nlos error mitigation for toa-based localization via convex relaxation

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 8, AUGUST 2014 4119

NLOS Error Mitigation for TOA-BasedLocalization via Convex Relaxation

Gang Wang, Member, IEEE, H. Chen, Youming Li, and Nirwan Ansari, Fellow, IEEE

Abstract—In this paper, we address the time-of-arrival (TOA)based localization problem in an adverse environment, whereline-of-sight (LOS) signal propagation between the source and thesensor is not readily available, in which case we have to resort tonon-line-of-sight (NLOS) signals. Two convex relaxation methods,i.e., the semidefinite relaxation (SDR) and the second-order conerelaxation (SOCR) methods, are proposed to mitigate the effect ofNLOS errors on the localization performance. We consider twoseparate cases in which the information of the NLOS status istotally unknown and perfectly known, respectively. The proposedmethods can be applied without knowing the distribution of NLOSerrors. Moreover, we propose a NLOS error mitigation methodthat is robust to detection errors, which are generated in theprocess of detecting NLOS paths. Simulation results show thatthe proposed convex relaxation methods outperform some existingstate-of-the-art methods.

Index Terms—Non-line-of-sight, robust localization, second-order cone programming (SOCP), semidefinite programming,time-of-arrival.

I. INTRODUCTION

LOCALIZATION by a wireless network has attracted muchattention in recent years since it finds wide applications,

such as emergency preparedness and response, intelligent trans-portation systems, target tracking, and others [1]. Some tradi-tional localization methods assume that the signal propagationbetween the source and the sensor is in line-of-sight (LOS) [2]–[4], which is not practical due to the obstruction of obstacles inadverse environments, such as in complex buildings and denseurban areas. This assumption can significantly degrade theperformance of the localization methods, which do not accountfor non-line-of-sight (NLOS) conditions. Thus, mitigation ofNLOS errors becomes an urgent task. In this paper, we focus

Manuscript received July 19, 2013; revised November 5, 2013, January 9,2014, and March 21, 2014; accepted March 22, 2014. Date of publicationMarch 31, 2014; date of current version August 8, 2014. This work wassupported in part by the National Natural Science Foundation of China un-der Grants 61201099 and 61301152, the Ningbo Municipal Natural ScienceFoundation under Grant 2012A610027, and the K. C. Wong Magna Fund inNingbo University. The associate editor coordinating the review of this paperand approving it for publication was D. Niyato.

G. Wang and Y. Li are with the College of Information Science andEngineering, Ningbo University, Ningbo 315211, China (e-mail: [email protected]; [email protected]).

H. Chen was with Southwest Jiaotong University, Chengdu 610031, China.N. Ansari is with the Advanced Networking Laboratory, Department of

Electrical and Computer Engineering, New Jersey Institute of Technology,Newark, NJ 07102 USA (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TWC.2014.2314640

on the NLOS mitigation problem in the time-of-arrival (TOA)based localization problem.

The NLOS mitigation problem has been extensively studiedin the last decade. One easy way of addressing this problemis to identify (detect) the NLOS paths, discard the NLOSmeasurements, and then use the LOS measurements to locatethe source [5], [6]. However, this incurs two drawbacks: 1) ifthe number of the LOS measurements is less than 3 in a 2-Dplane or 4 in a 3-D space, the source cannot be located ifany measurement is discarded; 2) the detection of NLOS pathsalways has a probability of missed detection or false alarm, andthis will degrade the localization performance. Even if the twodrawbacks are mitigated, using LOS measurements only is stillnot good enough since the NLOS measurements can be used toprovide some useful information on the source location [7].

The most popular method for localization is the maximumlikelihood (ML) method. Under NLOS conditions, one mustknow the exact distributions of both the NLOS errors andthe measurement noise to formulate the ML problem [8],[9]; obtaining these distributions is very difficult in practice.To partially alleviate the difficulty, Guvenc et al. [10], [11]proposed the weighted least squares (WLS) method, whichrequires only the first two moments of the measurement noiseand the NLOS errors. Another form of the WLS method is theresidual weighting (RW) method [12], which does not requirethe statistics of the NLOS errors. However, this method iscomputationally inefficient. Recently, the optimization basedmethods become popular. In [13], a constrained WLS methodwas proposed, in which the NLOS measurements are used toform a region in which the source lies. This method introducesa nuisance parameter to simplify the problem. However, thismethod ignores the known relationship between the sourcelocation and the nuisance parameter. Yang et al. [14] improvedthis method by explicitly incorporating the known relationship,which is referred to as the quadratic programming (QP) method.Venkatesh and Buehrer [7] proposed a linear programming (LP)based method, which uses the LOS measurements to obtain thecost function of the LP and uses the NLOS measurements toform a set of constraints on the source location. Yu and Guo[15] formulated a set of optimization problems based on the in-formation of the NLOS status and solved the problems using thesequential quadratic programming (SQP) algorithm. Lui et al.[16] proposed a maximum a posteriori (MAP) method whichassumes to know the prior information on the probability of theNLOS status, and the exact distribution of the NLOS errors.Chen et al. [17] proposed a semidefinite relaxation (SDR)method for cooperative sensor network localization underNLOS conditions, where three cases are considered based on

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4120 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 8, AUGUST 2014

the knowledge of the NLOS status and the distribution param-eters. Recently, localization using angle-of-arrival (AOA) andTOA measurements in the mixed LOS/NLOS environments hasbeen studied [18], [19]. In these methods, NLOS identificationis first performed, then the (single) reflection point is locatedusing AOA measurements, and finally the target node (TN)is located using TOA measurements. These methods requireboth the base nodes (BNs) and the TN to be equipped withmultiple antennas [18] or require cooperation between the BNsand the TN [19]. In summary, the existing methods, except theQP method in [14] and the third case in [17] (which, however,requires to know the measurement noise power), require one ofthe following conditions: 1) the statistics of the NLOS errorsare completely or partially known, and 2) the NLOS status foreach path or its probability is known.

