04610272

5758 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 12, DECEMBER 2008

On the Use of a Calibration Emitter forSource Localization in the Presence of

Sensor Position UncertaintyK. C. Ho, Senior Member, IEEE, and Le Yang, Student Member, IEEE

Abstract—Sensor position uncertainty is known to degrade sig-nificantly the source localization accuracy. This paper investigatesthe use of a single calibration emitter, whose position is knownto the sensor array, to reduce the loss in localization accuracydue to sensor position errors that are random. Using a Gaussiannoise model, we first derive the Cramér–Rao lower bound (CRLB)for a time difference of arrival (TDOA)-based source locationestimate with the use of a calibration source. The differentialcalibration technique that is commonly used in Global PositioningSystem through the use of a calibration source to mitigate theinaccuracy in satellite ephemeris data is analyzed. The analysisindicates that differential calibration in most cases cannot reachthe CRLB accuracy. The paper then proceeds to propose an alge-braic closed-form solution for the source location estimate usingboth TDOA measurements from the unknown and the calibrationsource. The proposed algorithm is shown analytically, under highsignal-to-noise ratio (SNR) and small sensor position noise, orunder moderate level of SNR and sensor position noise togetherwith distant unknown and calibration sources, to reach the CRLBaccuracy. Simulations are used to corroborate and support thetheoretical development.

Index Terms—Cramér–Rao lower bound (CRLB), sensor po-sition uncertainty, source localization, time difference of arrival(TDOA).

I. INTRODUCTION

T HE passive localization of an emitting source is a funda-mental problem in many signal processing applications

including tracking, navigation, and search and rescue [1]–[4].The research of source localization has experienced renewedinterest in recent years. One reason is due to the Federal Com-munication Commission (FCC)’s E911 requirement for a cel-lular system, where a wireless network operator is required tolocate a cell-phone user with an accuracy of 100 m at 67% prob-ability and 300 m at 95% probability for a network-based solu-tion [5], [6]. In addition, wireless sensor network is gaining pop-ularity and one of its promising applications is source localiza-tion. The sensor nodes often have power constraint and limitedcomputation capability. The emerging applications and the newsensor platforms coupled with various localization scenarios

Manuscript received August 17, 2007; revised June 19, 2008. First publishedAugust 29, 2008; current version published November 19, 2008. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Antonia Papendreou-Suppappola.

The authors are with the Department of Electrical and Computer Engi-neering, University of Missouri-Columbia, Columbia, MO 65211 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2008.929870

create new challenges for source localization (see [7]–[10] andreferences therein).

Traditionally, source localization assumes a platform withmany spatially separated sensors whose positions are known ex-actly. The sensors receive the emitted signal from an unknownsource and compute the positioning parameters. Typical posi-tioning parameters include time of arrival (TOA), time differ-ence of arrival (TDOA), angle of arrival (AOA), as well as fre-quency difference of arrival (FDOA) if there is relative motionbetween the source and the sensors [11]. A set of nonlinear equa-tions that relate the unknown source location and the positioningparameters is then used to identify the source location. To limitthe scope in this paper, we shall consider the localization of asingle stationary source using TDOA measurements only.

A large amount of literature is available for source localiza-tion using TDOAs. Some methods are iterative and require goodinitial solution guesses [12], [13], and some are closed-formsolutions [11], [14]–[21]. Most of these methods are able toreach the Cramér–Rao lower bound (CRLB) accuracy, and theirbehaviors could be different depending on the signal-to-noiseratio (SNR) condition. All of these works, however, assume thesensor positions are known exactly. In the emerging applica-tions for source localization, the sensors could be airborne suchas unmanned aerial vehicles (UAVs) whose positions will not beknown precisely. Also, the sensors in sensor networks are typ-ically deployed randomly in a field and their locations may notbe accurate. Even though their positions can be measured at thebeginning, the sensors may drift over time. The sensor positionuncertainty, even a small amount, can lead to significant degra-dation in the source localization accuracy [22].

Recently, Ho et al. [22] performed a theoretical study on theamount of degradation of a localization algorithm that assumesthe sensor positions are perfectly known, but in fact have errors.They also proposed a novel algorithm that takes the statisticaldistributions of the sensor positions into account to improve thelocalization accuracy. This new algorithm was shown analyti-cally to reach the CRLB for distant sources at sufficiently highSNR. The localization accuracy, on the other hand, could still bequite far from the case where the sensor positions are perfectlyknown.

This paper considers the use of a single calibration sourcewhose position is known exactly, to improve the source localiza-tion accuracy in the presence of random sensor position errors.The idea of using a calibration source is not new. In fact, it hasbeen used in the global positioning systems (GPS) to mitigatethe effect of uncertainties in satellite positions, and various er-

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HO AND YANG: ON THE USE OF A CALIBRATION EMITTER FOR SOURCE LOCALIZATION 5759

rors caused by satellite clock mismatch, tropospheric and iono-spheric layers. In GPS literature, this technique is called differ-ential calibration (DC) [23], and the resulting GPS system iscalled differential GPS (DGPS) [24], [25]. In DGPS, the TOAfrom a GPS satellite to the calibration platform is subtractedfrom the TOA of the same satellite to the unknown location to beidentified, before the TOA is used to obtain the unknown loca-tion. It has been shown that DGPS can improve the localizationaccuracy tremendously [24], [25]. Although the DC techniquecan be effective in reducing the effect of sensor position uncer-tainty, whether it is able to achieve the best possible localizationaccuracy has not been examined.

