wang2009a.

wang2009a.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009
4777
Ranging Energy Optimization for Robust Sensor
Positioning Based on Semidefinite Programming
Tao Wang, Geert Leus, Senior Member, IEEE, and Li Huang, Member, IEEE
Abstract—Sensor positioning is an important task of location-aware wireless sensor networks. In most sensor positioning
systems, sensors and beacons need to emit ranging signals to
each other. Sensor ranging energy should be low to prolong
system lifetime, but sufficiently high to fulfill prescribed accuracy
requirements. This motivates us to investigate ranging energy
optimization problems. We address ranging energy optimization
for an unsynchronized positioning system, which features robust
sensor positioning (RSP) in the sense that a specific accuracy requirement is fulfilled within a prescribed service area. We assume
a line-of-sight (LOS) channel exists between the sensor and each
beacon. The positioning is implemented by time-of-arrival (TOA)
based two-way ranging between a sensor and beacons, followed
by a location estimation at a central processing unit. To establish
a dependency between positioning accuracy and ranging energy,
we assume the adopted TOA and location estimators are unbiased
and attain the associated Cramér–Rao bound. The accuracy requirement has the same form as the one defined by the Federal
Communication Commission (FCC), and we present two constraints with linear-matrix-inequality form for the RSP. Ranging
energy optimization problems, as well as a practical algorithm
based on semidefinite programming are proposed. The effectiveness of the algorithm is illustrated by numerical experiments.
Index Terms—Cramér–Rao bound, localization, semidefinite
programming, wireless sensor networks.
I. INTRODUCTION
IRELESS sensor networks (WSNs) enable a rich
variety of promising applications, and therefore have
attracted intensive research interest lately [1]. Typical WSNs
consist of untethered sensors randomly deployed to collect
application-specific measurements, as well as a few fusion
centers for in-network data processing. For most WSNs, sensor
positions have to be estimated first, because they are often
indispensable to annotate sensed data. For instance, temperature data produced by an environment-monitoring WSN are
W
Manuscript received December 16, 2008; accepted June 18, 2009. First published July 21, 2009; current version published November 18, 2009. The associate editor coordinating the review of this manuscript and approving it for
publication was Dr. Brian M. Sadler. Part of this paper was presented at International Conference on Acoustics, Speech and Signal Processing (ICASSP),
Taipei, Taiwan, April 19–24, 2009.
T. Wang was with the Faculty of Electrical Engineering, Mathematics and
Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands. He is also with the Stichting IMEC Nederland, Holst Centre, Eindhoven,
The Netherlands (e-mail: [email protected]).
G. Leus is with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands
(e-mail: [email protected]).
L. Huang is with the Stichting IMEC Nederland, Holst Centre, Eindhoven,
The Netherlands (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2009.2028211
useless unless we know where they are sensed. Furthermore,
some WSN applications, e.g., target tracking, require the sensor
positions to be known a priori [2]. Especially for large-scale
WSNs, the sensor positions are required for self-organization,
such as naming [3], routing [4], as well as ciphering [5], just to
name a few. Therefore, sensor positioning has been a research
topic of particular interest over the past few years.
To support location-aware WSNs, sensor positions must be
estimated reliably with prescribed accuracy requirements fulfilled. Perhaps the most well-known requirement is the one defined by the Federal Communication Commission (FCC) [6].
More specifically, it requires the location estimation error to
with probability higher than ,
have a length smaller than
and
have a prescribed value. Although this
where both
requirement has been introduced to regulate the localization of
mobile users, it can be considered to prescribe an accuracy requirement for sensor positioning as well.
In practice, it is costly to equip each sensor with a global positioning system (GPS). Instead, a few beacons, which are sensors
or fusion centers with known positions, are encompassed in
most WSNs for locating the rest of the sensors in two steps. In
the first step, beacons and sensors are scheduled to emit ranging
signals to each other, and some signal parameters related to
sensor positions are measured. Possible parameters include
time-of-arrival (TOA), time-difference-of-arrival (TDOA), received-signal-strength (RSS), as well as angle-of-arrival (AOA)
[7], [8]. In the second step, these measurements are transformed into distance or bearing information, from which the
sensor positions are estimated by a specific algorithm with
the use of the beacons’ positions. In practice, all the above
position-dependent parameters carry information about the
sensor position, and an optimal localization algorithm should
make use of all information related to the sensor position.
However, the complexity may be too high especially when
the localization task is partly carried out by the sensor. As a
result, practical two-step algorithms usually estimate one or
several dominant position-dependent parameters in the first
step, and then estimate the sensor position in the second step.
This incurs a loss of position-related information, and therefore
practical algorithms have suboptimal performance in general.
