LNCS 4239 - Clock Offsets in TDOA Localization - Springer Link
Clock Offsets in TDOA Localization Nak-Seon Seong1 and Seong-Ook Park2 1
Electronics and Telecommunications Research Institute,161 Gajeong-dong, Yuseong-gu, Daejeon, 305-700, Korea 2 Department of Electronic Engineering, Information and Communications University, 119 Munjiro, Yuseong-gu, Daejeon, 305-732, Korea [email protected], [email protected]
Abstract. Time based TDOA localization systems require time or clock synchronization between receivers such as cellular base stations, satellites, and sensor nodes. Imperfection of time synchronization causes degradation in positioning accuracy. However if we know about its characteristics and how to estimate the clock offsets, the localization system can be properly calibrated to provide good quality of services. In this paper, hence, we present how to derive a localization error vector with independent clock offset, and illustrate its effect on the positioning errors, and then, provide a simple method of TDOA clock offset estimation from the observation of error vectors.
1 Introduction Many researches on source localizations have been addressed in many applications including sonar, radar, mobile communications, and, more recently, wireless sensor networks [1]-[6]. The approaches to the solution of the source location problem include iterative least-squares [7], [8] and maximum likelihood (ML) estimation [9]. Closed-form solutions have been given in [10]-[11], using “spherical intersection” and “spherical interpolation” methods, respectively. Fang gave an exact solution when the number of TDOA measurements is equal to the number of unknowns [12]. Meanwhile, Chan suggested a non-iterative general solution which is an approximation of maximum likelihood (ML) estimator of achieving optimum performance for arbitrarily placed sensors [5], [9]. But most of the previous works have focused on location estimation techniques. Major sources of errors in time-based localization detection schemes are the receiver time synchronization offset, the wireless multi-path fading channel, and the non-line-of-sight (NLOS) transmission. In theses, although NLOS problem is most significant, time synchronization offset (or clock offset) is primarily required to be resolved, which is our focus in this paper. Even a few nano-seconds of clock offset can make position error of up to several meters in triangulation, which is ferocious performance for high resolution positioning systems. However, it can be mitigated by system calibration by way of observing localization error vectors and estimating the time differences of clock offset. Literatures about estimating and canceling the clock offsets in wireless sensor networks and how to self-calibrate the unknown clock offsets was recently addressed [15]-[16]. Patwari suggested joint ML estimation of clock-offsets and node H.Y. Youn, M. Kim, and H. Morikawa (Eds.): UCS 2006, LNCS 4239, pp. 111 – 118, 2006. © Springer-Verlag Berlin Heidelberg 2006
112
N.-S. Seong and S.-O. Park
coordinates [15], and Rydström, more recently, two ML estimators of linear preprocessing operations to remove the effect of unknown clock-offsets from the estimation problem in [16]. In this paper, first, a matrix equation with independent parameters of clock offset is derived for the estimation, and the effects on the localization is illustrated, and then, finally, a simple method of how to extract the difference of clock offsets in threereceiver case is presented.
2 Effects of Clock Offset 2.1 Theory The localization error depends on the all of the clock offsets in the receivers. Most previous works and articles provide TDOA techniques on time difference, however, this paper presents simple equation with independent clock offset parameters, and then illustrate how they causes in localization. To obtain the matrix equation, we first investigate effect of clock offset in a single receiver, and then all of the contributions from three receivers are put together.
Fig. 1. Two-dimensional TDOA source localization geometry with three receivers
It is assumed that localization geometry is two-dimensional to simplify the problem, and that there are three receivers at locations (xi,yi), and that signals propagate in line-of-sight (LOS) without multi-paths as in Fig. 1. All the three receivers are to be synchronized to the reference clock. When the true source location is at P(x,y), the range Ri (i=1,2,3) between the source and the sensor i is given by Ri =
(xi − x )2 + (yi − y)2 = CTi
(1)
where C is the signal propagation speed and Ti is the signal flight time from the source to the receiver i according to the distance. Hence, the range difference Rij between the receivers i and j is
Clock Offsets in TDOA Localization
113
R ij = R i − R j = (x i − x) 2 + (yi − y)2 − (x j − x) 2 + (y j − y)2
(2)
so that Rij is a function of x and y R ij = f(x, y).
