Efficient Anchor Power Allocation for Location-Aware Networks

Abstract

Many future wireless applications rely on the availability of position information for mobile wireless nodes (agents). Such information can be obtained through ranging and communication between agents and fixed infrastructure (anchors). Since the transmission power of the anchors affects network lifetime, throughput, and interference, in this paper we will investigate the problem of power allocation among anchors. We start with a formulation of minimizing the squared position error bound (SPEB), which characterizes the limits of location accuracy. However, the problem is difficult to solve due to its non-convexity. To address this issue, we leverage the insights obtained from the geometric interpretation of localization information and formulate the objective to maximum directional position error bound (DPEB). We first solve the problem for the single-agent case, and then extend the algorithm for the multiple-agent case. The simulation results show that the proposed scheme achieves close-to-optimal solution but with much lower computational complexity.

Full text

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Efficient Anchor Power Allocation for
Location-Aware Networks
William Wei-Liang Li∗, Student Member, IEEE, Yuan Shen†, Student Member, IEEE,
Ying Jun (Angela) Zhang∗, Member, IEEE, and Moe Z. Win†, Fellow, IEEE
∗Department of Information Engineering, The Chinese University of Hong Kong
Shatin, New Territory, Hong Kong
Email: {wlli, yjzhang}@ie.cuhk.edu.hk
†Laboratory for Information and Decision Systems, Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Email: {shenyuan, moewin}@mit.edu
Abstract—Many future wireless applications rely on the avail-
ability of position information for mobile wireless nodes (agents).
Such information can be obtained through ranging between
agents and fixed infrastructure (anchors). Since transmission
power efficiency affects network lifetime, throughput, and inter-
ference, in this paper we will investigate the power allocation
among anchors for localization. We start with a formulation
of minimizing the squared position error bound (SPEB), which
characterizes the limits of localization accuracy. Since this prob-
lem is difficult to solve due to its non-convexity, we leverage the
insights obtained from the geometric interpretation of localization
information and formulate the objective to maximum directional
position error bound (mDPEB). We first optimize anchor power
allocation for a single agent, and then extend the algorithm
to the multiple-agent case. The simulation results show that
the proposed scheme achieves close-to-optimal solution but with
much lower computational complexity.
Index Terms—Localization, resource allocation, squared posi-
tion error bound (SPEB), second-order cone program (SOCP).
I. INTRODUCTION
Position information is of pivotal importance for future
wireless networks, since there is an increasing number of
location-based applications and services, such as cellular po-
sitioning, search and rescue work, and blue-force tracking in
battle fields. Traditional localization techniques based on the
global positioning system (GPS) usually fail to provide reliable
position information in harsh and indoor environments due to
the inability of GPS signals to penetrate most obstacles. To
address this problem, localization systems based on wideband
transmission have been introduced in GPS-challenged envi-
ronments, and such systems are able to provide high-accurate
and reliable localization performance [1], [2].
Wireless localization is referred to as determining the po-
sitions of agents based on the measurements with respective
This research was supported, in part, by the GRC grant (Project number
419509) established under the University Grants Committee (UGC) of Hong
Kong Special Administrative Region, the National Science Foundation under
Grant ECCS-0901034, the Office of Naval Research Presidential Early Career
Award for Scientists and Engineers (PECASE) N00014-09-1-0435, and the
MIT Institute for Soldier Nanotechnologies.
A
C
B
D
anchor
agent
Fig. 1: Location-aware networks: the anchors (A, B, C, and
D) transmit wireless signals to the agent.
to the nodes with position knowledge (anchors). A location-
aware network is illustrated in Fig. 1. Fundamental limits
of wideband localization has been derived in [3], [4] where
the localization performance is characterized in terms of the
squared position error bound (SPEB). Thus, a common goal
of localization algorithms is to minimize such performance
metric.
