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Signal Processing 87 (2007) 3096–3100

www.elsevier.com/locate/sigpro

Accurate three-step algorithm for joint source position andpropagation speed estimation

Jun Zheng, Kenneth W.K. Lui, H.C. So�

Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China

Received 29 December 2006; received in revised form 25 May 2007; accepted 2 June 2007

Available online 10 July 2007

Abstract

A popular strategy for source localization is to utilize the measured differences in arrival times of the source signal at

multiple pairs of receivers. Most of the time-difference-of-arrival (TDOA) based algorithms in the literature assume that

the signal transmission speed is known which is valid for in-air propagation. However, for in-solid scenarios such as

seismic and tangible acoustic interface applications, the signal propagation speed is unknown. In this paper, we exploit the

ideas in the two-step weighted least squares method [1] to design a three-step algorithm for joint source position and

propagation speed estimation. Simulation results are included to contrast the proposed estimator with the linear least

squares scheme as well as Cramer–Rao lower bound.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Source localization; Propagation speed estimation; Time-difference-of-arrival; Weighted least squares

1. Introduction

Passive source localization using time-difference-of-arrival (TDOA) information from an array ofspatially separated sensors is an important problem insignal processing. In the TDOA method, the differ-ences in arrival times of the source signal at multiplepairs of sensors are measured. Each TDOA measure-ment defines a hyperbolic locus on which the sourcemust lie and the position is given by the intersectionof two or more hyperbolas for noise-free two-dimensional localization. Although there are numer-ous TDOA-based positioning algorithms in theliterature, such as [1–6], most of them assume that

e front matter r 2007 Elsevier B.V. All rights reserved

gpro.2007.06.014

ing author. Tel.: +852 2788 7780;

7791.

ess: hcso@ee.cityu.edu.hk (H.C. So).

the signal transmission speed is known which isvalid for in-air propagation. However, for in-solidscenarios such as seismic [7] and tangible interfacefor human–computer interaction [8] applications, thesignal propagation speed is unknown, and we need tofind it together with the source position for accurateestimation. In this paper, an efficient three-stepalgorithm for joint source position and propagationspeed estimation is derived by applying the ideasof [1], namely, employment of weighted least squares(WLS) and exploitation of the relationship betweenparameter estimates. Our major contributions are toaddress the positioning problem with unknownpropagation speed and exploit the nonlinear relation-ship between the source position and speed para-meters in the third step of the developed algorithm.

The rest of the paper is organized as follows. Theproposed three-step method is developed in Section 2.

.

ARTICLE IN PRESSJ. Zheng et al. / Signal Processing 87 (2007) 3096–3100 3097

It is proved that the first step solution is in factequal to the least squares (LS) algorithm of [6].Simulation results are included in Section 3 toevaluate the estimation performance of the three-step algorithm by comparing with the LS methodand Cram�er–Rao lower bound (CRLB). Finally,conclusions are drawn in Section 4.

2. Joint source position and propagation speed

estimation algorithm

In this section, we develop a three-step algorithmto jointly estimate source location and propagationspeed using TDOA measurements from M sensors.The discrete-time signal received at the ith sensorcan be expressed as

riðkÞ ¼ sðk �DiÞ þ qiðkÞ; i ¼ 1; 2; . . . ;M, (1)

where sðkÞ is the signal radiating from the source,and Di and qiðkÞ are the time-of-arrival and additivenoise, respectively, at the ith sensor.

Let ðx; yÞ and ðxi; yiÞ, i ¼ 1; 2; . . . ;M, be theunknown source location and known position ofthe ith sensor, respectively. Denote di;1 and Di;1 ¼

Di �D1 as the range difference and TDOA withrespect to the first sensor, respectively, then we havethe following relationship:

di;1 ¼ cDi;1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi � xÞ2 þ ðyi � yÞ2

q

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 � xÞ2 þ ðy1 � yÞ2

q,

i ¼ 2; 3; . . . ;M, ð2Þ

where c is the unknown propagation speed. Ourtask is to find ðx; yÞ and c with the use of fDi;1g andfðxi; yiÞg. Following the idea of [3,4], we rearrangeand square (2) to yield

2ðxi � x1Þxþ 2ðyi � y1Þyþ 2Di;1uþD2i;1v

¼ x2i � x2

1 þ y2i � y2

1; i ¼ 2; 3; . . . ;M, ð3Þ

where u ¼ c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 � xÞ2 þ ðy1 � yÞ2

qand v ¼ c2 are

introduced to make a linear representation. Inmatrix form, we have

Ah ¼ b, (4)

where

A ¼

2ðx2 � x1Þ 2ðy2 � y1Þ D22;1 2D2;1

..