In this paper, we begin the study by evaluating the CRLB ofa source location estimate when the TDOAs from a calibrationsource are available. The insight gained from the CRLB indi-cates that the calibration source can dramatically reduce the ef-fect of sensor position uncertainty and greatly improve the lo-calization accuracy. We then continue to analyze the DC tech-nique. Interestingly enough, DC is found to be unable to achievethe CRLB accuracy in general. The sub-optimum performanceof DC then motivates us to develop a new algorithm that canreach the CRLB accuracy. The newly proposed algorithm is al-gebraic and closed-form. It is based on weighted least-squaresminimization in several stages. The algorithm starts with somefixed weighting matrix to generate a solution. A few iterations(one to two) are then used beginning with this initial solutionto improve the weighting matrices and generate a better solu-tion. In fact, the accuracy of the proposed solution is provedanalytically to attain the CRLB accuracy for Gaussian TDOAand sensor position noises when two mild conditions are satis-fied. The paper is closed by the use of computer simulations tocorroborate the theoretical development.

In the sequel of this paper, bold face lower case letters rep-resent column vectors, and bold face upper case letters denotematrices. If contains noise, is used to denote the truevalue of . The paper contains a number of symbols. The sym-bols and notations are summarized in Table II of Appendix I forquick reference.

The rest of the paper is organized as follows. Section II for-mulates the problem and introduces the symbols and notationsused. The CRLB for the problem is evaluated in Section III,and the performance analysis of DC is presented in Section IV.Section V develops the proposed solution and proves analyti-cally that under some conditions it attains the CRLB accuracyfor Gaussian noise. Section VI confirms the theory through sim-ulations, and Section VII concludes the paper.

II. PROBLEM FORMULATION

The localization scenario is shown in Fig. 1. A point sourceat unknown position radiates a signal thatis captured by sensors located at

. The TDOAs of the received signals with respectto the signal at a reference sensor, say sensor 1, are estimated.After multiplying by the known signal propagation speed , wehave the measurements

(1)

Fig. 1. Source localization scenario. The circles represent sensors, the triangledenotes the unknown source whose position is to be estimated, and the squareis the calibration source whose position is known.

where , and is the estimated TDOA betweensensor pair and 1. is the true TDOA, and

(2)

where is the Euclidean norm. is times the TDOAnoise. is often referred to as range difference of arrival(RDOA) in some literature. TDOA and RDOA will be usedinterchangeably in this paper since they are equivalent. Fornotation simplicity, we collect to form the

RDOA vector as

where , whose elements are , is assumed to be a zero-mean Gaussian vector with covariance matrix . Note that theRDOA noise power is inversely proportional to the SNR of thesignals received at the sensors.

The true sensor positions are not known and only noisyversions of them, denoted by , are available. Mathematically,we have

(3)

where is the position error in . The collection of formsthe 1 sensor position vector

where and .We shall model as a zero-mean Gaussian vector with covari-ance matrix .

In order to reduce the effects of sensor position errors, theTDOAs of the signal from a calibration source at known position

to the sensors are also measured, giving thecalibration RDOA vector

where is the estimatedcalibration TDOA between sensor pair and


is thetrue value of , and

(4)

is a zero-mean Gaussian vector with covariance matrix .The noise vectors and are assumed to be independentwith one another. We further assume here that the calibrationTDOA measurements are acquired immediately after the TDOAmeasurements from are taken so that and depend onthe same sensor positions [see (2) and (4)] if is changing withtime. The objective is to obtain the source location based onthe TDOA measurements from and from .

III. CRLB

The CRLB establishes a lower bound on the error covariancematrix for any unbiased estimate of a parameter vector, and itis equal to the inverse of the Fisher information matrix (FIM)that is created from the probability density function (PDF) ofthe underlying problem [26]. Under the localization scenariopresented in Section II, the PDF that is parameterized on theunknowns and is

(5)

where we have applied the fact that and are Gaussiandistributed and independent with one another. is a constantthat does not depend on the unknowns. Taking logarithm of theabove equation, applying partial derivatives with respect to theunknowns twice, negating the sign and then taking expectationyield

(6)

where

(7a)

(7b)

and

(7c)

We next evaluate the partial derivatives. The ( -1)th row ofis , and from (2) we have

(8)

where represents the unit vector from to , and it isequal to

The ( -1)th row of is , and againfrom (2)

(9)

Similarly, the ( -1)th row of is, from (4)

(10)

The upper left 3 3 block of the inverse of the FIM is the CRLBof , where is an unbiased estimate of . Applying the par-titioned matrix inverse formula [27] to (6) yields

CRLB

(11)

where is the CRLB of when is known. It is shown inAppendix II that the second term in the second line of the aboveequation is positive definite when , which indicates thatusing a calibration source is not able to reduce the CRLBback to the one without sensor position uncertainty.

We now compare (11) with the CRLB of without a calibra-tion source. We shall denote it as CRLB , and it is given in(9) of [22]. The difference between CRLB and CRLBlies in , where the application of a calibration sources intro-duces an extra component

We shall express (7c) as , where is equal to the sumof the first two terms of (7c) on the right. Applying the matrixinversion Lemma [27] to , substitutingback into (11) and simplifying, we arrive at

CRLB CRLB (12)

where


TABLE ITRUE POSITIONS (IN METERS) OF SENSORS

and is the Cholesky decomposition of , i.e.,.

