05208736

IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 11, NO. 3, THIRD QUARTER 2009 107

A Survey on TOA Based Wireless Localization andNLOS Mitigation Techniques

Ismail Guvenc, Member, IEEE, and Chia-Chin Chong, Senior Member, IEEE

Abstract—Localization of a wireless device using the time-of-arrivals (TOAs) from different base stations has been studiedextensively in the literature. Numerous localization algorithmswith different accuracies, computational complexities, a-prioriknowledge requirements, and different levels of robustnessagainst non-line-of-sight (NLOS) bias effects also have beenreported. However, to our best knowledge, a detailed unifiedsurvey of different localization and NLOS mitigation algorithmsis not available in the literature. This paper aims to give acomprehensive review of these different TOA-based localizationalgorithms and their technical challenges, and to point outpossible future research directions. Firstly, fundamental lowerbounds and some practical estimators that achieve close tothese bounds are summarized for line-of-sight (LOS) scenarios.Then, after giving the fundamental lower bounds for NLOSsystems, different NLOS mitigation techniques are classifiedand summarized. Simulation results are also provided in orderto compare the performance of various techniques. Finally, atable that summarizes the key characteristics of the investigatedtechniques is provided to conclude the paper.

Index Terms—Cramer-Rao Lower Bound, Location Estima-tion, NLOS Mitigation, Positioning, Time-of-Arrival.

I. INTRODUCTION

RECENTLY, location awareness has received great dealof interest in many wireless systems such as cellular

networks, wireless local area networks, and wireless sensornetworks due its capability to provide wide range of add-onapplications. Location-based services such as location basedadvertisement, location based social networking, and E911emergency services have become more important in orderto enhance the future lifestyle. For example, the locationbased advertisement allows users to selectively receive promo-tional advertisement by strategically placing messaging nearwhere buyer behavior can be most immediately influenced.For instance, a user will receive electronics sales items andcoupons only when he/she is entering a shopping mall. Onthe other hand, location based social networking may furtherenhance the Internet based social networking services suchas Facebook, Friendsters, MySpace, etc. by allowing usersforming groups based on their social preference and interest.For the E911 emergency services, user will be able to makeemergency call that allows local authority to track and locatethe user position under both indoor and outdoor scenarioswith high accuracy. The aforementioned example applications

Manuscript received 9 December 2007; revised 2 June 2008.The authors are with DOCOMO Communications Laboratories USA, 3240

Hillview Avenue, Palo Alto, CA 94304, USA (e-mail: [email protected], [email protected]).Digital Object Identifier 10.1109/SURV.2009.090308.

offered by location awareness will enable ubiquitous andcontext aware network services which necessitate the locationof the wireless device to be accurately estimated.

Even though location estimation problems have been inves-tigated extensively in the literature in the last few decades,there are still some open issues that remain unresolved.One of the key challenges in localization is the efficiencyand preciseness of the estimation in dense cluttered non-line-of-sight (NLOS) scenarios. NLOS scenarios occur whenthere is an obstruction between transmitter (TX) and receiver(RX) which are commonly encountered in modern wirelesssystem deployment for both indoor (e.g., residential, office,shopping malls, etc.) and outdoor (e.g., metropolitan, urbanarea, etc.) environments. In such circumstances, the use of theglobal positioning system (GPS) becomes impractical if notimpossible.

Several previous works have been reported in the literature(e.g., [1]–[5]) that provide extensive review on localizationusing angle-of-arrival (AOA), time-of-arrival (TOA), time-difference-of-arrival (TDOA), and received-signal-strength(RSS) techniques. However, none of these works investigatethe impact of NLOS mitigation techniques in detail in or-der to improve the performance degradation incurred by theblockage of the direct path. In this paper, we provide acomprehensive survey for TOA based localization techniqueswhich are applicable for both LOS and NLOS scenarios. Forother techniques such as AOA, TDOA, and fingerprint-basedmethods, interested readers are referred to [6]–[8] and thereferences therein.

The goal of this paper are two folds. Firstly, to providea unified overview of different TOA based localization tech-niques and related NLOS mitigation approaches. Secondly,to study the trade-offs and inter-relations among the variouslocalization techniques. Note that specific techniques requiredto estimate the TOA of the first arriving path for TOA-basedranging are outside the scope of this paper; interested readersare referred to [9]–[17].

The paper is organized as follows. Section II briefly re-views different location estimation techniques; namely, TOA,TDOA, AOA, RSS, and pattern-matching based approaches.In Section III, the TOA based localization scenario is outlinedand the system model as well as the problem definitionare presented. For the rest of the sections, Section IV andSection V are dedicated to LOS scenarios while Sections VI-X are dedicated to NLOS scenarios. Section IV providesfundamental lower bounds for LOS systems and summarizessome of the key maximum likelihood (ML) based techniques

1553-877X/09/$25.00 c© 2009 IEEE

108 IEEE COMMUNICATIONS SURVEYS & TUTORIALS, VOL. 11, NO. 3, THIRD QUARTER 2009

that can achieve close to these fundamental bounds. Section Vprovides a unified discussion of different least squares (LS)location estimation techniques for TOA-based systems. Sec-tion VI provides fundamental lower bounds and some MLtechniques for NLOS systems, Section VII is on LS techniquesin NLOS scenarios, Section IX is on robust estimators inNLOS scenarios, and Section X is a review of identifyand discard type of NLOS mitigation techniques. SectionXI studies the impact of FT distribution on the localizationaccuracy, and finally, Section XII summarizes the availabletechniques and provides some concluding remarks.

II. OVERVIEW OF DIFFERENT LOCALIZATIONTECHNIQUES

In general, there are two phases towards realizationof highly accurate location awareness applications. Firstly,through accurate ranging i.e., an action of estimating the dis-tance between two nodes, and secondly, through localizationi.e., an action of determining the exact location of an unknownnode from known nodes (anchors) by means of the intersectionof three or more measured ranges from known nodes. In thissection, a brief overview of five commonly used localizationtechniques will be discussed. A summary is also presented inTable I.For the TOA based technique, the distance information is

extracted from the propagation delay between a TX and anRX. This technique can be further classified into TOA one-way ranging and TOA two-way-ranging. The former requiresperfect synchronization between TX and RX, while the latterdoes not require synchronization between TX and RX. Thistechnique is more common in cellular networks since thereceiving nodes are typically synchronized to base stations.For the TDOA based technique, the difference between

TOAs in several RXs is used to reconstruct a TX’s position.This could either be based on the difference in the times atwhich a single signal from a single node arrives at three ormore nodes, or based on the difference in the times at whichmultiple signals from a single node arrive at another node.This requires highly precise synchronization between the RXs,but not precise synchronization between TX and RXs. TDOAbased technique is more commonly used in wireless sensornetworks.For the AOA based technique, the distance between nodes

is reconstructed from the angle between them. The majordisadvantage of this technique is that it requires new hardwarei.e., adoption of antenna arrays and a minimum distancebetween the antenna elements which results additional costsand larger node sizes. Furthermore, this technique is highlysensitive to multipath, NLOS conditions, and array precision.For the RSS based technique, the distance is measured

based on the attenuation introduced by the propagation of thesignal from TX to RX. Since the relation between distanceand attenuation depends on channel behavior, an accuratepropagation model is required for reliable distance estimation.The main advantage of this technique is its low cost andthe fact that most RXs are capable of estimating the RSS.However, the MT mobility and unpredictable variations inchannel behavior can occasionally lead to large errors in

Fig. 1. Illustration of a simple scenario for wireless localization.

distance evaluation. Also, this technique is very susceptibleto noise and interference. Thus, the RSS technique is not anaccurate method, and its adoption is confined to applicationsthat require coarse estimation.Finally, for pattern-matching based techniques, fingerprint

information of the measured radio signals at different geo-graphical locations are utilized for position estimation. Suchinformation should be location-sensitive and can be collectedduring a training (off-line) phase in a database. During thereal-time (on-line) phase, the fingerprint information can beused to locate the mobile node. A fingerprint database can besimply composed of received signal strengths from/at differentreference nodes and at different mobile locations. However,it may as well capture more detailed fingerprint information,such as the mean excess delay, root mean square (RMS) delayspread, maximum excess delay, total received power, numberof multipath components etc. A challenge with such systemsis that due to changes in the channel and environment, thefingerprint database may become unreliable, and may need tobe updated frequently.

III. SYSTEM MODEL FOR TOA-BASED LOCATIONESTIMATION

Consider a wireless network as in Fig. 1 where there areN fixed terminals (FTs)1, x = [x y]T is the estimate of themobile terminal (MT) location, xi = [xi yi]T is the positionof the ith FT, di is the measured distance between the MTand the ith FT commonly modeled as

di = di + bi + ni = cti, i = 1, 2, ..., N , (1)

where ti is the TOA of the signal at the ith FT, c is the speedof light, di is the real distance between the MT and the ith FT,ni ∼ N (

0, σ2i

)is the additive white Gaussian noise (AWGN)

with variance σ2i , and bi is a positive distance bias introduced

1FT is usually a base station in cellular networks, an anchor node in sensornetworks, or an access point in wireless local area networks.

GUVENC and CHONG: A SURVEY ON TOA BASED WIRELESS LOCALIZATION AND NLOS MITIGATION TECHNIQUES 109

TABLE IOVERVIEW OF DIFFERENT LOCALIZATION ALGORITHMS.

