Accuracy Studies for TDOA and TOA LocalizationRegina Kaune

Dept. SDFFraunhofer FKIE/University of Bonn

Wachtberg, Germanyregina.kaune@fkie.fraunhofer.de

Abstract—In sensor networks, passive localization can beperformed by exploiting the received signals of unknown emitters.In this paper, the Time of Arrival (TOA) measurements areinvestigated. Often, the unknown time of emission is eliminatedby calculating the difference between two TOA measurementswhere Time Difference of Arrival (TDOA) measurements areobtained. In TOA processing, additionally, the unknown timeof emission is to be estimated. Therefore, the target state isextended by the unknown time of emission. A comparison isperformed investigating the attainable accuracies for localizationbased on TDOA and TOA measurements given by the Cramer-Rao Lower Bound (CRLB). Using the Maximum Likelihoodestimator, some characteristic features of the cost functionsare investigated indicating a better performance of the TOAapproach. But counterintuitive, Monte Carlo simulations do notsupport this indication, but show the comparability of TDOAand TOA localization.Keywords: TOA, TDOA, CRLB, multilateration, sensornetworks.

I. INTRODUCTION

Localization in sensor networks using passive measure-ments from an unknown emitter can be based on varioustypes of measurements like Angle of Arrival (AOA), Timeof Arrival (TOA), Frequency of Arrival (FOA) or ReceivedSignal Strength (RSS). In this paper, the focus is on mul-tilateration which is based on TOA or Time Difference ofArrival (TDOA) measurements of transmitted signals. Severaldistributed, time-synchronized sensors measure the TOA ofa signal emitted by the unknown target. Often, the unknowntime of emission is eliminated by calculating the differencebetween two TOA measurements where TDOA measurementsare obtained. Usually, in a passive scenario, correlating thesignals received at two different sensors delivers the TDOAmeasurements. Alternative, knowing some features of thesignal, TOA measurements can be gained at a single sensor,[4], which can be processed to TDOA measurements.

TDOA localization is called hyperbolic positioning as il-lustrated in Fig. 1. A single noiseless TDOA measurementlocalizes the emitter on a hyperboloid or a hyperbola withthe two sensors as foci. TOA measurements define spheres orcircles as possible emitter positions (green circles in Fig. 1).For TOA localization, the parameter vector which is to beestimated is extended by the unknown time of emission.

TDOA and TOA localization has already been analyzed inmany different aspects: an overview of hyperbolic positioningand passive localization is given in [10], [11]. Algorithmsusing a moving sensor pair are described in [6], [7], [8], [12].

x

x(1) x(2)

x(3)

x

x(1) x(2)

x(3)

−10 0 10 20 30

−20

−10

0

10

20

x [km]

y[km]

TDOATOA

Fig. 1. Multilateration using TDOA and TOA measurements with hyperbolaeand circles respectively as possible emitter locations

Closed form formulations in sensor networks can be found in[13] - [15]. An analysis of achievable localization accuracy isconsidered in [3], [16], [9].

In recent time, the focus of discussion is on direct use ofTOA measurements without TDOA processing. Prerequisitefor the direct TOA approach is the availability of TOA mea-surements: in active scenarios, it is no problem to determinethe TOAs at a single sensor due to the knowledge on theemitted signal. In passive localization, there are differenttypes of scenarios. Firstly, for non-cooperative scenarios, TOAmeasurements cannot be provided at a single sensor. Thereason is, that signals need to be transmitted to a referencesensor or a fusion center to obtain TDOA measurementsby cross correlation. Secondly, if there exists knowledge onthe signal structure, like in cooperative or semi-cooperativepassive scenarios, the TOA estimation at a single sensor ispossible, see [4]. [1] investigates the error characteristics ofTDOA and TOA localization for a cooperative scenario: thesuperior performance for the TOA compared to the TDOAprocessing is shown for localization of mobile nodes usinga fixed reference sensor, a large sensor number and emittersin the inner area between sensors. [2] describes the TOAprocessing for a semi-cooperative scenario.