In this paper, we solve the NLOS error mitigation problemby using convex relaxations, i.e., the SDR and the second-order cone relaxation (SOCR). In recent years, the convexrelaxation methods have been successfully applied to severalsource and sensor network localization problems [20]–[25]. Forthe localization problem studied in this paper, we propose tojointly estimate the source location and another parameter thatis related to the NLOS errors to partially mitigate the NLOSerrors. Owing to the non-convexity of the formulation, weapproximately solve it by applying SDR or SOCR to obtainthe convex semidefinite program (SDP) or second-order coneprogram (SOCP). Based on whether the NLOS status of eachpath is known or not, we consider two separate cases: theNLOS status is entirely unknown and it is perfectly known,respectively. Both methods do not require any statistics of theNLOS errors. Therefore, the proposed methods can be usedregardless of the distribution of the NLOS errors.

In practice, there always exists a probability of detection er-rors in identifying the NLOS status, including missed detectionand false alarm. This problem has not been well studied inthe literature. We, to the best of our knowledge, for the firsttime propose a localization method that is robust to detectionerrors. Unlike the convex relaxation methods stated above,which jointly estimate the NLOS errors and source location, therobust method takes the NLOS errors as nuisance parametersand estimates the source location only. Moreover, it neitherrequires to perform NLOS detection, nor requires to know thestatistics of NLOS errors, and only requires to know the upperbound of the NLOS errors.

The following notations are adopted throughout the paper.Bold face lower case letters and bold face upper case lettersdenote the vectors and matrices, respectively. ‖ · ‖ denotes the�2-norm. In denotes the n× n identity matrix, and 1n denotesthe n× 1 column vector. Aij denotes the (i, j)th entry of thematrix A, and tr(A) denotes the trace of matrix A. ai denotesthe ith entry of the vector a.

The rest of the paper is organized as follows. In Section II,we introduce the TOA measurement model under NLOS con-ditions. The proposed SDR and SOCR methods are presentedin Sections III and IV, respectively. The robust localizationmethod is presented in Section V. Simulations are conducted toverify the performance of the proposed methods in Section VI,and conclusions are drawn in Section VII.

II. PROBLEM FORMULATION

Consider a wireless sensor network (WSN) with N sensorsand one unknown source, where the locations of sensors arerespectively denoted by s1, . . . , sN , and the location of theunknown source is denoted by x (whose dimension is k × 1,k = 2 or 3). The range measurements are denoted by

di = ‖x− si‖+ ni, i ∈ El (1)

without considering the NLOS error, and

di = ‖x− si‖+ ei + ni, i ∈ En (2)

with considering the NLOS error, where ni is the measurementnoise, ei is the NLOS error, and El and En are the indexsets corresponding to the measurements of LOS and NLOS,respectively. We assume that ni follows a zero-mean Gaussiandistribution N (0, σ2

i ). Without loss of generality, we assumethat El = {1, . . . , Nl} and En = {Nl + 1, . . . , N}, where Nl isthe number of the LOS measurements and Nn = N −Nl is thenumber of the NLOS measurements.

In the literature, the NLOS error is assumed to be positiveand follows various distributions, e.g., uniform, Gaussian, andexponential distributions. In this paper, we do not assumeany distribution, i.e., our method can be applied under theassumption of any distribution. The only assumption here is thatthe NLOS error is much greater than the measurement noise. Itis worth noting that this assumption is made in most existingworks.

III. NLOS MITIGATION VIA SEMIDEFINITE RELAXATION

In this section, we present the SDR method for the NLOSmitigation problem.

A. Unknown NLOS Status

First, we consider the most difficult case that no informationon the NLOS status is known. Our idea is to estimate thesource location as well as the NLOS errors using both LOSand NLOS measurements, i.e., without discarding any NLOSmeasurements. To this end, we first rewrite (1) and (2) in theuniform form

di − ei = ‖x− si‖+ ni, i = 1, . . . , N (3)

where for i ∈ El, ei = 0.Based on (3), the ML estimation of x and ei(i = 1, . . . , N)

can be formulated as

minx,{ei}

N∑i=1

(di − ‖x− si‖ − ei)2

σ2i

s.t. ei ≥ 0, i = 1, . . . , N (4)

where we impose a set of known constraints ei ≥ 0, i = 1,. . . , N .

Mathematically, Problem (4) yields a trivial solution. Any

solution [xT , e1, . . . , eN ]T

that satisfies di − ‖x− si‖ ≥ 0,

WANG et al.: NLOS ERROR MITIGATION FOR TOA-BASED LOCALIZATION 4121

i = 1, . . . , N is the optimal solution to (4) because the optimalvalue of the objective function, 0, is achieved if we set ei =di − ‖x− si‖. This indicates that Problem (4) is meaninglessif we do not have enough LOS measurements and do not knowthe NLOS status, or we do not have other information on ei,i = 1, . . . , N . As a result, we estimate only one NLOS errorinstead of all NLOS errors. As pointed out in [15], this NLOSerror plays the role of a “balancing” parameter that partiallymitigates the NLOS effect. Hence, instead of solving (4), wesolve the following problem

minx,e

N∑i=1

(di − ‖x− si‖ − e)2

σ2i

(5)

where e is seen as the “balancing” parameter.In the ideal case, the NLOS errors are completely mitigated,

i.e., for i = 1, . . . , N , we have ei = ei (Here, ei is the truevalue of the ith NLOS error). Thus, in the ideal case, the sourcelocation estimate is the optimal value of the following problem:

minx

N∑i=1

(di − ‖x− si‖ − ei)2

σ2i

. (6)

Note that (6) is different from (4) since the NLOS errors arecompletely mitigated in (6).

In the following, we examine how close (5) is in approximat-ing (6). Define the approximation error as

pΔ=

[N∑i=1

(di−‖x−si‖−ei)2

σ2i

−N∑i=1

(di−‖x−si‖−e)2

σ2i

]2.

(7)

Under the assumption of ei � ni, we have di − ‖x− si‖ ≈ ei.Replacing di − ‖x− si‖ in (7) with ei, we have the followingapproximation:

p ≈[

N∑i=1

(ei − e)2

σ2i

]2. (8)

Note that if the ith measurement is LOS, i.e., ei = 0, (8) stillholds.