(3)
and ΔR
ij
=
∂f ∂x
Δx +
∂f ∂y
Δy
(4)
where ΔR denotes range estimation error and Δx , Δy are source location errors in x and y coordinates respectively. Substituting the partial derivatives in (4) and combing (2), (3), and (4) gives −1 [(xi − x)Δx + (yi − y)Δy] Ri
ΔR ij = +
(5)
1 [(x j − x) Δx + (y j − y)Δy]. Rj wZO'S 'P TDOA 323
TDOA 331
w' O'S 'P w OS P
TDOA131
TDOA 223
wYGO'S 'P
{kvhYZ
{kvhZX
wXGO'S 'P TDOA112
{kvhXY
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Fig. 2. Shift of hyperbolic curves due to time synchronization offset at receiver 1
Since Rij can be obtained from time difference of two propagation delays of Ti, Tj, it has a relation of R ij = CTij = C(Ti − Tj ) , hence, when the offsets are small, the variation of range difference is given by
114
N.-S. Seong and S.-O. Park ΔR ij = C( ΔTi − ΔT j ) = C(ε i - ε j )
(6)
where εi, εj denote clock offset of receiver i, j from global or local reference clock. To find individual contribution to the position error by a single clock offset at a receiver, we need to first start with εi 0, εj=0 in (6) and then all the contributions will
≠
be summed up. When ε1 ≠ 0, ε 2 = ε 3 = 0 , then, in Fig. 2, the two location curves TDOA 12
, TDOA 31 move to TDOA 112 , TDOA131 respectively along the TDOA 23
curve. From the two new curves, a new intersection P1(x',y') is obtained. Combining (5) and (6) gives a set of linear equations about u1 = [Δx 1 Δy1 ] and ε1 as T
[
−(x1 − x) R1 [
+
−(x 2 − x) R2
(x 2 − x) R2 +
]Δx1 + [
(x 3 − x) R3
−(y1 − y)
]Δx 1 + [
R1
+
−(y 2 − y) R2
(y 2 − y) R2 +
]Δy1 = Cε1
(y 3 − y) R3
]Δy1 = 0
(7)
(8)
where suffix 1 in Δx , Δy means the error from time synchronization offset at receiver 1. Combining (7) and (8) gives a matrix equation with a single independent parameter ε1 B12 ⎤ ⎡ Δx 1 ⎤ ⎡ε 1 ⎤ ⎥⎢ ⎥ = C⎢ ⎥ B 23 ⎦ ⎣ Δy1 ⎦ ⎣0⎦
⎡ A12 ⎢ ⎣A 23
where Aij =
B ij =
− (x i − x) (x j − x) + Ri Rj
− (y i − y) (y j − y) + (i = 1,2,3). Ri Rj
(9)
From (9), we can obtain the effect of clock offset due to ε 1 on the location estimation as ⎡ A12 ⎡ Δx 1 ⎤ ⎢ ⎥ = C⎢ ⎣A 23 ⎣ Δy 1 ⎦
B12 ⎤ ⎥ B 23 ⎦
−1
⎡ε 1 ⎤ C ⎡ B 23 ⎤ ⎢ ⎥ε 1 ⎢ ⎥= ⎣ 0 ⎦ D1 ⎣ − A 23 ⎦
(10)
where D1 is defined as A12 B 23 − A 23 B12 . Similarly, the error vectors u 2 = [Δx 2
Δy 2 ]T and u 3 = [Δx 3
Δy 3 ]T due to ε 2 ,
ε 3 can be also obtained as in (11) and (12) ⎡ Δx 2 ⎤ C ⎡ B 31 ⎤ ⎢ ⎥ε 2 ⎢ ⎥= y Δ D ⎣ 2⎦ 2 ⎣ − A 31 ⎦
(11)
Clock Offsets in TDOA Localization ⎡Δx 3 ⎤ C ⎡ B12 ⎤ ⎢ ⎥= ⎢ ⎥ε 3 ⎣Δy 3 ⎦ D 3 ⎣ − A12 ⎦
115
(12)
where D 2 , D 3 are defined as A 23 B 31 − A 31 B 23 and A 31B12 − A12 B 31 respectively. Since each error vector u i (i=1,2,3) is independent, the integrated location error of the source can be expressed as sum of individual error vector, S b = u1 + u 2 + u 3 , and we obtain as in (14) ⎡ B 23 ⎢D Δx ⎡ ⎤ ⎢ 1 C = ⎢ ⎥ ⎢ − A 23 ⎣Δy ⎦ ⎢ ⎣ D1
B 31 D2 − A 31 D2
⎤ ⎥ ⎡ε1 ⎤ ⎥ ⎢ε ⎥. 2 − A12 ⎥ ⎢ ⎥ ⎥ ⎢⎣ε 3 ⎥⎦ D3 ⎦ B12 D3
