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3182 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 5, SEPTEMBER 2007

NLOS Mitigation Using Linear Programming inUltrawideband Location-Aware NetworksSwaroop Venkatesh, Student Member, IEEE, and R. Michael Buehrer, Senior Member, IEEE

Abstract—In this paper, we propose a linear-programming (LP)approach to the problem of nonline-of-sight (NLOS) mitigationin ad hoc ultrawideband wireless networks. The locations of“unlocalized” nodes can be estimated using range or distanceestimates from location-aware “anchor” nodes. In the absence ofLOS between the unlocalized and anchor nodes, e.g., in indoornetworks, the NLOS range estimates can be significantly biased.The direct incorporation of these biased range estimates intopractical location estimators, such as the least squares estimator,without the mitigation of these bias errors, can potentially leadto severe degradation in the accuracy of node-location estimates.On the other hand, with certain geometries of anchor nodes,NLOS range estimates can be used to improve the accuracy oflocation estimation. Furthermore, discarding the biased rangeestimates may not be a viable option, as the number of rangeestimates available may be limited. We present a novel NLOS-biasmitigation scheme based on LP, that 1) allows us to incorporateNLOS range information into location estimation, but 2) does notallow NLOS bias errors to degrade node-localization accuracy.

Index Terms—Line of sight (LOS), location estimation, mobilead hoc networks, non-LOS (NLOS) environment, time-of-arrival(TOA) estimation, ultrawideband (UWB), wireless networks.

I. INTRODUCTION

THE ENVISIONED applications for ad hoc wireless net-works increasingly rely on the automatic and accurate

location of deployed terminals or nodes, and as a result, thereis a rapidly increasing demand in location-based functionality.In sensor networks, particularly for environmental applications[2] such as water-quality monitoring, precision agriculture, andindoor air-quality monitoring, the available sensing data maybe rendered useless by the absence of accurate sensor-locationestimates. The availability of accurate location estimates of

Manuscript received May 10, 2006; revised September 15, 2006 andDecember 4, 2006. This work was supported in part by the Office of NavalResearch (ONR) under Grant N00014-05-1-179 and in part by the NationalScience Foundation (NSF) under Grant 0515019. This paper was presentedin part at the 2006 Information Processing for Sensor Networks (IPSN) Con-ference, Nashville, TN, April 19–21, 2006, and the 2006 IEE Conference onUltrawideband, London, U.K., April 19–21, 2006. The review of this paperwas coordinated by Dr. R. Klukas.

S. Venkatesh was with the Mobile and Portable Radio Research Group,Bradley Department of Electrical and Computer Engineering, Virginia Poly-technic Institute and State University, Blacksburg, VA 24061 USA. He is nowwith Marvell Semiconductor, Sunnyvale, CA 94086 USA (e-mail: [email protected]).

R. M. Buehrer is with the Mobile and Portable Radio Research Group,Bradley Department of Electrical and Computer Engineering, Virginia Poly-technic Institute and State University, Blacksburg, VA 24061 USA (e-mail:[email protected]).

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

Digital Object Identifier 10.1109/TVT.2007.900397

nodes in wireless ad hoc networks can help reduce configura-tion requirements and device cost in addition to enhancing per-formance in communication [3] and routing [4]. Additionally,accurate location estimation or “localization” enables applica-tions such as inventory management, intrusion detection [5],and traffic monitoring. Furthermore, tracking networks [6] havenumerous applications such as locating emergency workers inbuildings [7], command-and-control of military personnel, andthe guidance of robots in remote or hazardous locations.

The design of such ad hoc “location-aware” networks typi-cally requires the capability of peer-to-peer range or distancemeasurement [8]. A node whose location is unknown can esti-mate its location based on range measurements from location-aware “anchors,” whose locations are known or estimateda priori. Range estimates from anchor nodes can be obtainedusing received-signal strength (RSS) or time-of-arrival (TOA)estimation techniques [8], [9]. Impulse-based ultrawideband(UWB) or impulse radio is an excellent physical-layer solutionfor indoor location-aware networks [10], [11] due to its robust-ness in dense multipath environments [12], its ability to fuseaccurate position location with low-data-rate communication[6], [9], and its covertness for tactical applications. Due to thefine time resolution afforded by UWB signals, the estimation ofdistances between nodes in UWB networks typically relies onthe estimation of the TOA of either the earliest [9], [13], [14] orthe strongest [15] multipath components. Localization accura-cies on the order of tens of centimeters have been demonstratedin line-of-sight (LOS) scenarios for UWB TOA-based rangingand positioning applications [9], [14].

However, in dense multipath propagation environments, par-ticularly indoors or in urban scenarios, the LOS path betweennodes may be obstructed, as illustrated in Fig. 1(a). As a result,in non-LOS (NLOS) conditions, TOA-based range estimatesare positively biased with high probability, since the first ar-riving multipath component travels a distance that is in excessof the true LOS distance. A similar effect is seen in the case ofRSS-based range estimates, where the received-signal power isreduced due to the obstruction of the LOS path. These effectsresult in range estimates that are often much larger than the truedistances, and therefore, in NLOS scenarios, the accuracy ofnode-location estimates can be adversely affected.

The problem of location estimation with biased NLOS rangeestimates has been considered before, mostly in the context ofcellular communications [16], [17], where it has been shownthat the bias errors in NLOS range estimates lead to largeerrors in the computation of a node’s location. Similar obser-vations have been made with UWB signals [15], where it wasdemonstrated that the NLOS bias errors can be on the order

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VENKATESH AND BUEHRER: NLOS MITIGATION USING LINEAR PROGRAMMING IN UWB LOCATION-AWARE NETWORKS 3183

Fig. 1. (a) NLOS problem: In the absence of LOS in TOA-based ranging between two nodes A and B, the estimated range is larger than the true distancebetween A and B. (b) Hypothesis testing of the rms delay spread for the nature of the UWB channel: The histogram of the rms delay spread for the LOS andNLOS scenarios. We see that, since the probability density function of the rms delay spread for the LOS and NLOS cases are well separated, the performance ofNLOS identification using this approach is not sensitive to the value of the threshold selected.

of several meters and are typically much larger than the range-measurement errors in LOS scenarios.

Broadly speaking, the literature on the NLOS problem fallsin two categories: NLOS identification and NLOS mitigation.The former deals with the problem of distinguishing betweenLOS and NLOS range information, whereas the latter typicallydeals with the reduction of the adverse impact of NLOS rangeerrors on the accuracy of location estimates, assuming that theNLOS range estimates have been identified. Several statisticalNLOS-identification techniques [18] for cellular systems havebeen discussed previously, which exploit prior knowledge ofNLOS delays [19] and a series of measurements to reduce thebias in NLOS range estimates. In the case of UWB signals, LOSand NLOS signals can be identified by quantifying the temporaldispersion in received-signal energy [12]. UWB measurementsresults indicate that [12], [20] NLOS signals tend to havemuch higher values of delay-spread statistics, such as the rmsdelay spread and mean-excess delay. Given the received-signalsamples [12], [21], we can distinguish between LOS and NLOSsignals by “hypothesis testing” [22] the delay-spread statistics,as illustrated in Fig. 1(b). Consequently, in this paper, we focuson the problem of NLOS mitigation, assuming that we areable to accurately distinguish between LOS and NLOS rangeestimates.

Equipped with the ability to perfectly distinguish betweenLOS and NLOS range estimates, the Cramer–Rao lower bound(CRLB) analysis presented in [23] and [24] characterizedthe performance of the minimum-variance unbiased estimator(MVUE) [22] of a node’s location, given a mixture of (unbi-ased) LOS and (biased) NLOS range estimates. This analysisshowed that, in the absence of a priori statistical informationon the NLOS range estimates, the MVUE discards the biasedNLOS range estimates and utilizes only LOS range informationwhile estimating sensor locations. However, as will be demon-

strated in the following sections, in the case of practical noneffi-cient [22] estimators, such as the commonly used least squares(LS) estimator [25], discarding NLOS range information doesnot necessarily improve performance. Additionally, for 2-D(3-D) location estimation, the LS estimator requires at leastthree (four) range estimates in order to obtain an unambiguoussolution. In ad hoc networks, limited connectivity with anchorsmay imply that we may not have the luxury of discarding anyrange estimates. This suggests that, in general, given a mixtureof LOS and NLOS range estimates, we may be required to usethe entire set of available range information in order to estimatea node’s location.

