Position estimation using ultra-wideband timedifference of arrival measurements

J. Xu, M. Ma and C.L. Law

Abstract: Positioning enables many applications such as emergency services, target identificationand tracking, health monitoring and geographic routing and so on. Time difference of arrival(TDOA) ranges measured using ultra-wideband signals are used in the position estimationprocess. A TDOA error-minimising localisation method has been proposed to estimate thelocations of blind nodes and its performance is investigated in both LOS and NLOS propagations.Theoretical lower bound on the variance of the blind node positions has also been derived for theLOS situation.

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1 Introduction

Ultra-wideband signals (UWB) have recently drawnincreasing interests from industries as well as academiabecause of their unique characteristics [1, 2]. UWB isdefined by federal communication commission (FCC) as afractional bandwidth of more than 20% or absolute band-width of more than 500 MHz. UWB provides high temporalresolution with a large bandwidth. Therefore UWB canachieve a high ranging accuracy and hence good positioningprecision. UWB radios are robust to multipath interference,and this characteristic provides robustness against multipathfading and makes UWB particularly attractive for densemultipath environment [3–6]. In [7], the theoreticalranging accuracy for UWB has been investigated for differ-ent signals types proposed by the IEEE 802.15.3a and IEEE802.15.4a Task Groups. Many ranging techniques withUWB radio have been proposed [8–10]. In [8], a UWBTOA ranging scheme was suggested, which implements asearch algorithm for the detection of a direct path signalin the presence of dense multipath. In [9], a novel methodwas proposed based on the estimation on arrival time ofthe first multipath of UWB signals under the existenceof the LOS path in a multipath environment. Anenhanced-TDOA measurement method was proposed in[10], which offered accurate ranges by separating LOSand NLOS time difference measurements.TDOA is one of the commonly used ranging techniques

in position estimation. It estimates the difference in thearrival times of the signals between the synchronised refer-ence nodes. TDOA ranging does not require knowledge ofabsolute time of the transmission, and therefore only syn-chronicity of the reference nodes is necessary forTDOA-based positioning. Once the time difference hasbeen estimated using UWB signals, we can obtain theTDOA range measurements by multiplying the estimatedtime difference with radio speed. Each TDOA range deter-mines a hyperbola. In 2-D plane, the position of the blind

# The Institution of Engineering and Technology 2008

doi:10.1049/iet-smt:20060089

Paper first received 20th July 2006 and in revised form 8th April 2007

The authors are with the School of Electrical and Electronic Engineering,Nanyang Technological University, Nanyang Avenue, Singapore

E-mail: emdma@ntu.edu.sg

IET Sci. Meas. Technol., 2008, 2, (1), pp. 53–58

node (node that has no knowledge of its location) can beestimated if there are at least three reference nodes (nodeswith knowledge of their locations). The intersection pointof the hyperbolas is the position of the blind node.It is not so easy to solve the hyperbola equations obtained

from TDOA measurements since the equations are non-linear. Several algorithms have been proposed to solve it.Chan’s method is one of the well-known algorithms [11].It takes advantage of redundant measurements and canachieve high accuracy. It rearranges a set of nonlinearequations that are formed based on TDOA measurements,produces an approximate of maximum likelihood estimatorand gives a closed-form, non-iterative solution. However, itcannot work efficiently if the measurement has large errors,such as NLOS error. The Taylor series (TS) method or itsvariations is also used very frequently [12, 13]. It is an itera-tive method and the hyperbola equations are linearisedusing a TS expansion. This method requires an accurateinitial position guess that is not practical in many situations.An improved Taylor algorithm was proposed in [14], whichuses least squares algorithm to give the initial position esti-mate, and estimates the blind node position using the TSmethod.In this paper, we propose an error-minimising localis-

ation method which uses TDOA ranges measured usingUWB signals to estimate the positions of the blind nodes.All the blind nodes in the network are considered a group,and their locations are estimated simultaneously. Thismethod compares the measured TDOA ranges with thosecalculated from the node coordinates. And then, the pos-itions of the blind nodes are estimated by minimising theerror sum. Cramer–Rao lower bound (CRLB) is derivedfor Gaussian-distributed TDOA range measurements. Weinvestigate the performance of the error-minimisingmethod for both LOS and NLOS propagations. Chan’sand TS methods have been used as references to show theadvantage of our proposed method. From the simulationresults, TDOA error-minimising method shows better per-formance than Chan’s and TS methods for the samenumber of reference nodes and the same propagationenvironment.The remainder of the paper is organised as follows:

Section 2 describes the model of the measured rangeerrors for both LOS and NLOS situations. Section 3 pre-sents our localisation scheme. Section 4 derives the CRLB

of the position estimation error. Section 5 presents thesimulation results. Finally Section 6 concludes the paper.

