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Journal ArticleDOI

Best linear unbiased estimator approach for time-of-arrival based localisation

06 Jun 2008-Iet Signal Processing (IET)-Vol. 2, Iss: 2, pp 156-162

AbstractA common technique for source localisation is to utilise 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 optimisation, least squares calibration (LSC) and linear least squares (LLS) are two computationally simple positioning alternatives which reorganise 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 utilise weighted LS estimation. In the 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 analysed. Simulation results are also included to show the comparative performance of the BLUE-LSC, BLUE-LLS, LSC, LLS and constrained weighted LSC methods with Crame-r-Rao lower bound.

Topics: Least squares (59%), Linear least squares (56%), Best linear unbiased prediction (55%), Nonlinear system (51%)

Summary (2 min read)

1 Introduction

  • Source localization using measurements from an array of spatially separated sensors has been an important problem in radar, sonar, global positioning system [1], mobile communications [2], multimedia [3] and wireless sensor networks [4].
  • For two-dimensional positioning, each noise-free TOA provides a circle centered at the sensor on which the source must lie.
  • It is computationally intensive and sufficiently precise initial estimates are required to obtain the global solution.
  • 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 their theoretical development.
  • Finally, conclusions are drawn in Section 4.

2 Best Linear Unbiased Estimator based Positioning

  • The authors 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.
  • For simplicity, the authors assume line-of-sight propagation between the source and all sensors such that each ni is a zero-mean white process with known variance σ2i [14].

2.1 BLUE-LSC Algorithm

  • BLUE [12] is a linear estimator which is unbiased and has minimum variance among all other linear estimators.
  • 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.
  • Since {di} are unknown, they will be substituted by {ri} in practice.

2.2 BLUE-LLS Algorithm

  • The estimator of (9) has minimum variance according to the data model of (8).
  • This is analogous to TOA-based and TDOA-based positioning where the former estimation performance bound is lower than that of the latter if the TDOAs are obtained from substraction between the TOAs [15]-[16].
  • In the following, the authors will prove that as long as the (M − 1) equations belong to the independent set, the BLUE-LLS estimator performance will agree with the covariance matrices given by (6) and (12).
  • Their suboptimality is then illustrated by contrasting with the CRLB.

2.3 Relationship and Performance

  • From (17), (20) and (22), the authors easily see that the estimation performance of the BLUE-LLS and BLUELSC algorithms is essentially identical.
  • Assuming that {ni} are Gaussian distributed, comparison of (12) and the CRLB for positioning is made as follows.
  • Denote the corresponding Fisher information matrix by D−1, which has the form of [14].

2.4 Complexity Analysis

  • Finally, the computational complexity of the linear equation based algorithms is investigated.
  • The numbers of multiplications and additions, denoted by M and A , respectively, required in the BLUELSC, LSC, BLUE-LLS and LLS algorithms are provided in Table I which clearly shows the calculation breakdown.
  • Note that the Gaussian elimination is employed for performing the matrix inverse operation.
  • Excluding the computationally extensive task of solving the Lagrange multiplier corresponding to the constraint of x2 + y2 = R, the CWLSC method needs (16M + 24) multiplications and (10M +7) additions.
  • The authors see that the former is preferable because it is more computationally attractive.

3 Numerical Examples

  • Computer simulations have been conducted to evaluate the performance of the BLUE-LSC and BLUELLS algorithms by comparing with the LLS, LSC and CWLSC [14] algorithms as well as CRLB.
  • It is seen that the CWLSC scheme has the best estimation performance as its MSPE attains the CRLB when the average noise power is less than 70 dBm2 where m is referenced to one meter or σ = 103.5m.
  • The theoretical development of (6) or (11) is again confirmed for sufficiently small noise conditions and the suboptimality as well as equivalence of the BLUE-LSC and BLUE-LLS methods are demonstrated.
  • As a result, the estimation performance of the methods differs at each trial because the positioning accuracy varies with the relative geometry between the source and sensors.

4 Conclusion

  • Best linear unbiased estimator (BLUE) versions of the least squares calibration (LSC) and linear least squares (LLS) time-of-arrival based positioning algorithms have been examined.
  • It is proved that various realizations of the BLUE-LLS approach are indifferent as long as the equations which correspond to the independent set are employed, and their estimation performance is identical to that of the BLUE-LSC algorithm.
  • In spite of the suboptimality of the BLUE approach, its estimation accuracy can be close to Cramér-Rao lower bound particularly when the source is located inside the region bounded by sensor coordinates.
  • Furthermore, the computational requirement of the BLUE-LSC algorithm is similar to that of the standard LSC and LLS methods and is significantly less than that of the constrained weighted LSC estimator which provides optimal positioning accuracy for sufficiently small noise conditions.

