Achieving Asymptotic Efficient Performance for Squared Range and Squared Range Difference Localizations

2836 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 11, JUNE 1, 2013

Achieving Asymptotic Efficient Performance forSquared Range and Squared Range Difference

LocalizationsShanjie Chen, Student Member, IEEE, and K. C. Ho, Fellow, IEEE

Abstract—The estimation of a source location using directly therange or range difference measurements is difficult and requiresnumerical solution, which is caused by the highly non-linearrelationship between the measurements and the unknown. Wecan obtain a computationally efficient and non-iterative algebraicsolution by squaring the measurements first before solving forthe unknown. However, a recent study has shown that such asolution is suboptimum in reaching the CRLB performance andthe localization accuracy could be significantly worse in somelocalization geometries. This paper demonstrates that when rangeweighting factors are introduced to the squared measurements,the resulting solution will be able to reach the CRLB accuracy.Both the squared range and squared range difference cases areconsidered, and the mean-square error (MSE) and the bias of theresulting solutions are derived. The asymptotic efficiency of theproposed cost functions are proven theoretically and validatedby simulations. The effects of range weighting factors on thelocalization performance under different sensor number, noisecorrelation, and localization geometry are examined. Introducingrange weightings to the squared range measurements increasesthe bias but it is negligible in the MSE. Having range weightingsin the squared range difference measurements improves both theMSE and bias.

Index Terms—Efficient estimator, localization, TDOA, TOA.

I. INTRODUCTION

L OCALIZATION has been a very important and funda-mental research topic in GPS, radar, sonar, and especially

in mobile communications and sensor networks over the pastfew years [1]–[10]. Localization of a signal source is often ac-complished by using a number of sensors that measure the radi-ated signal from the source. The positioning parameters such astime of arrivals (TOAs) or time differences of arrival (TDOAs)are obtained from the received signals. The relationship betweenthe source location and the positioning parameters is exploitednext to determine where the source is. This paper considers therange based localization technique that uses TOAs or TDOAs.Most of the research effort has been concentrating on the es-

timation of the source location from the positioning parameters.The criteria to evaluate a location estimator include compu-tational efficiency, estimation accuracy and robust behaviors

Manuscript received July 16, 2012; revised December 22, 2012; acceptedFebruary 22, 2013. Date of publication March 26, 2013; date of current versionMay 09, 2013. The associate editor coordinating the review of this manuscriptand approving it for publication was Prof. Minyue Fu.The authors are with the Electrical and Computer Engineering Department,

University of Missouri, Columbia, MO 65211 USA (e-mail: [email protected]; [email protected]).Digital Object Identifier 10.1109/TSP.2013.2254479

under different localization geometries. The relationship be-tween the source location and the positioning parameters iswell-defined. It is, however, highly non-linear. The non-linearrelationship complicates the localization task considerably. TheMaximum Likelihood Estimator (MLE) [11], which is knownto be asymptotically efficient, is found to be not trivial to usein practice because it has to be realized by numerical iterativesearch. The Likelihood function is non-quadratic and couldhave local minima, and good initial solution guesses close to thetrue solution are required to achieve good performance. Conse-quently, many researchers have been working on closed-formalgebraic solution for the localization problem that can avoidthe initialization and complexity issues as in the MLE.Nearly all algebraic solutions use the approach of squaring

the measurements first, then introduce an auxiliary variable anddefine a constraint relating the extra variable and the source po-sition to obtain a location estimate. For example, [12] and [13]derive closed-form solutions through subtracting the squaredmeasurements to eliminate the auxiliary variable. [14], on theother hand, ignores the constraint and solves the squared mea-surement equation using linear least squares (LS). [15] and [16]express the source location in terms of the auxiliary variableand solution finding reduces to determining the auxiliary vari-able that minimizes the corresponding cost function. Althoughsimple and computationally attractive, these solutions do notreach the Cramer-Rao Lower Bound (CRLB) accuracy.Recently, [17] utilizes the squared range least squares

(SR-LS) cost function and imposes the constraint explicitlyduring the minimization process to obtain a global minimumsolution through the generalized trust region subproblems(GTRS) technique [18]. Good localization accuracy, sometimesreaching the CRLB for uncorrelated Gaussian noise, has beenreported over other competing methods. Subsequently, [19]has shown that even for the fundamental case of Gaussianuncorrelated noise, the approach of squaring the range (TOA)measurements to solve the localization problem will not be ableto yield the same performance as using the measurements di-rectly, e.g., fromMLE, unless under some special and restrictedlocalization geometries. Indeed, the asymptotic localizationerror from SR-LS relative to the CRLB could be unbounded forsome configurations. The work here continues this fundamentalresearch subject on the implication for localization perfor-mance when the measurements are squared before obtaining asolution.Our research studies presented here indicate the approach of

squaring the range measurements for localization is able to pro-vide the CRLB performance asymptotically regardless of thegeometry (except the source is very close to a sensor), when