A semidefinite-programming approach to node localizationbased on connectivity information was investigated in [26],and a quadratic-programming approach with NLOS range es-timates was discussed in [27], but these approaches result inhigh computational complexity [28]. The residual weightingalgorithm was proposed in [29], whose advantage is that NLOSidentification is not required a priori. However, this algorithmimplicitly assumes that the range-measurement noise is muchsmaller than the NLOS-bias introduced in order to inherentlydistinguish between the residuals from LOS and NLOS rangeestimates. More importantly, it relies on the availability of alarge number of range estimates, several of which are LOS,so that the set of range estimates finally selected to compute anode’s location results in the smallest residual error. However,in indoor networks, situations may arise where only NLOSrange estimates are available while estimating a node’s location.

The statistics of the NLOS bias errors depend on the spatialdistribution of scatterers in the propagation environment. Giventhe scattering model for the environment, the statistics of TOAmeasurements can be obtained, and well-known techniquessuch as maximum a posteriori and maximum-likelihood es-timation can be employed to mitigate the effects of NLOS



errors [30]. However, in an ad hoc wireless network, the dis-tribution of scatterers in the propagation environment may notbe available, and therefore, we assume that the statistics of theNLOS bias errors are unknown a priori.

In this paper, we present a novel computationally efficientlinear-programming (LP) approach that effectively incorporatesboth LOS and NLOS range information into the estimation of anode’s location. An LP approach was briefly mentioned for thecase of NLOS range estimates in [28] but was not pursued. Wedemonstrate that this low-complexity LP approach can be gen-eralized to handle a mixture of LOS and NLOS range estimates(with the “only LOS” and “only NLOS” range-information sce-narios as subcases) without discarding any range information.The main advantages of this approach are as follows.

1) The statistics of the NLOS bias errors are not assumed tobe known a priori.

2) No range information is discarded.3) It outperforms the LS estimator, given a mixture of LOS

and NLOS range estimates.4) It can be generalized and extended to handle degenerate

cases with insufficient (< 3) LOS range estimates.

In the proposed approach, we leverage the following featuresof UWB TOA-based range estimation: 1) the range bias er-rors in NLOS conditions are always positive and significantlylarger in magnitude than the range-measurement errors in LOSconditions [15] and 2) NLOS range estimates are readily dis-tinguished from LOS range estimates through channel iden-tification [21]. It is important to point out that, although thedevelopment of the approach is from the perspective of UWBlocation-aware networks, the approach itself is generally validfor any location-aware system with the mentioned features.For the purpose of analytical simplicity and clear exposition,the development and insights that follow largely pertain to2-D location estimation but can easily be extended to 3-Dlocalization scenarios.

This paper is organized as follows. In Section II, we discussthe impact of NLOS bias errors on the accuracy of LS-locationestimates and the motivation for new approaches. Section IIIdiscusses the LP approach to incorporating LOS range esti-mates, NLOS range estimates, and, finally, a mixture of LOSand NLOS range estimates, into node location estimation. InSection IV, we discuss some extensions and a series of subcasesthat need to be addressed in order to generalize the proposedapproach. Simulation results are presented in Section V, wherewe evaluate the performance of the proposed method in termsof node-localization accuracy. We also present two practicalscenarios (including a 3-D indoor scenario) in Section VI,where the efficacy of the proposed approach is highlighted. Ourconclusions are presented in Section VII.

II. IMPACT OF NLOS BIAS ERRORS

A. Notation, Models, and Assumptions

Suppose that an “unlocalized” node’s (unknown) locationis x = [x y]T . Let A denote the set of anchor locations, andL ∈ A denote the set of locations of anchors which provideLOS range estimates, with cardinality mL = |L|. The known

locations of the LOS anchors are denoted by xLj, j =1, 2, . . . ,mL. Similarly, N ∈ A represents the set of locationsof anchors that provide NLOS range estimates, with mN =|N |, and the known set of locations of the NLOS anchors isrepresented by xNj, j = 1, 2, . . . ,mN .

The LOS range estimates rLj, j = 1, 2, . . . ,mL, are mod-eled as unbiased Gaussian [2] estimates of the actual internodedistances RLj = ‖x − xLj‖

rLj = RLj + nLj , j = 1, 2, . . . ,mL (1)

where nLj represents zero-mean Gaussian range-measurementnoise in the jth LOS range estimate: nLj ∼ N (0, σ2Lj). Therange-measurement noise variance σ2Lj can be modeled as [31]

σ2Lj = KERβL

Lj (2)

where βL is the LOS path-loss exponent, and KE is a pro-portionality constant (governed by the transmit power and thereceiver noise floor) that determines the accuracy of rangeestimation. This model arises due to the fact that the accuracyof TOA-based range estimates can be shown [32], [33] tobe inversely proportional to the received signal-to-noise ratio(SNR) (or, more generally, the signal-to-interference-and-noiseratio) assuming matched-filter detection. The above model forthe accuracy of range estimates [31] applies to both TOA- andRSS-based range estimates when β = 2. The vector of LOSrange estimates is denoted by rL = [rL1rL2 · · · rLmL

]T(1×mL),

where [·](1×n) denotes a row vector of length n.The NLOS range estimates are assumed to be positively

biased Gaussian estimates [19] of the true distances

rNj = RNj + nNj + bNj , j = 1, 2, . . . ,mN (3)

where RNj = ‖x − xNj‖, nNj ∼ N (0, σ2Nj), σ2Nj =KER

βN

Nj , and bNj are the NLOS bias errors, and βN is theeffective path-loss exponent under NLOS conditions. Weassume that the bias errors are always positive: bNj > 0, ∀j.The bias errors bNj and the range-measurement noise nNj areassumed to be independent random variables. Although wemake no assumptions about the statistical distribution [24] ofthe NLOS bias errors in the following development, for thepurpose of simulation, we assume that bias errors are uniformlydistributed: bNj ∼ U(0, Bmax), where Bmax represents themaximum possible bias.1 Additionally, we assume that the biaserrors are, with high probability, much larger than the range-measurement noise Bmax σNj , j = 1, 2, . . . ,mN . Finally,without loss of generality, we assume that the coordinate axesare selected such that x ≥ 0.

B. Discarding NLOS Range Estimates

As we shall show in a later section, when we have at leastthree range estimates, the LS estimator [25] can be used tocompute an estimate x of the unlocalized node’s location x.

1Practically speaking, Bmax would depend on the propagation environmentand physical-layer parameters, such as the transmit power.



Fig. 2. (a) This example shows several instances of the LS location estimate x, one for each realization of the range estimates, for 1) (top) only mL = 3 LOSestimates and 2) (bottom) including mL = 3 LOS and mN = 1 NLOS range estimates. The NLOS range estimate is treated exactly like an LOS range estimateand directly incorporated into the LS solution in the bottom figure. In this case, the addition of the biased NLOS range estimate degrades localization accuracywith respect to µΩ and σΩ. (b) In this case, the addition of the biased NLOS range estimate improves localization accuracy with respect to µΩ and σΩ. In bothcases, x = [3 3]T, βL = 2, βN = 2.5, KE = 0.1, and Bmax = 4 m.

We define the localization error, a measure of the accuracy ofthe location-estimate x, as

Ω = ‖x − x‖2 (in square meters). (4)

It must be noted that Ω is a random variable, with differentinstances corresponding to different realizations of the range-measurement noise, bias errors, and anchor locations. There-fore, we characterize the accuracy of location estimates throughthe mean µΩ and standard deviation σΩ of the localization errordefined in (4); smaller values of both µΩ and σΩ indicate moreaccurate node-location estimates on the average.

When mN = 0 and mL ≥ 3, the LS estimator providesaccurate estimates of a node’s location [25]. However, whenmN > 0, we need effective ways of incorporating NLOS in-formation into the estimation procedure. The CRLB analysispresented in [23] and [24] showed that, in the absence ofprior statistical information on the NLOS range estimates,the MVUE discards the biased NLOS range estimates andutilizes only LOS range information while computing locationestimates. However, this approach may not be optimal whenusing practical estimators, such as the LS estimator, that donot achieve the CRLB. Fig. 2(a) and (b) shows the impact ofdirectly (without mitigation of the bias errors) incorporatingNLOS range estimates into the LS solution for two specificscenarios. For the specific distribution of anchors shown inFig. 2(a), directly incorporating the NLOS ranges into the LSsolution without any mitigation of the bias in the range estimatecan degrade the average localization accuracy defined in termsof µΩ and σΩ. However, in some cases and, in particular,for the example shown in Fig. 2(b), introducing the NLOSrange estimate directly into LS location estimation can improveperformance in terms of µΩ and σΩ.