We find e by minimizing the approximation error

e ≈ argmine

[N∑i=1

(ei − e)2

σ2i

]2

= argmine

N∑i=1

(ei − e)2

σ2i

=

(N∑i=1

eiσ2i

)/(N∑i=1

1

σ2i

). (9)

Based on (9), we can find an upper bound on e. Since thetrue range between the source and the ith sensor, ‖x− si‖, isgenerally greater than the absolute value of the noise |ni|, wehave ei < di. Hence, from (9), we have an upper bound on e:e < (

∑Ni=1(di/σ

2i ))/(

∑Ni=1(1/σ

2i )). This bound is reasonable

since it is obtained based on the optimal e, e. Imposing thisconstraint and the explicit constraint e > 0 to (5), we have

minx,e

N∑i=1

(di − ‖x− si‖ − e)2

σ2i

s.t. 0 < e <

(N∑i=1

di/σ2i

)/(N∑i=1

1

σ2i

). (10)

Recall that we have assumed that the NLOS error is muchgreater than the measurement noise. Based on this assumption,we can obtain a set of constraints on the source location, likethose in [14]. Since ei � ni, we have

‖x− si‖ ≤ di, i = 1, . . . , N. (11)

Squaring both sides of (11) yields

‖x− si‖2 ≤ d2i , i = 1, . . . , N (12)

which implies

‖x‖2 − 2sTi x ≤ d2i − ‖si‖2, i = 1, . . . , N. (13)

Writing all inequalities in (13) into the matrix form gives

H

[x

‖x‖2]≤ f (14)

where

H =

⎡⎣ −2s1 1

......

−2sN 1

⎤⎦ f =

⎡⎢⎣

d21 − ‖s1‖2...

d2N − ‖sN‖2

⎤⎥⎦ . (15)

Adding (14) to (10), we have

minx,e

N∑i=1

(di − ‖x− si‖ − e)2

σ2i

s.t. 0 < e <

(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)

H

[x

‖x‖2]≤ f . (16)

It is worth noting that adding (14) to (10) is helpful for improv-ing the localization performance [7].

Problem (16) is very difficult to solve since it is noN - convex.In this paper, we relax (16) into an SDP problem, which isconvex and can be solved efficiently using the interior pointmethods.

Problem (16) can be equivalently written as

ming,x

(Bg − d)TQ−1(Bg − d)

s.t. g = [‖x− s1‖, . . . , ‖x− sN‖, e]T

0 < gN+1 <

(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)

H[xT ‖x‖2

]T ≤ f , (17)


where Q = diag{σ21 , . . . , σ

2N}, B = [IN 1N ], and d = [d1,

. . . , dN ]T .Problem (17) can be further written as

minG,g,r,x

tr

{C

[G ggT 1

]}s.t. Gii = r − 2sTi x+ ‖si‖2, i = 1, . . . , N[

G ggT 1

]� 0, rank{G} = 1

‖x‖2 = r,

gN+1 > 0

GN+1,N+1 <

[(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)]2

H[xT r]T ≤ f (18)

where C =

[BTQ−1B −BTQ−1d−dTQ−1B dTQ−1d

]. Note that we do not

put an upper bound on gN+1 since from [G g; gT 1] � 0, wehave g2N+1 ≤ GN+1,N+1, which implies if the upper bound ofGN+1,N+1 is determined, so will be that of gN+1.

Furthermore, according to the Cauchy-Schwarz inequality,we have

Gij = ‖x− si‖‖x− sj‖≥∣∣(x− si)

T (x− sj)∣∣ = ∣∣r − (si + sj)

Tx+ sTi sj∣∣ .(19)

Adding (19) to (18), dropping the rank-1 constraint andrelaxing ‖x‖2 = r as ‖x‖2 ≤ r, we can obtain the SDP:

minG,g,r,x

tr

{C

[G ggT 1

]}s.t. Gii = r − 2sTi x+ ‖si‖2, i = 1, . . . , N

Gij ≥∣∣r − (si + sj)

Tx+ sTi sj∣∣

i, j = 1, . . . , N, i > j,[G ggT 1

]� 0,

[Ik xxT r

]� 0

gN+1 > 0,

GN+1,N+1 <

[(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)]2

H[xT r]T ≤ f . (20)

Note that adding (19) can improve the localization perfor-mance. According to [25], only when the off-diagonal elementsof C are negative, rank{G} = 1 holds. If this is true, adding(19) is not helpful for performance improvement. However, thiscondition does not hold for this problem, and thus adding (19)is necessary.

B. Known NLOS Status

In this section, we consider the case that the NLOS statusis perfectly identified. To utilize this information and improve

the performance, we propose to address the LOS and NLOSmeasurements separately. For LOS measurements, we have

di = ‖x− si‖+ ni, i ∈ El. (21)

Squaring both sides of (21) gives

d2i = ‖x− si‖2 + 2‖x− si‖ni + n2i

= ‖x− si‖2 + 2dini − n2i , i ∈ El. (22)

With simple manipulations, we have

d2i − ‖x− si‖22di

≈ ni, i ∈ El (23)

where n2i is neglected. n2

i can be neglected only when 2di ismuch greater than ni. This implies that only when 2di is compa-rable to ni, i.e., the source is sufficiently close to the ith sensor,neglecting n2

i will cause performance degradation. However,in practice, if this happens, we usually do not need to locatethe source using localization algorithms. Thus, neglecting thesecond-order terms is reasonable. Indeed, this has been appliedin many existing works, such as [26]–[28].

Based on (2) and (23), we can formulate another localizationproblem under NLOS conditions:

minx,e

∑i∈El

(d2i − ‖x− si‖2

)24d2iσ

2i

+∑i∈EN

(di − ‖x− si‖ − e)2

σ2i

s.t. 0 < e <

Nn∑j=1

(dNl+j/σ

2Nl+j

)Nn∑j=1

(1/σN2

l+j

) (24)

where we still jointly estimate the source location and oneNLOS error as earlier.

Similarly, we can add the following constraints to (24)

Hn

[xT ‖x‖2

]T ≤ fn (25)

where

Hn =

⎡⎢⎣−2sNl+1 1

......

