IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 1121
[7] J. Bitzer, K.-D. Kammeyer, and K. U. Simmer, “An alternative imple-
mentation of the superdirective beamformer,” in Proc. IEEE Workshop
Application Signal Process. Audio Acoust., New Paltz, NY, Oct. 1999.
[8] C. Marro, Y. Mahieux, and K. U. Simmer, “Analysis of noise reduction
and dereverberation techniques based on microphone arrays with post-
filtering,” IEEE Trans. Speech Audio Processing, vol. 6, pp. 240–259,
May 1998.
[9] S. Nordholm, I. Claeson, and M. Dahl, “Adaptive microphone array em-
ploying calibration signals: an analytical evaluation,” IEEE Trans. An-
tennas Propagat., vol. 8, pp. 975–981, Aug. 1992.
[10] K.-C. Huarng and C.-C. Yeh, “Performance analysis of derivative
constraint adaptive arrays with pointing errors,” IEEE Trans. Antennas
Propagat., vol. 40, pp. 975–981, Feb. 1992.
[11] S. Gannot, D. Burshtein, and E. Weinstein, “Theoretical performance
analysis of the general transfer function GSC,” Technion—Israel Inst.
Techno., Haifa, Israel, CCIT Rep. 381, May 2002.
[12]
, “Theoretical analysis of the general transfer function GSC,” in
Proc. Int. Workshop Acoust. Echo Noise Contr., Darmstadt, Germany,
Sept. 2001.
[13] S. Nordholm, I. Claesson, and P. Eriksson, “The broadband wiener solu-
tion for Griffiths-Jim beamformers,” IEEE Trans. Signal Proc., vol. 40,
pp. 474–478, Feb. 1992.
[14] O. Shalvi and E. Weinstein, “System identification using nonstationary
signals,” IEEE Trans. Signal Proc., vol. 44, pp. 2055–2063, Aug. 1996.
Least Squares Algorithms for Time-of-Arrival-Based
Mobile Location
K. W. Cheung, H. C. So, W.-K. Ma, and Y. T. Chan
Abstract—Localization of mobile phones is of considerable interest in
wireless communications. In this correspondence, two algorithms are de-
veloped for accurate mobile location using the time-of-arrival measure-
ments of the signal from the mobile station received at three or more base
stations. The first algorithm is an unconstrained least squares (LS) esti-
mator that has implementation simplicity. The second algorithm solves a
nonconvex constrained weighted least squares (CWLS) problem for im-
proving estimation accuracy. It is shown that the CWLS estimator yields
better performance than the LS method and achieves both the Cramér–Rao
lower bound and the optimal circular error probability at sufficiently high
signal-to-noise ratio conditions.
Index Terms—Mobile terminals, positioning algorithms, time-of-arrival.
I. INTRODUCTION
Mobile location has received significant interest since the first ruling
of the Federal Communications Commission for detection of emer-
gency calls in the United States in 1996 [1]. In addition to emergency
Manuscript received September 9, 2002; revised June 24, 2003. This work
was supported by a Grant from the Research Grants Council of the Hong Kong
Special Administrative Region, China under Project CityU 1119/01E. The as-
sociate editor coordinating the review of this manuscript and approving it for
publication was Prof. Randolph L. Moses.
K. W. Cheung and H. C. So are with the Department of Computer Engineering
and Information Technology, City University of Hong Kong, Kowloon, Hong
Kong (e-mail: gary.cheung@student.cityu.edu.hk; ithcso@cityu.edu.hk).
W.-K. Ma is with the Department of Electrical and Electronic Engineering,
the University of Melbourne, Parkville, Australia (e-mail: w.ma@ee.mu.oz.au).
Y. T. Chan is with the Department of Electronic Engineering, The
Chinese University of Hong Kong, Shatin, N.T., Hong Kong (e-mail:
wkma@ee.cuhk.edu.hk; ytchan@ee.cuhk.edu.hk).
Digital Object Identifier 10.1109/TSP.2004.823465
management, mobile position information will also be useful in intel-
ligent transport systems, location billing, interactive map consultation,
and monitoring of the mentally impaired [2]–[6].
Wireless location systems usually require two or more base stations
(BSs) to intercept a mobile station (MS) signal. Common location ap-
proaches are based on time-of-arrival (TOA), received signal strength
(RSS), time-difference-of-arrival (TDOA), or angle-of-arrival (AOA)
measurements determined from the MS signals received at the BSs
[6]–[10]. In this correspondence, we focus on mobile positioning using
the TOA information.