LocalizationTechnique

Summary and Characteristics Strength and Weakness Usage and Applicability

TOA Uses distance information between FT andMT.

One-way ranging requires perfect synchro-nization, while two-way ranging does not.

More common in cellular networks.

TDOA Difference between TOAs in several FTs areutilized.

Needs highly precise synchronization be-tween MTs, while not precise synchroniza-tion between FTs and MTs.

More common in wireless sensor networks.

AOA Uses the angle information to construct thelines between MT and FTs and use theirintersection to find MT location

Requires new hardware (antenna arrays).This means additional costs and larger nodesizes.

More appropriate for FTs rather than MTsdue to large size. Otherwise MT size has tobe able to accommodate an antenna array.

RSS Distance is estimated based on the attenua-tion introduced by propagation of the signalfrom FT to MT.

An accurate propagation model is neededfor reliable distance estimation. It is lowcost due to most RX being able to estimateRSS. MT mobility and channel variationmay yield large errors.

Since it has low-precision characteristic, typ-ically used in applications which requirecoarse estimate.

PatternMatching

Fingerprint information of measured radiosignal at different geographical locations areutilized.

Needs an off-line training stage to obtain adatabase. Also, this database may be unreli-able if the channel and environment changeswith time.

Mostly used in wireless local area networkswith RSS as the metric used in the database.Also considered for cellular systems.

due to the blockage of direct path given by

bi =

{0 , if ith FT is LOS ,

ψi , if ith FT is NLOS .(2)

For NLOS FTs, the bias term ψi was modeled in dif-ferent ways in the literature such as exponentially dis-tributed [18], [19], uniformly distributed [20], [21], Gaussiandistributed [22], constant along a time window [23], or basedon an empirical model from measurements [24], [25]. Typi-cally, the model depends on the wireless propagation channeland the specific technology under consideration (e.g., cellularnetworks, wireless sensor networks, etc.).Let

d = d(x) = [d1, d2, ..., dN ]T , (3)

be a vector of actual distances between the MT and the FTs,

d = [d1, d2, ..., dN ]T , (4)

be a vector of measured distances, and

b = [b1, b2, ..., bN ]T , (5)

be a bias vector. Also let

Q = E[nnT ] = diag[σ21 , σ

22 , ..., σ

2N ]T , (6)

to denote the covariance of noise vector n = [n1, n2, ..., nN ]T

with the assumption that all the noise terms are zero mean andindependent Gaussian random variables.In the absence of noise and NLOS bias, the true distance

di between the MT and the ith FT defines a circle around theith FT corresponding to possible MT locations

(x− xi)2 + (y − yi)2 = d2i , i = 1, 2, ..., N , (7)

where all the circles intersect at the same point, and solvingthese expressions jointly gives the exact MT location. How-ever, the noisy measurements and NLOS bias at different FTsyield circles which do not intersect at the same point (see Fig.1), resulting in the inconsistent equations as follows

(x− xi)2 + (y − yi)2 = d2i , i = 1, 2, ..., N . (8)

In order to have more compact expressions throughout thepaper, we further define the following terms

s = x2 + y2 , ki = x2i + y2

i . (9)

The problem of TOA-based location estimation can bedefined as the estimation of the MT’s location x from the noisy(and possibly biased) distance measurements in (4) given theFT locations xi; in other words, given the set of equationsin (8). Various localization techniques were proposed in theliterature in order to estimate the MT location from (8) inLOS and NLOS scenarios. In the following sections, we willreview these algorithms and discuss their trade-offs.

IV. LOS SCENARIOS: FUNDAMENTAL LIMITS AND MLSOLUTIONS

In this section, we will overview the fundamental lowerbounds and ML type of algorithms for LOS scenarios (i.e.,bi = 0 for all i). First, we define below the ML algorithmthat maximizes the conditional probability of the measureddistances d. Then, the Cramer-Rao lower bound (CRLB) willbe derived in the following section. Two other sub-optimumML type of algorithms asymptotically achieving the CRLBwill also be described.

A. Maximum Likelihood Algorithm

In the absence of NLOS bias (i.e., bi = 0 for all i), theconditional probability density function (PDF) of d in (4)given x can be expressed as follows [26], [27]

P (d|x) =N∏

i=1

1√2πσ2

i

exp{− (di − di)2

2σ2i

}(10)

=1√

(2π)Ndet(Q)exp

{− J

2

}, (11)

where

J =[d− d(x)

]T

Q−1[d− d(x)

], (12)


with Q as given in (6). Then, the ML solution for x is theone that maximizes P (d|x), i.e.,

x = arg maxx

P (d|x) . (13)

Note that solving for x from (13) requires a search overpossible MT locations which is computationally intensive.For the special case of σ2

i = σ2 for all i, the ML solutionin (13) is equivalent to minimizing J. In order to find theminimum value of J, the gradient of J with respect to x isequated to zero, yielding [26]

N∑i=1

(di − di)(x− xi)di

= 0 , (14)

N∑i=1

(di − di)(y − yi)di

= 0 , (15)

which are non-linear equations. Hence, x can not be solvedin closed form from (14) and (15) using a linear least squares(LS) algorithm. Also, both (14) and (15) depend on di, whichare unknown. Even though a closed form ML solution isnot possible, approximate and iterative ML techniques can bederived as will be discussed in Section IV-C and Section IV-D,which may asymptotically achieve the CRLB.

B. Cramer-Rao Lower Bounds

Given the conditional PDF of d as in (10), we may derivethe CRLB for TOA based location estimation. The CRLB, ingeneral, can be defined as the theoretical lower bound on thevariance of any unbiased estimator of an unknown parameter.The CRLB for the TOA based location estimation mainlydepends on the following factors:

• Positions of the FTs (xi),• True position of the MT (x), and• Measurement noise variances (σ2

i ).The CRLB is calculated using the Fisher information matrix

(FIM), whose elements are defined as

[I(x)]ij = −E[∂2lnP (d|x)

∂xi∂xj

]. (16)

Then, using the PDF given in (10), the FIM can be calculatedas [27], [28]

I(x) =

⎡⎢⎣

∑Ni=1

(x−xi)2

σ2i d2

i

∑Ni=1

(x−xi)(y−yi)σ2

i d2i∑N

i=1(x−xi)(y−yi)

σ2i d2

i

∑Ni=1

(y−yi)2

σ2i d2

i

⎤⎥⎦ ,

(17)

=

⎡⎢⎣

∑Ni=1

cos2(αi)σ2

i

∑Ni=1

cos(αi)sin(αi)σ2

i∑Ni=1

cos(αi)sin(αi)σ2

i

∑Ni=1

sin2(αi)σ2

i

⎤⎥⎦ ,

(18)

where αi defines the angle from the ith FT to the MT, andthe CRLB is given by I−1(x). Thus, for an estimate x of theMT location obtained with any unbiased estimator, we have

Ex

[(x − x)(x − x)T

] ≥ I−1(x) . (19)

The CRLB can be related to another important measurementmetric referred to as the geometric dilution of precision

5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

CRLB

x (meter)

y (m

eter

)

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

(meter)

Fig. 2. The CRLB for a simple localization scenario where there are fourFTs at the corners of a square room of size 40 × 40 meters.

(GDOP) in the literature. For identical noise variances σ2i =

σ2 at different FTs, the GDOP can be defined as

GDOP =RMSEloc

RMSErange=σloc

σ, (20)

where RMSEloc and RMSErange are the root mean squareerror (RMSE) of the location estimate and the range estimate,respectively, and σloc is the standard deviation of the locationestimate. GDOP depends highly on the positions of the FTsand the MT location. While GDOP values smaller than threeare usually preferable, values larger than six may imply a verybad geometry of the FTs. If the employed location estimatorcan achieve the CRLB, the GDOP is given by

GDOP =

√trace

[I−1(x)

]σ

, (21)

=√

trace[I−1(x)

], (22)

where

I(x) =

[ ∑Ni=1 cos2(αi)

∑Ni=1 cos(αi)sin(αi)∑N

i=1 cos(αi)sin(αi)∑N

i=1 sin2(αi)

].

(23)

The relation between the achievable localization accuracy andthe geometry between the locations of the MT and the FTs isapparent from (22).1) Simulation Results: The CRLBs for a simple wireless

localization scenario at different MT locations is illustrated inFig. 2. Four FTs are positioned at [0, 0] m, [0, 20] m, [20, 0] m,and [20, 20] m and we have σ2 = 0.5 for all the FTs. TheCRLB becomes lower when the MT is closer to the center ofthe room. Also, at four specific MT locations, the FIM in (16)becomes singular and does not have a matrix inverse, whichexplains the white spots in Fig. 2.In order to see the typical values that GDOP may take

in wireless localization, some example node topologies aresimulated in Fig. 3. Three MT locations are considered,namely [5, 5]m, [25,−25]m, and [−50, 50]m. FT-1 location is


−50 −40 −30 −20 −10 0 10 20 30 40 50−50

−40

−30

−20

−10

0

10

20

30

40

50

meters

met

ers

MT LocationsFT−1 Location

PlacenewFTs

(25,−25)

(5,5)

(−50,50)

2π/Nmax

(a) Topology for GDOP simulations.

