. RESEARCH PAPER .

SCIENCE CHINAInformation Sciences

doi: 10.1007/s11432-013-4800-2

c© Science China Press and Springer-Verlag Berlin Heidelberg 2014 info.scichina.com link.springer.com

Source localization and calibration using TDOA andFDOA measurements in the presence of sensor

location uncertainty

LI JinZhou, GUO FuCheng∗ & JIANG WenLi

School of Electronics Science and Engineering, National University of Defense Technology,Changsha 410073, China;

Received August 14, 2012; accepted November 20, 2012

Abstract Source localization accuracy is very sensitive to sensor location error. This paper performs analysis

and develops a solution for locating a moving source using time difference of arrival (TDOA) and frequency

difference of arrival (FDOA) measurements with the use of a calibration emitter. Using a Gaussian random

signal model, we first derive the Cramer-Rao lower bound (CRLB) for source location estimate in this scenario.

Then we analyze the differential calibration technique which is commonly used in Global Positioning System.

It is indicated that the differential calibration cannot attain the CRLB accuracy in most cases. A closed-form

solution is then proposed which takes a calibration emitter into account to reduce sensor location error. It is

shown analytically that under some mild approximations, our approach is able to reach the CRLB accuracy.

Numerical simulations are included to corroborate the theoretical developments.

Keywords source localization, calibration, time difference of arrival (TDOA), frequency difference of arrival

(FDOA), sensor location uncertainty, Cramer-Rao lower bound (CRLB)

Citation Li J Z, Guo F C, Jiang W L. Source localization and calibration using TDOA and FDOA measurements

in the presence of sensor location uncertainty. Sci China Inf Sci, 2014, 57, doi: 10.1007/s11432-013-4800-2

1 Introduction

Passive source localization has received significant attention due to its numerous applications in surveil-

lance and navigation [1,2]. There are many typical location parameters including time of arrival (TOA),

angle of arrival (AOA), time difference of arrival (TDOA). If there is relative motion between source and

sensors, frequency differences of arrival (FDOA) can be utilized to improve the estimate accuracy [3].

This paper focuses on both TDOA and FDOA localization for a single moving source.

It is difficult to find the source location due to the high nonlinearity of TDOA and FDOA equations.

Over the years, many approaches have been proposed for the problem, including the iterative Taylor-

series method [4], two-step least squares [5,6], linear-correction least squares [7], multidimensional scaling

[8] and other methods [9]. However, all these algorithms assume the sensor locations to be known exactly

which is unavailable in practice. To analyze the effect of sensor location errors, Ho et al. [10] performed a

∗Corresponding author (email: gfcly@21cn.com)

2 Li J Z, et al. Sci China Inf Sci

theoretical study on the amount of degradation of localization accuracy. They also proposed an algorithm

that takes the sensor error into account to reduce the estimation error. Although such a solution can

reach the CRLB performance, the localization accuracy is much worse than the one when the sensor

location error is absent.

Recently, Ho et al. [11] investigated the use of a calibration emitter to reduce the loss in TDOA

localization accuracy due to the uncertainty in sensor locations. This part of work was extended to a

more practical scenario where the locations of calibration emitters are also unknown [12]. Then they

generalized the results to a case where multiple calibration emitters are available.

This paper extends the work in [11] which considers the localization of a stationary source only. In

this paper, our major contributions include: (i) Derive the CRLB of localization scenario in this paper.

There are many CRLBs in [6,10,11]. We analyze the relations between these CRLBs and compare their

differences. (ii) Analyze the disadvantage of DC method commonly used in GPS system [13]. Ho et al.

[11] has also analyzed the disadvantage of DC method using only TDOA measurement. In this paper,

we take a similar approach to analyze the DC method while both TDOA and FDOA measurements are

available. (iii) By adding the FDOA measurements to the work in [11], we proposed an closed-form

solution for source position and velocity estimation with the use a calibration emitter. The paper is

closed by computer simulation to corroborate the theoretical development.

2 Location scenario

We shall consider the general case of a moving source localization using TDOAs and FDOAs from the

source and calibration emitter. There are M sensors with positions soi = [xoi , yoi , z

oi ]

T and velocities soi =

[xoi , yoi , z

oi ]

T. The position of unknown source is uo = [xo, yo, zo]T and the velocity is uo = [xo, yo, zo]T.

The calibration emitter is positioned at c = [xc, yc, zc]T and with velocity c = [xc, yc, zc]

T.