2 Range error modelling

Multipath and NLOS are the major sources of rangingerrors. However, UWB signal does not suffer multipathfading because of its unique characteristics. Hence onlysystem measurement noise and NLOS errors are consideredhere. The system measurement noise is normally modelledas a zero-mean random variable. The LOS TDOA rangeRm,n of reference node m and n to the querying node i canbe modelled as

(Rm,n)los ¼ (kpi � pmk � kpi � pnk)þ 1m,n (1)

where Rm,n is the range difference from node m and n to thequerying blind node i, and pm and pn are coordinates of tworeference nodes. And pi is the coordinate of the blind node.1m,n is the measurement noise, normally assumed to be zeromean Gaussian 1m,n � N(0, sm,n

2 )We now have a look at the variance in the Gaussian

measurement noise and find how accurate the rangemeasurements can be achieved. For a single path, thereceived signal is given by

r(t) ¼ As(t � t)þ n(t) (2)

where A is the path amplitude, t is the arrival delay, s(t) isthe transmitted signal and n(t) is the addictive whiteGaussian noise (AWGN) with spectral density N0/2.The arrival time can be estimated using maximum likeli-

hood (ML) estimator. It has been derived that the theoreticalbound of TOA for single path is [15]

s2toa �

c2

2� SNR � F2(3)

where

F2 ¼

Ð1�1

(2pF)2jS(F)j2 dFÐ1�1

jS(F)j2 dF

is the mean square bandwidth, S(F) is the Fourier transformof s(t) � SNR ¼ Eb/No, represents the signal to noise ratio(SNR) and C is the speed of light.As we can see from (3), the variance in the TOA ranges is

inversely proportional to SNR, which is intuitively reason-able since the detection of stronger signals is easy. Theinequality also shows the increase in TOA accuracy withthe bandwidth, which makes UWB a good candidate fortime-based ranging.We know that the TDOA can be seen as the difference

between the two TOA measurements. Thus, we can obtainthe bound of the TDOA measurement [16]

s2tdoa ¼ 2s2

toa �c2

SNR � F2(4)

and it is obvious that the accuracy of TDOA measurementsalso increases with higher bandwidth and stronger signals.For NLOS, some obstacles may exist between the nodes

and LOS is no longer present. It is known that TDOA is actu-ally the difference of TOAs that are estimated relative to acommon reference time but independent on the actual trans-mission time [17]. For TOA in NLOS environment, a posi-tive bias is introduced in the measurement as a result ofthe delay in received signal because the LOS between thetwo nodes is blocked. Its magnitude of the positive biasdepends on the propagation environment. If we measure

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TDOA ranges in the same environment with same obstacles,the same amount of bias will be present for a UWB pulse tra-velling between the nodes. Thus, we can reasonably approxi-mate the TDOA range value of blind node i as

(Rm,n)nlos ¼ (kpi � pmk þ bm þ 1m)� (kpi � pnk þ bn þ 1n)

(5)

where Rn is the TOA range between node n and i, 1n isGaussian measurement noise and bn is a positive biaspresent when LOS is blocked.Generally, the NLOS bias is modelled as exponential dis-

tributed. In [15], a simpler model based on UWB indoormeasurements is proposed where the NLOS bias is mod-elled as Gaussian with a positive mean. The negative partis relatively small and it is actually negligible.