In wireless networks, transmission power affects network
lifetime, throughput, and interference. In previous literature,
transmission power control has been investigated in wireless
cellular and ad hoc networks [5], [6]. However, fewer works
have addressed the problem of power resource allocation for
localization, especially for the case when there are multiple
agents. The authors in [7] formulated several optimization
problems of anchor power allocation for different localization
scenarios. While the work provided the optimal solutions for
special cases based on geometric interpretation proposed in
[4], a general global optimal solution is not available due to
the non-convexity of the original problem.
In this paper, we leverage the insights obtained from the
geometric interpretation of localization information and cir-
cumvent the difficulty of non-convex optimization on the
SPEB. Instead, we formulate the optimization objective by
978-1-61284-231-8/11/$26.00 ©2011 IEEE
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using a new metric, called directional position error bound
(DPEB). Geometrically, this new metric is the projection
of SPEB on one dimension. The mDPEB characterizes the
maximum position error of an agent over all directions. The
new formulation is proved to be convex, and hence, facilitates
the design of efficient power allocation schemes in location-
aware networks. The major contributions of this paper are as
follows:
• We formulate a convex optimization problem by min-
imizing the mDPEB instead of the SPEB. The new
formulation is a second-order cone program (SOCP), of
which the optimal solution can be efficiently solved by
the off-the-shelf optimization tools.
• We design an anchor power allocation scheme for
multiple-agent cases. A single-stage optimization prob-
lem is formulated which is shown to be convex, and later
a two-stage iterative algorithm is proposed to improve
the computational efficiency while achieving similar lo-
calization performance.
The rest of the paper is organized as follows. In Section II,
the system model is described and a formulation based on the
SPEB is presented. In Section III, a new optimization objective
is introduced by exploiting the geometric interpretation of
localization information. Based on the new metric, we design
the anchor power allocation schemes for both single-agent
and multiple-agent cases in Section IV. In Section V, the
performance of the proposed schemes is evaluated through
simulations. Finally, the paper is concluded in VI.
II. PROBLEM FORMULATION
In this section, we first describe the system model, and
introduce a performance metric of location-aware networks.
Based on the proposed metric, we give the formulation of
anchor power allocation problem.
A. System Model
Consider a location-aware network consisting Na agents
and Nbanchors. The anchors are the fixed infrastructure with
known positions, and the agents are the mobile nodes. The
position of each agent is determined based on the point-
to-point measurements from Nb anchors. Denote the set of
agents by Na = {1,2,...,Na}, and the set of anchors by
Nb= {1,2,...,Nb}. The position of node k is denoted by
pk= [xkyk]T. The distance between agent k and anchor j is
given by
dkj= ?pk− pj?.
The angle from anchor j to agent k is given by
φkj= arctan
?yk− yj
xk− xj
?
.
The agent estimates its position based on the wideband
waveform signals transmitted from the anchors. We consider
the measurements between anchors and agents do not interfere
each other, i.e., perfect medium access control is assumed.
Without loss of generality, we assume that agents and anchors
are perfectly synchronized.
B. Position Error Bound
We define the SPEB [3], a measure that characterizes the
localization accuracy, as
P(pk) ? tr?J−1
where Je(pk) is the equivalent Fisher information matrix
(EFIM) for agent k’s position. Using the information inequal-
ity [8], we can show that the localization error is bounded by
the SPEB, i.e.,
E??ˆ pk− pk?2?≥ P(pk),
where ˆ pkis an estimate of the position pk.
It has been shown in [3] that the EFIM can be written as a
2 × 2 matrix,
Je(pk) =
j∈Nb
where qkj = [cosφkj sinφkj]T. In (2), λkj is defined as
ranging information intensity (RII) of agent k with respect to
anchor j, which is given by
e (pk)?,
(1)
?
λkj· qkjqT
kj,
(2)
λkj= ξkj·Pkj
d2
kj
,
(3)
where ξkjis a positive coefficient determined by the channel
property and the effective bandwidth of s(t), and Pkj is the
power of the transmitted waveform.