The TDOA and FDOA measurements between sensor pair i and 1 are

di1 =ri1vc

=1

vc(roi1 + ni1), i = 2, 3, . . . ,M, (1)

di1 =fovcri1 =

fovc

(roi1 + ni1), i = 2, 3, . . . ,M, (2)

where vc is the signal propagation speed, fo is signal carrier frequency, and roi1, r

oi1 are true range difference

and range rate difference between sensor pair i and 1

roi1 = roi − ro1 = ‖uo − soi ‖ − ‖uo − so1‖ , (3)

roi1 = roi − ro1 =(uo − soi )

T(uo − soi )

‖uo − soi ‖− (uo − so1)

T(uo − so1)

‖uo − so1‖. (4)

The collections of TDOAs and FDOAs form r = [r21, r31, . . . , rM1]T = ro+n, r = [r21, r31, . . . , rM1]

T =

ro + n, where n and n are zero mean Gaussian distributed with covariance matrices Qu and Qu.

The true sensor positions soi and velocities soi are not known, and only noisy versions of them, denoted

by si and si, are available.

si = soi +Δsi, (5)

si = soi +Δsi, (6)

where s = [s1; s2; . . . ; sM ], Δs = [Δs1; Δs2; . . . ; ΔsM ], s = [s1; s2; . . . ; sM ], Δs = [Δs1; Δs2; . . . ; ΔsM ].

Δs and Δs are zero mean Gaussian distributed with covariance matrices Qs and Qs.

The TDOAs and FDOAs from the calibration emitter are denoted by

di1,c =ri1,cvc

=1

vc(roi1,c + ni1,c), i = 2, 3, . . . ,M, (7)

di1,c =fovcri1,c =

fovc

(roi1,c + ni1,c), i = 2, 3, . . . ,M. (8)

Li J Z, et al. Sci China Inf Sci 3

where roi1,c and roi1,c are true range difference and range rate difference.

roi1,c = roi,c − ro1,c = ‖c− soi ‖ − ‖c− so1‖ , (9)

roi1,c = roi,c − ro1,c =(c− soi )

T(c− soi )

‖c− soi ‖− (c− so1)

T(c− so1)

‖c− so1‖. (10)

Let rc = [r21,c, r31,c, . . . , rM1,c]T = ro

c + nc and rc = [r21,c, r31,c, . . . , rM1,c]T = ro

c + nc, where nc and

nc are zero mean Gaussian distributed with covariance matrix Qc and Qc.

(1), (3), (5), (7) and (9) are similar to those in [11]. We present them here for the integrality of this

paper.

3 CRLB

The CRLB is the lowest possible variance that an unbiased estimator can achieve. This section first

derives the CRLB of a moving source localization based on a calibration emitter and then compares it

with the CRLB when calibration emitter is absent. First, let us define α = [rT, rT]T, β = [sT, sT]T and

γ = [rTc , r

Tc ]

T. θ = [uT, uT]T is the unknown parameter. Hence, the logarithm of the probability density

function for the data vector is

ln f(m;ϕ) = ln f(α;ϕ) + ln f(β;ϕ) + ln f(γ;ϕ)

= K − 1

2(α − αo)TQ−1

α (α−αo)− 1

2(β − βo)TQ−1

β (β − βo)− 1

2(γ − γo)TQ−1

γ (γ − γo), (11)

where K is a constant that does not depend on the unknowns, Qα = diag{Qu, Qu}, Qβ = {Qs, Qs} and

Qγ = diag{Qc, Qc}.The CRLB of ϕ is equal to [14]

CRLB(ϕ) = −E

[∂ ln f(m;ϕ)

∂ϕ∂ϕT

]−1

=

[X Y

Y T Z

]−1

. (12)

CRLB(ϕ) is a 6 + 6M matrix, where

X = −E

[∂2 ln p

∂θo∂θoT

]=

(∂αo

∂θo

)T

Q−1α

(∂αo

∂θo

), Y = −E

[∂2 ln p

∂θo∂βoT

]=

(∂αo

∂θo

)T

Q−1α

(∂αo

∂βo

),

Z = −E

[∂2 ln p

∂βo∂βoT

]=

(∂αo

∂βo

)T

Q−1α

(∂αo

∂βo

)+Q−1

β +

(∂γo

∂βo

)T

Q−1γ

(∂γo

∂βo

). (13)

The partial derivatives are given in Appendix B. Applying the partitioned matrix inverse formula [15]

into (12) yields

CRLB(θ) = X−1 +X−1Y (Z − Y TX−1Y )−1Y TX−1. (14)

Note that X−1 is the CRLB of the source location vector θ when sensor locations are accurate [6]. If

sensor locations are inaccurate and without the use of the calibration emitter, this is the case in [10], the

CRLB will be

CRLB(θ)o = X−1 +X−1Y (Z − Y TX−1Y )−1Y TX−1, (15)

where

Z =

(∂αo

∂βo

)T

Q−1α

(∂αo

∂βo

)+Q−1

β . (16)