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CHEN AND HO: SQUARED RANGE DIFFERENCE LOCALIZATIONS 2837

suitable weightings are introduced to the squared measurementsbefore solving for the solution. The resulting cost function istermed as squared range weighted least squares (SR-WLS). Theweightings are formed from the measurements. The efficiencyof the SR-WLS solution is shown analytically through smallnoise analysis. The asymptotic bias of the source location es-timate from SR-WLS is found to be larger than that of SR-LS.However, the bias is relatively insignificant compared to vari-ance.We also investigate the possible performance loss by squaring

the range difference measurements when solving for the sourcelocation using least squares. The analysis shows that squaredrange difference least squares (SRD-LS) is not able to reach theCRLB performance in general, and the performance could bevery worse in some cases.Another advance in our study is that by introducing suitable

weightings to the squared range difference measurements be-fore least squares minimization, denoted as squared range dif-ference weighted least squares (SRD-WLS), the performanceloss will be compensated for and the CRLB performance canbe reached. This is again shown analytically and supported bysimulations. Different from the range measurement case, theproposed SRD-WLS has smaller bias than SRD-LS when thenumber of sensors is not near critical.The study in this paper is different from the previous investi-

gations. [19] only examined the case of squared range localiza-tion for 2D under uncorrelated noise. We advance the study to3D with correlated noise. In addition to variance that [19] usedas performance indicator, we also examine the localization bias.Furthermore we extend the study to the squared range differencecase in which the previous work has not considered. Most im-portant, we propose the range weightings that allow the reachingof asymptotic efficient performance when the measurements aresquared.The investigation here provides justification of using the

squared ranges and squared range differences for obtaining asource location estimate, when suitable weightings are used. Italso includes many insights about the effect of weighting underdifferent localization geometries, sensor numbers and noisecorrelations.In the next section, we shall describe the localization

scenario, the TOA and TDOA measurements and the cor-responding localization CRLBs. Section III introduces theproposed squared range weighted least squares (SR-WLS) costfunction, derives the mean square error (MSE) matrix andbias of its location estimate, and shows that SR-WLS achievesasymptotically the CRLB performance. Section IV examinesthe proposed squared range difference weighted least squares(SRD-WLS) cost function and provides the MSE and biasanalysis. Section V presents the simulation results to comparethe performance between SR-LS and SR-WLS, and betweenSRD-LS and SRD-WLS, and Section VI concludes the paper.

II. RANGE BASED LOCALIZATION

A. Localization Scenario

The purpose of localization is to determine the source po-sition using measurements from a number of sensors. In thispaper, we shall consider two types of measurements, TOA andTDOA, that are based on the ranges between the source and thesensors. We shall use TOA and range, and TDOA and range

difference interchangeably because they differ from each otherby a constant scaling factor only. The source position is andthe ith sensor position is , where is thenumber of sensors. and are vectors of Cartesian co-ordinates where for 2D and for 3D localization.The true distance between the source and the th sensor is

, and is the Euclidean norm.For TOA localization, the true distance is corrupted by

additive noise and the measurements are

(1)

In compact form,

(2)

where and . Thenoise vector is zero-mean Gaussian withcovariance matrix . The average measurement noise power is

. We are interested in estimating from .For TDOA localization, the true range difference of the

source to the th and the 1st sensor is

(3)

whose noisy version from measurement is

(4)

The 1st sensor is the reference to obtain the range difference.The vector form is

(5)

where and .The noise vector is zero-meanGaussian with covariancematrix . The averagemeasurementnoise power is . TDOA localization usesto obtain the source position.For convenient purpose in later Sections, we shall define

some notations as follows:

(6)

is the unit vector pointing from the th sensor to the sourceand is the difference between and . and are di-agonal matrices containing the ranges of the source to differentsensors. Furthermore, is used to denote a vector of unity andis an identity matrix. is a diagonal matrix formed by theelements in and the operator is element-by-element multi-plication.

B. CRLB

We shall use the CRLB [20] as a benchmark to examine theperformance of a location estimator. The CRLB is for unbiasedestimator and the localization problem is non-linear that could


yield a biased solution. Hence we shall use the CRLB over smallerror region only where the bias is negligible compared to vari-ance. The CRLB of a source location estimate for TOA posi-tioning is [21]

(7)

The CRLB for TDOA positioning is [22]

(8)

III. WEIGHTED SQUARED RANGE LOCALIZATION

A. Cost Function and Solution

The range least squares (R-LS) method finds by mini-mizing the cost function

(9)

Under Gaussian noise with , R-LS gives the MaximumLikelihood estimate and achieves the CRLB performance. Thecost function is difficult to solve due to the squareroot operation in the Euclidean distance. One has to rely on gridsearch, or iterative numerical solution where performance couldbe highly dependent on initializations.As an alternative to simplify the solution finding, many

closed-form solutions can be obtained by minimizing theSR-LS cost function

(10)