3 4 5 6 7 8 9 10

100

101

N: Number of FTs

GD

OP

[5,5], T1[25,−25], T1[−50,50], T1[5,5], T2[25,−25], T2[−50,50], T2[5,5], T3[25,−25], T3[−50,50], T3

(b) GDOP versus the number of FTs for different topologies T1, T2, andT3.

Fig. 3. GDOPs for different topologies and MT locations.

fixed to [20√

2, 0] m and new FTs are added counter-clockwisearound the illustrated circle. Three topologies are considered,namely, T1, T2, and T3, and for each topology, a new FT isplaced at an increment of 2π/Nmax, π/Nmax, and π/2Nmax

radians, respectively, where Nmax = 10 denotes the maximumnumber of FTs. The GDOP is less than two for an MT locatedat [5, 5] m for all the topologies, and becomes better as moreFTs are deployed. For an MT located at [−50, 50] m, GDOPis worst, and it may be as large as 10 for T3. The results showthat as we increase the number of FTs, it is possible to haveGDOP values smaller than 1, which implies that the standarddeviation of the location estimate becomes smaller than thestandard deviation of the distance measurements.

C. Two-Step ML Algorithm

While the CRLB gives a lower bound on the best achievableaccuracy, it may practically be difficult to approach it inrealistic scenarios. One of the earlier TOA based techniques

in the literature that approaches the CRLB under certainconditions is introduced in [29]. It is a two-step ML algorithm,where the ML solution for the MT location estimate can beobtained as

θ =12(AT

1 Ψ−1A1)−1AT1 Ψ−1p1 , (24)

where A1 and p1 are as defined in (35), and

θ = [x, y, s]T , (25)

Ψ = E[ψψT ] = BQB , (26)

ψ = p1 − A1θ , (27)

B = diag{2d1, 2d2, ..., 2dN} . (28)

Since the elements of B are unknown distances, an approxi-mate solution is obtained by using the measurements di insteadof actual distances di in B for obtaining an initial solution.Then, a more accurate solution is obtained using this initialsolution to re-calculate B, and few iterations are shown to besufficient for convergence.

D. Approximate ML Algorithm

Another ML based technique that approaches to the CRLBis proposed in [26]. It was shown in Section IV-A that thesolution of the MT location using the ML method requires theknowledge of true distances di. With the assumption that σ2

i =σ2 ∀i, [26] proposes an approximate ML (AML) solution thatachieves the CRLB in most scenarios. They use the identity

di − di =d2

i − d2i

di + di

, (29)

in solving (14) and (15) and obtain the following matrixequation [26]

2

[ ∑gixi

∑giyi∑

hixi

∑hiyi

] [x

y

]=

[ ∑gi(s+ ki − d2

i )∑hi(s+ ki − d2

i )

],

(30)

where s and ki are as in (9) and all the summations are from1 to N , with

gi =x− xi

di(di + di), hi =

y − yi

di(di + di). (31)

Note that other than s, the weights gi and hi also dependon x in the above relations, which is (obviously) unknown.Hence, the AML technique first obtains a rough initial estimate(e.g., using the estimator in (37)) of x to compute gi and hi.Then, (30) is solved using a LS algorithm and x is obtainedin terms of s, which results in a quadratic expression. A rootselection routine is then used to select the appropriate solutionfor x. The new solution is re-used to compute new gi and hi,and few iterations of the algorithm can achieve results closeto the CRLB.

V. LOS SCENARIOS: LEAST-SQUARES TECHNIQUES

In this section, first, we will review non-linear least squares(NLS) techniques for estimating the position of an MT. Then,some linearization techniques will be briefly discussed.


A. Non-Linear Least Squares

The NLS is a well known technique for the estimationof an unknown parameter when its probability distribution isnot known. It is also one of the common techniques for theestimation of an MT’s location, which is given by [30]

x = arg minx

{Res(x)

}(32)

= arg minx

{N∑

i=1

βi

(di − ||x − xi||

)2}, (33)

where Res(x) is the residual error corresponding to MTlocation x. Some weights βi can be used to characterize thereliability of each link, which yields a weighted least squares(WLS) solution. If no reliability information is available,βi = 1 for all i. Minimizing the non-linear expressionin (33) requires numerical search methods such as the steepestdescent [31] or the Gauss-Newton techniques [2], which maybe computationally costly and require good initialization inorder to avoid converging to the local minima of the lossfunction [2].

B. Matrix Representation of the Non-Linear Model

We may represent the non-linear expressions in (8) in matrixform. After some manipulation, we may write them as [32]

A1θ =12p1 , (34)

where

A1 =

⎡⎢⎢⎢⎢⎢⎢⎣

x1 y1 −0.5

x2 y2 −0.5

......

xN yN −0.5

⎤⎥⎥⎥⎥⎥⎥⎦,

θ =

⎡⎢⎢⎣x

y

s

⎤⎥⎥⎦ ,p1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

k1 − d21

k2 − d22

...

kn − d2N

⎤⎥⎥⎥⎥⎥⎥⎥⎦. (35)

with s = x2 + y2 being a part of the vector of unknownvariables. In Section V-E, we will show how it can be usedas a constraint to solve for x.After some further mathematical manipulation

of (34) and (35), we may obtain an alternative LS solution asfollows

A2x =12p2 , x =

12(AT

2 A2)−1AT2 p2 , (36)

where

A2 =

[x1 x2 ... xN

y1 y2 ... yN

]T

, p2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

s+ k1 − d21

s+ k2 − d22

...

s+ kn − d2N

⎤⎥⎥⎥⎥⎥⎥⎥⎦.

(37)

Note that the above LS solution obtains x in terms of s,which results in a quadratic expression. Hence, a root-selectionmethod can be employed to find x [26]. However, due to theinconsistency of equations, (36) is inaccurate. Nevertheless,it may be used as an initial location estimate to enhance thelocalization performance of more accurate (yet more complex)algorithms as will be discussed in the later sections.

C. Linearization of NLS Solution Through Taylor’s SeriesExpansion

The non-linear function d(x) in (3) can be linearized arounda reference point x0 using Taylor series expansion. If thehigher order terms are neglected, we have [33]

d(x) ≈ d(x0) + H0(x − x0) , (38)

where the Jacobian matrix of d(x) around x0 is given by

H0 =

[ ∂d1∂x

∂d2∂x . . . ∂dN

∂x

∂d1∂y

∂d2∂y . . . ∂dN

∂y

]T

x=x0

. (39)

Note that the reference point x0 should be chosen sufficientlyclose to the true location in order for (38) to be valid. Bysubstituting (38) into (33), we have a linear system whichcan be written in a matrix form and solved using a linearLS estimator. A more accurate iterative technique may usethis LS estimate as an intermediate estimate, plug it into (38)to re-linearize the system around it, and iterate until conver-gence [31].

D. An Alternative Linear Least Squares Solution

The non-linear model discussed in previous sections con-tains the parameter s which is quadratic in x and y. In order toobtain a linear model, an alternative technique was proposedin [34] for canceling out these non-linear terms. By fixingexpressions for the rth FT in (8), subtracting it from the restof the equations for i = 1, 2, ..., N (i �= r), and re-arrangingthe terms, we have the following linear model

A3x =12p3 , (40)

with

A3 =

⎡⎢⎢⎢⎢⎢⎢⎣

x1 − xr y1 − yr

x2 − xr y2 − yr

......

xN − xr yN − yr

⎤⎥⎥⎥⎥⎥⎥⎦,p3 =

⎡⎢⎢⎢⎢⎢⎢⎣

d2r − d2

1 − kr,1

d2r − d2

2 − kr,2

...

d2r − d2

N − kr,N

⎤⎥⎥⎥⎥⎥⎥⎦,

(41)

where kr,i = kr − ki, and r is the reference FT that is usedto obtain the linear model. Note that the non-linear terms x2

and y2 in p2 of (36) are canceled out in p3. From the aboveexpressions, the LS solution for x can be written as (call itLLS-1)

x =12(AT

3 A3)−1AT3 p3 . (42)

Note that while (8) defines a circle around each FT, the x2 andy2 terms are canceled in (40), resulting in linear expressions


Fig. 4. The illustration of circles and lines for the non-linear and linearmodels, respectively.

that can be seen as the lines connecting the intersection points(if any) of the pairs of circles. An illustration of the geometricrepresentations of the non-linear and linear models is given inFig. 4.In order to get more insight about the accuracy of LLS-1

estimator, it is worth to analyze the perturbation in the vectorp3. By replacing (1) in p3 and assuming that bias terms arezero, we have

p3 = p(c)3 + p(n)

3 , (43)

where the constant and noisy components are given by

p(c)3 =

⎡⎢⎢⎢⎢⎢⎢⎣

d2r − d2

1 − kr + k1

d2r − d2

2 − kr + k2

...

d2r − d2

N − kr + kN

⎤⎥⎥⎥⎥⎥⎥⎦, (44)

p(n)3 =

⎡⎢⎢⎢⎢⎢⎣

2drnr − 2d1n1 + n2r − n2

1

2drnr − 2d2n2 + n2r − n2

2

...2drnr − 2dNnN + n2

r − n2N

⎤⎥⎥⎥⎥⎥⎦ , (45)

Note that while we got rid of the quadratic s term, the numberof noisy terms in p(n)