. ... ..

. ...

2ðxM � x1Þ 2ðyM � y1Þ D2M ;1 2DM ;1

26664

37775,

h ¼

x

y

v

u

26664

37775,

and

b ¼

x22 � x2

1 þ y22 � y2

1

..

.

x2M � x2

1 þ y2M � y2

1

2664

3775.

In the presence of fqiðkÞg, the TDOA measurementsare noisy which can be modelled as

Di;1 ¼ D0i;1 þ ni;1; i ¼ 2; 3; . . . ;M, (5)

where fD0i;1g denote the noise-free TDOA’s while

each TDOA estimation error ni;1 is characterized byriðkÞ and r1ðkÞ.

Based on (4), the standard LS estimate of h,denoted by h1, is simply

h1 ¼ ðATAÞ�1ATb, (6)

where T and �1 denote the transpose operator andmatrix inverse, respectively. It is noteworthy that (6)is in fact identical to the solution of [6], although wework on the hyperbolic equations from TDOAmeasurements while circular equations from time-of-arrival information are considered in the latter.

For more accurate estimation, we propose toutilize the ideas of [1], namely, employing WLSand exploiting the relationship between x, y, u andv, in the following two steps. For sufficiently smallnoise conditions, the measured error vector in (4),denoted by e, can be approximated as [1]

e ¼ b� Ah

� 2½ðuþ vD02;1Þn2;1 ðuþ vD0

3;1Þn3;1 � � �

�ðuþ vD0M ;1ÞnM ;1�

T. ð7Þ

The covariance matrix for e, denoted by U2, is then

U2 ¼ EfeeTg � 4B2QB2, (8)

where E denotes expectation operator, B2 ¼

diagðuþ vD02;1; uþ vD0

3;1; . . . ; uþ vD0M ;1Þ and Q is

the covariance matrix for fni;1g which can bedetermined using the power spectra of sðkÞ andfqiðkÞg [1,9]. For simplicity, we assume that thesource signal and noises in (1) are white processesand the signal-to-noise ratios at all friðkÞg are

ARTICLE IN PRESSJ. Zheng et al. / Signal Processing 87 (2007) 3096–31003098

identical. In doing so, Q will be proportional to

2 1 � � � 1

1 2 � � � 1

..

. ... . .

. ...

1 1 � � � 2

266664

377775

and we will substitute this matrix forQ in our study.With the use of (8), the WLS estimate of h, denotedby h2, is [10]

h2 ¼ ðATU�12 AÞ�1ATU�12 b. (9)

Note that the technique of WLS has already beenutilized in localization [1,2], although A and U2 in(9) are of different forms because we have theunknown parameter of speed as well.

To compute U2 in practice, fD0i;1g in B2 are

replaced by fDi;1g in (5) and v and u areapproximated by

v ¼DTd

DTD

� �2

(10)

and

u ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðð½h1�1 � x1Þ

2þ ð½h1�2 � y1Þ

2Þ

q, (11)

where ½h1�1 and ½h1�2 represent the first and second

elements of h1, which are the LS estimates of x and

y, respectively. The D ¼ ½D2;1;D3;1; . . . ;DM ;1�T and

d ¼ ½d2;1; d3;1; . . . ; dM ;1�T are the TDOA vector

and range difference vector, respectively, where di;1

is computed as di;1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi � ½h1�1Þ

2þ ðyi � ½h1�2Þ

2q

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 � ½h1�1Þ

2þ ðy1 � ½h1�2Þ

2q

. It is noteworthy that

choosing the above initial estimates of u and v isbased on the fact that the location estimate in (6) ismore accurate than the speed estimate [5]. Thecovariance matrix for the WLS estimate in (9) is [10]

covðh2Þ ¼ ðATU�12 AÞ�1. (12)

The relationship between x, y, u and v has notbeen exploited so far. In the third step, we utilizetheir relationship

u2 ¼ vððx� x1Þ2þ ðy� y1Þ

2Þ. (13)

We first define the matrices

H ¼

1 0 0

0 1 0

0 0 1

½h2�3 ½h2�3 0

266664

377775; ! ¼

ðx� x1Þ2

ðy� y1Þ2

v

264

375

and

p ¼

ð½h2�1 � x1Þ2

ð½h2�2 � y1Þ2

½h2�3

½h2�24

2666664

3777775.

