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Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2011, Article ID 172902, 8 pagesdoi:10.1155/2011/172902

Research Article

Total Least Squares Method for Robust Source Localization inSensor Networks Using TDOA Measurements

Yang Weng,1 Wendong Xiao,2 and Lihua Xie3

1 College of Mathematics, Sichuan University, Chengdu 610064, China2 Networking Protocols Department, Institute for Infocomm Research, Singapore 1386323 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798

Correspondence should be addressed to Wendong Xiao, [email protected]

Received 31 March 2011; Accepted 15 June 2011

Copyright © 2011 Yang Weng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The TDOA-based source localization problem in sensor networks is considered with sensor node location uncertainty. A totalleast squares (TLSs) algorithm is developed by a linear closed-form solution for this problem, and the uncertainty of the sensorlocation is formulated as a perturbation. The sensitivity of the TLS solution is also analyzed. Simulation results show its improvedperformance against the classic least squares approaches.

1. Introduction

Driven by many practical applications such as environmentmonitoring, traffic management in intelligent transporta-tion, healthcare for the older and disabled, cyber-physicalsystem (CPS) has emerged as a new advanced system tolink the virtual cyber world with the real physical world[1, 2]. Sensor network is crucial to enable CPS by efficientmonitoring and understanding the physical world. The mainpurpose of a sensor network is to monitor an area, includingdetecting, identifying, localizing, and tracking one or moreobjects of interest. These networks may be used by the mili-tary in surveillance, reconnaissance, and combat scenarios oraround the perimeter of a manufacturing plant for intrusiondetection. The problem of source localization involves theestimation of the position of a stationary transmitter frommultiple noisy sensor measurements, which can be time-of-arrival (TOA), time-difference-of-arrival (TDOA), or angle-of-arrival (AOA) measurements or a combination of them[3–5].

Source localization methods using TDOA measurementslocate the source at the intersection of a set of hyperboloids.Finding this intersection is a highly nonlinear problem [6, 7].Over the years, many iterative numerical algorithms havebeen proposed for the problem, including the maximumlikelihood estimation methods [8, 9] and the constrainedoptimization methods [10, 11]. In these approaches, linear

approximation and iterative numerical techniques have tobe used to deal with the nonlinearity of the hyperbolicequations. However, it is difficult to select a good initialguess to avoid a local minimum for them; therefore theconvergence to the optimal solution cannot be guaranteed.The closed-form solution methods are widely used since noinitial solution guesses are required and have no divergenceproblem compared with the iterative techniques [12–16]. Forreal-time application in WSNs, the iterative procedure foriterative algorithm is time consuming while the closed-formmethod is computational efficient.

The aforementioned approaches need the precise loca-tion of sensors. In practice, the receiver locations may not beknown exactly. For example, in sensor network applications,the receivers can be with airplanes or unmanned aerialvehicles (UAVs) whose positions and velocities may not beprecisely known. Hence, the inaccuracy in receiver locationsneeds to be taken into account in practical applications whichis challenging and difficult as the estimation performance ofsource location can be very sensitive to the accurate knowl-edge of the receiver positions and a slight error in a receiver’slocation can lead to a big error in the source locationestimate. In [17], a closed-form solution is proposed thattakes the receiver error into account to reduce the estimationerror. The proposed solution is computationally efficient anddoes not have the divergence problem as in the iterativetechniques. In [18], the maximum likelihood formulation

2 International Journal of Distributed Sensor Networks

of source localization problem is given and an efficientconvex relaxation for this nonconvex optimization problemis proposed. A formulation for robust source localizationin the presence of sensor location errors is also proposed.Both the above methods assume that the measurement noiseis Gaussian and characterize the uncertainty by stochasticapproach with the perturbation being white and Gaussian.

However, such white and Gaussian assumptions areunrealistic in many practical applications [19–21]. Usually,if the Gaussian assumption are not met, the maximum-likelihood-based results under Gaussian assumption maylead to poor estimation performance, which means that theestimation performance is sensitive to the exact knowledgeof the parameters of the system (see, e.g., [22]). Thesefacts motivate us to further research on robust sourcelocalization method without any distribution assumption formeasurements noise and sensor location error.

