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详细说明：chan算法CHAN AND HO: A SIMPLE AND EFFICIENT ESTIMATOR FOR HYPERBOLIC LOCATION SYSTEM
y=yo. In the next iteration, Io and yo are then set to o+A where
and yo +Ay. The whole process is repeated again until A and
Ay are sufficiently small. This method has the difficulties of
T21-K2+K1
requiring a close enough starting point and large computations.
h=1一K+K
Moreover, convergence is not guaranteed
KM+Kl
An alternative [6]-19] is to first transform( 6)into another
M,1
set of equations. From (6), T2 =(ri, I +r1)2 so that(5)
92172,1
can
G
买3,133,1T3,1
be rewritten as
(11)
21+2;1+2=K12-2x;x-2y+r2+y2.(8)
1yM,1
when (4)is used to express Ti, l as ri, 1
十c;,1 and noting
Subtracting (5)at i= 1 from(8), we obtain
from(6)that rp=ri, +rf, w is found to
r21+2r
2x;13-2y,1y+Ha-k
y=cBn+0.5cn⊙
B=diag{n,r3,…,r}
(12)
The symbols fi, I and yi, 1 stand for ai -f1 and yi-yi The symbol O represents the Schur product(clement-by
respectively. Note that(9)is a sct of lincar equations with element product). The TDOA found by generalized cross-
unknowns a, y and T1. To solve for r and g, [6]eliminates r'l correlation with Gaussian data is asymptotically normally
from(9)and produces(M-2) linear equations in z and y The distributed when the signal-to-noise ratio(SNR)is high [1]
source position is then computed by Ls. On the other hand, It follows that the noise vector n in Hahn and Tretter's 12
[8] first solves a and y in terms of r1. The intermediate result estimator is also asymptotically normal. The covariance matrix
is inserted back to(9)to generate equations in the unknown r1 of ab can therefore be evaluated. In practice, the condition
only. Substituting the computed ri value that minimizes the Ls CI, I ii is usually satisfied. When ignoring the second
term on the right of (12), yb becomes a Gaussian random vector
This is termed the spherical-interpolation (SI) method [8]. with covariance matrix given by
The preceding two solutions are shown to be mathematically
equivalent 6]. They are, however, not optimum and the
亚一Eψy2]=c2BQB
(13)
weighting matrices required in Ls are not easy to determine. a
new estimator for position fixing which is capable of achieving
The elements of za are related by (5), which means that (11)
optimum performance is next given
is still a set of nonlinear equations in two variables a and y.
The approach to solve the nonlinear problem is to first assume
that there is no relationship among C, y and T1. They can then
A. Arbitrary array
be solved by LS. The final solution is obtained by imposing
1)Three Sensors(M=3): With three sensors, I and y can the known relationship(5)to the computed result via another
be solved in terms of Ti frum (9). That is
s computation. This two step procedure is an approximation
of a true ml estimator for emitter location. B
y/2,1
the elenents of za independent, the ml estimate of za is
T31y3,1
Za =arg mint(h-Gaza)- (h-Gaza)1
+K1
(GIYGaGay-h
2 r3. -K3+
which is also readily recognized as the generalized ls solution
Inserting this intermediate result into(S)at i=1 gives a of( 1).y is not known in practice as B contains the true
quadratic in r1. Substitution of the positive root back into(10)distances between source and receivers. Further approximation
produces the solution. On some occasions, therc may be two is necessary in order to make the problem solvable. When the
positive roots that produce two different answers The solution source is far from array, each r i is close to ru so that B A r I,
ambiguity can be resolved by restricting the transmitter to lie where r designates the range and I is an identity matrix of
within the region of interest. This answer is equivalent to the size M-1. Since scaling of y does not affect the answer,an
one in [7
approximation of (14a)is
za≈(GaQ-aa)cQ1h
(14b)
equations in (9)will not meet at the same point and the to obtain an initial solution to estimate B. The final answer is
proper answer is the(a, y )that best fit these equations. Let then computed from(14a). Although(14a)can be iterated to
za =[zd, ri] be the unknown vector, where zp
,y. provide an even better answer, simulations show that applying
With TDOA noise, the error vector derived from(9)is
(14a)once is sufficient to give an accurate result.