In the TOA method, the one-way propagation time of the signal trav-
eling between the MS and each of the BSs is measured, and this pro-
vides a circle centered at the BS on which the MS must lie. The TOA
measurements are then converted into a set of circular equations, from
which the MS position can be determined with the knowledge of the
BS geometry. A straightforward approach for determining the MS po-
sition is to solve the nonlinear equations [9] relating these measure-
ments directly, but it is computationally intensive. Apart from the direct
methodology, another common technique [10]–[12] that avoids solving
the nonlinear equations is to linearize them, and then, the solution is
found iteratively. However, this approach requires an initial estimate
and cannot guarantee convergence to the correct solution unless the ini-
tial guess is close to it. To allow real-time implementation and ensure
global optimization, we adopt the idea of the spherical interpolation
(SI) in TDOA-based location [13] that reorganizes the nonlinear hyper-
bolic equations into a set of linear equations by introducing an inter-
mediate variable, which is a function of the source position. However,
the SI estimator solves the linear equations directly via least squares
(LS) without using the known relation between the intermediate vari-
able and the position coordinate. To improve the location accuracy of
the SI approach, Chan and Ho have proposed [14] to use a two-stage
weighted LS to solve for the source position by exploiting this relation
implicitly, whereas [15] incorporates the relation explicitly by mini-
mizing a constrained LS function based on the technique of Lagrange
multipliers. According to [15], these two modified algorithms are re-
ferred to as the quadratic correction least squares (QCLS) and linear
correction least squares (LCLS), respectively.
The rest of the paper is organized as follows. In Section II, the model
for the TOA measurements is described. Two important performance
measures of location accuracy, namely, the Cramér–Rao lower bound
(CRLB) [16] and circular error probability (CEP) [12] are then
reviewed. In Section III, we first derive a simple LS TOA-location
algorithm via the introduction of a range parameter. An improved
algorithm, which weighs the LS function and exploits the relation
between the range variable and position coordinate, is then devised.
Performance of the developed algorithms is analyzed in Section IV.
Simulation results are presented in Section V to evaluate the location
estimation performance of the two methods. Finally, conclusions are
drawn in Section VI.
II. TOA M
EASUREMENT MODEL AND
PERFORMANCE MEASURES
It is assumed that a reliable non-line-of-sight detection algorithm,
such as [17] or [18], has first been employed to eliminate the measure-
ments with large errors. As a result, all measurements we utilize for
mobile location come from line-of-sight (LOS) propagation. Let
[
x; y
]
be the MS position to be determined and the known coordinate of the
i
th BS be
[
x
i
;y
i
]
;i
=1
;
2
;
...
;M
, where
M
is the total number of re-
ceiving LOS BSs. The distance between the MS and the
i
th BS, which
is denoted by
d
i
, is given by
d
i
= (
x
0
x
i
)
2
+(
y
0
y
i
)
2
;i
=1
;
2
;
...
;M:
(1)
1053-587X/04$20.00 © 2004 IEEE
1122 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
In the absence of measurement error, the one-way propagation time
taken for the signal to travel from the MS to the
i
th BS, which is denoted
by
t
i
,is
t
i
=
d
i
c
;i
=1
;
2
;
...
;M
(2)
where
c
is the speed of light. The range measurement based on
t
i
in the
presence of disturbance, which is denoted by
r
i
, is modeled as
r
i
=
d
i
+
n
i
=
(
x
0
x
i
)
2
+(
y
0
y
i
)
2
+
n
i
i
=1
;
2
;
...
;M
(3)
where
n
i
is the noise in
r
i
or range error at the
i
th BS. For ease of anal-
ysis, we assume that each measurement error
n
i
is a zero-mean white
Gaussian process with known variance
2
i
. (The zero-mean assumption
is valid as long as the multipath effect, if any, has been circumvented
[19]. Although the parameters
f
2
i
g
are usually unknown in practice,
they can be determined for a particular signaling type in the TOA-based
location system by channel measurement. Once the variance estimates
have been obtained, we consider them to be constants for all TOA mea-
surements taken.)