Generally speaking, it is observed that discarding the NLOSrange estimates does not result in poor performance when thegeometry of LOS anchor nodes has certain properties, bestdescribed by the geometric dilution of precision (GDOP) [2],where a large GDOP (as defined in [2]) implies poor localiza-tion accuracy. It has been observed that when the GDOP of LOSanchors is large, the presence of an additional NLOS range esti-mate results in an improvement in performance: The addition ofan NLOS node typically reduces the effective GDOP, and thiscompensates for the inaccuracy of the NLOS range estimate.

These two examples show that 1) directly incorporatingNLOS range estimates into existing practical estimators with-out reducing the impact of bias errors can adversely affectlocalization accuracy; however, 2) we do not wish to discardthe NLOS range estimates, since their use could improve theperformance of practical estimators under certain conditions.Indeed, in indoor networks, we may have more NLOS rangeestimates than LOS range estimates. Therefore, what is desiredis a method that allows the “soft activation” of NLOS rangeinformation: The NLOS range estimates are not incorporateddirectly but are used in conjunction with LOS range estimateswhen LOS range estimates alone do not guarantee accuratenode location estimates. In the following sections, an LP ap-proach that achieves this goal is described.

III. LP APPROACH

In this section, we show that the problem of node localiza-tion, given LOS range information, can be cast into the form ofa linear program. We then modify the linear program to utilizeadditional NLOS range information, resulting in a method thatutilizes a mixture of LOS and NLOS range estimates to estimatea node’s location.



A. LOS Range Estimates

The LOS range estimates, which are modeled as unbiasedestimates of the true ranges, can be used to define conditionssatisfied by the unknown node location x. We can write

‖x − xLi‖ = rLi ⇒ (x− xLi)2 + (y − yLi)2

= r2Li, i = 1, 2, . . . ,mL. (5)

These relations are nonlinear equations in x and y and representthe fact that x lies on a circle of radius rLi whose center is xLi.This system of equations can be linearized by extracting thedifference of each of these equations from the others, forminga linear system of M =

(mL

2

)distinct equations

aijx+ bijy = cij , i, j = 1, 2, 3, . . . ,mL, i < j

where

aij =xLi − xLj

bij = yLi − yLj

cij =

(x2Li − x2Lj

)+(y2Li − y2Lj

)−(r2Li − r2Lj

)2

. (6)

Each of these M equations can be viewed as representingthe lines connecting the intersection points (if any) of pairsof circular constraints defined in (5). As the range estimatesdefined in (1) are noisy, in general, rLi = RLi, and solvingthese equations simultaneously may not yield a unique solution.Resorting to an error-minimization approach, for every poten-tial solution x and for every equation, we can define the residualerror as

eij = aijx+ bijy − cij , i, j = 1, 2, 3, . . . ,mL, i < j.(7)

The final estimate x can be selected such that an objectivefunction Z, such as the sum of the residual-error squares, isminimized

x = arg minx

Z = arg minx

∑i

∑j,j>i

e2ij .

This is the equivalent to the LS approach defined in [25]. It isimportant to note that 1) although the system of equations in (6)is linear, the objective function Z is nonlinear in x and y, and2) we require mL ≥ 3 to form an unambiguous solution.2 TheLS solution is given by

x = (ATA)−1ATc (8)

where

A =[a12 a13 · · · a1mL

a23 · · · a(mL−1)mL

b12 b13 · · · b1mLb23 · · · b(mL−1)mL

]T2×M

(9)

2In the 3-D case, we require mL ≥ 4.

and

c = [ c12 c13 · · · c1mLc23 · · · c(mL−1)mL

]T1×M .(10)

Looking at (7), we see that the set of variables eij plays therole of unconstrained “slack variables” [34] in the system ofM equations. Therefore, this linear system of equations can beconverted to a linear program [34] if the objective function Z isa linear function of the unknowns. Specifically, if we define

Z∆=∑

i

∑j,j>i

|eij |

and then replace the unconstrained variable eij by e+ij − e−ij ,e+ij ≥ 0, e−ij ≥ 0, we can write an alternative linearized objec-tive function, that is to be minimized, as

Z∆=∑

i

∑j,j>i

(e+ij + e−ij

). (11)

It must be noted that, in the optimal solution that minimizes Z,only one term among e+ij , e−ij will be equal to |eij |, with theother being zero [34]. The constraints are then given by

aijx+ bijy−e+ij +e−ij =cij , i, j = 1, 2, 3, . . . ,mL, i < j.(12)

Since there are now 2M nonnegative slack variables, the vectorz of (2M + 2) variables can be written as z = [x y εT]T, where

ε =[e+12 e

−12 e

+13 e

−13 · · · e+(mL−1)mL

e−(mL−1)mL

]T1×2M

.

(13)

Thus, the linear program can be formulated in standard form[34] as

minZ = fTLz

such that

[A|J ]z = c,z ≥ 0 (14)

where A and c were, respectively, defined in (9) and (10)

J =

−1 1 0 0 · · · · · · 0 00 0 −1 1 · · · · · · 0 0...

...0 0 0 0 0 0 −1 1

M×2M

(15)

and fL = [0T2×1 1T2M×1]T. Here, 0k×l represents a k × l ma-

trix of zeros, and 1k×l represents a k × l matrix of ones.It can be verified [1] that the linearization of the objective

function does not result in significant degradation of the lo-calization accuracy. Therefore, we now have a linear programthat can be used to solve for a node’s location given LOSrange estimates. In this linear program, the objective functionZ, which is defined in (11), is a function of the distances of apoint x to the straight lines given in (6). If we use NLOS rangeestimates in a similar manner, by incorporating them into theobjective function, we could potentially degrade the accuracy



of the location estimate. Instead, as described in the followingsection, we can use the NLOS range estimates to constrainthe feasible region for x without affecting the objective func-tion defined using LOS range estimates, thereby limiting thepossibility of large errors, particularly when the number of LOSrange estimates is small.

B. NLOS Range Estimates

As the bias errors in the NLOS range estimates are alwayspositive and are assumed to be much larger than the range-measurement noise, we know each NLOS range estimate rNi

defined in (3) is, with high probability, larger than the truerange RNi, i = 1, 2, . . . ,mN . Based on this observation, wecan convert the NLOS range estimates into inequalities fori = 1, 2, . . . ,mN

‖x − xNi‖ ≤ rNi ⇒ (x− xNi)2+ (y − yNi)2≤ r2Ni. (16)

These inequalities imply that the feasible region for x lies in theinterior of each of the circular constraints defined by (16). Notethat this assumption cannot be made if the standard deviationof the zero-mean measurement noise and the positive bias in(3) are comparable. Once again, these are nonlinear constraintson x and y. However, these constraints can be relaxed to thefollowing linear constraints, as suggested in [28]:

x− xNi ≤ rNi, −x+ xNi ≤ rNi

y − yNi ≤ rNi, −y + yNi ≤ rNi, i = 1, 2, . . . ,mN .

(17)

This essentially relaxes the circular constraints to rectangularconstraints, as shown in Fig. 3(a). It is readily seen that the newrectangular feasible region contains the original (convex) fea-sible region formed by the intersection of the original circularregions. We can now write the above four constraints for the ithNLOS range estimate in standard form [34]

x− xNi + u1i = rNi, −x+ xNi + u2i = rNi

y − yNi + v1i = rNi, −y + yNi + v2i = rNi

u1i, u2i, v1i, v2i ≥ 0, i = 1, 2, . . . ,mN (18)

where u1i, u2i, v1i, and v2i are the slack variables corre-sponding to the ith NLOS range estimate. Defining wi =[ui1 u2i vi1 v2i ]T1×4 and zi = [x y wT

i ]T as the vectorsof variables corresponding to the ith NLOS range estimate, wecan express the above equations in matrix form as

[B1| I4×4]zi = ri, zi ≥ 0

where

B1 =

1 0−1 00 10 −1

, ri =

rNi + xNi

rNi − xNi

rNi + yNi

rNi − yNi

and In×n denotes an n× n identity matrix. We can now stackthe constraints corresponding to each of the mN NLOS range

estimates to form a system of N = 4mN equations as follows:

[BIN×N ]z = r, z ≥ 0

where

B = [BT1 BT

1 · · ·BT1 ]T(2×N)

r = [ rT1 rT2 · · · rTmN]T(1×N) (19)

w = [wT1 wT

2 · · · wTmN

]T(1×N) (20)

with the vector of variables being defined as

z = [x y wT]T(1×(N+2)).