The covariance matrix of za is obtained by evaluating the
nb=h-G,zo
expectations of za and zu zd from (14a). The calculation
908
IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL 42. NO. 8, AUGUST 1994
is quite involved because the matrix Ga contains random The matrix y' is not known since it contains the true values
quantities Ti. 1. We compute the covariance matrix by using Nevertheless, B can be approximated by using the values
the perturbation approach. In the presence of noise, ri, 1= in za, Go in(17)approximated by Ga, and B in(13
1+ CTi, 1. Ga and h can be expressed as Ga=Ga +AGa approximated by the values computed from(14b). If the source
andh=h°+△h. Since goz=h°,(11) implies that
is distant, then cov(za)N cr(Gu Q ga and (22a)
reduces to
y=△h△GazO
(15)
et za -zo Aza. Then from (14a)
zl≈(GnB-caQ1GaB′1Ga)
(Gm+△Ga)业-(c+△Ga)(z+△za)
(GaB′-GaQ1GaB′)h'.(22b)
(G+AGa)y (h+ Ah).(16)Notice that the matrix G! is constant. By taking expectations
Retaining only the linear perturbation terms and then using of za and zaza, the covariance matrix of zais
(12)and(15), Aza and its covariance matrix is
cov(za=(Ga yG
△za=c(Gav-lG)-lc亚-1Bn
cov(za)=E4a4zd-(Goy-IGo)-( 17) The final position estimate is then obtained from z 'as
here the square error tem in( 12)has been ignored and( 13)
has been used to give cov(za)
The solution of za assumes that s, y and ri are independent
y
But they are related by (5)at i= l. The remaining question or
is how to incorporate this relationship to give an improved
estimate. When bias is ignored( this is justified when the noises
in the tDoas are small), the vector za is a random vector
with its mean centred at the true value and covariance matrix
given by(17). Hence the elements of za can be expressed as The proper solution is selected to be the one which lics in the
region of interest. If one of the coordinates of z! is close to
+e2, zn.3
18
ero, the square root in(24 )may become imaginary. In such
where e1, e2 and es are estimation errors of za. Subtracting a case, the imaginary component is set to zero. To find the
the first two components of za by i and yi, and then squaring covariance matrix of position estimate we express our final
the elements gives another set of equations
solution in the form r=r+ex and y=y+ey. from the
definition of za in(19), it follows that
v=h'-G
where
(m.-1)2=2(0-x1)
n2-(30-m)2=2(0-/xg
2-J1
G
3
The errors ex and Cy are relatively small compared with ao
and yo. Ignoring e2 and e2, and upon using(13),(17), (21)
(19) and(23), the covariance matrix of zp is found to be
where yl' is a vector denoting inaccuracies in za. Substituting
Φ=cov(zn)=:B
Za)B
(18) into (19)gives
)e1+i≈2(x0-x1)e1
cBGBGMBQBGOBGB-I
v1=2(
v2=2(0-mn)e2+e2≈2(y0-m)e2
2793+≈27e3
(20)
The approximation is valid as the errors ei,i= 1, 2, 3 are
B=/0
0(y0-y)
small. This is another approximation to the true maximum
likelihood procedure. The covariance matrix of y is therefore In summary, (14a), (22a), and( 24) are the solution equ
y='v=4B'cov(za B
ons. Since the weighting matrices in (14a)and(22a) are
proper approximation is necessary to find the an
B′=diag{x0-x1,y-,"}
swer. When the source is far from array,(14b),(22b), and
Since a is Gaussian, it follows that w/ is also
aussian. Inus
(24)are used. For a near-field source (14b)is first used to
the ML estimate of z is
give an approximation of B. equations(14a),(22a), and(24)
then give the solution. Position accuracy is assessed through
zn=(GnWG)-G′-h
(22a) the covariance matrix in (26).