3 increased. In particular, we may claimthat the accuracy of the above algorithm will degrade as theMT moves away from rth FT due to the distance-dependentnoise terms at each link. If p(n)

3 → 0, all the lines in Fig. 4will intersect at a single point. For theoretical derivation ofthe MSE for LLS-1, the reader is referred to [35].1) Averaging Techniques: The LLS-1 estimator for (40)

utilizes the measurements di, i = 1, . . . , N , only through theterms d2

r − d2i , for i = 1, . . . , N and i �= r. Therefore, the

measurement set for LLS-1 effectively becomes

di = d2r − d2

i , i = 1, . . . , N, i �= r . (46)

Note that since the effective measurements in (46) becomedifferent than the measurements in (1), as also implied byFig. 4, the corresponding CRLB will be different, which arederived in [36].In another LLS approach proposed in [20] (call it LLS-2),

N × (N − 1)/2 linear equations are obtained by subtractingeach individual equation from all of the other equations. Inother words, in the LLS-2 technique, the following observa-tions are employed for position estimation:

dij = d2i − d2

j , i, j = 1, 2, . . . , N, i < j . (47)

Similar to the LLS-1, the linear LS solution as in (42) is usedin order to obtain the position of the target node in the LLS-2technique.In a third LLS technique proposed in [37] (call it LLS-3),

instead of obtaining the difference of the equations directlyas in the LLS-1 and LLS-2 approaches, the average of themeasurements is obtained first, and this average is subtractedfrom all the equations resulting in N linear relations. Then,the linear LS solution as in (42) is obtained for the position ofthe target node. The observation set employed in the LLS-3technique can be expressed as

di = d2i −

1N

N∑j=1

d2j , i = 1, 2, . . . , N . (48)

2) Reference FT Selection: The LLS-1 solution in (40)selects an arbitrary FT as the reference FT. However, observ-ing the noisy terms in p(n)

3 given in (45), all the rows of thevector p(n)

3 depend on the true distance between the MT andthe reference FT. If the FT is away from the MT location, thisimplies that all the elements of vector p3 will be more noisy,degrading the localization accuracy. Hence, how the referenceFT is selected may considerably affect the estimator’s meansquare error (MSE). A simple method to select the referenceFT for improved location accuracy in LOS scenarios is tochoose it so that its measured distance is the smallest amongall the distance measurements. The index of the reference FTthat has the smallest measured distance is given by [38]

r = arg mini{di} , i = 1, 2, . . . , N . (49)

Then, the matrix A3 and the vector p3 can be obtained usingthe selected reference FT (FT-r), and we refer the resultingestimator as LLS with reference selection (LLS-RS). Forexample, in Fig. 1, FT-1 is used to obtain the linear modelfrom non-linear expressions since d1 is the minimum amongall the measured distances.3) Utilizing the Covariance Matrix: While the reference

FT selection discussed above improves the location accuracy,it does not account for the correlation between the rows of thevector p(n)

3 , which become correlated during the linearizationprocess. As discussed in [39], the optimum estimator inthe presence of correlated observations is given by an MLestimator. First, consider the following modification of therelationship in (43) for a LOS scenario

p3 = A3x + p(n)3 , (50)


where x is the actual location of the MT, and hence p(c)3 =

A3x. Then, based on (50), the MLE2 for this linear modelcan be written as [39]

x = (AT3 C−1A3)−1AT

3 C−1p3 , (51)

whereC = Cov(p(n)3 ) is the covariance matrix of vector p(n)

3 .When all the FTs are in LOS, the covariance matrix of vectorpn can be derived as

C = 4d2rσ

2 + 2σ4 + diag{4σ2d2

1 + 2σ4, ...,

4σ2d2i + 2σ4, ..., 4σ2d2

N + 2σ4}, (52)

with i ∈ {1, 2, ..., N}, i �= r, and where diag{λ1, ..., λN} is adiagonal matrix obtained by placing λi on the diagonal of an(N − 1)× (N − 1) zero matrix ∀i. Note that since di are notavailable in practice, the noisy measurements di can be usedto evaluate the covariance matrix.4) Simulation Results: Monte-Carlo simulations are per-

formed in order to compare the different LLS estimators. Asin Fig. 2, four FTs are positioned on the corners of a squareroom. The simulation results in Fig. 5(a) show the 2-D MSEfor LLS-1 technique, where the FT-1 is used as a referenceFT. We observe that the MSE tends to be smaller when theMT is closer to FT-1. For the simulation results in Fig. 5(b),the MT location x is changed with 10 meter intervals within[−40, 40] m both in x and y directions, yielding a 9×9 grid ofpossible MT locations. The MSE of different techniques aresimulated at each location on the grid, and then averaged overall the MT locations on the grid. The results show that theLLS-1 performs worst compared to all the other techniques.The LLS-2 and LLS-3 techniques perform slightly better thanLLS-1, and their MSEs are identical. However, they are bothbeaten by the LLS-RS technique. The MLE performs slightlybetter than that of LLS-RS and very close to the CRLB.

E. Constrained Weighted Least Squares

A constrained weighted least squares (CWLS) approachwas presented in [32] which operates on (34) to find the MTlocation3. More specifically, it uses the relationship betweens and x as a constraint on the LS solution, and developsa solution based on a Lagrange multiplier. The constraintoptimization problem is formulated as [32]

θcw = arg minθ

(A1θ − p1)T W(A1θ − p1) , (53)

subject to the constraint s = x2 + y2, i.e.,

qT θ + θT Pθ = 0 , (54)

where A1, θ, and p1 are as in (35), and

P =

⎡⎢⎢⎣

1 0 0

0 1 0

0 0 0

⎤⎥⎥⎦ , and q =

⎡⎢⎢⎣

0

0

−1

⎤⎥⎥⎦ . (55)

2Note that in order to have the MLE as in (51), the elements of pn

should be zero-mean and Gaussian distributed random variables. While thereare some non-Gaussian terms (i.e., the noise-square terms) in pn, they areassumed to be negligible, or fit closely to a Gaussian distribution to obtainthe MLE.3For a more detailed discussion on different constrained LS algorithms for

AOA, RSS, TDOA, and hybrid techniques, the reader is referred to [40].

In order to determine the optimal weighting matrix W, theauthors examine the disturbance in p1. At high signal-to-noiseratios (SNRs), we have [32]

d2i = (d+ ni)2 ≈ d2

i + 2dini for i = 1, 2, ..., N , (56)

which implies a disturbance εi = d2i − d2

i =2dini, and can be represented in vector form as ε =[2d1n1, 2d2n2, ..., 2dNnN ]T . The covariance matrix of thedisturbance is given by [32]

Ψ = E[εεT ] = BQB , (57)

with B = diag(2d1, 2d2, ..., 2dN ), and the optimal weightingmatrix becomes W = Ψ−1. Note that it depends on theactual distances {di} between the MT and the FTs whichare unknown, and an approximate weighting matrix can beobtained using B = diag(2d1, 2d2, ..., 2dN ), instead of B.The CWLS problem in (53) and (54) can then be solved by

minimizing the Lagrangian as follows [32]

L(θ, λ) = (A1θ − p1)T Ψ−1(A1θ − p1) + λ(qT θ + θT Pθ) ,(58)

where λ is a Lagrange multiplier. It was shown in [32] thateither a global or a local solution to (58) is given by

θcw = (AT1 Ψ−1A1 + λP)−1

(AT

1 Ψ−1p1 − λ

2q), (59)

with λ being determined from a five-root equation.In another related work, the authors propose a covariance

shaping LS (CSLS) technique for location estimation, whichyields good performance compared to other LS estimators atlow SNRs [41].

VI. NLOS SCENARIOS: FUNDAMENTAL LIMITS AND MLSOLUTIONS

In typical environments, especially in indoor scenarios, itmay be possible that the LOS between the MT and someof the FTs may be obstructed (i.e., bi > 0 for some i).These NLOS FTs may seriously degrade the localizationaccuracy. Simplest way of NLOS mitigation is achieved byidentifying and discarding the NLOS FTs, and estimating theMT location by using one of the LOS techniques discussed inthe previous section. However, there is always the possibilityof false-alarms (identifying a LOS FT as NLOS) and missed-detections (identifying an NLOS FT as LOS) which degradethe localization accuracy. In this section, we review alternativeNLOS mitigation techniques reported in the literature. Firstly,the ML based techniques and the CRLBs in NLOS scenarioswill be discussed.

A. ML Based Algorithms

ML approaches for NLOS mitigation were discussedin [19], [23], which require prior knowledge regarding thedistribution of NLOS bias. For example, [19] provides an MLsolution for the position of the MT with the assumption thatthe NLOS bias bi is exponentially distributed with parameterλi. Since in most cases NLOS bias is much larger than


5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40

Linear LS Estimator MSE (Sim)

x (meter)

y (m

eter

)

0.5

1

1.5

2

2.5

3

3.5

(meter)

(a) 2-D MSE of the LLS-1 in a LOS scenario.

0.5 1 1.50.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Ave

reag

e M

SE

(m

2 )

Noise variance (m2)

LLS−1LLS−RSMLELLS−2LLS−3Original CRLB

(b) Simulation results for different linear LS estimators in LOS scenario.

