MULTIPLE TARGETS DETECTION AND LOCALIZATION
BASED ON BLIND ESTIMATION IN WIRELESS
SENSOR NETWORK
Peng Zhang, Xiaoyong Deng, Huimin Wen and Jifu Guo
Beijing Transportation Research Center, Liuliqiao South Road, Beijing, China
Keywords: Multi-target, Path fading, Bind estimation, Independent component analysis.
Abstract: Observations of sensors are modeled as mixed signals in multiple targets scenario. Each element of mixing
matrix represents the power decay of a pair of target and sensor, and each column preserves the waveform
formed by the corresponding target respectively. Making use of blind estimation algorithms, we get the
estimation of mixing matrix. Target locations are then estimated using the least squares method.
1 INTRODUCTION
Research on single target localization and tracking
approaches in wireless sensor network has been
carried out for a decade, and effective algorithms
have been proposed (Savarese et al., 2001; Taff,
1997). Some existing approaches aiming at multiple
targets apply sensor arrays (Nehorai et al., 1994),
which are different from adhoc sensor networks as
the latter are with unstable topological structure of
sensors. Research corresponding to multiple targets
scenario has just emerged in recent years and some
methods have been proposed, most of which are
under the framework of maximum likelihood and
expectation-maximization like methods (Xiao et al.,
2005; Krasny et. al., 2001).
In this paper, by taking into consideration of the
statistical properties of targets, we use the
independent component analysis to estimate the
number of target, and make use of blind separation
algorithms to solve the mixing matrix which
describes the overlap of the multiple targets in each
sensor’s measurement. A target localization
algorithm based on the least squares methods is then
obtained.
In sectionⅡ, the system model, mixing model
and assumption of sources are presented. In section
Ⅲ, a source detection method is given to estimate
the target number. In section Ⅳ , algorithms of
sources separation and estimation of mixing matrix
are introduced. Finally, the target locations are
estimated in section V.
2 SYSTEM MODEL
2.1 Source Model
Whatever signals are transmitted by certain form of
wave, e.g. acoustic, radio, earth wave, etc, signals
produced by sources are carried by waves from
sources to receivers. Due to physical essences,
signals are described as stochastic processes with
statistical properties. Whether the waves are
generated by sources or reflected by sources, some
inherited properties of sources are loaded on carriers
inevitably.
Source signal can be modeled as
() ( ) ( )
∑
∞
−∞=
−=
k
sT
nTtgksts (1)
where
(
)
ks is the source signal. It can be modelled
as a zero mean stationary process with a non-
singular covariance matrix.
()
tg
T
is unit amplitude
rectangular pulse of width
s
T .
2.2 Fading Model
For wireless radio, sonar, or earth waves, signals
generally suffer from two major sorts of fading, one
is caused by space condition, e.g. loss of distance,
multi-path and the other caused by relative motion
between transmitter and receiver.
Signals suffers from different sorts of path
fading, here we only take into consideration of
61
Zhang P., Deng X., Wen H. and Guo J..
MULTIPLE TARGETS DETECTION AND LOCALIZATION BASED ON BLIND ESTIMATION IN WIRELESS SENSOR NETWORK.
DOI: 10.5220/0003464700610064
In Proceedings of the International Conference on Wireless Information Networks and Systems (WINSYS-2011), pages 61-64
ISBN: 978-989-8425-73-7
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

signal power decays related to path length in free
space. Let
()
df denote the path fading coefficient,
the received signal by a sensor can be expressed as
()
(
)()
tsdftr = (2)
where
d is the distance between source and
sensor. One of the generally used models is
()
()
2
1
1
+
=
d
df
After all, how to model the path fading does not
affect the source detection and separation
essentially.
2.3 Mixing Model
Suppose there are
m
sources and
n
sensors in the
sensor field, generating signal
()
ks
j
, mj ,,1 "= and
observation records
()
kx
i
, ni ,,1 "= , mn > ,
respectively. Let
() () ()
[]
Τ
= tstst
m
,,
1
"s
and
(
)
(
)
(
)
[]
Τ
= txtxt
n
,,
1
"x
denote the source and sensor observation vector,
respectively. The observation vector
()
tx can be
expressed as the sum of linear combination of
signals and noises as
() () ()
ttt nAsx += (3)
where
()
mn
ji
a
×
∈=
,
A is the mixing matrix
between sources and sensors.
() ( )
n
i
nt ∈=n is an
additive Gaussian noise with zero mean, covariance
matrix
I
2
n
σ , and I is the identity matrix. Note that
power decays and Doppler shift are independent to
each other,
()
(
)
jiji
dfta
,,
=
(4)
where,
ji
d
,
is the distance between the thj source
and the
thi sensor.