CHAN AND HO: A SIMPLE AND EFFICIENT ESTIMATOR FOR HYPERBOLIC LOCATION SYSTEM
1909
B. Linear arra
can be replaced by a matrix of diagonal elements 1 and 0. 5
The above formulas(10).(14), and(22)are valid if the for all other elements(see(44)below). If the sensor positions
matrices involved are full rank. When sensors are arranged have significant uncertainties, we can also incorporate these
linearly, the matrices containing si and yi will be singular uncertainties to the matrix so that more weights are given
because the sensor positions satisfy yi ari b, i
to the equation containing the more reliable sensor position
1, 2, .., M, where a and b are some constants. Rewrite(9)as
(c;,32)
he covariance matrix of position estimate contains the
uncertainty information in localization. In particular, the po-
2 ci, 1(a+a3)-2ri, 1 1=rf, -Ki+Ki.(27) Sition mean-square error(MSE) is equal to the sun of the
diagonal elements of Another commonly used measure of
Equation (27)is linear in w=(a+ay) and r1. Using the first localization accuracy is the circular error probable( CEP)[12]
stage procedure illustrated in Section II-a gives a solution It is defined as the radius of the circle that has its centre at the
similar to(14a)
mean and contains half the realizations of the random position
estimate. Recall that since the noise vector n is Gaussian, A2p
z=(Gy-G)-G业-h
is also Gaussian distributed. The ceP is thus related to 4
2,172,1
Details of its computation from can be found in [11][12]
t ay
3,1T3,1
ZI
(28)
III. COMPARISON WITH THE CRLB
M1M,1
The Cramer-Rao inequality [14] sets a lower bound for the
variance of any unbiased parameter estimators. Hence it is of
The vector h is defined in(11). A second stage is unnecessary interest to compare the estimator with the optimum
since T1 and r+ ay are two independent variables. To obtain The CRLB of the localization problem is derived in the
the position estimate, substitute the computed Ti and i= Appendix. It is given by
w-ay into (5) at2= l to produce a quadratic in y. The
location estimate is
22(GtQ-GD)
-E土√E2-4AC
where the matrix Go is defined in (7)with(, 3, r:)
2A
-i =w-ay
We shall first consider the arbitrarily distributed array case
where
The corresponding position covariance matrix is given in(26)
Denote the(i, j)th element of a matrix R as [Rli j. Then from
A=1+a2,E=-2(-6),C=K1-21m+-1.(29)(11,(12)and(21)
If the array is on the x-axis and if(e1, 31) is at the origin
a=b=u1= K1=0 and(28)reduces to the simple form
IB-GB′-1]-1,2
y=tvri-w2 and a=w. This solution is identical to the
1
70
given in L4」
BG B
3
To calculate the covariance matrix, we perturb the random
rOro
quantities in(28)and proceed as before to obtain
Hence from the definition of G and Bin(19)and(26), and
E[△z1△z]=(Gpc9)-1
(30) noting that Ii, 1 =i-31 and n,
B-GB′-GaB"]-l,1
According to(5),△m,△yand△: are related by△r1=
(x0-x1)
1
1
△x+(y1-y)△y}/r.Hnce
△x
G
T△z
△a
Similarly
and the required covariance matrix is
B-lGB-GaB"1-12=(G9-1,2
(35)
Thus
重=T1(G0rycB)-T-
H-Q-1B-G81)
B-GOB′-1GB"=G
(32)
(36)
Comparison between(26)and (33)reveals that the position
The proposed solution requires the knowledge of Tdoa estimate with arbitrary array can achieve the optimum perfor
covariance matrix Q which may not be known in practice. mance and is therefore efficient, when the measurement errors
If the noise power spectral densities are similar at sensors, it in TDOA's are small