Fig. 5. Comparison of the MSEs of different linear LS location estimation techniques averaged over a grid.

the Gaussian measurement error, the following simplifyingassumption was made

di = di +

{bi , i = 1, 2, ..., NNL ,

ni , i = NNL + 1, ..., N,(60)

where NNL is the number of NLOS FTs, and NL = N −NNL denotes the number of LOS FTs. Thus, a simplified MLsolution is given as follows [19]

x = arg minx

{NNL∑i=1

λi(di − di) +N∑

i=NNL+1

(di − di)2

2σ2i

}.

(61)

It is also possible to obtain the exact decision rule by con-sidering the summation of Gaussian and exponential randomvariables, which has the following probability density function

P (x) = λ exp(− λ

(x− λσ2/2

))Q(λσ − x

σ

), (62)

where Q(α) = 1√2π

∫∞α exp(−x2/2)dx denotes the Q-

function, and the exact ML solution becomes [19]

x = arg minx

{NNL∑i=1

λi

(di − di − λiσ

2i /2

)

−NNL∑i=1

log[Q

(λiσi − di − di

σi

)]+

N∑i=NNL+1

(di − di)2

2σ2i

}.

(63)

Another NLOS mitigation technique for TOA based systemsbased on the ML approach is introduced in [23]. The authorsconsider several hypothesis for different sets of FTs, and then,utilizing the ML principle, the best set (that is assumed to becomposed of LOS FTs) is selected for location estimation.The hypothesis index estimate for the best FT set is derivedas [23]

i = arg mini

[lnγ−1(i) +

∑k∈SLOS

i

Ntrn

2σ2

k(i)σ2

k

], (64)

where SLOSi denotes the ith set of MTs which are hypothe-

sized to be LOS, and γ(i) are assigned according to the a-priori probability of each hypothesis (equivalent to 1 if noinformation available). Also,

σ2k(i) =

1Ntrn

Ntrn∑m=1

(tk,m − ||x(i) − xk||)2 , (65)

denotes the estimated variance of theNtrn TOA measurementstied with the kth FT under the ith hypothesis, and tk,m denotesthe mth TOA measurement at the kth FT. Note that the aboveapproach requires buffering of Ntrn TOA measurements forthe purpose of obtaining the noise statistics at a particular FT.Once the set of LOS FTs is selected using the ML principle,the MT location is estimated using only these FTs and theML algorithm. Simulation results in [23] show that this yieldsbetter accuracy than residual weighting (Rwgh) techniqueintroduced in [18] (to be discussed in Section VII-B), andslightly worse than when only the true LOS FTs are used inlocalization. Also, at low SNR and small NLOS bias values,simulation results imply it may be better not to employ anyNLOS mitigation in order not to degrade the accuracy.

B. Cramer-Rao Lower Bound

In NLOS scenarios, the CRLB depends on if there isany prior information available about the NLOS bias. Firstconsider that there is no prior information about the NLOSbias, except that only the NLOS FTs are assumed to beperfectly known. Then, an extended version of the FIM in (19)is given by [42]

I(xb) = AI(d)AT , (66)

where

xb = [x, y, b1, b2, ...bNNL]T , (67)


is an (NNL +2)×1 vector of unknown parameters (includingthe NLOS bias values) with

A =∂d∂xb

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

∂d1∂x

∂d2∂x . . .

∂dNNL

∂x . . . ∂dN

∂x∂d1∂y

∂d2∂y . . .

∂dNNL

∂y . . . ∂dN

∂y∂d1∂b1

∂d2∂b1

. . .∂dNNL

∂b1. . . ∂dN

∂b1...

.... . .

.... . .

...∂d1

∂bNNL

∂d2∂bNNL

. . .∂dNNL

∂bNNL. . . ∂dN

∂bNNL

⎤⎥⎥⎥⎥⎥⎥⎥⎦,

(68)

and

I(d) = Ed

[∂lnf(d|d)

∂d

(∂lnf(d|d)

∂d

)T ]. (69)

As discussed in [42], A can be written in terms of its LOSand NLOS components as

A =[

ANL AL

INNL 0NNL,NL

], (70)

where INNL and 0NNL,NL are an identity matrix of sizeNNL×NNL and a zero matrix of size NNL×NL, respectively,and

ANL =[

cosα1 cosα2 . . . cosαNNL

sinα1 sinα2 . . . sinαNNL

], (71)

AL =[

cosαNNL+1 cosαNNL+2 . . . cosαN

sinαNNL+1 sinαNNL+2 . . . sinαN

].

(72)

Similarly, I(d) can be written in terms of its NLOS andLOS components as [42]

I(d) =[

ΛNL 00 ΛL

], (73)

where ΛNL = diag(σ−21 , ..., σ−2

NNL) and ΛL =

diag(σ−2NNL+1, ..., σ

−2NN

). After some manipulation, I(xb)can be obtained as [42]

I(xb) =[

ANLΛNLATNL + ALΛLAT

L ANLΛNL

ΛNLATNL ΛNL

].

(74)

Note that (74) depends both on NLOS and LOS signals.However, it was further proven in [42] that the CRLB forthe MT location is given by

E[(xb − xb)(xb − xb)T

]≥ I−1(xb) =

(ALΛLAT

L

)−1

.

(75)In other words, the CRLB exclusively depends on theLOS signals if the NLOS FTs can be accurately identified.Hence, the ML estimator that can achieve the CRLB inNLOS scenarios first identifies the NLOS FTs, discards thesemeasurements, and then obtains the location estimate usingthe LOS FTs, as illustrated in Fig. 6(a).If there is further side information related to the statistics

of the NLOS bias vector b, a better positioning accuracy can

Fig. 6. Illustration of block diagrams for (a) ML estimator and (b) MAPestimator in NLOS scenarios. In part (a), without loss of generality, it isassumed that the first NL measurements are the LOS measurements.

be obtained. Then, the generalized CRLB (G-CRLB) can bewritten as [42]

E[(xb − xb)(xb − xb)T

]≥

(I(xb) +

[0 00 Ω

] )−1

(76)

=

([ANLΛNLAT

NL + ALΛLATL ANLΛNL

ΛNLATNL ΛNL + Ω

])−1

,

(77)

where Ω = diag(σ21 , ..., σ

2NNL

), and σ2i can be interpreted

as the variance4 of bi. As an upper bound on the G-CRLB,when the variances σ2

i are infinitely large, Ω → 0, and G-CRLB is reduced to the CRLB (since there is practically noinformation available on bi). The estimator that asymptoticallyachieves the G-CRLB is given by the maximum a-posteriori(MAP) estimator, and it employs the statistics of the NLOSbiases as illustrated in Fig. 6(b).

VII. NLOS SCENARIOS: LEAST SQUARES TECHNIQUES

The LS techniques for location estimation can be tuned tosuppress the NLOS bias effects, e.g., through some appropriateweighting. In this section, weighted least squares approachesas well as the residual weighting algorithm will be brieflyreviewed.

A. Weighted Least Squares

A simple way to mitigate the effects of NLOS FTs is togive less emphasis to corresponding NLOS terms in the LSsolution. In [19], [30], with the assumption that the variancesof the distance measurements are larger for NLOS FTs, the

4For Gaussian distributed NLOS bias, it is strictly the variance of the NLOSbias.


TABLE IISTEPS OF THE RESIDUAL WEIGHTING ALGORITHM IN [18].

1) Form Ncb =PN

i=3 N Ci range measurement combinations, whereN Ci denotes the total number of combinations with i FTs selectedfrom a total of N FTs. Also let {Sk|k = 1, 2, ..., Ncb} denote the setof FTs for the kth combination.

2) For each set of combinations Sk , compute an intermediate LS locationestimate as follows

xk = arg minx

nRes(x; Sk)

o, (83)

where Res(x; Sk) is the residual error when only the FTs in set Sk

are used for calculating the MT location. Also define the normalizedresidual

Res(xk ;Sk) = Res(xk ;Sk)/|Sk | , (84)

where |Sk| denotes the size of Sk .3) Find the final location estimate by weighting the intermediate location

estimates with their corresponding normalized residual errors:

x =

PNcbk=1 xk

hRes(xk;Sk)

i−1

PNcbk=1

hRes(xk ;Sk)

i−1. (85)

inverses of these variances are used as a reliability metricβi in (33). This is actually derived from the ML solution asfollows. For the ML algorithm, the location estimate is givenby

xML = arg maxx

p(d|x) , (78)

where

p(d|x) = pn(d − d|x) . (79)

If the noise is Gaussian distributed, we have

pn(n) =1√

2πσi

exp(− n2

2σ2i

). (80)

Then, the joint probability function becomes

p(d|x) =1

(2π)N/2∏N

i=1 σi

exp

(−

N∑i=1

(di − ||x − xi||)22σ2

i

).

(81)

Upon further manipulation of (81), the ML solution becomesequivalent to

xML = arg minx

N∑i=1

(di − ||x − xi||)2σ2

i

, (82)

which is equivalent to the WLS solution for βi = 1/σ2i .

However, for a static MT, the variance of TOA mea-surements may not be significantly different for LOS andNLOS FTs. Still, the bias in NLOS distance measurementsmay degrade the localization accuracy. Hence, in [43], [44]an alternative weighting technique is proposed, which usescertain statistics of the multipath components of the receivedsignals. In particular, kurtosis, mean excess delay, and root-mean-square (RMS) delay spread of the received signal areused to evaluate the likelihood value of the received signal tobe LOS. The likelihood values are then used to evaluate theweighting parameters βi.