Comparing (14) and (15), the difference lies in Z and Z. We express Z in (13) as Z = Z+Z. Applying

the matrix inversion lemma [15] to ((Z − Y TX−1Y ) + Z)−1, and substituting it back into (14) yields

CRLB(θ)o − CRLB(θ) = X−1Y ΓY TX−1, (17)

4 Li J Z, et al. Sci China Inf Sci

Table 1 True position (in meters) and velocity (in meter/second) of sensors

Sensor No. xi yi zi xi yi zi

1 300 100 150 30 −20 20

2 400 150 100 −30 10 20

3 300 500 200 10 −20 10

4 350 200 100 10 20 30

5 −100 −100 −100 −20 10 10

6 200 −300 −200 20 −10 10

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20

30

40

50

60

70

10lg

(pos

ition

CR

LB

(m

))

−80 −70 −60 −50 −40 −30 −20 −10 00

20

40

60

80

10lg

(vel

ocity

CR

LB

(m

/s))

CRLB1CRLB2CRLB3CRLB4

10lg(σs2 (m2))

10lg(σs2 (m2))

CRLB1CRLB2CRLB3CRLB4

(a)

(b)

Figure 1 Comparison of the CRLBs with and without a calibration emitter for an unknown source. The line with CRLB1

is X−1 in (14), representing source localization without sensor location errors. The line named CRLB2 is the CRLB in (15),

representing source localization with sensor location errors and without using calibration emitter. CRLB3 is the CRLB

in (14) using calibration emitter 1. CRLB4 is the CRLB in (14) using calibration emitter 2. (a) Position estimation; (b)

velocity estimation.

where Γ = A−1P (I + PTA−1P )−1PTA−1, A = Z − Y TX−1Y , P =(

∂γo

∂βo

)Lc. Lc is the Cholesky

decomposition of Q−1γ , Q−1

γ = LcLTc . The right-hand side of (17) is a positive semidefinite matrix which

has been verified in [11]. Thus, having a calibration emitter will not degrade the best localization accuracy

of the unknown source location.

The derivations of (14) and (17) are similar to the results in [11]. But we take the FDOA measurements

into account and the source is also moving in this paper.

We consider the same localization geometry used in [10]. The sensor true positions and velocities

are shown in Table 1. The unknown source is located at [2000, 2500, 3000]T m with a velocity of

[−20, 15, 40]T m/s. There are two calibration emitters. Calibration 1 is located at [1500, 1550, 1500]T m

with a velocity of [−22, 18, 42]T m/s. Calibration 2 is located at [−600,−650,−550]T m with a velocity

of [−20, 20, 40]T m/s. Q is an (M − 1) × (M − 1) matrix with diagonal elements equal to 1 and other

elements equal to 0.5. Define covariance matrices Qr = Q× 10−4, Qc = Q× 10−4, Qr = Q× 10−5 and

Qc = Q× 10−5. Let Rs = σ2s diag{1, 1, 1, 2, 2, 2, 3, 3, 3, 10, 10, 10, 20, 20, 20, 30, 30, 30} m2, Rs = 0.5Rs.

Figure 1 compares the CRLBs of different localization scenarios. The position and velocity of the first

calibration emitter are close to the source, while the position difference of calibration 2 and source is

great. As we can observe from the figure, when the position difference of calibration emitter and source

is small, the estimate accuracy of position and velocity can both be improved significantly. For example,

Li J Z, et al. Sci China Inf Sci 5

at the sensor position error power of σ2s = 10−2, the increase in CRLB of using calibration 2 for position

u is 8.61 dB and that for velocity u is 5.90 dB, while the increase in CRLB of using calibration 1 for

position u is 17.10 dB and that for velocity u is 17.66 dB.

4 Analysis of difference calibration

This section examines the MSE of DC approach for source localization using TDOAs and FDOAs. DC

is a simple approach to exploiting the measurements from calibration emitter to improve the localization

accuracy. DC subtracts the calibration TDOAs and FDOAs from the corresponding TDOAs and FDOAs

of the unknown source, and then uses the differences to estimate the source position and velocity.