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 42 NO.8, AUGUsT 1994
In the linear array case, using the definitions of B, Gi and
T in(12),(28)and (31), it can be verified that
B-IGUT= Go
37)
It is evident from ( 32) that the covariance matrix i is identical
to o of (33). As a consequence our estimator is also efficient
for linearly distributed sensors. This is consistent with the
results given in L4」
IV. COMPARISON WITH PREVIOUS WORK
We shall next compare our technique with those in the
literatu
A. Linear Array
In the special case of a linear array of three sensors, Carter
·: sensor
source
[1] has derived an exact formula for source range and bearing
We next show that the solution of Section II-B will give the
r: range
e: bearing
same answer
Fig. 2. Localization with a linear three sensor array
The localization geometry is shown in Fig. 2. The sensor
positions are numbered as(=1=0, 1=0,(22=-L1, y2
O)and(c3= L2, 33=0)so that ri corresponds to the range the weights is crucial to achieve optimum performance but
r of the object. Substituting a=0,f2, 1=-L1, 3, 1=L2, expressions for the weights are, in general, very complicated
K1=0, K2=Li and K3=L2 into(27)yields
[3]. Moreover, the solution is correct only for distant source
Recently, Abel and smith [4] deduced an explicit closed
L
(38)form solution for the problem which is simple to compute
r3.1
and achieves CRlB around small error region. In addition
Solving (38)for r and r gives
no distant source assumption is required. Our result is indeed
identical to theirs when the coordinate axes are chosen to be
L11-("4)2+
()2
2(2+;
(39) In sonar, the quantities of interest are often range and
bearing instead of the a and y coordinates of the emitter.
and the bearing angle
The optimum bounds for range and bearing variances have
been derived by Bangs and Schultheiss [16]. They assume
6= cos
A=CosI
L2-231-731
(40) spatially incoherent noise fields and a distant source. It is
also possible to derive the variances of range and bearing
for our estimator. As the estimator has already been shown
ilts given in [1](where the sensors to be efficient around small error region, the corresponding
are numbered as(-1=0, y1=0),(E2= L1, 92=0)and variances will be the CRlB. It will then be shown that with
(a3= L1 +L2, 13=0). Accordingly, our T2, and Ta, 1 are the two assumptions made in [16], we shall come to Bangs
equal to -T2, 1 and r3, 2 in [1]. is equal to
and Schultheiss's solution
2212-7212一T2731r32
without loss of generality, let the y-coordinate of all sensors
2{r2,L2+r31L1}
zero. In terms of range T and bearing 8, a is equal to r cos g
and hence△=-rsin60△+cs0△;. We arrange the
and y is obtained from Vr2-r2
sensors so that T1 is the source range. Then
With three or more sensors a traditional method is the
focused beamforming derived by Carter [1,115]. The sensor
△
A2I a
outputs are prehiltered and time shifted according to de
=\△T1
H
lays chosen by some hypothesized range and bearing. The
sin
H
COs
A0
processed sensor outputs are summed, filtered, squared
(42)
averaged to give a final output. The sourcc range and bearing
estimate is the pair which maximizes the final output. This
From(30), we have
technique requires an extensive search in the 2-D range and
bearing space. Another approach is by Hahn[ 1-[31 in which
-E[△2门]-mQPr(4
the Tdoa vector d is first estimated. The source range and