In particular, [17] shows that the global minimum of this costfunction can be efficiently solved through the utilization of theGTRS technique [18].However, the large-sample analysis performed in [19] finds

that SR-LS has worse asymptotic localization accuracy thanR-LS in general and there are localization geometries for whichthe performance difference between them is unbounded.We propose a new cost function based on

by introducing the weighting factors

(11)

The rationale behind this new cost function is that when thenoise relative to the source range is small and is close tosuch that , we have

(12)

which is the given in (9). Hence it is expected thatthe new squared range weighted least squares (SR-WLS) costfunction has similar performance as R-LS when we are nearthe solution. SR-WLS will remain to enjoy the computational

efficiency as in SR-LS because no square root appears in thecost function.Another reason for introducing the weights in the pro-

posed cost function (11) is that squaring the ranges will em-phasize the TOA measurements from the sensors that are far-ther away from the source. The SR-LS cost function (10) willtherefore yield a solution that puts more emphasis to the mea-surements from the far than from the near sensors. To compen-sate this undesirable effect, we use the weights inversely pro-portional to the square of the distances. The net effect is that itequalizes the contributions of measurements from far and nearsensors when obtaining the solution.The aforementioned cost functions correspond to uncorre-

lated noise covariance matrix . It is natural to extendour investigation to the general case for Gaussian noise witharbitrary non-singular covariance matrix . Let’s rep-resent its inverse as , then we have

(13)

(14)

and

(15)

where (13) is the Maximum Likelihood (ML) cost function.The analysis on the MSE and bias will be based on (14) and

(15). We shall show that SR-WLS can asymptotically achievethe CRLB performance, while SR-LS cannot.

B. Analysis and Comparison With CRLB

We shall perform the small noise analysis of the solutions ob-tained from the SR-LS and SR-WLS cost functions. The anal-ysis is up to second order noise terms, and it gives asymptoticperformance when the signal-to-noise ratio (SNR) of the signalmeasurements to obtain TOAs is high or the signal observationperiod is long.Let be the global minimum of . Also, let us assume is

smooth around such that the derivatives of at exist up tothird order. Obviously, the gradient of at is zero. Using theTaylor series expansion around the true value

...

where

is the th element of is length of and. The approximation comes from truncating the


expansion up to the second order term. Note that is sym-metric. Rearranging gives

... (16)

To examine the MSE, we multiply (16) by its transpose andtake expectation, giving

(17)

where the second term in (16) has been ignored. This is becausethe last term in (16) will contribute to the component in theMSE which is negligible for small noise analysis. Since con-tains only noise, retaining up to term reduces (17) to

(18)

where . Equation (18) is indeed the formulagiven in [23] to obtain the asymptotic MSE as . It isalso the formula [19] used to perform large-sample analysis ofSR-LS.The bias can be obtained by taking the expectation of (16),

... (19)

where and we have made approximation inthe last term to keep up to the second order noise componentsonly. Note that is the MSE matrix.The MSE and bias formulae (18) and (19) are accurate up

to the second order noise terms and the approximations comefrom ignoring some higher order terms. For simplicity the twoformulae are considered exact in the following analysis.We shall evaluate the terms in (18) and (19) to obtain theMSE

and bias. In general, contains the 1st order, the 2nd order,and higher order noise components. contains a fixed termindependent of noise, as well as the 1st order, the 2nd order, andhigher order noise terms. Evaluating (18) only needs the linearnoise component of . However, obtaining the first term of (19)requires up to the 2nd order noise portion of and up to the 1storder noise portion of . Hence we retain only up to the 2ndorder noise terms of and up to the 1st order noise terms of .We shall use small-o notation to providemore precise order of

approximation in the followings. In our representation,means .

The analysis below concentrates on the MSE. The bias studyis a little tedious and the details are provided in Appendix A.1) Asymptotic Performance of SR-LS: The first derivative of

in (14) is

(20)

where we have used the symmetric property that .Then

(21)

where

(22)

Therefore,

(23)

We continue by calculating the second derivative offrom (20),

(24)

Hence

(25)

whose asymptotic value as is

(26)

Consequently, we obtain from (18) the asymptotic MSE ma-trix of SR-LS as

(27)

In general, (27) is not equal to the CRLB unless for a few spe-cial cases. One such a case is when all ’s are equal suchthat , where is positive constant. in (22) becomes

and (27) reduces to ,which is the CRLB in (7).The bias of the SR-LS solution is evaluated by using (26),

(27), (64) and (65) into (19).2) Asymptotic Performance of SR-WLS: The first derivative

of is

(28)

From series expansion,

(29)

Using (29) in (28) gives

(30)

Hence

(31)


The second derivative of from (28) is

(32)

and therefore

(33)

Its asymptotic value as is(34)

As a result, putting (31) and (34) into (18) gives the asymp-totic MSE matrix of SR-WLS as

(35)

is exactly equal to the CRLB (7) for TOA localiza-tion and the solution of the SR-WLS cost function is asymptot-ically efficient.The bias of the SR-WLS solution is obtained by applying

(34), (35), (67) and (68) to (19).