For gaining TDOA measurements, sensors must be paired.

408

There exist different TDOA measurement sets. Firstly, TDOAmeasurements can be received using a common referencesensor. Dependent on the choice of reference sensor, differentmeasurement sets are obtained. In algorithms, often the caseof a fixed reference sensor is assumed, for example [1].Secondly, considering all possible sensor pairings yields thefull measurement set with a larger number of measurements. Inthis paper, a comparison is performed using different referencesensors for a sensor network of three stationary sensors.The transfer to a larger network is analogue. The attainableaccuracies for the full measurement set using all possiblesensor pairs and measurement sets with fixed but differentreference sensors are computed and analyzed.

The contribution of the paper is a comparison of the errorbounds of localization for TDOA and TOA processing usingthe Cramer-Rao Lower Bound (CRLB). For a network ofthree sensors, the equality of the theoretical lower bound isshown for TOA and TODA localization with a fixed referencesensor. Additionally, the choice of reference sensor assumingconstant variances for the measurements, does not impact theattainable accuracies. A Maximum Likelihood (ML) estima-tor is implemented to estimate the emitter state where theunknown time of emission is treated as nuisance parameter.Characteristic features of the cost functions give reason toexpect a superior performance for TOA compared to TDOAlocalization. But Monte Carlo simulations show the equalestimation performance of TOA and TDOA localization.

The paper is organized as follows: in Section II, the princi-ples of TOA and TDOA localization are outlined. Section IIIinvestigates the attainable accuracies in terms of the CRLB.The description of the estimation algorithm for TDOA andTOA processing follows in Section IV. Characteristic featuresof the cost functions are evaluated in Section V. Monte Carlosimulations analyze the performance of the two algorithms,Section VI. In Section VII, the main points of the paper aresummarized.

II. SCENARIOS

An unknown stationary emitter x = (x, y)T ∈ R2 sendsa signal which several time-synchronized stationary sensorsx(i), i = 1, . . .M , receive. A passive scenario is considered.Therefore, sensors do not send signals to illuminate the regionof interest. They work passively and exploit the signal receivedfrom the emitter.

A. TOA scenario

At a single sensor, the TOA of the signal is determined.The TOA is composed of the emitting time t0 and the timethe signal needs to propagate the distance between emitter andsensor i:

TOA = t0 +ric, (1)

where

ri = ||x− x(i)||

=√(x− x(i))2 + (y − y(i))2, i = 1, . . . ,M (2)

denotes the Euclidean distance between the emitter and sensori and c is the speed of light.

By multiplication the TOA measurements in the time do-main with the speed of light c the range measurement functionis obtained:

hi = r0 + ri, (3)

where r0 = c t0 is the unknown time of emission multipliedwith c.

Knowing or estimating the time of emission, a single TOAmeasurement localizes the emitter on a sphere, for two-dimensional localization on a circle. Fig. 1 shows a typicalTOA scenario with three stationary sensors, green circles illus-trate the geometrical interpretation of the TOA measurements.

In practice, TOA measurements are noisy. The TOA mea-surement noise is modeled as additive white Gaussian noisewith standard deviation σi, for i = 1, . . . ,M and assumed tobe uncorrelated from each other and from time step to timestep:

zi = hi(x) + v, v ∼ N (0, σ2i ). (4)

Due to the Gaussian measurement noise, the Likelihood func-tion p(zi|x) for a single TOA measurement is given by:

p(zi|x) =1√2πσi

exp(− 1

2σ2i

(zi − hi(x))2). (5)

The Likelihood function for the complete measurement setzt = (z1, . . . , zM )

T is expressed by:

p(zt|x) =1√|2πRt|

exp(−1

2(zt − ht(x))

TR−1t (zt − ht(x))

), (6)

where | · | indicates the determinant,ht(x) = (h1(x), . . . , hM (x))

T is the measurementfunction and the covariance matrix is given by:

Rt =

σ21 0 . . . 0

0 σ22

. . ....