C. Problem Formulation
Since the RII (3) is related to the transmission power of
anchors, the anchor power allocation will affect the SPEB
in localization systems. In the following, we consider the
objective of minimizing the total SPEB of all the agents. First,
we rewrite the SPEB of agent k based on (1) and (2) as
function of anchor transmission power:
Pk({Pkj}) =
2?
j∈NbξkjPkj/d2
kj
?
j∈Nb
?
j?∈Nbξkjξkj?PkjPkj? sin2(φkj− φkj?)/d2
The problem of anchor power allocation is formulated as
follows:1
?
s.t.
0 ≤ Pkj≤¯Pkj,
?
?
The first constraint (5) gives the upper bound of the transmis-
sion power from anchor j to agent k. The second constraint (6)
kjd2
kj?.
min
{Pkj}
k∈Na
Pk({Pkj})
(4)
∀k ∈ Na,∀j ∈ Nb
∀j ∈ Nb
(5)
k∈Na
Pkj≤ Ptot
?
j,
(6)
k∈Na
j∈Nb
Pkj≤ Ptot.
(7)
1In real systems, since the distance dkjand the angle φkjare usually not
perfectly known, the estimated values can be used for anchor power allocation.
For example, this algorithm can be applied when the agents have the prior
position estimates and need to refine their position information. The case with
imperfect topology information is addressed in our ongoing work.
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y
x
√μ1,k
√μ2,k
βk
Fig. 2: Geometry interpretation of the EFIM of agent k.
implies that the total transmission power of anchor j to localize
all the agents must be bounded by Ptot
power of all the anchors is bounded by Ptotwhich is shown
in the last constraint (7).
It is difficult to directly solve the above problem due to the
non-convexity of the objective function Pk({Pkj}). Instead,
we first explore the geometric interpretation of the localization
information in the following section.
j. The total transmission
III. DIRECTIONAL DECOUPLING OF SPEB
In this section, we will put forth a geometric interpretation
for the EFIM, and present the definition of DPEB and mDPEB.
A. Geometric Interpretation for EFIM
The EFIM (2) can be written, by eigen analysis, as [3]
Je(pk) = Uβk
?μ1,k
0
0
μ2,k
?
UT
βk
where μ1,kand μ2,kare the two eigenvalues of EFIM (μ1,k≥
μ2,k), given by,
⎛
j=1
μ1,k,μ2,k=1
2
⎝
Nb
?
λkj±
?????
?Nb
j=1λkjcos2φkj
?Nb
j=1λkjsin2φkj
?????
⎞
⎠,
(8)
and Uβkis a rotation matrix with angle βk, given by
?cosβk
Geometrically, the EFIM of agent k can be viewed as an
information ellipse given by {x ∈ R2: xTJ−1
(Fig. 2) [4], where 2√μ1,k and 2√μ2,k represent the major
axis and minor axis, respectively. Such an ellipse provides us
the insight that the localization information can be decoupled
into two orthogonal directions. Consequently, we only need
to consider the calculation of the SPEB in two decoupled
directions.
Uβk=
−sinβk
cosβk
sinβk
?
.
e (pk)x = 1}
B. DPEB and mDPEB
Definition 1. The DPEB of agent k is defined as
P(pk;u) ? uT[J−1
e ]u
(9)
where u,u⊥ ∈ R2satisfy the constraint uT
and ?u,u⊥? = 0.
⊥(ˆ pk− pk) = 0
It has been shown in [4] that the SPEB of agent k is the
sum of the DPEBs in any two orthogonal directions, i.e.,
P(pk) = P(pk;u) + P(pk;u⊥).
Now, we choose u∗
[−sinβk cosβk]T. By substituting them into (9), we obtain
the DPEBs in two orthogonal directions, which are simply
expressed as P(pk;u∗) = 1/μ1,k and P(pk;u∗
Thus, the SPEB of agent k can be written as
= [cosβk
sinβk]Tand u∗
⊥
=
⊥) = 1/μ2,k.
P(pk) =
1
μ1,k
+
1
μ2,k.
Definition 2. The mDPEB is defined as
[P(pk;u)]max?
1
μ2,k.
Since 1/μ1,k ≤ 1/μ2,k, a larger portion of position error
is attributed to the smaller eigenvalue μ2,k. The reason can
also be understood via the geometric interpretation of EFIM.