The CRLB gives a lower bound on variance attainable by any un-
biased estimators, and thus, it can be served as a benchmark to con-
trast with the mean square error of positioning algorithms. We show in
Appendix A that the CRLBs for
x
and
y
are as in (4) and (5), shown
at the bottom of the page. In addition to the CRLB, the CEP [12] is
another approximate but simple performance measure of location ac-
curacy. It is defined as the radius of the circle that has its center at the
mean and contains half the realizations of the location estimates. If the
location estimator is unbiased, the CEP is a measure of the uncertainty
in the location estimate relative to the actual MS location. Therefore,
the smaller the CEP, the more reliable the estimator should be. Note
that an ellipse, which is characterized by its angle of rotation from the
x-axis, and the major and minor axes, can generally describe the con-
tour that contains half the realizations of estimates better than the CEP
circle. The complete procedures for computing this ellipse, as well as
the CEP using the ML location estimate in Gaussian noise, can be found
in [12]. Since the ML method should give optimum location estimates,
the CEP using the ML location estimate is the optimal CEP.
III. M
OBILE LOCATION
ALGORITHMS
In this section, we develop two TOA-based mobile location algo-
rithms using the SI principle. A simple LS mobile location estimator is
first derived as follows. Without measurement errors, (3) becomes
r
i
= (
x
0
x
i
)
2
+(
y
0
y
i
)
2
;i
=1
;
2
;
...
;M:
(6)
Squaring both sides of (6) yields
r
2
i
=
R
2
0
2
xx
i
0
2
yy
i
+(
x
2
i
+
y
2
i
)
)
x
i
x
+
y
i
y
0
0
:
5
R
2
=
1
2
x
2
i
+
y
2
i
0
r
2
i
;i
=1
;
2
;
...
;M
(7)
where
R
=
x
2
+
y
2
is the range variable introduced in order to
reorganize (6) into a set of linear equations in
x; y
and
R
2
. Equation
(7) can be expressed in matrix form as
A
=
b
(8)
where
A
=
x
1
y
1
0
0
:
5
.
.
.
.
.
.
.
.
.
x
M
y
M
0
0
:
5
;
=
x
y
R
2
;
and
b
=
1
2
x
2
1
+
y
2
1
0
r
2
1
.
.
.
x
2
M
+
y
2
M
0
r
2
M
:
In the presence of measurement errors,
can be estimated using the
standard LS
^
=argmin
(
A
0
b
)
T
(
A
0
b
)
=(
A
T
A
)
0
1
A
T
b
(9)
where
=[
x;
y;
R
2
]
T
represents an optimization variable vector.
For better performance, we can add a weighting matrix
W
to (9) and
restrict
to satisfy the basic relationship
R
=
x
2
+
y
2
:
(10)
This leads to the following constrained optimization problem:
^
cw
= arg min
(
A
0
b
)
T
W
(
A
0
b
)
(11)
subject to
q
T
+
T
P
=0
(12)
where
P
=
100
010
000
and
q
=
0
0
0
1
:
Here, (12) is a matrix characterization of the relation in (10).
Let us study the disturbance in
b
, which will lead to a suggestion on
the choice of
W
[14]. For sufficiently small measurement error or high
signal-to-noise ratio (SNR) conditions, the squared value of
r
i
can be
approximated as
r
2
i
=(
d
i
+
n
i
)
2
d
2
i
+2
d
i
n
i
;i
=1
;
2
;
...
;M:
(13)
As a result, the disturbance between the true and measured squared
distances is
"
i
=
r
2
i
0
d
2
i
2
d
i
n
i
;i
=1
;
2
;
...
;M:
(14)
In vector form,
f
"
i
g
are expressed as
"""
=[2
d
1
n
1
;
2
d
2
n
2
;
...
;
2
d
M
n
M
]
T
:
(15)
The covariance matrix of the disturbance is thus of the form
9
=
E """"
T
=
BQB
(16)
CRLB
(
x
)=
M
i
=1
(
y
0
y
)
[(
x
0
x
) +(
y
0
y
) ]
M
i
=1
(
x
0
x
)
[(
x
0
x
) +(
y
0
y
) ]
M
i
=1
(
y
0
y
)
[(
x
0
x
) +(
y
0
y
) ]
0
M
i
=1
(
x
0
x
)(
y
0
y
)
[(
x
0
x
) +(
y
0
y
) ]
2
(4)
CRLB
(
y
)=
M
i
=1
(
x
0
x
)
[(
x
0
x
) +(
y
0
y
) ]
M
i
=1
(
x
0
x
)
[(
x
0
x
) +(
y
0
y
) ]
M
i
=1
(
y
0
y
)
[(
x
0
x
) +(
y
0
y
) ]
0
M
i
=1
(
x
0
x
)(
y
0
y
)
[(
x
0
x
) +(
y
0
y
) ]
2
(5)
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 1123
where
E
[
1
]
is the expectation operator,
B
=
diag
(2
d
1
;
2
d
2
;
...