.... . . . . . 0

0 . . . 0 σ2M

. (7)

B. TDOA scenario

TDOA measurements are obtained by calculating the dif-ference between two TOA measurements and eliminating theunknown time of emission. Therefore, the sensors must bepaired to get TDOA measurements. Multiplication with thespeed of light c yields the measurement functions of rangedifferences:

hij = ri − rj , i, j ∈ {1, . . .M} ∧ j 6= i (8)

Assuming additive white Gaussian noise uncorrelated fromtime step to time step and from each other, the measurementequations follow:

zij = hij(xk) + vij ,

i, j ∈ {1, . . . ,M} ∧ i 6= j, vij ∼ N (0, σ2i + σ2

j ) (9)

409

where the measurement noise vij = vi + vj is composed ofthe noises at the two associated sensors and has the covarianceσ2i + σ2

j .For TDOA processing, different measurement sets are to be

distinguish. Using a common reference sensor, M differentmeasurement sets consisting of M − 1 measurements canbe obtained. The full measurement set contains

∑M−1l=1 l =

M(M−1)2 measurements, here, all possibilities of sensor pairing

are performed. Let sensor i be the reference sensor. Then, themeasurements using a fixed reference sensor are defined by:

zij = hij(xk) + vij ,

i, j ∈ {1, . . . ,M} ∧ j 6= i, (10)

where i is fixed.The measurements of the full measurement set are given

by:

zij = hij(xk) + vij ,

i = {1, . . .M}, j ∈ {2, . . . ,M} ∧ j > i, (11)

where all sensor pairings are considered.A single TDOA measurement defines a hyperboloid of

possible target locations with the two sensors as foci. If sensorsand emitter are located in the same plane, the emitter is locatedon a hyperbola. In Fig. 1, the green hyperbolic curves representthe full TDOA measurement set.

The measurement equation depends only on the emitterposition, and the known positions of the sensors enter asparameters. Therefore, we have a two-dimensional localizationproblem. The two-dimensional position vector of the emitteris to be estimated.

In the following, a sensor network of three stationarysensors is considered. Let sensor 1 be the reference sen-sor. The Likelihood function for the measurement vec-tor z∆t1 = (z12, z13)

T and the measurement functionh∆t1 = (h12, h13)

T can be formulated as:

p(z∆t1|x) =1√

|2πR∆t1|exp

(−1

2(z∆t1 − h∆t1(x))

TR−1∆t1(z∆t1 − h∆t1(x))

),

(12)

where R∆t1 is the covariance matrix of the measurement set.According to [3], the standard deviation can be dependent onthe distances between emitter and sensors, but this distancedependency is here neglected. In this paper, the case ofconstant standard deviations with different values at singlesensors is assumed:

R∆t1 =

(σ21 + σ2

2 σ21

σ21 σ2

1 + σ23

). (13)

Using sensor 2 as reference sensor, the covariance matrix hasfollowing form:

R∆t2 =

(σ22 + σ2

1 σ22

σ22 σ2

2 + σ23

). (14)

The covariance for the measurement vector using sensor 3 is:

R∆t3 =

(σ23 + σ2

1 σ23

σ23 σ2

3 + σ22

). (15)

For three sensors, the full measurement setz∆tfull = (z12, z13, z23)

T consists of three measurements.The associated covariance matrix is a 3× 3 matrix:

R∆tfull =

σ21 + σ2

2 σ21 σ2

2

σ21 σ2

1 + σ23 σ2

3

σ22 σ2

3 σ22 + σ2

3

. (16)

III. CRLB ANALYSIS

The Cramer-Rao lower bound (CRLB) provides a lowerbound on the covariance that is asymptotically achievable byany unbiased estimation algorithm based on the measurementvector z, [5], [3]. It is calculated from the inverse of the FisherInformation Matrix (FIM) J. Let the emitter location x ∈ R2

be the parameter of interest and x an estimate of it obtainedfrom the measurement vector z. Then the error covarianceE[(x− x)(x− x)T

]is bounded below by:

E[(x− x)(x− x)T

]≥ J−1, (17)

J = E[∇x ln p(z|x)(∇x ln p(z|x))T

], (18)

where E [·] determines the expectation value.