The second term in (12) represents the performance improve-ment due to the introduction of a calibration source. In the spe-cial case that is zero, then and the secondterm in (12) would become zero. In other words, there is no im-provement in the localization accuracy. This is not unexpectedbecause implies that the calibration TDOA measure-ments are so noisy and become useless. In fact, the second termin (12) is a positive semidefinite matrix. This can be verified bynoting that this term has a symmetric structure and the matrix

does not have full column rank. Thus, having a calibrationsource will not degrade the best localization accuracy of the un-known source . In fact, the improvement could be significantat typical calibration TDOA noise level, as is illustrated by thefollowing example.

Let us consider the same localization geometry used in[22], where the true locations of the sensors are shown inTable I. The unknown source is distant from the sensors and islocated at m. The effects of two differentcalibration sources are examined, one closer to the unknownsource at m and the other farther awayat m. Please refer to Section VI for thedetails on the noise parameter settings. Fig. 2 plots the trace ofthe CRLB from (11), the trace of CRLB and the traceof the CRLB without sensor position error , as a functionof , where is the receiver position noise power andis the RDOA noise power. As we can observe from the figure,the lower bound of source localization error without sensorposition uncertainty is the lowest and is independent of thevalue of . On the other hand, the lower bound with sensorposition uncertainty grows significantly as increases.The interesting point is that the application of one calibrationsource can improve significantly the localization accuracy.Using the farther away calibration source can provide at least5-dB improvement in the localization accuracy compared to thecase without the calibration source, when dB. Forthe calibration source that is closer to the unknown source, itcan offer even greater improvement in accuracy (at least 9.5 dBwhen dB). In general, we find that the closerthe calibration source is to the unknown source, the larger theimprovement will be.

IV. ANALYSIS OF DIFFERENTIAL CALIBRATION

Before going into the details of DC, let’s first examine howthe sensor position errors affect the source localization accu-racy. From (2), the TDOA measurements are dependent on the

Fig. 2. Comparison of the CRLBs with and without a calibration source for anunknown distant stationary source. (1) The best achievable accuracy withouta calibration source [��CRLB�� from (9) in [22]], (2) the best achievableaccuracy with a calibration source when � � �� m[��CRLB�� from (11)], (3) the best achievable accuracy with a calibra-tion source when � � �� m [��CRLB�� from (11)],(4) the best achievable accuracy when the sensor positions are known exactly[�� from (7a)].

true locations of the sensors. Since are not known and onlyare available, we substitute from (3) intoand apply the Taylor-series expansion up to first-order term toobtain

(13)

where is the unit vector from to . Expressing interms of its noisy measurement as and substi-tuting (13) into (2) yield the solution equation that relates themeasurements and the unknown source location:

(14)

where

(15)

is the equation error which determines the localization accuracy.It is interesting to see from (15) that the effect of the sensorposition error is equivalent to increasing the TDOA noise, wherethe contribution from the noise in is its projection along thedirection from to . This is shown pictorially in Fig. 3(a).(15) also indicates that is directly proportional to the sensorposition errors . When the sensor position noise powers arelarge, the variance of the error term will be big as well andcorrespondingly, the localization accuracy will be worse.

DC is a simple method to exploit the calibration TDOA mea-surements to improve the localization accuracy. DC subtractsdirectly the calibration TDOA from the corresponding TDOAof the unknown source and uses the difference to estimate thesource location by considering the noisy sensor positions asaccurate.


Fig. 3. (a) Effect of the sensor position error on the source localization accu-racy. (b). Principle of differential calibration.

The positioning equation for DC is

Substituting and , and applyingthe Taylor-series expansion similar to (13) in each term on theright side of the above equation yield the solution equation

(16)

where

(17)

If we consider the term inside the square bracket in (16) aspseudo TDOA measurement, then (16) is in the same form astraditional TDOA equation with sensors located at . Hence,the traditional TDOA localization algorithm can be applied toobtain the emitter location.

In the overall error , as illustrated in Fig. 3(b), the con-tribution of is the difference in the projections of along

and . Alternatively, we have, from the dot-productrelationship,

If is close to will be very small. Hence, thecontribution of the error from to could be reduced usingDC. The drawback of DC is that the effective noise power due toTDOA measurements is the sum from the unknown source andfrom the calibration source, as this is apparent from the first twoterms of given in (17). Thus, DC is effective in improvingthe localization performance only when the calibration sourceis close to the unknown source and the TDOA measurementsfrom the calibration source are much more accurate than thosefrom the unknown source.