bearing are deduced from the weighted sum of ranges and which is a general formula for arbitrary source and noise fields
bearings obtained from the associated TDOA's for every In spatially incoherent noise fields of identical receiver noise
triplet ((c1, y1),(, yi)(C,, 1i)),i * j. The selection of power spectra, the vector TDOA covariance matrix Q in(3)
CHAN AND HO: A SIMPLE AND EFFICIENT ESTIMATOR FOR HYPERBOLIC LOCATION SYSTEM
can
bc simplified to
10.505·0.5
9-105105
0.5
M
0.50.50.51
M-1
1
1M-1-1
1-1
22S(d)2/N(a)2
+MS(u)/(
e
When the definitions of H, Gi and y given in(42),(28)and
(0,0)
(13)are substituted into(43), we have(45), at the bottom of
,1
he next
If the distant source assumption is imposed
a. sensor
source
0
r: range
e: bearing
and as illustrated in Fig 3
Fig 3. Sensors and source geomet
wi, I cos 80 +rii hi
B Localization with Arbitrary Array
(1
For arbitrarily distributed sensors, spherical-intersection
i sin 6
2
(SX)[7], spherical-interpolation(Si)[6],[81-191, penalty
=2sin
function SI [9], DAC [10], and Taylor-series [111-112
21
are commonly used methods. Although the first four have
1 sin-e
,1
46) Computational advantage over the last, their solution is less
accurate
SX assumes known T1 and solves u and y in terms of r
Since;1≤r0,[-]1,2≈0. It follows that
from (9).()is then used to find ri and hence a, y. Since
T1 is assumed to be constant in the first step, the degree of
≈{9e2sn26∑xa-)2
freedom to minimize the second norm of yb is reduced. The
solution obtained is therefore nonoptimum as demonstrated
in [8]. SI assumes the three variables a, y and r1 in(9)to
be independent, and eliminates ri from those equations. The
140∑∑
(47) answer obtained is of course nonoptimum as the relationship
(5)among w,y and r1 is ignored. Penalty function si is an
improvement on SI by incorporating (5), but a suitable choice
where o2 and 0? denote the variances of source bearing of the parameter a which controls the contributions of (5)
and range. They are identical to those given by Bangs and to the error criterion is unknown. Our method, on the other
Schultheiss [151-[16
hand incorporates this constraint to the solution by using a
∑∑(
12=
SIn
{(M-1∑
r;1(r,1cos6+T21)
∑∑10
12=g2∑∑
os60+
(45)
r0
19L2
lEEE TRANSACTIONS ON SIGNAL PROCESSING VOL 42. No,8, AUGUST 19y4
TABLE I
COMPARISON OF MSE FOR THE SL. TAYLOR-SERIES AND PROFOSED METHODS: ARBITRARY ARRAY AND NEAR SOURCE
MSE
M=3
M=6
M=7
M=8
15768
0.15970.1480
0.1229
0.1164
0.1148
0.1103
ABCDE
2.17260706b0.14600.134
0.1057
0.1034
0.09462
0.7282
0.14560.13590.1159
0.1077
0.1055
009697
2.1726069860.14510.13370.11410.105
0.103
0.09480
19794068840.1451013340.11430.1054
0.1032
0.09432
A: SI method. B: Taylor series method. C: proposed method, ((146),(220),(24)I. D: proposed method,i(146),
(14a),(22a),(24)),E: theoretical MSE of the new method= CRLB
LS computation with appropriate weighting. It follows that for different methods. For simplicity, we assume that the signal
the aptimum estimate is obtained
and noises in(1)are white random processes and that the snr
DAC consists of dividing the sensor measurements of all sensor inputs are identical. Consequently, the covariance
(TDOA,'s in our case) into groups, each having a size equal matrix Q is found from(3)to be ad for diagonal elements and
to the number of unknowns. The unknown parameters are 0.5 od for all other elements, where o d is the TDOA variance.