In the information ellipse, the localization information in the
direction of minor axis is less than that in the direction of
major axis. Due to the reciprocal, the SPEB is dominated by
the smaller eigenvalue μ2,k. Thus, in order to improve the
localization performance, it is more helpful to increase the
localization information in the direction of μ2,k. This insight
allows us a new objective for anchor power allocation, i.e., to
maximize the localization information in the direction of μ2,k.
IV. OPTIMIZATION OF ANCHOR POWER ALLOCATION
Since the larger portion of SPEB is determined by its
mDPEB, we will consider the minimization of mDPEB as our
objective. By such, we will show the anchor power allocation
problem can be formulated as a convex optimization problem.
A. Single-agent Case
We first consider that the network consists of Nbanchors
and one agent. For simplicity, we omit k in the subscripts of all
the notations. The optimization problem for the single-agent
case is formulated as follows:
Ps:min
{Pj}
s.t.
1
μ2
0 ≤ Pj≤¯Pj,
Nb
?
(10)
∀j
(11)
j=1
Pj≤ Ptot.
(12)
Note that the constraints in (5)–(7) have been reduced to (11)–
(12), since we only have the constraints on the power of a
single anchor and on the total power of all the anchors.
Moreover, the minimization of 1/μ2 is equivalent to the
maximization of μ2. By substituting the λj = ξjPj/d2
jinto
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(8), the problem (10)–(12) can be equivalently written as
max
{Pj,r}
Nb
?
j=1
ajPj− r
?????
Pj≤ Ptot,
(13)
s.t.
r ≥
0 ≤ Pj≤¯Pj,
Nb
?
?Nb
j=1bjPj
?Nb
j=1cjPj
?????
(14)
∀j
(15)
j=1
(16)
where the coefficients aj = ξj/d2
cj= ξj/d2
j, bj = ξj/d2
jcos2φj, and
jsin2φj.
Proposition 1. The problem (13)–(16) is formulated as an
SOCP.
The proof is straightforward. The constraint (14) defines a
second-order cone given by
Q = {(r,x) ∈ R × R2: r ≥ ?x?}
where x = [?Nb
Note that since SOCP is a well-known convex optimization
problem, the optimal solution of Pscan be obtained efficiently
by many off-the-shelf optimization tools for solving SOCP [9].
j=1bjPj
?Nb
j=1cjPj]T. Therefore, we have
formulated the problem for single-agent case into an SOCP.
B. Multiple-agent Case
We then consider the location-aware network consisting
multiple agents (Na> 1). Similar to the formulation in (4)–
(7) which minimizes the total SPEB, we consider the total
mDPEB as the objective now:
Pm:min
{Pkj}
s.t.
Na
?
0 ≤ Pkj≤¯Pkj,
Na
?
Na
?
k=1
1
μ2,k
(17)
∀k,∀j
k=1
Pkj≤ Ptot
j,
∀j
k=1
Nb
?
j=1
Pkj≤ Ptot.
Proposition 2. The formulation Pmis a convex problem.
Proof: Since all the constraints are linear, we only need to
show the convexity of the objective. Note that μ2,kis a concave
function of Pkj, due to the convexity of the Euclidean norm.
The function 1/μ2,k is convex and decreasing in μ2,k, and
hence, 1/μ2,k is convex in Pkj. As the linear combination
preserves the convexity, the objective function in (17) is a
convex function of Pkj.