It is important to note that, in the above analysis, no objectivefunction was defined based on the NLOS range estimates,and only a feasible region for x was derived. The feasibleregion can further be constrained by including the tangentsat the intersection points of the circular constraints definedin (16) to reduce the size of the feasible region, as shown inFig. 3(b). In the following section, we integrate the constraintsand objective function obtained using LOS range estimates withthe NLOS constraints defined above, for the problem of nodelocation estimation given any mixture of LOS and NLOS rangeestimates, such that mL ≥ 3, mN ≥ 0.

C. Combining the LOS and NLOS Range Information

Based on the above sections, given mL ≥ 3 LOS rangeestimates and mN ≥ 0 NLOS range estimates, we can combinethem into a single linear program. We define the vector ofvariables as

z = [x y ε w]T(1×(2M+N+2))

where ε and w are, respectively, defined in (13) and (20). Theobjective function Z is defined as

Z = fTz

where fT = [0 0 12M×1 0N×1]1×(2M+N+2). The completelinear program is then formulated as

minZ = fTz such that

Dz = g, z ≥ 0

where

D =[

A| J 0M×N

B| 02M×N IN×N

](M+N)×(2+2M+N)

g =[

cr

](2+2M+N)×1

.

In the above equations, the matrices A, J , and B are, respec-tively, defined in (9), (15), and (19), and the vectors c and r aredefined in (10) and (19), respectively.

It must be pointed out that, in the above linear program,LOS range information is primarily used to define the objective



Fig. 3. (a) Linearization of mN = 3 NLOS constraints: The NLOS circular constraints are converted to rectangular constraints. (b) Tangents at the intersectionscan be used to further reduce the size of the feasible region.

function, whereas the NLOS range information is used onlyto define the feasible region. This allows the NLOS rangeestimates to “assist” in improving the accuracy of locationestimates by limiting the size of the feasible region but doesnot allow the NLOS bias errors to adversely affect node-localization accuracy, since the NLOS range information playsno part in defining the objective function. The efficacy ofthe proposed method is demonstrated through simulations inSection V. The above approach works for any mixture of LOSand NLOS range estimates, provided mL ≥ 3, mN ≥ 0. Inthe following section, we discuss some special subcases wherethere are insufficient LOS and NLOS range estimates to applythe approach described above.

IV. EXTENSIONS, SPECIAL CASES, AND ANALYSIS

In the previous section, we formulated the “basic-LP,” whichutilizes NLOS constraints to generate a rectangular feasibleregion to limit the potential values for x. Intuitively, reduc-ing the size of the feasible region can limit large values ofthe localization error, improving average localization accuracy.In this section, we consider extensions of the basic LP thatutilize additional constraints to further reduce the size of thefeasible region. We also consider several special cases wherethe number of LOS range estimates is not sufficient to applythe conventional LS solution. We then present an analyticalperspective on the impact of the size of the feasible region onthe performance of the LP approach.

A. Piecewise Linear Feasible Region

Fig. 3(b) illustrates the use of the tangents at the points ofintersection of the circular constraints in reducing the size of thefeasible region. Given mN > 1 NLOS constraints, there are amaximum of L = 2

(mN

2

)points of intersection. Consequently,

as a tangent to each circle can be drawn at each intersectionpoint, there are, at most, 2L tangents that can be drawn toconfine the feasible region. If the points of intersection of the

NLOS circular constraints, which is defined in (16), are denotedby si = [sxi syi]T, i = 1, 2, . . . , L, then a feasible region canbe constructed using the convex hull [34] of these points, whichis given by

F =

x

∣∣∣∣∣x =L∑

i=1

λisi, 0 ≤ λi ≤ 1, ∀i. (21)

It must be noted that, as shown in Fig. 3(b), the above feasibleregion F is different from the original feasible region discussedin Section III-C and can either be used instead of or in additionto the previously defined feasible region. In order to incorporatethe above constraints into the original linear program, we canrestructure the constraints in (21) into the following form:

x−L∑

i=1

λisxi =0

y −L∑

i=1

λisyi =0, 0≤λi≤1, i = 1, 2, . . . , L.

The vector of variables, in this case, is

z = [x y lT]T1×(L+2), l = [λ1 λ2 · · · λL]T1×L.

The above constraints can be written in matrix form using

Hz = 02×1, z ≥ 0, l ≤ 1L×1

where

H =[

1 0 −sx1 −sx2 · · · −sxL

0 1 −sy1 −sy2 · · · −syL

]= [I2×2 S2×L]

and

S2×L =[−sx1 −sx2 · · · −sxL

−sy1 −sy2 · · · −syL

].



B. Extension to LOS Estimates

In the development of the basic LP, while LOS range esti-mates are used to define the objective function and the feasibleregion of the linear program, the NLOS range estimates con-tribute solely in constraining the feasible region. In the absenceof NLOS range estimates (mN = 0, mL ≥ 3), it has beenverified [1] that the performance of the LP approach in termsof the mean and standard deviation of the localization error issimilar to that of the LS estimator. However, given only LOSrange estimates, we can artificially introduce a bias to create“artificial” NLOS range estimates, which can then be used toconfine the feasible region. The underlying assumption here isthat a rough estimate of (or at least an upper bound on) thestandard deviation of the LOS range estimates is known, suchthat a large enough bias (sufficiently larger than the standarddeviation of the LOS range estimates) can be determined. Thus,even in the absence of “true” NLOS range estimates, we havea method of restricting the feasible region, thereby providingperformance gains relative to the LS approach.

If rL is the given vector of LOS range estimates, introducinga sufficiently large bias results in the “artificial NLOS” vectorof range estimates

rL = rL + b

where b denotes the vector of introduced bias terms. This servesas the vector of NLOS range estimates when mN = 0, and thelinear program is formulated exactly as in the case of the basicLP, which is developed in Section III-C, by replacing rNj withrLj . When mN > 0, rL is concatenated with rN , resulting inconstraints given by (19), where B, r, and w are, respectively,replaced by

B = [BT1 BT

1 · · · BT1 ]T(2×N)

r = [ rT rTL ]T(1×N) (22)

w = [wT1 wT

2 · · · wTmN+mL

]T(1×N) (23)

and N = 4(mN +mL). In the following section, we present aformulation of the linear program, complete with the extensionsdiscussed above.

C. Complete Linear Program

Based on the above sections, we can now formulate thecomplete “extended LP.” In this case, the feasible region is de-termined by 1) both NLOS and “artificial NLOS” constraints, aswell as 2) the tangents at intersections of NLOS and “artificialNLOS” circular constraints. The formulation of the extendedlinear program and the vector of the variables are as follows:

z = [x y ε w l]T(1×(2+2M+N+L))

where ε and w are, respectively, defined in (13) and (23). Inthis case, l = [λ1 λ2 · · · λL ]T

1×L, L = 2

(mN+mL

2

). The

objective function Z is defined as

Z = fTz

where fT = [0 0 12M×1 0N×1 0L×1]1×(2+2M+N+L). Thelinear program is then formulated as

minZ =fTz such that

Dz = g, z ≥ 0, l ≤ 1L×1

where

D =

AM×2| JM×2M 0M×N 0M×L

BN×2| 0N×2M IN×N 0N×L

I2×2| 02×2M 02×N S2×L

(M+N+2)×(2+2M+N+L)

g =

c

r0L×1

(2+2M+N+L)×1.

D. Special Cases

The extended linear program described above can be usedto accurately estimate a node’s location for a large number ofcases with a mixture of LOS and NLOS range estimates, whenmL ≥ 3, mN ≥ 0. In order to generalize the scheme to handleother degenerate cases, we consider the exceptions as follows.

Case I—mL = 0, mN > 0: If no LOS range estimates areavailable, then the LP approach in the form discussed above willnot be applicable since the NLOS range estimates are not usedto define an objective function. An example of this situationwith mN = 3 is shown in Fig. 4(a). In this case (if mN ≥ 3),we could either use the LS estimator without the mitigation ofbias errors or simply use the centroid of the feasible region as alocation estimate

x =1n

n∑i=1

vi

where vi, i = 1, 2, . . . , n are the vertices of the feasible region.In the case of the basic LP, the feasible region is rectangular(n = 4) and it is straightforward to show that

x =12

[minixNi + rNi + maxixNi − rNiminiyNi + rNi + maxiyNi − rNi

].