Assumptions
(A1)
A
has full column rank, i.e.
()
m
=
Arank .
(A2)
() (){}
tsts
m
,,
1
" are uncorrelated.
(A3) There exists a
0>τ such that
() ( )
(
)
miτtstsE
ii
,,1,0
*
"=≠+ .
(A4) Sampling rate satisfies
D
ft >Δ1 .
(A5) Source location is approximated to be
unchanged in
s
T , and observation window
s
TtN <Δ , where N is the sample number in
observation window.
3 SOURCES DETECTION
Under the above assumptions, the covariance matrix
of
(
)
tx can be given by
(
)
(
)()
{
}
ttE
H
xxR
x
=0
IAA
s
2
)0(R
n
H
σ +=
(5)
where
H
denotes conjugate transpose,
(
)
(
)
{
}
ttE
H
ss
s
=)0(R
is the nonsingular covariance
matrix of
(
)
ts . For assumption (A2),
(
)
(
)
{
}
tsts
m
,,
1
" are uncorrelated, )0(R
s
are
diagonal.
(
)
0
x
R is full rank, so it can be
diagonalized.
(
)
(
)
H
n
diag UUR
x
λλλ ,,0
21
"=
(6)
where
],[
,,21 n
uuuU
"
=
is an nn× matrix.
ni
i
,,1, "
=
u are eigenvectors of
(
)
0
x
R .
ni
i
,,1, "= λ are eigenvalues of
(
)
0
x
R ,
2
121
,
nn
m
m
σλλλλλ ==>≥≥≥
+
"" , and
,α
2
nii
σλ += mi
i
,,1,α "
=
are eigenvalues of
H
AAR
s
)0( .
Source number m can be determined from the
multiplicity of the smallest eigenvalue of
(
)
0
x
R .
According to MDL information theoretic criteria
(
Wax et al., 1989), value of m is estimated as m
ˆ
,
where
m
ˆ
is chosen such that
(
)
(
)()
{
}
1,,0min
ˆ
−
=
nMDLMDLmMDL " (7)
with
(
)
()
() nkvLkMDL
Nkn
k
,log +−=
−
where
(
)
()
∑
+=
−
+=
−
∏
=
n
ki
i
kn
i
n
ki
k
kn
L
1
/1
1
1 λ
λ
(
)
(
)
Nknknkv log2,
2
1
−=
The noise variance
2
n
σ can be estimated as
∑
+=
−
=
n
mi
i
mn
n
1
1
2
ˆ
λσ (8)
and the eigenvalues of
H
AAR
s
)0( can be estimated
as
2
α
nii
σλ −= (9)
WINSYS 2011 - International Conference on Wireless Information Networks and Systems
62

4 SOURCES SEPARATION AND
ESTIMATION OF MIXING
MATRIX
From (5)
()()
IAARIAARR
sxx
22
s
)0()0(R)0()0(
n
H
H
n
HH
σσ ++=
IAARAAR
ss
2
)0()0(
n
HHH
σ +=
)0()0(
H
xx
RR=
(10)
Hence,
()
0
x
R is a normal matrix. According to
matrix theory, there must exist an unitary matrix,
which can diagonalize
()
0
x
R , so U is also an
unitary matrix that satisfies
IUU =
H
. Denote
H
AARR
sA
)0()0( = . )0(
A
R can be diagonalized as
(
)
H
m
diag
ssA
UUR α,α,α)0(
21
"= (11)
where
],[
,,21 m
uuuU
s "
= .
Define
(
)
H
s
UT
2
1
2
1
2
1
m21
α,α,α
−−−
≡ "diag .
Multiply equation (5) by
T from the left, then it can
be expressed as
()
(
)()
ttt wBsy += (12)
where
() ()
tt Txy =
TAB =
() ()
tTntw
=
For
IBB =
H
, B is an unitary matrix.
()
(
)( )
()
ττ
H
Et t=+
y
Ryy
()
(
)()()()()
(
)
() ( )
()
() ( )
()
()
H
HH
HH
H
ttE
ttE
ttttE
BBR
BssB
BsBs
wBswBs
s
τ
τ
τ
ττ
=
+=
+=
++++=
For Assumptions (A3),
()
τ
s
R is diagonal.
()
τ
y
R can be obtained from the statistics of
observation of
()
tx .