3 Position estimation

Consider a network with N nodes, where r nodes are refer-ence nodes and the remaining (N2 r) are blind nodes(Fig. 1). The reference nodes which are assumed to be syn-chronised measure the TDOA of the UWB signals, and thetime differences are then converted to range measurements.For simplicity, we restrict ourselves to two dimensions. Inour UWB localisation system, we assume that the radiorange of each reference node is large enough so that itcan communicate with all the other nodes. Rm,n is therange difference from node m and n to the querying blindnode i, which equals radio speed times measured TDOA,Rm,n ¼ c � ttdoa � pi, pm and pn are positions of blindnode i and reference nodes m, n, respectively.Then the Euclidean distances between the nodes im and in

are kpi2 pmk and kpi2 pnk. And hence their difference is(kpi2 pmk2 kpi2 pnk). If Rm,n is accurate without errorand pi, pm and pn are the true coordinates of the threenodes, then Rm,n ¼ (kpi2 pmk2 kpi2 pnk). However, infact, Rm,n is always with noise, as mentioned in Section 2,and among the three nodes i, m and n, at least the coordinateof the querying node i is unknown. ThereforeRm,n and (kpi2pi2 pmk2 kpi2 pnk) are most likely not equal. Therefore,we have to minimise the difference between Rm,n and (kpi2pi2 pmk2 kpi2 pnk) in order to find the positions of theblind nodes. All the blind nodes are considered as a group.And a cost function (6) can be generated. Conjugate gradient

Fig. 1 Network with four reference nodes and five blind nodes

IET Sci. Meas. Technol., Vol. 2, No. 1, January 2008

method can be used to do the minimisation [18].

f (p) ¼XNi¼rþ1

Xr�1

m¼1

Xrn¼mþ1

(kpi � pmk � kpi � pnk � Rm,n)2

(6)

We determined the performance of the positioningmethod with root mean square error (RMSE), which isobtained by comparing estimated positions with actual pos-itions. Lower RMSE means better performance. RMSE isexpressed as

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN�ri¼1 k(pest)i � (pactual)ik

2

N � r

s(7)

where pest and pactual are estimated and actual coordinates,respectively.

4 Theoretical analysis

CRLB is a theoretical lower bound of the variance in theposition estimates and shows the smallest positioningerror that can be achieved [15]. Although the ranging accu-racy degrades with the distances, especially in multi-pathenvironments, the ranging errors used in the CRLB deri-vation are normally assumed to be distance independent.Here, we derives and analyses the theoretical boundCRLB for TDOA error-minimising positioning. CRLB isdefined as the inverse of the Fisher information matrix(FIM). C p̂ is the covariance matrix of the position estimatorof the blind node

C p̂ ¼ E[(p̂� p)(p̂� p)T] (8)

where p ¼ [p1, p2]T ¼ [x, y]T and E[.] is the expected value.

I(x, y) represents the FIM and can be found using

I(x, y) ¼ E@ ln(p(rjp)

@p

� �@ ln(p(rjp)

@p

� �T" #

(9)

where r ¼ [r2,1, r3,1, . . . , rM,1]T is a vector of the TDOA

ranges, p(rjp) is the PDF function of the ranges based onthe blind node position. The covariance matrix and theFIM satisfy the following inequality

C p̂ � I�1(p) � 0 (10)

and hence the positioning error bound satisfies

s2blind � tr[I�1(p)] (11)

where tr[.] is the trace of a square matrix, defined to be thesum of the diagonal elements.We assume that the measured TDOA ranges are Gaussian

with mean dm,n and variance s2.

Rm, n � N (dm,n, s2) (12)

where

dm,n ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x� xm)

2 þ (y� ym)2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x� xn)

2 þ (y� yn)2

qThen, the joint conditional PDF function is

p(rjx, y) ¼Yr�1

m¼1

Yrn¼mþ1

1

2ps2exp �

(Rm,m � dm,n)2

2s2

!(13)

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We can express the FIM as

I(x, y) ¼I11 I12I21 I22

� �(14)

The bound of the variance in the blind node position canbe obtained as

s2blind ¼ var(x)þ var(y) �

I22

I11I22 � I212þ

I11

I11I22 � I212

s2blind �

s2 Pr�1m¼1

Prn¼mþ1 (Am � An)

2hþPr�1

m¼1

Prn¼mþ1 (Bm � Bn)

2i

Pr�1m¼1

Prn¼mþ1 (Am � An)

2

�Pr�1

m¼1

Prn¼mþ1 (Bm � Bn)

2

�Pr�1

m¼1

Prn¼mþ1 (Am � An)(Bm � Bn)

h i2

(15)

where

Ai ¼(x� xi)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(x� xi)2 þ (y� yi)