B. Residual Weighting Algorithm

The Rwgh algorithm proposed in [18] is based on theobservation that the residual error Res(x) is typically largerif NLOS FTs are used when estimating the MT location. Byassuming that there are more than three FTs available, theRwgh estimates the MT location as detailed in Table II.It was shown in [18] through simulations that Rwgh per-

forms better than choosing the location estimate with theminimum residual error (MRE). A sub-optimal version ofRwgh algorithm that has lower computational complexity wasproposed in [45]. In that paper, instead of considering allthe combinations of the FTs (which may be very large ifN is large), first, all the combinations with (N − 1) FTsare considered in order to calculate the intermediate locationestimates and the corresponding residuals. Then, among Ndifferent combinations, the FT which is not employed in thebest estimator (i.e., corresponding to the combination withthe smallest residual error) is discarded. The process iteratesuntil a certain pre-determined stopping rule is reached (such aswhen a minimum number of FTs is reached, or, if the changein the residual error is small).

VIII. NLOS SCENARIOS: CONSTRAINED LOCALIZATIONTECHNIQUES

In this section, a different class of NLOS mitigation al-gorithms which utilize some constraints associated with theNLOS measurements will be briefly reviewed.

A. Constrained LS Algorithm and Quadratic Programming

The two-step ML algorithm discussed in Section IV-C isnot robust to NLOS effects. In [46], a quadratic programming(QP) technique for NLOS environments is developed. Themathematical programming is formulated as follows

θcw = arg minθ

(A1θ − p1)T Ψ−1(A1θ − p1) , (86)

s.t. A1θ ≤ p1 , (87)

where A1, p1, and θ are as in (35). Note that (86) and (87)constitute a constrained LS (CLS) algorithm that can besolved using quadratic programming techniques5. The intuitiveexplanation of the CLS is that (86) finds a WLS solutionto the MT location, while the constraint (87) relaxes theequality (which holds in LOS scenarios) into an inequalityfor the NLOS scenarios. In [46], a further refining stage isalso introduced to incorporate the dependency between s andx.

B. Linear Programming

In [20], [47], a linear programming approach was intro-duced which assumes perfect a-priori identification of LOSand NLOS FTs. As opposed to the identify&discard (IAD)type algorithms (to be discussed in Section X), it does notdiscard NLOS FTs, but uses them to construct a linear feasibleregion for the MT location. The location estimate is obtainedusing a linear programming technique that employs only the

5E.g., using the quadprog function in Matlab.


Fig. 7. Illustration of the constrained linear LS technique. The NLOS FTsare used to determine the feasible region. For original (non-linear) model, thefeasible region is obtained from the intersections of the circles. For the linearmodel, the feasible region is obtained from the intersections of squares.

LOS FTs, with the constraint that it has to be within thefeasible region.First, [20] re-expresses the LS location estimator in (42)

using linear programming with all the LOS FTs. The goal isto minimize a linear objective function with respect to a certainconstraint, and it is shown that the performance is very similarto that of (42).If the ith FT is identified to be NLOS, we have the non-

linear constraint

||x − xi|| ≤ di, (88)

which were used in [46] to formulate a quadratic programmingtechnique for two-step ML algorithm (without the assumptionthat the NLOS FTs are accurately identified). In order tolinearize the constraints in (88), we can relax them as [48]

x− xi ≤ di , −x+ xi ≤ di , (89)

y − yi ≤ di , −y + yi ≤ di , (90)

where i = 1, 2, ..., N . In essence, the above is equiv-alent to relaxing the circular constraints into rectangular(i.e. square) constraints for the purpose of linearizing theequations, as illustrated in Fig. 7. By defining some slackvariables6, (89) and (90) can be converted to equalities thatdefine the feasible region. Then, the MT location is estimatedby using only the LOS FTs for minimizing the objectivefunction with the constraint that, the location estimate shouldbe within the feasible region obtained using both the NLOSand LOS FTs.

C. Geometry-Constrained Location Estimation

In [49], a geometry-constrained location estimation (GLE)was proposed, which uses the two-step ML technique inSection IV-C with some additional parameters to incorporatethe geometry of the FTs (only a scenario with three FTs was

6Slack variable is a variable which is used to turn an inequality into anequality (e.g., x < 5 ⇒ x + s = 5).

considered). Let the intersection points of the three circlesin (8) be defined as xA = [xA, yA]T , xB = [xB , yB]T ,xC = [xC , yC ]T . Then, a constrained cost function, whichis referred as the virtual distance is defined as [49]

γ =

√13

[||x − xA||2 + ||x − xB ||2 + ||x − xC ||2

]. (91)

For an expected position xe of the MT, we can calculate theexpected virtual distance using (91) as γe = γ + ne, withne denoting the noise in the expected virtual distance. Thecoordinates for xe are chosen as [49]

xe = ω1xA + ω2xB + ω3xC , (92)

where the weights are obtained as

ωi =σ2

i

σ21 + σ2

2 + σ23

,with i = 1, 2, 3 . (93)

Basically, it is assumed that for NLOS FTs, the measurementvariance will be larger. Hence, the weights defined in (93) willmove xe towards the center of the NLOS FT circle.These geometric constraints are then incorporated into the

two-step ML algorithm in Section IV-C by updating A1 andp1 in (35) as follows [49]

A1 =

[A1

γx γy −0.5

], and p1 =

[p1

γk − γ2e

],

(94)

where

γx =13(xA + xB + xC) , (95)

γy =13(yA + yB + yC) , (96)

γk =13(x2

A + x2B + x2

C + y2A + y2

B + y2C) . (97)

The geometric constraints are also incorporated into othervariables as

B = diag(d1, d2, d3, γ) , (98)

n = [n1, n2, n3, nγ ]T . (99)

Then, the two-step ML algorithm in [29] is employed to solvefor the MT location using the variables updated with thegeometric constraints.

D. Interior Point Optimization

In [33], an interior point optimization (IPO) method wasproposed to find the optimum location estimate in the presenceof NLOS bias. Using the linearization of the system usingTaylor’s series approximation as discussed in Section V-C, alinearized measurement vector is defined as follows [33]

y = H0x + b + n , (100)

where H0 is as defined in (39). If NLOS bias is neglected,the bias-free position estimate is given by [33]

x = (HT0 Q−1H0)−1HT

0 Q−1y . (101)


If the bias vector b is known, a more accurate bias-freelocation estimate is given by [33]

x = x + Vb , (102)

where

V = −(HT0 Q−1H0)−1HT

0 Q−1 , (103)

is a bias correction matrix. However, in reality, b is unknownand has to be estimated. In order to estimate b from (100),the observed bias metric is defined as [33]

z = y − H0x , (104)

which can be simplified to z = Sb+w, where S = I+H0V,and the bias noise is given by

w = H0(x − x) − n . (105)

Then, the following constrained optimization problem is de-fined to estimate the NLOS bias errors [33]

b = arg minb

(z − Sb)T Q−1w (z − Sb) , (106)

s.t. bi ∈ Bi , i = 1, 2, ..., N , (107)

where Bi = [li, ui] are the a-priori information for the rangeof bi lower-bounded by li ≥ 0 and upper-bounded by ui, andQw is the covariance matrix of w.In order to solve the constrained optimization problem

in (106) and (107), an IPO technique was used in [33]. Inparticular, (106) and (107) are modified as

b = arg minb

(z − Sb)T Q−1w (z − Sb) , (108)

s.t. gi(bi) − si = 0, and si > 0 , i = 1, ..., N , (109)

where si is a slack variable, and gi(bi) is a barrier functionthat satisfies gi(bi) > 0 ∀bi ∈ [li, ui]. A generally usedsmooth second order function that satisfies the requirementis gi(bi) = (ui − bi)/(bi − li). Then, (108) and (109) aresolved by minimizing the following Lagrangian [33]

L(b, λ, s) = (z − Sb)T Q−1w (z − Sb)

− μ

N∑i=1

lnsi − λT (g(b) − s) , (110)

where g(b) and s are obtained upon stacking gi(bi) and si,respectively, into N × 1 vectors. Note that the logarithmicbarrier function

μ

N∑i=1

lnsi , (111)

ensures that si = gi(bi) > 0 and bias error is always within[li, ui].The solution to (110) can be obtained by differentiat-

ing (110) with respect to b, λ, and s, and solving themtogether to obtain b. Once an estimate of the bias vectorb is obtained, the authors employ the bias correction matrixin (103) to calculate the bias-free location using (102). Thesimulation results reported in [33] show that better accuraciescan be obtained through IPO compared to Rwgh and iterativeLS algorithms in NLOS scenarios.

IX. NLOS SCENARIOS: ROBUST ESTIMATORS FORLOCALIZATION

Robust estimators are commonly used to suppress theimpact of outliers in a given data, and different classes ofrobust estimators have already been used in the literaturefor NLOS mitigation purposes. In below, few of the popularrobust estimators considered for NLOS mitigation are brieflyreviewed.