However, they are preferred for WSNs where the reduction of
implementation complexity is a big concern.
To locate sensors with high accuracy, TOA measurements
are preferred, since TDOA measurements require multiple synchronized beacons, RSS measurements need an accurate pathloss model, and antenna arrays are necessary for AOA measurements [7]. Thanks to the superior penetration and resolution
capability of ultrawideband (UWB), TOA-based ranging using
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UWB pulses has been regarded as the most promising candidate for WSNs. Especially for unsynchronized WSNs, TOAbased two-way ranging (TWR) is proposed and supported by
the IEEE 802.15.4a standard [9]. Recently, a variety of work on
TOA-based ranging has been reported in [10]–[17]. Particularly,
the associated CRBs have been derived to benchmark the performance of a particular algorithm [7], [18]–[21].
As for the second step, two types of positioning algorithms
have been developed. There is a rich literature on noncollaborative positioning algorithms, which consider only the aforementioned ranging between the sensor and the beacons [22]–[31].
Recently, collaborative positioning algorithms, which employ
additional inter-sensor ranging for performance improvement,
have been developed [32]–[37]. The associated CRBs have been
derived in [8], [38]–[40], and [41]–[44] to benchmark the performance of a noncollaborative and a collaborative algorithm,
respectively.
For the above two-step algorithms, positioning accuracy improves if the ranging energy of sensors and beacons is enhanced.
In real scenarios, a beacon might have a reliable power supply
and its ranging energy can be easily increased, but the ranging
energy of an untethered sensor must be reduced in order to prolong system lifetime. Therefore, the positioning accuracy is actually dominated by the sensor ranging energy, which should be
small but sufficiently high to fulfill prescribed accuracy requirements. This motivates us to investigate the following ranging
energy optimization problem: how to allocate the ranging energy to sensors and beacons, so that the sensor ranging energy
is minimized and specific accuracy requirements are fulfilled as
well?
We will address this problem for an unsynchronized robust
sensor positioning (RSP) system, which consists of power-supplied beacons connected to a central processing unit (CPU), as
well as sensors randomly deployed within a prescribed service
area. We assume a line-of-sight (LOS) channel exists between
the sensor and each beacon. The positioning is implemented by
TOA-based TWR between a sensor and the beacons, followed
by a location estimation at the CPU with a noncollaborative algorithm. In particular, this system features RSP, in the sense
that a specific accuracy requirement is fulfilled within a prescribed service area. To reduce the implementation complexity,
the ranging energy of both the sensor and the beacons is fixed
and determined during the system design phase. In order to establish a mathematical dependency between the positioning accuracy and the ranging energy, we assume the adopted TOA estimator achieves its CRB with an unbiased Gaussian distribution,
and the positioning CRB is achieved by an unbiased location
estimator. The motivation behind these assumptions is twofold.
One is that although our assumptions seem too optimistic in
practice, the associated positioning CRB has a very attractive
mathematical structure which lends the proposed problems to
be efficiently solved by semidefinite programming (SDP). The
other is that the optimal ranging energy allocation for a real scenario is actually lower bounded by the result produced by SDP
for the optimistically assumed scenario. Therefore, our assumptions lead to useful optimization results, which can not only be
found efficiently by SDP, but also provide a sense of how much
energy should be allocated at least.
Fig. 1. An exemplary sensor positioning system, where the circular area and
the small circles represent the service area and sensors, respectively.
The rest of this paper is organized as follows. In the next
section, we will derive the performance of the considered TOA
and location estimators. Next, Section III will present the RSP
constraints when the location estimate is Gaussian distributed,
and when such statistical knowledge is unavailable. After that,
we will propose ranging energy optimization problems and a
practical algorithm based on SDP in Section IV. In Section V,
we will illustrate the effectiveness of the proposed algorithm by
numerical experiments. Finally, we will wrap up this paper by
some conclusions in Section VI.
II. SYSTEM SETUP AND PERFORMANCE ANALYSIS
In this section, we will first describe a two-dimensional (2D)
RSP system. Next, a TOA-based TWR procedure is introduced
and its performance is derived. Finally, the performance of the
adopted positioning algorithm, which is actually the positioning
CRB, is derived.
A. System Setup
We study a 2D RSP system with
beacons deployed and
connected to a CPU through wired or radio links
.
is at a known coordinate
,
Beacon
, and we will consider locating a sensor at an unknown
within a prescribed service area .
coordinate
The distance between the sensor and beacon
is denoted as
, where
represents the -norm operator. In addition, the clocks of the sensor and the beacons are
unsynchronized but run at the same pace. We assume that the
two-sided power spectral density (PSD) of the additive white
Gaussian noise (AWGN) at the sensor and the beacons is
and
, respectively. For illustration purposes, Fig. 1 shows
an exemplary system with a circular service area and a few
sensors.