;
2
d
M
)
, and
Q
=
diag
(
2
1
;
2
2
;
...
;
2
M
)
. The optimum
weighting matrix for (11) is given by
W
=
9
0
1
. Since it depends
on the unknown
f
d
i
g
, we use the approximate value of
9
^
BQ
^
B
,
where
^
B
=
diag
(2
r
1
;
2
r
2
;
...
;
2
r
M
)
.
We now consider solving the constrained weighted least squares
(CWLS) problem in (11) and (12), which is equivalent to minimizing
the Lagrangian [15]
L
(
;
)=(
A
0
b
)
T
9
0
1
(
A
0
b
)+
(
q
T
+
T
P
)
(17)
where
is the Lagrange multiplier. We show in Appendix B that a
minimum point for the CWLS problem, either global or local, is given
by
^
cw
=(
A
T
9
0
1
A
+
P
)
0
1
A
T
9
0
1
b
0
2
q
(18)
where
is determined from the five-root equation
c
3
f
3
0
2
c
3
g
3
+
2
i
=1
c
i
f
i
1+
i
0
2
2
i
=1
c
i
g
i
1+
i
+
2
i
=1
e
i
f
i
i
(1 +
i
)
2
0
2
2
i
=1
e
i
g
i
i
(1 +
i
)
2
0
2
2
i
=1
c
i
f
i
i
(1 +
i
)
2
+
2
4
2
i
=1
c
i
g
i
i
(1 +
i
)
2
=0
(19)
and
f
c
i
g
;
f
e
i
g
;
f
f
i
g
, and
f
g
i
g
;i
=1
;
2
;
3
have been defined in
Appendix B. The desired
is found by the following procedure.
a) Obtain the five roots of (19) using a root-finding algorithm. Dis-
card any complex roots because the Lagrange multiplier is al-
ways real for real optimization problems.
b) Put the real
’s back to (18), and obtain subestimates of
^
cw
.
c) The subestimate that yields the smallest objective value of
(
A
0
b
)
T
W
(
A
0
b
)
is taken as the globally optimal CWLS solution.
Efficient numerical methods for root finding can be found in [20], and
interested readers may refer to it. Note that we can follow the argument
of [15] by finding the
whose value is closest to zero only in order to
save computation.
IV. P
ERFORMANCE ANALYSIS
In this section, the bias and variance of the proposed location algo-
rithms under sufficiently high SNR conditions are analyzed. Based on
(13), we define
b
=
1
2
x
2
1
+
y
2
1
0
d
2
1
.
.
.
x
2
M
+
y
2
M
0
d
2
M
and
~
b
=
0
d
1
n
1
.
.
.
d
M
n
M
:
The vectors
b
and
~
b
are the noise-free and disturbance components of
b
, respectively. The LS solution of (9) can then be written as
^
=(
A
T
A
)
0
1
A
T
(
b
+
~
b
)
:
(20)
Since
E
[
~
b
]
is zero, taking the expected value of (20) yields
E
[
^
]=
(
A
T
A
)
0
1
A
T
b
=
, which indicates the unbiasedness of the LS al-
gorithm under sufficiently small noise conditions.
On the other hand, we follow [15] to derive the bias of the CWLS
algorithm, which has the form
E
[
^
cw
]
0
=
0
2
(
A
T
9
0
1
A
)
0
1
q
+
1
n
=1
(
0
(
A
T
9
0
1
A
)
0
1
P
)
n
0
2
1
n
=1
(
0
(
A
T
9
0
1
A
)
0
1
P
)
n
(
A
T
9
0
1
A
)
0
1
q
(21)
for
<
1
=
k
(
A
T
9
0
1
A
)
0
1
P
k
. Although the CWLS estimator is bi-
ased, the bias magnitude will be very small as
should close to zero for
sufficiently high SNR conditions, and this is also illustrated via simu-
lation results in the following section. It is noteworthy that Huang
et al.
[15] have also demonstrated that the constrained algorithms, namely,
the QLCS and LCLS methods, possess negligible biases.
