2568 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

Positioning for NLOS Propagation: AlgorithmDerivations and Cramer–Rao Bounds

Honglei Miao, Student Member, IEEE, Kegen Yu, Member, IEEE, andMarkku J. Juntti, Senior Member, IEEE

Abstract—Mobile positioning has drawn significant attentionin recent years. Nonline-of-sight (NLOS) propagation error isthe dominant error source in mobile positioning. Most previousresearch in this area has focused on NLOS identification and mit-igation. In this paper, we investigate new positioning algorithmsto take advantage of the NLOS propagation paths rather thancanceling them. Based on a prior information about the NLOSpath, a geometrical approach is proposed to estimate mobile loca-tion by using two NLOS paths. On top of this, the least-squares(LS)-based position estimation algorithm is developed to takemultiple NLOS paths into account, and its performance in termsof root mean-square error (RMSE) is analyzed. A general LS algo-rithm considering both LOS and NLOS paths is also derived, andthe maximum likelihood-based algorithm is presented to jointlyestimate the mobile’s and scatterers’ positions. The Cramer–Raolower bound on the RMSE is derived for the benchmark of theperformance comparison. The performance of the proposed algo-rithms is evaluated analytically and is done via computer simu-lations. Numerical results demonstrate that the derived analyticalresults closely match the simulated results.

Index Terms—Cramer–Rao lower bound (CRLB), mobilelocation, nonline-of-sight (NLOS) propagation.

I. INTRODUCTION

W IRELESS positioning has received increasing atten-tion over the past decade [1]–[5]. Due to the dif-

ferent applications and the associated requirements on theequipment, e.g., physical dimension, power consumption, lo-cation accuracy, operating environment, etc., three kinds ofradio location systems are commonly applied, i.e., satellitepositioning systems like global positioning system (GPS) andGALILEO, cellular- and wireless-local-area-network-based po-sitioning systems, and sensor-network-based positioning sys-tems, which are often based on the ultrawideband (UWB)radio. Some applications need the combination of the above

Manuscript received October 5, 2005; revised April 24, 2006. This paper waspresented in part in the IEEE International Conference on Acoustics, Speech,and Signal Processing (ICASSP 2006), Toulouse, France, on May 14–19. Thiswork was supported by the Academy of Finland and by the Nokia Foundation.The review of this paper was coordinated by Prof. W. Su.

H. Miao was with the Centre for Wireless Communications, University ofOulu, 90014 Oulu, Finland. He is now with the Nokia Technology Platform,FI-00180 Helsinki, Finland (e-mail: honglei.miao@nokia.com).

K. Yu was with the Centre for Wireless Communications, Universityof Oulu, 90014 Oulu, Finland. He is now with the Wireless TechnologiesLaboratory, CSIRO ICT Centre, Marsfield NSW 2122, Australia (e-mail:kegen.yu@csiro.au).

M. J. Juntti is with the Centre for Wireless Communications, University ofOulu, 90014 Oulu, Finland (e-mail: markku.juntti@ee.oulu.fi).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.899948

positioning systems, e.g., the cooperation of the GPS and acellular network. The GPS system significantly improves theefficiency of fleet management, taxi and delivery drivers, andintelligent transportation systems [6], [7]. Location technol-ogy in a cellular network can be employed to improve ra-dio resource and mobility management [8]–[14]. The FederalCommunication Commission has released an order in the U.S.that mandates all the wireless service providers to providean accurate subscriber location information for Enhanced-911(E-911) services [15]. Due to its high-accuracy, low-power,and low-cost implementation, the UWB-radio-based position-ing systems have been applied to numerous applications suchas logistics, medical applications, family communications,search and rescue, control of home appliances, and so on[16], [17].

The position of a mobile station (MS) can be estimated frompassive measurements of the arrival times, direction of arrival,or Doppler shifts of propagation waves received at variousfixed stations (FSs) [18]. Sun et al. [19] investigated a numberof positioning techniques from a system point of view. Thepositioning problems for cellular networks were considered in[20] and [21]. A UWB-radio-based localization problem wasinvestigated in [22]. Radio location can be implemented in oneof two ways, that is, either the MS transmits a signal whichFSs use to determine its location or the FSs transmit signalsthat the MSs use to calculate their own positions, e.g., like inthe GPS.

There are several fundamental positioning approaches, i.e.,the time-based scheme which measures the time of arrivalor time difference of arrival (TDOA) [18], [23], the angle-of-arrival (AOA)-based scheme [18], [24], and the hybridTDOA/AOA approach [25], which combines the TDOA mea-surements at the MS with the AOA measurements at theFS. All these algorithms assume that the line-of-sight (LOS)propagation paths exist between the MS and the FSs. In thepresence of non-LOS (NLOS) propagation, the major position-ing errors result from the measurement noise and the NLOSpropagation error, which is the dominant factor [20]. With theassumption that the NLOS corrupted measurements consist ofonly a small portion of the total measurements, NLOS errormitigation techniques have been extensively examined in theliterature [26]–[29]. Since NLOS corrupted measurements areinconsistent with LOS expectations, first, all of those detectthe NLOS measurements and then ignore them as outliers.However, most of them suffer from multiple NLOS errorsdue to the difficulty of detecting the NLOS measurements. Toovercome the drawback of the preceding approaches, Cong

0018-9545/$25.00 © 2007 IEEE

MIAO et al.: POSITIONING FOR NLOS PROPAGATION: ALGORITHM DERIVATIONS AND CRAMER–RAO BOUNDS 2569

and Zhuang [30] proposed an algorithm which is capable ofdetecting multiple NLOS errors by virtue of the prior informa-tion about the NLOS state at all possible MS locations.

So far, most researches on NLOS errors focus on the NLOSerror mitigation techniques, i.e., how to detect the errors andremove their impacts. However, the open literature shows onlya few positioning techniques which take benefit from the NLOSpropagation paths. In a rich scattering environment, most ofthe propagation paths between the MS and FSs are NLOS;occasionally, the first path between the MS and the home FScan be assumed as LOS. Qi et al. [31] presented a positioningmethod which is able to employ suitable multipaths in thetime domain, with strong paths and small variance of NLOS-induced path-length errors. To improve the accuracy of thepositioning algorithm only using the LOS paths, the algorithmrelies on prior knowledge about the statistics of the NLOS-induced path-length errors. In this paper, we present a novelpositioning algorithm which takes advantage of the NLOSpaths by assuming prior knowledge about parameters of eachNLOS path. Each path is characterized by three parameters (α,β, and d), where α stands for the angle of departure (AOD), βdenotes the AOA, and d defines the distance of the propagationpath. Therefore, NLOS paths are distinguished in the space-time domain rather than the time domain in [31]. While thegoal of the algorithm in [31] is to improve the accuracy ofpositioning by taking the NLOS paths into account, the onethat is used in this paper can be viewed as a “stand-alone”algorithm that is capable of localizing the MS by only using theNLOS paths.

Although our concern is not on how to estimate the para-meters (α, β, and d) at the FS, it is important to note thatit is indeed feasible. The AOA β can be estimated at the FSthat is equipped with an antenna array by applying variousalgorithms, like multiple signal classification (MUSIC) [32]or estimation of signal parameters via rotational invariancetechniques [33]. The distance d can be obtained from thetime delay estimate τ ; such a method requires accurate syn-chronization between the FSs and MS clocks [34], [35]. Ingeneral, the time delay estimation problem is a fundamentalone that has been studied for many years; the study in [35]and [36] presents a number of techniques to address this prob-lem. Moreover, the time delay and the AOA can be jointlyestimated [37], [38]. The most challenging task is to obtainthe AOD estimate of the propagation path at the FS. Thesubsequent three kinds of schemes can be applied to solve thisproblem.