We assume a LOS channel exists between the sensor and
and attenbeacon , which incurs a propagation delay
. Here represents the signal propagation speed and
uation
, where and refer to the path gain at 1 m
and the path-loss coefficient, respectively. We assume and
are both known by system designers.
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During the ranging phase, the sensor and beacon
are
scheduled to broadcast respectively the ranging signals
and
. In practice, a ranging signal can be the preamble
part of a communication signal between the sensor and beacon
[9]. Note that these signals are not necessarily UWB in
general. Finally, we assume the channel remains unchanged
are
during the ranging phase, and both the signal TOA and
regarded as unknown deterministic parameters by the adopted
TOA and location estimators.
It has been shown in [7], [20], and [45] that the CRB of TOA
estimation depends on the root-mean-square (RMS) angular frequency of the adopted ranging signals. We assume all beacon
ranging signals have the same RMS angular frequency. To facilitate the following derivations, we define the RMS angular
and
respectively as
frequency of
[18]–[20] for scenarios similar to the one considered here. For
self-consistency purposes, we put a brief derivation of the CRB
is Gaussian disin Appendix A, where we show that
tributed as
, and
is expressed by
(4)
Similarly,
has
, where
the Gaussian
is evaluated as
distribution
(5)
and
are independent since they
We can see that
is disare generated from independent signals. Therefore,
tributed as
, where
is given by
(6)
(1)
(7)
(2)
where
and
, respectively.
represent the spectrum of
and
and
. In fact, reprewith
sents the TOA estimation accuracy of the sensor relative to that
(
means that the TOA estiof beacon when
mation of the sensor is more accurate).
C. Performance of the Positioning Algorithm
B. TOA-Based TWR and Performance
TOA-based TWR proceeds as follows [9]. First, the CPU
schedules the beacons to broadcast ranging signals sequentially,
so that they are separated when arriving at the sensor. Let’s say
broadcasts
of energy
at time
. At the
beacon
sensor, the LOS signal’s TOA
is estimated and denoted
. After the sensor has generated all the TOA estimates,
as
of energy
at time
to the beacons. At
it broadcasts
is estimated and debeacon , the LOS signal’s TOA
noted as
. We assume hardware calibration is perfectly
and
accomplished by the sensor and beacon , so that
are precisely known by the sensor and beacon , respectively.
and
are recorded by the
It is important to note that
sensor with its internal clock, while
and
are recorded
by beacon with its internal clock. Finally, all the processing
delays
produced by the sensor are first transmitted through data packets to the beacons, and then sent to
generated by
the CPU. Meanwhile, the total delay
beacon is also transmitted to the CPU, which evaluates
as
(3)
Ideally, if
and
are precisely estimated,
is
. However, the estimation performance is degraded
equal to
by the AWGN. It is well known that the variance of any unbiased estimator is lower-bounded by the CRB. We assume each
TOA estimator for the TWR achieves the CRB with an unbiased Gaussian distribution. In fact, this can be asymptotically
accomplished by the maximum-likelihood (ML) estimator [10],
[18]. The CRB has been derived in [21] when the distribution
is known a priori, and in [7],
of the propagation delay and
After TWR, a set of independent measurements
is available at the CPU for estimating . For any unbiased losets a lower-bound to its covarication estimator, the CRB
ance. The CRBs for similar scenarios have been derived in [8]
and [38]–[40]; however, they did not consider the presence of
as nuisance parameters, and thus the results reported
can be
there can’t be used here. In Appendix B, we show
evaluated according to
(8)
, and
. Here,
can be
regarded as the effective energy that combines the joint effect
and
on
. Note that
, and
of
is an increasing function of both
and
.
One may assume that the location estimate follows the
. Such an assumption is attracGaussian distribution
tive because the Gaussian distribution has salient mathematical
properties and therefore can ease theoretical analysis. For
instance, this assumption was adopted for localization performance analysis in [8] and [46]. In addition, the Gaussian-distribution assumption can be justified since the CRB can be
achieved with an unbiased Gaussian distribution by the ML
estimator, provided that we have a large set of independent
and identically distributed (i.i.d.) ranging measurements [45].
Although such a large set is usually unavailable in practice, it
has been shown in [25] that the estimator based on constrained
weighted least-squares could approach the CRB if the ranging
accuracy is sufficiently high, and the distribution of the location
estimate resembles an unbiased Gaussian distribution.
where
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On the other hand, although the location estimators
based on approximate ML [24], [27], and multidimensional
scaling (MDS) [31] can also achieve the CRB, they do not
necessarily follow a Gaussian distribution.
