This paper is a postprint of a paper submitted to and accepted for publication in Electronics Letters
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
1
Range-based Source Localization with a
Pure Reflector in the Presence of
Multipath Propagation
Kenneth W. K. Lui and H. C. So
Department of Electronic Engineering
City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Email : [email protected], [email protected]
Index terms : positioning algorithm, non-line-of-sight propagation, time-of-arrival
Abstract : Source localization can be achieved by employing the time-of-arrival (TOA) measurements
between the source and several spatially separated sensors. In this Letter, we develop a simple and
accurate algorithm for TOA-based positioning in the presence of multipath propagation. With the use of
a pure reflector, the multipath TOA information can be effectively utilized. Simulation results show that
the performance of the proposed method approaches Cramér-Rao lower bound when the measurement
errors are sufficiently small.
This paper is a postprint of a paper submitted to and accepted for publication in Electronics Letters
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
2
Introduction: The problem of source localization using measurements from a number of spatially separated
sensors has many important application areas such as radar, sonar, communications and sensor networks
[1]–[3]. A typical positioning approach is to utilize the time-of-arrival (TOA) measurements [2], that is,
the line-of-sight (LOS) path propagation delays between the source and sensors. The measured TOAs
can be straightforwardly converted to distance estimates between them by multiplying the former by the
speed of signal propagation, from which the source position is computed with the known sensor array
geometry.
Nevertheless, multipath propagation often occurs in practical situations which means that apart from
the LOS paths, non-line-of-sight (NLOS) TOA information is available. For example, acoustic multipath
signals come from bottom bounces or reflections from the ocean surface in sonar [1] while radio multipath
signals occur where there are reflections at the ionosphere [4]. Recently, Seow and Tan [3] have proposed
a least squares based geometrical localization approach which utilizes both LOS and NLOS measurements
to achieve higher positioning accuracy than its LOS-only counterpart. Assuming that the reflector surface
is pure [1], the problem of two-dimensional TOA-based location in the presence of multipath propagation
is tackled in this Letter. By representing the reflector as a straight-line equation, we first derive the
positions of the virtual sensors, and then follow the hyperbolic localization approach [5] to develop a
two-step weighted least squares (WLS) position estimator using the multipath TOA information. It is
shown that the estimation accuracy of the proposed scheme attains Cramér-Rao lower bound (CRLB)
when the measurement errors are sufficiently small.
Proposed Method: Consider an array of M sensors with known positions at xi = [xi , yi ]T , i =
1, 2, · · · , M , receiving signal from an active source at unknown position, namely, xs = [xs , ys ]T , where
T
denotes the transpose operator. For simplicity but without loss of generality, we assume that the NLOS
multipath signals undergo one-bound scattering [3]. The discrete-time signal received at the ith sensor,
denoted by ri (k), is then expressed as
ri (k) = αi,1 s(k − Di,1 ) + αi,2 s(k − Di,2 ) + ηi (k),
i = 1, 2, · · · , M,
(1)
where s(k) is the known signal source, αi,1 and Di,1 represent the gain and propagation delay of the LOS
This paper is a postprint of a paper submitted to and accepted for publication in Electronics Letters
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
3
path at the ith sensor, respectively, and αi,2 and Di,2 are the corresponding parameters for the NLOS
path. The ηi (k) is the additive measurement error which is assumed uncorrelated zero-mean white process
with variance σi2 .
Consider optimum multipath TOA estimation has been achieved using the nonlinear least squares
estimator and we denote the corresponding estimates as {D̂i,1 } and {D̂i,2 }. Multiplying {D̂i,1 } by the
known speed of propagation yields M distance measurements based on the LOS paths, denoted by di :
di = kxs − xi k2 + qi ,
i = 1, 2, · · · , M,
(2)
where k · k2 represents the 2-norm and the disturbance qi is characterized by the multipath parameters
and σi2 . In order to make use of {D̂i,2 }, we derive the virtual sensor positions corresponding to them as
follows.
As depicted in Figure 1, we represent the reflector as y = mx + c where m and c are the slope and
y-intercept, respectively, and let xi+M be the virtual sensor of xi . The perpendicular distance between
the reflector and xi or xi+M , denoted by h, is:
mxi − yi + c mxM+i − yM+i + c .
√
=
h= √
m2 + 1 m2 + 1
(3)
Equating the expressions of (3) with opposite signs yields:
mxM+i − yM+i + mxi − yi + 2c = 0.
(4)
Noting that the slope of the line passing through xi and xi+M is −m, we have:
yM+i − yi
= −m.
xM+i − xi
(5)
Solving (4)–(5), the virtual sensor position is


 mxi 

xM+i = 


yi /m − 2c
(6)
Employing {xM+i } and {D̂i,2 }, we have another M distance measurements as in (2), denoting as di+M
with disturbance qi+M , i = 1, 2, · · · , M .
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and is subject to Institution of Engineering and Technology Copyright.
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Following [5], we first linearize di , i = 1, 2, · · · , 2M , by squaring both sides of (2) and introduce a
dummy variable R = xTs xs to yield:
e1 = h1 − G1 z
(7)
where


 2kxs − x1 k2 q1 




 2kx − x k q 

s
2 2 2 
,
e1 = 


..