1) The space-alternating-generalized-expectation-maximi-zation algorithm can be employed to estimate the AOAand the AOD at the FS [39], [40] when both the FSand the MS are equipped with multiple antennas. Asaddressed in [41], typically, the FS has been perceivedas more easily affordable for the extra cost and spaceof the antenna array than the MS. However, as the MSsare gradually evolving to become sophisticated wirelessInternet access devices rather than just pocket telephones,the stringent size and complexity constraints are be-coming somewhat more relaxed.

2) When the system operates in the time-division-duplexmode, the channels for the forward and reverse links aresymmetric. If the MS is able to estimate the AOAs of for-ward channels which correspond to the AODs of reversechannels, the MS can feed back the AOA information tothe FS. The AOA estimation at the MS can be achievedby exploiting either the antenna array or Doppler estimatewith a known velocity [18].

3) When the MS is equipped with a directional antenna, thesignal transmission of the reverse link is designed in sucha manner that the received signal at the specified time slotimplies a predefined AOD. This method is also feasiblefor signals without a carrier, e.g., the pulse-position-modulated UWB system.

Based on these parameters and the geometrical relationshipof the possible MS location implied from each NLOS path, anovel positioning algorithm is proposed, and its performance isanalyzed. It is interesting to note that the proposed algorithmis also able to take LOS paths into account as a special caseshown in Appendix A. To summarize, the main contributionsof this paper are threefold.

1) Possible region and position of the MS: Based on themeasurement (α, β, and d) of an NLOS path, we derivethe possible region of the MS. It turns out to be a linesegment. With the aid of two NLOS paths, the positionestimate of the MS is derived as the intersection of twolines. We assume that all the measurements are corruptedby a Gaussian noise, and the root mean-square error(RMSE) of each coordinate of the position estimate isanalyzed.

2) Least squares (LS) algorithm considering multiple NLOSpaths: Aiming at the minimum equation-error norm [23],the LS algorithm is proposed to estimate the positionof the MS in the presence of multiple NLOS paths. Inaddition, LOS paths are incorporated into the general LSalgorithm. The RMSE of the coordinate estimate is alsoanalyzed and compared to the simulation results.

3) Maximum likelihood (ML) algorithm and Cramer–Raolower bound (CRLB) for joint MS and scatterers’ positionestimation: Based on the knowledge of the parametersof multiple NLOS paths, we derive the ML algorithmfor joint MSs and scatterers’ position estimation. Theperformance of the algorithm in terms of RMSE is beingstudied using analysis and computer simulations. TheCRLB for the variance of the MS position estimation isderived as well for the benchmark of the performancecomparison. Because the ML estimation is a nonlinearoptimization problem, iterative methods are required tofind the solution, and the above LS solution can beemployed as the initial searching point.

The rest of this paper is organized as follows. Section IIincludes the system model and notations. Section III derives thepossible region of the MS with the knowledge about the NLOSpath, and the geometric approach is addressed to estimate theposition of the MS. The LS algorithm employing multipleNLOS paths is discussed in Section IV. The ML algorithmand CRLB for joint MS and scatterers’ position estimation

2570 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

Fig. 1. System model in the NLOS scenario.

are derived in Section V. Numerical results are presented inSection VI, and the conclusion is drawn in Section VII.

II. SYSTEM MODEL

The notations used in this paper are as follows. Uppercaseand lowercase boldface letters denote matrices and vectors,respectively, (.)T denotes the transpose, and |A| denotes thedeterminant of matrixA. Fig. 1 shows the system model wherewe concentrated in positioning a single mobile. Let Nf denotethe number of FSs which perceive the transmitted signal fromthe MS. As shown in Fig. 1, no LOS paths exist between the MSand any connected FS.1 Each propagation path is parameterizedby three (α, β, and d), i.e., the AOD α, the AOA β, andthe distance d of the propagation path from the MS to thecorresponding FS. We assume that each FS has knowledge onthe number of paths corresponding to the MS and parametersassociated with each path. Depending on the requirement ofthe positioning accuracy and the positioning algorithm to beperformed, the proposed algorithm can be employed in eitherthe FS or the information processing center (IPC) which con-trols several FSs, such as the radio network controller (RNC)in a cellular network. If the positioning algorithm is performedin the home FS, only the propagation paths corresponding tothat particular FS can be utilized. On the other hand, if thepositioning of the MS is performed in the IPC or RNC, theproposed algorithm would take advantage of the informationabout either all the propagation paths or only the strongestpath related to each connected FS. In summary, the proposedalgorithm can be employed in three setups which are describedin Table I.

For simplicity, we will focus on the third setup in Table I,where the positioning algorithm is run in the IPC, taking onlythe strongest paths corresponding to each controlled FS intoaccount. However, it is straightforward to apply the proposedidea to the other two scenarios mentioned in Table I.

1As shown in Appendix A, LOS can be included as a special case.

III. GEOMETRIC APPROACH WITH TWO FSS

A. Algorithm Derivation

To set the stage for the proposed algorithm, we need tofind out the possible region of the MS when the FS has theinformation on the AOD α, AOA β, and distance d of thestrongest path. However, it needs to have knowledge aboutthe position of the scatterer ps = (xs, ys)T. As shown in Fig. 2,pf = (xf , yf)T defines the position of the FS, and r stands forthe distance between the FS and the scatterer. Therefore, thepossible coordinates of the scatterer are

ps = pf + r

(sinβ

cos β

), r ∈ (0, d) (1)

and the coordinates p = (x, y)T of the MS are expressed asfollows:

p = ps − (d − r)(

sin α

cos α

), r ∈ (0, d). (2)

It follows by substituting (1) to (2) so that the possible positionof the MS can be described as the following straight-lineequation2 of the slope–intercept form:

y = k(α, β)x + b(α, β, d) (3)

where

k(α, β) =cos α + cos β

sin α + sin β(4)

b(α, β, d) = −k(α, β)(xf − d sin α) + yf − d cos α. (5)

This implies that, if we have the knowledge about two propa-gation paths originating from the MS, the position of the MScan be estimated as the intersection of two lines which arederived from (3). For example, assume that the strongest pathsfrom the MS to two FSs located at the coordinates (xf1, yf1)and (xf2, yf2) are parameterized by θ1 = (α1, β1, d1)T andθ2 = (α2, β2, d2)T, respectively. Let ϑ = (θT

1 ,θT2 )T be the

parameter vector of both paths. The coordinates p of the MScan be determined as the intersection of the following two linearequations:

li : y = kix + bi, i = 1, 2 (6)

where ki = k(αi, βi), bi = b(αi, βi, di), and i = 1, 2. If k1 �=k2, it follows that the coordinates p(ϑ) = (x(ϑ), y(ϑ))T of theMS become

p(ϑ) =(

b2 − b1

k1 − k2,k1b2 − k2b1

k1 − k2

)T

. (7)

For simplicity of the exposition, we define pG = (xG, yG)T =p(ϑ) as the coordinate estimates in the presence of parameterϑ estimation errors.

2Precisely speaking, the possible region of the MS should be a line segmentinstead of the line of infinite length; however, the line expression simplifies thederivation of the proposed algorithm without the loss of the accuracy of thepositioning algorithm.