The derivation begins by defining quantities parameterized on the source location vector

gi1(θo) = ‖uo − si‖ − ‖uo − s1‖ ,

gi1,c(c) = ‖c− si‖ − ‖c − s1‖ ,

gi1(θo) =

(uo − si)T(uo − si)

‖uo − si‖ − (uo − s1)T(uo − s1)

‖uo − s1‖ ,

gi1,c(c) =(c− si)

T(c− si)

‖c− si‖ − (c − s1)T(c− s1)

‖c − s1‖ . (18)

Submitting si = soi +Δsi into ‖uo − si‖ and expanding through Taylor-series yields

‖uo − si‖ � roi −(uo − soi )

TΔsiroi

, (19)

where the second and higher order are ignored. Hence

ri1 − gi1(θo) = ni1 + aT

i Δsi − aT1 Δs1. (20)

Similarly, we have

ri1,c − gi1,c(c) = ni1,c + aTi,cΔsi − aT

1,cΔs1, (21)

ri1 − gi1(θo) = ni1 + aT

i Δsi + bTi Δsi − aT1 Δs1 − bT1 Δs1, (22)

ri1,c − gi1,c(c) = ni1,c + aTi,cΔsi + bTi,cΔsi − aT

1,cΔs1 − bT1,cΔs1, (23)

where ai, bi, ai,c and bi,c are defined in Appendix B. The DC equations are

{ri1 − ri1,c = roi1 − roi1,c + ni1 − ni1,c,

ri1 − ri1,c = roi1 − roi1,c + ni1 − ni1,c,i = 2, . . . ,M. (24)

Substitution of (20)–(23) into (24) yields

eiΔ= ri1 − ri1,c − gi1(θ

o) + gi1,c(c) = ni1 − ni1,c +(aTi − aT

i,c

)Δsi −

(aT1 − aT

1,c

)Δs1, (25)

eiΔ= ri1 − ri1,c − gi1(θ

o) + gi1,c(c)

= ni1 − ni1,c +(aTi − aT

i,c

)Δsi +

(bTi − bTi,c

)Δsi −

(aT1 − aT

1,c

)Δs1 −

(bT1 − bT1,c

)Δs1. (26)

If the position of the calibration is close to the source,∥∥aT

i − aTi,c

∥∥ will be very small. Hence, the

contribution to TDOAs from sensor error could be reduced using DC, as has been pointed out in [11].

Nevertheless, (26) indicates that the contribution to FDOAs is relative to both the position difference and

velocity difference between calibration emitter and unknown source. As the range of velocity difference

is smaller than that of position difference, the position difference can significantly affect the equivalent

FDOA in (26), so it is not sensitive to the velocity difference between calibration emitter and unknown

source. In a word, for the case of a moving source localization using TDOAs and FDOAs, the position

6 Li J Z, et al. Sci China Inf Sci

CRLB without a calibration emitter

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20

30

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50

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lg(p

ositi

on C

RL

B (

m))

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40

60

80

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(vel

ocity

CR

LB

(m

/s))

10lg(σs2 (m2))

10lg(σs2 (m2))

CRLB with a calibration emitterTheoretical MSE of DC

CRLB without a calibration emitterCRLB with a calibration emitterTheoretical MSE of DC

(a)

(b)

Figure 2 Comparison of the CRLBs with the theoretical MSE of using DC for the first calibration emitter. (a) Position

estimation; (b) velocity estimation.

CRLB without a calibration emitter

−80 −70 −60 −50 −40 −30 −20 −10 0

10lg

(pos

ition

CR

LB

(m

))

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60

80

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(vel

ocity

CR

LB

(m

/s))

10lg(σs2 (m2))

10lg(σs2 (m2))

CRLB with a calibration emitterTheoretical MSE of DC

CRLB without a calibration emitterCRLB with a calibration emitterTheoretical MSE of DC

0

20

40

60

80

(a)

(b)

Figure 3 Comparison of the CRLBs with the theoretical MSE of using DC for the second calibration emitter. (a) Position

estimation; (b) velocity estimation.

difference between calibration emitter and unknown source is the primary factor which degrades the

location estimate accuracy with DC method. Another drawback of DC is that the effective noise power

due to TDOAs and FDOAs is the sum from the unknown source and the calibration emitter.

The mean squares error (MSE) of source localization using DC method is derived as (49) in Appendix C.

cov(ϕ)D = (HoTD WDHo

D)−1. The performance degradation of using DC with respect to the CRLB is (51)

in Appendix C. cov(ϕ)D = CRLB(ϕ)+CRLB(ϕ)HoTΥT(I −ΥHoCRLB(ϕ)HoTΥT

)−1ΥHoCRLB(ϕ).

The localization geometry is the same as the one used in generating Figure 1. Figure 2 gives the results

using the first calibration emitter whose position and velocity are both close to the unknown source. The

Li J Z, et al. Sci China Inf Sci 7

accuracy of using DC method is comparable with the CRLB of using the calibration emitter. Figure 3

gives the simulation of the second calibration emitter whose position is far away from the unknown source.

It is clear from Figure 3 that the accuracy of DC method is always worse than the CRLB of the method

without using a calibration emitter.