C. Special Geometries

Under independent and identically distributed (i.i.d.) mea-surement noise such that , [19] lists four classes of spe-cial geometries to demonstrate that SR-LS is worse than R-LSin general. Reference [19] also generates random geometries toillustrate the distribution of the MSE ratio between the solu-tions from SR-LS and R-LS. Since the proposed SR-WLS hasthe same asymptotic accuracy as R-LS up to the second ordernoise term, we can essentially draw the same conclusion whencomparing SR-LS with SR-WLS.In the situation where the source signal is not available or

when there is an unknown but constant time offset in the signalsacquired, TOA cannot be used to locate the source and TDOAwill be applied instead. TDOA does not require the source signalto be known since it can be obtained through cross-correla-tion. Furthermore, the constant time offset is irrelevant becauseTDOA measures the time difference. In fact, the performanceof TDOA localization is the same as TOA positioning with anunknown common clock bias (time offset) [24]. We shall nextexamine the localization performance using the squared rangedifferences.

IV. WEIGHTED SQUARED RANGE DIFFERENCE LOCALIZATION

A. Cost Function and Solution

For measurement noise with arbitrary non-singular covari-ance matrix , the cost function for range difference leastsquares (RD-LS) is

(36)

where is the element in the th row and thcolumn of . The RD-LS solution cor-responds to the MLE and has the CRLB performance when thenoise is Gaussian. Equation (36) is difficult to solve and quiteoften the squared range difference least squares (SRD-LS) costfunction is used instead,

(37)

where

can be obtained from in (14) by re-placing with . The SRD-LS cost function usedin [17] is a special case of (37) when the noise is i.i.d. Generallyspeaking, the solution of SRD-LS is worse than that of RD-LS.Motivated by the result that SR-WLS can asymptotically

achieve the CRLB accuracy, we propose to add weightingfactor to the SRD-LS and generate the squared rangedifference weighted least squares (SRD-WLS) cost function

(38)

The rationale for using the weighting is the same as forSR-WLS. In particular, the weights are used to compensate forthe artificial effect of emphasizing more the TDOA measure-ments from the far sensors than from the near sensors causedby squaring the measurements. The in the weighting factor isapproximated by

(39)

where is from some initial source position estimate which willbe elaborated later.

B. Analysis and Comparison With CRLB

We shall perform the small noise analysis of the solutionsobtained from SRD-LS and the proposed SRD-WLS.1) Asymptotic Performance of SRD-LS: The first derivative

of is

(40)

Evaluating at the true source location gives

(41)

where

(42)


Thus

(43)

We continue by calculating second derivative offrom (40),

(44)

where . Retaining up to linear error termsgives

(45)

where . Its asymptotic value asis

(46)

Finally, we obtain from (18), (43) and (46) that the asymptoticMSE matrix of SRD-LS is

(47)

Generally speaking, (47) is larger than the CRLB. One excep-tion is that all 's , are equal. In such a case,is proportional to an identity matrix and putting in (42), (47)

becomes the CRLB in (8) for TDOA localization. This specialcase corresponds to the localization geometry where all sensors,except the first, are lying on a circle and the source is at thecenter.The bias of the SRD-LS solution is obtained using (46), (47),

(70) and (71) in (19).2) Asymptotic Performance of SRD-WLS: The first deriva-

tive of is

(48)

and is obtained from (39). Assuming is relatively accurateand has small error , where is a matrix determinedby the algorithm to obtain , the use of Taylor series expansiongives , where

. Therefore,and the error in is . Approximating

as gives

(49)

Hence

(50)

The second derivative of from (48) is

(51)

and

(52)


Fig. 1. Special geometries for comparison of SRD-LS and SRD-WLS. (a) Case1. (b) Case 2. (c) Case 3.

Its asymptotic value as is

(53)

Consequently, putting (50) and (53) into (18) gives the asymp-totic MSE matrix of SRD-WLS as

(54)

is exactly identical to the CRLB (8) for TDOA lo-calization, meaning that the solution from the SRD-WLS costfunction is asymptotically efficient.The bias of the SRD-WLS solution is evaluated by putting

(53), (54), (73) and (74) into (19).The MSE performance of the proposed SRD-WLS cost func-

tion is independent of the choice of reference sensor over thesmall noise region. This is because from the relationship

, using sensor as reference instead of sensor 1 isequivalent to applying a linear transformation to the TDOA datavector in (5) through the pre-multiplication of an invertible

matrix. The same linear transformationapplies to the TDOA covariance matrix and the gradientmatrix . Since the transformation is linear and invertible, wewill have the same MSE matrix given in (54) for the source lo-cation estimate. It should be noted though the bias could dependon the choice of the reference sensor because it is caused by thenon-linearity in the estimation.