(13)
From (13), the source position error vector S b is obtained in a simple matrix form of S b = CHTb , where S b = [Δx
Δy]T , Tb = [ε1 ε 2
⎡ B 23 ⎢D H=⎢ 1 ⎢ − A 23 ⎢ ⎣ D1
B 31 D2 − A 31 D2
ε 3 ]T , and
⎤ ⎥ ⎥. − A12 ⎥ ⎥ D3 ⎦ B12 D3
(14)
Hence, the final location error ΔL is given by ΔL = Δx 2 + Δy 2 . 2.2 Results and Discussions
Fig. 3 shows how time error of a receiver affects on the source localization. In this paper, we assume that three receivers are placed in triangular shape. We can find the fact that when a receiver is out of synchronization from a global or local reference clock but the others are not, then the localization error contours shift to the direction of the defective receiver. This means localization accuracy near the defective receiver is ironically better than other service area with perfect receivers. Direction of the error vectors reverses depending on the polarity of the clock offset [Fig. 3-(b), (c)]. However, Fig. 3-(d)-(i) gives us additional information about the movement of accuracy region. In Fig. 3-(d)-(f), error contours shift to a perfectly synchronized receiver when only one receiver is synchronized but the others have clock offsets of the same magnitudes. Fig. 3-(g)-(i), however, illustrates that the accuracy geometry moves to the opposite direction of Fig. 3-(d)-(f) when the polarities of the clock offsets of the two receivers are different each other. Hence, summing up, the information above puts the fact that localization accuracy of geometry moves the direction where a receiver has a different clock offset from the others with similar magnitudes whether or not the receiver is in a good synchronization with the reference clock.
116
N.-S. Seong and S.-O. Park
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Fig. 3. Localization errors due to clock offsets
3 Estimation of Clock Offsets In the previous section, we derived a TDOA error vector equation in a form of independent clock offset parameters as in (14). But, since matrix H is not a square matrix, we can not directly obtain Tb from (14). Using (5)-(6) again gives a simple form of solution in difference of time offsets as in (15) ⎡ −(x 1 − x) (x 2 − x) + ⎢ R1 R2 ⎢ ⎢ − (x 2 − x) + (x 3 − x) ⎢ R2 R3 ⎣
−(y1 − y) (y 2 − y) ⎤ + ⎡ε − ε 2 ⎤ R1 R 2 ⎥⎥ ⎡ Δx ⎤ = C⎢ 1 ⎥ − (y 2 − y) (y 3 − y) ⎥ ⎢⎣ Δy ⎥⎦ ⎣ε 2 − ε 3 ⎦ + ⎥ R2 R3 ⎦
(15)
Clock Offsets in TDOA Localization
117
So we obtain ΔTb = C −1GS b where ⎡ε 1 − ε 2 ⎤ ΔTb = ⎢ ⎥ ⎣ε 2 − ε 3 ⎦
and ⎡ −(x 1 − x) (x 2 − x) + ⎢ R R2 1 G=⎢ ⎢ − (x 2 − x) + (x 3 − x) ⎢ R R3 2 ⎣
−(y1 − y) (y 2 − y) ⎤ + R1 R 2 ⎥⎥ . − (y 2 − y) (y 3 − y) ⎥ + R2 R 3 ⎥⎦
(16)
Measurement of S b in several positions gives more accurate estimation of difference of clock offsets Tˆ b for the receivers. Since the estimation matrix is given by E[ΔTb ] = C −1GE[S b ] , the solution for estimated difference of clock offsets is −1
ΔTˆ b = C GSˆ b .