Based on the above considerations, we assume is unbi, while leaving the assumpased and its covariance attains
tion about the distribution of open. In the next section, we
will derive constraints for RSP when is Gaussian distributed,
as well as when the knowledge about the distribution of is
unavailable.
location-dependent functions. We can see that a higher accuracy
and increasing .
level is expected by reducing
In the following subsections, we will propose constraints for
RSP when is Gaussian distributed, as well as when the knowledge about the distribution of is unavailable.
A. Gaussian Distribution of the Location Estimate
In this case, a sufficient condition to satisfy the accuracy re,
quirement at has been introduced in [8] as
where
is the minimal eigenvalue of
, and
is the threshold translated from
and
according to
D. Comments
It is important to examine more closely the assumptions
we made. First, we assumed the TOA estimation CRB can
be achieved. However, it was shown in [47], [48] that this
CRB is in nature a local bound and not tight for scenarios
with low signal-to-noise ratio (SNR). Alternatively, one may
consider the improved Ziv–Zakai bound (ZZB) as a better
metric. Second, we assumed a LOS channel exists between the
sensor and each beacon, which may be too optimistic for real
scenarios. Especially in indoor or dense urban environments,
the LOS path may not exist and even if it exists, there may exist
strong multipath interference on the LOS signal. Third, we
assumed each ranging estimate is only degraded by the TOA
estimation error and thus Gaussian distributed. In practice, it
may be further degraded by the random behavior of queuing
the ranging signals at the sensor and beacons.
Our assumptions are motivated by the fact that if we take
other bounds, such as the ZZB, or more realistic assumptions,
finding the optimal ranging energy allocation for the proposed
problems is generally very difficult. Although our assumptions
seem too optimistic in practice, the associated CRB computed
from (8) has a very attractive mathematical structure which
lends the proposed problems to be efficiently solved by SDP.
Furthermore, the optimal ranging energy allocation for a real
scenario is actually lower bounded by the result produced by
SDP for the optimistically assumed scenario. Therefore, our
assumptions lead to useful optimization results, which can not
only be found efficiently by SDP, but also provide a sense of
how much energy should be allocated at least.
Note that it is very attractive to investigate the gap between
the optimal energy allocation under our assumptions and that for
real scenarios, when more knowledge is available about the statistics of the channels, the signal queuing, and the implemented
TOA and position estimation algorithms. Intuitively, this gap
should be translated from the localization performance loss due
to the impairments present in a real scenario but ignored by our
assumptions. This topic is part of our future work.
III. CONSTRAINTS FOR RSP
To support location-awareness reliably, we require the sensor
to be located with sufficiently high accuracy. To this end, we
impose an RSP requirement on the considered system using the
FCC definition. More specifically, we require that for every
within , the location estimation error
falls into the
with probability higher than
origin-centered circle of radius
, i.e.,
,
. Note that
and
can be prescribed according to the accuracy level expected by
(9)
This sufficient condition can be justified as follows. Since is
distributed as
, it was shown in [46] that falls within
an ellipse expressed by
with probability . This ellipse has a major principal axis
. When
of length
is satisfied,
is contained within the circle
, which means that
. Therefore,
is sufficient to fulfill the accuracy requirement at .
Note that
is equivalent to the linear matrix
inequality (LMI)
, where is a 2
2 identity matrix, and
means that
is a positive
semidefinite matrix. According to the convex optimization
theory, the set of ’s satisfying this LMI is convex [49]. With
this constraint, convex optimization methods can be used
to effectively solve the optimization problems proposed in
Section IV.
To satisfy the RSP requirement, we require the considered
system to fulfill this sufficient condition for every position
within , namely
,
.
B. Unknown Distribution of the Location Estimate
In this case, a sufficient condition to satisfy the accuracy re. In the folquirement at will be derived as
lowing, we will show that can be translated from
and
according to
(10)
This sufficient condition can be derived according
to the Chebyshev’s inequality, which is of the form
for the considered system
denotes the trace of
and is equal
[50]. Here,
to the mean square error of . Note that this inequality
holds for any particular distribution of , and thus can be
used to develop a sufficient condition when the knowledge about the distribution of
is unavailable. As a result,
, and it is sufficient to
satisfy the accuracy requirement if
, or
equivalently
. Referring to (8), each
eigenvalue of
is greater than
, which means
that each eigenvalue of
is upper bounded by
.
As a result,
. Therefore, a stronger
sufficient condition is
, or equivalently
.
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Note that
is a sufficient condition as
well. However, the associated set of feasible ’s forms a nonconvex set in general, and this will make it difficult to solve the
optimization problems proposed in Section IV. Therefore, we
choose the stronger condition
.