We now obtain the maximum achievable localization accu-racy using DC. Let be a vector containingall measurements, and be the unknown pa-rameter vector. Using the functional forms from (2) and (4) that

relate the unknowns to the measurements, the Taylor-series ex-pansion of at up to linear term is

(18)

where is the value of evaluated at is the gradientmatrix given by

(19)

and the partial derivatives are given in (8)–(10).Let be the matrix equal to

(20)Then the DC method simply uses

(21)

instead of to obtain an estimate of . Let and. Then, premultiplying (18) by gives

(22)

The best linear unbiased estimator (BLUE) [26] of fromthe linear model (22) is

(23)

where is the inverse of the covariance matrix of the mea-surements after DC. Specifically, we have

(24)

where (21) has been used and is the covariance matrix of, which is a block diagonal matrix with the diagonal blocks

equal to and . Subtracting on both sides of (23),multiplying by its transpose, taking expectation and using (24)yield the best localization accuracy for DC,

(25)

We now compare cov with the CRLB of . Appendix IIIshows that they are related by

CRLB (26)

where is defined in (76). Applying the matrix inversionLemma [27] to (26) gives

CRLB CRLB

CRLB CRLB (27)

The second term is the performance degradation of DC with re-spect to the CRLB. From its symmetric structure, we can show


Fig. 4. Comparison of the CRLB�� with the best MSE of � using DC fora distant unknown source, (a) the calibration source is close to the unknownsource, (b) the calibration source is far away from the unknown source. (1)The best achievable accuracy without a calibration source [��CRLB�� from (9) in [22]]: solid line, MSE from simulation using the solution in [22]:square symbol, (2) the best achievable accuracy with a calibration source[��CRLB�� from (11)], (3) the best achievable accuracy with DC fromtheory [�� from (25)].

that it is positive semidefinite. Hence, the DC technique gener-ally cannot achieve the CRLB accuracy.

To gain some insights about the performance of DC, we eval-uate, in Fig. 4, the trace of , where is the upperleft 3 3 block of in (27). is actuallythe minimum achievable mean-square error (MSE) of an unbi-ased estimate of using DC. The localization geometry is thesame as the one used in generating Fig. 2. Fig. 4(a) gives the re-sults when the calibration source m iscloser to the unknown source, and Fig. 4(b) is when the calibra-tion source m is farther away fromthe unknown source. The theoretical MSE without a calibrationemitter [ CRLB from (9) in [22]] and the MSE of thetarget estimates generated by the solution proposed in [22] forthe case where there is no calibration emitter are also shownfor comparison. Note that the results of the algorithm in [22]denoted by the square symbols in curve (1) attain the CRLB ac-curacy when 20 dB.

It is clear from both Fig. 4(a) and (b) that when the sensorposition error is small relative to the RDOA measurementnoise ( 15 dB), DC is always 3 dB worse than theCRLB as well as the practical estimator from [22] and thisperformance degradation persists regardless of the change inthe calibration source position. This is because from (17), thecontribution of the sensor position error to the equationerror becomes negligible when the sensor positions areaccurate, and the localization accuracy is mainly determined bythe doubling of RDOA noise power after DC when .

On the other hand, when 10 dB, the sensor positionerror dominates and the localization accuracy of DCdepends heavily on the calibration source position. Fig. 4(a)shows that when the calibration source is closer to the unknownsource, the localization accuracy of DC approaches the CRLBaccuracy. This is because from (17), the effects of sensorposition errors can be greatly reduced by DC. Nevertheless,when the calibration source is located farther away from theunknown source, DC is not able to reach CRLB all the time,as is illustrated in Fig. 4(b). In fact, the minimum MSE fromDC is even 5 dB worse than that of the case where there isno calibration source. This is mainly due to the increase inRDOA measurement noise and the large distance between thecalibration source and the unknown emitter, which decreasesDC’s ability to mitigate the sensor position errors.

We would like to clarify that the calibration emitter, no matterit is distant from or close to the sensors, provides additionalinformation to the localization problem. It should not degrade,if not improve, the localization accuracy when this informationis used properly. This is supported by the CRLB analysis in(12). The worst performance of DC shown in Fig. 4(b) whenthe calibration emitter is far away from the unknown target isdue to the DC method which does not exploit the calibrationmeasurements properly.

V. CLOSED-FORM LOCALIZATION ALGORITHM

We shall develop in this section an algebraic closed-formsolution to the localization problem in the presence of sensorposition errors where a single calibration source is available.Compared to iterative solutions such as numerical maximumlikelihood (ML), the proposed solution does not require initialguesses close to the true solution and would not suffer from localconvergence problem. An interesting property of the proposedsolution is that it is able to achieve the CRLB accuracy at suf-ficiently high SNR. As a byproduct of the algorithm, we couldalso obtain improved sensor positions. We shall first describethe proposed algorithm, then an analysis will follow to showthat the localization accuracy attains the CRLB under two mildconditions.

A. Algorithm

The proposed solution has three stages. The first stage esti-mates the sensor position errors and improves the sensor posi-tions, using the calibration TDOA measurements from . Theremaining two stages of the proposed solution follow the ap-proach in [22] to obtain an estimate of the unknown target loca-tion using the updated sensor positions. In the second stage, wetransform the nonlinear solution equation to a pseudolinear oneby introducing a nuisance parameter. After solving the pseudo-linear equation, the unknown source location estimate is refinedwith the help of the estimated value of the nuisance parameterin the third stage.