−2sN 1

⎤⎥⎦ , fn =

⎡⎢⎣d2Nl+1 − ‖sNl+1‖2

...d2N − ‖sN‖2

⎤⎥⎦ .

(26)

Similar to (18), (24) can be equivalently written as

mingn,x,{ci}

∑i∈El

ci + tr

{Cn

[gng

Tn gn

gTn 1

]}

s.t.

(d2i − ‖x− si‖2

)24d2iσ

2i

≤ ci, i ∈ El

gn = [‖x− sNl+1‖, . . . , ‖x− sN‖, e]T

gn,Nn+1 > 0

Gn,{N+1,N+1} <

⎡⎣∑Nn

j=1

(dNl+j/σ

2Nl+j

)∑Nn

j=1

(1/σ2

Nl+j

)⎤⎦2

Hn[xT ‖x‖2]T ≤ fn, (27)


where Cn=

[BT

nQ−1n Bn −BT

nQ−1n dn

−dTnQ

−1n Bn dT

nQ−1n dn

], Qn=diag{σ2

Nl+1,

. . . , σ2N}, Bn = [INn

1Nn], and dn = [dNl+1, . . . , dN ]T .

Problem (27) can be further written as

minGn,gn,r,x,{ci}

∑i∈El

ci + tr

{Cn

[Gn gn

gTn 1

]}

s.t.(d2i − r + 2sTi x− ‖si‖2

)2 ≤ 4cid2iσ

2i , i ∈ El,

Gn,jj = r − 2sTNl+jx+ ‖sNl+j‖2, j = 1, . . . , Nn,[Gn gn

gTn 1

]� 0, rank{Gn} = 1

‖x‖2 = r,

gn,Nn+1 > 0,

Gn,{N+1,N+1} <

⎡⎣∑Nn

j=1

(dNl+j/σ

2Nl+j

)∑Nn

j=1

(1/σ2

Nl+j

)⎤⎦2

Hn[xT r]

T ≤ fn. (28)

In a similar manner, we can relax (28) into the following SDP:

minGn,gn,r,x,{ci}

∑i∈El

ci + tr

{Cn

[Gn gn

gTn 1

]}

s.t.(d2i − r + 2sTi x− ‖si‖2

)2 ≤ 4cid2iσ

2i , i ∈ El

Gn,jj = r − 2sTNl+jx+ ‖sNl+j‖2, j = 1, . . . , Nn

Gn,mj ≥ |r − (sNl+m + sNl+j)Tx+ sTNl+msNl+j |

m, j = 1, . . . , Nn, m > j[Gn gn

gTn 1

]� 0,

[Ik xxT 1

]� 0

gn,Nn+1 > 0

Gn,{N+1,N+1} <

⎡⎣∑Nn

j=1

(dNl+j/σ

2Nl+j

)∑Nn

j=1

(1/σ2

Nl+j

)⎤⎦2

Hn[xT r]T ≤ fn. (29)

C. Feasibility Problem

It is worth noting that the constraints in (14) (or (25)) ispossibly not satisfied due to the fact that some paths are LOS orthe NLOS errors are small. Hence, we need to identify whetherthe constraints are satisfied before solving the SDP. To this end,we solve the following feasibility problem when the NLOSstatus is unknown:

find x, r

such that ‖x‖2 = r

H[xT r]T ≤ f . (30)

For the relaxed version of the original problem (18), i.e., (20),we solve the following feasibility problem:

find x, r

such that ‖x‖2 ≤ r

H[xT r]T ≤ f . (31)

If (31) is infeasible, we replace the constraints in (31) by theconstraint:

∑Ni=1 hi[x

T r]T ≤

∑Ni=1 fi, where hi and fi are

the ith row of H and the ith element of f , respectively.In some very rare cases, we cannot find feasible points un-

der the constraint∑N

i=1 hi[xT r]

T ≤∑N

i=1 fi. In such cases,we remove the constraints formed by NLOS measurements.Since these cases often occur when the number of NLOSmeasurements is very small or the NLOS errors are very small,removing these constraints would not cause much performancedegradation.

Similarly, when the NLOS status is unknown, we solve thefollowing feasibility problem:

find x, r

such that ‖x‖2 ≤ r

Hn[xT r]

T ≤ fn. (32)

If (32) is infeasible, we perform a similar procedure like thatwhen the NLOS status is unknown.

The SDR method when the NLOS status is unknown issummarized in Algorithm 1, and that when the NLOS statusis known is similar to Algorithm 1.

Algorithm 1 NLOS Mitigation Using SDR When the NLOSStatus is Unknown

Input:{di}: range measurements;{σ2

i }: variances of the measurement noise;{si}: sensor locations;

1: solve problem (31);2: if (31) is feasible, solve (20);3: if (31) is infeasible, solve the following problem

find x, r

such that ‖x‖2 ≤ rN∑i=1

hi[xT r]

T ≤N∑i=1

fi. (33)

4: if (33) is infeasible, solve (20) by removing H[xT r]T ≤

f ; otherwise, solve (20) by replacing H [xT r]T ≤ f

with∑N

i=1 hi[xT r]

T ≤∑N

i=1 fi.Output: the solution of (20), x, is the estimate of the source

location.

IV. NLOS MITIGATION VIA SECOND-ORDER

CONE RELAXATION


Squaring both sides of (3) yields

(di − ei)2 = ‖x− si‖2 + 2‖x− si‖ni + n2

i . (34)

With simple manipulations, we have

(di − ei)2 − ‖x− si‖2

2‖x− si‖≈ ni (35)


where n2i is negligible since it is generally much smaller than

2‖x− si‖ni.Based on (35), the approximate ML estimation of x and

ei(i = 1, . . . , N) can be formulated as

minx,{ei}

N∑i=1

[(di − ei)

2 − ‖x− si‖2]2

4σ2i ‖x− si‖2

. (36)

We can obtain a localization problem similar to (16)

minx,e

N∑i=1

[(di − e)2 − ‖x− si‖2

]24σ2

i ‖x− si‖2

s.t. 0 < e <

(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)

H

[x

‖x‖2]≤ f . (37)

Note that imposing the constraints are reasonable since (36) isa close approximation to (4) [only the second-order noise termsare neglected in (35)].