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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.

Citations
More filters

Journal ArticleDOI
TL;DR: It is proved that the performance of the improved LLS estimator achieves Cramer-Rao lower bound at sufficiently small noise conditions and the variances of the position estimates are derived and confirmed by computer simulations.
Abstract: A conventional approach for passive source localization is to utilize signal strength measurements of the emitted source received at an array of spatially separated sensors. The received signal strength (RSS) information can be converted to distance estimates for constructing a set of circular equations, from which the target position is determined. Nevertheless, a major challenge in this approach lies in the shadow fading effect which corresponds to multiplicative measurement errors. By utilizing the mean and variance of the squared distance estimates, we devise two linear least squares (LLS) estimators for RSS-based positioning in this paper. The first one is a best linear unbiased estimator while the second is its improved version by exploiting the known relation between the parameter estimates. The variances of the position estimates are derived and confirmed by computer simulations. In particular, it is proved that the performance of the improved LLS estimator achieves Cramer-Rao lower bound at sufficiently small noise conditions.

181 citations


Cites background or methods from "Best linear unbiased estimator appr..."

  • ...The first one is a best linear unbiased estimator (BLUE) [ 10 ], [21], which is shown to be equivalent to [19] and [20] but with smaller computational requirement, while the second is its improved version by exploiting the known relation between the position and range estimates....

    [...]

  • ...[ 10 ] H.-A. Loeliger, J. Dauwels, J. Hu, S. Koral, L. Ping, and F. R. Kschis-...

    [...]

  • ...Following [ 10 ], it can be proved that both BLUEs are equivalent in terms of estimation performance but the proposed LLS scheme is more computationally efficient....

    [...]

  • ...The RSS is different from the TOA and TDOA which are proportional to range [8]–[ 10 ] and range difference [11], [12], respectively....

    [...]


Book ChapterDOI
01 Jan 2012
TL;DR: This chapter contains sections titled: Introduction Measurement Models and Principles for source Localization Algorithms for Source Localization Performance Analysis for LocalizationAlgorithms and Conclusion.
Abstract: Time of arrival (TOA), time difference of arrival (TDOA), time sum of arrival (TSOA), received signal strength (RSS), and direction of arrival (DOA) of the emitted signal are commonly used measurements for source localization. This chapter introduces two categories of positioning algorithms based on TOA, TDOA, TSOA, RSS, and DOA measurements. The first category works on the nonlinear equations directly obtained from the nonlinear relationships between the source and measurements. Corresponding examples, namely, nonlinear least squares (NLS) and maximum likelihood (ML) estimators, are presented. The second category attempts to convert the equations to linear. The chapter discusses the linear least squares, weighted linear least squares (WLLS), and subspace approaches. It develops the mean and variance expressions for any positioning method which can be formulated as an unconstrained optimization problem. The Cramer‐Rao lower bound (CRLB), which is a lower bound on the variance attainable by any unbiased location estimator using the same data, is also discussed.

106 citations


Book ChapterDOI
06 Sep 2011

90 citations


Journal ArticleDOI
TL;DR: A low-complexity algorithm is proposed, which is based on the linearized equations from TOA measurements and applies a weighted least square (WLS) criterion in a computationally efficient way to closely approach the LS solution in estimation performance.
Abstract: Joint synchronization and localization using time of arrival (TOA) measurements is a very important research topic for many wireless ad hoc sensor network applications. For such TOA based joint synchronization and localization, the least square (LS) criterion and its corresponding solution have been shown to exhibit optimum estimation performance but generally at a very high computational complexity. Due to its importance and difficulty, in this paper we consider the issue how to approach the estimation performance of such LS solution at low computational complexity: We propose a low-complexity algorithm, which is based on the linearized equations from TOA measurements and applies a weighted least square (WLS) criterion in a computationally efficient way to closely approach the LS solution in estimation performance; Via analyzing and simulating its estimation performance we evidently demonstrate the proposed algorithm of its superior trade-off between estimation performance and computational complexity. The proposed algorithm is also applicable to similar application areas involving TOA base joint timing and positioning.

82 citations


Cites methods from "Best linear unbiased estimator appr..."

  • ...For TOA based joint synchronization and localization three linearization based algorithms with closed form solutions are straightforward directly from the counterparts for the TDOA or TOA based localization, see [5][10] and the references therein: One algorithm is achieved by regarding the linearizationcaused extra variable as a totally irrelevant variable, i....

    [...]