First Stage: Since the location of the calibration source isknown, the difference between the measured RDOA and the pre-dicted RDOA using the erroneous sensor positions provides in-formation about the difference between the actual and assumedsensor positions, and the difference can be exploited to improvethe sensor positions. The calibration RDOA measurement is


, where is the true value and is the noise.After substituting (4), we arrive at

The predicted RDOA using the erroneous sensor positions is. Using the error model (3) and applying

the Taylor-series expansion up to linear error term, we have theapproximation . The differencebetween the measured RDOA and the predicted one is

(28)

Expressing (28) in matrix form gives

(29)

where is an vector with its th elementequal to is anmatrix and its th row is

(30)

and is the calibration RDOA measurement noise vector withcovariance matrix . In (29), is considered as the unknownto be solved. We know a priori that is a zero-mean Gaussian-distributed random vector with covariance matrix . Fromthe Bayesian Gauss–Markov theorem [26], the linear minimumMSE (LMMSE) estimator of , denoting it as , is

(31)

and the resulting covariance matrix of is, when ignoring thenoise in

(32)

With the estimated sensor position errors given in (31), weobtain the improved sensor positions as

(33)

and it has a covariance matrix equal to (32). It can be easily seenfrom (32) that covis positive semidefinite. Hence, with the calibration RDOA mea-surements, we have an updated sensor position vector at least asgood as, if not better than, the original one to obtain the un-known source location.

Second Stage: According to (2), the true RDOA is relatedto the true ranges and as . Squaring bothsides, substituting and

and simplifying yield

(34)

Substituting the true RDOA value by asobtained from (1), we have

(35)

where the second-order error terms containing have been ig-nored. and are the true sensor positions that are not known.The true distance between the unknown source and sensor 1, ,depends on the unavailable as well. Stage 1 provides an im-proved sensor position vector using the calibration TDOAs. Weshall express (35) in terms of the improved sensor positionsand given in (33).

Using the Taylor-series expansion of up to linear errorterm, we arrive at

(36)

where . Also,. Thus, (35) becomes

(37)

where

(38)

In (37), the unknowns are and and they are correlated,because both are dependent on the unknown source location .If we assume and are independent variables, (37) is linearwith respect to the unknowns and can be easily solved. Note that

and are in fact dependent. It is the purpose of applyingstage 3 later to exploit their relationship to improve the sourcelocation estimate.

Defining be the unknown vector, we havethe linear matrix equation from (37)

(39)

where

...

......

(40)

The equation error vector is .From (38), it can be rewritten in terms of and as

(41)


where [see (42) and (43), shown at the bottom of the page].The covariance matrix of is and that of is given

in (32). and are independent. Hence, the covariancematrix of is, when ignoring the noise in and

(44)

The weighted least-squares (WLS) solution to (39) is

(45)

where the weighting matrix is defined as

(46)

We next evaluate the accuracy of by examining its co-variance matrix. Premultiplying the estimation error

with and applying (39) and (45) yield

Hence, the estimation error vector is equal to. Then, if we assume that both

the RDOA measurement noise and sensor position error aftercorrection using (33) are sufficiently small so that canbe considered as fixed, the covariance matrix of can beapproximated by1

(47)

where the definition of in (46) has been applied.Third Stage: In this stage, the relation between and is

explored to improve the localization accuracy of the unknownsource. This is achieved by constructing a weighted LS problemusing the solution obtained in stage 2.

1A more concrete approach to derive �� is applying the per-turbation method. It can be verified that up to first order error term,�� , where the �� th row of � is�� .

We collect the first three elements of as . Notethat is an estimate of . Expressing it as

and retaining up to linear error term give

(48)

where denotes the error vector in , andrepresents the Schur product (element by element multiplica-

tion). The fourth element of is an estimate of and hence,it can be expressed as , where isthe estimation error. Squaring both sides and using the defini-tion of below (36) yield

(49)

where the second-order error term has been ignored.Hence, from (48) and (49), we have

(50)

where

(51)

is a vector of unity and

(52)

The error vector can be expressed in terms of the error in, as

(53)

where is defined as

(54)

......

. . ....

(42)

and

......

.... . .

...(43)


The weighted LS solution to (50) is

(55)

The weighting matrix is chosen to be

(56)

where (47) has been substituted and the noise in and isignored. Following a similar approach that gives , wecan obtain the covariance matrix of as

(57)

Having computed from (55), the final unknown sourceposition estimate is obtained from the mapping according to thedefinition of in (52) as

(58)

where is the sign function defined as 1 when andotherwise. The purpose of the diagonal matrix is to remove

the sign ambiguity due to the square root operation.To examine the localization accuracy, we first note that

. Then, substituting yields

(59)

where the second-order term has been ignored, andis defined as

(60)

Post-multiplying (59) by its transpose and taking expectation,the covariance matrix of the unknown source position estimate

from the proposed three-stage algebraic solution is, after ig-noring the noise in

(61)

We now summarize the steps to compute the source locationestimate: i) find an estimate of the sensor position error vectorby (31); ii) obtain an improved sensor position vector using (33);iii) compute from (45), where is defined in (44) and (46);iv) find using (55), where is given in (56); and v) obtainthe unknown source location estimate from (58).

The evaluation of the weighting matrices and re-quires the true source location and true sensor positions and theyare not known. In practice, following a similar approach used in[22], we first set to be and apply (45) to obtain aninitial estimate of . It is then used together with the updatedsensor positions to generate and an improved estimate of

is computed. To obtain an even better , we can repeat theabove process several times (one to three times are sufficient).The first three elements of the resulting are used to form .Through extensive computer simulations, we find that the per-formance degradation due to the use of approximated and

described above is insignificant.