A. TOA accuracy

In TOA processing, additionally the unknown time of emis-sion t0 is to be estimated. Therefore, for localization, theparameter of interest is the extended position state of theemitter:

x(+) = (t0,xT )T ∈ R3 (19)

Given the measurement vector zt, the CRLB for TOAlocalization for one time step is computed from the inverseof the Fisher information for TOA, a 3× 3 matrix:

Jt =∂hTt∂x(+)

R−1t

∂ht

∂x(+)(20)

The Jacobian of the measurement function is:

∂ht

∂x(+)=

∂h1

∂t0∂h1

∂x∂h1

∂y∂h2

∂t0∂h2

∂x∂h1

∂y∂h3

∂t0∂h3

∂x∂h1

∂y

=

cx−x1

r1

y−y1

r1

c x−x2

r2

y−y2

r2

c x−x3

r3

y−y3

r3

.

(21)The computation of the FIM follows as:

Jt =

c c cx−x1

r1x−x2

r2x−x3

r3y−y1

r1

y−y2

r2

y−y3

r3

1σ2

10 0

0 1σ2

20

0 0 1σ2

3

c x−x1

r1

y−y1

r1

c x−x2

r2

y−y2

r2

c x−x3

r3

y−y3

r3

=3∑i=1

c2

σ2i

cσ2i

x−xi

ricσ2i

y−yiri

cσ2i

x−xi

ri1σ2i

(x−xi)2

r2i

1σ2i

(x−xi)(y−yi)r2i

cσ2i

y−yiri

1σ2i

(x−xi)(y−yi)r2i

1σ2i

(y−yi)2r2i

.

(22)

410

The Fisher information can be expressed by:

Jt =

J11 J12 J13J21 J22 J23J31 J32 J33

=

(Jt Jt,pos

Jpos,t Jpos

), (23)

where Jpos is the Fisher information of the position space,Jt The FIM of the time space and the others are the crossterms. According to [1], the CRLB of the position space canbe computed using the matrix inversion lemma. The time ofemission is treated as nuisance parameter. The resulting CRLBfor the position space is the same as for TDOA localization.

It can be shown that Jpos = J∆ti, i = 1, . . . , 3.

B. TDOA accuracy

For TDOA localization, different measurement sets are tobe considered: the full measurement set and measurement setsusing a fixed reference sensor.

Analogue to (20), the FIM for TDOA localization (one timestep) is given by (this equation can be applied to all differentmeasurements sets using the appropriate indices):

J∆t =∂h∆t

∂xR−1∆t

∂h∆t

∂x(24)

The Jacobian of the measurement set with sensor 1 as refer-ence sensor is:

∂h∆t1

∂x=

(∂r12

∂x∂r12

∂y∂r13

∂x∂r13

∂y

)

=

(x−x1

r1− x−x2

r2

y−y1

r1− y−y2

r2x−x1

r1− x−x3

r3

y−y1

r1− y−y3

r3

), (25)

The FIM is computed as:

J∆t1 =

(F11 F12

F21 F22

), (26)

F11 =1

l

3∑i=1

∑j 6=i

σ2j

(x− xi)2ri

− 1

l

2∑i=1

3∑j=i+1

∑k 6=i,j

2σ2k

(x− xi)(x− xj)rirj

(27)

F22 =1

l

3∑i=1

∑j 6=i

σ2j

(y − yi)2ri

− 1

l

2∑i=1

3∑j=i+1

∑k 6=i,j

2σ2k

(y − yi)(y − yj)rirj

(28)

F12 = F21 (29)

=1

l

3∑i=1

∑j 6=i

σ2j

(x− xi)(y − yi)ri

− 1

l

3∑i=1

∑j 6=i

∑k 6=i,j

σ2k

(x− xi)(y − yj)rirj

(30)