We can equivalently rewrite (37) as

minx,e,{ai},{bi},u,v

N∑i=1

(ai − bi)2

4σ2i bi

s.t. 0 < e <

(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)

e2 = u,

ai = d2i − 2die+ u, i = 1, . . . , N

‖x‖2 = v

bi = v − 2sTi x+ ‖si‖2, i = 1, . . . , N

H[xT v]T ≤ f . (38)

Similarly, we relax (38) into an SOCP problem. To this end,we relax e2 = u and ‖x‖2 = v into e2 ≤ u and ‖x‖2 ≤ v,respectively, thus yielding the following problem

minx,e,{ai},{bi},u,v

N∑i=1

(ai − bi)2

4σ2i bi

s.t. e > 0

e2 ≤ u <

[(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)]2

ai = d2i − 2die+ u, i = 1, . . . , N

‖x‖2 ≤ v

bi = v − 2sTi x+ ‖si‖2, i = 1, . . . , N

H[xT v]T ≤ f (39)

which can be equivalently written in the epigraph form

minx,e,{ai},{bi},{ci},u,v

N∑i=1

ci

s.t.(ai − bi)

2

4σ2i bi

≤ ci, i = 1, . . . , N

e > 0

e2 ≤ u <

[(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)]2

ai = d2i − 2die+ u, i = 1, . . . , N

‖x‖2 ≤ v,

bi = v − 2sTi x+ ‖si‖2, i = 1, . . . , N

H [xT v]T ≤ f (40)

where ci(i = 1, . . . , N) are introduced to form a linear objec-tive function.

Problem (40) can be further written into an SOCP

minx,e,{ai},{bi},{ci},u,v

N∑i=1

ci

s.t.∥∥[4σ2

i bi−ci; 2(ai−bi)]∥∥≤4σ2

i bi+ci, i = 1, . . . , N

e > 0

‖[e;u− 1/4]‖ ≤ u+ 1/4

u <

[(N∑i=1

diσ2i

)/(N∑i=1

1

σ2i

)]2

ai = d2i − 2die+ u, i = 1, . . . , N

‖[x; v − 1/4]‖ ≤ v +1

4bi = v − 2sTi x+ ‖si‖2, i = 1, . . . , N

H [xT v]T ≤ f . (41)


For NLOS measurements, we have a similar result to (35) asfollows:

(di − ei)2 − ‖x− si‖2

2‖x− si‖≈ ni, i ∈ En. (42)

To partially mitigate the NLOS error, we similarly estimate thesource position and one NLOS parameter as that in Section IV-A.Thus, we can obtain a similar optimization problem to (37) asfollows:

minx,e

∑i∈El

(d2i − ‖x− si‖2

)24σ2

i d2i

+∑i∈En

[(di − e)2 − ‖x− si‖2

]24σ2

i ‖x− si‖2

s.t. 0 < e <

∑Nn

j=1

(dNl+j/σ

2Nl+j

)∑Nn

j=1

(1/σ2

Nl+j

)Hn

[x

‖x‖2]≤ fn (43)

where the constraint in (25) is incorporated.


In a similar manner, we can relax (43) into the followingSOCP:

minx,e,{ai},{bi},{ci}, u,v

N∑i=1

ci

s.t.∥∥[d2i − bi; 4σ

2i d

2i ci − 1/4

]∥∥ ≤ 4σ2i d

2i ci + 1/4, i ∈ El∥∥[4σ2

i bi − ci; 2(ai − bi)]∥∥ ≤ 4σ2

i bi + ci, i ∈ Ene > 0

‖[e;u− 1/4]‖ ≤ u+ 1/4

u <

⎡⎣∑Nn

j=1

(dNl+j/σ

2Nl+j

)∑Nn

j=1

(1/σ2

Nl+j

)⎤⎦2

ai = d2i − 2die+ u, i ∈ En‖[x; v − 1/4]‖ ≤ v + 1/4

bi = v − 2sTi x+ ‖si‖2, i = 1, . . . , N

Hn[xT v]

T ≤ fn. (44)

Note that it is also required to solve the feasibility problem(31) or (32) before solving the SOCP.

The SOCR method when the NLOS status is unknown issummarized in Algorithm 2, and that when the NLOS statusis known is similar to Algorithm 2.

Algorithm 2 NLOS Mitigation Using SOCR When theNLOS Status is Unknown

Input:{di}: range measurements;{σ2

i }: variances of the measurement noise;{si}: sensor locations;

1: solve problem (31);2: if (31) is feasible, solve (41);3: if (31) is infeasible, solve (33);4: if (33) is infeasible, solve (41) by removing H[xT r]

T ≤f ; otherwise, solve (41) by replacing H[xT r]

T ≤ f

with∑N

i=1 hi[xT r]

T ≤∑N

i=1 fi.Output: the solution of (41), x, is the estimate of the source

location.

V. ROBUST LOCALIZATION USING

SEMIDEFINITE RELAXATION

In the above, we have assumed that the NLOS paths areperfectly detected without any detection errors. However, de-tection errors are inevitable in practice. Motivated by [21], wepropose a localization method that is robust to detection errorsin this section. This method only requires to specify the upperbound of NLOS errors, and neither requires to perform NLOSdetection, nor requires to know the statistics of NLOS errors.