Given the convexity of Pm, its optimal solution can be
solved by interior point method [10]. However, the computa-
tional complexity of solving Pmis much higher than that of
solving an SOCP in Ps. In location-aware networks, it is more
important to obtain the power allocation decision in real time,
Algorithm 1 Anchor power allocation algorithm for multiple
agents
Require: the distance dkjand the angle φkjbetween anchor
j (j = 1,...,Nb) and agent k (k = 1,...,Na)
1: Initialize i ← 1 and Pi
2: repeat
3:
Solve the first-stage problem PI
optimal solution ?∗
4:
Set ?i
5:
Solve the second-stage problem PII
optimal solution P∗
k
6:
Set i ← i + 1, and Pi
k=1(Pi
k
)2≤ ? is satisfied
8: Set Pkj← Pi
k← Ptot/Na
mwhich gives the
kjand optimal objective value μ∗
kj, and μi
2,k
2,k
kj← ?∗
2,k← μ∗
mwhich gives the
k← P∗
k
7: until?Na
k− Pi−1
k?i
kj
and hence we design an efficient algorithm by decomposing
the problem into two subproblem, which can be solved with
much lower complexity.
Let Pkj= ?kjPkwhere Pk> 0 is the total power assigned
for localizing agent k, and ?kj∈ [0,1] is a fractional number
denoting the percentage of Pkallocated to anchor j. Then, we
can define
¯ μ2,k? μ2,k/Pk
?
=
Nb
j=1
akj?kj−
?????
?Nb
j=1bkj?kj
?Nb
j=1ckj?kj
?????.
By introducing the two variables ?kjand Pk, we can separate
the problem Pminto two stage problems. First, we consider,
given the total power Pk to localize agent k, the optimal
allocation of Pk to each anchor. In particular, the first-stage
problem is formulated as follows:
PI
m:max
{?kj}
¯ μ2,k
s.t.
0 ≤ ?kj≤
Nb
?
¯Pkj
Pk
,
∀j
j=1
?kj≤ 1.
The optimal solution and the optimal objective value of PI
are denoted by ?∗
formulated for agent k, and hence, there are totally Naproblem
to solve on the first stage.
In the second stage, we allocate the total power Pk for
localizing agent k. The parameters μ2,k (k ∈ Na) and ?kj
(k ∈ Na, j ∈ Nb) in PII
and the optimal solution ?∗
particular, we formulate the problem as:
m
kjand μ∗
2,k. Note that the above problem is
mare fed by the objective value μ∗
kjobtained from solving PI
2,k
m. In
PII
m:min
{Pk}
s.t.
Na
?
Pk≥ 0,
k=1
1
¯ μ2,kPk
∀k
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−10−8−6 −4−20246810
−10
−8
−6
−4
−2
0
2
4
6
8
10
(m)
(m)


Anchor
Agent
Fig. 3: The topology of location-aware network consisting 10
anchors (red square) and 1 agents (blue dot), where the anchors
are uniformly distributed in the square region.
Na
?
Na
?
kis obtained, we plug it into the
kjand μ∗
k=1
?kjPk≤ Ptot
j,
∀j
k=1
Pk≤ Ptot.
Once the optimal solution P∗
problem PI
iteratively until the termination criterion is satisfied. The ter-
mination criterion can be chosen based on the distance of the
variables in two adjacent iterations, e.g.,?Na
negligibly small, it means that the algorithm has converged
into a stable point. The detailed algorithm is described in the
Algorithm 1.
Compared with directly solving Pm, using Algorithm 1
to solve PI
reason is that PI
by off-the-shelf optimization tools. Moreover, the Na first-
stage problems PI
agent, since each agent itself does not require any information
from other agents. Thus, the computation efficiency can be
improved by Na times due to the distributed computation
among the agents.
mto solve ?∗
2,k. We proceed the process
k=1(Pi
k−Pi−1
k
)2≤
? where ? is a small constant. When the distance become
mand PII
mis more efficient in computation. One
mis an SOCP, which can be quickly solved
mcan be independently solved by the Na
V. SIMULATION RESULTS
In this section, we investigate the localization performance
by the anchor power allocation schemes proposed in Section
IV. We analyze the networks which consist of a single agent
and multiple agents, respectively. The total anchor power for
localization is normalized to Ptot= 1. The individual power
constraints at each anchor for localizing one agent is set as
¯Pkj= 0.4, and for localizing all the agents is set as Ptot
j
= 0.8.