Fig. 2. CRLB plot for TDOA and TOA localization respectively

with

l =

2∑i=1

3∑j=i+1

σ2i σ

2j . (31)

This FIM is obtained for all measurement sets using a fixed butdifferent reference sensor. Therefore, the choice of referencesensor does not influence the theoretical lower bound for thevariance of an unbiased estimator. Using different referencesensors yields the same CRLB. Fig. 2 shows the CRLB for thescenario of Section VI using a network of three sensors. For agrid of possible emitter positions in the plane, the associatedCRLB matrix (for TDOA = TOA) is computed. The squareroot of the trace is shown. For better visibility, values largerthan 1000 m have been cut off. The color bar gives the CRLBvalues. Sensor positions are marked with white triangles.

The Jacobian of the measurement function vector of the fullmeasurement set is:

∂hT∆tfull

∂x=

∂h12

∂x∂h12

∂y∂h13

∂x∂h13

∂y∂h23

∂x∂h23

∂y

(32)

=

x−x1

r1− x−x2

r2

y−y1

r1− y−y2

r2x−x1

r1− x−x3

r3

y−y1

r1− y−y3

r3x−x2

r2− x−x3

r3

y−y2

r2− y−y3

r3

. (33)

The computation of the FIM for the full measurement setgives different entries as for the measurement set with a fixedreference sensor. There is an information gain in processingthe full TDOA set indicating a smaller theoretical lowerbound. The diagonal entries of the FIM of the full TDOA

411

measurement set are given by:

F full11 =

1

l

3∑i=1

∑j 6=i

σ2j

(x− xi)2ri

+2

lσ22

(x− x2)2r2

+1

l

3∑i6=2

∑j 6=i

(σ22σ

2j

σ2i

(x− xi)2ri

+σ22σ

2j

σ2i

(x− x2)2r2

)

− 1

l

∑i6=2

∑j 6=i

2(σ22 + σ2

j )(x− x2)(x− xi)

r2ri

− 1

l

3∑i6=2

∑j 6=i

σ22σ

2j

σ2i

(x2 − xi)2r2ri

(34)

F full22 =

1

l

3∑i=1

∑j 6=i

σ2j

(y − yi)2ri

+2

lσ22

(y − y2)2r2

+1

l

3∑i6=2

∑j 6=i

(σ22σ

2j

σ2i

(y − yi)2ri

+σ22σ

2j

σ2i

(y − y2)2r2

)

− 1

l

∑i6=2

∑j 6=i

2(σ22 + σ2

j )(y − y2)(y − yi)

r2ri

− 1

l

3∑i6=2

∑j 6=i

σ22σ

2j

σ2i

(y2 − yi)2r2ri

. (35)

Analyzing the diagonal entries of the FIM of the fullmeasurement set, the main difference to the FIM using afixed reference sensor are 6 additional positive terms and6 additional negative terms (double terms are taken intoaccount twice). The impact on the CRLB is dependent onthe geometry and the values for the standard deviations andleads to different CRLBs for the full TDOA measurement setand the measurement set using a fixed reference sensor.

IV. ESTIMATION ALGORITHM

Due to the nonlinear measurement equations, nonlinearestimation methods are needed to solve the localization prob-lem. There are several nonlinear methods like the MaximumLikelihood (ML) estimator, nonlinear forms of the KalmanFilter, like the Extended and the Unscented KF or for TDOAlocalization using a sufficient number of sensors, closed formformulations.

For TOA localization, an extended parameter vector is touse. For the purpose of comparing the TOA and TDOAresults, the extended dimension, the time of emission, is treatedas nuisance parameter. The MLE offers a good comparablesolution for the TOA and the TDOA measurement sets.

A. TOA Localization

In TOA processing, additionally the unknown time of emis-sion t0 is to be estimated. The target state is extended by theunknown time of emission. Therefore, the extended positionvector x(+), see (19), is to be estimated.

