On Best Linear Unbiased Estimator Approach
for Time-of-Arrival based Localization
Frankie K. W. Chan, H. C. So, Jun Zheng and Kenneth W. K. Lui
Department of Electronic Engineering, City University of Hong Kong
Tat Chee Avenue, Kowloon, Hong Kong
Email: k.w.chan@student.cityu.edu.hk, hcso@ee.cityu.edu.hk, junzheng@cityu.edu.hk
50469990@student.cityu.edu.hk
January 30, 2008
Keywords: time-of-arrival, fast algorithm, position estimation, weighted least squares
Abstract
A common technique for source localization is to utilize the time-of-arrival (TOA) measurements
between the source and several spatially separated sensors. The TOA information defines a set of
circular equations from which the source position can be calculated with the knowledge of the
sensor positions. Apart from nonlinear optimization, least squares calibration (LSC) and linear
least squares (LLS) are two computationally simple positioning alternatives which reorganize the
circular equations into a unique and non-unique set of linear equations, respectively. As the LSC and
LLS algorithms employ standard least squares (LS), an obvious improvement is to utilize weighted
LS estimation. In this paper, it is proved that the best linear unbiased estimator (BLUE) version
of the LLS algorithm will give identical estimation performance as long as the linear equations
correspond to the independent set. The equivalence of the BLUE-LLS approach and the BLUE
variant of the LSC method is analyzed. Simulation results are also included to show the comparative
performance of the BLUE-LSC, BLUE-LLS, LSC, LLS and constrained weighted LSC methods with
Cram´er-Rao lower bound.
1
This paper is a postprint of a paper submitted to and accepted for publication in IET – Signal Processing
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
1 Introduction
Source localization using measurements from an array of spatially separated sensors has been an im-
portant problem in radar, sonar, global positioning system [1], mobile communications [2], multimedia
[3] and wireless sensor networks [4]. One commonly used location-bearing parameter is the time-of-
arrival (TOA) [2],[4], that is, the one-way signal propagation or round trip time between the source
and sensor. For two-dimensional positioning, each noise-free TOA provides a circle centered at the
sensor on which the source must lie. By using M ≥ 3 sensors, the source location can be uniquely
determined by the intersection of circles. In practice, the TOA measurements are noisy which implies
multiple intersection points and thus they are usually converted into a set of circular equations, from
which the source position is estimated with the knowledge of the signal propagation speed and sensor
array geometry.
Commonly used techniques for solving the circular equations include linearization via Taylor-series
expansion [5] and steepest descent method [6]. Although this direct approach can attain optimum
estimation performance, it is computationally intensive and sufficiently precise initial estimates are
required to obtain the global solution. On the other hand, an alternative approach which allows
real-time computation and ensures global convergence is to reorganize the nonlinear equations into a
set of linear equations by introducing an extra variable that is a function of the source position. It
is noteworthy that this idea is first introduced in [7]-[8] for time-difference-of-arrival (TDOA) based
localization. The linear equations can then be solved straightforwardly by using least squares and
the corresponding estimator is referred to as the least squares calibration (LSC) method [9], or by
eliminating the common variable via subtraction of each equation from all others, which is referred
to as the linear least squares (LLS) estimator [10]-[11]. In this work, we will focus on relationship
development between the the best linear unbiased estimator (BLUE) [12] versions of the LSC and
LLS algorithms. Our contributions do not lie on new positioning algorithm development as the BLUE
technique for localization applications has already been proposed in the literature [13]. Our major
findings include (i) All BLUE realizations of the LLS algorithm have identical estimation performance
as long as the (M − 1) linear equations correspond to the independent set [10]; (ii) The covariance
matrices of the position estimates in the BLUE-LLS scheme with the independent set and the BLUE
version of the LSC algorithm are identical. By comparing with Cram´er-Rao lower bound (CRLB)
for TOA-based localization [14], it is then shown that they are suboptimal estimators, and this result
is different from the iterative BLUE estimator of [13] which gives maximum likelihood estimation
performance; and (iii) Among the BLUE-LLS and BLUE-LSC algorithms, the latter is preferable as
it involves lower computational complexity. Note that the research results can also be applied to
source localization systems with received signal strength [2] measurements as they employ the same
trilateration concept where the propagation path losses from the source to the sensors are measured
2
This paper is a postprint of a paper submitted to and accepted for publication in IET – Signal Processing
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
to give their distances.
The organization of this paper is as follows. In Section 2, we first develop the weighted versions
of the LSC and LLS methods based on BLUE. The equivalences between various forms of the BLUE-
LLS solutions within the independent set and the BLUE-LSC estimate are then proved. Furthermore,
their suboptimality and computational requirement will be discussed. Simulation results are included
in Section 3 to evaluate the estimation performance of the BLUE-LSC and BLUE-LLS algorithms
by comparing with the LSC, LLS and constrained weighted LSC [14] methods as well as verify our
theoretical development. Finally, conclusions are drawn in Section 4.
2 Best Linear Un biased Estimator based Positioning
In this Section, we first present the signal model for TOA-based localization. The BLUE-LSC and
BLUE-LLS algorithms are then devised from the LSC and LLS formulations, respectively. Their
relationship, estimation performance and computational complexity are also provided.