MIAO et al.: POSITIONING FOR NLOS PROPAGATION: ALGORITHM DERIVATIONS AND CRAMER–RAO BOUNDS 2571

TABLE ISETUP OF THE POSITIONING ALGORITHM

Fig. 2. Possible region of the MS.

B. RMSE Analysis

Let αoi , β

oi , do

i , i = 1, 2 denote the actual AODs, AOAs,and distances of the propagation paths of interest. We as-sume that the estimated parameters αi, βi, and di, i = 1, 2are independently Gaussian-distributed random variables [42],3

i.e., αi ∼ N (αoi , σ

2αi

), βi ∼ N (βoi , σ2

βi), and di ∼ N (do

i , σ2di

).Indeed, the following RMSE analysis can be straightforwardlyextended to the case of the correlated Gaussian parameterestimates by substituting the diagonal estimation error covari-ance matrix with a dense matrix whose off-diagonal elementscharacterize the correlation of different parameter estimates.For simplicity of the exposition, we focus on the independentGaussian distribution. We define θo

i = (αoi , β

oi , do

i )T, i = 1, 2

as the parameter vector of actual values. Let ϑo = (θo1T,θo

2T)T

denote the actual parameter vector of two paths. When thevariances of the estimated parameters are small, according tothe Taylor expansion, the estimated coordinates of the MS canbe approximated as follows:

p(ϑ) ≈ p(ϑo) + ∇p(ϑo)(ϑ− ϑo) (8)

where ∇p(ϑo) =(∇x(ϑo)

∇y(ϑo)

)∈ R

2×6 is the Jacobian matrix forp(ϑ) at ϑo.

Let po = (x(ϑo), y(ϑo))T denote the true coordinates ofthe MS. We denote ko

1 = k(αo1, β

o1) and ko

2 = k(αo2, β

o2) as

slopes corresponding to accurate linear equations. Let bo1 =

b(αo1, β

o1 , do

1) and bo2 = b(αo

2, βo2 , do

2) denote the x-intercepts ofaccurate linear equations. The above gradient vectors ∇x(ϑo)

3The MUSIC estimator was shown to be Gaussian-distributed for sufficientlylarge measurements.

and ∇y(ϑo) can be achieved from the following partialderivatives:

∂x(ϑo)∂α1

=∂b(θo

1)

∂α (ko2 − ko

1) + (bo1 − bo

2)∂k(θo

1)

∂α

(ko1 − ko

2)2(9)

∂x(ϑo)∂α2

=∂b(θo

2)

∂α (ko1 − ko

2) + (bo2 − bo

1)∂k(θo

2)

∂α

(ko1 − ko

2)2(10)

∂x(ϑo)∂β1

=∂b(θo

1)

∂β (ko2 − ko

1) + (bo1 − bo

2)∂k(θo

1)

∂β

(ko1 − ko

2)2(11)

∂x(ϑo)∂β2

=∂b(θo

2)

∂β (ko1 − ko

2) + (bo2 − bo

1)∂k(θo

2)

∂β

(ko1 − ko

2)2(12)

∂x(ϑo)∂d1

=∂b(θo

1)

∂d

ko2 − ko

1

(13)

∂x(ϑo)∂d2

=∂b(θo

2)

∂d

ko1 − ko

2

(14)

and (15)–(20), shown at the bottom of the next page, where

∂k

∂α=

−1 − cos(α − β)(sin α + sinβ)2

∂k

∂β=

∂k

∂α

∂b

∂α=

∂k

∂α(d sin α − xb) + kd cos α + d sin α

∂b

∂β=

∂k

∂β(d sin α − xb)

∂b

∂d= k sin α − cosα. (21)

Therefore, when the variances of the estimated parameters ϑare small, the covariance matrix RpG

= E((pG − po)(pG −po)T) of the estimation error of the coordinates p can bedescribed as

RpG≈ ∇p(ϑo)diag(σ2

α1, σ2

β1, σ2

d1, σ2

α2, σ2

β2, σ2

d2)∇pT(ϑo).

(22)

We denote the RMSE σG =√

Tr(RpG)/2 as the average

RMSE of each estimated coordinate of the MS.

IV. LS ALGORITHM WITH MULTIPLE FSS

A. Algorithm Derivation

The geometrical approach in Section III can be generalized tothe case where Nf FSs located at (xf,i, yf,i), i ∈ {1, 2, . . . , Nf}can receive the signal from the MS. Let θ = (θT

1 , . . . ,θTNf

)T

and θo = (θo1T, . . . ,θo

Nf

T)T denote the parameter vector esti-mates and the parameter vector of actual values, respectively. In

2572 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

general, the RMSE of the estimated position can be improvedby exploiting all the observations of different FSs. Let bi andki denote the x-intercept and the slope of the linear equationassociated with the strongest path received by the ith FS,respectively. Let bo

i and koi be the actual counterparts of bi and

ki, respectively. To achieve the minimum equation-error norm,the LS coordinate estimate pLS = (xLS, yLS)T of the MS canbe obtained by

(xLS, yLS) = arg min(x,y)

Nf∑i=1

(kix + bi − y)2. (23)

It follows straightforwardly that

xLS(θ) =∑Nf

i=1 bi

∑Nfi=1 ki − Nf

∑Nfi=1 biki

Nf

∑Nfi=1 k2

i −(∑Nf

i=1 ki

)2

yLS(θ) =∑Nf

i=1 bi + xLS

∑Nfi=1 ki

Nf

=∑Nf

i=1 bi

∑Nfi=1 k2

i −∑Nf

i=1 ki

∑Nfi=1 biki

Nf

∑Nfi=1 k2

i −(∑Nf

i=1 ki

)2 . (24)

It is interesting to note that (xLS, yLS) is equal to (xG, yG)4 ifNf = 2. The possible region of the MS that is conducted froman NLOS path in the preceding discussion can be expressed bya linear equation which has a finite slope. There are two casesin which the slope k does not exist, that is, one is the LOS casewhere the possible region shrinks to a point instead of a linesegment, and the other is the case in which the possible regionis a line parallel to the y-axis. A more detailed discussion about

4In this particular case, the minimum equation-error norm in (23) is zero.

the LS algorithm dealing with two special cases is presented inAppendix A.

B. RMSE Analysis

In Section III-B, all estimated parameters are assumed tobe independently Gaussian-distributed random variables. Whenthe variances of the estimated parameters θ are small, we defineσ2

i = (σ2αi

, σ2βi

, σ2di

)T, i = 1, 2, . . . , Nf and pLS = pLS(θ) =(xLS(θ), yLS(θ))T, and the estimated coordinates (24) of theMS are described as follows:

pLS(θ) ≈ po + ∇pLS(θo)(θ − θo) (25)

where ∇pLS(θo) =(∇xLS(θo)

∇yLS(θo)

)∈ R

2×3Nf is the Jacobian ma-trix of pLS(θ) at θo. The gradient vectors ∇xLS(θo) and∇yLS(θo) are derived in Appendix B.

Therefore, the covariance matrix RpLS= E((pLS −

po)(pLS − po)T) of the estimation error of the coordinates(xLS, yLS) is

RpLS≈ ∇pLS(θo)diag(σ2

1, . . . ,σ2f )∇pT

LS(θo). (26)

In Section III-B, we denote the RMSE σLS =√

Tr(RpLS)/2

as the average RMSE of each estimated coordinate of the MS.