Case II—mL = 1, mN ≥ 2: This situation is illustrated inFig. 4(b) and (c), which represents two subcases. Since mL =1, we have a single LOS-equality constraint, and the lineariza-tion performed in (6) is not possible. In the first subcase[Fig. 4(b)], the circle formed using the LOS range estimaterL1 passes through the feasible region formed by the NLOSconstraints (and with LOS constraints in the extended LP). Insuch a case, the center of the arc of the circle within the feasibleregion can be selected as the solution. In the second subcase,illustrated in Fig. 4(c), the circular constraint formed using theLOS range estimate does not pass through the feasible region.In such a case, we can pick the vertex of the feasible region thatis closest to the circle as a potential solution, i.e.,

x = arg minvi

| ‖vi − xL1‖ − rL1| .



Fig. 4. Special cases, with mN ≥ 1 and mL < 3. (a) Case I: mL = 0 and mN = 3. The vertices of the feasible region are denoted by v1, v2, v3, and v4.(b) Case II: mL = 1, mN ≥ 2. Subcase 1: LOS constraint cuts through the feasible region generated using the NLOS constraints. (c) Case II: mL = 1, mN ≥ 2.Subcase 1: LOS constraint does not pass through the feasible region generated using the NLOS constraints. (d) Case III: mL = 2, mN ≥ 1, Subcase 2: Feasibleregion contains one of two intersection points of the circle. (e) Case III: mL = 2, mN ≥ 1. Subcase 3: Feasible region contains both intersection points of thecircle.

Case III—mL = 2, mN ≥ 1: In this case, since mL = 2,we have two circular constraints, and the linearization of LOSestimates is not particularly useful since a single linear con-straint is generated using the difference. Therefore, instead oftwo potential solutions corresponding to the intersections ofthe two circles, we have an infinite number of solutions. It iseasier to compute the two intersections of the circles, and ifthere are additional NLOS constraints, three subcases arise:

1) Neither of the intersection points lies inside the feasibleregion formed by the NLOS constraints; 2) only one of theintersection points lies inside the feasible region [Fig. 4(d)];and 3) both intersection points lie inside the feasible regionformed by the NLOS constraints [Fig. 4(e)]. When neitherintersection point lies inside the feasible region formed by theNLOS constraints, we can pick the intersection that lies closestto the centroid of the feasible region. In the second subcase,



Fig. 5. (a) Upper bound on the probability p that the true solution lies outside the feasible region generated using mN NLOS range estimates for different valuesof KE and Bmax. (b) Area A of the feasible region as mN increases for different values of KE and Bmax.

we can simply pick the intersection point that lies within thefeasible region. In the last subcase, in order to eliminate one ofthe solutions, consider the following approach: Since the NLOSrange estimates are larger than the true ranges, decrement theNLOS range estimates by small amounts until only one ofthe intersection points remains inside the feasible region. Theintersection point remaining in the feasible region is likely tobe closer to the true location than the intersection point that liesoutside the feasible region and can be selected as the locationestimate.

E. Impact of the Number of NLOS Range Estimates

The efficacy of the proposed approach depends on the factthat the NLOS-bias errors are much larger than the range-measurement noise, so that, the true node location x lies insidethe feasible region generated using the NLOS range estimates.Given that the true node location lies inside the feasible region,it is evident that reducing the size of feasible region improveslocalization accuracy by limiting the localization error. As thenumber of NLOS range estimates mN increases or as Bmax

decreases, two trends, which have opposing effects on thelocalization error, are expected: 1) The probability p that thetrue solution lies outside the feasible region formed by NLOSestimates increases and 2) the area A of the feasible regiondecreases. If the true solution lies inside the feasible region,then reducing the area of the feasible region also reduces themean and variance of the localization error. It is important topoint out that, even if the true location does not lie inside thefeasible region, this does not automatically imply a large lo-calization error (particularly when the feasible region is small),although the two events are correlated. This suggests that thearea of feasible region A may play a more significant role indetermining average localization error than the probability p.

The probability p that x lies outside the linear feasible regionformed using NLOS range estimates can be upper bounded

by the probability that the true distance ‖x − xNj‖ is largerthan the range estimate rNj for at least one value of j, j ∈1, 2, . . . ,mN. In such a case, the region of overlap betweenthe NLOS constraints does not contain the true location x. Fromthe model for NLOS range estimates, which is given in (3), pcan be bounded by

p ≤ 1 −mN∏j=1

qj (24)

where qj is given by

qj =PrrNj<RNj=PrbNj +nNj < 0, j=1, 2, . . . ,mN .

It must be emphasized that (24) is an upper bound for p,since the right-hand side of (24) represents the probabilitythat the true solution lies outside the original feasible regionformed using circular constraints, which is contained within thelinearized feasible region. Assuming that nNj ∼ N (0, σ2Nj)and bNj ∼ U(0, Bmax), we can show that (see Appendix)

qj =1 − 12erfc

(zj√2

)− 1√

2π

1 − e

−z2j

2

zj

zj =Bmax

σNj. (25)

Fig. 5(a) shows the variation of the probability p, whichis computed using (24) and (25), as mN increases, with thesimulation parameters used in Fig. 2(a) and (b). Fig. 5(b) showsthe decrease in the area of the feasible region as the number ofavailable NLOS range estimates increases. We see that, asBmax

increases, the area of the feasible region is larger, implyingreduced gains due to NLOS constraints on the feasible region,but the probability (1 − p) of the feasible region containingthe true location increases. Furthermore, we do observe that



when KE [in (2)] is small, the probability p is small, andtherefore, we are likely to see a considerable decrease in themean localization error as the number of NLOS range esti-mates increases. Furthermore, we expect that the extent of theimprovement obtained by restricting the feasible region withadditional NLOS range estimates diminishes as the number ofLOS range estimates increases.

As we will observe from the simulation results presentedin the following section, decreasing Bmax or increasing mN

reduces the area of the feasible region A and can significantlydecrease average localization error, despite the resulting in-crease in p.

V. SIMULATION RESULTS

In this section, we present simulation results that demonstratethat the proposed approach not only mitigates the effect ofNLOS-bias errors but utilizes the NLOS range information toimprove node-localization accuracy. In the following discussionof simulation results, the anchor nodes are randomly (uni-formly) distributed over a W ×W area, where W = 10 m. Theunknown node location of interest is x = [5 5]T (in meters).The values of βL and βN are assumed to be 2 and 2.5, respec-tively. We compare the performance of four location-estimationapproaches in terms of the mean and standard deviation of thelocalization error Ω: 1) the basic LP approach (“LP-Basic”)discussed in Section III-C, 2) the extended LP approach (“LP-Extended”) discussed in Section IV-C, 3) the LS estimator,utilizing only LOS range estimates, while discarding the NLOSrange estimates [“LS-(Pure-LOS)”], and 4) the LS estimator,utilizing both LOS and NLOS range estimates, without themitigation of NLOS bias errors [“LS-(LOS+NLOS)”]. For theextended LP approach, the standard deviation of LOS rangeestimates is assumed to be known, and bias errors equal to threetimes the standard deviation are added to generate the artificialNLOS range estimates.

For these four methods, the values of Ω are computed for alarge number (Niter = 20 000) of realizations of the measure-ment noise and bias errors for 50 sets of randomly generatedanchor locations. The mean µΩ and the standard deviation σΩof the localization error are shown in Fig. 6(a), for differentvalues of the proportionality constant KE defined in (2). In thissimulation, mL = 3, mN = 3, and Bmax = 8 m. As expected,for all location estimators, node-localization accuracy degradesas the variance of the range estimates increases (i.e., KE in-creases). The proposed LP approaches outperform the LS-basedschemes in terms of both the mean and standard deviation of thelocalization error and, therefore, on the average, generate moreaccurate node location estimates. In general, it is observed that,for the mentioned estimation procedures, µΩ and σΩ followidentical trends in terms of their variation with KE , even forlarger values of βN

µΩ,LP−Extended < µΩ,LP−Basic < µΩ,LS−Pure LOS

< µΩ,LS−(LOS+NLOS)

σΩ,LP−Extended < σΩ,LP−Basic < σΩ,LS−Pure LOS

< σΩ,LS−(LOS+NLOS). (26)

The variation of the mean localization error µΩ with KE ,while, respectively, increasing the maximum bias Bmax and thenumber of NLOS range estimates mN , is shown in Fig. 6(b)and (c). We observe that 1) the LP approaches outperform boththe LS approaches and are less sensitive to an increase in Bmax

and 2) the performance of the LP approaches improves as mN

increases. The former effect is due to the fact that the NLOSranges do not contribute to the objective function of the linearprogram, while the latter is because additional NLOS rangeestimates reduce the size of the feasible region, thereby limitinglarge values of the localization error. The LP approaches onceagain outperform the LS estimator that utilizes only LOS rangeestimates.