The first stage of the proposed algorithm corrects the sensorposition errors to provide better sensor positions for the esti-mation of the unknown source position in stage 2 and stage 3.Nevertheless, it can be shown that the accuracy of the improvedsensor position vector does reach the CRLB. Finding a solutionfor better estimating the sensor positions is a subject for futurestudy.

For the 3-D localization problem we consider in this paper,the minimum number of sensors required to obtain a unique es-timate of the unknown target position is four, i.e., we need atleast three RDOA measurements. This is necessary to ensure theFIM is invertible. This condition would not change with the ap-plication of a calibration emitter with known location, becausethe calibration measurements do not contain any information re-garding the unknown source position.

B. Analysis

We shall compare the source localization accuracy of the pro-posed algebraic solution with the CRLB given in (11). Thefollowing two conditions will be used in the analysis:

C1) and ;C2) .

The first condition states that the position error in sensor issmall compared to the range between the calibration source andthe sensor, and the range between the unknown source and thesensor. This condition is satisfied if the sensor position errorsare small, or the calibration source and the unknown source arefar away from the sensors. The second condition means that theRDOA measurement noise between the sensor pair and 1 is in-significant compared to the range from the unknown source tothe th sensor. This condition is valid when the RDOA measure-ments are accurate, or the unknown source are far away fromthe sensor array. In practice, these two conditions can be easilyfulfilled for distant unknown and calibration sources at mod-erate level of RDOA and sensor position noise. If the sourcesare close to the sensor array, the conditions can still be validwhen the error in the sensor positions is sufficiently small andthe SNR is high enough so that the noise component in RDOAis also small. Under the two conditions, we shall prove in thefollowing that the accuracy of the proposed solution is able toreach the optimum performance, that is, the CRLB.

The proof begins with the covariance matrix of the sourcelocation estimate of the proposed method. Under condition C1)which implies that the Taylor series approximations in (28) and(36) are valid, then the covariance matrix is given in (61). Whensubstituting (57), (56) in sequel and re-expressing definedin (44) and (46) using the matrix inversion Lemma, Appendix IVshows that the inverse of (61) is

(62)

where , and andare defined in (82) and (79), respectively.

The inverse of the CRLB is, from the first line of (11),

CRLB (63)


where , and are defined in (7). Note that (62) and (63)have the same structural form. Substituting (7) into (63) andcomparing with (62), it can be identified that if

(64a)

(64b)

and

(64c)

then (62) and (63) are identical. Indeed, Appendix V shows that(64) is true, under conditions C1) and C2). Consequently, weestablish the proof that

CRLB (65)

when the two conditions are satisfied.

VI. SIMULATIONS

We shall verify the performance of DC and the proposed al-gebraic closed-form solution through simulation. The localiza-tion geometry is the same as in Fig. 2 and 4. Besides the distantunknown source located at m, we also con-sider a near unknown source at m. The calibra-tion emitter can take two positions at mor m. In generating the simulationresults, the RDOA measurements are created by adding to thetrue values zero-mean white Gaussian noise with covariancematrices and . We set mand is equal to the matrix with diagonal elements equal to 1and all other elements equal to 0.5 [21]. The noisy sensor posi-tions are generated using a similar approach, where the covari-ance matrix of is , and is a identity matrix.

The implementation of the proposed three-stage algorithmfollows the steps as described in the two paragraphs below(61). The second stage is repeated three times to improve

and hence . The results for DC were generated byforming the pseudo RDOA measurements as inside the squarebracket in (16) and then applying the closed-form solution from[21] when pretending the noisy sensor positions are actual.The localization accuracy from simulation is presented as

, where represents the un-known source position estimate at ensemble andis the number of ensemble runs.

Fig. 5 plots the localization accuracy of the proposed solution[denoted by cross symbols in curve (2)] and the DC approach[denoted by diamond symbols in curve (3)] for the distantsource at m, where the calibration source islocated closer to the unknown source at m.For comparison purpose, we also show CRLB in solidline and in dashed line in the two curves. It isevident from the figure that the proposed algorithm reaches theCRLB accuracy when 35 dB. The simulation resultsfor DC also reach the theoretical best MSE very well. The pro-posed algebraic solution, however, can offer 3 dB improvementin localization accuracy over DC when the sensor position error

Fig. 5. Comparison of the localization accuracy of the proposed method andthe DC solution for a distant unknown source where the calibration source isclose to the unknown source. (1) The best achievable accuracy without a calibra-tion source [��CRLB�� from (9) in [22]]: solid line, (2) the best achievableaccuracy with a calibration source [��CRLB�� from (11)]: solid line, MSEof the proposed method from simulation: cross symbol, (3) the best achievableaccuracy with DC [�� from (25)]: solid line, MSE of the DC fromsimulation: diamond symbol, (4) the best achievable accuracy when the sensorpositions are known exactly [�� from (7a)].

Fig. 6. Comparison of the localization accuracy of the proposed method andthe DC solution for a distant unknown source where the calibration source isfar away from the unknown source. (1) The best achievable accuracy withouta calibration source [��CRLB�� from (9) in [22]]: solid line, (2) the bestachievable accuracy with a calibration source [��CRLB�� from (11)]: solidline, MSE of the proposed method from simulation: cross symbol, (3) the bestachievable accuracy with DC [�� from (25)]: solid line, MSE of theDC method from simulation: diamond symbol, (4) the best achievable accuracywhen the sensor positions are known exactly [�� from (7a)].

is small ( 10 dB). When the sensor position erroris large such that ( 10 dB), the performance of theproposed method and the DC technique is comparable.