V. ML POSITIONING ESTIMATION AND CRLB

A. Joint MS and Scatterers’ Position Estimation

Section IV improved the positioning accuracy of the MSby using multiple FSs. Indeed, all the observed parameters θprovide information about not only the position of the MS butthe positions of the associated scatterers as well. We define(xo

si, yosi), i = 1, 2, . . . , Nf as the accurate coordinates of the

∂y(ϑo)∂α1

=

(∂k(θo

1)

∂α bo2 − ko

2∂b(θo

1)

∂α

)(ko

1 − ko2) + (ko

2bo1 − ko

1bo2)

∂k(θo1)

∂α

(ko1 − ko

2)2 (15)

∂y(ϑo)∂α2

=

(∂b(θo

2)∂α ko

1 − bo1

∂k(θo2)

∂α

)(ko

1 − ko2) + (ko

1bo2 − ko

2bo1)

∂k(θo2)

∂α

(ko1 − ko

2)2 (16)

∂y(ϑo)∂β1

=

(∂k(θo

1)∂β bo

2 − ko2

∂b(θo1)

∂β

)(ko

1 − ko2) + (ko

2bo1 − ko

1bo2)

∂k(θo1)

∂β

(ko1 − ko

2)2 (17)

∂y(ϑo)∂β2

=

(∂b(θo

2)∂β ko

1 − bo1

∂k(θo2)

∂β

)(ko

1 − ko2) + (ko

1bo2 − ko

2bo1)

∂k(θo2)

∂β

(ko1 − ko

2)2 (18)

∂y(ϑo)∂d1

=ko2

∂b(θo1)

∂d

ko2 − ko

1

(19)

∂y(ϑo)∂d2

=ko1

∂b(θo2)

∂d

ko1 − ko

2

(20)

MIAO et al.: POSITIONING FOR NLOS PROPAGATION: ALGORITHM DERIVATIONS AND CRAMER–RAO BOUNDS 2573

scatterers. According to Fig. 2, the real parameter θoi depends

on the coordinates (xosi, y

osi) and (xo, yo) as follows:

αoi =

arc tan xosi−xo

yosi−yo xo

si ≥ xo, yosi ≥ yo

π + arc tan xosi−xo

yosi−yo xo

si ≥ xo, yosi < yo

π + arc tan xosi−xo

yosi−yo xo

si < xo, yosi < yo

2π + arc tan xosi−xo

yosi−yo xo

si ≥ xo, yosi ≥ yo

βoi =

arc tan xosi−xo

fiyosi−yo

fixo

si ≥ xofi, y

osi ≥ yo

fi

π + arc tan xosi−xo

fiyosi−yo

fixo

si ≥ xofi, y

osi < yo

fi

π + arc tan xosi−xo

fiyosi−yo

fixo

si < xofi, y

osi < yo

fi

2π + arc tan xosi−xo

fiyosi−yo

fixo

si ≥ xofi, y

osi ≥ yo

fi

doi =

√(xo

si−xo)2+(yosi−yo)2+

√(xo

si−xofi)

2+(yosi−yo

fi)2.

(27)

In Section III-B, all the estimated parameters are assumedto be independently Gaussian-distributed random variables. Wedefine p(θ;xo, yo, (xo

si, yosi)i=1,2,...,Nf ) as the parameterized

joint probability density function of all the observed parame-ters. Due to the assumption that all the observed parame-ters are independently Gaussian-distributed random variables,the log-likelihood function L(θ;xo, yo, (xo

si, yosi)i=1,2,...,Nf ) =

lnp(θ;xo, yo, (xosi, y

osi)i=1,2,...,Nf ) can be expressed as

L(θ;xo, yo, (xo

si, yosi)i=1,2,...,Nf

)

=Nf∑i=1

(ln

1(2π)3/2σαi

σβiσdi

− 12

((αi − αo

i )2

σ2αi

+(βi − βo

i )2

σ2βi

+(di − do

i )2

σ2di

)).

(28)

By ignoring the constant term in (28), we define the objectivefunction as

O(θ;xo, yo (xo

si, yosi)i=1,2,...,Nf

)

=12

Nf∑i=1

((αi − αo

i )2

σ2αi

+(βi − βo

i )2

σ2βi

+(di − do

i )2

σ2di

).

(29)

The ML-based joint of the MS and scatterers’ position estima-tor becomes(

xML, yML (xMLSi, yMLSi)i=1,2,...,Nf

)= arg min

xo,yo,(xosi

,yosi

)i=1,2,...,Nf

O(θ;xo, yo(xo

si, yosi)i=1,2,...,Nf

).

(30)

The minimization problem that is discussed earlier is a non-linear programming problem which can be solved by variousiterative procedures [43, Ch. 3], e.g., the steepest descent orconjugate gradient methods. All the iterative methods require

Fig. 3. NLOS propagation path from the MS to the FSs.

an initial searching point. A good starting point leads to theglobal minimum, whereas a badly selected one may result inconvergence to a local minimum. For example, the LS algo-rithm in Section IV provides a reasonable initialization for theML algorithm.5

B. Cramer–Rao Lower Bound (CRLB)

The CRLB is the lower bound on the variance of anyunbiased estimator for unknown parameters [44]. Let us de-fine ϕ = (xo, yo)T, xo

s = (xos1, x

os2, . . . , x

osN f

)T, yos = (yo

s1,

yos2, . . . , y

osN f

)T, ψ = (xosT,yo

sT), and ρ = (ϕT,ψT)T. The

Fisher information matrix (FIM) I(ρ) ∈ R(2Nf+2)×(2Nf+2) can

be expressed as

I(ρ) = E(

∂L(θ;ρ)∂ρ

∂L(θ;ρ)∂ρT

). (31)

The derivation of the FIM I(ρ) is detailed in Appendix C.It follows that CRLB(ρ) = I−1(ρ), and the RMSE σML =√

(E(x − xo)2 + E(y − yo)2)/2 of any unbiased positioningestimator has the lower bound

σML ≥√

(CRLB(ρ))11 + (CRLB(ρ))222

(32)

where (CRLB(ρ))ij denotes the (i, j)th entry of the matrixCRLB(ρ).

VI. NUMERICAL EXAMPLES

This section presents analytical and simulation results toillustrate the performance of the proposed LS and the MLalgorithms. Fig. 3 shows the distribution of the MS and FSsalong with the NLOS propagation paths. The positions of the

5Even though the LS estimator only produces the position of the MS, thepositions of the scatterers can also be straightforwardly calculated from (1)and (2).

2574 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

TABLE IICOORDINATES (IN METER) OF THE MS, FSs, AND SCATTERERS

Fig. 4. Analytical RMSE versus standard deviations of AOA β estimate andAOD α estimate for four FSs. (a) σd = 5 m. (b) σd = 10 m.

MS, FSs, and scatterers are given in Table II. We assume thatthe parameters associated with different propagation paths havethe same normalized estimation error variance, i.e., σ2

αi= σ2

α,σ2

βi= σ2

β , σ2di

= σ2d, for all i = 1, 2, 3, and 4.

A. LS Algorithm

We examine the effects of the standard deviations of esti-mated parameters, i.e., σα, σβ , and σd, on the RMSE of the LS-based positioning algorithm. We also examine the impact of thenumber of FSs, i.e., the number of employed NLOS paths, on itsperformance. As shown in Fig. 3, there is an NLOS propagationpath between the MS and each of the four FSs. When weonly employ two out of the four NLOS paths, there are

(42

)

Fig. 5. Analytical RMSE versus standard deviations of distance estimate andAOA (= AOD) estimate for four FSs.

combinations of two NLOS paths. Different combinations maylead to different RMSE performances, e.g., the performanceyielded by the FS pair (1, 2) could be different from that of theFS pair (3, 4). To obtain the average performance of employingthe same number of NLOS paths, all RMSE performancesunder different FS combinations are averaged out.