A. Robust Formulation

Assume that the upper bound of the NLOS errors is denotedas ρ and is known, i.e.,

0 ≤ ei ≤ ρ, i = 1, . . . , N (45)

which implies that∣∣∣ei − ρ

2

∣∣∣ ≤ ρ

2, i = 1, . . . , N =⇒

∥∥∥e− ρ

21N

∥∥∥ ≤ ρ

2

√N

(46)

with e = [e1, e2, . . . , eN ]T .By taking e as a nuisance vector, we formulate the localiza-

tion problem as

minx,g,e

(d− g − e)TQ−1(d− g − e)

s.t. gi = ‖x− si‖, i = 1, . . . , N. (47)

The worst-case robust formulation can be written as:

minx,g

supe

{(d− g − e)TQ−1(d− g − e)

}s.t. gi = ‖x− si‖, i = 1, . . . , N (48)

which is equivalent to

minx,g,η

η

s.t. supe{(d− g − e)TQ−1(d− g − e)} ≤ η

gi = ‖x− si‖, i = 1, . . . , N. (49)

The constraint

supe

{(d−g−e)TQ−1(d−g−e)|

∥∥∥e− ρ

21N

∥∥∥ ≤ ρ

2

√N}≤ η

(50)

implies that

∀e ∈{e|∥∥∥e− ρ

21N

∥∥∥ ≤ ρ

2

√N}

=⇒ (d− g − e)TQ−1(d− g − e)− η ≤ 0 (51)

i.e.,[e1

]T [IN −ρ

21N

−ρ21

TN 0

] [e1

]≤ 0

=⇒[e1

]T[Q−1 −Q−1(d− g)

−(d− g)TQ−1 q − η

][e1

]≤0

(52)

with q = (d− g)TQ−1(d− g).By using the S-procedure [29], (52) implies that there exists

λ ≥ 0 such that[Q−1 −Q−1(d− g)

−(d− g)TQ−1 q − η

]� λ

[IN −ρ1N

2

−ρ1TN

2 0

](53)

i.e.,[λIN −Q−1 Q−1(d− g)− λρ

2 1N

(d− g)TQ−1 − λρ2 1T

N −q + η

]� 0.

(54)


Thus, the robust formulation can be rewritten as

minx,g,η,λ

η

s.t.

[λIN −Q−1 Q−1(d− g)− λρ

2 1N

(d− g)TQ−1 − λρ2 1T

N −q + η

]� 0

q = (d− g − e)TQ−1(d− g − e)

gi = ‖x− si‖, i = 1, . . . , N, λ ≥ 0. (55)

Applying SDR, we have the following SDP:

minG,g,x,r,η,λ

η

s.t.

[λIN−Q−1 Q−1(d−g)− λρ

2 1N

(d−g)TQ−1− λρ2 1T

N −q + η

]�0

q = tr(Q−1G)− 2dTQ−1g + dTQ−1d[G ggT 1

]� 0,

[Ik xxT r

]� 0

Gii = r − 2sTi x+ ‖si‖2, i = 1, . . . , N

Gij ≥ |r − (si + sj)Tx+ sTi sj |,

i, j = 1, . . . , N, i > j, λ ≥ 0

H[xT r]T ≤ f (56)

where the constraint H[xT r]T ≤ f is added for the purpose

of performance improvement.

B. Robustness Analysis

According to the Schur complement, we can equivalentlyrewrite the first constraint in (55) into

q +

[(d− g)TQ−1 − λρ

21TN

](λIN −Q−1)−1×

[Q−1(d− g)− λρ

21N

]≤ η (57)

which can be equivalently written as:

q+

(d−g− λρ

2Q1N

)T

(λQ2−Q)−1

(d−g− λρ

2Q1N

)≤η.

(58)

Minimizing η in (55) is equivalent to minimizing the termon the left-hand side of (58) since (55) is written in theepigraph form. From (58), we see that there is an ad-ditional term (d− g − (λρ/2)Q1N )T (λQ2 −Q)

−1(d− g −

(λρ/2)Q1N ) as compared with the non-robust localizationproblem (which minimizes q). This term can be seen as aregularization term. The value of the regularization term isdetermined by the values of ρ, λ, and Q, which can be regardedas regularization factors. ρ and Q are assumed to be known,and λ is obtained by solving the relaxed version of the robustformulation (55), i.e., the SDP (56). Thus, after solving (56),the regularization term is automatically determined.

VI. SIMULATION RESULTS

In this section, simulations are conducted to verify the per-formance of the proposed methods. We compare the perfor-mance of the proposed methods with the existing state-of-the-art LP [7], QP [14], and min-max algorithm (MMA) [30].1

The performance of a randomly guessed location in the local-ization area (denoted by “Random Guess”) is also presentedfor comparison. In the following, we use “SDP”, “SOCP”, and“SDP-Robust” to represent the SDR, the SOCR, and the robustmethod, respectively.

We assume that both the sensor and the source locationsare randomly chosen from a region of size 10× 10 m2. Weperform M independent Monte Carlo (MC) runs, and in eachrun, we locate a randomly located source using a random sensorgeometry. Following [7], the measurement noise is assumed tofollow the Gaussian distribution with zero mean and identicalvariance σ2 for each sensor, i.e., σ2

1 = σ22 = · · · = σ2

N = σ2.The NLOS error is assumed to follow the uniform distributionU(0, Bmax). If not specified, Bmax is set as Bmax = 5 in thefollowing. The performance is evaluated using the root meansquare error (RMSE), which is computed by

RMSE =

√√√√ 1

M

M∑i=1

‖xi − xi‖2. (59)

Here, xi and xi are the estimate of the source location andthe true source location in the ith run, respectively. In thesimulations, we set M = 3000. The SDP and SOCP are solvedusing the MATLAB toolbox CVX [31], and the solver isSeDuMi [32].


In this section, we consider the case when the NLOS status isunknown. We consider three scenarios: 1) the number of NLOSmeasurements is small; 2) the number of NLOS measurementsis large; and 3) the fraction of NLOS measurements varies.

1) Scenario 1: In this scenario, we assume that the num-ber of sensors is N = 5, and the numbers of NLOS and theLOS measurements are 1 and 4, respectively, i.e., Nn = 1 andNl = 4. The performance of the proposed convex relaxationmethods and the QP method in [14] is compared in Fig. 1.From this figure, we see that QP performs better than the othermethods. This is attributed to the following reasons: 1) when thenumber of LOS measurements is large, the objective functionin the QP method is not subject to much mismatch caused byNLOS errors and the QP can be solved exactly (see [14] fordetails), while SDP, SOCP, and SDP-Robust are relaxationsof the original problems; and 2) the case, where there are nofeasible points in (31), frequently occurs in SDP, SOCP, andSDP-Robust, leading us to use the relaxed constraints (seeSection III-C), while this infeasibility problem does not exist

1Although MMA is not designed for NLOS mitigation, it can be tailoredto do so. In our simulations, we add the constraint (14) (when NLOS statusis unknown) or (25) (when NLOS status is known) to the original min-maxproblem.