Considering that
E
[
^
cw
]
, the variances of the MS location es-
timate
[^
x
cw
;
^
y
cw
]
for the CWLS algorithm are derived as follows. We
first notice that the solution for the constrained optimization problem
in (11) and (12) is essentially
^
cw
=
arg min
f
x;
y
g
J
cw
(22)
where
J
cw
=
M
i
=1
w
i
x
i
x
+
y
i
y
0
0
:
5(
x
2
+
y
2
)
0
1
2
x
2
+
y
2
0
r
2
i
2
(23)
because
R
2
should satisfy
R
2
=
x
2
+
y
2
according to (10). For sim-
plicity, we consider
W
=
diag
(
w
1
;w
2
;
...
;w
M
)
, which agrees with
our uncorrelated noise assumption. By extending a standard variance
analysis technique [21], [22] to multiple parameter estimation, the vari-
ances of
^
x
cw
and
^
y
cw
, when they are located in a reasonable prox-
imity to
(
x; y
)
, are given by (see Appendix C) (24) and (25), shown
at the bottom of the page. In Appendix C, we show that (24) and (25)
are equivalent to the CRLB in (4) and (5), respectively, and this indi-
cates that the CWLS algorithm is optimal under sufficiently high SNR
conditions.
In a similar but more tedious manner, the variances of the MS loca-
tion estimate
[^
x;
^
y
]
for the LS algorithm have also been derived. Since
their expressions are much more complicated and cannot give us any
insightful findings, we omit them in this paper.
var
(^
x
cw
)
E
@J
@
x
2
E
@ J
@
y
2
0
2
E
@ J
@
x@
y
E
@ J
@
y
E
@J
@
x
@J
@
y
+
E
@J
@
y
2
E
@ J
@
x@
y
2
E
@ J
@
x
E
@ J
@
y
0
E
@ J
@
x@
y
2
2
(24)
var
(^
y
cw
)
E
@J
@
y
2
E
@ J
@
x
2
0
2
E
@ J
@
x@
y
E
@ J
@
x
E
@J
@
x
@J
@
y
+
E
@J
@
x
2
E
@ J
@
x@
y
2
E
@ J
@
x
E
@ J
@
y
0
E
@ J
@
x@
y
2
2
(25)
1124 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004
Fig. 1. Mean square range errors at
[
x; y
] = [1000
;
2000]
m.
Fig. 2. Mean absolute relative errors at
[
x; y
] = [1000
;
2000]
m.
V. S IMULATION
RESULTS
Computer simulations had been conducted to evaluate the perfor-
mance of the proposed TOA-based location algorithms by comparing
with the iterative LS technique [11] and CRLB. For [11], we used the
actual MS position as the initial estimate in the iterative procedure
to ensure global convergence. The CEPs of the LS and CWLS
estimators were also studied and contrasted with those of the ML
estimates. We considered a five-BS geometry with coordinates
[0
;
0]
m,
[3000
p
3
;
3000]
m,
[0
;
6000]
m,
[
0
3000
p
3
;
3000]
m, and
[
0
3000
p
3
;
0
3000]
m. All results were averages of 1000 independent
runs.
Fig. 1 plots the mean square range errors (MSREs) of the LS, CWLS,
and iterative LS methods versus the average noise power at
[
x; y
]=
[1000
;
2000]
m, where the noise refers to the range error. The theo-
retical range variance of the LS estimator at sufficiently high SNRs
and that of the CWLS method or the CRLB were also included. The
MSRE was defined as
E
[(
x
0
^
x
)
2
+(
y
0
^
y
)
2
]
, and it has a linear rela-
tionship with the geometric dilution of precision (GDOP) [6], whereas
the average noise power was given by
(1
=M
)
M
i
=1
2
i
, where
2
i
was
chosen such that all
2
i
=d
2
i
were kept identical. It can be seen that the
Fig. 3. MS estimate distribution of the LS estimator at
[
x; y
] = [1000
;
2000]
m for average noise power of 25 dB m
.
Fig. 4. MS estimate distribution of the CWLS estimator at
[
x; y
]=
[1000
;
2000]
m for average noise power of 25 dB m
.
performance of the CWLS method approached the CRLB, which veri-
fied its optimality as well as our analysis and outperformed the LS and
iterative LS estimators by approximately 5 and 2 dB m
2
, respectively,
for the whole range of noise powers. Moreover, there was good agree-
ment between the MSRE and theoretical variance of the LS method.
It is noteworthy that the average noise power range was reasonable for
practical mobile location applications [23]. The mean absolute relative
errors (MAREs) of the proposed methods and iterative LS estimator,
which were defined as
j
(
E
[^
x
]
0
x
)
=x
j
+
j
(
E
[^
y
]
0
y
)
=y
j
, are shown
in Fig. 2. We see that all methods had comparable MAREs, which im-
plies that they had similar empirical biases. From Figs. 1 and 2, we
know that these biases were in fact very small when comparing with
their variances.