The results in Fig. 6(b) and (c) are in agreement with theanalysis presented in Section IV-E, where we argued that eitherdecreasing Bmax or increasing mN results in the reduction ofthe size of the feasible region and the average localization error,despite the increase in the probability that the true solution liesoutside the feasible region. We also note that the extended-LP approach outperforms the basic-LP approach, due to theavailability of a larger number of constraints on the feasibleregion, at the cost of higher complexity.

From Fig. 6(d), we see that the improvement in performanceof the LP-based approaches over the LS-(Pure LOS) approachdecreases as the number of LOS range estimates increases.For instance, when KE = 0.1, the percentage improvement ofthe extended LP method over the LS-(Pure LOS) estimatordecreases from 78% when mL = 3 to 65% when mL = 4.Therefore, when the number of LOS range estimates is large(mL 3), discarding the NLOS range estimates is likely tohave negligible impact on localization accuracy, and the LS-(Pure LOS) estimator can be used to generate accurate locationestimates.

Fig. 7(a)–(c) compares the performance of the extended LPapproach with the LS-(LOS+NLOS) approach in scenarioswhere the LS-(Pure LOS) approach cannot be applied due toan insufficient number of LOS range estimates (mL < 3), i.e.,the special cases discussed in Section IV-D. In each of thesecases, we see that the LP approach substantially outperformsthe LS approach in terms of µΩ and σΩ.

The simulation results presented thus far assumed that theNLOS-bias errors were uniformly distributed between zero andBmax. Fig. 7(d) compares the performance of the approacheswhen the NLOS-bias errors are exponentially distributed [35],which has been observed in UWB NLOS measurements [12],[15]. For the sake of comparison, the mean NLOS bias forboth uniform and exponential distributions was set at Bmax/2,and the mean localization error was computed for 100 ran-dom anchor locations, each with a large number of range-measurement-noise realizations. We see that, even when thebias errors are exponentially distributed, the LP-based ap-proaches provide significant reduction in the average localiza-tion error as compared to the LS-based approaches. In fact,the LP-based approaches perform approximately 20% betterwhen the bias errors are exponentially distributed than whenthey are uniformly distributed. With exponentially distributedbias errors, the probability of obtaining smaller bias values ishigher than with uniformly distributed bias errors; smaller bias



Fig. 6. (a) Mean µΩ and the standard deviation σΩ of the localization error Ω are plotted versus KE . Here, mL = 3, mN = 3, and Bmax = 10 m.(b) Mean µΩ of the localization error Ω, mL = 3 and mN = 3. The maximum bias Bmax is increased from 5 to 10 m. (c) Mean µΩ of the localizationerror Ω, mL = 3 and Bmax = 10 m. The number of NLOS range estimates is varied from mN = 3 to mN = 6. (d) Mean µΩ of the localization error Ω,mN = 4, Bmax = 10 m, as the number of LOS range estimates is increased from mL = 3 to mL = 4.

values imply a smaller feasible region which, in turn, resultsin lower average localization error. Finally, this result suggeststhe possibility that the LP-based approaches can provide gainsin localization accuracy relative to LS estimators in general,when empirical statistical distributions of NLOS-bias errors areapplied.

VI. SIMULATION RESULTS: APPLICATION TO MOBILE

ad hoc LOCATION-AWARE NETWORKS

In the previous section, we investigated the average local-ization accuracy obtained in the case of a stationary unlo-calized node. In this section, we evaluate the efficacy of theproposed NLOS mitigation scheme in a mobile ad hoc network,where localization accuracy is a function of time. It must bepointed out that, in mobile scenarios, tracking algorithms canbe applied in order to mitigate the impact of NLOS-bias errors

[36]–[38]. However, the efficacy of tracking algorithms may belimited in indoor networks due to the fact that, in the case ofad hoc deployment of nodes in a typical indoor environment,the number of NLOS range estimates available can far exceedthe number of LOS range estimates throughout the durationof the mobile’s trajectory. As a result, the tracking algorithmis provided with biased range estimates at every instant oftime, and as a consequence, its ability to ameliorate presentestimates based on past location and range estimates is reduced.However, tracking algorithms can be used to exploit the spatialcorrelation between location and range estimates within themobile’s trajectory and can thus operate in conjunction withNLOS-identification and NLOS-mitigation schemes to improvelocalization accuracy.

In this simulation framework, the network is assumed tocomprise NA anchors distributed over an area (or volume)of interest and whose locations are known a priori. A single



Fig. 7. Some subcases where discarding the NLOS estimates is not possible when the LS estimator is used, as mL < 3. (a) Case I: mL = 0, mN = 4.(b) Case II: mL = 1, mN = 4. (c) Case III: mL = 2, mN = 4. (d) Comparison of the mean localization error obtained using different approaches when theNLOS-bias errors are 1) uniformly distributed and 2) exponentially distributed. The mean bias error for both distributions is normalized to Bmax/2, whereBmax = 10 m.

mobile node, whose time-varying location is denoted by thevector x(t), is assumed to travel through this region at a con-stant speed of v meters per second. The location of the mobilenode at time t = (n+ 1)Ts is generated using the followinglinear model:

xM ((n+ 1)Ts) =xM (nTs) + ve(nTs)

xM (0) =xM0 (27)

where xM0 is the location of the mobile at t = 0, and e(t)is the time-varying unit vector in the direction of the mobile’sinstantaneous motion.

The mobile is assumed to gather range estimates fromnearby anchors and estimate its location every Ts seconds.At time t = nTs, successful “ranging” is assumed to occurbetween the mobile node and the ith anchor if the effectiveSNR ξi(nTs) is larger than a threshold ξT . The effective SNR

ξi(nTs) on the link between the ith anchor and mobile nodeis modeled by

ξi(nTs) = KPPTR−βi (nTs)

where PT denotes the (constant) transmit power, Ri(nTs) =‖x(nTs) − xi‖ is the distance between the mobile node, andthe ith anchor and KP is a constant that subsumes the effectsof other physical-layer parameters. Depending on the locationsof the mobile node, anchors, and obstructions, at any instant oftime, the mobile node may successfully obtain range estimatesfrom both LOS and NLOS anchors. The path-loss exponentsfor the LOS and NLOS cases are assumed to be βL and βN ,respectively, βL < βN . Therefore, at a given instant of time,the mobile node receives range estimates from LOS and NLOSanchors that are, respectively, within distances

Rmax,L =(KPPT

ξT

) 1βL

Rmax,N =(KPPT

ξT

) 1βN



Fig. 8. Simulation of a 2-D location-aware network in an NLOS environment: Values of the simulation parameters used are the following: PT = 1 mW,KP = 1 × 105, ξT = 15 dB, βL = 2, βN = 3, KE = 0.1, and Bmax = 10 m. The total number of anchors is NA = 80, and the mobile node moves at aspeed of v = 2.5 m/s through a W × W m2 area, where W = 100 m. (a) True mobile trajectory and the mobile location estimates obtained using the LS- andLP-based location estimators. (b) Comparison of the different location estimators in terms of the localization error Ω versus time. We observe the trends suggestedby (26).

from the mobile’s current location. We assume that the natureof the propagation channel (LOS or NLOS) is identified foreach received range estimate and, for simplicity, that the rangeestimates are uncorrelated at different time instants. At time t,let the set of all anchor locations A be partitioned into the setof locations of all anchors which have LOS to the mobile ALt

and the set of locations of all anchors which do not have LOS tothe mobile AN t, such that ALt

⋃AN t = A,ALt

⋂AN t = φ.

Then, the subsets of LOS and NLOS anchors which providerange estimates at time t are, respectively, defined by

Lt = xA |xA ∈ ALt, ‖x(t) − xA‖ ≤ Rmax,L Nt = xA |xA ∈ AN t, ‖x(t) − xA‖ ≤ Rmax,N .

At t = nTs, the location-estimate x(t) is computed using LOSand NLOS range estimates from Lt and Nt, which is governedby (1) and (3), respectively. The localization error, as a functionof time, is computed using

Ω(t) = ‖x(t) − x(t)‖2

using the different estimation approaches for x(t).