To satisfy the RSP requirement, we require the considered
system to fulfill the proposed sufficient condition at every posi,
.
tion within , namely
.
We present the following theorem to compute
, where
Theorem 1:
denotes the minimal eigenvalue of
.
Proof: Let’s consider the following optimization problem:
C. Comments
(12)
The second RSP constraint is stronger than the first one, since
for any
. This is because
the first one is derived with extra knowledge about the error
ellipse of a Gaussian distribution. With the second constraint,
the RSP can be achieved no matter how is distributed. This
means that using the second constraint guarantees the RSP even
though the knowledge about the distribution of is unavailable.
To facilitate the discussion in the following sections, we for,
,
mulate a general RSP constraint as:
if we know is Gaussian distributed, or
where
if we have no knowledge about the distribution of .
and
are the only optimization variables
Note that
in (12). Alternatively, (12) can be regarded as a simapproaches infinity.
plification of (11) when every
, we can
With the equality constraint
easily find that the inequality constraint in (12) reduces to
. This means that
satisfies
each feasible
. As a result, the optimal objective
.
value is
We can express
and
alternatively as
(13)
IV. RANGING ENERGY OPTIMIZATION
In this section, we will first formulate a few ranging energy
optimization problems of interest under the RSP constraint proposed in Section III. Then, we will present a practical algorithm
to solve them based on semidefinite programming (SDP). Note
, are all treated as optimization variables
that ,
on which the equality constraints
,
are imposed.
A. Sensor Ranging Energy Optimization Problem
We first prove that
is a nondecreasing function
,
. Suppose
is increased by a positive
of
. As a consequence, ,
and
bevalue
come ,
and
,
is a positive semidefinite matrix,
respectively. Since
according to Corollary 4.3.3 in [51],
is nondecreasing
which justifies our claim. Since
with
, the RSP constraint can be fulfilled by increasing
entries of , which in turn is accomplished by enhancing
and
.
, where
It is important to note that
denotes the -norm of , and the equality
increases to be infinity. To prolong
holds when at least one
should be reduced as much as possible.
system life time,
is reduced too much, every entry of will
However, in case
be bounded by a too small
to fulfill the RSP constraint. This
for , above which RSP
motivates us to find a threshold
can be computed as the
becomes possible. Mathematically,
optimal objective value of the following optimization problem:
(11)
where
and
are the feasible sets of
for the problems
(11) and (12), respectively.
since
. Suppose now that
Apparently,
. This means that there exists at least one particular
that fulfills the constraints of (11) and
. We can
build a vector
. Obviously, each entry of
is no smaller than that of . Since
is nonde,
, and therecreasing with
is feasible for (12). This means
is a possible
fore
objective value of (12), but this value is smaller than
, and
is the minimal objective value of (12) is
thus the fact that
. Therefore,
violated. This contradicts the assumption
.
We can see that
is actually the minimal -norm of any
fulfilling the RSP constraint. This implies that at least one
has to be infinitely high, in case
is reduced to its lowest level
. It is interesting to observe that, if is increased from
to ,
has to be increased by
dBJ, where
. In effect, this increase can
be regarded as the extra sensor energy required to compensate
for the unavailable knowledge about the distribution of when
ideally approaches infinity.
It is also interesting to study the impact of the path loss
on
. In order to facilitate the following
coefficient
and,
discussion, let’s denote
thus,
. In fact,
where
. Suppose after
increases,
and
change to
and
, respectively. Since
reis positive semidefinite,
duces and each
holds according to Corollary 4.3.3 in [51]. Suppose
, we have
. This means that is
nonincreasing with . Therefore, the threshold sensor energy
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will not reduce, if the signal energy decays faster with respect
to the distance.
B. Sensor and Beacon Ranging Energy Optimization Problems
In real scenarios, we can not simply adopt the energy alloca,
for at least one beacon, because
tion
is usually constrained by a prescribed value
due to implementation difficulties, e.g., short ranging duration or limited
power due to power-amplifier nonlinearity. Note that the semust be no smaller than
in order to fulfill the
lected
since
RSP constraint. This is because
and
hold. Besides,
must be satisfied in order to fulfill the RSP constraint. Theremust be no smaller than
in order to fulfill the
fore,
RSP constraint.
and
The optimization problem now is to find the minimal
the associated
which meet the RSP constraint. Mathematically, this problem can be cast into the following form:
contains more than one element, since using any
other
, another solution with less beacon energy usage can
than
be constructed. For all those solutions, the optimal value of
remains fixed at
. Apparently, this
, and the gap
between
and
value is greater than
can be computed in dBJ as follows:
(18)
We can see that
reduces as
increases, which means
can approach
closer if
increases. In practice,
that
and
,
(18) can be used to establish a tradeoff between
which is helpful to guide ranging hardware design of beacons
,
increases
and sensors. We can also see that for a fixed
increases. This means that if changes from
to ,
as
and
becomes greater.
the gap between
Instead of using the solution in (17), another solution with a
more efficient use of beacon energy can be constructed by using
obtained by solving the following problem:
in (16) the
(19)
(14)
,
Let’s denote the optimal solution to (14) by
and . We present a general solution to the above problem with
the following theorem.