(17)
4 Conclusion We derived a new solution of independent clock offset matrix in TDOA localizations to see how each offset of a receiver affects on the source localization, and how to measure the difference of clock offsets by measuring position error data. Hence localization systems can be monitored and calibrated by simply observing error vectors and thus compensating for them.
References 1. Y. Rockah and P.M. Schultheiss, “Array Shape calibration using sources in unknown location Part I: Far-field source,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, No. 3, pp. 286-299, Mar. 1987. 2. Y. Rockah and P.M. Schultheiss, “Array Shape calibration using sources in unknown location Part II: Near-field source and estimator implementation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, No. 6 , pp. 724-735, Jun. 1987. 3. Y.M. Chen, J.H. Lee, and C.C. Yeh, “Two- dimensional angle of arrival estimation for uniform planar array with sensor position errors,” Proc. IEE Radar, Signal Processing, vol.140 No. 1, pp. 37-42, Feb. 1993. 4. W.H. Foy, “Position-location solution by Taylor series estimations,” IEEE Trans. Aerosp. Electron. Syst, vol. AES-12, pp.187-194, Mar. 1976. 5. Y.T. Chan and K.C. Ho, “An efficient closed-form localization solution from time difference of arrival measurements,” in Proc. IEEE ICASSP, ICASSP 94 vol. ii, , pp. II/393-II/396 Apr. 1994. 6. Xiang Ji and Hongyuan Zha, “Robust sensor localization algorithm in wireless ad-hoc sensor networks,” in Proc. IEEE ICCCN 2003, pp.527 – 532, Oct. 2003.
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7. R. Schmidt, “Least squares range difference location,” IEEE Transactions on Aerospace and Electronic Systems, 32, 1, Jan. 1996. 8. R. Schmidt, “A new approach to geometry of range difference location,” IEEE Transactions on Aerospace and Electronic Systems, AES-8, 3, Nov. 1972. 9. Y.T. Chan and K.C. Ho, “A simple and efficient estimator for hyperbolic location,” IEEE Transactions on Signal Processing, 42, 8 Aug. 1994. 10. H.C. Schau and A.Z. Robinson, “Passive source localization employing intersecting spherical surfaces from time-of-arrival differences,” IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-35, Aug. 1987. 11. J.O. Smith and J.S. Abel, “Closed-form least-squares source location estimation from range-difference measurements,” IEEE Transactions on Acoustics, Speech, and Signal Processing, ASSP-35, 12, Dec. 1987. 12. B.T. Fang, “Simple solutions for hyperbolic and related location fixes,” in IEEE Transactions on Aerospace and Electronic Systems, vol. 26, Issue 5, Sep. 1990. 13. A. Nasipuri and K.Li, “A Directionality Based Location Discovery Scheme far Wireless Sensor Networks,” SNA’O2, pp. 105-111, 2002. 14. E.D. Kaplan, “Understanding GPS: Principles and Applications,” Artech House Published, 1996. 15. M. Rydström, E.G. Strom, and A. Svensson, “Clock-offset cancellation methods for positioning in asynchronous sensor networks,” International Conference on Wireless Networks, Communications and Mobile Computing, vol.2, pp.981 - 986, June 2005. 16. N. Patwari, A.O. Hero III, M. Perkins, N.S. Correal, amd R.J. O'Dea, “Relative location estimation in wireless sensor networks,” IEEE Transactions on Signal Processing, vol.51, Issue 8, pp.2137 – 2148, Aug. 2003.