computed from each group and then appropriately combined The TDoa estimates are simulated by adding to the actu
to give the final answer. The solution uses stochastic approx- TDOA's correlated Gaussian random noises with covariance
imation and requires the Fisher information in each group to matrix given by Q
be sufficiently large. As a consequcncc, optimum pcrformance Tablc I compares the localization accuracy of the Sl, Taylor-
can be achieved only when the noise is small enough. This series and the new estimator for an arbitrary array with a
mplies a low noise threshold before performance deviates different number of sensors. The sensor positions are(T1
from the CRLB. Another difficulty is that if the number of 0, y1=0,(
4,/3
sensors is not equal to an integral multiple of the number of(34 =-2, 34=1)15 =7, y5=3),(a6
unknowns plus one, the TDOa estimates from thosc remaining (C7= 2, 37=5)
sensors cannot be utilized to improve solution accuracy. and(=10=1,110=8). The source is at(x=8. y=22)
Our estimator has no such restriction. Moreover, it does not The TDOa noise power is set to 0.001/c and the mse
require stochastic approximation and hence has a larger noise El(a -xo)2+(y-y 22] are obtained from the
average
threshold. This is verified by simulations in Section V
100 000 independent runs. The weighting matrices W and V
While the Taylor-serics method can give an accurate posi- in the SI method [8, eq. 12)] are both sct to Q-l. The initial
tion estimate al reasonable noise levels, the major drawback is position guess in the Taylor-series method is chosen to be
that it is iterative, and there is no guarantee for convergence. the true solution to allow for convergence. Simulations show
Our approach does not require iteration. If the source is far that at least three iterations are required for convergence. Two
from the array, we compute first the intermediate solution za ways are used to compute the solution in the new method. One
by(14b)and the final solution by(22b)followed by (24). It uses(14b),(22b), and(24). The other first employs (14b)to
does not require the commputation of l/Ti for i=1, 2, ,. M, deduce an estimate of B and then uses(14a)to calculate the
which is most costly for large M as square root and division intermediate solution za. The final result is obtained from(22a)
operations are required. Although computing 1/r: is necessary followed by(24). The matrix GO is replaced by Gu and the
for a near source, it is needed only once. In the Taylor-series true values in B' is substituted by the values in za in(22). The
method, calculation of 1/ri is a must for each iteration as can differences between the two results are small. This indicates
be seen in (7). Finally, n] special procedure is required to that in most cases, the simplified formulae( 14b)and (22b)are
detect divergence
sufficient. Among the three methods, SI performs worst and
Under the condition that the Taylor-series method is prop- our solution method gives a slightly smaller mse than the
rly converged, it is of interest to compare the accuracy of Taylor-series method. Note that the proposed method with the
the two techniques. The Taylor-series method assumes that simplified formulae still performs better than the SI method
he linearization enor is small. It has been illustrated in [17] Additionally, the SI method cannot produce a position estimate
that the higher order terms are significant in determining for the three sensor situation when the number of independent
the solution in bad geometric dilution of precision(GDoP) equations equals the number of unknowns. The error for SI is
situations. Even through the noise power is relatively small, most significant when there are four sensors. The theoretical
there is no guarantee that the obtained solution is accurate. It is mse given by the sum of the diagonal elements of in (26)
expected that our method works better in cases where GDOP which is identical to the CRLB, is also computed Notice that
is poor. In fact, our method guarantees optimum performance there is a close match between the predicted and the simulated
around small TDOA noise region and thus is always be values and the validity of(26)is confirmed. This also confirms
superior.