Theorem 2: The general solution to (14) is
(15)
(16)
, which belongs to the set
.
Proof: First of all, after some simple mathematical manipand
given above satisfy the
ulations we can see that the
must be no less than
equality constraint in (14). Note that
in order to fulfill the RSP constraint, and thus
is nonnegative. Besides, since
,
and hence
is also nonnegative. Therefore,
and
belong to the
feasible set of (14).
Second,
is the optimal value of , bebecomes smaller than that value, the RSP concause once
. The reason for
straint can not be satisfied since
,
this is that for any
when
and
, since
is
and
.
increasing with
Third, when
,
. To fulfill the RSP
must be equal to
and thus must belong
constraint,
can be found by (16).
to . Therefore,
At least one solution to (14) exists and is given by
where
is the
th entry of
(17)
In fact, this solution is constructed with
in . This solution is the worst one if
where
is a weighting vector. We assume
it is designed under the constraints that
and
. Particularly, assigning a greater
repre. One special case
sents a stronger expectation to reduce
of interest is to set all entries of
to zero except for the th
and
if
. Using the associated
entry, i.e.,
’s by (16) reduce the
optimal for (19), the constructed
ranging energy of beacon to its minimal possible value with
the RSP constraint satisfied.
In general, no closed-form solution exists for (19).
is equivalent to the LMI
Nevertheless,
, and, thus, (19) actually belongs to the
class of SDP problems, which can be solved numerically with
convex optimization techniques [49].
C. A Practical Algorithm
To evaluate
and solve (19), the main difficulty lies in the
fact that is in general a continuous area. A practical algorithm
is realized by replacing with a discrete grid set
,
is the th grid point within . Then, we can evaluate
where
and solve (19) using numerical convex optimization software, such as Sedumi [52].
Suppose that is generated by sampling uniformly in both
. Ideally,
vertical and horizontal directions with a spacing
should approach zero as close as possible. But if
is too small,
the optimization problems will go beyond the processing ability
of the optimization software, since too many LMI constraints
are produced. This motivates us to choose a reasonably small
. To this end, let’s consider
and
, which devalue for
note the sets of fulfilling the RSP constraint over and ,
, and
should be suffirespectively. Obviously,
.
ciently small such that
, consider a square cell
To find a principle for choosing
of lateral length centered at a grid point
. Such a
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WANG et al.: RSP BASED ON SEMIDEFINITE PROGRAMMING
Fig. 2. An exemplary square cell C
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and its position relative to three beacons.
cell is shown in Fig. 2 for illustration purposes. Within this cell,
has the following property.
is approximately a concave function
Theorem 3:
, provided that
, where
of
refers to the distance of to the closest
holds at four corner
beacon. In addition, if
points
,
for every
.
,
,
Proof: When
and
and thus
hold for every
.
Therefore,
can be approximated as follows:
(20)
and
.
where
operator is an
Obviously, the function inside the last
affine function of parameterized by and , so
is approximately a pointwise minimum of this function over .
is approximately a concave function
Therefore,
[49].
of
In fact,
is the convex hull formed by its corner points
. This means that
, there exists a set
where
and
, such that
[49]. If
holds,
is satisfied
within
because of the concavity of
over
(21)
Based on Theorem 3, if the RSP constraint is fulfilled at all
corner points, the RSP constraint will be satisfied over the whole
. As a result,
should be chosen according to
Fig. 3. System setup and subregions.
. Under this condition, using any fulfilling
the RSP constraint over , the RSP constraint is satisfied over
as well, because the RSP constraint is satisfied within each cell
formed by four adjacent points in , and those cells cover .
To use the above sampling condition, we assume that all beacons lie outside . It is important to note that, if one beacon is
will be very small. In order to
very close to ,
reduce the size of , we can sample nonuniformly with the
following method. First, divide into subregions
.
Then we sample each region uniformly with a spacing
to produce a discrete set . Finally, is pro’s as:
. Based on a
duced by combining all
similar analysis as that for uniform sampling, we can then jus. Apparently,
, so has a size no greater
tify
than that produced by uniform sampling.