Electronics and Telecommunications Research Institute,161 Gajeong-dong, Yuseong-gu, Daejeon, 305-700, Korea 2 Department of Electronic Engineering, Information and Communications University, 119 Munjiro, Yuseong-gu, Daejeon, 305-732, Korea [email protected], [email protected]
Abstract. Time based TDOA localization systems require time or clock synchronization between receivers such as cellular base stations, satellites, and sensor nodes. Imperfection of time synchronization causes degradation in positioning accuracy. However if we know about its characteristics and how to estimate the clock offsets, the localization system can be properly calibrated to provide good quality of services. In this paper, hence, we present how to derive a localization error vector with independent clock offset, and illustrate its effect on the positioning errors, and then, provide a simple method of TDOA clock offset estimation from the observation of error vectors.
1 Introduction Many researches on source localizations have been addressed in many applications including sonar, radar, mobile communications, and, more recently, wireless sensor networks [1]-[6]. The approaches to the solution of the source location problem include iterative least-squares [7], [8] and maximum likelihood (ML) estimation [9]. Closed-form solutions have been given in [10]-[11], using “spherical intersection” and “spherical interpolation” methods, respectively. Fang gave an exact solution when the number of TDOA measurements is equal to the number of unknowns [12]. Meanwhile, Chan suggested a non-iterative general solution which is an approximation of maximum likelihood (ML) estimator of achieving optimum performance for arbitrarily placed sensors [5], [9]. But most of the previous works have focused on location estimation techniques. Major sources of errors in time-based localization detection schemes are the receiver time synchronization offset, the wireless multi-path fading channel, and the non-line-of-sight (NLOS) transmission. In theses, although NLOS problem is most significant, time synchronization offset (or clock offset) is primarily required to be resolved, which is our focus in this paper. Even a few nano-seconds of clock offset can make position error of up to several meters in triangulation, which is ferocious performance for high resolution positioning systems. However, it can be mitigated by system calibration by way of observing localization error vectors and estimating the time differences of clock offset. Literatures about estimating and canceling the clock offsets in wireless sensor networks and how to self-calibrate the unknown clock offsets was recently addressed [15]-[16]. Patwari suggested joint ML estimation of clock-offsets and node H.Y. Youn, M. Kim, and H. Morikawa (Eds.): UCS 2006, LNCS 4239, pp. 111 – 118, 2006. © Springer-Verlag Berlin Heidelberg 2006
112
N.-S. Seong and S.-O. Park
coordinates [15], and Rydström, more recently, two ML estimators of linear preprocessing operations to remove the effect of unknown clock-offsets from the estimation problem in [16]. In this paper, first, a matrix equation with independent parameters of clock offset is derived for the estimation, and the effects on the localization is illustrated, and then, finally, a simple method of how to extract the difference of clock offsets in threereceiver case is presented.
2 Effects of Clock Offset 2.1 Theory The localization error depends on the all of the clock offsets in the receivers. Most previous works and articles provide TDOA techniques on time difference, however, this paper presents simple equation with independent clock offset parameters, and then illustrate how they causes in localization. To obtain the matrix equation, we first investigate effect of clock offset in a single receiver, and then all of the contributions from three receivers are put together.
Fig. 1. Two-dimensional TDOA source localization geometry with three receivers
It is assumed that localization geometry is two-dimensional to simplify the problem, and that there are three receivers at locations (xi,yi), and that signals propagate in line-of-sight (LOS) without multi-paths as in Fig. 1. All the three receivers are to be synchronized to the reference clock. When the true source location is at P(x,y), the range Ri (i=1,2,3) between the source and the sensor i is given by Ri =
(xi − x )2 + (yi − y)2 = CTi
(1)
where C is the signal propagation speed and Ti is the signal flight time from the source to the receiver i according to the distance. Hence, the range difference Rij between the receivers i and j is
Clock Offsets in TDOA Localization
113
R ij = R i − R j = (x i − x) 2 + (yi − y)2 − (x j − x) 2 + (y j − y)2
(2)
so that Rij is a function of x and y R ij = f(x, y).