The h2 can be written in terms of h as

½h2�i ¼ ½h�i þ ei; i ¼ 1; 2; 3; 4, (14)

where feig are the estimation errors in (9). Now wehave

H! � p. (15)

Similar to (7), the error vector for (15) can beapproximated as

n ¼ p�H!

� ½2ðx� x1Þe1; 2ðy� y1Þe2; e3; 2ue4 � re3�T, ð16Þ

where r ¼ ðx� x1Þ2þ ðy� y1Þ

2. Following [1], thecovariance matrix for n, denoted by U3, is

U3 ¼ EfnnTg � B3 covðh2ÞB3 þ ½0 G�, (17)

where B3 ¼ diagð2ðx� x1Þ; 2ðy� y1Þ; 1; 2uÞ, 0

is a 4� 3 zero matrix and G ¼ ½�2rðx�

x1ÞEfe1e3g; �2rðy� y1ÞEfe2e3g; �rEfe23g; r2Efe23g�

4urEfe3e4g�T with Efeiejg corresponding to the ði; jÞ

entry of covðh2Þ in (12). In practice, the values of x, y

and u are approximated by the estimates of (9). The

WLS estimate of !, denoted by !, is then

! ¼ ðHTU�13 HÞ�1HTU�13 p. (18)

The final estimates of the location and speed are thencomputed as

x ¼ �

ffiffiffiffiffiffiffiffi½!�1

qþ x1,

y ¼ �

ffiffiffiffiffiffiffiffi½!�2

qþ y1,

c ¼

ffiffiffiffiffiffiffiffi½!�3

q. ð19Þ

The values of x and y which are closest to the

corresponding estimates in h2 are chosen as theposition estimate. Note that when the square root isa complex number, the imaginary component will beset to zero, and this happens when ðx1; y1Þ is veryclose to ðx; yÞ.

The proposed three-step algorithm for jointsource position and propagation speed estimation

ARTICLE IN PRESS

10

J. Zheng et al. / Signal Processing 87 (2007) 3096–3100 3099

is summarized as follows:

5 (dB

)

LS

(i) Compute the LS solution using (6).

rror

proposed

(ii) 0n

e CRLB

Compute the second step solution of (9) withthe use of (10) and (11).

sitio

(iii)

-5

po

Compute the third step solution using (18) and(19) with the use of h2.

-80 -75 -70 -65 -60

-20

-15

-10

mean s

quare

σ2 (dB)

Fig. 1. Mean square position errors.

-80 -75 -70 -65 -60

45

50

55

60

65

70

75m

ean s

quare

speed e

rror

(dB

) LSproposedCRLB

σ2 (dB)

Fig. 2. Mean square speed errors.

3. Simulation results

Computer simulations are conducted to evaluatethe proposed three-step algorithm for source localiza-tion and speed estimation by comparing it with theLS solution of (6) or [6], as well as CRLB [5]. Weconsider a tangible acoustic interface application ofinteractive displays [11]. Five sensors are placed on a1m� 1m pane of glass with coordinates (0.5, 0.5)m,(0, 0)m, (1, 0)m, (1, 1)m, and (0, 1)m while theunknown source position is located at (0.2, 0.1)m.Note that the acoustic propagation speed in solid isdependent on the material of medium as well as thetype of tactile interaction. Here we assume a 1-cmthick pane with a knuckle tap, the propagation speedis set to 1200ms�1 [11]. For this relatively high speed,the TDOA’s become much smaller than the locationcoordinates. As a result, the values in the last twocolumns of A are significantly less than those of thefirst two columns, which will make A a badly scaledmatrix and result in inaccuracy of the solution. Toavoid this problem, we multiply both sides of (2) by103. This operation scales up x; y;xi; yi and Di;1 by103. In doing so, the elements in the first, second andfourth columns will be multiplied by 103 while thoseof the third column will be multiplied by 106 and thusthe condition number of A will be decreased. Thesolution of ðx; yÞ in (19) is scaled down accordingly toobtain our solution. The same scaling operation isalso applied to the LS method in [6]. The noise-freeTDOA’s are added by the correlated Gaussian noiseswith covariance matrix given by Q with diagonalelements equal 2s2 and all other elements equal s2.All simulation results are averages of 5000 indepen-dent runs.