Fig. 1. Comparison of RMSE using different methods when the NLOS statusis unknown: Nn = 1 and Nl = 4.

Fig. 2. Comparison of RMSE using different methods when the NLOS statusis unknown: Nn = 4 and Nl = 1.

in QP since the constraint in QP is looser (note that in QP, itonly has one constraint H[xT r]

T ≤ f [14]).2) Scenario 2: In this scenario, the numbers of NLOS and

the LOS measurements are assumed to be 4 and 1, respectively,i.e., Nn = 4 and Nl = 1. The results are shown in Fig. 2, fromwhich we see that SDP-Robust performs the best, and bothSDP-Robust and SDP perform better than QP. SOCP performsbetter than QP when the noise is not very large. This is due tothe fact that the mismatch in the objective function of the QPbecomes very large when there are more NLOS measurements.When the noise is small, the performance of QP is worse thanthat when the noise is large. This strange observation can beexplained as follows. If the noise is large and negative, theNLOS error can be partially offset, which makes the linearconstraints generated by the NLOS errors tighter.

3) Scenario 3: In this scenario, we fix the number of sen-sors and change the number of the NLOS measurements. Thenumber of sensors is fixed as N = 10, and the number of NLOSmeasurements varies from 5 to 10, and the standard deviation of

Fig. 3. RMSE versus the fraction of NLOS measurements when the NLOSstatus is unknown.

the measurement noise is fixed as 0.5 m. The results are shownin Fig. 3, from which we see that the increase of the NLOSmeasurements improves the performance of SDP-Robust. Notethat the only factor that affects the performance of SDP-Robustis the upper bound of the NLOS error ρ. Apparently, ρ is toolarge when the number of NLOS measurements is small, whichwould degrade the performance of SDP-Robust. In comparison,the performance of SDP, SOCP, and MMA almost remainsrelatively stable. This indicates that the capability of NLOSmitigation of the three methods is rather consistent. As thenumber of NLOS measurements increases, the “balancing”parameter would also increase to partially offset the errors inNLOS measurements. Moreover, QP performs the best whenthe number of NLOS measurements is small, which complieswith the results in Fig. 1.


In this section, we consider the case when the NLOS statusis perfectly known. Similarly, we also consider three scenarios.

1) Scenario 4: Fig. 4 plots the performance of several meth-ods when Nn = 1 and Nl = 4. From this figure, we see thatSOCP and SDP perform comparably, and both perform betterthan QP, LP, and MMA. The performance is mostly determinedby LOS measurements if the number of LOS measurements ismuch greater than that of NLOS measurements. Note that whenthe NLOS status is known, the first summation term in (24)is exactly the same as that in (43), and this is also true whenthey are relaxed. This leads to similar performance betweenSDP and SOCP. SDP-robust performs badly in this scenariobecause 1) it does not use the prior information of NLOSstatus, and 2) it uses an improper upper bound ρ for NLOSerrors. In comparing Fig. 1 and Fig. 4, we see that QP performseven worse when the NLOS status is known. This reveals thatthe performance of QP is rather dependent on the constraintsimposed by the NLOS measurements, since when the NLOSstatus is known, the number of the constraints imposed by theNLOS measurements is smaller.


Fig. 4. Comparison of RMSE using different methods when the NLOS statusis known: Nn = 1 and Nl = 4.

Fig. 5. Comparison of RMSE using different methods when the NLOS statusis known: Nn = 4 and Nl = 1.

2) Scenario 5: We further evaluate the performance of theproposed methods when the number of NLOS measurementsis large, i.e., Nn = 4 and Nl = 1. Fig. 5 shows the simulationresults, from which we see that SDP-Robust performs the best,and SDP performs better than SOCP, QP, and MMA. Theperformance of LP is not shown here because LP gives onlya very coarse estimate of the source location when the numberof LOS measurements is less than 3.

It is worth noting that the comparison of Figs. 1 and 4 (orFigs. 2 and 5) reveals that the information of the NLOS statuscan improve the localization performance.

3) Scenario 6: In this scenario, the parameter setup is thesame as that in Scenario 3, except that we assume the NLOSstatus is known. Fig. 6 plots the performance curves of dif-ferent methods. From this figure, we see that SDP and SOCPperform the best when the number of NLOS measurementsis small, while SDP-Robust performs the best in a moredense NLOS environment, which complies with the results inFigs. 4 and 5.

Fig. 6. RMSE versus the fraction of NLOS measurements when the NLOSstatus is known.

Fig. 7. Effect of missed detection on the performance when Bmax = 5.

C. Sensitivity to NLOS Detection Errors

In the last subsection, we have assumed that the NLOS statusis perfectly known. However, this is not true in practice. In thissubsection, we examine the effect of NLOS detection errors onthe localization performance.

We consider two types of detection errors: missed detectionand false alarm. In this subsection, we assume that the numbersof both the NLOS and LOS measurements are 3, i.e., Nn =Nl = 3.

1) Scenario 7: In this scenario, we consider the effect ofmissed detection, i.e., NLOS measurements are detected asLOS measurements. Assume that two NLOS measurementsare detected as LOS. Two cases that the bound of the NLOSerrors is Bmax = 5 and Bmax = 8, respectively, are examined.The simulation results are shown in Figs. 7 and 8, respectively.From the two figures, we see that SDP-Robust is very robustregardless of large or small NLOS errors. SDP and SOCP alsoperform very well, while other methods are quite sensitive tomissed detection.


Fig. 8. Effect of missed detection on the performance when Bmax = 8.

Fig. 9. Effect of false alarm on the performance when Bmax = 5.

2) Scenario 8: In this scenario, we consider the effect offalse alarm, i.e., LOS measurements are detected as NLOSmeasurements. Figs. 9 and 10 show the simulation results whenthe bound of the NLOS errors is Bmax = 5 and Bmax = 8,respectively. From the figures, we see that SDP-Robust is quiterobust to false alarm, while other methods are more or lesssensitive.