In this paper, we will develop a total least squares (TLSs)algorithm for location estimation of a stationary source.TLS is a least squares data modeling technique in whichobservational errors on both dependent and independentvariables are taken into account. The uncertainty of thesensor location is formulated as a perturbation on the givensensor location. The sensitivity of the TLS solution willalso be analyzed to show the superiority of our proposedalgorithm. Compared with the existing methods which needthe Gaussian assumption for both measurements noise andsensor location error, the TLS approach does not dependon any assumed distribution of the noise and errors.Simulation results support the above analysis and show goodperformance of the proposed method.

The rest of this paper is organized as follows. In Section 2,a linear closed-form solution is given for source localiza-tion problem using TDOA measurements. The total leastsquares method for source localization with sensor locationuncertainty is given in Section 3, and the correspondingsensitivity analysis is derived in Section 4. Simulation resultsare presented in Section 5 to show the improved performanceof the proposed method against the classic least squaresapproaches. Concluding remarks are made in Section 6.

2. A Linear Closed-Form Solution

Assume that sensor i is located at point Si = {xi, yi, zi}.Denote the unknown source location by S = {x, y, z}. Let cbe the signal propagation speed and N the number of sensornodes distributed in the network. A reference node, denotedas S0, exists in the field. We define the distance from thesource to sensor i as Di. The TDOA that we derive fromthe data, τ0i, for each sensor node i ∈ [1,N] relative to thereference sensor S0 is

τ0i = 1c‖S− S0‖ − 1

c‖S− Si‖ = 1

c(D0 −Di). (1)

Denote the distance difference of arrival (DDOA) data for theith sensor as

d0i = τ0i × c = D0 −Di. (2)

We have

D20 −D2

i = ‖S− S0‖2 − ‖S− Si‖2,

= (x − x0)2 +(y − y0

)2 + (z − z0)2,

−(

(x − xi)2 +

(y − yi

)2 + (z − zi)2)

,

= x20 − x2

i + 2x(xi − x0) + y20 − y2

i + 2y(yi − y0

),

+ z20 − z2

i + 2z(zi − z0),

= D20 − (D0 − d0i)

2,

= 2D0d0i − d20i.

(3)

Group all the known terms together and denote

bi = 12

[x2

0 − x2i + y2

0 − y2i + z2

0 − z2i + d2

0i

]. (4)

Rearranging and substituting give

(x0 − xi)x +(y0 − yi

)y + (z0 − zi)z + d0iD0 = bi, (5)

which is a linear model for unknown parameters x, y, z,and D0. Stacking the N sensor measurements, we have thelinear system in matrix form

AX = b, (6)

where

A =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

x0 − x1 y0 − y1 z0 − z1 d01

x0 − x2 y0 − y2 z0 − z2 d02

......

......

x0 − xN y0 − yN z0 − zN d0N

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

,

X =

⎡

⎢⎢⎢⎢⎢⎢⎣

x

y

z

D0

⎤

⎥⎥⎥⎥⎥⎥⎦

, b =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

b1

b2

...

bN

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

.

(7)

When the location of sensor nodes can be precisely knownand the TDOA measurements are noise-free, linear system(6) is compatible. The solution is unique while the datamatrix A is of full rank.

In [16], a noise ε is considered at the right side of (6):

AX = b + ε. (8)

The least squares (LSs) problem seeks to

minimize : ‖b− bls‖,

subject to : bls ∈R(A),(9)

where R(A) denote the vector space spanned by the columnvector of matrix A. Once a minimizing bls is found, then any

International Journal of Distributed Sensor Networks 3

X satisfying AX = bls is called an LS solution and b − bls thecorresponding LS correction. The unique LS solution can beobtained while the data matrix A is of full rank:

Xls =(ATA

)−1ATb. (10)

The ordinary LS problem amounts to perturbing theobservation vector b by a minimum amount b − AXls.The underlying assumption is that errors only occur in thevector b and that the matrix A is exactly known. However,the TDOA measurements always have noise, that is, d0i

is perturbed by a noise. Hence, both sides of (5) haveperturbation according to term d0i. Besides, in practice, thereceiver locations may not be known exactly. As a result,the inaccuracy in receiver locations needs to be taken intoaccount in practical environments. Therefore, consideringonly the perturbation at the right side of (6) is not enough.We should consider the perturbations at both sides.