(3)
and ΔR
ij
=
∂f ∂x
Δx +
∂f ∂y
Δy
(4)
where ΔR denotes range estimation error and Δx , Δy are source location errors in x and y coordinates respectively. Substituting the partial derivatives in (4) and combing (2), (3), and (4) gives −1 [(xi − x)Δx + (yi − y)Δy] Ri
ΔR ij = +
(5)
1 [(x j − x) Δx + (y j − y)Δy]. Rj wZO'S 'P TDOA 323
TDOA 331
w' O'S 'P w OS P
TDOA131
TDOA 223
wYGO'S 'P
{kvhYZ
{kvhZX
wXGO'S 'P TDOA112
{kvhXY
2 TDOA12
Fig. 2. Shift of hyperbolic curves due to time synchronization offset at receiver 1
Since Rij can be obtained from time difference of two propagation delays of Ti, Tj, it has a relation of R ij = CTij = C(Ti − Tj ) , hence, when the offsets are small, the variation of range difference is given by
114
N.-S. Seong and S.-O. Park ΔR ij = C( ΔTi − ΔT j ) = C(ε i - ε j )
(6)
where εi, εj denote clock offset of receiver i, j from global or local reference clock. To find individual contribution to the position error by a single clock offset at a receiver, we need to first start with εi 0, εj=0 in (6) and then all the contributions will
≠
be summed up. When ε1 ≠ 0, ε 2 = ε 3 = 0 , then, in Fig. 2, the two location curves TDOA 12
, TDOA 31 move to TDOA 112 , TDOA131 respectively along the TDOA 23
curve. From the two new curves, a new intersection P1(x',y') is obtained. Combining (5) and (6) gives a set of linear equations about u1 = [Δx 1 Δy1 ] and ε1 as T
[
−(x1 − x) R1 [
+
−(x 2 − x) R2
(x 2 − x) R2 +
]Δx1 + [
(x 3 − x) R3
−(y1 − y)
]Δx 1 + [
R1
+
−(y 2 − y) R2
(y 2 − y) R2 +
]Δy1 = Cε1
(y 3 − y) R3
]Δy1 = 0
(7)
(8)
where suffix 1 in Δx , Δy means the error from time synchronization offset at receiver 1. Combining (7) and (8) gives a matrix equation with a single independent parameter ε1 B12 ⎤ ⎡ Δx 1 ⎤ ⎡ε 1 ⎤ ⎥⎢ ⎥ = C⎢ ⎥ B 23 ⎦ ⎣ Δy1 ⎦ ⎣0⎦
⎡ A12 ⎢ ⎣A 23
where Aij =
B ij =
− (x i − x) (x j − x) + Ri Rj
− (y i − y) (y j − y) + (i = 1,2,3). Ri Rj
(9)
From (9), we can obtain the effect of clock offset due to ε 1 on the location estimation as ⎡ A12 ⎡ Δx 1 ⎤ ⎢ ⎥ = C⎢ ⎣A 23 ⎣ Δy 1 ⎦
B12 ⎤ ⎥ B 23 ⎦
−1
⎡ε 1 ⎤ C ⎡ B 23 ⎤ ⎢ ⎥ε 1 ⎢ ⎥= ⎣ 0 ⎦ D1 ⎣ − A 23 ⎦
(10)
where D1 is defined as A12 B 23 − A 23 B12 . Similarly, the error vectors u 2 = [Δx 2
Δy 2 ]T and u 3 = [Δx 3
Δy 3 ]T due to ε 2 ,
ε 3 can be also obtained as in (11) and (12) ⎡ Δx 2 ⎤ C ⎡ B 31 ⎤ ⎢ ⎥ε 2 ⎢ ⎥= y Δ D ⎣ 2⎦ 2 ⎣ − A 31 ⎦
(11)
Clock Offsets in TDOA Localization ⎡Δx 3 ⎤ C ⎡ B12 ⎤ ⎢ ⎥= ⎢ ⎥ε 3 ⎣Δy 3 ⎦ D 3 ⎣ − A12 ⎦
115
(12)
where D 2 , D 3 are defined as A 23 B 31 − A 31 B 23 and A 31B12 − A12 B 31 respectively. Since each error vector u i (i=1,2,3) is independent, the integrated location error of the source can be expressed as sum of individual error vector, S b = u1 + u 2 + u 3 , and we obtain as in (14) ⎡ B 23 ⎢D Δx ⎡ ⎤ ⎢ 1 C = ⎢ ⎥ ⎢ − A 23 ⎣Δy ⎦ ⎢ ⎣ D1
B 31 D2 − A 31 D2
⎤ ⎥ ⎡ε1 ⎤ ⎥ ⎢ε ⎥. 2 − A12 ⎥ ⎢ ⎥ ⎥ ⎢⎣ε 3 ⎥⎦ D3 ⎦ B12 D3
(13)
From (13), the source position error vector S b is obtained in a simple matrix form of S b = CHTb , where S b = [Δx
Δy]T , Tb = [ε1 ε 2
⎡ B 23 ⎢D H=⎢ 1 ⎢ − A 23 ⎢ ⎣ D1
B 31 D2 − A 31 D2
ε 3 ]T , and
⎤ ⎥ ⎥. − A12 ⎥ ⎥ D3 ⎦ B12 D3
(14)