V. NUMERICAL EXPERIMENTS
For illustration purposes, numerical experiments have been
conducted using the system setup shown in Fig. 3. Suppose
is a square area centered at (0,0) with lateral length 2 m. There
,
,
are three beacons located at
, respectively. Note that the unit for all those
and
,
values is meter. The system parameters are set as:
,
, and
. We will set
in the following experiments unless otherwise stated. All computations are performed
with Matlab v7.1 and Sedumi v1.1 on a laptop equipped with
an AMD Turion CPU of speed 2.2 GHz and a memory of 2 G
Bytes.
to be fixed at 6 cm.
For the RSP constraint, we prescribe
from 0.7
As we expect a better accuracy level by increasing
to 0.95,
increases from 28.3 to 32.2 dB as shown in Fig. 4.
to , the increase of
in dBJ is
When changes from
, which is shown in Fig. 4 as well. In
equal to
effect, this increase is the extra sensor energy required to compensate for the unavailable knowledge about the distribution of
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009
Fig. 4. The computed results for and =
when R
= 6 cm.
Fig. 5. Number of constraints and computation time.
, when
attains its lowest level
and at least one
approaches infinity. It is shown that this extra energy is between
4.4 and 8.2 dBJ, and a higher demands a higher extra energy.
which corIn the following experiments, we choose
responds to an extra energy of about 5.6 dBJ.
, should be replaced with a discrete set .
To evaluate
Both uniform and nonuniform sampling are performed to produce . For the uniform sampling, we use a spacing
where
. For the nonuniform sampling, we
divide into four subregions
by
and
as
is sampled uniformly with a spacing
shown in Fig. 3, and
where
. Here, is a parameter for tuning the sampling spacing. During the evaluation, we
the computed
remains essentially
find that when
and
,
unchanged as 4.32 and 1.14 dBJ for
respectively. This implies that
is quite close to
when
is below 1%. When the nonuniform sampling is used instead of
Fig. 6. The computed results for E
Fig. 7. The computed results for
, respectively.
=
E
= with respect to .
with respect to
E
when
=
and
the uniform sampling, the number of constraints and computation time is reduced by about 17% as shown in Fig. 5. Hence, we
replace by produced with the nonuniform sampling using
for the following experiments.
, we have evaluated
In order to show the effect of on
when varies from 2 to 4, and the results are shown in
increases with . This means for
Fig. 6. It is shown that
will increase when signal energy decays faster
a fixed ,
with respect to distance.
increases from 5 to 12 dBJ, we have computed
When
when
and
, respectively. The results are
increases,
reduces.
shown in Fig. 7. It is shown that as
,
is significantly higher than when
.
When
, which
In the following experiments, we choose
corresponds to
and 0.60 dBJ, or equivalently
and 1.75 dBJ, when
and
, respectively.
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WANG et al.: RSP BASED ON SEMIDEFINITE PROGRAMMING
4785
should be increased from 7.11 to 9.79 dBJ. These energy
increases correspond to the extra sensor and beacon energy required to compensate for the unavailable knowledge about the
distribution of , when the energy of beacon 1 is used most
efficiently.
VI. CONCLUSION
Fig. 8. The computed energy allocation to E and E
when
w = [1; 0; 0]
.
We have addressed ranging energy optimization for an unsynchronized positioning system, which features RSP in the sense
that a specific accuracy requirement is fulfilled within a prescribed service area. We assume a LOS channel exists between
the sensor and each beacon. The positioning is implemented by
TOA-based TWR between a sensor and beacons, followed by
a location estimation at a CPU. To establish a dependency between positioning accuracy and ranging energy, we assume the
adopted TOA and location estimators are unbiased and attain the
associated CRB. The accuracy requirement has the same form as
that defined by the FCC, and we have presented two constraints
with LMI form for RSP. Under these constraints, ranging energy
optimization problems, as well as a practical algorithm based on
SDP have been proposed. We have illustrated the effectiveness
of the algorithm by numerical experiments. Although only a 2D
system is considered here, the proposed methods can be easily
extended to a 3D RSP system. Besides, these methods are not
just for UWB based positioning systems, but can be generally
applied to other systems as well, as long as the system models
and assumptions considered here hold for those systems.
APPENDIX A
DERIVATION OF THE CRB FOR ESTIMATING
Fig. 9. Error ellipses
8
and circles
8
Let’s consider the estimation of
at the sensor. The received signal can be expressed as
, where
is the AWGN with PSD
. Let’s stack
all unknown parameters into a vector
. The
log-likelihood function of can be expressed as [10]
for randomly chosen points.