C. Special Geometries

We have proven SRD-WLS achieves the CRLB accuracyunder small noise condition and has better estimation accuracythan SRD-LS. We would like to examine how much perfor-mance gain SRD-WLS can achieve compared to SRD-LS andhow the improvement varies with different geometries. Forthe sake of brevity, we consider 2D localization with sensornumber . In the following, we construct three classes ofspecial geometries and use the MSE matrices for SRD-LS andSRD-WLS in (47) and (54) to compare their performance.The special geometries are shown in Fig. 1:Case 1: All the sensors except the first (reference) are lo-

cated on a circle with radius and centered at , as shownin Fig. 1(a).

Case 2: The sensors are at, and the source is at the origin, as

depicted in Fig. 1(b).Case 3: The sensors are at

, and the source is at the origin, asillustrated in Fig. 1(c). The sensor (anchor) arrangement in thisform is quite common in sensor networks.We shall look at the SRD-WLS performance relative to

SRD-LS for uncorrelated and correlated noise, using the ratio.

1) Uncorrelated Noise:The TDOA noise can be uncorrelated when the TDOAs are

estimated one by one at different times.Case 1: In this case,

(55)

and both SRD-LS and SRD-WLS achieve the CRLB per-formance and . This result can be explained by thefact that under small noise, the weighting factorin the SRD-WLS cost function becomes a constant since

. Therefore the SRD-WLS cost function differsfrom the SRD-LS cost function by a constant scaling factoronly and hence they have the same performance.Case 2: For this configuration,

(56)

where . When becomes unity becausethe geometry is the same as in Case 1. When increases, how-ever, tends to 3/2. In this special case, SRD-WLS can provideup to 1.76 dB improvement over SRD-LS.Case 3: In this situation,

(57)

This is a very interesting case because as .In other word, as and move away from the source,SRD-WLS provides infinite performance gain over SRD-LS.Note that is bounded for ,

(58)

It is because as that makes theperformance improvement infinite.2) Correlated Noise:The TDOA noise is correlated with such a covariance matrix

when the TDOAs are estimated jointly and the signals receivedat the sensors have i.i.d. noise and identical SNR [2].Case 1: In this case,

(59)


The reason that SRD-LS and SRD-WLS have the same perfor-mance as CRLB is the same as before. RD-LS, SRD-LS andSRD-WLS have identical performance under small noise situa-tion.Case 2: In this scenario with

(60)

It can be proved that and the maximum2.23737 is attained at . For arbitrary , wehave not proven the boundedness of , but from simulation andover the region with a resolution of 1,we find that , which is quite similar to the resultwith . The performance improvement is larger than thesituation when the noise is uncorrelated.Case 3: In this configuration,

(61)

which indicates as . Note thatis bounded for at the value when

(62)

The infinite performance gain is resulted fromas increases.

V. SIMULATION

In this section, we shall validate the theoretical study and ex-amine the performance differences in MSE and bias betweenSR-LS and SR-WLS, and between SRD-LS and SRD-WLS.Since we focus on the solution accuracy of different cost func-tions and not the methods of solving them, we use the Gauss-Newton method with the true source location as the initialguess to obtain the numerical solutions. For SRD-WLS, we usethe solution from SRD-LS to do the initialization and to ob-tain in (39) for the purpose of generatingfor the weighting. The corresponding error in is

.We consider two different localization scenarios to examine

the influence from the weighting factors. The first scenario fixesthe source location and varies the sensor positions to create anumber of random geometries. In particular, the source is atthe origin and the sensors are randomly placed with uniformdistributions in x and y coordinates inside the unit circle. Thesecond scenario varies the source location and fixes the sensorpositions for creating another set of random geometries. In par-ticular, the source is randomly placed with uniform distribu-tions in x and y coordinates inside the circle of radius 0.98 andthe sensors are allocated uniformly in the unit circle, i.e.,

.In the simulation, the number of ensemble runs for each ge-

ometry is 1,200; the noise covariance matrix iswith different nonnegative value .

A. Comparison of SR-LS and SR-WLS

We use uncorrelated noise with covariance matrixfor the comparison unless stated otherwise.

Fig. 2. Average MSE for SR-LS and SR-WLS in the first scenario, .

Fig. 3. Average bias for SR-LS and SR-WLS in the first scenario, .

1) Average MSE and Bias: Fig. 2 shows the average MSEresult for over 4,000 randomly generated geometriesfor the first scenario. SR-WLS has 1.14 dB improvement overSR-LS and achieves the CRLB for less than 0.1. Beyond,the bias in SR-WLS dominates performance, causing the MSEbelow the CRLB. SR-WLS has larger bias than SR-LS as shownin Fig. 3, the difference is 9.20 dB for small noise. The increasein bias may be justified to achieve smaller MSE because the biasis negligible compared to the MSE. If we increase to 10, theaverage MSE improvement of SR-WLS over SR-LS increasesto 1.21 dB, and the average bias difference rises to 19.05 dB.Although the average improvement in MSE is not much, theimprovement at some common geometries is significant as willbe shown in Fig. 7.For the second scenario, the average results with 4,000

random geometries are similar to those in Figs. 2 and 3. When, SR-WLS has 1.22 dB improvement in average MSE

at the expense of 21.10 dB increase in average bias. When, SR-WLS has 1.28 dB improvement in average MSE

and the increase in bias is 8.78 dB.