We illustrate in Fig. 6 the results for the same distant sourcebut the calibration source is placed farther away from it at

m. The simulation results of the pro-posed algorithm and the DC method again agree very well withthe theoretical performance when the variance of the sensor po-sition error is not large ( 20 dB). The proposed methodcan always provide more accurate source location estimates


Fig. 7. Comparison of the localization accuracy of the proposed method andthe DC solution for a near unknown source where the calibration source is closeto the unknown source. (1) The best achievable accuracy without a calibrationsource [��CRLB�� from (9) in [22]]: solid line, (2) the best achievable ac-curacy with a calibration source [��CRLB�� from (11)]: solid line, MSE ofthe proposed method from simulation: cross symbol, (3) the best achievable ac-curacy with DC [�� from (25)]: solid line, MSE of the DC methodfrom simulation: diamond symbol, (4) the best achievable accuracy when thesensor positions are known exactly [�� from (7a)].

than DC in this simulation. In particular, the proposed methodoffers a 3-dB improvement over DC when the sensor positionerror is small ( 10 dB), while it achieves at least10-dB improvement when 10 dB. Finally, because thecalibration source is farther away from the unknown emitter inthis simulation, the thresholding effect occurs earlier in bothalgorithms compared to the case where the calibration sourceis closer to the unknown emitter. Here, the thresholding effectrefers to the sudden deviation of the localization accuracy fromthe CRLB as the variance of the sensor position error increases,and this is a consequence due to the nonlinear nature of theestimation problem.

Figs. 7 and 8 give the results for the near source locatedat m, where the calibration source is locatedcloser to the unknown source at m and far-ther to it at m. The localization accuracyis generally better for a near source than a distant source. This isbecause when the source is near to the sensor array, the sensorsare far apart relative to the distance between the target and thesensor array. The localization geometry is more regular and thegeometric dilution of precision (GDOP) value would be smallercompared to the case of a distant source. Hence, the localiza-tion accuracy for a near source is usually better under the sameRDOA measurement noise level. It is clear from both figuresthat the proposed method reaches the CRLB and no obviousthresholding effect takes place. The simulation results of DCalso match the theoretical MSE well. The observations from thetwo figures are consistent with the previous distant source casethat the proposed algorithm remains to provide performancegain over DC, and that having a calibration source can greatlyreduce the effect of inaccurate sensor positions on the localiza-tion accuracy of the unknown source.

Fig. 8. Comparison of the localization accuracy of the proposed method and theDC solution for a near unknown source where the calibration source is far awayfrom the unknown source. (1) The best achievable accuracy without a calibra-tion source [��CRLB�� from (9) in [22]]: solid line, (2) the best achievableaccuracy with a calibration source [��CRLB�� from (11)]: solid line, MSEof the proposed method from simulation: cross symbol, (3) the best achievableaccuracy with DC [�� from (25)]: solid line, MSE of the DC methodfrom simulation: diamond symbol, (4) the best achievable accuracy when thesensor positions are known exactly [�� from (7a)].

VII. CONCLUSION

This paper investigated the use of a single calibration emitterwhose position is known to improve the TDOA-based localiza-tion accuracy of an unknown source in the presence of randomsensor position errors. Our study began with Gaussian errormodel and established the CRLB that provides the best achiev-able localization accuracy when a calibration source is present.We then showed that the commonly used differential calibrationtechnique, although simple, is not able to reach the CRLB ac-curacy in general. Next, an algebraic closed-form solution wasdeveloped, where the calibration TDOAs are used to correct thesensor positions before they are applied to obtain the unknownsource location estimate. Under two mild conditions, the pro-posed algorithm was shown analytically to reach the CRLB ac-curacy of the unknown source location estimate. Simulationshave confirmed our theoretical developments and examined thebehavior of the DC technique and the proposed algorithm.

APPENDIX ISYMBOLS AND NOTATIONS

The symbols and notations used are summarized in Table II.

APPENDIX II

To show that the second term in (11) is positive definite, weonly need to prove that the big matrix in the middle of the term,

, is positive definite because isfull rank. The proof is composed of two parts. In the first part,we show that has full column rank. In the second part, thepositive definiteness of the matrix is established.

We first note that becauseis invertible, must have a


TABLE IISYMBOLS AND NOTATIONS

full column rank of 3. Furthermore, it can be observedfrom the definition of given in (9) thatthe row rank of is , or equiva-lently speaking, has full column rankof . Hence, using the definition of in (7b),

is a full columnrank matrix. That is, if is a nonzero 3 1 vector, is anonzero 3 1 vector.

We proceed to show that is positive definite.From the definitions of in (7b) and in (7c), we have

(66)


where is a posi-tive definite matrix when . Applying the definition ofin (7a) and the Cholesky decomposition of ,we can express the big matrix in the middle of the first term onthe right side of (66) as

(67)

where . The positive semidefinitenessof is ob-vious from (67), because is a projectionmatrix. Hence, the first term on the right side of (66) is also pos-itive semidefinite and as a result, from the definition of

is a positive definite matrix when .Applying the above results, we have for any nonzero 3 1

vector ,

(68)

because is a nonzero vector. In other words,is a positive-definite matrix.