3. Total Least Squares Solution

The definition of the total least squares method is motivatedby the asymmetry of the least squares method that b iscorrected while A is not. Provided that both A and b aregiven data, it is reasonable to treat them symmetrically.One important application of TLS problem is parameterestimation in errors-in-variables models, that is, consideringthe measurements in A [23, 24]. We assume that the mmeasurement in A and b by

AX = b, (11)

where A = A + ΔA, b = b + Δb, and AX = b, is compatible.The total least squares (TLSs) problem seeks to

minimize :∥∥∥[A; b]− [Atls; btls]

∥∥∥F

,

subject to : btls ∈R(Atls),(12)

where ‖ · ‖F denotes the Frobenius norm of matrix A, thatis,

‖A‖F =√√√√√

m∑

i=1

n∑

j

∣∣∣ai j

∣∣∣

2 =√

trace(ATA). (13)

Once a minimizing [Atls; btls] is found, then any X satisfying

AtlsX = btls is called a TLS solution and [A; b]− [Atls; btls] thecorresponding TLS correction [25].

When A is of full rank, the closed-form expression of thebasic TLS solution can be obtained as the following:

Xtls =(ATA− σ2

n+1I)−1

AT b, (14)

where σn+1 is the smallest singular value of [A; b]. It canbe proved that the TLS solution Xtls estimates the trueparameter A+b consistently, that is, Xtls converges to thesolution of AX = b as the number of measurements tends toinfinity, where A+ denotes the Moore-Penrose pseudoinverse

of matrix A. This property of TLS estimates does not dependon any assumed distribution of the errors. Note that the LSestimates are inconsistent in this case.

In the following, the algorithm computation complexityis analyzed by considering the number of floating-pointoperations (FLOPS). The calculation of FLOPS is brieflydescribed as follows: additions and multiplications count asone FLOP each. Adding matrices of sizes m × n requires mnFLOPS. Multiplying matrices of sizes m×k and k×p requiresmnk FLOPS. Matrix inverse of size n× n requires n3 FLOPS.Singular value decomposition of matrix with size n × mrequires nm2 FLOPS. The computation overhead of threeclosed-form algorithms is investigated, that is, least squares(LSs) method in [16], two-stage weighted least squares(WLSs) method in [17] and the proposed TLS method.For N-deployed nodes and p dimension of source locationparameter, the numbers of FLOPS in the LS, WLS, and theproposed TLS algorithms are (2p + 3)(p + 1)N + (p + 1)3,(2p + 3)(2p + 2)N + 2(p + 1)3, and (2p + 5)(p + 2)N +(p + 2)2(p + 3), respectively. It is to show that LS methodis the most computationally efficient, whereas WLS methodis two-stage least square and TLS method has singular valuedecomposition for matrix. It is clear that they all havecomparable computation complexity O(N) since p = 3 orp = 4 for source localization problem.

4. Sensitivity Properties of the Solution

In this section, we will first examine how perturbations inA and b affect the solution X . In this analysis the conditionnumber of the matrixA plays a significant role. The followingdefinition generalizes the condition number of a squarenonsingular matrix [26].

Definition 1. Let A ∈ Rm×n have rank r. The conditionnumber of A is

κ(A) = ‖A‖ · ∥∥A+∥∥ = σ1

σr, (15)

where σ1, σ2, . . . , σr are the singular values of A by decreasingorder.

Matrices with small condition numbers are said tobe well conditioned while the ones with large conditionnumbers are said to be ill-conditioned. Analyzing the effectof perturbation on the solution of linear system AX = b, weintroduce the following lemma [27].

Lemma 1. If rank(A) = rank(A) and ‖ΔA‖‖A+‖ < 1, then∥∥∥A+

∥∥∥ ≤ ‖A+‖

1− ‖A+‖‖ΔA‖ . (16)

Theorem 1. Denote Xls = A+b as the LS solution of

unperturbed system AX = b, and Xls = A+b is the LS solution

of perturbed system AX = b with A = A+ΔA, b = b+Δb. Onefurther assumes that ‖ΔA‖ ≤ ε‖A‖, ‖Δb‖ ≤ ε‖b‖, κ(A) = κ.Then, one has

∥∥∥Xls − Xls

∥∥∥

‖Xls‖ = O(εκ). (17)


Proof. Noting that A and A are full rank and (I −AA+)b = 0since b ∈R(A), we have

Xls − Xls = A+b− A+b

= A+(b + Δb)− A+b

= A+Δb +(A+ − A+

)b

= A+Δb +(A+ + A+AA+ − A+AA+ − A+

)b

= A+Δb +(A+ + A+AA+ − A+AA+ − A+AA+

)b

= A+Δb + A+(I − AA+)b− A+(A− A

)A+b

= A+Δb− A+ΔAXls.(18)