Fig. 4. Empirical probability density functions of bias ratio between SR-LSand SR-WLS with different sensor numbers in the first scenario.

Fig. 5. Average of MSE ratio between SR-LS and SR-WLS with differentsensor numbers and noise correlation factors in the first scenario.

2) Distribution of MSE Ratio and Bias Ratio: We next ex-amine the distributions of the theoretical MSE ratio and the biasratio between SR-LS and SR-WLS over different geometries forthe first scenario. The MSE ratio distribution for uncorrelatednoise has been given in [19] and it tends to concen-trate around for large enough sensor number. Herewe only show the bias ratio distribution obtained from 200,000random geometries.Fig. 4 shows the empirical probability density functions of

the logarithmic bias ratiounder 5 different sensor numbers. First, we notice that canbe higher than 0, meaning that for some geometries SR-WLScan have smaller bias. Second, the bias ratio tends to concentratearound , i.e., with large enough sensornumber .3) Comparison Under Different Sensor Numbers and

Noise Correlations: We consider the sensor number rangingfrom 3 to 50 and the noise correlation factor varying frominteger values of 0 to 5. Fig. 5 shows the average of the MSE

Fig. 6. Average of bias ratio between SR-LS and SR-WLS with differentsensor numbers and noise correlation factors in the first scenario.

Fig. 7. MSE and bias comparison of SR-LS and SR-WLS under Special Case4 in [19] with .

ratio over 40,000 randomgeometries of the first scenario as increases. The minimumaverage ratio is 1.26, which occurs when and

. For and , the average keeps increasingbefore and then tends to a constant value with largersensor number . For , the average varies slightlyat the beginning and then stabilizes. When a further increases,contrary to that in and , the average is largerfor small sensor number than the steady value for large .It is also noticeable that the MSE improvement from SR-WLSover SR-LS is more significant as the noise becomes morecorrelated. For , the minimum average is 1.8.Fig. 6 shows the corresponding results as in Fig. 5 for the

average of the bias ratio .The average is smaller than 0.6 in each curve and SR-LS hassmaller bias. For uncorrelated noise , the average de-creases before , increases between and ,and finally converges to 0.24. From to , each curvedecreases with increasing sensor number .We also notice thatlarger produces smaller average .


Fig. 8. AverageMSE for SRD-LS and SRD-WLS in the first scenario, .

Fig. 9. Average bias for SRD-LS and SRD-WLS in the first scenario, .

4) Special Case With Large MSE Ratio : Fig. 7 shows thesimulation result of the geometry from Special Case 4 in [19]with , where the source is at (0, 0) and the sensors areat and . This geometry is known to be badfor SR-LS. We can see that the MSE of SR-WLS achieves theCRLB, and has more than 18 dB improvement over SR-LS fornoise power below dB. In addition, the bias in SR-WLS isabout 45 dB smaller than that in SR-LS for noise power below

dB.

B. Comparison of SRD-LS and SRD-WLS

We use the following particular setting unless stated other-wise. The noise covariance matrix is , cor-responding to the noise correlation factor equal to 1. For thefirst scenario, the distances between the sensors and source areat least 0.1, i.e., all the sensors are uniformly distributed in theannulus which is inside the unit circle and outside the circle withradius 0.1. The purpose is to avoid the bad geometries where thesensors are too close to the source.

Fig. 10. Average MSE for SRD-LS and SRD-WLS in the second scenario,.

Fig. 11. Average bias for SRD-LS and SRD-WLS in the second scenario,.

1) Average MSE and Bias: Fig. 8 shows the average MSEresult for over 4,000 random geometries under the firstscenario. The average MSE for SRD-WLS is 1.14 dB smallerthan that of SRD-LS and achieves the CRLB accuracy. UnlikeSR-WLS, SRD-WLS does not introduce obvious bias comparedto SRD-LS as can be seen in Fig. 9. If we increase to 10, theaverage improvement is 1.43 dB in MSE and 1.08 dB in bias.The SRD-WLS improvement at some common geometries issignificant as will be shown in Fig. 16.The results for the second scenario averaged over 4,000

random geometries are shown in Figs. 10–11. When ,SRD-WLS has 3.96 dB improvement in average MSE and 1.79dB improvement in average bias. For , the improve-ment in MSE and bias increases to 5.93 dB and 6.62 dB.2) Distribution of MSE Ratio and Bias Ratio: The distribu-

tions of the theoretical MSE ratio and the bias ratio are gener-ated from 200,000 random geometries from the first scenario.Fig. 12 shows the empirical probability density functions of the


Fig. 12. Empirical probability density functions of MSE ratio betweenSRD-LS and SRD-WLS with different sensor numbers in the first scenario.