2q , Bi ¼

(y� yi)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(x� xi)

2 þ (y� yi)2

qIt is clear that the right-hand side of the inequality (15) is

proportional to the variance in the TDOA measurements,which means that the lower bound on the positioning errorof the blind node increases with standard deviation of theTDOA range errors. Furthermore, since Ai and Bi are coeffi-cients and Ai

2þ Bi

2 ¼ 1, Ai and Bi can be expressed as cos(bi)and sin(bi), respectively, where bi is the angle between theblind node and the reference node i with respect to x-axis.Therefore it can be concluded that the CRLB of the variancein the blind node position is closely related to the geometryof the network, that is, CRLB varies with the positions of thereference nodes and blind nodes.Fig. 2 shows the CRLB for standard deviation s ¼ 1 m in

a network with four reference nodes placed at the fourcorners of a 1 � 1 m area. It is obvious that the blindnodes in the convex hull have lower theoretical positioningerrors with minimum value of 0.5 m.In the above case, it is assumed that all the possible TDOA

ranges are used, that is, r(r2 1)/2 TDOA ranges, for eachblind node. And the range errors are assumed to be indepen-dent and Gaussian distributed. If only the ranges relative toone reference node are used (e.g. R2,1, R3,1, . . . , Rr,1), the

Fig. 2 Plot of CRLB, r ¼ 4, s ¼ 1 m, total r(r2 1)/2 withTDOA ranges

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corresponding CRLB can be derived similar to (15)

s2blind �

s2 Prm¼2 (Am � A1)

2þPr

m¼2 (Bm � B1)2

� �Pr

m¼2 (Am � A1)2Pr

m¼2 (Bm � B1)2

�Pr

m¼2 (Am � A1)(Am � A1)� �2

(16)

Fig. 3 is the contour plot of (15) for the case where fourreference nodes are placed in the same way as shown Fig. 2,and only the ranges relative to one reference node (0, 0) areused. It can be observed that the contour plot is not asregular as that in Fig. 2. Its minimum value is larger thanthe case where all possible TDOA ranges are used to deter-mine the nodes positions. Same as Fig. 2, lower CRLBvalues can be obtained within the convex hull.As mentioned earlier, Chan’s method is a well-known

way to solve hyperbolic localisation problems. The CRLBobtained by Chan’s method has been derived in [11], andFig. 4b illustrates its contour plot. The pattern of thecontour plot shown in Fig. 4 agrees well with that inFig. 4a. The difference between the two figures is that thelower bound in Fig. 4a is smaller than that in Fig. 4b forthe same location, which indicates that, theoretically, theproposed error-minimising estimator should performbetter than Chan’s method. The above theoretical analyseswill be further verified in the next section.

Fig. 3 Contour plot of CRLB for error-minimising method,r ¼ 4, s ¼ 1 m, total (r2 1) TDOA ranges used

2008

Fig. 4 Contour plot of CRLB

a Error-minimising methodb Chan’s method, r ¼ 4, s ¼ 1 m

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5 Measurement and simulation

5.1 Measurement campaign

The measurements were done at the Positioning andWireless Technology Center (PWTC), which is located infourth floor of Research Techno Plaza in the campus ofNanyang Technological University, Singapore. The rangesare obtained by sending UWB signals from one transmitterto multiple receivers placed at different locations. The planlayout of the measurement campaign is shown in Fig. 5. Thekernel of measurement setup is an in-house-designed pulsegenerator generating UWB signal at a pulse repetition fre-quency (PRF) of 2 MHz.The probability of the ranging errors can be expressed by

a combination of Gaussian distribution function and

Fig. 5 Floor plan for the measurement campaign

exponential distribution function

f (x) ¼ a1

sffiffiffiffiffiffi2p

p exp�x2

2s2

� �þ (1� a)

1

bexp

�x

b

� �(17)

where s ¼ 0.071 m is standard deviation of the Gaussiandistribution function, b ¼ 5.235 m is mean of exponentialdistribution function, and a ¼ 0.767 m is the weightcoefficient.The values of above parameters can be obtained by fitting

the measured range error PDF shown in Fig. 6. Then TDOAranges can be represented by (3) and range noise because oflink blockage can be modelled with the PDF function givenin (17).