A. Huber M-Estimator

The M-estimators, which are “ML type” of estimators, are aclass of robust estimators that have been considered for NLOSmitigation purposes. As discussed in the previous sections, theML algorithm tries to maximize a function of the form

N∏i=1

f(xi) , (112)

which is equivalent to minimizing∑N

i=1 − log f(xi). In thepresence of outliers, ML algorithm fails to yield accurateresults. A generalized form of the ML algorithm is referredas the M -estimator, which was introduced by Huber in 1964,and aims to minimize

N∑i=1

ρ(xi) , (113)

where ρ(.) is a convex function. For the Huber M -estimator,the ρ(.) is defined as [50]

ρ(ν) =

{ν2/2 |ν| ≤ ξ ,

ξ|ν| − ξ2/2 |ν| > ξ ,(114)

which is not strictly convex, and therefore, minimization of theobjective function yields multiple solutions which are close toeach other [50].In [51],M -estimator was used to estimate the MT location,

which yields

x = arg minx

{N∑

i=1

ρ((di − ||x − xi||

)/σi

)}. (115)

Simulation results in [51] show thatM -estimator outperformsthe conventional LS estimator, especially for large NLOS biaserrors. If bootstrapping technique is used in conjunction withM -estimator, the accuracy can be improved further [51].

B. Least Median Squares

In [37], a least median squares (LMS) technique wasproposed for NLOS mitigation, which is one of the mostcommonly used robust fitting algorithms. It can tolerate ashigh as 50% of outliers in the absence of noise. The locationestimate of the MT using the LMS solution is given by [37]

x = arg minx

{medi

(di − ||x − xi||

)2}, (116)

where medi(θ(i)) is the median of θ(i) over all possiblevalues of i. Since calculation of (116) is computationallyintensive, [37] proposes a lower-complexity implementation


that uses random subsets of xi to obtain several candidate x,and the one with the least median of residue is selected as thesolution. It should be noted that [37] uses LMS algorithm inthe context of security, where some of the measurements maybe outliers in the presence of certain attacks to the localizationsystem. A parallel approach that uses LMS algorithm forNLOS mitigation purposes is reported in [52].

C. Other Robust Estimation Options

Besides the M-estimation and LMS techniques discussedabove, there are some other robust estimation techniques [53]which, to our best knowledge, have not been consideredin detail for NLOS mitigation purposes. For example, leasttrimmed squares (LTS) aims to minimize the sum of squaresfor the smallest n of the residuals. S-estimators, on theother hand, is quiet robust to outliers; however, it is notas efficient as the M-estimator. It attempts to find a lineminimizing a robust estimate of the scale of the residuals.Finally, MM-estimator can be considered as an M-estimator,which starts at the coefficients given by the S-estimator. Withmore computational complexity compared to the other twotechniques, MM-estimator has both good robustness and goodefficiency.There are also some other parametric approaches for robust

regression, where, the normal distribution is replaced with aheavy tailed distribution in order to model the outliers. Forexample, t-distribution can be a good distribution for modelingthe outliers in a Bayesian robust regression algorithm [54].Alternatively, a mixture of zero-mean Gaussian distributions(i.e., a contaminated normal distribution) with different vari-ances can be considered to model the noise in the presence ofNLOS bias, e.g.

ni ∼ (1 − ε)N (0, σ2) + εN (0, cσ2) , (117)

where the second Gaussian distribution is intended to capturethe outliers, and ε < 1 is a small number (typically smallerthan 0.1) that characterizes the impact of the outliers togetherwith the constant term c.

X. NLOS SCENARIOS: IDENTIFY AND DISCARD BASEDTECHNIQUES

As discussed in the beginning of Section VI, one of thesimplest techniques to mitigate the NLOS effects is to identifythe NLOS FTs and discard them during localization (i.e.,find the MT location using only the LOS FTs). In fact, theML estimator of [42] illustrated in Fig. 6(a) and the MLestimator of [23] discussed in Section VI-A are also IADtype of estimators. Moreover, the WLS technique discussedin Section VII-A also becomes an IAD type of techniqueif the weights are set to 0 for NLOS FTs, and 1 for LOSFTs. A common problem in all these approaches is accurateidentification of the NLOS FTs. In this section, we willreview another IAD based technique which uses a residualtest algorithm to identify the NLOS FTs.

A. Residual Test Algorithm

Residual test (RT) algorithm proposed in [28] falls into thegroup of algorithms where the MT is localized using only theLOS FTs. Hence, the NLOS FTs have to be correctly identifiedand discarded, which is achieved as follows. Firstly, usingthe AML algorithm discussed in Section IV-D, by employingdifferent combinations of FTs,7

S0 =N∑

i=3

NCi , (118)

different location estimates xk are computed (k ∈{1, 2, ..., S0}).For each k, the squares of the normalized residuals8 are

computed as

χ2x(k) =

[xk − xS0 ]2

Ix(k), χ2

y(k) =[yk − yS0 ]2

Iy(k), (119)

where Ix(k) and Iy(k) are obtained from (17), and are theCRLBs9 for x and y dimensions for the kth hypothesis.Without loss of generality, S0 is the case when all the FTsare used in localization.If all the FTs in the kth hypothesis are LOS, from (119), we

have χx(k) ∼ N (0, 1) and χy(k) ∼ N (0, 1). Hence, χ2x(k)

and χ2y(k) have centralized Chi-square distribution with one

degree of freedom. On the other hand, if there is at least oneNLOS FT in the kth hypothesis, both random variables havenon-centralized Chi-square distributions with non-centralityparameter depending on the NLOS bias.This observation suggests that if the PDF of χ2

x(k) andχ2

y(k) (which should ideally be the same) can be identifiedcorrectly, this allows for determining if all the FTs are LOSor not. This can simply be achieved by a threshold test.For example, for seven FTs case as in [28], an appropriatethreshold to characterize the Chi-square PDF for the LOS caseis determined to be 2.71. If the area under the PDF on theright of this threshold is larger than 0.1, the random variableis identified to be non-centralized Chi-square distributed, andotherwise, it is centralized Chi-square distributed.In case the PDF is determined to be non-centralized Chi-

square, this implies that there is at least one NLOS FT. Then,the algorithm forms NCN−1 = N sets of FTs, with (N − 1)FTs in each set. For each set,

N−1∑i=1

N−1Ci , (120)

estimates of xk are obtained. If any of these sets are found tobe distributed according to centralized Chi-square distributionusing the threshold test, then the number of LOS FTs is (N−1), and the FTs within that particular set are used to estimatethe MT location. Otherwise, the algorithm iterates until thereare at most three LOS FTs. Since three FTs do not provide

7The summation in (118) starts from 3 since at least 3 FTs are requiredfor location estimation in 2-D.8Note that the definition of the residual in RT algorithm is different than

the residual in Rwgh algorithm.9The computation of CRLB requires the actual MT location x. Since

it is not available, xS0 is used as an approximation to x to compute anapproximate CRLB.


sufficient number of realizations for a reliable RT, delta testprocedure is proposed, which takes two FTs first, and thencombines them with one of the rest of the FTs to check ifall three FTs are LOS. The simulation results show that theproposed technique outperforms the Rwgh [18] and CLS [46]algorithms, and can achieve the CRLB if the number of LOSFTs is larger than half of total number of FTs.

XI. IMPACT OF FT DISTRIBUTION ON THE LOCALIZATIONACCURACY

Before concluding the survey, one last important issueto be discussed relates to the impact of FT distribution10

on the localization accuracy. Impact of the node locationson the accuracy has been analyzed in terms of achievablefundamental lower bounds in [21], [56]. In [56], the authorsconclude that, for anchor-free localization11, if the nodesare distributed within a rectangular region of L1 × L2, theachievable accuracy improves as L1 → L2 (i.e., when theregion converges to a square). Nevertheless, for large numberof nodes, the impact of the network shape on the achievableaccuracy becomes insignificant.In [21], an iterative algorithm called RELOCATE is pro-

posed for optimally placing the reference nodes. For a fixedposition of the target node, it optimally places the referencenodes so as to minimize the Cramer-Rao bound. Extension ofthe algorithm for multiple locations of the target node (suchas a walking path within a building) is also presented.Practical aspects of three dimensional placement of the FTs

are evaluated in [57] using well known optimal solutions. Anexample scenario for placing four FTs within a cubic room isconsidered. Placing all the FTs on a planar surface (e.g., fourdifferent corners of the room’s ceiling) yields a relatively lowhorizontal dilution of precision (HDOP) but a large verticaldilution of precision (VDOP)12. On the other hand, if thetarget nodes are placed in an “as good as possible” tetrahedronconfiguration, the HDOP is relatively smaller while the VDOPis significantly smaller compared to the planar configuration.Optimum geometries of the FTs for different number of

FTs are derived in [58]. In general, the FTs are placed on ageometry whose corners are “equally” distributed on a unitspherical surface. The five solutions to this problem for N =4, 6, 8, 12, 20 correspond to a tetrahedron, octahedron, cube,icosahedron, and dodecahedron, respectively, which are alsoreferred as Platonic solids. Also, any superposition of centeredPlatonic solids yields another optimum geometry [58].In [59], [60], the authors analyze the relation between

the localization probability and node distribution. First, thenodes are classified as L-nodes and NL-nodes. L-nodes areassumed to know their location, and NL-nodes are assumedunaware of their location (and need to localize themselves).The distributions of the L-nodes and NL-nodes in a two

10Other than the FT distribution, other nuisance parameters may also havea considerable impact on the localization accuracy. For example, the readeris referred to [55] for a detailed discussion on the effects of FT height andpath loss exponent on the achievable localization accuracies in a log-normalfading channel.11No node knows its own location, but only the inter-node distance

measurements are known.12HDOP and VDOP are expressions for GDOP in horizontal and vertical

domains, respectively.

dimensional domain Sdom ⊆ R2 are modeled through Poissonpoint processes ρL and ρNL, respectively. Then, [59] derivesthe probability that a randomly chosen NL-node over Sdom

gets localized, as well as the probability of whole networkof NL-nodes being localized for a log-normal shadow fadingscenario.The NL-node localization failure probability over a circular

domain of radius Rcir, with per-node radio coverage radiusdrad < Rcir, total number of NL-nodes kNL, and total numberof nodes N is shown to be tightly bounded as follows in [60]

PF ≥[1 − (1 − a)b2

]N−3[1 + b2(1 − a)(N − 3)

+ b4(1 − a)2(N − 1)(N − 2)

2

], (121)

where a = 1 − kNL/N is the fraction of the NL-nodes tototal number of nodes and b = drad/Rcir. Extensions to log-normal shadowing and analysis of transition thresholds arealso provided.