Fig. 13. Empirical probability density functions of bias ratio between SRD-LSand SRD-WLS with different sensor numbers in the first scenario.

MSE ratio under 6 dif-ferent sensor numbers. From to , the distribu-tion of becomes more concentrated, and the probability thatis near 1 or larger than 2.6 becomes smaller. At , this

distribution has mode near and has higherprobability than . As keeps increasing, the distribu-tion of moves toward the right hand side. It does not seemto have an upper limit of and we can always have larger im-provement in through increasing sensor number .Fig. 13 shows the empirical probability den-

sity functions of the logarithmic bias ratiounder 5 different

sensor numbers. For , SRD-LS has slightly smallermean bias value. However, as increases, the distributionsskew to the right and SRD-WLS gives smaller bias on theaverage.3) Comparison Under Different Sensor Numbers and

Noise Correlations: We vary the sensor number from 4to 50 and noise correlation factor from integer values of 0 to

Fig. 14. Average of MSE ratio between SRD-LS and SRD-WLS with dif-ferent sensor numbers and noise correlation factors in the first scenario.

Fig. 15. Average of bias ratio between SRD-LS and SRD-WLSwith differentsensor numbers and noise correlation factors in the first scenario.

5 in the first scenario and obtain the theoretical results from4,000 random geometries. In Fig. 14, the minimum of theaverage of is 1.2, which occurs when and . Foruncorrelated noise , the average increases under smalland settles to 1.32. For the other values, after it

keeps increasing with larger sensor number , indicating theMSE improvement of SRD-WLS over SRD-LS can alwaysstrengthen as the number of sensors increases. Furthermore,larger gives larger improvement.Fig. 15 gives the average value of as the sensor number

increases from 15 to 50. We do not give the result forbecause can be bigger or smaller than 1 with nearly equalchances and its average value may not give proper indicationabout the relative bias between SRD-LS and SRD-WLS. Foruncorrelated noise , the average value of is near 2 forthe sensor numbers tested. For between 1 and 5, it is largerthan 1.8 and tends to increase with sensor number not less than15.4) Special Case With Large MSE Ratio : Fig. 16 shows

the simulation result of Case 3 under correlated noise in


Fig. 16. MSE comparison of SRD-LS and SRD-WLS under Case 3 with cor-related noise, ; where dashed line and represent theoretical and simu-lated bias of SRD-LS, and dotted line and represent theoretical and simulatedbias of SRD-WLS.

Subsection IV.C2 with . When the noise power is below0.01, SRD-WLS offers 18.07 dB improvement in MSE overSRD-LS and achieves the CRLB performance. The simulationuses the solution from SRD-LS to generate for the weightingof SRD-WLS. Even though the SRD-LS solution is very in-accurate, SRD-WLS is able to maintain good results whenis not larger than 0.01. At , the thresholding effectin the SRD-LS solution occurs, which makes the performanceof SRD-WLS deviates quickly from the CRLB. If we useinstead in SRD-WLS to obtain for the weighting, we stillhave good performance at . The bias of SRD-WLSis 28.57 dB smaller than that of SRD-LS when noise level isbelow .The average MSE improvements of SR-WLS and SRD-WLS

are not obvious as shown in Figs. 2 and 8 when the sensornumber is small. However, under some unfavorable geometriessuch as the ones used in Fig. 7 and in Fig. 16, significant per-formance gains are observed. It is expected that SR-WLS andSRD-WLS will be beneficial and offer considerable better re-sults when the localization geometries are not favorable.It should be noted that the proposed SR-WLS and SRD-WLS

cost functions require the measurement noise covariance matrixto be known, as in the case of typical WLS, in order to reach theoptimum CRLB performance.

VI. CONCLUSION

In this paper, we propose the introduction of range weightingfactors to the SR-LS and SRD-LS cost functions to improvetheir solution accuracy. The resulting cost functions, calledSR-WLS and SRD-WLS, maintain the attractive computa-tional efficiency of SR-LS and SRD-LS to obtain global andclosed-form solutions, while overcoming the disadvantage ofSR-LS and SRD-LS that have suboptimum performance andyield very large localization errors under some geometries.The location MSE and bias obtained from the solutions ofSR-WLS and SRD-WLS are analyzed theoretically and theyare contrasted with those from SR-LS and SRD-LS. We alsoelaborate the performance differences between SR-LS andSR-WLS, and between SRD-LS and SRD-WLS in terms of

localization geometry, sensor number and noise correlation.The proposed SR-WLS is shown to yield a solution reachingthe CRLB accuracy under Gaussian noise and has better MSEperformance than SR-LS. However, SR-WLS has larger bias ingeneral compared to SR-LS. Nevertheless, the bias is relativelysmall and negligible compared to the MSE. If needed, the biascan be estimated and subtracted from the location estimate ofSR-WLS to reduce the bias. The proposed SRD-WLS is alsoproved to be able to yield an efficient solution and has lowerMSE than SRD-LS. Unlike the case of range measurements,SRD-WLS often has smaller solution bias than SRD-LS as well,when the number of sensors is not critical. In both SR-WLS andSRD-WLS, the MSE improvement over SR-LS and SRD-LSincreases as the number of sensors or the amount of noisecorrelation increases. The study in the paper illustrates theadvantage of the proposed weighting in improving localizationaccuracy, especially for range difference based localization.The performance gain from range weighting factors in