2468 10 121416
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Number of anchors
SPEB (m2)


Scheme 1
Scheme 2
Uniform Allocation
Fig. 4: Compare the SPEB in single-agent network resulted by
three power allocation schemes: Scheme 1 (minimize SPEB),
Scheme 2 (minimize mDPEB), and uniform allocation.
First, we consider a location-aware network with a single
agent. The anchors are uniformly distributed in a squared
region, i.e., pj ∼ U([−10,10] × [−10,10]). An example of
the network topology is given in Fig. 3, which consists of
one agent and ten anchors. To study the effect of number of
anchors on the SPEB, we fixed the position of the agent in
the origin. In Fig. 4, we plot the curves of SPEB with the
different number of anchors. A tendency of decrease in SPEB
is observed, as the number of anchors increases from 2 to 16.
This is reasonable since the agent can gather more position
information when there are more anchors.
In Fig. 4, we compare the SPEB resulted by the power
allocation via minimizing mDPEB formulated in Ps, with
the other two different power allocation schemes. One is
to minimize the SPEB obtained based on the result in [7].
The other is the uniform power allocation, which equally
assigns Ptotover all the anchors. The results show that the
minimization of mDPEB outperforms the uniform allocation
by about 40%, and achieve the SPEB which is close to the one
obtained by minimizing SPEB. Since Pshas been formulated
into a SOCP, it can be solved efficiently by the standard
optimization solver, e.g., CVX [11].
Next, we investigate the network with multiple agents.
Four anchors are placed with fixed locations in a square
region, and the multiple agents are uniformly distributed in
the region (see Fig. 5). In Fig. 6, we plot the curves of both
the SPEB and the mDPEB with different number of agents,
i.e., N ∈ {1,2,...,10}. We observe that both the SPEB
and the mDPEB increase linearly with the number of agents.
This is reasonable since the total anchor power is fixed in the
simulation.
In Fig. 6, we compare both the SPEB and the mDPEB
solved by the two-stage optimization (PI
the results obtained from one-stage optimization (Pm) and
mand PII
m) with
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−10−8−6 −4 −202468 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
(m)
(m)


Anchor
Agent
Fig. 5: The topology of location-aware network consisting 4
anchors (red square) and 10 agents (blue dot), where the agents
are uniformly distributed in the squared region.
the uniform power allocation, respectively. The solution of
one-stage optimization can be considered as a benchmark for
that of two-stage optimization, since we have proved that
the one-stage problem Pm is convex and its global optimal
solution can be achieved. The results show that the SPEB and
the mDPEB solved by two-stage optimization matches well
with that solved by one-stage optimization. Meanwhile, both
these two schemes outperform the uniform power allocation
by about 20% as shown by the figure. Moreover, we choose
? = 10−3, and it shows that PII
Since the two-stage optimization allows parallel computation
in distributed agents, and also the standard optimization tools
can be used, it is more suitable to be applied in realistic
systems.
mis solved within 2-3 iterations.
VI. CONCLUSION
In location-aware networks, energy efficiency is a critical
issue which is not only related to the network lifetime but also
affects the localization accuracy. In this paper, we investigated
the power allocation among anchors for localization based
on the performance metric SPEB. We showed that the direct
minimization of SPEB is intractable due to its non-convexity.
Instead, we formulated the problem to minimize the mDPEB
by exploring the geometric interpretation of localization in-
formation. We showed that the new formulation is an SOCP
of which the optimal solution can be efficiently obtained. We
then extended the algorithm into multiple-agent networks by
minimizing the total mDPEB of all the agents. Moreover, in
order to improve the computational efficiency, we decomposed
the problem into a two-stage iterative optimization. The local-
ization performance of the proposed schemes are evaluated
through simulations.
12345678910
0
0.05
0.1
0.15
0.2
0.25
0.3
Number of agents
Square position error (m2)


Two-stage optimization
One-stage optimization
Uniform allocation
SPEB
Maximum DPEB
Fig. 6: Compare the SPEB (resp., mDPEB) in multiple-agent
network by uniform power allocation, the allocation via one-
stage optimization (Ps), and the allocation via two-stage
optimization (PI
mand PII
m).
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