Comparing Figs. 7–10, we have a strange observation:when Bmax = 5, missed detection causes graceful performancedegradation as compared to false alarm, while when Bmax = 8,the conclusion is opposite, i.e., false alarm causes gracefulperformance degradation as compared to missed detection. Thiscan be explained as follows. Note that the detection errorswill cause two kinds of mismatches: mismatch in the objectivefunction and mismatch in the constraints formed by NLOSmeasurements. When the NLOS error is small, i.e., the casewhen Bmax = 5, the effect of mismatch in the constraintswould dominate that of mismatch in the objective function.On one hand, the mismatch in the objective function causedby small NLOS errors is small; on the other hand, if LOS

Fig. 10. Effect of false alarm on the performance when Bmax = 8.

measurements are detected as NLOS, the constraints formedby the NLOS measurements will be much tighter than the trueconstraints. This may cause the infeasibility problem, whichdegrades the performance. When the NLOS error is large, i.e.,the case when Bmax = 8, the constraints formed by NLOSmeasurements are loose, and the mismatch in the constraintsdoes not have a significant effect even if false detections oc-cur. However, the mismatch in the objective function causedby NLOS errors will become large, especially when NLOSmeasurements are detected as LOS.

D. Summary

We summarize the results of the eight localization scenariosunder NLOS conditions as follows:

1) The proposed methods perform worse than the existingmethod only when the number of NLOS measurements isquite small.

2) The known information of the NLOS status is helpful forimproving the localization performance.

3) The degree of the effect of missed detection and falsealarm depends on the levels of NLOS errors. If theNLOS errors are large, missed detection causes moreperformance degradation; otherwise, false alarm exertsmore damage.

4) The SDP-Robust method performs very well in all exam-ined scenarios expect Scenario 4.

The above results give us some guidelines in choosing meth-ods for different environments. For instance, if the source lies ina sparse NLOS environment and the NLOS status is unknown,QP is the best choice; if it lies in a sparse NLOS environmentand the NLOS status is known, SDP and SOCP are the bestchoices; if it lies in a dense NLOS environment, SDP-Robustwould be the best choice.

VII. CONCLUSION

In this paper, we have considered the TOA-based localizationproblem under NLOS conditions, and proposed two convex


relaxation methods, i.e., the SDR and the SOCR method.We have considered two separate cases: known and unknownNLOS status. In both cases, the distribution of the NLOS errorsis not required. Furthermore, we have considered the impactof NLOS detection errors, and proposed a localization methodthat is robust to detection errors. Simulation results show thesuperior performance of the proposed methods over the existingmethods. Moreover, the results give us some guidelines inchoosing methods for different environments.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersand the Editor for their valuable comments, which have greatlyimproved the quality of the paper. The first author wouldalso like to thank Prof. Anthony M.-C. So for the helpfuldiscussions.

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Gang Wang (M’13) received the B.Eng. degreefrom Shandong University, Jinan, China, and thePh.D. degree from Xidian University, Xi’an, China,both in electrical engineering, in 2006 and 2011,respectively.

He joined Ningbo University, Ningbo, China, inJanuary 2012. His research interests are in the area oftarget localization and tracking in wireless networks.

H. Chen received the B.S. and M.S. degrees fromSouthwest Jiaotong University, Chengdu, China, andthe Ph.D. degree from the University of Tokyo,Japan.

His research interests include Wireless Localiza-tion, Wireless Sensor Networks, and Statistical Sig-nal Processing.


Youming Li received the B.S. degree in com-putational mathematics from Lanzhou University,Lanzhou, China, in 1985, the M.S. degree in compu-tational mathematics from Xi’an Jiaotong University,Xi’an, China, in 1988, and the Ph.D. degree in elec-trical Engineering from Xidian University, Xi’an,China, in 1995.

From 1988 to 1998, he worked in the Departmentof Applied Mathematics, Xidian University, Xi’an,where he was an Associate Professor. From 1999to 2004, he worked in the School of Electrical and

Electronics Engineering, Nanyang Technological University, DSO NationalLaboratories, Singapore; and the School of Engineering, Bar-Ilan University,Israel, respectively. Since 2005, he has been with Ningbo University, Ningbo,China, where he is currently a Professor. His research interests are in cognitiveradio and wireless/wireline communications.

Nirwan Ansari (S’78–M’83–SM’94–F’09) receivedthe B.S.E.E. (summa cum laude with a perfect GPA)degree from NJIT, the M.S.E.E. degree from theUniversity of Michigan, Ann Arbor, MI, USA, andthe Ph.D. degree from Purdue University, WestLafayette, IN, USA.

He joined NJIT in 1988, where he is Professor ofElectrical and Computer Engineering. He has alsoassumed various administrative positions at NJIT. Hehas been a Visiting (Chair/Honorary) Professor atseveral universities. His current research focuses on

various aspects of broadband networks and multimedia communications. Hehas served on the editorial/advisory board of nine journals. He was elected toserve on the IEEE Communications Society (ComSoc) Board of Governorsas a member-at-large (2013–2015), as well as the IEEE Region 1 Board ofGovernors as the IEEE North Jersey Section Chair. He has chaired ComSoctechnical committees, and has actively organized numerous IEEE Internationalconferences/symposia/workshops, assuming leadership roles such as Chair orTPC Chair.

Prof. Ansari is the author of Computational Intelligence for Optimization(Springer 1997) with E.S.H. Hou, Media Access Control and Resource Allo-cation for Next Generation Passive Optical Networks (Springer, 2013) withJ. Zhang, and is the editor of Neural Networks in Telecommunications (Springer1994) with B. Yuhas. He has also (co)-authored over 450 publications, over onethird of which were published in widely cited refereed journals/magazines. Hehas been granted more than 20 U.S. patents. He has also guest-edited a numberof special issues, covering various emerging topics in communications and net-working. He is frequently selected to deliver keynote addresses, distinguishedlectures, and tutorials. Some of his recent recognitions include a couple of bestpaper awards, several Excellence in Teaching Awards, the Thomas Alva EdisonPatent Award (2010), the NJ Inventors Hall of Fame Inventor of the Year Award(2012), the NCE Excellence in Research Award (2014), and designation as aComSoc Distinguished Lecturer (2006–2009).

nlos error mitigation for toa-based localization via convex relaxation

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