IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 10, OCTOBER 2013 949

Robust Localization With MinimumNumber of TDoA Measurements

Sanvidha C. K. Herath, Member, IEEE, and Pubudu N. Pathirana, Senior Member, IEEE

Abstract—This letter looks at the theoretical conditions under-pinning unique localization of an emitter using Time-Delay-of-Ar-rival(TDoA) from minimum number of sensors in 2-D and 3-Dspace. A discussion is carried out on the unique localization regionwith the TDoA measurements subjected to a bounded error. Forboth 2-D and 3-D, error bounds have been found, beyond which,there is no existence of the unique solution region.

Index Terms—Localization ambiguity, TDoA.

I. INTRODUCTION

L OCALIZING a radiating emitter can be carried out frompassive measurements of, arrival times, received signal

strength, directions of arrival or Doppler shifts associated withelectromagnetic waves received at various sensor locations[1]–[5]. Localization based on TDoA technology is currentlyapplicable in numerous applications including intelligent trans-port system (ITS), resource management and performanceenhancement in mobile cellular networks, electromagneticradar and acoustic-based systems. TDoA-based systems maybe used to estimate the location of a wireless emitter or audiosource, where a considerable amount of work exists, [6]–[10].Time-difference-of-arrival (TDoA) systems, generally lo-

calize an emitter by processing signal arrival-time measuredat three or more sensors in 2-D space and four or more sen-sors in 3-D space. In the absence of noise and interference,the arrival-time measurements at two sensors are combinedto produce a relative arrival time that, confines the possibleemitter location to a hyperbola in 2-D and a hyperboloid in3-D, with the two sensors as foci. Emitter location is estimatedfrom the intersections of two or more independently generatedhyperbolas in 2-D and the intersections of three or more inde-pendently generated hyperboloids in 3-D [1].If two hyperbolas or three hyperboloids are considered,

they can have either one or two points of intersection. In theseinstances there are some regions in the space which gives an

Manuscript received May 12, 2013; revised July 16, 2013; accepted July17, 2013. Date of publication July 23, 2013; date of current version July 30,2013. This work was supported by the Commonwealth of Australia, throughthe Cooperative Research Centre for Advanced Automotive Technology (Au-toCRC). The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Alireza Seyedi.The authors are with the Networked Sensing and Control Laboratory, School

of Engineering, Deakin University, Geelong, VIC 3217, Australia (e-mail:sanvidha.herath@deakin.edu.au; pubudu@deakin.edu.au).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LSP.2013.2274273

unique solution to an emitter location [11]. The geometry ofthe space is related to the sensor positions. This unique solutionregion gradually reduces with the increasing measurementerror. Usually, the location ambiguity occurred by two pointsof intersection may be resolved by using a priori informationabout the location or an additional sensor to construct anadditional hyperbola/hyperboloid. Error bounds have beenfound for both two-dimensional and and three-dimensionallocalization systems after which, there is no existence of theunique solution region. Our analysis is aimed at minimizing thecost and the complexity in TDoA localization systems.

II. LOCALIZATION OF AN EMITTER

Lets consider an emitter in -dimensional space,number of sensors with denoting the signal arrivaltime at sensor . Here, . is the error bound. Thearrival delay with respect to the reference sensor is

, . Here and. The corresponding range difference is

, where is the velocity of signal propagation. Letthe spatial coordinate vectors be: ,

and , where is the referencesensor position, is the th sensor position and the unknownemitter position is . The range between the th sensor and theemitter can be written as, . The distance betweenthe reference sensor and the emitter is . Then the pathdifference can be written as, , which yields

[11].For a general case of sen-

sors, following matrices can be defined as,

, andwhere, . In matrix notation

and solving for emitter position,.

Consider a weighting matrix for the sensors. This isintroduced as the range differences are not measured to the sameaccuracy. Then,

(1)Define the new vectors,

and. Then, and the estimation for the

source position is obtained as

(2)

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950 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 10, OCTOBER 2013

Substituting in , the following quadratic equationcan be obtained,

(3)

where .

A. Solution Space

In 3-D space, if at least four sensors are not coplanar and thereis a subset of three sensors which are not collinear, then the ma-trix has full rank and it is possible to solve the quadratic equa-tion in (3). However, depending on the sensor-emitter configu-ration, (3) will lead to two possible solutions. It can be shownthat (3) leads to an unique solution if and in 2-D planeand 3-D space, at least, 3 sensors are not co-linear and 4 sensorsare no co-planer respectively [11], [12]. Also, it can be shownthat, generally, 3 and 4 non-collinear sensors are needed for aunique localization of a target in 2-D plane and 3-D space, re-spectively, but an additional sensor is needed for both cases toresolve the ambiguity in some situations.Theorem 1: The space where a unique solution for

minimum number of TDoA measurements (3 sensors for 2-Dand 4 sensors for 3-D) is given by,

(4)

where is the -dimensional region for .Proof: in (3) can also be written as,

If , fur-ther simplification will lead to

(5)

describes an ellipse in 2-D and ellipsoid in 3-D re-spectively. is the transformation of the 2-D or 3-D region into an ellipse or an ellipsoid respectively. In this transformation,

corresponds to the unique solution region in D. Regionbounded by the intersection of all the ellipsoids cor-responds to the region in which always guarantees a uniquesolution for the emitter position. Then it can be stated that forminimum number of TDoA measurements, the unique solutionregion in D is given by (4).

B. Error Upper Bound for Unique Solution

When the error bound increases the unique solution regiongiven by (4) decreases. Then there is a maximum bound for theerror before which, there is a unique solution region foran emitter for given sensor positions. occurs at for

.

Fig. 1. Unique solution area for 2-D.

At this instant one of the ellipsoids generated touches theorigin making . Hence (5) becomes

(6)

Since is a symmetric matrix, using eigen decomposition,

(7)

where is an orthogonal matrix with the columns which areeigen vectors of and . Now (7) can be written as

(8)

where

are the eigen values of .Equation (8) refers to a rotated ellipsoid of (6) where the prin-

cipal diagonals coincide with the coordinate axes. Then the min-imum distance between the origin and the ellipsoid is given by

. Hence, the at the minimum shift,

(9)

Then

(10)

For a given sensor configuration if , there is a regionin which a unique solution can always be guaranteed.

III. COMPUTER SIMULATIONS

Fig. 1 depicts the unique solution area for three time-of-ar-rival sensors positioned in 2-D at , and . Fig. 2shows the , which is an ellipse.

HERATH AND PATHIRANA: ROBUST LOCALIZATION WITH MINIMUM NUMBER OF TDoA MEASUREMENTS 951

Fig. 2. for 2-D.

Fig. 3. Unique solution region for 3-D.

Fig. 4. for 3-D.

A unique solution region for 3-D is shown in Fig. 3, where thesensors are positioned at , , and .The corresponding is depicted in Fig. 4. In Fig. 5, the uniquesolution regions for and

are shown. It can be seen thatthese two regions marginally touches each other at this errorbound which agrees with our analysis.

Fig. 5. Unique solution region for 3-D with errors.

IV. CONCLUSION

A theoretical analysis has been provided in this letter whichis required for unique localization of an emitter using minimumnumber of TDoA measurements with bounded error. Errorbounds have been found for both 2-D and 3-D after which,there is no existence of the unique solution region. Morecomplex analysis can be carried out, specially in the geometryof the sensor positions for robust localizations based on thisdiscussion.

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