SR-WLS or SRD-WLS is quite effective when the source isnear or inside the sensors array such as in sensor networkapplications. If the source is too close to a sensor, the sourcerange could be close to zero and adding a small constant tothe range before forming the weight may be needed to avoidnumerical instability. The effect of weighting becomes lesssignificant when the source is far away from the sensors, sinceall the distances between the source and sensors are closeand SR-WLS and SRD-WLS will reduce back to SR-LS andSRD-LS.

APPENDIX ABIAS ANALYSIS

In this Appendix, we will provide the bias analysis formulasfor SR-LS and SR-WLS, and for SRD-LS and SRD-WLS.

A. Bias of SR-LS

When the noise is small, we have from (25) using the Neu-mann expansion [25]

(63)

Applying (21) and maintaining up to second order noise term,

(64)

where .In the second bias component, and

are given in (26) and (27). From (24),

(65)

where the column vector has 1 in the th element and 0 inothers. Notice that (65) doesn’t contain noise and

.


Putting (26), (27), (64) and (65) into (19) gives the bias ofSR-LS.

B. Bias of SR-WLS

Starting from (33),

(66)

is given in (30). Hence up to the second order noiseterms

(67)

where the column vector , and; similarly, the column vector

, and .and are in (34) and (35). From (32),

(68)

As a result, we can obtain the bias of SR-WLS from (19).

C. Bias of SRD-LS

According to (45),

(69)

Using in (41) and keeping up to the second noise terms

(70)

where .and are in (46) and (47). From (44),

(71)

where . The bias for SRD-LS asshown in (19) can be evaluated.

D. Bias of SRD-WLS

From (52) and under small noise assumption,

(72)

Using given in (49) and keeping up to the secondorder noise terms

(73)

where the column vector ,and ; simi-larly, the column vector , and

.and are in (53) and (54). Using (51),

(74)

The bias of SRD-WLS can now be found from (19).

ACKNOWLEDGMENT

The authors thank the reviewers for providing valuable com-ments and suggestions to improve the paper.

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Shanjie Chen (S’11) received the B.S. degree inelectrical engineering from the Nanjing AgriculturalUniversity, Nanjing, China, in 2007, and M.S.degree in electrical engineering from the Universityof Science and Technology of China, Hefei, China,in 2010.He is currently working toward the Ph.D. degree

with the Department of Electrical and ComputerEngineering, University of Missouri, Columbia. Hiscurrent research interests include source localizationand sensor node self-localization.

K. C. Ho (M’91–SM’00–F’09) was born in HongKong. He received the B.Sc. degree with First ClassHonors in electronics in 1988 and the Ph.D. degreein electronic engineering in 1991, both from the Chi-nese University of Hong Kong.He was a Research Associate in the Royal Military

College of Canada from 1991 to 1994. He joinedBell-Northern Research, Montreal, Canada in 1995as a member of scientific staff. He was a facultymember with the Department of Electrical Engi-neering, University of Saskatchewan, Saskatoon,

Canada, from September 1996 to August 1997. Since September 1997, he hasbeen with the University of Missouri where he is currently a Professor withthe Electrical and Computer Engineering Department. His research interestsare in sensor array processing, source localization, subsurface object detection,wireless communications, and the development of efficient signal processingalgorithms for various applications. He is an inventor of 19 patents in theUnited States, Europe, Asia, and Canada on geolocation, and signal processingfor mobile communications.Dr. Ho is the Associate Rapporteur of ITU-T Q16/SG16: Speech en-

hancement functions in signal processing network equipment and was theRapporteur of ITU-T Q15/SG16: Voice Gateway Signal Processing Functionsand Circuit Multiplication Equipment/Systems from 2009-2012. He is theEditor of the ITU-T Standard Recommendations G.160: Voice EnhancementDevices and G.168: Digital Network Echo Cancellers. He is the Chair ofthe Sensor Array and Multichannel Technical Committee in the IEEE SignalProcessing Society. He has been serving his second Associate Editor term ofthe IEEE TRANSACTIONS ON SIGNAL PROCESSING since January 2009. He wasan Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from2003 to 2006 and the IEEE SIGNAL PROCESSING LETTERS from 2004 to 2008.He received the Junior Faculty Research Award in 2003 and the Senior FacultyResearch Award in 2009 from the College of Engineering at the University ofMissouri.

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