.






2kxs − x2M k2 q2M

d21
xT1 x1


−






 d2 − xT x



2
2
2

,
h1 = 

..


.






T
d2M − x2M x2M
−2xT1



 −2xT

2
G1 = 
 .
 ..



−2xT2M

1


1

,
.. 
.



1
 
xs 

z=
 
R
which is valid for sufficiently small noise conditions. In the first step, we solve for z without utilizing
the relationship between xs and R according to WLS:
h2 = (GT1 W1 G1 )−1 GT1 W1 h1 ,
(8)
−1
where h2 = [ẑ1 , ẑ2 , ẑ3 ]T is the estimate of z, W1 = E{e1 eT1 }
≈ (BQB)−1 with E{·} is the
expectation operator, B = diag(d1 , d2 , · · · , d2M ) is the diagonal matrix constructed from {di } and
Q = cov(q) is the covariance matrix of q = [q1 , q2 , · · · , q2M ]T . In the second step, we relate xs and R
in h2 by constructing
e2 = h2 − G2 zp
(9)
where

ẑ12
x2s

−







e2 =  ẑ22 − ys2 
,




ẑ3 − x2s − ys2

1


G2 = 
0


1

0


1
,


1


zp = 



x2s 
ys2
and the WLS solution of zp is
ẑp = (GT2 W2 G2 )−1 GT2 W2 h2
(10)
−1
where W2 = E{e2 eT2 }
≈ (B1 cov(h2 )B1 )−1 with B1 = diag(ẑ12 , ẑ22 , 0.5). Finally, the position
p
√
estimate is computed as x̂s = diag(sgn([ẑ1 , ẑ2 ])) ẑp with · is the element-wise square root operator
and sgn is the sign function.
This paper is a postprint of a paper submitted to and accepted for publication in Electronics Letters
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
5
Results and Conclusion: To evaluate the source localization performance of the proposed method,
computer simulations have been conducted. We consider that there are three receivers or M = 3,
located at x1 = [10, 5]T m, x2 = [0, 10]T m and x3 = [−10, 5]T m while the reflective surface is
characterized by y =
√
√
3x − (5 3 + 10). The source is positioned at x = [5, −10]T m. According
to (6), the virtual sensor positions are x4 = [19.3239, −4.7640]T m, x5 = [24.0986, −15.2360]T m and
x6 = [19.3239, −25.7080]T m. The noise covariance matrix Q is obtained by assuming that s(k) is white
and all {σi2 } are identical. The mean square position error (MSPE) is employed as the performance
measure and all results are based on averages of 1000 independent runs.
Figure 2 plots the MSPE versus σ 2 = 1/(2M )
P2M
i=1
E{qi2 } which is the average disturbance in
{di }. We also include the results without utilizing the multipath information and CRLB for performance
comparison. It is seen that there is approximately 2 dB improvement over the scheme which only uses
LOS measurements. We also observe that the MSPE of the proposed method attains CRLB when σ 2 ≥ 4
dB.
To conclude, we have developed a two-dimensional range-based positioning algorithm which effectively
utilizes multipath information with the use of a pure reflector. It is demonstrated that the proposed method
achieves optimum source localization performance for sufficiently small measurement errors.
R EFERENCES
[1] YUAN, Y.X., CARTER, G.C., and SALT, J.E.: “Near-optimal range and depth estimation using a vertical array in a correlated
multipath environment,” IEEE Transaction on Signal Processing, 48, (2), pp.317–330, Feb. 2000
[2] PATWARI, N., ASH, J.N., KYPEROUNTAS S., HERO III, A.O., MOSES, R.L., and CORREAL, N.S.: “Locating the nodes:
cooperative localization in wireless sensor networks,” IEEE Signal Processing Magazine, 2005, 2, (4), pp.54–69
[3] SEOW, C.K., and TAN, S.Y.: “Non-line-of-sight localization in multipath environments,” IEEE Transactions on Mobile
Computing, 7, (5), pp.647–660, May 2008
[4] HORING, H.C.: “Comparison of the fixing accuracy of single-station locators and triangulation systems assuming ideal
shortwave propagation in the ionosphere,” IEEE Proceedings – Pt. F, 137, (3), pp.173–176, Jun. 1990
[5] CHAN, Y.T., and HO, K.C.: “A simple and efficient estimator for hyperbolic location,” IEEE Transactions on Signal Processing,
42, (8), pp.1905–1915, Aug. 1994
This paper is a postprint of a paper submitted to and accepted for publication in Electronics Letters
and is subject to Institution of Engineering and Technology Copyright.
The copy of record is available at IET Digital Library.
6
Fig. 1.
Illustration of xi and xM +i
30
Mean Square Position Error (dB)
20
without virtual sensors
with virtual sensors
CRLB
10
0
−10
−20
−30
−40
−40
−30
−20
−10
2
σ (dB)
Fig. 2.
Mean square position error versus noise power
0
10
20