1) Effects of standard deviations of parameter estimates:Fig. 4(a) and (b) shows the analytical RMSE of the LS-based positioning algorithm by virtue of all four NLOSpaths versus (σα, σβ) when the standard deviation of thedistance σd is 5 and 10 m, respectively. It was foundthat RMSE performance is more sensitive to σα than σβ

in the environment shown in Fig. 3. Fig. 5 shows theanalytical RMSE performance with respect to (σd, σβ)when σα = σβ . It is shown that the standard deviationsσα and σβ have a larger effect on the RMSE than σd.

2) Effects of the number of FSs: Figs. 6–8 show the an-alytical and simulated RMSE performance of the LSpositioning algorithm when two, three, or four NLOSpaths are exploited. Fig. 6 shows how the number ofFSs and σα affect the RMSE performance, given σβ andσd. Figs. 7 and 8 show the effect of the number of FSsemployed along with σβ and σd, respectively. They showthat, when more NLOS paths are used, the LS algorithmcan achieve a higher accuracy. All the simulations areperformed with 1000 independent runs. The fact that thesimulation results are close to the corresponding analyti-cal ones confirms that the approximate analytical RMSEexpressions derived in Section IV-B are accurate when thestandard deviations of parameter estimates are reasonablysmall.

MIAO et al.: POSITIONING FOR NLOS PROPAGATION: ALGORITHM DERIVATIONS AND CRAMER–RAO BOUNDS 2575

Fig. 6. RMSE versus standard deviations of the AOD α estimate, where σd =5 m and σβ = 2o.

Fig. 7. RMSE versus standard deviations of the AOA β estimate, where σd =5 m and σα = 2o.

B. ML Algorithm and CRLB

Figs. 6–8 also show the RMSE performance of the ML-based algorithm and the corresponding CRLBs against σα,σβ , and σd, respectively. The MATLAB function fmincon.m isemployed to find the ML solution of (30). All the simulationsof the ML algorithm employ the solution yielded by the LSalgorithm as the initial searching point. It shows that the MLalgorithm does improve the performance significantly at thecost of the increased computational complexity. It is also foundout that the LS algorithm is capable of providing a good startingpoint for the ML algorithm. It is interesting to note that theperformance of the ML algorithm is close to the CRLB. Thisdemonstrates that four NLOS paths, which are furnished byfour FSs, are enough to make the ML algorithm an efficientestimator, which is able to achieve the optimal performance,i.e., the CRLB.

Fig. 8. RMSE versus standard deviations of the distance estimate, whereσα = σβ = 2o.

VII. CONCLUSION

In this paper, we considered the position estimation of theMS in the NLOS scenario. With knowledge about the AOA,the AOD, and the distance associated with each NLOS path, ageometric approach of calculating the intersection of two linesegments which were derived as possible regions for the MSwas presented. To achieve the minimal equation-error norm,the LS algorithm was proposed to exploit multiple NLOSpaths to improve the positioning accuracy. The ML algorithmwas further developed to improve the performance of the LSalgorithm, and it was able to estimate the positions of bothMS and scatterers. The CRLB was also derived to benchmarkthe performance of the positioning algorithms. It was shownboth analytically and through computer simulations that theproposed algorithms are able to estimate the MS position onlyby employing the NLOS paths, and the ML algorithm canachieve the optimal RMSE performance, i.e., the CRLB.

Several interesting related open research problems still re-main. As mentioned in the introduction, the task of obtainingknowledge about the NLOS path, i.e., the AOA and the AODat the FS, is rather challenging in practice; in particular, it isextremely hard to obtain the angle information in the impulseradio-based systems. Another interesting problem is how tochoose the NLOS paths in the presence of multiple NLOSpaths. It is noted in Section VI that different combinationsof NLOS paths lead to different performances. This wouldenable a more detailed performance-complexity tradeoff in realpositioning systems.

APPENDIX ADERIVATION OF THE LS ALGORITHM

UNDER THE GENERAL SCENARIO

As mentioned in Section IV-A, there are two cases where(3) does not hold due to the nonexistence of the slope k. Thishappens if and only if

sin α + sin β = 0. (A-1)

2576 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

Fig. 9. LOS case. (a) α − β = π. (b) β − α = π.

Because the range of α and β is [0, 2π), the solutions of (A-1)are described as

|α − β| =π (A-2)

α + β =2π. (A-3)

It is interesting to note that (A-2) corresponds to the LOS case,as shown in Fig. 9. Therefore, (A-2) implies a new method ofidentifying the LOS path. It is obvious that the possible regionof the MS in the LOS case is a point having the followingcoordinates:

xLOS =xf + d sin β,

yLOS = yf + d cos β. (A-4)

Equation (A-3) corresponds to the possible region which is aline parallel to the y-axis; this is shown in Fig. 10, when thepossible region becomes

x⊥ = xf − d sin α. (A-5)

We assume that Nf paths consist of N1 LOS paths, N2 NLOSpaths from which derived possible regions are lines with slopesof finite values, and N3 NLOS paths from which the derivedpossible regions are lines parallel to the y-axis, Nf ≥ Ni ≥

Fig. 10. Possible region. A line parallel to the y-axis.

0, i = 1, 2, and 3. The LS-based position estimate (xLS, yLS)of the MS considering this general case can be derived as

(xLS, yLS) = arg min(x,y)

N1∑i=1

((x − xLOSi)2 + (y − yLOSi)2

)

+N2∑i=1

(kix + bi − y)2 +N3∑i=1

(x − x⊥i)2 (A-6)

where (xLOSi, yLOSi), i = 1, 2, . . . , N1, and x⊥i, i =1, 2, . . . , N3 can be calculated from (A-4) and (A-5),respectively. We define the objective function as

O(x, y) =N1∑i=1

((x − xLOSi)2 + (y − yLOSi)2

)

+N2∑i=1

(kix + bi − y)2 +N3∑i=1

(x − x⊥i)2. (A-7)

The LS solution (xLS, xLS) of (A-6) can be obtained by

xLS =∂O(x, y)

∂x= 0 (A-8)

yLS =∂O(x, y)

∂y= 0. (A-9)

It can be derived that

∂O(x, y)∂x

= x

N2∑i=1

ki + y(N1 − N2) +N2∑i=1

bi −N1∑i=1

yLOSi

(A-10)

∂O(x, y)∂y

= x

(N1 + N3 +

N2∑i=1

k2i

)− y

N2∑i=1

ki

+N2∑i=1

kibi −N1∑i=1

xLOSi −N3∑i=1

x⊥i. (A-11)

MIAO et al.: POSITIONING FOR NLOS PROPAGATION: ALGORITHM DERIVATIONS AND CRAMER–RAO BOUNDS 2577

We denote G, E, and F as shown by (A-12)–(A-14), shown atthe bottom of the page. According to the Cramer’s Rule, the LSsolution (xLS, xLS) can be calculated from

xLS =E

G

yLS =F

G. (A-15)

Equation (A-15) shows the LS-based position estimate of theMS which takes all the propagation paths, including LOS andNLOS paths, into account.