Figs. 1 and 2 compare the positioning and speedestimation accuracy, respectively, of the proposedmethod with the LS approach and CRLB for differentvalues of s2. The mean square error (MSE) is used asthe performance measure. The MSE of the positionestimate is defined as E ðx� xÞ2 þ ðy� yÞ2

� �where x

and y denote the estimates of x and y, respectively. Onthe other hand, the MSE of the speed estimate is

defined as E ðc� cÞ2� �

where c is the estimate of c.From Figs. 1 and 2, it is observed that the MSE’s ofthe proposed method are very close to CRLB, and aresmaller than those of the LS method, for both locationand speed estimation.

4. Conclusion

A three-step algorithm for joint source localiza-tion and propagation speed estimation is developedwith the use of TDOA measurements. Two inter-mediate variables are introduced to make a linearrepresentation of the nonlinear TDOA equations.The LS solution in the first step provides the initialestimates. The second step refines the estimationby employing WLS while the final step further

ARTICLE IN PRESSJ. Zheng et al. / Signal Processing 87 (2007) 3096–31003100

improves the estimates by another WLS via utilizingthe relationship between the source position, speedand intermediate variables. For sufficiently smallnoise conditions, it is shown that the accuracy of theproposed method approaches CRLB and outper-forms the LS method of [6].

Acknowledgement

The work described in this paper was supportedby a grant from CityU (Project no. 7002132).

References

[1] Y.T. Chan, K.C. Ho, A simple and efficient estimator for

hyperbolic location, IEEE Trans. Signal Process. 42 (August

1994) 1905–1915.

[2] K.W. Cheung, H.C. So, Y.T. Chan, W.-K. Ma,

A constrained least squares approach to mobile positioning:

algorithms and optimality, EURASIP J. Appl. Signal

Processing 2006 (2006) 1–23, Article ID 20858.

[3] J.O. Smith, J.S. Abel, Closed-form least-squares source

location estimation from range-difference measurements,

IEEE Trans. Acoust. Speech Signal Process. 35 (December

1987) 1661–1669.

[4] B. Friedlander, A passive localization algorithm and its

accuracy analysis, IEEE J. Ocean. Eng. 12 (January 1987)

234–245.

[5] C.W. Reed, R. Hudson, K. Yao, Direct joint source

localization and propagation speed estimation, in: Proceed-

ings of the ICASSP’99, vol. 3, March 1999, pp. 1169–1172.

[6] A. Mahajan, M. Walworth, 3-D position sensing using the

differences in the time-of-flights from a wave source to

various receivers, IEEE Trans. Robotics Automation 17

(February 2001) 91–94.

[7] J.C. Chen, K. Yao, R.E. Hudson, Source localization and

beamforming, IEEE Signal Process. Magazine 19 (2) (March

2002) 30–39.

[8] G. De Sanctis, D. Rovetta, A. Sarti, G. Scarparo and

S. Tubaro, Localization of tactile interactions through

TDOA analysis: geometric vs. inversion-based method, in:

Proceeding of 2006 European Signal Processing Conference,

Florence, Italy, September 2006.

[9] W.R. Hahn, S.A. Tretter, Optimum processing for delay-

vector estimation in passive signal arrays, IEEE Trans.

Inform. Theory 19 (September 1973) 608–614.

[10] S.M. Kay, Fundamentals of Statistical Signal Processing:

Estimation Theory, Prentice-Hall, Englewood Cliffs, NJ,

1993.

[11] J.A. Paradiso, et al., Passive acoustic sensing for tracking

knocks atop large interactive displays, in: Proceedings of

IEEE International Conference on Sensors, vol.1, Orlando,

Florida, June 2002, pp. 521–527.