APPENDIX IIIPROOF OF (26)

We shall begin with cov . Substituting (20) into (24) andsimplifying yield

From (25) and using then can be ex-pressed as

(69)

where is the gradient matrix defined in (19) and

The inverse of CRLB is equal to the FIM given in (6). It isstraightforward to see from (6) that FIM can be rewritten as

CRLB (70)

where is defined below (24). Therefore, subtracting (69) from(70) gives

CRLB (71)

where

Applying matrix inversion Lemma [27] to sim-plifies to

(72)

Similarly, we have

(73)

Moreover, it is easy to verify that

Noting that is symmetric and positive defi-nite, we can decompose it as

(74)

where is a lower triangular matrix. Substituting (74) into (72)and (73) gives

(75)

where

(76)

Substituting (75) into (71) would yield (26).

APPENDIX IVPROOF OF (62)

The covariance matrix of the source location estimate fromthe proposed algorithm is given in (61), which is dependent on

. Substituting in (57) and in (56), we have

(77)

The weighting matrix is defined in (44) and (46). Sinceis a diagonal matrix with full rank, its inverse exists so thatcan be rewritten as

(78)


For notation simplicity, we define

(79)

and

(80)

Applying matrix inversion Lemma [27] to the big matrix in themiddle of (78) yields

(81)

Putting (81) into (77) gives (62), where is defined as

(82)

APPENDIX VPROOF OF (64)

We shall evaluate first. Putting and defined in(60), (51) and (54) to (82), we have

(83)

and are defined in (42) and (40), and the th rowof the matrix product is

(84)

In the first element, we have used the approximation .This is valid under the first condition C1), because from thedefinition of below (36), we have

In the second element, the condition C2) is used so that. Consequently, the th row of

is

(85)

can be expressed in terms of as, using (33),

Also, the term is insignificant compared to from thecondition C1). Thus, we have

(86)

and the th row of can be approximated by

(87)

It can be seen that is the same as the th rowof that is given in (8). As a result, we have underconditions C1) and C2) that

(88)

is defined in (79). After substituting (42) and (43), theth row of is

(89)

where represents a zero vector of appropriate size. Now ap-plying condition C1) so that and (86), the above equa-tion reduces to

(90)

(90) is identical to (9) except for a negative sign. As a result, wehave, under condition C1),

(91)

which is exactly (64b).The proof of (64c) is straightforward. Under condition C1),

it can be verified that

(92)

Applying the above approximation in the definition of in(30) then gives

(93)

which is valid under condition C1). The proof of (64) is nowcompleted.


ACKNOWLEDGMENT

The authors would like to thank the reviewers for providingvaluable comments and suggestions that have resulted in im-proving the quality of the paper.

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K. C. Ho (S’89–M’91–SM’00) was born in HongKong. He received the B.Sc. degree (First-ClassHons.) in electronics in 1988 and the Ph.D. degreein electronic engineering in 1991 from the ChineseUniversity of Hong Kong.

He was a Research Associate in the Royal MilitaryCollege of Canada from 1991 to 1994. He joinedBell-Northern Research, Montreal, QC, Canadain 1995 as a Member of Scientific Staff. He wasa faculty member in the Department of ElectricalEngineering at the University of Saskatchewan,

Saskatoon, Canada, from September 1996 to August 1997. Since September1997, he has been with the University of Missouri, Columbia, where he iscurrently a Professor in the electrical and computer engineering department.His research interests are in statistical signal processing, source localization,subsurface object detection, wavelet transform, wireless communications, andthe development of efficient adaptive signal processing algorithms for variousapplications, including landmine detection and echo cancellation. He is theinventor/co-inventor of three U.S. patents, three Canadian patents, two patentsin Europe, and four patents in Asia on mobile communications and signalprocessing.

Dr. Ho has served as an Associate Editor of the IEEE TRANSACTIONS ON

SIGNAL PROCESSING from 2003 to 2006, and the IEEE SIGNAL PROCESSING

LETTERS from 2004 to 2008. He received the Junior Faculty Research Awardfrom the College of Engineering of the University of Missouri, Columbia, in2003. He has been active in the development of the ITU Standard Recommen-dation G.168 since 1995. He is the editor of the ITU Standard RecommendationsG.168: Digital Network Echo Cancellers and G.160: Voice Enhancement De-vices.

Le Yang (S’08) was born in Sichuan, China. He re-ceived the B.Eng. (with honors) and M.Sc. degreesin electrical engineering from the University of Elec-tronic Science and Technology of China (UESTC),Chengdu, China, in 2000 and 2003, respectively. Heis currently working towards the Ph.D. degree at theUniversity of Missouri, Columbia.

Between 2003 and 2004, he was a Lecturer atUESTC. From 2004 to 2005, he was with the Uni-versity of Victoria, Victoria, BC, Canada, workingon diversity techniques and performance evaluation

of communication systems. In January 2006, he transferred to McMasterUniversity, Hamilton, ON, Canada, where he focused his research on artificialneural networks and computational neuroscience. Since January 2007, he hasbeen with the University of Missouri, Columbia. His current research interestsinclude sensor networks, passive localization, tracking, and signal detection.