Effects of anchor placement on mean-CRB for
localization
N. Salman, H. K. Maheshwari, A.H. Kemp, M. Ghogho
School of Electronic and Electrical engineering,
University of Leeds.
Leeds LS2 9JT, UK
email: {elns, elhkm, a.h.kemp, m.ghogho@leeds.ac.uk}
Abstract—
In this paper we discuss the lower bound i.e. the Cramer-rao-bound
(CRB) on the accuracy of localization of nodes in a wireless network.
We investigate the effects of anchor (or base station) placement on
optimal target node positioning. The optimal and worst anchor positions
are determined through extended simulation by comparing their mean
CRB. Furthermore the ramifications of an additive and multiplicative
noise model on the mean CRB are explored. Finally, the least squares
(LS) method for localization is used and its performance is compared
with the lower bound for optimal anchor positions.
I. I NTRODUCTION
In recent years, there has been a great interest in research towards
positioning of wireless devices. Precise and accurate localization is of
great importance in military, health, environment, and commercial ap-
plications [1]. For outdoor positioning, the global positioning system
(GPS) is most widely used, but it exhibits suboptimal performance in
harsh propagation environments (i.e. inside buildings, underground
and between heavy vegetative cover) due to the absence of a line
of sight (LoS). Also, due to the high power requirements of GPS,
it has become impractical to employ GPS technology in low power
networks such as wireless sensor networks (WSNs). On the other
hand, cellular based positioning systems such as the enhanced 911
(E911) have also been recently implemented. With the E911 service,
the position of the calling party can be pinpointed in cases when
the caller cannot locate itself (e.g. on a remote highway or during a
kidnapping).
Two widely used methods for range estimation are the time-
of-arrival (ToA) and the received signal strength (RSS). Various
techniques have been developed to solve the trilateration distance
equations. These include the LS methods [2], the weighted LS
method [3] and the maximum likelihood (ML) approach [4]. However
the performance of these algorithms is bounded by the CRB. The
CRB puts a lower bound on the variance of any unbiased estimator.
The CRB for localization is dependent on the geometry of the
anchors and the target node. However the results in [6] are based
on the additive noise model (aNm) while a modified CRB based
on the multiplicative noise model (mNm) is proposed in [7]. In this
paper, we investigate the optimal anchor placement for both models.
We begin by giving a review of the two noise models in section-II.
In section III we discuss the CRB for localization. Our simulation
results are presented in section-IV. In section-V the LS method is
simulated and compared with the lower bound, which is followed by
the conclusion.
II. SIGNAL MODELS
Let us consider a network consisting of N anchor nodes whose
locations θ
i
=(x
i
,y
i
)
T
for i =1, ..., N are known, this can be done
by placing these anchors at predefined spots or their position can
be determined via GPS. It is desired to determine the location of a
target node θ =(x, y)
T
. Then the estimated distance between each
anchor and the target node can be modeled either by the aNm or
the mNm. The aNm is a widely accepted signal model, however the
0 10 20 30 40 50
-10
0
10
20
30
40
50
60
Range (m)
Estimated Range (m)
Range (m)
σ
2
= 2
σ
2
= 4
Figure 1. Simulation of estimated range for aNm
10 20 30 40 50
0
10
20
30
40
50
60
Range (m)
Estimated Range (m)
κ = 0.005
Range (m)
η=1.83
η=1.87
η=2
η=2.2
10 20 30 40 50
0
10
20
30
40
50
60
Range (m)
Estimated Range (m)
κ = 0.008
Range (m)
η=1.83
η=1.87
η=2
η=2.2
Figure 2. Simulation of estimated range for mNm
mNm is more suitable for practical propagation channels. The two
noise models are discussed in the following subsections.
A. Additive noise model
The signal received at the target node from the i
th
anchor is given
by
r
i
(t)= A
i
s(t − τ
i
)+ n
i
(t) (1)
where A
i
is the amplitude or attenuation of the signal, τ
i
is the
propagation delay and n
i
(t) is the thermal noise. The delay τ
i
that
is dependent on the distance between the anchor and the target node
is given by
τ
i
(x, y, ℓ
i
)=
1
c
(x − x
i
)
2
+(y − y
i
)
2
+ ℓ
i
(2)
where c is the speed of the electromagnetic wave
(
c ≃ 3 × 10
8
)
and ℓ
i
is non line of sight (NLoS) bias. Since in the present work,
we only consider the LoS case, thus ℓ
i
=0.
From Eq. (2), we note that the distance between i
th
anchor and
the target node is given by
d
i
= cτ
i
(3)
To include distances from N anchors, Eq. (3) is given in vector
form:
2011 The 10th IFIP Annual Mediterranean Ad Hoc Networking Workshop
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