The ML estimate is that value of the extended positionvector x(+) which maximizes the likelihood function or rather

minimizes:

x(+) = argminx+

3∑i=1

(zi − hi)2σ2i

(36)

It is realized with the simplex method due to Nelder and Meadas numerical iterative search algorithm. The two-dimensionalinitialization point is extended by the time of emission accord-ing to:

t0 =z1c− ||xinit − x(1)||

c. (37)

B. TDOA Localization

For TDOA localization, the full measurement set and themeasurement set using a fixed reference sensor are processedin a ML estimator (using the appropriate indices, the equationcan be applied to all TDOA measurement sets):

x = argminx

(z∆t − h∆t(x))TR−1∆t (z∆t − h∆t(x)). (38)

As in TOA localization, the same numerical iterative searchalgorithm is applied. The initialization point of the ML es-timator is generated randomly from a Gaussian distributioncentered at the true emitter position with standard deviationof 500 m in x and y direction.

V. ANGLES OF INTERSECTION

While the CRLB describe the theoretical lower boundfor localization, here, some practical aspects are taken intoconsideration. Using a nonlinear estimation method like theMLE, the global maximum of the cost function must be found.Therefore, the characteristics of the cost function are essential.The shape of the error ellipses depend on the value of theintersection angles of the emitter lines. In two-dimensionalTOA localization, emitter lines are circles around the sensorpositions. In TDOA localization, emitter lines are hyperbolaewith the sensor positions as foci. The magnitude of the anglesbetween two curves may influence the performance of theML estimator. Intersecting circles and hyperbolae, the anglesof intersection can be computed using the tangents in theintersection points.

A. TOA angles

The determination of the angles between the TOA circlescan be performed in two steps:• Computation of lines between the centers of the circles,

that is the sensor positions, and the intersection points ofcircles.

• Intersection of lines and calculation of the angles of in-tersection; angles must be in the interval [0, π2 ] in radiansor [0◦, 90◦] in degrees. An angle near 0◦ means poorperformance, an angle of 90◦ implies the best informationgain and good performance. If the angle is in the interval[90◦, 180◦], the opposite angle is calculated.

Fig. 3(a) shows angles of intersection plotted in the planefor a network of three sensors. For a grid of possible emitterpositions in the plane, the angles of intersecting circles are

412

(a) TOA

(b) TDOA

Fig. 3. Mean values of angles of intersection between circles and hyperbolaerespectively

computed. For each grid point, the mean value of the threeassociated angles is plotted in the plane. Sensor positions areassumed as in the first scenario of Section VI and markedwith white triangles. The color bar gives the values of meanangles in degree. For visualization purpose, values of angleshave been cut off to 60◦. In the inner area, between sensorpositions, the mean values of angles are high. Compared toTDOA, there exist a large region of large angles.

B. TDOA angles

The algorithm of determining angles between two TDOAhyperbolae is as follows:

• Computation of tangents at intersection points of twoTDOA hyperbolae.

• Intersection of tangents and calculation of the anglesof intersection; angles must be in the interval [0, π2 ] inradians or [0◦, 90◦] in degrees, see above.

Angles of intersection in the plane are illustrated in Fig. 3(b).Results are computed using the full measurement set with3 measurements. The field of large angles is much morelimited than for TOA.

The results of the study on the angular size let expect asuperior performance of the TOA localization compared to theTDOA localization. The superiority of the TOA processing isdescribed in [1] for a large network of twelve sensors. In [1],the TDOA measurement set is gained using a fixed referencesensor, emitters are only in the inner area between sensors andlocalization of mobile emitters is performed. The present paperanalyzes two different scenarios in Section VI. For TDOA, notonly the measurement set using a fixed reference sensor is in-vestigated, but also the full measurement set. Counterintuitiveto the results of the angle study, in simulations, TDOA andTOA localization show almost equal localization performance.