Therefore, ‖Xls − Xls‖ = ‖A+Δb − A+ΔAXls‖. Dividing by‖Xls‖ at both sides, we have

∥∥∥Xls − Xls

∥∥∥

‖Xls‖ =∥∥∥A+Δb− A

+ΔAXls

∥∥∥

‖Xls‖

≤ A+Δb

‖Xls‖ +∥∥∥A+ΔA

∥∥∥

≤∥∥∥A+

∥∥∥(‖Δb‖‖Xls‖ + ‖ΔA‖

)

≤∥∥∥A+

∥∥∥(ε‖b‖‖Xls‖ + ε‖A‖

)

≤ ‖A+‖1− ‖A+‖‖ΔA‖

(ε‖b‖‖Xls‖ + ε‖A‖

)

≤ ‖A+‖1− ‖A+‖ε‖A‖

(ε‖b‖‖Xls‖ + ε‖A‖

)

= 11− εκ

(ε‖A+‖‖b‖‖A+b‖ + εκ

)

= 11− εκ

(ε‖A+‖‖AA+b‖

‖A+b‖ + εκ)

≤ 11− εκ

(ε‖A+‖‖A‖‖A+b‖

‖A+b‖ + εκ)

= 2εκ1− εκ

= O(εκ).

(19)

From the theorem we can see that, only when theperturbations are sufficiently small, the LS solution is a goodestimator of the true solution of AX = b.

Additionally, we will show when the TLS solution hasbetter performance than the LS solution for the perturbed

model AX ≈ b. Denote the singular value decomposition(SVD) of A by

A = UΣVT , (20)

where

U = [u1,u′2, . . . ,um], U ′ ∈Rm, UTU = Im,

V = [v1, v′2, . . . , vn], V ′ ∈Rn, VTV = In,

Σ = diag(σ1, σ2, . . . , σn) ∈Rm×n, σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0,(21)

denote the singular value decomposition (SVD) of A by

A + ΔA = A = U ′Σ′V′T , (22)

where

U ′ = [u′1,u′2, . . . ,u′m], U ′ ∈Rm, U ′TU ′ = Im,

V ′ = [v′1, v′2, . . . , v′n], V ′ ∈Rn, V ′TV ′ = In,

Σ′ = diag(σ ′1, σ ′2, . . . , σ ′n

) ∈Rm×n,

σ ′1 ≥ σ ′2 ≥ . . . ≥ σ ′n ≥ 0,

(23)

and denote the SVD of [A; b] by

[A + ΔA; b + Δb, A; b

]= UΣVT , (24)

where

U = [u1, u2, . . . , um], U ∈Rm, UTU = Im,

V = [v1, v2, . . . , vn], V ∈R(n+1), VTV = In+1,

Σ = diag(σ1, σ2, . . . , σn, σn+1) ∈Rm×n+1,

σ1 ≥ σ2 ≥ · · · ≥ σ(n+1) ≥ 0.

(25)

The accuracy of TLS and LS solutions related to the per-turbation effects on singular values and associated singularsubspaces [23]. Several papers have analyzed the bounds onthe perturbation effects related to singular subspaces [28–30]. The most interesting results for sensitivity analysis ofTLS solution are given in [30].

Definition 2. Denote two subspaces as L and M. For anyunitary invariant norm, the distance between two subspacesis defined as the sine of the largest canonical angle

dist(L,M) = ‖sinΘ(L,M)‖ = ‖(I − PM)PL‖, (26)

where PM and PL are the projection operators [29, 30].

Theorem 2. Let the SVD of A ∈ Rm×n,m ≥ n, be given by(20), rank(A) = r. Add perturbations ΔA to A, and let SVD ofA be given by (22). If σ ′r − σr+1 > 0, then

dist(R(A), R

(A))≤ ‖ΔA‖

σ ′r − σr+1. (27)

Theorem 2 is a special case of the generalized sin θ-theoremin [30].


Recall the perturbed system AX ≈ b; LS projects A and binto the singular subspaces

L(u′1,u′2, . . . ,u′m

) =R(A)

, (28)

while the TLS projects A and b into the singular subspaces

L(u1, u2, . . . , um) =R([A; b

]). (29)

Furthermore, the real source location X∈R(A), since AX =b. Therefore, it is important to analyze the distance between

R(A), R([A; b]) and R(A). Assumeing that A and A havefull rank, we can obtain the following corollary according toTheorem 2.