Journal ArticleDOI
TL;DR: Two computationally attractive localization methods based on the weighted least squares approach are devised for locating an unknown-position source using received signal strength (RSS) measurements in an accurate and low-complexity manner.
Abstract: Locating an unknown-position source using received signal strength (RSS) measurements in an accurate and low-complexity manner is addressed in this paper Given that the source transmit power is unknown, we employ the differential RSS information to devise two computationally attractive localization methods based on the weighted least squares (WLS) approach The main ingredients in the first algorithm development are to obtain the unbiased estimates of the squared ranges and introduce an extra variable The second method improves the first version by implicitly exploiting the relationship between the extra variable and source location through a second WLS step The performance of the two estimators is analyzed in the presence of zero-mean white Gaussian disturbances Numerical examples are also included to evaluate their localization accuracy by comparing with the maximum likelihood approach and Cramer-Rao lower bound

63 citations


References
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Abstract: (1995). Fundamentals of Statistical Signal Processing: Estimation Theory. Technometrics: Vol. 37, No. 4, pp. 465-466.

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Journal ArticleDOI
TL;DR: Using the models, the authors have shown the calculation of a Cramer-Rao bound (CRB) on the location estimation precision possible for a given set of measurements in wireless sensor networks.
Abstract: Accurate and low-cost sensor localization is a critical requirement for the deployment of wireless sensor networks in a wide variety of applications. In cooperative localization, sensors work together in a peer-to-peer manner to make measurements and then forms a map of the network. Various application requirements influence the design of sensor localization systems. In this article, the authors describe the measurement-based statistical models useful to describe time-of-arrival (TOA), angle-of-arrival (AOA), and received-signal-strength (RSS) measurements in wireless sensor networks. Wideband and ultra-wideband (UWB) measurements, and RF and acoustic media are also discussed. Using the models, the authors have shown the calculation of a Cramer-Rao bound (CRB) on the location estimation precision possible for a given set of measurements. The article briefly surveys a large and growing body of sensor localization algorithms. This article is intended to emphasize the basic statistical signal processing background necessary to understand the state-of-the-art and to make progress in the new and largely open areas of sensor network localization research.

2,909 citations


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TL;DR: Three noniterative techniques are presented for localizing a single source given a set of noisy range-difference measurements, and in one case the maximum likelihood bearing estimate is approached.
Abstract: Three noniterative techniques are presented for localizing a single source given a set of noisy range-difference measurements. The localization formulas are derived from linear least-squares "equation error" minimization, and in one case the maximum likelihood bearing estimate is approached. Geometric interpretations of the equation error norms minimized by the three methods are given, and the statistical performances of the three methods are compared via computer simulation.

688 citations


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Journal ArticleDOI
TL;DR: The proposed maximum-likelihood location estimator for wideband sources in the near field of the sensor array is derived and is shown to yield superior performance over other suboptimal techniques, including the wideband MUSIC and the two-step least-squares methods.
Abstract: In this paper, we derive the maximum-likelihood (ML) location estimator for wideband sources in the near field of the sensor array. The ML estimator is optimized in a single step, as opposed to other estimators that are optimized separately in relative time-delay and source location estimations. For the multisource case, we propose and demonstrate an efficient alternating projection procedure based on sequential iterative search on single-source parameters. The proposed algorithm is shown to yield superior performance over other suboptimal techniques, including the wideband MUSIC and the two-step least-squares methods, and is efficient with respect to the derived Cramer-Rao bound (CRB). From the CRB analysis, we find that better source location estimates can be obtained for high-frequency signals than low-frequency signals. In addition, large range estimation error results when the source signal is unknown, but such unknown parameter does not have much impact on angle estimation. In some applications, the locations of some sensors may be unknown and must be estimated. The proposed method is extended to estimate the range from a source to an unknown sensor location. After a number of source-location frames, the location of the uncalibrated sensor can be determined based on a least-squares unknown sensor location estimator.

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TL;DR: Subscriber radio location techniques are investigated for code-division multiple-access (CDMA) cellular networks, with measured times of arrival (ToA) and angles of departure (AoA) considered.
Abstract: Subscriber radio location techniques are investigated for code-division multiple-access (CDMA) cellular networks. Two methods are considered for radio location: measured times of arrival (ToA) and angles of arrival (AoA). The ToA measurements are obtained from the code tracking loop in the CDMA receiver, and the AoA measurements at a base station (BS) are assumed to be made with an antenna array. The performance of the two methods is evaluated for both ranging and two-dimensional (2-D) location, while varying the propagation conditions and the number of BS's used for the location estimate.

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Frequently Asked Questions (1)
Q1. What are the contributions in "On best linear unbiased estimator approach for time-of-arrival based localization" ?

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. 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.