XII. CONCLUSION

In this paper, an extensive survey of different TOA basedlocalization and NLOS mitigation techniques is presented.While some algorithms can perform close to the CRLB, theymay require high computational complexities and availabilityof different prior information. For example in NLOS situa-tions, prior information regarding the NLOS bias may notbe available in many scenarios. In Table III, we provide abrief summary of different techniques, as well as their com-plexities and requirements. Practical and efficient localizationtechniques in the presence of NLOS bias still requires furtherresearch. The authors believe that this survey will serve asa valuable resource for evaluating the merits and trade-offsof the different available techniques towards developing moreefficient and practical NLOS mitigation algorithms.

ACKNOWLEDGEMENT

The authors would like to thank Dr. Fujio Watanabe fromDOCOMO USA Labs and Dr. Sinan Gezici from BilkentUniversity for fruitful discussions, to Mr. Hiroshi Inamurafrom NTT DOCOMO Japan for his continuous support, andto anonymous reviewers and the editor Dr. Nelson Fonsecafor their insightful comments.

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TABLE IIIOVERVIEW OF TOA BASED LOCALIZATION ALGORITHMS IN LOS AND NLOS SCENARIOS.

Algorithm Name References Summary Complexity, A-priori Knowledge etc.

ML Type Algo-rithms (LOS)

ML Algorithm [19], [23] The x that maximizes the joint probability of thedistance measurements is taken as the locationestimate.

Requires a comprehensive search over possibleMT locations. Requires the knowledge of PDFsfor distance measurements.

Two-Step ML [29] The location estimate can be expressed in closed-form as in (24).

Requires iteration: first, B is evaluated basedon the measured distances, an initial locationestimate is obtained, and this is used to refineB.

Approximate ML [26] Uses the relation in (30) to obtain x in terms ofs. The resulting quadratic relation is solved byemploying a root selection routine.

Needs an initial location estimate (with a simplerestimator) to evaluate (31). May also need toiteratively update

LS Algorithms(LOS)

Non-Linear LS [2], [30],[31]

Finds the x that minimizes the residual error asin (33).

Requires a search over possible MT locations.May employ techniques such as Gauss-Newtonor Steepest Descent for faster convergence.

Linear LSthrough Taylor’sSeries Expansion

[34] Employs the Jacobian matrix in (39) in order toobtain the linear model in (38). Then solves itthrough a simple linear LS estimator.

Requires an accurate initial estimate x0 for lin-earization. May need to iterate for improvedaccuracy.

LLS-1, LLS-2,LLS-3

[34] [37][20]

Cancels out the non-linear x2 and y2 terms in (8)by simple subtraction operations to obtain thelinear model. Then employs linear LS estimator.

LLS-1 does not appropriately selects the refer-ence FT for linearization. Averaging techniquesLLS-2 and LLS-3 have same accuracy, but maystill use undesired FTs in linearization.

LLS-RS [38] Selects the FT with smallest measured distanceas a reference for linearization in LLS-1.

May not work well in NLOS scenarios.

LLS-MLE [38] Utilizes the covariance matrix of observations inthe linear model to obtain the MLE.

Requires the noise variance information, whichis assumed identical at all the FTs.

ConstrainedWeighted LeastSquares

[32] Uses the constraints in (54) to solve for the WLSformulation in (53).

Need to solve for the Langrange multiplier λfrom the 5-root expression in (59).

ML Type Algo-rithms (NLOS)

ML AlgorithmUtilizing NLOSStatistics

[19] [42] In [19], x that maximizes the joint probabilitydensity function of the observations in NLOSscenarios is selected. MAP estimator utilizing theNLOS bias statistics is introduced in [42].

The probability density function of the NLOSbias and the distance measurements are assumedknown, and requires a search over possible MTlocations.

IAD based MLAlgorithm

[23] [42] Uses the ML principle to discard the NLOS FTs.Then, only the LOS FTs are used in locationestimation.

Need to collect Ntrn TOA measurements at eachFT to capture the noise statistics [23]. There mayalways be a possibility of mis-identification of theLOS FTs.

LS Algorithms(NLOS)

Weighted LS [43], [44] Uses some appropriate weights (e.g., using thevariance of the distance measurements, or thestatistics of the multipath components) to assignless reliability to NLOS FTs.

For a static MT, variance information may not bevery different for LOS and NLOS FTs.

Residual Weight-ing Algorithm

[18] Different possible combinations of FT locationsare considered. Then, each of the correspondinglocation estimates are weighted with the inversesof the residual errors to obtain the final locationestimate.

Needs to solve for Ncb =PN

i=3 N Ci locationestimates for different hypothesis before weight-ing them.

ConstrainedLocalizationTechniques(NLOS)

Constrained LSwith QP

[46] Two-step ML technique is used to obtain anestimate of the MT, with a quadratic constraintgiven as in (87).

May have high computational complexity.

Constrained LSwith LP

[20] [47] The NLOS FTs are used to obtain a feasibleregion composed of squares. Then, the LOS FTsare used to solve for the MT location via LLS-1 technique so that the solution is within thefeasible region.

Linear constraints yield a less complex (yet acoarser) solution compared to the quadratic con-straints.

GeometryConstrainedLocalization

[49] A constraint related to the intersection points ofthe circles is incorporated into the two-step MLalgorithm.

Slightly more complex than the two-step MLalgorithm.

Interior Point Op-timization

[33] First estimate NLOS bias values with IPO. Thenuse the NLOS bias estimates in a WLS solution(linearized using Taylor’s series approximation).

Bias estimation through IPO may be computa-tionally complex.

Robust Estima-tors (NLOS)

M-estimators [51] Employs a convex function ρ(ν) to capture theeffects of NLOS bias values in an “ML-type” ofestimator.

Need to tune ρ(ν) appropriately. Better alter-natives such as the S-estimators (more robust)and MM-estimators (both robust and efficient)are available.

Least Median ofSquares

[37] The location that minimizes the LMS of theresidual is selected as a location estimate.

Robust up to %50 of outliers. More computa-tionally complex than the NLS.

Identify andDiscardTechniques(NLOS)

Residual Test Al-gorithm

[28] Identifies and discards the NLOS FTs. The resid-ual errors are normalized by the CRLBs, andresulting variables are checked to find if theyare centralized or non-centralized Chi-square dis-tributed.

Computationally complex due to testing numer-ous hypothesis, Delta-test etc.


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Ismail Guvenc received the B.S. degree in electricaland electronics engineering from Bilkent University,Turkey, in 2001, the M.S. degree in electrical andcomputer engineering from University of New Mex-ico in 2002, and the Ph.D. degree in electrical en-gineering from University of South Florida in 2006.He was with Mitsubishi Electric Research Labs be-tween January and August, 2005, where he workedon UWB ranging and positioning. Since June 2006,he has been with the DOCOMO USA Communi-cations Laboratories, Palo Alto, CA. His research

interests are related to UWB communications, UWB ranging/localization, andnext generation wireless systems. He has published more than 30 internationalconference and journal papers, several standardization contributions, and abook chapter. He has several pending US patent applications and has co-authored a book on UWB ranging and localization. He is a member of theIEEE.

Chia-Chin Chong received the B.Eng (Hons) andPh.D. degrees from The University of Manchester,Manchester, UK and The University of Edinburgh,Edinburgh, UK, in 2000 and 2003, respectively,and currently a senior researcher at the DOCOMOUSA Labs. Her research interests include channelmeasurements and modeling, UWB systems, rang-ing and positioning techniques, 4G cellular systems,and relaying and cooperative communications. Shehas published more than 80 international journals,conference papers, and standard contributions. Dr.

Chong has received numerous awards including the IEEE InternationalConference on Ultra-Wideband Best Paper Award, DOCOMO USA LabsPresident Award and The Outstanding Young Malaysian Award, all in 2006,and URSI Young Scientist Award in 2008. She currently serves as an Editorfor the IEEE Transactions on Wireless Communications and has served onthe Technical Program Committee (TPC) of various international conferencesincluding the TPC Co-Chair for the Wireless Communications Symposium ofthe IEEE International Conference on Communications (ICC) 2008. She isalso the Chair for the DG-EVAL Channel Model standardization group withinthe ITU-R WP5D for IMT systems. She is a senior member of the IEEE.

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