that our estimator achieves the CRLB and that replacing the
V. SIMULATION RESULTS
true values in B. Go and b by their noisy versions does not
affect the result much
Simulations are performed to corroborate the theoretical In the case of a linear array, the results are given in
development and to compare the relative localization accuracy Table Il. Only the proposed and Taylor-series method are
CHAN AND HO: A SIMPLE AND EFFICIENT ESTIMATOR FOR HYPERBOLIC LOCATION SYSTEM
1913
TABLE II
COMPARISON OF MSE FOR THE PROPOSED AND TAYLOR-SERIES METHODS: LINEAR ARRAY AND NEAR SOURCE
MSE
M=3
M=4
M=5
M=6
M=7
M=8
M=10
8.257412836
0.3566
0.12220061550.02854
001746
0.009540
BCDE
8.25741.1131035560.12250.061990028840.017720.009726
825741.l117035450.12190.061480028520.01746
0.009541
191.1000035480.12190.061230028400.017500.009599
B: Taylor series method. C: proposed method. [(28)with v=Q, (29)). D: proposed method,((28)with v=Q2,
(28),(29). E: theoretical MSE of the new method CRLB
compared because si fails due to singularity problem. The
TABLE III
sensor coordinates are(r=-(i-1),i=o when i is odd
COMPARISON OF MSE FOR THE SI, TAYLOR-STRIES AND PROPOSED
METHODS ARBITRARY ARRAY AND DISTANT SOURCE
(a=i, yi= 0)when i is even, i= 1, 2
source position is the same as before. The TDOa variance is MSE M=4 M=5 M-6 M=7 M=8M=9 M=10
set to 0.0001/ c2. The solution is computed by(28),(29).TheA
101881213.9348.1240.40418740.113723
localization mse decreases as the number of sensor increases
346861475744.3838643863365533.80
Again, che differences in the results by using the approximation C
348.741448444063841384736.5033.87
业=Qin(28)and
ng yE
328821439444.0638543853364
are small. The theoretical MSE, which is the same as CRLB, A: SI method. B: Taylor series method. C: proposed method, ((146), ( 22b),
are also evaluated from(32). They are in close agreement with (24)J-E: theoretical Mse of the new method crlB
the simulated values. again, the Taylor-series method
almost identical results as the new method
When the source is far away al position(o=-50, yo
250), the MSE for random and linear arrays are given in Tables
COMPARISON OF MSE FOR THE PROPOSED AND TAYLOR-SERIES
METHODS: LINEAR ARRAY AND DISTANT SOURCE
III and IV respectively. The sensor positions are the same
as before and the TDOA noise power is o2=0.00001/c2 MSE M=4 M=5M=6 M=7M-8M=9 M=lo
in both cases. Since the source range is large, the resultsB
179713432681595069.1634.5618.5610.85
btained from the simplified and actual formulae are found c
158831409.6615392684134418.5610.86
to be identical. The source position cannot be estimated if E 1437. 25 408.17 154.05'68.06 34.25 18.57 10.90
there are only three sensors due to large position variations. B: Taylor series method. C: proposed method, ((28)with y =Q, (29)).E
As shown in Table Ill. the proposed method performs much theorelical MSE of the new teod CrLB
better than Si and slightly better than Taylor-series. Indeed
the new method attains the crlb for both the random and
linear array scenarios. In the case of a linear array, thcrc arc requires an extremely low noise power to make the stochastic
some interesting observations. When M is small(4 or 5), the approximation valid. The result is that DaC has a low noise
proposed method has significantly less mse than the Taylor- threshold
series method. The larger MSE in the Taylor-series method is To give a comparison of the threshold
ct between
due to linearization errors. With a distant source, this error is Dac and our method, the mse for a fixed
onfiguration
Formance is observed. It must be emphasized is illustrated in Fig 4. Both methods perform well at low
Lhat the Taylor-series method has been given a most favorable noise level. Thresholding effect occurs in dAC when a2
itial guess, namely, the true position. In practice, this is not 0.000016. On the other hand, our method follows closely
possible and solution divergence may occur
with the CRLB until od=00001 (six times larger then
It has been shown in [10] that DAC is an optimum estimator that in DAC). It is also interesting to see that although
around small crror region. The mcthod was tcsted in an deviation from CRLB starts at high noise level, the MSE
arbitrary array in the same conditions as before. We found that does not jump to a large value as ce
mpared to DAC. It
DAC perform equally well with our method in the near-field is mentioned in [18] that the bhattachayya bound is tighter
case. In the far-field situation, DAc gives an mse in the order than CRLB for nonlinear parameter estimation. Hence it is
of 10, even through the noise in the second case is smaller. more realistic to compare our estimator's performance with