APPENDIX BDERIVATION OF THE GRADIENT VECTORS

OF THE LS POSITION ESTIMATES

The gradient vector ∇xLS(θo) in (25) is calculated as

∂xLS(θo)∂αi

=∂xLS(θo)

∂ki

∂k (θoi )

∂α

+∂xLS(θo)

∂bi

∂b (θoi )

∂α, i = 1, 2, . . . , Nf

∂xLS(θo)∂βi

=∂xLS(θo)

∂ki

∂k (θoi )

∂β

+∂xLS(θo)

∂bi

∂b (θoi )

∂β, i = 1, 2, . . . , Nf

∂xLS(θo)∂di

=∂xLS(θo)

∂bi

∂b (θoi )

∂d, i = 1, 2, . . . , Nf (B-1)

where ∂k/∂α, ∂k/∂β, ∂b/∂α, ∂b/∂β, and ∂b/∂d are ob-tained from (21). Let P =

∑Nfi=1 ko

i , Q =∑Nf

i=1(koi )2, W =∑Nf

i=1 boi k

oi , and Z =

∑Nfi=1 bo

i . Then, ∂xLS(θo)/∂ki and∂xLS(θo)/∂bi, i = 1, 2, . . . , Nf are expressed using (B-2),

shown at the bottom of the page. As in (B-1), the gradient vector∇yLS(θo) in (25) becomes

∂yLS(θo)∂αi

=∂yLS(θo)

∂ki

∂k (θoi )

∂α

+∂yLS(θo)

∂bi

∂b (θoi )

∂α, i = 1, 2, . . . , Nf

∂yLS(θo)∂βi

=∂yLS(θo)

∂ki

∂k (θoi )

∂β

+∂yLS(θo)

∂bi

∂b (θoi )

∂β, i = 1, 2, . . . , Nf

∂yLS(θo)∂di

=∂yLS(θo)

∂bi

∂b (θoi )

∂d, i = 1, 2, . . . , Nf (B-3)

where ∂yLS(θo)/∂ki, and ∂yLS(θo)/∂bi, i = 1, 2, . . . , Nf areexpressed as

∂yLS(θo)∂ki

=xLS(θo) + P ∂xLS(θo)

∂ki

Nf

∂yLS(θo)∂bi

=1 + P ∂xLS(θo)

∂bi

Nf. (B-4)

APPENDIX CDERIVATION OF THE FIM

We partition the FIM I(ρ) as follows:

I(ρ) =(A BBT C

)(C-1)

where

A = E(

∂L(θ;ρ)∂ϕ

∂L(θ;ρ)∂ϕT

)∈ R

2×2

B = E(

∂L(θ;ρ)∂ϕ

∂L(θ;ρ)∂ψT

)∈ R

2×2Nf

C = E(

∂L(θ;ρ)∂ψ

∂L(θ;ρ)∂ψT

)∈ R

2Nf×2Nf . (C-2)

G =∣∣∣∣

∑N2i=1 ki N1 − N2

N1 + N3 +∑N2

i=1 k2i −

∑N2i=1 ki

∣∣∣∣ (A-12)

E =∣∣∣∣

∑N1i=1 yLOSi −

∑N2i=1 bi N1 − N2∑N1

i=1 xLOSi +∑N3

i=1 x⊥i −∑N2

i=1 kibi −∑N2

i=1 ki

∣∣∣∣ (A-13)

F =∣∣∣∣

∑N2i=1 ki

∑N1i=1 yLOSi −

∑N2i=1 bi

N1 + N3 +∑N2

i=1 k2i

∑N1i=1 xLOSi +

∑N3i=1 x⊥i −

∑N2i=1 kibi

∣∣∣∣ (A-14)

∂xLS(θo)∂ki

=Nfb

oi (P

2 − NfQ) + Z(NfQ + P 2) − 2Nf (koi PZ + W (P − Nfk

oi ))

(NfQ − P 2)2

∂xLS(θo)∂bi

=P − Nfk

oi

NfQ − P 2(B-2)

2578 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

To calculate the I(ρ), we need the following expressions:

∂αoi

∂xo= − ∂αo

i

∂xosi

=yo − yo

si

(xosi − xo)2 + (yo

si − yo)2

∂αoi

∂yo= − ∂αo

i

∂yosi

=xo

si − xo

(xosi − xo)2 + (yo

si − yo)2(C-3)

∂βoi

∂xo=

∂βoi

∂yo

= 0∂βo

i

∂xosi

=yosi − yfi

(xosi − xfi)2 + (yo

si − yfi)2

∂βoi

∂yosi

=xfi − xo

si

(xosi − xfi)2 + (yo

si − yfi)2(C-4)

and

∂doi

∂xo=

xo − xosi√

(xosi − xo)2 + (yo

si − yo)2,

∂doi

∂yo=

yo − yosi√

(xosi − xo)2 + (yo

si − yo)2

∂doi

∂xosi

=xo

si − xo√(xo

si − xo)2 + (yosi − yo)2

+xo

si − xfi√(xo

si − xfi)2 + (yosi − yfi)2

∂doi

∂yosi

=yosi − yo√

(xosi − xo)2 + (yo

si − yo)2

+yosi − yfi√

(xosi − xfi)2 + (yo

si − yfi)2. (C-5)

For simplicity of the exposition, we define L(θ) .= L(θ;ρ).The partial derivatives of L(θ;ρ) with respect to xo, yo, xo

s ,and yo

s become

∂L(θ)∂xo

=Nf∑i=1

((αi − αo

i )σ2

αi

∂αoi

∂xo+

(di − doi )

σ2di

∂doi

∂xo

)

∂L(θ)∂yo

=Nf∑i=1

((αi − αo

i )σ2

αi

∂αoi

∂yo+

(di − doi )

σ2di

∂doi

∂yo

)

∂L(θ)∂xo

si

=(αi − αo

i )σ2

αi

∂αoi

∂xosi

+(βi − βo

i )σ2

βi

∂βoi

∂xosi

+(di − do

i )σ2

di

∂doi

∂xosi

, i = 1, 2 . . . , Nf

∂L(θ)∂yo

si

=(αi − αo

i )σ2

αi

∂αoi

∂yosi

+(βi − βo

i )σ2

βi

∂βoi

∂yosi

+(di − do

i )σ2

di

∂doi

∂yosi

, i = 1, 2 . . . , Nf . (C-6)

Due to the above expressions and the assumption that all theobserved parameters θ are independently distributed Gaussianrandom variables, we can obtain the following second-ordermoments:

E(

∂L (θ)∂xo

)2

=Nf∑i=1

(1

σ2αi

(∂αo

i

∂xo

)2

+1

σ2di

(∂do

i

∂xo

)2)

E(

∂L (θ)∂yo

)2

=Nf∑i=1

(1

σ2αi

(∂αo

i

∂yo

)2

+1

σ2di

(∂do

i

∂yo

)2)

E(

∂L (θ)∂xo

∂L (θ)∂yo

)

=Nf∑i=1

(1

σ2αi

(∂αo

i

∂xo

)(∂αo

i

∂yo

)+

1σ2

di

(∂do

i

∂xo

)(∂do

i

∂yo

))

(C-7)

E(

∂L (θ)∂xo

∂L (θ)∂xo

si

)

=1

σ2αi

(∂αo

i

∂xo

)(∂αo

i

∂xosi

)

+1

σ2di

(∂do

i

∂xo

)(∂do

i

∂xosi

), i = 1, 2, . . . , Nf

E(

∂L (θ)∂yo

∂L (θ)∂xo

si

)

=1

σ2αi

(∂αo

i

∂yo

)(∂αo

i

∂xosi

)

+1

σ2di

(∂do

i

∂yo

)(∂do

i

∂xosi

), i = 1, 2, . . . , Nf

E(

∂L (θ)∂xo

∂L (θ)∂yo

si

)