For
() () () ()
ττττ
yyyy
RRRR
y
H
y
H
= , ()
τ
y
R
is a normal
matrix and must have an eigen-decomposition of the
following form
() ()
H
VVRR
sy
ττ = (13)
where
V is an unitary matrix. Although
waveform of each column is preserved by unitary
transformation, the inherited indeterminacies of
blind estimation associated with the magnitude of
sources and the order in which sources are arranged
are inevitable.
B and V are related by
VEJB
=
(14)
where
E is a nonsingular diagonal matrix and J is a
permutation matrix. Substitute
TAB = , the maxing
matrix can be estimated as
VEJTA
+
=
ˆ
(15)
where superscript + denotes the Moore-Penrose
pseudoinverse.
5 LOCALIZATION OF SOURCES
In (15), each element of E multiplies na arbitrary
coefficient to each column of
A
ˆ
,
respectively.
J exchanges the columns of A
ˆ
.
Without loss of generality, we assume
IE
=
and IJ
=
, that will not affect the estimation
result as shown in the following.
A
ˆ
can be rewritten
as
VTA
+
=
ˆ
(16)
Assume that
mj
j
,,1,
ˆ
"=a
is the
thi
column
of
A
ˆ
. As in (6)
]
ˆˆ
,
ˆ
[
ˆ
,,,,2,1 jmjji
aaa
"
=a
represents the
proportion between fading coefficients of the
thi source to each sensor, where the magnitude of
j
a
ˆ
represents the proportion between path fading
coefficients
(
)
nidp
ji
,,1,
,
"
=
, and the angle of
j
a
ˆ
represents the proportion between Doppler
shifts
(
)
nifg
ji
,,1,
,
"=
.
A. Location Estimation
(
)( ) ( )
jnjjjmjj
dpdpdpaaa
,,2,1,,1,1
:::
ˆ
::
ˆ
:
ˆ
"" =
(17)
where
⋅ denotes magnitude of a complex
number. Introduce a reference coefficient
ρ
,(18)
can be expressed as
(
)
niaρdp
jiji
,,2,1
ˆ
,,
"== (18)
Then
ji
d
,
can be expressed by the inverse
function of
ji
a
,
ˆ
(
)
(
)
jiji
aρpd
,
1
,
ˆ
−
= (19)
MULTIPLE TARGETS DETECTION AND LOCALIZATION BASED ON BLIND ESTIMATION IN WIRELESS
SENSOR NETWORK
63

Substitute
ji
d
,
by coordinate of source, we can get
()()
()
()()
2
,
1
22
ˆˆˆ
jiijij
aρpyyxx
−
=−+− (20)
where
[
]
mjyx
jj
,,2,1,
ˆ
,
ˆ
"=
are the coordinates of
the
thj source, and
[]
niyx
ii
,,2,1,, "= are the
coordinates of the
thi sensor.
By substituting
ni ,,2,1 "=
into (21), and
subtracting each other, we have that
()()
()
()()
()
()()
1,,2,1
ˆˆ
2
1
ˆˆ
2
,
1
2
,1
122
1
22
1
11
−=
⎟
⎠
⎞
⎜
⎝
⎛
−+−+−=
−+−
−
+
−
++
++
ni
aρpaρpyyxx
yyyxxx
jijiiiii
iijiij
"
The above equations can be expressed as
[]
mjyx
jjjj
,,2,1
ˆ
,
ˆ
"==
Τ
eC (21)
where,
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−−
−−
=
−− 11
1212
,
,
nnnn
j
yyxx
yyxx
#C
()
()()
()
()()
()
()()
()
()()
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−+−+−
−+−+−
=
−
−−
−−
−−
++
2
,1
1
2
,
12
1
22
1
2
2
,1
1
2
,2
122
1
22
1
ˆˆ
ˆˆ
2
1
jnjnnnnn
jjiiii
j
aρpaρpyyxx
aρpaρpyyxx
#e
The least squares estimate of source position can be
given as
[]
mjyx
jjjj
,,2,1
ˆ
,
ˆ
"==
+
Τ
eC (22)
Introduce (22) to (20),
ρ
can be solved. Return
ρ
to
(22), the source position is obtained.
6 CONCLUSIONS
Path fading is introduced to model the multiple
target network. Based on blind estimation, a range
free multiple target localization algorithm was
presented. This development is especially applicable
in fast, time-varying environments, where multiple
targets maneuver quickly and randomly.
ACKNOWLEDGEMENTS
This work was supported by the Beijing Excellent
Talents Foundation (2009D01000700001).
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Xiao, S., Hu Y., and Silverman, H, 2005. Maximum
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