Furthermore, when the sensor location error is small relative to the TDOA and FDOA measurement

noise (σ2s = 10−8), DC is always 3 dB worse than the CRLB for both position and velocity estimates in

Figure 2. This is because from (25) and (26), the contribution of sensor position error Δsi and velocity

error Δsi to the equation error ei and ei is negligible when the sensor position and velocity are accurate.

The doubling of TDOAs noise and FDOAs noise dominate in the equation error ei and ei, which degrade

the estimate accuracy.

5 New calibration algorithm

This paper extends the work in [11] to a moving source localization situation using TDOA and FDOA

measurements. The proposed solution has two stages. The first stage, named calibration stage, uses

the TDOAs and FDOAs from calibration emitter to improve sensor positions and velocities. The second

stage is localization stage which utilizes the improved sensor locations and TDOA/FDOA measurements

from the unknown source to estimate the source position and velocity. The last stage is the same as that

developed in [10], so we do not present it here.

Since the position and velocity of the calibration emitter are known, we can utilize the TDOAs and

FDOAs to estimate the sensor positions and velocities. Rewrite (20) and (22) as

ri1,c − gi1,c(c) = ni1,c + aTi,cΔsi − aT

1,cΔs1,

ri1,c − gi1,c(c) = ni1,c + aTi,cΔsi + bTi,cΔsi − aT

1,cΔs1 − bT1,cΔs1, (27)

where the defining of gi1,c(c) and gi1,c(c) is in (18). Expressing (27) in a matrix form, we have

hc = Gcψ + n, (28)

where hc is a 2(M − 1)× 1 matrix, and the elements of the matrix are ri1,c − gi1,c(c) and ri1,c − gi1,c(c).

ψ = [Δs; Δs], n = [nc; nc].

Gc =

[D1 O

D2 D1

]. (29)

The unknown parameter ψ is a zero mean and Gaussian distributed random vector with covariance

matrix Qβ . By the Bayesian Gauss-Markov theorem [14], the linear minimum MSE (LMMSE) estimator

of ψ is

ψ =(Q−1

β +GTc QγGc

)−1GT

c Qγhc. (30)

The best localization accuracy of ψ is

cov(ψ − ψ) =(Q−1

β +GTc QγGc

)−1. (31)

With the estimated sensor position and velocity errors given in (30), we obtain the improved sensor

positions and velocities as

βc = β − ψ = βo + ψ − ψ. (32)

The covariance of βc is given in (31). Hence, comparing with original covariance matrix yields

cov(β)− cov(βc) = Qβ − (Q−1

β +GTc QγGc

)−1. (33)

Obviously, the matrix cov(β)−cov(βc) is positive semidefinite with the use of the matrix inversion lemma

[15]. Hence, with the calibration TDOAs and FDOAs, we have updated sensor positions and velocities at

least as good as, if not better than, the original one to obtain the unknown source location. This result

is consistent with that in [11].

8 Li J Z, et al. Sci China Inf Sci

In the localization stage, we first utilize the improved sensor positions and velocities to replace the

original ones. Then, we can use the same algorithm in [10] to obtain a closed-form solution to the source

location. The approach can be found in [10].

We now analyze the theoretical covariance of the proposed solution, and compare it with the CRLB.

The accuracy of the two-step least squares method [10] has been shown theoretically to reach the CRLB

under some mild approximations.

MSE(θ)o = X−1 +X−1Y(Z − Y TX−1Y

)−1Y TX−1, (34)

where X and Y are defined in (13), and Z is defined in (16).

Hence, if we first utilize a calibration emitter to improve the sensor locations, and then use the two-step

least squares method in [10], the total localization accuracy will be

MSE(θ) = X−1 +X−1Y(�

Z − Y TX−1Y)−1

Y TX−1, (35)

where�

Z =

(∂αo

∂βo

)T

Q−1α

(∂αo

∂βo

)+(cov(ψ − ψ)

)−1. (36)

Putting (31) into (36) yields�

Z = (∂αo/∂βo)TQ−1α (∂αo/∂βo) +Q−1

β +GTc QγGc.

With the partial derivatives (39), (40) and (41) given in Appendix B, we can show thatGc = −∂γo/∂βo.

So�

Z is equivalent to Z in (13), and (35) is equal to (14). So the proposed algorithm can reach the CRLB

accuracy under some mild approximations.

6 Simulations

The localization geometry is the same as the one used in generating Figure 2. Figure 4 and Figure 5 are

the simulation results of the two calibration emitters respectively. Every figure contains three localization

methods. The first method is the localization accuracy without using calibration emitter [10] (denoted

by CRLB1 and MSE1 in figures). The second method is the localization accuracy utilizing the proposed

method to improve the sensor locations (denoted by CRLB2 and MSE2). The last method is the estimate

accuracy using DC algorithm to improve the sensor locations (denoted by CRLB3 and MSE3). Every

figure plots the simulation results and the theoretical accuracy for each method. The results are the

average of 5000 independent runs.

