipse propagationmodel within a cellular radio network, The 21stInternational Conference on Advanced Information Networking andApplications Niagara Falls, 21 - 23 May 2007.(IEEE, Piscataway, 2007), pp. 459–466

24. M Akar, U Mitra, Variations on optimal and suboptimal handoff control forwireless communication systems. IEEE J. Selected Areas Commun.19(6), 1173–1185 (2001)

25. L Mihaylova, D Angelova, S Honary, D Bull, N Canagarajah, B Ristic,Mobility tracking in cellular networks using particle filtering. IEEE Trans.Wireless Commun. 6(10), 3589–3599 (2007)

26. K Achutegui, J Rodas, CJ Escudero, J Miguez, Amodel-switching sequentialMonte Carlo algorithm for indoor tracking with experimental RSS data. TheInternational Conference on Indoor Positioning and Indoor Navigation (IPIN),Zurich, 15–17 September 2010.(IEEE, Piscataway, 2010), pp. 1–8

27. J Turkka, M Renfors, Path loss measurements for a non-line-of-sightmobile-to-mobile environment The 8th International Conference on ITSTelecommunications, Phuket, 24–24 October 2008.(IEEE, Piscataway, 2008), pp. 274–278

28. S Srinivasa, M Haenggi, Path loss exponent estimation in large wirelessnetworks. (CERN Document Server, Meyrin, 2008)

29. J Shirahama, T Ohtsuki, RSS-based localization in environments with differentpath loss exponent for each link. IEEE Vehicular Technology Conference,Singapore, 11–14May 2008. (IEEE, Piscataway, 2008), pp. 1509–1513

30. I Popescu, D Nikitopoulos, P Constantinou, I Nafornita, ANN predictionmodels for outdoor environment. IEEE 17th International Symposium onPersonal, Indoor andMobile Radio Communications, Helsinki, 11–14September 2006. (IEEE, Piscataway, 2006), pp. 1–5

31. T Rappaport,Wireless Communications. (Prentice Hall PTR, New Jersey,2002)

32. V Erceg, LJ Greenstein, SY Tjandra, SR Parkoff, A Gupta, B Kulic, AA Julius,An empirically based path loss model for wireless channels in suburbanenvironments. IEEE J. Selected Areas Commun. 17(7), 1999–1205 (1999)

33. K Martinez, JK Hart, R Ong, Environmental sensor networks. Computer37, 50–56 (2004)

34. LJ Greenstein, V Erceg, Gain reductions due to scatter on wireless pathswith directional antennas. IEEE Commun. Lett. 3(6), 169–171 (1999)

35. PF Wilson, PB Papazian, MG Cotton, Y Lo, Advanced antenna test bedcharacterization for wideband wireless communications, NTIA Technicalreport. (US Department of Commerce, Washington DC, 1999)

36. TA Rahman, GB Ser, Propagation measurement in IMT-2000 band inMalaysia environment. TENCON 2000. 1, 77–81 (2000)

37. S-W Wang, I Wang, Effects of soft handoff, frequency reuse and non-idealantenna sectorization on CDMA system capacity. 43rd IEEE VehicularTechnology Conference, Secaucus, 18–20May 1993. (IEEE, Piscataway, 1993),pp. 850–854

38. A Vavoulas, N Vaiopoulos, DA Varoutas, A Chipouras, G Stefano,Performance improvement of fixed wireless access networks byconjunction of dual polarization and time domain radio resourceallocation technique. Int. J Commun. Syst. 24, 483–491 (2010)

39. J Park, JW Jang, S-G Park, W Sung, Capacity of sectorized distributednetworks employing adaptive collaboration from remote antennas. IEICETrans. E93-B, 3534–3537 (2010)

40. D Runyon, Optimum directivity coverage of fan-beam antennas. IEEEAntennas Propagation Mag. 44(2), 66–70 (2002)

41. SM Kay, Fundamentals of Statistical Signal Processing: Estimation Theory.(Prentice Hall PTR, New Jersey, 1993)

42. N Yamashita, M Fukushima, On the rate of convergence of theLevenberg-Marquardt methods. Comput Suppl. 15, 237–249 (2001)


43. J-Y Fan, Y-X Yuan, On the quadratic convergence of theLevenberg-Marquardt method without nonsingularity assumption.Computing. 74, 23–39 (2005)

44. K Madsen, HB Nielsen, O Tingleff,Methods for Non-linear Least SquaresProblems. (Informatics and Mathematical Modelling Technical Universityof Denmark, Lyngby, 2004)

45. K Pahlavan, A Levesque,Wireless Information Networks. (John Wiley andSons, Inc., New York, 1995)

46. X Li, RSS-based location estimation with unknown pathloss model. IEEETrans. Wireless Commun. 5(12), 3626–3633 (2006)

doi:10.1186/1687-1499-2013-178Cite this article as: Zeytinci et al.: Location estimation using RSS mea-surements with unknown path loss exponents. EURASIP Journal on WirelessCommunications and Networking 2013 2013:178.

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