Corollary 1. One has

dist(R(A), R

(A))≤ ‖ΔA‖

σ ′n,

dist(R(A), R

([A; b

]) )≤ ‖ΔA‖

σn.

(30)

The interlacing theorem (see [31]) implies that

σ ′1 ≥ σ1 ≥ · · · ≥ σ ′n ≥ σn. (31)

Therefore,

‖ΔA‖σn

≤ ‖ΔA‖σ ′n

. (32)

From Corollary 1 we can see that the upper bound ofthe distance between R(A) and R(A) is smaller than theupper bound of distance between R([A; b]) and R(A). Thisimplies that the TLS solution is expected to be closer toits corresponding unperturbed subspace. Hence, the TLSsolution is expected to be more accurate than the LS solution.

5. Simulations

5.1. Simulation Setup. In this section, simulations are carriedout to show the effectiveness of the proposed method. Twoscenarios are investigated for their effects on localizationperformance, including Gaussian distribution and truncatedGaussian distribution for measurement noise and sensornode location error. In our simulation, unless otherwisespecified, sensors are randomly deployed in a 5000 m ×5000 m area with uniform distribution. An example is shownin Figure 1, where 30 nodes are uniformly distributed. Thesource is denoted as a triangle with red and the referencenode is denoted as a square in blue. The exact location ofthe reference node can be known.

In order to show the improved performance of ourproposed method with the existing closed-form method, weinvestigate LS method in [16] and WLS method in [17].Root mean square error (RMSE) is used as the criterionfor localization performance. The following simulations areperformed with 500 Monte Carlo trials. In all figures, the redsolid line with square, black dash line with circle, and bluedotted dash line with triangle represent the results of TLS,WLS, and LS solutions, respectively.

5000

4000

3000

2000

1000

00 1000 2000 3000 4000 5000

NodesSourceReference node

Figure 1: Simulation scenario.

12

10

8

6

4

2

010 20 30 40 50 60 70 80 90 100

LSTLSWLS

RM

SE(m

)

σn

Figure 2: Localization errors versus noise variance (Gaussian case).

5.2. Gaussian Case. In this case, we consider the Gaussiandistribution for both measurement noise and sensor nodelocation error. The effect on estimation performance fordifferent measurement noise variance value is investigated.Measurement noise variance (σn) is changed in the range10–100. The variance of perturbation (σp) is set to be aconstant number 25. The localization performance variationwith noise variance is depicted in Figure 2. As noise varianceincreases, localization performance degrades in all LS, WLS,and TLS algorithms. The TLS algorithm outperforms the LSand WLS algorithms.


35

30

25

20

15

10

5

05 10 15 20 25 30 35 40 45 50

RM

SE(m

)

LS

TLS

WLS

σp

Figure 3: Localization errors versus perturbation variance (Gaus-sian case).

In the second simulation, the effect on estimation perfor-mance with different variance of perturbation is investigated.The variance of perturbation is set to be varied from 10 to100 and sensors are deployed uniformly. The measurementnoise variance is set to be 50. The localization performancevariation with variance of perturbation is depicted inFigure 3. The TLS algorithm outperforms the LS and WLSsignificantly with high-level variance of perturbation.

The estimation performance of three algorithms is alsoinvestigated with different numbers of deployed nodes.Figure 4 shows the RMSEs versus the number of nodes. Fiftynodes are uniformly generated in the field. Localization startsby using five randomly selected nodes for each algorithm,followed by adding more nodes of five in a group until allfifty nodes are used. The number of FLOPS is consideredto evaluate the algorithm computational complexity withvariation of the number of nodes. Results are given inFigure 5 with 2-dimensional source location parameter and5–50 nodes. It can be seen that LS required the fewest FLOPSwhile TLS required the most FLOPS with SVD of matrix.Although the estimation performance is improved withmore nodes for all algorithms, the computation overheadis also increased. t is to show that LS method is the mostcomputationally efficient, whereas WLS method is two-stageleast square method and TLS method has singular valuedecomposition for matrix.

5.3. Truncated Gaussian Case. Since the white and Gaussianassumptions are unrealistic in many applications, we con-sider the truncated Gaussian distribution for both measure-ment noise and sensor node location error in this case. Let Xbe a random variable with zero mean Gaussian distribution

5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

Number of nodes

LSTLSWLS

RM

SE(m

)Figure 4: Localization errors versus number of nodes (Gaussiancase).