Figure 4 gives the results using the first calibration emitter whose position and velocity are both close

to the unknown source. It is obvious that the proposed algorithm reaches the CRLB accuracy for both

position and velocity estimate. The simulation results of DC also reach the theoretical best MSE very

well. The performance of the proposed method and the DC method is comparable when the sensor

location error is large (σ2s > 10−5). However, when the sensor location error is small (σ2

s < 10−7), the

proposed algebraic solution can offer 3 dB improvement in localization accuracy over DC. The reason has

been analyzed in (26). In DC method, the effective noise power due to TDOAs and FDOAs is the sum

from the unknown source and the calibration emitter. So the performance cannot attend the CRLB when

sensor errors are small. Moreover, both the DC and the proposed method are superior to the algorithm

without using calibration emitter when sensor noise is not negligible.

Figure 5 gives the simulation of the second calibration emitter whose position is far away from the

unknown source. It is clear from Figure 5 that simulation results of DC method are always worse than the

CRLB without using a calibration emitter. The proposed method outperforms both DC and the algorithm

without using a calibration emitter. The reasons have been analyzed in (26) and (34). In DC method, the

position difference between calibration emitter and unknown source is the primary factor which degrades

the location estimate accuracy. However, the proposed method can attend the CRLB performance in

this localization scenario. For the position estimate, the improvements of the proposed method are 14 dB

compared with the DC and 8.2 dB with the algorithm without using a calibration emitter when the

sensor position error is σ2s = 10−3. The improvements are 11.5 dB and 5.7 dB respectively for velocity

estimate.

Li J Z, et al. Sci China Inf Sci 9

−80 −70 −60 −50 −40 −30 −20 −10 0

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CRLB1CRLB2CRLB3MSE1MSE2MSE3

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lg(p

ositi

on M

SE (

m))

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(vel

ocity

MSE

(m

/s))

10lg(σs2 (m2))

CRLB1CRLB2CRLB3MSE1MSE2MSE3

(a)

(b)

Figure 4 Comparison of the localization accuracy of the proposed method and the DC solution using the first calibration

emitter. CRLB1 and MSE1 are theory accuracy and simulation accuracy respectively for source localization without using

calibration emitter. CRLB2 and MSE2 are theory accuracy and simulation accuracy respectively for source localization using

proposed method with calibration emitter. CRLB3 and MSE3 are theory accuracy and simulation accuracy respectively

for source localization using DC method with calibration emitter. (a) Position estimation; (b) velocity estimation.

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))10

lg(v

eloc

ity M

SE (

m/s

))

10lg(σs2 (m2))

10lg(σs2 (m2))

CRLB1CRLB2CRLB3MSE1MSE2MSE3

CRLB1CRLB2CRLB3MSE1MSE2MSE3

(a)

(b)

Figure 5 Comparison of the localization accuracy of the proposed method and the DC solution using the second calibration

emitter. CRLB1 and MSE1 are theory accuracy and simulation accuracy respectively for source localization without using

calibration emitter. CRLB2 and MSE2 are theory accuracy and simulation accuracy respectively for source localization using

proposed method with calibration emitter. CRLB3 and MSE3 are theory accuracy and simulation accuracy respectively

for source localization using DC method with calibration emitter. (a) Position estimation; (b) velocity estimation.

7 Conclusion

Source localization using TDOA and FDOA measurements requires very precise knowledge of sensor

locations. A small error in the sensor locations can lead to a significant decrease in source localization

10 Li J Z, et al. Sci China Inf Sci

accuracy. This paper investigated the use of a calibration emitter to improve the TDOA/FDOA-based

localization accuracy of an unknown moving source in the presence of random sensor location errors. We

first deduced the CRLB of a moving source location using a calibration emitter, and compared it with

the CRLB without using a calibration emitter. Next, we showed that the commonly used differential

calibration technique is not able to reach the CRLB accuracy in general. To overcome the drawback,

we developed a new calibration algorithm. The solution accuracy is shown theoretically and verified by

simulations to reach the CRLB under some mild approximations.

Acknowledgements

The work was supported in part by National High-tech R&D Program of China (863 Program) (Grant No.

2011AA7072043), National Defense key Laboratory Foundation of China (Grant No. 9140C860304), and Fund of

Innovation, Graduate School of National University of Defense Technology (Grant No. B120406). The authors

gratefully acknowledge financial support from China Scholarship Council.