Hence, the final location error ΔL is given by ΔL = Δx 2 + Δy 2 . 2.2 Results and Discussions
Fig. 3 shows how time error of a receiver affects on the source localization. In this paper, we assume that three receivers are placed in triangular shape. We can find the fact that when a receiver is out of synchronization from a global or local reference clock but the others are not, then the localization error contours shift to the direction of the defective receiver. This means localization accuracy near the defective receiver is ironically better than other service area with perfect receivers. Direction of the error vectors reverses depending on the polarity of the clock offset [Fig. 3-(b), (c)]. However, Fig. 3-(d)-(i) gives us additional information about the movement of accuracy region. In Fig. 3-(d)-(f), error contours shift to a perfectly synchronized receiver when only one receiver is synchronized but the others have clock offsets of the same magnitudes. Fig. 3-(g)-(i), however, illustrates that the accuracy geometry moves to the opposite direction of Fig. 3-(d)-(f) when the polarities of the clock offsets of the two receivers are different each other. Hence, summing up, the information above puts the fact that localization accuracy of geometry moves the direction where a receiver has a different clock offset from the others with similar magnitudes whether or not the receiver is in a good synchronization with the reference clock.
116
N.-S. Seong and S.-O. Park
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Fig. 3. Localization errors due to clock offsets
3 Estimation of Clock Offsets In the previous section, we derived a TDOA error vector equation in a form of independent clock offset parameters as in (14). But, since matrix H is not a square matrix, we can not directly obtain Tb from (14). Using (5)-(6) again gives a simple form of solution in difference of time offsets as in (15) ⎡ −(x 1 − x) (x 2 − x) + ⎢ R1 R2 ⎢ ⎢ − (x 2 − x) + (x 3 − x) ⎢ R2 R3 ⎣
−(y1 − y) (y 2 − y) ⎤ + ⎡ε − ε 2 ⎤ R1 R 2 ⎥⎥ ⎡ Δx ⎤ = C⎢ 1 ⎥ − (y 2 − y) (y 3 − y) ⎥ ⎢⎣ Δy ⎥⎦ ⎣ε 2 − ε 3 ⎦ + ⎥ R2 R3 ⎦
(15)
Clock Offsets in TDOA Localization
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So we obtain ΔTb = C −1GS b where ⎡ε 1 − ε 2 ⎤ ΔTb = ⎢ ⎥ ⎣ε 2 − ε 3 ⎦
and ⎡ −(x 1 − x) (x 2 − x) + ⎢ R R2 1 G=⎢ ⎢ − (x 2 − x) + (x 3 − x) ⎢ R R3 2 ⎣
−(y1 − y) (y 2 − y) ⎤ + R1 R 2 ⎥⎥ . − (y 2 − y) (y 3 − y) ⎥ + R2 R 3 ⎥⎦
(16)
Measurement of S b in several positions gives more accurate estimation of difference of clock offsets Tˆ b for the receivers. Since the estimation matrix is given by E[ΔTb ] = C −1GE[S b ] , the solution for estimated difference of clock offsets is −1
ΔTˆ b = C GSˆ b .
(17)
4 Conclusion We derived a new solution of independent clock offset matrix in TDOA localizations to see how each offset of a receiver affects on the source localization, and how to measure the difference of clock offsets by measuring position error data. Hence localization systems can be monitored and calibrated by simply observing error vectors and thus compensating for them.
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