Let (x, y)and(x
i
,y
i
), i =1, 2,...,M, be the unknown source position and the known coordinates
of the ith sensor, respectively. With known signal propagation speed, the range measurements between
the source and sensors are straightforwardly determined from the corresponding TOA measurements,
which are modelled as
r
i
= d
i
+ n
i
,i=1, 2,...,M (1)
where d
i
=
(x − x
i
)
2
+(y − y
i
)
2
is the noise-free range and n
i
is the noise in r
i
. For simplicity, we
assume line-of-sight propagation between the source and all sensors such that each n
i
is a zero-mean
white process with known variance σ
2
i
[14].
2.1 BLUE-LSC Algorithm
BLUE [12] is a linear estimator which is unbiased and has minimum variance among all other linear
estimators. In order to employ the BLUE technique, we need to restrict the parameters to be estimated
linear in the data. It is suitable for practical implementation as only the mean and covariance of the
data are required and complete knowledge of the probability density function is not necessary. The
BLUE version of the LSC estimator is derived as follows.
Squaring both sides of (1), we have [9]:
x
i
x + y
i
y − 0.5R =
1
2
x
2
i
+ y
2
i
−r
2
i
+ m
i
,i=1, 2,...,M (2)
where m
i
= n
2
i
/2+d
i
n
i
and R = x
2
+ y
2
is the introduced variable to reorganize (1) into a set of
linear equations in x, y and R. To facilitate the development, we express (2) in matrix form:
Aθ + p = b (3)
3
This paper is a postprint of a paper submitted to and accepted for publication in IET – Signal Processing
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
where
A=
⎡
⎢
⎢
⎢
⎣
x
1
y
1
−0.5
.
.
.
.
.
.
.
.
.
x
M
y
M
−0.5
⎤
⎥
⎥
⎥
⎦
θ =
⎡
⎢
⎢
⎣
x
y
R
⎤
⎥
⎥
⎦
p=
⎡
⎢
⎢
⎢
⎣
−m
1
.
.
.
−m
M
⎤
⎥
⎥
⎥
⎦
and
b=
1
2
⎡
⎢
⎢
⎢
⎣
x
2
1
+ y
2
1
− r
2
1
.
.
.
x
2
M
+ y
2
M
− r
2
M
⎤
⎥
⎥
⎥
⎦
For sufficiently small noise conditions, p ≈ [−d
1
n
1
···−d
M
n
M
]
T
and E{r
2
i
}≈d
2
i
, i =1, 2, ···,M,
where
T
denotes transpose operation and E is the expectation operator. Hence we have E{b}≈Aθ
which corresponds to the linear unbiased data model. Using the information that p is approximately
zero-mean and its covariance matrix, denoted by C
p
, is a diagonal matrix of the form:
C
p
≈
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
d
2
1
σ
2
1
0 ··· 0
0 d
2
2
σ
2
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
00··· d
2
M
σ
2
M
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(4)
The BLUE for θ based on (3), denoted by
ˆ
θ,isthen[12]:
ˆ
θ =(A
T
C
−1
p
A)
−1
A
T
C
−1
p
b (5)
where
−1
represents matrix inverse. Note that the LSC estimate is given by (5) with the substitution
of C
p
= I
M
where I
M
is the M ×M identity matrix, without utilizing the mean and covariance of the
data. Since {d
i
} are unknown, they will be substituted by {r
i
} in practice. The covariance matrix for
ˆ
θ, denoted by C
θ
,is[12]:
C
θ
≈ (A
T
C
−1
p
A)
−1
(6)
4
This paper is a postprint of a paper submitted to and accepted for publication in IET – Signal Processing
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
where the variances for the estimates of x and y aregivenbythe(1, 1) and (2, 2) entries of C
θ
,
respectively. It is worthy to mention that the same weighting matrix of C
−1
p
has been proposed in
[14], which can be considered as a constrained weighted least squares calibration (CWLSC) algorithm
with utilizing the constraint of x
2
+ y
2
= R. We expect that the BLUE-LSC algorithm is inferior to
the CWLSC scheme as the parameter relationship in θ is not exploited.
2.2 BLUE-LLS Algorithm
On the other hand, subtracting the first equation of (2) from the remaining equations, R can be
eliminated and we get (M − 1) equations:
(x
i
− x
1
)x +(y
i
− y
1
)y =
1
2
x
2
i
+ y
2
i
− x
2
1
− y
2
1
− r
2
i
+ r
2
1
+ m
i
− m
1
,i=2, 3,...,M (7)
Expressing (7) in matrix form yields
Gφ + q = h (8)
where
G=
⎡
⎢
⎢
⎢
⎣
x
2
− x
1
y
2
− y
1
.
.
.
.
.
.
x
M
− x
1
y
M
− y
1
⎤
⎥
⎥
⎥
⎦
φ =
⎡
⎣
x
y
⎤
⎦
q=
⎡
⎢
⎢
⎢
⎣
m
1
− m
2
.
.
.
m
1
− m
M
⎤
⎥
⎥
⎥
⎦
and
h=
1
2
⎡
⎢
⎢
⎢
⎣
x
2
2
+ y
2
2
− x
2
1
− y
2
1
− r
2
2
+ r
2
1
.
.
.
x
2
M
+ y
2
M
− x
2
1
− y
2
1
− r
2
M
+ r
2
1
⎤
⎥
⎥
⎥
⎦
Following the development of the BLUE-LSC algorithm, the BLUE-LLS estimate for φ based on (8),
denoted by
ˆ
φ,is:
ˆ
φ =(G
T
C
−1
q
G)
−1
G
T
C
−1
q
h (9)
5
This paper is a postprint of a paper submitted to and accepted for publication in IET – Signal Processing
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.