References

1 Cater G C. Time delay estimation for passive sonar signal processing. IEEE Trans Acoust Speech Signal Process,

1981, 29: 462–470

2 Weinstein E. Optimal source localization and tracking from passive array measurements. IEEE Trans Acoust Speech

Signal Process, 1982, 30: 69–76

3 Li J Z, Guo F C, Jiang W L. A linear-correction least-squares approach for geolocation using FDOA measurements

only. Chin J Aeronaut, 2012, 25: 709–714

4 Foy W H. Position-location solution by Taylor-series estimation. IEEE Trans Aerosp Electron Syst, 1976, 12: 187–194

5 Chan Y T, Ho K C. A simple and efficient estimator for hyperbolic location. IEEE Trans Signal Process, 1994, 42:

1905–1915

6 Ho K C, Xu W. An accurate algebraic solution for moving source location using TDOA and FDOA measurements.

IEEE Trans Signal Process, 2004, 52: 2453–2463

7 Huang Y T, Benesty J, Elko G W, et al. Real-time passive source localization: a practical linear-correction least-squares

approach. IEEE Trans Speech Audio Process, 2001, 9: 943–956

8 Wei H W, Peng R, Wan Q, et al. Multidimensional scaling analysis for passive moving target localization with TDOA

and FDOA measurements. IEEE Trans Signal Process, 2010, 58: 1677–1688

9 Musicki D, Kaune R, Koch W, et al. Mobile emitter geolocation and tracking using TDOA and FDOA measurements.

IEEE Trans Signal Process, 2010, 58: 1863–1874

10 Ho K C, Lu X N, Kovavisaruch L, et al. Source localization using TDOA and FDOA measurements in the presence of

receiver location errors: analysis and solution. IEEE Trans Signal Process, 2007, 55: 684–696

11 Ho K C, Yang L. On the use of a calibration emitter for source localization in the presence of sensor position uncertainty.

IEEE Trans Signal Process, 2008, 56: 5758–5772

12 Yang L, Ho K C. Alleviating sensor position error in source localization using calibration emitters at inaccurate

locations. IEEE Trans Signal Process, 2010, 58: 67–83

13 Pattison T, Chou S I. Sensitivity analysis of dual-satellite geolocation. IEEE Trans Aerosp Electron Syst, 2000, 36:

56–71

14 Kay S M. Fundamentals of Statistical Signal Process, Estimation Theory. New Jersey: Prentice-Hall, 1993

15 Scharf L L. Statistical Signal Process: Detection, Estimation and Time Series Analysis. Addison-Wesley, 1991

Appendix A Summary of symbols and notations

The symbols and notations used are summarized in the following table.

Symbol Explanation

M Number of sensors

si Noisy position of ith sensor,soi = [xoi , yoi , z

oi ]

T

si Noisy velocity of ith sensor,soi = [xoi , yoi , z

oi ]

T

uo Position of the unknown source, uo = [xo, yo, zo]T

Li J Z, et al. Sci China Inf Sci 11

uo Position of the unknown source, uo = [xo, yo, zo]T

c Position of the known calibration emitter, c = [xc, yc, zc]T

c Velocity of the known calibration emitter, c = [xc, yc, zc]T

s Sensor position vector, s = [s1; s2; . . . ; sM ]

s Sensor velocity vector, s = [s1; s2; . . . ; sM ]

Δs Sensor position error vector, Δs = [Δs1;Δs2; . . . ; ΔsM ]

Δs Sensor velocity error vector, Δs = [Δs1;Δs2; . . . ; ΔsM ]

di1 TDOA measurement between the unknown source and sensor i and 1

di1 FDOA measurement between the unknown source and sensor i and 1

di1,c TDOA measurement between the calibration emitter and sensor i and 1

di1,c FDOA measurement between the calibration emitter and sensor i and 1

n TDOA measurement noise vector between the unknown source and sensors

n FDOA measurement noise vector between the unknown source and sensors

nc TDOA measurement noise vector between the calibration emitter and sensors

nc FDOA measurement noise vector between the calibration emitter and sensors

Qs Covariance matrix of s

Qs Covariance matrix of s

Qr Covariance matrix of r

Qr Covariance matrix of r

Qc Covariance matrix of c

Qc Covariance matrix of c

Qα Covariance matrix of α, α = [rT, rT]T

Qβ Covariance matrix of β, β = [sT, sT]T

Qγ Covariance matrix of γ, γ = [rTc , r

Tc ]T

m Vector containing all measurements, m = [αT,βT,γT]T

CRLB(θ) CRLB of position and velocity with a calibration emitter

CRLB(θ)o CRLB of position and velocity without a calibration emitter

X−1 CRLB of position and velocity without sensor location error

V Differential matrix of DC

Ho Gradient matrix of the Taylor-series expansion of m

ψ Sensor position and velocity error estimate

Appendix B Definition of the partial derivatives

The CRLB formula with the use of a calibration emitter is given in (14). For notation simplicity, we define