A. Two-Dimensional Application: Indoor Tracking

Consider a W ×W m2 area of interest with W = 100 m,containing NA = 80 randomly (uniformly) distributed anchors.Several obstructions are randomly dispersed over the area ofinterest. At time t, if the path between the mobile’s true locationx(t) and a given anchor contains any portion of the obstruc-tions, then the range estimates from that anchor are assumed tobe given by (3); otherwise, the range estimates are given by (1).The density of nodes in this simulation was selected such thatthe total number of range estimates (mL +mN ) received ateach instant of time lies between three and eight. The mobile isassumed to move at a speed v = 2.5 m per second in a direction

e(0) = (1/√

2)[1 1]T at t = 0 and, subsequently, reflects offthe boundary of any obstacle encountered.

Fig. 8(a) compares the true location of a mobile node with thelocation estimates computed using the LS and LP approaches.The mobile node estimates its location based on range in-formation from anchors every Ts = 1 s between t = 0 andt = 50 s. We see that the LP-based NLOS-mitigation schemesoutperform the two LS-based schemes. This is more evident inFig. 8(b), which compares the localization error achieved by theestimation approaches versus time. We see that, on the average,the LP-based NLOS-mitigation schemes achieve much higherlocalization accuracy than the LS-based location estimators.

B. Three-Dimensional Application: Indoor Tracking WithGround Sensors

Consider a building with three floors, each floor having di-mensionsW ×W ×H ,W = 100 m, andH = 10 m, as shownin Fig. 9(a). Let a total of NA = 108 anchors be deployeduniformly over the three floors on the ground, i.e., (NA/3) =36 anchors on each floor. The coordinates of the anchors, whichare in meters, are generated using

xA =

[L · u1 L · u2 (g + F ·H)]T

where u1 and u2 are uniformly distributed random variablesover [0, 1], g is a zero-mean Gaussian random variable withunit variance, and F = 0, 1, 2 is the index of the floor. Asingle mobile node is assumed to be present on the floor F = 1,moving linearly with a speed v = 1.5 m/s from an initiallocation xM0 = [5 5 H]T (in meters) at t = 0. The motion ofthe mobile is defined using (27), where

e(nTs) =12

[cos(0.2πnTs) sin(0.2πnTs)1]T .



Fig. 9. In this 3-D indoor tracking simulation, W = 50 m, H = 10 m, NR = 108, xM0 = [0 0 10]Tm, and v = 1.5 m/s. The remaining simulation parametersare the same as those in Fig. 8(a) and (b). The time axis is assumed to run from t = 0 to t = 25 s, with Ts = 0.5 s. (a) Illustration of the trajectory of the mobilenode and the distribution of anchor nodes in a building. (b) x and y components of the localization error Ω(t) versus time t. (c) z component of the localizationerror Ω(t) versus time t. (d) Total localization error Ω(t) as a function of time t. The time-averaged localization error is also shown for comparison.

It is assumed that all anchors on floor F = 1 have LOSlinks to the mobile and provide unbiased range estimates.Furthermore, anchors from floors F = 0, 2 are NLOS withrespect to the mobile node and, therefore, provide biased rangeestimates.

In this scenario, all the LOS anchors are approximately lo-cated on the plane z = H . As a consequence, it is expected thatthe LS-(Pure LOS) approach will have poor location resolutionalong the z-direction, relative to the x and y directions. Insuch a case, it is intuitive that the NLOS-mitigation approacheswould play a key role in determining localization accuracyalong the z-direction, as the NLOS anchors are located arounddifferent planes z = 0 and z = 2H , and can provide improvedresolution in the z-direction. This intuition is confirmed inFig. 9(b) and (c), where we see that any scheme that utilizesNLOS information provides better location resolution in thez-direction. From Fig. 9(c), we see that the LS-(Pure LOS)

method has very poor localization accuracy in the z-coordinateof the mobile’s location. Due to the presence of a large numberof LOS anchors along the x–y plane at z = H , the discussedapproaches perform similarly in terms of localization accu-racy in the x and y coordinates of the mobile, as shown inFig. 9(b).

Fig. 9(d) compares the different schemes in terms of the totallocalization error Ω(t) as the mobile moves through the volumeof interest at a speed v between t = 0 and t = 25 s. The mobileattempts to compute its location every Ts = 0.5 s. Once again,we see that the LP-based localization schemes outperform theLS-based schemes in terms of the time-averaged localizationerror by an order of magnitude. Note that this is a particularlyinteresting scenario since 3-D localization is crucial to severalapplications but is known to be difficult in indoor scenarios.This example highlights the potential benefit of the proposedapproach in practical scenarios.



VII. CONCLUSION

In this paper, we described a novel LP approach to theproblem of wireless localization in NLOS environments. Themain motivation for the development of this method was that,in typical indoor wireless networks, it is likely that we would berequired to compute a node’s location using a mixture of LOSand NLOS range estimates. Using the LOS range estimates todefine the objective function and the NLOS range estimates torestrict the feasible region for the linear program, we showedthat NLOS range information could be used to improve node-localization accuracy without incurring performance degrada-tion due to NLOS-bias errors. The performance of this approachis found to be comparable to the LS estimator when onlyLOS range estimates are provided and substantially better thanthe LS estimator when a mixture of LOS and NLOS rangeestimates is provided, particularly when the number of LOSrange estimates is small. Relative to LS estimation, it wasfound that the LP method is less sensitive to increase in NLOS-bias errors and that increasing the number of NLOS rangeestimates improves node-localization accuracy. Furthermore,the proposed approach can be applied to the general (andpractically significant) 3-D location-estimation problem andto a number of degenerate cases with insufficient LOS rangeestimates.

APPENDIX

EXPRESSION FOR qj , j = 1, 2, . . . ,mN

The term qj is the probability that the node’s true locationlies inside the jth NLOS circular constraint

qj = PrrNj > RNj = PrbNj + nNj > 0

j =1, 2, . . . ,mN .

Assuming that nNj ∼ N (0, σ2Nj) and bNj ∼ U(0, Bmax)

qj =

Bmax∫0

PrnNj > −b(

1Bmax

)db

=1

Bmax

Bmax∫0

∞∫

−b

1√2πσ2Nj

exp

(− x2

2σ2Nj

)dx

db

=1 − 12Bmax

Bmax∫0

erfc

(b√

2σNj

)db

=1 − 12erfc

(zj√2

)− 1√

2π

1 − e

−z2j

2

zj

,

zj =Bmax

σNj.

ACKNOWLEDGMENT

The authors would like to thank Dr. B. Fraticelli of theDepartment of Industrial Systems Engineering, Virginia Tech,Blacksburg, for her valuable comments and insight.

REFERENCES

[1] S. Venkatesh and R. M. Buehrer, “A linear programming approach toNLOS error mitigation in sensor networks,” in Proc. 5th Int. Conf. IPSN,Apr. 2006, vol. 1, pp. 301–308.

[2] N. Patwari, A. O. Hero, M. Perkins, N. S. Correal, and R. J. O’Dea,“Relative location estimation in wireless sensor networks,” IEEE Trans.Signal Process., vol. 51, no. 8, pp. 2137–2148, Aug. 2003.

[3] C. S. Taggart, Y. Viniotis, and M. L. Sichitiu, “Minimizing average net-work delay for ultra-wideband wireless networks,” in Proc. IEEE WirelessCommun. and Netw. Conf., Mar. 2005, vol. 4, pp. 2276–2280.

[4] B. Radunovic and J.-Y. Le Boudec, “Optimal power control, scheduling,and routing in UWB networks,” IEEE J. Sel. Areas Commun., vol. 22,no. 7, pp. 1252–1270, Sep. 2004.

[5] R. L. Moses, D. Krishnamurthy, and R. Patterson, “An auto-calibrationmethod for unattended ground sensors,” in Proc. ICASSP, May 2002,pp. 2941–2944.

[6] G. Schiavone, P. Wahid, E. V. Doorn, R. Palaniappan, and J. Tracy, “Targetdetection and tracking using a UWB sensor web,” in Proc. IEEE Antennasand Propag. Soc. Int. Symp., Jun. 2004, vol. 2, pp. 1287–1290.

[7] S. J. Ingram, D. Harmer, and M. Quinlan, “Ultra-wideband indoor posi-tioning systems and their use in emergencies,” in Proc. PLANS, Rome,Italy, Apr. 2004, pp. 706–715.

[8] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero, R. L. Moses, andN. S. Correal, “Locating the nodes,” IEEE Signal Process. Mag., vol. 22,no. 4, pp. 54–69, Jul. 2005.

[9] J.-Y. Lee and R. A. Scholtz, “Ranging in a dense multipath environmentusing an UWB radio link,” IEEE J. Sel. Areas Commun., vol. 20, no. 9,pp. 1677–1683, Dec. 2002.