This is because DAC requires a stochastic approximation the Bhattachayya bound at low SNR. It can be seen in Fig. 4
which is only valid when the transmitter position variance is that our method performs only a little worse than the second
mall. It can be seen from(33)and the definition of Gt in order Bhattachayya bound (BHB2). This diffcrcncc is probabl
()that transmitter position variance is roughly proportional due to the fact that we have neglected second order error
to the squared range of the object. Hence a far-field emitter terms in our algorithm. It is expected that our estimator
1914
EEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 42, No. 8, AUGUST 1994
APPENDIX
Hahn and Tretter's [2] estimator is an implementation
of the ML estimator for a vector of tDoa known to bc
asymptotically Gaussian with covariance matrix given by Q
Hence the conditional probability density function of d is
P(d zp)=
new method
丌)-1)/2QP
DAC
CHLB
Xp
d -=rQ
A1)
BHB2
0
22
18
where r=[v2, 1, 73, 1, ..., TM 1]2 is a function of zp. The
10og(c小
transmitter position can be expressed as a nonlinear function
Fig. 4. Comparison of threshold effect between the new method and DAc. of d, i. e, a= fi(d)and y= f2(d). Using Taylor-series
expansion of a and y around the true TDOA vector, it can
be verified that both the bias and variance of transmitter
will approach BHB2 if those second order crror tcrms arc if variations in TDoA's are small so that the bias square is
taken into account. Many simulations has been tried with
insignificant compared with the variance, the CRLB of zp is
different array geometry and source locations(near-field and
far-field ). They all verify that our estimator enters its large
given by [14
error region at a smaller SNr than DAC and follows closely
the bhb2
F
Inp(d
VI. CONCLUSION
Inp(d(z,
(A2
A new appmach for localizing a source from a set of
hyperbolic curves defined by TDOA measurements is pro
posed. By introducing an intermediate variable, the nonlinear
equations relating TDOA estimates and source position can
The partial derivative of In p(d zp) with respect to
be transformed into a set of equations which are linear
in the unknown parameters and the intermediate variable
A Ls then gives their solution. By exploiting the kn
Inp(d (zr)=-0-Q(d
nown
rclation bctwccn the intermediate variable and the position
coordinates, a second weighted ls gives the final solution Hence
for the position coordinates. The covariance matrix of the
T
position estimate was derived and found to meet the CRlB
Q
(A4)
The estimator is asymptotically efficient for arbitrary or linear
array configurations. Indeed, it is an approximation of the ml
estimator for hyperbolic position fix when the TDOa error is
where ar/azp is found from the definition of r to be G
small. In the case of a linear array, the solution reduces to
that of 14]. A comparison with other localization methods was
ACKNOWLEDGMENT
conducted. The proposed method has the same simplicity as The authors thank the anonymous reviewers for their helpful
the si method but performs significantly better, particularly comments and suggestions
when the number of sensors is small. It has a higher noise
threshold than the dac method and provides an explicit
REfERENCES
solution form which is not available in existing optimum
estimators. Finally, the new technique offers a computational
1] G. C. Carter, Time delay estimation for passive sonar signal process
ing, /EFF Trans. Acoust., Speech, Signal Processing, voL. ASSP-29
advantage over the Taylor-series technique and eliminates the
463-470,June1981.
convergence problem
[2
R. Hahn and S. A. Tretter, "Optimum processing for delay-vector
In this paper, we have only considered TDOA estimation
estimation in passive signal arrays, " IEEE Trans. inform Theory, vol
-19,pp.608614,sept.1973
error.In practical localization system, sensor position un- [3]WR. Hahn, Optimum signal processing for passive sonar range and
certainty is often encountered [19]. If the variances of the
earing estimation, J. Acoust. Soc. Am., vol. 58, PP. 201-207. July
uncertainties of individual sensors are known, it will not [4]J S. Abel and I. O. Smith, "Source range and depth estimation from
be difficult to incorporate the reciprocal of the variances as
multipath range difference measurements, "IEEE Trans. Acoust., Speech
Signal processing, vol 37, Pp. 1157-1165, Aug. 1989
weights in the weighted [S to give an ML estimator( see [5] B. T Fang, Simple solutions for hyperbolic and related position fixes
discussion after (32))
TEEE Trans. Aerosp. Electron. Syst., vol. 26, Pp. 748-753, Sept. 1990
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