(M − 1)× 3M matrices C1, C2, D1 and D2 below, where the ith rows of C1, C2, D1 and D2 are⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

C1(i, :) =[−aT

1 ,0T3(i−1)×1,a

Ti+1,0

T3(M−i−1)×1

],

C2(i, :) =[−bT1 ,0

T3(i−1)×1, b

Ti+1,0

T3(M−i−1)×1

],

D1(i, :) =[−aT

1,c,0T3(i−1)×1,a

Ti+1,c,0

T3(M−i−1)×1

],

D2(i, :) =[−bT1,c,0

T3(i−1)×1, b

Ti+1,c,0

T3(M−i−1)×1

],

(B1)

⎧⎪⎪⎨

⎪⎪⎩

ai =(uo − so

i )

roi, bi =

(uo − soi )

roi− roi

ro2i(uo − so

i ),

ai,c =(c− so

i )

roi,c, bi,c =

(c− soi )

roi,c− roi,c

ro2i,c(c− so

i ).

(B2)

Let E3M×3 = [I3×3, . . . , I3×3]T. Then the partial derivatives of α, β and γ with respect to the unknown

parameters are

(∂αo

∂θo

)

=

[C1E O(M−1)×3

C2E C1E

]

, (B3)

12 Li J Z, et al. Sci China Inf Sci

(∂αo

∂βo

)

=

[−C1 O(M−1)×3M

−C2 −C1

]

, (B4)

(∂γo

∂βo

)

=

[−D1 O(M−1)×3M

−D2 −D1

]

. (B5)

Appendix C The MSE of source localization using DC method

We now derive the mean squares error (MSE) of source localization using DC method. Let m = [αT,βT,γT]T

be a vector containing all measurements, and let ϕo = [θoT,βoT]T be the unknown parameter vector. Using the

Taylor-series expansion of m at ϕo, we have

m = mo +Ho(ϕ− ϕo), (C1)

where mo is the true value of m, and Ho is the gradient matrix given by

Ho =

⎡

⎢⎢⎢⎣

(∂αo

∂θo

)

2(M−1)×6

(∂αo

∂βo

)

2(M−1)×6M

O6M×6 I6M×6M

O2(M−1)×6

(∂γo

∂βo

)

2(M−1)×6M

⎤

⎥⎥⎥⎦. (C2)

The partial derivatives are given in Appendix B.

Let V be the (8M − 2)× (10M − 4) matrix.

V =

⎡

⎢⎣

I(M−1)×(M−1) O(M−1)×(M−1) O(M−1)×6M −I(M−1)×(M−1) O(M−1)×(M−1)

O6M×(M−1) O6M×(M−1) I6M×6M O6M×(M−1) O6M×(M−1)

O(M−1)×(M−1) I(M−1)×(M−1) O(M−1)×6M O(M−1)×(M−1) −I(M−1)×(M−1)

⎤

⎥⎦ . (C3)

Then the DC method can be simplified into

mD = V m = [r − rc;β; r − rc] . (C4)

Let moD = V mo and Ho

D = V Ho. Then, premultiplying (C1) by V yields

mD = moD +Ho

D(ϕ− ϕo). (C5)

The best linear unbiased estimator (BLUE) [14] of ϕo from linear model (C5) is

ϕ = ϕo + (HoTD WDHo

D)−1HoTD WD(mD −mo

D), (C6)

where WD is the inverse of the covariance matrix of the measurements of (C5)

W−1D = E

[(mD −mo

D)(mD −moD)T

]= V

⎡

⎢⎣

Qα O O

O Qβ O

O O Qγ

⎤

⎥⎦V T. (C7)

The MSE for DC method is

cov(ϕ)D = (HoTD WDHo

D)−1. (C8)

We now compare cov(ϕ)D and CRLB in (12), using a similar analysis to that in [11] and get

cov(ϕ)−1D = CRLB(ϕ)−1 −HoTΥTΥHo, (C9)

where Υ = [LTQ−1α ,O,LTQ−1

γ ]. L is the Cholesky decomposition of (Q−1α +Q−1

γ )−1, i.e., (Q−1α +Q−1

γ )−1=LLT.

Applying the matrix inversion lemma [15] to (C9) yields

cov(ϕ)D = CRLB(ϕ) + CRLB(ϕ)HoTΥT(I −ΥHoCRLB(ϕ)HoTΥT

)−1

ΥHoCRLB(ϕ). (C10)

The last term in (C10) is positive semidefinite from its symmetric structure, which represents performance

degradation of using DC method with respect to the CRLB.