5.2 Simulation

We perform simulation experiment to evaluate the proposedTDOA estimation method using Microsoft Visual Cþþ andMatLab. The reference nodes are placed at the edges of a50 � 50 m region. The blind nodes are randomly placedinside. It is assumed that the radio range of the referencenodes covers the whole area and the reference nodes areall synchronised. The initial positions of the blind nodesare guessed as long as they are within the rectangulararea. For LOS propagation, we assume that the rangeerrors are Gaussian with zero mean and standard deviationof 0.1 m, Rm,n � N (dm,n, s

2), where dm,n is the actual rangedifference. For NLOS case, we assume that the UWBsignals sent from one of the reference nodes located at the

Fig. 6 Histogram of NLOS ranging errors for TOA

Fig. 7 LOS: RMSE against number of blind nodes with differentnumber of reference nodes

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corners have NLOS to all the blind nodes as a result of anobstacle that is close to that reference node as shown inFig. 1, while the remaining reference nodes have clearLOS to the blind nodes.In the simulation, the number of reference nodes has been

set to increase from four to eight. And the number of blindnodes has been set to increase from 5 to 150. Fig. 7 illus-trates the relationship between the RMSE and the numberof blind nodes for the LOS case. It is observed thatRMSE almost keeps constant and is independent of thenumber of the blind nodes for the same number of referencenodes. Therefore we can obtain the positions of a group ofblind nodes simultaneously using error-minimising methodwithout reducing the positioning accuracy. Fig. 8 shows theresults of RMSE for both LOS and NLOS. It shows thatRMSE for LOS is much smaller compared with NLOSRMSE. This is because for a given positioning method,the only source of error is derived from the accuracy ofrange measurements, and the NLOS error is the major onein TDOA measurements. For both LOS and NLOS cases,the accuracy of location estimation increases as thenumber of reference nodes increases. It can be observedthat the positioning error for LOS is very close to thelowest CRLB bound for LOS when the TDOA error-minimising is used. Fig. 9 shows the number of iterationsrequired for convergence for different number of blindnodes, and it is obvious that NLOS case required more iter-ations to converge.

Fig. 9 Iterations against the number of blind nodes for error-minimising method

Fig. 8 RMSE against the number of reference nodes for bothLOS and NLOS

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The TDOA error-minimising method has been comparedwith Chan’s and TS methods for both LOS and NLOS cases.Chan’s algorithm is a classical method for TDOA-basedposition estimation, and many recent proposals of hyper-bolic location algorithms have been compared with it.Hence, we follow the same way to use Chan’s method asthe major reference to show the advantages of our proposedmethod. The TS method is also an effective and well-knownmethod in TDOA-based positioning, which has also beenused frequently. Chan’s and TS methods are qualified astwo typical hyperbolic position estimation methods, andthus our proposed method is reasonable to compare withthem. In the simulations, 50 blind nodes are placed in the50 � 50 m rectangle. And the number of reference nodesis increased from four to eight. As shown in Fig. 10,given the same number of reference nodes, TDOA error-minimising localisation method can achieve better accuracycompared with Chan’s and TS methods. It complies with thetheoretical result derived in Section 4. For NLOS, locationsof up to 16 nodes cannot be determined with four referencenodes using Chan’s method (RMSE . 5 m), whereas theerror-minimising method can estimate the locations of allthe blind nodes with RMSE value of less than 0.9 m.Therefore we can conclude that error-minimising methodis more robust to ranging errors compared with Chan’smethod. For TS method, it has been mentioned in Section 1that its main disadvantage is that it requires a good initialguess of the blind node position. Thus, in the simulation,we set the initial guess of TS method as the actual positionof the blind nodes, and it comes out that our proposedmethod still has better performance.

6 Conclusions

The issue of location positioning using UWB rangingmeasurements has been addressed in this paper. Themajor contributions in this paper are that (1) the TDOAranges for both LOS and NLOS propagations have beenmodelled; (2) the TDOA error-minimising method hasbeen proposed and the positions of the blind nodes have

Fig. 10 TDOA error-minimising localisation against Chan’smethod and TS method for both LOS and NLOS

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been estimated by the proposed method; (3) the effect ofrange errors on the positioning estimation accuracy hasbeen investigated by the derivation of CRLB. The simu-lation results have demonstrated that the positions of agroup of blind nodes can be estimated simultaneouslywithout reducing the positioning accuracy. It has alsobeen observed that the accuracy of TDOA error-minimisingmethod is very close to the derived CRLB value for LOS.