Figs. 3 and 4 show the distributions of the MS position estimates ob-
tained by the LS and CWLS methods, respectively, at an average noise
power of 25 dB m
2
. The circles in the figures were centered at the ac-
tual MS location of
[1000
;
2000]
m and included half of the location
estimates. The radii or CEPs in Figs. 3 and 4 were found to 19.90 and
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 4, APRIL 2004 1125
Fig. 5. Mean square range errors at
[
x; y
]=[200
;
400]
m.
Fig. 6. Mean absolute relative errors at
[
x; y
] = [200
;
400]
m.
10.57 m. This means that the CWLS estimator outperformed the LS
method by approximately 9 m in terms of CEP. Furthermore, the CEP
for the ML estimator was calculated as 10.45 m, and thus, the opti-
mality of the CWLS method is again demonstrated.
The above test was repeated for
[
x; y
]=[200
;
400]
m, and the re-
sults are shown in Figs. 5 to 8. In Fig. 5, we see that the CWLS al-
gorithm attained the CRLB and was superior to the LS and iterative
LS methods by about 4 and 2 dB m
2
, respectively, when the average
noise power was less than 55 dB m
2
. The performance of the CWLS
method deviated from the CRLB for larger noise powers because the
high SNR assumption for it broke down, although this assumption still
held for the remaining methods. This finding also agreed with Fig. 6,
where it is observed that the CWLS algorithm had larger MAREs than
those of the LS and iterative LS estimators. In Fig. 7, the CEP of the
LS method was found to be 18.64 m, whereas in Fig. 8, we used an
ellipse to include half of the location estimates because using a circle
would introduce large errors. The semi-major and semi-minor axes of
the ellipse were measured as 17.51 and 1.92 m, respectively, which are
close to those of the ellipse for ML estimation, which were computed
as 17.26 and 1.76 m.
Fig. 7. MS estimate distribution of the LS estimator at
[
x; y
] = [200
;
400]
m
for average noise power of 25 dB m
.
Fig. 8. MS estimate distribution of the CWLS estimator at
[
x; y
]=
[200
;
400]
m for average noise power of 25 dB m
.
VI. CONCLUSION
Two time-of-arrival (TOA)-based location algorithms are developed
from the spherical interpolation (SI) approach, which reorganizes non-
linear equations to linear equations via introduction of an intermediate
variable. The first least squares algorithm directly extends the SI using
the TOA measurements. The second constrained weighted least squares
(CWLS) method is an improved version of the first algorithm with the
use of weighting matrix and constraint. We have shown that the CWLS
approach can attain the Cramér–Rao lower bound and optimal circular
error probability under sufficiently small noise conditions.
A
PPENDIX A
Let
u
=[
u
1
;u
2
]
T
=[
x; y
]
T
. The CRLB for the
k
th parameter of
u
;k
=1
;
2
is computed from [16]
CRLB
(
u
k
)=[
I
0
1
(
u
)]
k;k
;k
=1
;
2
(A.1)
![Fig. 5. Mean square range errors at [x; y] = [200; 400]m.](/figures/fig-5-mean-square-range-errors-at-x-y-200-400-m-32q1j30g.webp)
![Fig. 8. MS estimate distribution of the CWLS estimator at [x; y] = [200; 400]m for average noise power of 25 dB m .](/figures/fig-8-ms-estimate-distribution-of-the-cwls-estimator-at-x-y-36oavjc6.webp)
![Fig. 6. Mean absolute relative errors at [x; y] = [200; 400]m.](/figures/fig-6-mean-absolute-relative-errors-at-x-y-200-400-m-2vdkfwn1.webp)
![Fig. 7. MS estimate distribution of the LS estimator at [x; y] = [200; 400]m for average noise power of 25 dB m .](/figures/fig-7-ms-estimate-distribution-of-the-ls-estimator-at-x-y-1povm71s.webp)
![Fig. 1. Mean square range errors at [x; y] = [1000;2000]m.](/figures/fig-1-mean-square-range-errors-at-x-y-1000-2000-m-1mdnwc6v.webp)
![Fig. 4. MS estimate distribution of the CWLS estimator at [x; y] = [1000;2000] m for average noise power of 25 dB m .](/figures/fig-4-ms-estimate-distribution-of-the-cwls-estimator-at-x-y-y3mnlr0p.webp)