5 10 15 20 25 30 35 40 45 50102

103

104

Com

puta

tion

cost

Number of nodes

LSWLSTLS

Figure 5: Computation cost versus number of nodes.

truncated in the interval x ≤ ασ ; its probability densityfunction (pdf) is given by

p(x) =

⎧⎪⎪⎨

⎪⎪⎩

b√

2piσ

(

exp−x2

2σ2− exp

−(αx)2

2σ2

)

, |x| ≤ ασ ,

0, |x| ≥ ασ ,(33)


10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

σn

LSTLSWLS

RM

SE(m

)

Figure 6: Localization errors versus noise variance (truncatedGaussian case).

5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

σp

LSTLSWLS

RM

SE(m

)

Figure 7: Localization errors versus perturbation variance (trun-cated Gaussian case).

where b is a normalizing constant, σ2 is the variance, and α isthe factor to extend the interval that the random variable Xlies at.

Compared with the Gaussian case, similar results canbe obtained and are shown in Figures 6, 7, and 8. Theestimation performance for the WLS algorithm distinctlydegrades significantly. This is because the weight matrix forthis WLS method is calculated according to the Gaussian

5 10 15 20 25 30 35 40 45 502

4

6

8

10

12

14

16

18

20

LS

TLS

WLS

Number of nodes

RM

SE(m

)Figure 8: Localization errors versus number of nodes (truncatedGaussian case).

assumption for the measurements noise and sensor nodelocation error. The proposed TLS algorithm still maintainsgood estimation performance.

6. Conclusions

In this paper, the TDOA model for source localization insensor networks has been considered. The total least squares(TLSs) algorithm has been developed for location estimationof a stationary source with sensor location uncertaintyin which the uncertainty of the sensor location has beenformulated as a perturbation. The sensitivity of the TLSsolution has also been analyzed to show the advantages of ourproposed algorithm. Compared with the existing methodswhich need the Gaussian assumption for both measurementsnoise and sensor location error, the TLS approach does notdepend on any assumed distribution of the noise and errors.Simulation results show the superior performance of theproposed method.

References

[1] A. Hande, D. Yunus, and E. Cem, “Wireless healthcaremonitoring with RFID-enhanced video sensor networks,”International Journal of Distributed Sensor Networks, vol. 2010,Article ID 473037, p. 10, 2010.

[2] F. Xia, L. Ma, J. Dong, and Y. Sun, “Network qos managementin cyber-physical systems,” in Proceedings of the InternationalConference on Embedded Software and Systems Symposia(ICESS ’08), pp. 302–307, IEEE, 2008.

[3] Y. Chan, H. Hang, and P. Ching, “Exact and approximate max-imum likelihood localization algorithms,” IEEE Transactionson Vehicular Technology, vol. 55, no. 1, pp. 10–16, 2006.


[4] E. Weinstein, “Optimal source localization and tracking frompassive array measurements,” IEEE Transactions on Acoustics,Speech, and Signal Processing, vol. 30, no. 1, pp. 69–76, 1982.

[5] R. O. Schmidt, “Multiple emitter location and signal param-eter estimation,” IEEE Transactions on Antennas and Propaga-tion, vol. 34, no. 3, pp. 276–280, 1986.

[6] J. Chen, K. Yao, and R. Hudson, “Source localization andbeamforming,” IEEE Signal Processing Magazine, vol. 19, no.2, pp. 30–39, 2002.

[7] Y. Huang, J. Benesty, G. Elko, and R. Mersereati, “Real-timepassive source localization: a practical linear-correction least-squares approach,” IEEE Transactions on Speech and AudioProcessing, vol. 9, no. 8, pp. 943–956, 2001.

[8] W. Hahn and S. Tretter, “Optimum processing for delay-vector estimation in passive signal arrays,” IEEE Transactionson Information Theory, vol. 19, no. 5, pp. 608–614, 1973.

[9] M. Wax and T. Kailath, “Optimum localization of multiplesources by passive arrays,” IEEE Transactions on Acoustics,Speech, and Signal Processing, vol. 31, no. 5, pp. 1210–1218,1983.

[10] K. Cheung, H. So, W. Ma, and Y. Chan, “A constrainedleast squares approach to mobile positioning: algorithms andoptimality,” Eurasip Journal on Applied Signal Processing, vol.2006, Article ID 20858, p. 23, 2006.