VI. SIMULATION RESULTS

Two different localization scenarios are analyzed. For bothscenarios, a grid of stationary emitter locations in the plane isinvestigated. In the first scenario, emitter localization is per-formed for a small sensor network of three stationary sensors.A large number of the analyzed emitters lies outside the innerarea of the sensors where the expected localization accuracy islow. The second scenario analyzes a large network consistingof eight sensors. The inner area between the emitters coversmost of the observation area.

Measurements are obtained using the stationary sensornetwork. TOA measurements are generated by adding whiteGaussian noise with a standard deviation of 15 m in therange domain (50 nsec in the time domain) to the true values.Standard deviations of different sensors are assumed to beequal. TDOA measurements are obtained by calculating thedifference between two TOA measurements. In this way, TOAand TDOA measurements underlie the same noise conditionsand localization results are comparable.

The Root Mean Square Error (RMSE), the squared distanceof the estimate to the true target location x is computed asperformance measure. It is averaged over N , the number ofMonte Carlo runs. Let x(k) be the estimate of the kth run.Then, the RMSE at is computed as:

RMSE =

√√√√ 1

N

N∑k=1

(x− x(k))T (x− x(k)). (39)

For each grid point, 100 independent Monte Carlo runs arecomputed.

Fig. 4 illustrates the results for the sensor network usingthree sensors. Localization results are similar to the resultsof the CRLB analysis, see Fig. 2. Values for RMSE are cutoff to 1000 m. The colorbar gives the localization results inm. White triangles designates the sensor positions. For TDOAlocalization, the full measurement set is used, the results forthe measurement set with fixed reference sensor show onlylittle differences, as well as the comparison of TDOA andTOA processing.

413

(a) TDOA using the full measurement set

(b) TOA

Fig. 4. TDOA and TOA localization results

For the sake of simplicity, the ML estimator for the largernetwork is implemented as least squares algorithm where thecovariance matrix of the measurement set is neglected. Thelocalization results for the large sensor network consisting of8 sensors are presented in Fig. 5. Most of the observationarea lies in the inner region between sensor positions wherethe expected localization accuracy is high. The differencesbetween TDOA and TOA localization are not significant.TDOA localization based on the full measurement set showsin the inner area between sensors better performance than thelocalization using a fixed reference sensor. Overall, TDOA andTOA localization yield roughly the same results.

These results do not support the expectations resulting fromthe angle study. One reason for the surprising performances isthe larger dimensionality of the parameter vector for the TOAoptimization process. The TOA cost function is minimizedover three dimensions, but the angle study only investigatesthe projection in the position space. Analyzing the process of

(a) TDOA using the full measurement set

(b) TDOA using a fixed reference sensor

(c) TOA

Fig. 5. TDOA and TOA localization results using eight sensors

414

optimization, the number of iterations for TOA processing islarger than for TDOA processing. This indicates a larger sen-sitivity of the TOA optimization to the initialization point. Forfuture research, it will be interesting to investigate additionalreasons for the equal performance of TDOA and TOA. Thefusion of TDOA and TOA and scenarios with known time ofemission might be the topic of further analysis.

VII. CONCLUSIONS

In this paper, accuracy studies for TDOA and TOA lo-calization are performed. For a network of three sensors,the CRLB are computed using the TDOA and the TOAmeasurement set. The theoretical lower bounds are identicalfor TOA localization and for TDOA localization using a fixedreference sensor. Additionally, the choice of reference sensordoes not influence the CRLB. Assuming constant variancewith different values at various sensors, the equality of theCRLB for different reference sensors is shown. Using the fullTDOA measurement set, more information is processed forlocalization. The theoretical lower bounds are smaller than forTDOA localization with a fixed reference sensor.

While the theoretical bounds show no difference in TOA andTDOA localization with a fixed reference sensor, in practice,considerations like characteristics of the cost function mustbe taken into account. A study on the angles between twoemitter lines may indicate difficulties of finding the maximumof cost function in TDOA processing. But counterintuitive,simulations do not support the superiority of the TOA local-ization but show almost equal performance of TOA comparedto TDOA localization.

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