(22)
To find a better beacon energy allocation than
,
we construct a solution to (14) by (16) using the that solves
subject to the RSP con(19). Let’s say we want to minimize
.
straint, so we prescribe the weighting vector as
The associated problem (19) was solved with Sedumi v1.1, and
’s constructed with the produced
are shown in Fig. 8
the
and
, respectively. The consumed CPU
when
computation time is 11.06 and 11.67 s, respectively. It is shown
is 7.11 and 9.79 dBJ, respectively, while
that the computed
and
are always 10 dBJ.
To show the effectiveness of the energy allocation
4.15 dBJ,
7.11 dBJ,
10 dBJ, and
10 dBJ
when
, we randomly select a set of points within . We
plot the error ellipse
and the circle
for each point, using
the given energy allocation. It is shown in Fig. 9 that each ellipse is enclosed by the associated circle, which indicates that
the RSP constraint is indeed satisfied for those randomly chosen
to , the energy allocation to
points. When changes from
should be increased from 4.15 to 1.75 dBJ, and that to
where keeps unchanged when changes.
The Fisher information matrix (FIM) is
, where
represents the ensemble average
is the Hessian matrix of
with
operator, and
can
respect to . Under the assumptions made in Section II,
be reduced to
(23)
where
is the energy of the differential of
. The CRB can then be evaluated as
, and thus
(24)
Since we assume the above CRB is achieved with an
is distributed as
unbiased Gaussian distribution,
.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009
APPENDIX B
DERIVATION OF THE CRB FOR THE LOCATION ESTIMATOR
. Note
Let’s consider the estimation of using
that
play the role of nuisance parameters. The
log-likelihood function of the unknown parameter vector
can be expressed as
The Fisher information matrix (FIM)
can be evaluated as
, where
is the Hessian matrix
of
with respect to . By some arrangements, we can show
:
that the following equalities hold
(25)
Using the above equalities,
can be reduced to
(26)
where
is the Hessian matrix of
with respect to
,
is an all-zero
matrix, and
is the submatrix at the right-bottom corner. It is shown in [45] that
(27)
where
is the gradient vector of
with respect to
(28)
Inserting (28) into (27), we can show that
(29)
The CRB for the location estimator is actually the 2 2 sub. By some rearrangement,
matrix at the upper-left corner of
can be evaluated according to (8).
ACKNOWLEDGMENT
The authors would like to thank D. Neirynck, F. Shu, and
G. Dolmans at Holst Centre for their suggestions. The authors
would also thank the anonymous reviewers for their valuable
comments.
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Tao Wang received the B.E. and Do.E. degrees in
electronic engineering from Zhejiang University,
China, in June 2001 and June 2006, respectively.
Since August 2000 to February 2001, he had an
internship on the design and implementation of a
microcontrol unit with Motorola Electronics Ltd.
Suzhou Branch, China. In October 2004 to March
2005, he was a visiting student with the Institute
for Infocomm Research, Singapore, working on
ultrawideband antenna design. His current research
interests are in the design of statistical signal
processing algorithms, as well as optimization of wireless localization and
communication systems.
Geert Leus (M’01–SM’05) was born in Leuven,
Belgium, in 1973. He received the Electrical Engineering and Ph.D. degrees in applied sciences from
the Katholieke Universiteit Leuven, in June 1996
and May 2000, respectively.
He has been a Research Assistant and a Postdoctoral Fellow of the Fund for Scientific Research—Flanders, Belgium, from October 1996 until
September 2003. During that period, he was with
the Electrical Engineering Department, Katholieke
Universiteit Leuven. Currently, he is an Associate
Professor with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands. During
summer 1998, he visited Stanford University, Stanford, CA, and from March
2001 until May 2002, he was a Visiting Researcher and Lecturer with the
University of Minnesota, Minneapolis. His research interests are in the area of
signal processing for communications.
Dr. Leus received a 2002 IEEE Signal Processing Society Young Author Best
Paper Award and a 2005 IEEE Signal Processing Society Best Paper Award.
He is the Chair of the IEEE Signal Processing for Communications Technical
Committee, and an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL
PROCESSING and the EURASIP Journal on Applied Signal Processing. In the
past, he has served on the Editorial Board of the IEEE SIGNAL PROCESSING
LETTERS and the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.
Li Huang (S’04–M’08) received the B.Eng. degree
in electronics and information engineering in 2002
from Huazhong University of Science Technology
(HUST), Wuhan, China, and the Ph.D. degree from
both the National University of Singapore (NUS)
and the Technical University Eindhoven (TU/e), The
Netherlands, in 2008.
Since 2006, he has been with IMEC-NL, Holst
Centre, Eindhoven, working on system-level design
for ultralow power wireless systems. His main research interests include statistical signal processing,
channel modeling, medium access control (MAC) design, wireless communications, and magnetic and optical recording.
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