[11] K. Yang, J. An, X. Bu, and G. Sun, “Constrained total least-squares location algorithm using time-difference-of-arrivalmeasurements,” IEEE Transactions on Vehicular Technology,vol. 59, no. 3, Article ID 5342476, pp. 1558–1562, 2010.

[12] B. Friedlander, “A passive localization algorithm and itsaccurancy analysis,” IEEE Journal of Oceanic Engineering, vol.12, no. 1, pp. 234–245, 1987.

[13] H. C. Schau and A. Z. Robinson, ““Passive source localizationemploying intersecting spherical surfaces from time-of-arrivaldifferences,” IEEE Transactions on Acoustics, Speech, and SignalProcessing, vol. 35, no. 8, pp. 1223–1225, 1987.

[14] J. O. Smith and J. S. Abel, “Closed-form least-squares sourcelocation estimation from range-difference measurements,”IEEE Transactions on Acoustics, Speech, and Signal Processing,vol. 35, no. 12, pp. 1661–1669, 1987.

[15] Y. T. Chan and K. C. Ho, “Simple and efficient estimator forhyperbolic location,” IEEE Transactions on Signal Processing,vol. 42, no. 8, pp. 1905–1915, 1994.

[16] M. D. Gillette and H. F. Silverman, “A linear closed-formalgorithm for source localization from time-differences ofarrival,” IEEE Signal Processing Letters, vol. 15, pp. 1–4, 2008.

[17] K. C. Ho, X. Lu, and L. Kovavisaruch, “Source localizationusing TDOA and FDOA measurements in the presence ofreceiver location errors: analysis and solution,” IEEE Transac-tions on Signal Processing, vol. 55, no. 2, pp. 684–696, 2007.

[18] K. Yang, G. Wang, and Z. Q. Luo, “Efficient convex relaxationmethods for robust target localization by a sensor networkusing time differences of arrivals,” IEEE Transactions on SignalProcessing, vol. 57, no. 7, pp. 2775–2784, 2009.

[19] S. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory, Prentice Hall, 1993.

[20] A. Renaux, P. Forster, E. Boyer, and P. Larzabal, “Uncondi-tional maximum likelihood performance at finite number ofsamples and high signal-to-noise ratio,” IEEE Transactions onSignal Processing, vol. 55, no. 5, pp. 2358–2364, 2007.

[21] K. K. Mada, H. C. Wu, and S. S. Iyengar, “Efficient and robustEM algorithm for multiple wideband source localization,”IEEE Transactions on Vehicular Technology, vol. 58, no. 6, pp.3071–3075, 2009.

[22] U. Theodor, U. Shaked, and C. E. de Souza, “A gametheory approach to robust discrete-timeH∞-estimation,” IEEETransactions on Signal Processing, vol. 42, no. 6, pp. 1486–1495,1994.

[23] S. van Huffel and J. Vandewalle, The Total Least SquaresProblem: Computational Aspects and Analysis, Society forIndustrial Mathematics, 1991.

[24] X. Zhang, Matrix Analysis and Application, Tsinghua Univer-sity Press and Springer, 2004.

[25] I. Markovsky and S. van Huffel, “Overview of total least-squares methods,” Signal Processing, vol. 87, no. 10, pp. 2283–2302, 2007.

[26] G. Golub and C. van Loan, Matrix Computations, JohnsHopkins University Press, Baltimore, Md, USA, 1996.

[27] P. Wedin, “Perturbation theory for pseudo-inverses,” BITNumerical Mathematics, vol. 13, no. 2, pp. 217–232, 1973.

[28] G. Stewart, “Error and perturbation bounds for subspacesassociated with certain eigenvalue problems,” SIAM Review,vol. 15, no. 4, pp. 727–764, 1973.

[29] C. Davis and W. M. Kahan, “The rotation of eigenvectors by aperturbation. III,” SIAM Journal on Numerical Analysis, vol. 7,no. 1, pp. 1–46, 1970.

[30] P. Wedin, “Perturbation bounds in connection with singularvalue decomposition,” BIT, vol. 12, no. 1, pp. 99–111, 1972.

[31] R. C. Thompson, “Principal submatrices IX: interlacinginequalities for singular values of submatrices,” Linear Algebraand Its Applications, vol. 5, no. 1, pp. 1–12, 1972.

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