7 Acknowledgment

The authors would like to thank Mr. Xu Chi for providingthe measurement data.

8 References

1 Fontana, R.J.: ‘Recent system applications of short-pulseultra-wideband (UWB) technology’, IEEE Trans. Microw. TheoryTech., 2004, 52, (9), (Part 1) pp. 2087–2104

2 Win, M.Z., and Scholtz, R.A.: ‘Impulse radio: how it works’, IEEECommun. Lett., 1998, 2, (2), pp. 36–38

3 Win, M., and Scholtz, R.: ‘On the performance of ultrawide bandwidthsignals in dense multipath environment’, IEEE Commun. Lett., 1998,2, (2), pp. 51–53

4 Fontana, R.J., and Gunderson, S.J.: ‘Ultra-wideband precision assetlocation system’. Digest of Papers. 2002 IEEE Conf. on UltraWideband Systems and Technologies, 21–23 May 2002, pp. 147–150

5 Gezici, S., Tian, Z., Giannakis, G.B., Kobayashi, H., Molisch, A.F.,Poor, H.V., and Sainoglu, Z.: ‘Localization via ultra-widebandradios: a look at positioning aspects for future sensor networks’,IEEE Signal Process. Mag., 2005, 22, (4), pp. 70–84

6 Ingram, S.J., Harmer, D., and Quinlan, M.: ‘UltraWideBand indoorpositioning systems and their use in emergencies’. Position Locationand Navigation Symp., 26–29 April 2004, pp. 706–715

7 Cardinali, R., De Nardis, L., Di Benedetto, M.-G., and Lombardo, P.:‘UWB ranging accuracy in high- and low-data-rate applications’,IEEE Trans. Microw.Theory Tech., 2006, 54, (4), pp. 1865–1875

8 Chung, W.C., and Ha, D.S.: ‘An accurate ultra wideband (UWB)ranging for precision asset location’. Proc. IEEE Conf. UltraWideband Systems and Technologies, November 2003, pp. 389–393

9 Lee, J.-Y., and Scholtz, R.A.: ‘Ranging in a dense multipathenvironment using an UWB radio link’, IEEE J. Sel. AreasCommun., 2002, 20, (9), pp. 1677–1683

10 Bocquet, M., Loyez, C., and Benlarbi-Delai, A.: ‘Usingenhanced-TDOA measurement for indoor positioning’, IEEEMicrow. Wirel. Compon. Lett., 2005, 15, (10), pp. 612–614

11 Chan, Y., and Ho, K.C.: ‘A simple and efficient estimator forhyperbolic location’, IEEE Trans. Signal Process., 1994, 42, (8),pp. 1905–1915

12 Foy, W.H.: ‘Position-location solutions by Taylor-series estimation’,IEEE Trans. Aerosp. Electron. Syst., 1976, 12, (2), pp. 187–194

13 Kovavisaruch, L., and Ho, K.C.: ‘Modified Taylor-series method forsource and receiver localization using TDOA measurements witherroneous receiver positions’. IEEE Int. Symp. on Circuits andSystems, May 2005, vol. 3, pp. 2295–2298

14 Xiong, J.Y., Wang, W., and Zhu, Z.L.: ‘An improved Tayloralgorithm in. TDOA subscriber position location’. Proc. ICCT, April2003, vol. 2, pp. 981–984

15 Kay, S.M.: ‘Fundamentals of statistical signal processing: estimationtheory’ (Prentice-Hall)

16 Richard, G.W.: ‘Electronic intelligence: the interception of radarsignals’ (Artech House Publishers)

17 Gezici, S., Tian, Z., Giannakis, G.B., Kobayashi, H., Molisch, A.F.,Poor, H.V., and Sainoglu, Z.: ‘Localization via ultra-widebandradios: a look at positioning aspects for future sensor networks’,IEEE Signal Process. Mag., 2005, 22, (4), pp. 70–84

18 Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T.:‘Numerical recipes in C: the art of scientific computing’(Cambridge University Press)

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