=1

σ2αi

(∂αo

i

∂xo

)(∂αo

i

∂yosi

)

+1

σ2di

(∂do

i

∂xo

)(∂do

i

∂yosi

), i = 1, 2, . . . , Nf

E(

∂L (θ)∂yo

∂L (θ)∂yo

si

)

=1

σ2αi

(∂αo

i

∂yo

)(∂αo

i

∂yosi

)

+1

σ2di

(∂do

i

∂yo

)(∂do

i

∂yosi

), i = 1, 2, . . . , Nf (C-8)

MIAO et al.: POSITIONING FOR NLOS PROPAGATION: ALGORITHM DERIVATIONS AND CRAMER–RAO BOUNDS 2579

and

E

(∂L (θ)∂xo

si

∂L (θ)∂xo

sj

)

=

{ 0 i �= j1

σ2αi

(∂αo

i

∂xosi

)2

+ 1σ2

βi

(∂βo

i

∂xosi

)2

+ 1σ2

di

(∂do

i

∂xosi

)2

i = j

E

(∂L (θ)∂yo

si

∂L (θ)∂yo

sj

)

=

{ 0 i �= j1

σ2αi

(∂αo

i

∂yosi

)2

+ 1σ2

βi

(∂βo

i

∂yosi

)2

+ 1σ2

di

(∂do

i

∂yosi

)2

i = j

E

(∂L (θ)∂xo

si

∂L (θ)∂yo

sj

)

=

0 i �= j1

σ2αi

(∂αo

i

∂xosi

)(∂αo

i

∂yosi

)+ 1

σ2βi

(∂βo

i

∂xosi

)(∂αo

i

∂yosi

)+ 1

σ2di

(∂do

i

∂xosi

)(∂αo

i

∂yosi

)i = j.

(C-9)

The FIM I(ρ) can be derived by substituting (C-3) and(C-7)–(C-9) into (C-2) and (C-1).

REFERENCES

[1] A. H. Sayed and N. R. Yousef, “Wireless location,” inWiley Encyclopediaof Telecommunications, J. Proakis, Ed. New York: Wiley, 2003.

[2] Location-Based Services: Finding Their Place in the Market, Feb. 2003,San Jose, CA: In-Stat/MDR. Tech. Rep.

[3] Y. Zhao, “Standardization of mobile phone positioning for 3G systems,”IEEE Commun. Mag., vol. 40, no. 7, pp. 108–116, Jul. 2002.

[4] J. Hightower and G. Borriello, “Location systems for ubiquitous comput-ing,” Computer, vol. 34, no. 8, pp. 57–66, Aug. 2001.

[5] M. Mauve, J. Widmer, and H. Hartenstein, “A survey on position-basedrouting in mobile ad-hoc networks,” IEEE Netw., vol. 15, no. 6, pp. 30–39,Nov. 2001.

[6] R. Jurgen, “Smart cars and highways go global,” IEEE Spectr., vol. 28,no. 5, pp. 26–36, May 1991.

[7] W. Collier and R. Weiland, “Smart cars, smart highways,” IEEE Spectr.,vol. 31, no. 4, pp. 27–33, Apr. 1994.

[8] I. F. Akyildiz and W. Wang, “The predictive user mobility profileframework for wireless multimedia networks,” IEEE/ACM Trans. Netw.,vol. 12, no. 6, pp. 1021–1035, Dec. 2004.

[9] V. Pandey, D. Ghosal, and B. Mukherjee, “Exploiting user profiles to sup-port differentiated services in next-generation wireless networks,” IEEENetw., vol. 18, no. 5, pp. 40–48, Sep./Oct. 2004.

[10] J. Ye, J. Hou, and S. Papavassiliou, “A comprehensive resource man-agement framework for next generation wireless networks,” IEEE Trans.Mobile Comput., vol. 1, no. 4, pp. 249–264, Oct.–Dec. 2002.

[11] M. Chiu and M. A. Bassiouni, “Predictive schemes for handoff prioritiza-tion in cellular networks based on mobile positioning,” IEEE J. Sel. AreasCommun., vol. 18, no. 3, pp. 510–522, Mar. 2000.

[12] A. Giordano, M. Chan, and H. Habal, “A novel location-based serviceand architecture,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile RadioCommun., Tokyo, Japan, Sep. 27–29, 1995, vol. 2, pp. 853–857.

[13] H. Hashemi, “Pulse ranging radiolocation technique and its application tochannel assignment in digital cellular radio,” in Proc. IEEE Veh. Technol.Conf., St. Louis, MO, May 19–22, 1991, vol. 1, pp. 675–680.

[14] R. Narasimhan and D. Cox, “A handoff algorithm for wireless systemsusing pattern recognition,” in Proc. IEEE Int. Symp. Pers., Indoor, MobileRadio Commun., Boston, MA, Sep. 8–11, 1998, vol. 1, pp. 335–339.

[15] Revision of the Commissions Rules to Ensure Compatibility With En-hanced 911 Emergency Calling Systems, RM-8143, Jul. 26, 1996,Washington, DC: FCC. CC Docket 94-102.

[16] K. Yu, H. Saarnisaari, J.-P. Montillet, A. Rabbachin, I. Oppermann, andG. Abreu, “Localization,” in Ultra-Wideband Wireless Communications

and Networks, S. Shen, M. Guizani, R. C. Qui, and T. Le-Ngoc, Eds.New York: Wiley, 2006, pp. 279–304.

[17] I. Oppermann, L. Stoica, A. Rabbachin, Z. Shelby, and J. Haapola, “UWBwireless sensor networks: UWEN—A practical example,” IEEE Com-mun. Mag., vol. 42, no. 12, pp. 527–532, Dec. 2004.

[18] D. J. Torrieri, “Statistical theory of passive location systems,” IEEE Trans.Aerosp. Electron. Syst., vol. AES-20, no. 2, pp. 183–198, Mar. 1984.

[19] G. Sun, J. Chen, W. Guo, and K. J. R. Liu, “Signal processing techniquesin network-aided positioning,” IEEE Signal Process. Mag., vol. 22, no. 4,pp. 12–23, Jul. 2005.

[20] J. Caffery, Jr. and G. L. Stüber, “Subscriber location in CDMA cellu-lar networks,” IEEE Trans. Veh. Technol., vol. 47, no. 5, pp. 406–416,May 1998.

[21] F. Gustafsson and F. Gunnarsson, “Mobile positioning using wirelessnetworks,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 41–53,Jul. 2005.

[22] S. Gezici, Z. Tian, G. B. Giannakis, H. Kobayashi, A. F. Molisch,H. V. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios,”IEEE Signal Process. Mag., vol. 22, no. 4, pp. 70–84, Jul. 2005.

[23] J. O. Smith and J. S. Abel, “Closed-form least-squares source lo-cation estimation from range-difference measurements,” IEEE Trans.Acoust., Speech, Signal Process., vol. ASSP-35, no. 12, pp. 1661–1669,Dec. 1987.

[24] S. Sakagami, S. Aoyama, K. Kuboi, S. Shirota, and A. Akeyama, “Vehicleposition estimates by multibeam antennas in multipath environments,”IEEE Trans. Veh. Technol., vol. 41, no. 1, pp. 63–68, Feb. 1992.

[25] L. Cong and W. Zhuang, “Hybrid TDOA/AOA mobile user locationfor wideband CDMA cellular systems,” IEEE Trans. Wireless Commun.,vol. 1, no. 3, pp. 439–447, Jul. 2002.