[10] L. Stoica, A. Rabbachin, H. O. Repo, T. S. Tiuraniemi, and I. Oppermann,“An ultrawideband system architecture for tag based wireless sensornetworks,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1632–1645,Sep. 2005.

[11] P. Withington, H. Fluhler, and S. Nag, “Enhancing homeland security withadvanced UWB sensors,” IEEE Microw. Mag., vol. 4, no. 3, pp. 51–58,Sep. 2003.

[12] R. Buehrer, W. Davis, A. Safaai-Jazi, and D. Sweeney, Ultra-WidebandPropagation Measurements and Modeling—DARPA NETEX Final Re-port, Jan. 2004, Blacksburg, VA: Virginia Tech. [Online]. Available:http://www.mprg.org/people/buehrer/ ultra/darpa_netex.shtml

[13] I. Guvenc and Z. Sahinoglu, “Threshold selection for UWB TOA esti-mation based on kurtosis analysis,” IEEE Commun. Lett., vol. 9, no. 12,pp. 1025–1027, Dec. 2005.

[14] Z. N. Low, J. H. Cheong, C. L. Law, W. T. Ng, and Y. J. Lee, “Pulsedetection algorithm for line-of-sight (LOS) UWB ranging applications,”IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 63–67, 2005.

[15] B. Denis, J. Keignart, and N. Daniele, “Impact of NLOS propagation uponranging precision in UWB systems,” in Proc. IEEE Conf. Ultra WidebandSyst. and Technol., Nov. 16–19, 2003, pp. 379–383.

[16] T. Silventoinen and M. Rantalainen, “Mobile station emergency locatingin GSM,” in Proc. IEEE Int. Conf. Pers. Wireless Commun., Feb. 19–21,1996, pp. 232–238.

[17] J. Caffery and G. Stuber, “Overview of radiolocation in CDMA cellularsystems,” IEEE Commun. Mag., vol. 36, no. 4, pp. 38–45, Apr. 1998.

[18] J. Borras, P. Hatrack, and N. B. Mandayam, “Decision theoretic frame-work for NLOS identification,” in Proc. VTC, May 18–21, 1998, vol. 2,pp. 1583–1587.

[19] M. P. Wylie and J. Holtzman, “The non-line of sight problem in mobile lo-cation estimation,” in Proc. 5th IEEE Int. Conf. Universal Pers. Commun.,1996, vol. 2, pp. 827–831.

[20] S. S. Ghassemzadeh, L. J. Greenstein, T. Sveinsson, A. Kavcic, andV. Tarokh, “UWB delay profile models for residential and commercialindoor environments,” IEEE Trans. Veh. Technol., vol. 54, no. 4, pp. 1235–1244, Jul. 2005.

[21] S. Venkatesh and R. M. Buehrer, “NLOS identification in ultra-wideband systems based on received signal statistics,” Proc. Inst. Electr.Eng.—Microw., Antennas Propag.: Special Issue on Antenna Systemsand Propagation for Future Wireless Communications, Aug. 2007.to be published.



[22] S. M. Kay, Fundamentals of Statistical Processing, 2nd ed, vol. I.Englewood Cliffs, NJ: Prentice-Hall, 1993.

[23] Y. Qi and H. Kobayashi, “Cramer–Rao lower bound for geolocationin non-line-of-sight environment,” in Proc. ICASSP, May 2001, vol. 3,pp. III-2473–III-2476.

[24] Y. Qi, H. Kobayashi, and H. Suda, “Analysis of wireless geolocation ina non-line-of-sight environment,” IEEE Trans. Wireless Commun., vol. 5,no. 3, pp. 672–681, Mar. 2006.

[25] J. J. Caffery, “A new approach to the geometry of TOA location,” in Proc.IEEE Veh. Technol. Conf., Sep. 2000, vol. 4, pp. 1943–1949.

[26] L. Doherty, K. S. J. Pister, and L. E. Ghaoui, “Convex position estimationin wireless sensor networks,” in Proc. INFOCOM, Apr. 2001, vol. 3,pp. 1655–1663.

[27] X. Wang, Z. Wang, and R. O’Dea, “A TOA-based location algorithmreducing the errors due to non-line-of-sight (NLOS) propagation,” IEEETrans. Veh. Technol., vol. 52, no. 1, pp. 112–116, Jan. 2003.

[28] E. G. Larsson, “Cramer–Rao bound analysis of distributed positioning insensor networks,” IEEE Signal Process. Lett., vol. 11, no. 3, pp. 334–337,Mar. 2004.

[29] P.-C. Chen, “A non-line-of-sight error mitigation algorithm in locationestimation,” in Proc. Wireless Commun. and Netw. Conf., Sep. 21–24,1999, vol. 1, pp. 316–320.

[30] S. Al-Jazzar, J. Caffery, and H. R. You, “A scattering model based ap-proach to NLOS mitigation in TOA location systems,” in Proc. IEEEVTC—Spring, May 2002, vol. 2, pp. 861–865.

[31] Y. Qi and H. Kobayashi, “On relation among time delay and sig-nal strength based geolocation methods,” in Proc. IEEE GLOBECOM,Dec. 2003, vol. 7, pp. 4079–4083.

[32] J. Zhang, R. R. A. Kennedy, and T. D. Abhayapala, “ Cramer–Rao lowerbounds for the time delay estimation of UWB signals,” in Proc. IEEE Int.Conf. Commun., Jun. 20–24, 2004, vol. 6, pp. 3424–3428.

[33] 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.

[34] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming andNetwork Flows. New York: Wiley, 1995.

[35] B. Alavi and K. Pahlavan, “Bandwidth effect on distance error model-ing for indoor geolocation,” in Proc. IEEE PIMRC, Sep. 2004, vol. 3,pp. 2198–2202.

[36] B. Denis, L. Ouvry, B. Uguen, and F. Tchoffo-Talom, “AdvancedBayesian filtering techniques for UWB tracking systems in indoor envi-ronments,” in Proc. IEEE ICU, Sep. 2005, vol. 1, pp. 638–643.

[37] C. Wann, Y. Chen, and M. Lee, “Mobile location tracking withNLOS error mitigation,” in Proc. IEEE GLOBECOM, Nov. 2002, vol. 2,pp. 1688–1692.

[38] J. M. Huerta and J. Vidal, “Mobile tracking using UKF, time measuresand LOS–NLOS expert knowledge,” in Proc. IEEE ICASSP, Mar. 2005,vol. 4, pp. iv/901–iv/904.

Swaroop Venkatesh (S’03) received the B.Tech.and M.Tech. degrees in electrical engineering fromthe Indian Institute of Technology, Madras, India,in 2003, specializing in the design of communi-cation systems, and the Ph.D. degree in electricalengineering from the Mobile and Portable RadioResearch Group, Virginia Polytechnic Institute andState University, Blasksburg, in 2007, working on ul-trawideband localization, communication, and radarsystems.

He is currently with Marvell Semiconductor,Sunnyvale, CA, where he is involved in the design of next-generation wirelesssolutions. His interests include wireless-communication system design andmodeling, multiple-antenna systems, and multiple-access communication.

R. Michael Buehrer (S’89–M’91–SM’04) receivedthe B.S.E.E. and M.S.E.E. degrees from the Uni-versity of Toledo, Toledo, OH, in 1991 and 1993,respectively. He received the Ph.D. degree fromVirginia Polytechnic Institute and State University(Virginia Tech), Blacksburg, in 1996, where he stud-ied under the Bradley Fellowship.

From 1996 to 2001, he was with Bell Laboratories,Murray Hill, NJ, and Whippany, NJ. While at BellLaboratories, his research focused on code-divisionmultiple-access systems, intelligent antenna systems,

and multiuser detection. He was named a distinguished member of the Tech-nical Staff in 2000 and was a cowinner of the Bell Laboratories President’sSilver Award for research into intelligent antenna systems. In 2001, he waswith Virginia Tech as an Assistant Professor with the Bradley Departmentof Electrical Engineering where he is currently working with the Mobile andPortable Radio Research Group, as an Associate Professor. His current researchinterests include position location networks, ultrawideband, spread spectrum,multiple-antenna techniques, interference avoidance, and propagation mod-eling. In 2003, he was named Outstanding New Assistant Professor by theVirginia Tech College of Engineering. He has coauthored approximately 22journal and 65 conference papers and is the holder of ten patents in the area ofwireless communications.

Dr. Buehrer is currently an Associate Editor for IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON VEHICULAR

TECHNOLOGIES, and IEEE TRANSACTIONS ON SIGNAL PROCESSING.


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