[26] L. Xiong, “A selective model to suppress NLOS signals in angle-of-arrivalAOA location estimation,” in Proc. IEEE Int. Symp. Pers., Indoor, MobileRadio Commun., Boston, MA, Sep. 8–11, 1998, vol. 1, pp. 461–465.

[27] P. Chen, “A nonline-of-sight error mitigation algorithm in location estima-tion,” in Proc. IEEE Wireless Commun. Netw. Conf., New Orleans, LA,Sep. 21–24, 1999, vol. 1, pp. 316–320.

[28] L. Cong and W. Zhuang, “Non-line-of-sight error mitigation in TDOAmobile location,” in Proc. IEEE Global Telecommun. Conf., San Antonio,TX, Nov. 25–29, 2001, vol. 1, pp. 680–684.

[29] S. AI-Jazzar, J. Caffery, and H. You, “A scattering model based approachto NLOS mitigation in TOA location systems,” in Proc. IEEE Veh. Tech-nol. Conf., Birmingham, AL, May 6–9, 2002, vol. 2, pp. 861–865.

[30] L. Cong and W. Zhuang, “Nonline-of-sight error mitigation in mobilelocation,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 560–573,Mar. 2005.

[31] Y. Qi, H. Suda, and H. Kobayashi, “On time-of-arrival positioning in amultipath environment,” in Proc. IEEE Veh. Technol. Conf., Los Angeles,CA, Sep. 26–29, 2004, vol. 5, pp. 3540–3544.

[32] R. O. Schmidt, “Multiple emitter location and signal parameter estima-tion,” IEEE Trans. Antennas Propag., vol. AP-34, no. 3, pp. 276–280,Mar. 1986.

[33] R. Roy and T. Kailath, “ESPRIT—Estimation of signal parameters viarotational invariant techniques,” IEEE Trans. Acoust., Speech, SignalProcess., vol. 37, no. 7, pp. 984–995, Jul. 1989.

[34] M. Ng and I. Lu, “A novel spread spectrum-based synchronization andlocation determination method for wireless system,” IEEE Commun. Lett.,vol. 3, no. 6, pp. 177–179, Jun. 1999.

[35] N. R. Yousef, A. H. Sayed, and L. M. A. Jalloul, “Robust wireless locationover fading channels,” IEEE Trans. Veh. Technol., vol. 52, no. 1, pp. 117–126, Jan. 2003.

[36] G. C. Carter, Ed., “Special issue on time-delay estimation,” IEEETrans. Acoust., Speech, Signal Process., vol. AP-29, no. 3, pp. 461–462,Jun. 1981.

[37] A. van der Veen, M. Vanderveen, and A. Paulraj, “Joint angle and delay es-timation using shift-invariance techniques,” IEEE Trans. Signal Process.,vol. 46, no. 2, pp. 405–418, Feb. 1998.

[38] A. L. Swindlehurst, “Time delay and spatial signature estimation usingknown asynchronous signals,” IEEE Trans. Signal Process., vol. 46, no. 2,pp. 449–462, Feb. 1998.

[39] B. H. Fleury, X. Yin, P. Jourdan, and A. Stucki, “High-resolution channelparameter estimation for communication systems equipped with antennaarrays—Invited paper,” in Proc. 13th IFAC Symp. SYSID, Rotterdam,The Netherlands, Aug. 2003.

[40] H. Miao, M. Juntti, and K. Yu, “2-D unitary ESPRIT based joint AOAand AOD estimation for MIMO system,” in Proc. 17th IEEE PIMRC,Helsinki, Finland, Sep. 2006, pp. 1–5.

[41] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “Fromtheory to practice: An overview of MIMO space-time coded wireless

2580 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 281–302,Apr. 2003.

[42] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, andCramer–Rao bound,” IEEE Trans. Acoust., Speech, Signal Process.,vol. 37, no. 5, pp. 720–741, May 1989.

[43] A. L. Peressini, F. E. Sullivan, and J. J. Uhl, Jr., The Mathematics ofNonlinear Programming. New York: Springer-Verlag, 1988.

[44] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, Jun. 1993.

Honglei Miao (S’03) received the B.S.E.E. degreefrom the Huazhong University of Science and Tech-nology, Wuhan, China, in June 1996, the M.S.E.E.degree from the China Academy of Telecommunica-tion Technology, Beijing, China, in March 1999, andthe Ph.D. degree from the University of Oulu, Oulu,Finland, in May 2007.

From 1999 to 2002, he was an R&D Engineerat Datang Mobile Company, Ltd., Beijing. From2002 to 2005, he was with the Centre for WirelessCommunications, University of Oulu. He was with

Elektrobit as a Senior Design Engineer in 2006. He is currently with NokiaTechnology Platform, Helsinki, Finland, as a Specialist. His research interestsare signal processing and information theory for wireless communications.

Kegen Yu (S’01–M’04) was born in Linchuan,China. He received the B.Eng. degree in electri-cal engineering from Jilin University, Changchun,China, and the GradDip degree from ShandongUniversity, China, the M.Eng. degree in electricalengineering from the Australian National University,Canberra, Australia, and the Ph.D. degree in elec-trical engineering from the University of Sydney,Sydney, Australia.

After completing the B.Eng. degree, he spent twoyears in industry, working on electronic instrument

maintenance. Later, he became a Lecturer and taught a number of undergrad-uate courses in electrical engineering at Nanchang University, Jiangxi, China.Upon completing the Ph.D. degree, he joined the Centre for Wireless Com-munications, University of Oulu, Oulu, Finland, as a Postdoctoral ResearchFellow. During this period, he participated in a number of projects, includingthe EU Sixth Framework project PULSERS. Since November 2005, he hasbeen with the Wireless Technologies Laboratory, CSIRO ICT Centre, NewSouth Wales, Australia. He is a coauthor of three positioning-related bookchapters in UWB Theory and Practice (Wiley, 2004), Ultra-Wideband WirelessCommunications and Networks (Wiley, 2006), and Ultra Wideband WirelessCommunication (Wiley, 2006). His current research interests include wirelesslocation and tracking and wireless sensor networks.

Dr. Yu is a recipient of the E-mail Metering Prize for initiating industrycontacts.

Markku J. Juntti (S’93–M’98–SM’04) received theM.Sc.Tech. and Dr.Sc.Tech. degrees in electrical en-gineering from University of Oulu, Oulu, Finland, in1993 and 1997, respectively.

He was with the Telecommunication Laboratoryand Centre for Wireless Communications, Universityof Oulu, from 1992 to 1998. From 1994 to 1995, hewas a Visiting Scholar at Rice University, Houston,TX. In 1999 to 2000, he was with Nokia Networks.He has been a Professor of telecommunications at theUniversity of Oulu since 2000. His research interests

include communication and information theory and signal processing for wire-less communication systems and their application in wireless communicationsystem design. He is an author or coauthor of some 150 papers published ininternational journals and conference records, as well as in the book WCDMAfor UMTS (Wiley).

Dr. Juntti is an Associate Editor of the IEEE TRANSACTIONS ON

VEHICULAR TECHNOLOGY. He was the Secretary of the IEEE Communica-tion Society, Finland Chapter, in 1996–1997 and the Chairman in 2000–2001.He was the Secretary of the Technical Program Committee (TPC) of the 2001IEEE International Conference on Communications (ICC’01) and the Co-Chairof the TPC of the 2004 Nordic Radio Symposium. He was the Co-Chair ofthe TPC of the 2006 IEEE International Symposium on Personal, Indoor, andMobile Radio Communications (PIMRC 2006).