Published in IET Signal Processing
Received on 26th November 2009
Revised on 15th March 2010
doi:10.1049/ietspr.2009.0284
ISSN 17519675
State estimation with quantised sensor information in
wireless sensor networks
J.Xu J.X.Li
Department of Automation,Shanghai Jiao Tong University,Shanghai 200240,People’s Republic of China
Email:xujian20011@163.com
Abstract:The problemof state estimation with quantised measurements is considered for general vector statevector observation
model in wireless sensor networks (WSNs),which broadens the scope of sign of innovations Kalman ﬁltering (SOIKF) and
multiplelevel quantised innovations Kalman ﬁlter (MLQIKF).Adhering to the limited power and bandwidth resources WSNs
must operate with,this paper introduces a novel decentralised unscented Kalman ﬁltering (UKF) estimators based on
quantised measurement innovations.In the quantisation approach,the region of a measurement innovation is partitioned into
L contiguous,nonoverlapping intervals.After quantised,the measurement information is broadcasted by using a variable
number of bytes coding method.A ﬁltering algorithm for general vector statevector observation case is developed based on
the quantised measurement information.Performance analysis and Monte Carlo simulations reveal that under the same
bandwidth constraint condition,the performance of novel quantised UKF tracker,indeed better than those of SOIKF and
MLQIKF in error covariance matrix (ECM) and root meansquare error (RMSE) and almost identical to these of an UKF
based on analogueamplitude observations.
1 Introduction
Quantisation was a wellstudied topic in digital signal
processing some decades ago [1,2],where the underlying
reason was the ﬁnite computation precision in micro
processors.Recently,wireless sensor networks (WSNs)
have attracted much attention because of their signiﬁcant
applications in environmental monitoring,intelligent
transportation and space exploration,military surveillance
etc.see [3–8].In WSNs,the target may be a signal source,
an animal,or a vehicle etc.Because of the bandwidth
constraint,each sensor is only able to transmit a ﬁnite
number of bits.This leads that observations have to be
quantised before transmission.Thus,these new reasons
motivate a revisit of the area in WSNs.
Obviously,the quantisation will result in the loss of large
amount of information.In a system with quantised
measurements,because of the nonlinearity of the
quantisation,estimating the system state is a nonlinear and
nonGaussian estimation problem even if the system is
linear and Gaussian [9].Much of the early work,for
example,[1,2,10],devised approximate point estimators of
the Kalmantype ﬁlter based on the optimal conditional
mean estimator which in general requires numerical
integration for implementation.Sviestins and Wigren [11]
derived an exact density ﬁlter based on solving the Fokker–
Planck equation and Bayes’ rule for a special case of the
problem and under somewhat restrictive assumptions.
Recently,Karlsson and Gustafsson [12] and Sukhavasi and
Hassibi [13] applied particle ﬁltering that practically
amounts to repeatedly using of Bayes’ rule.But,quantising
sensor measurements can lead to large quantisation noises
when the observed values are large which then leads to
poor estimation accuracy.In [14,15],this limitation is
overcame by developing an elegant distributed estimation
approach based on quantising the innovation to one bit (the
socalled sign of innovation or SOI).There,the quantisation
ﬁlter for vector statevector observation case is discussed.
However,the computing process of optimal gain matrix,
state estimation and estimation covariance matrix need to be
carried out N times in the correctional step,where N is the
dimension of vector observations.In the work of You et al.
[16] and Msechu et al.[6],the quantisation ﬁlter for vector
statescalar observation case is generalised to handle
multiple quantisation levels,but the quantisation ﬁlter for
vector statevector observation case is not given [6,Section
IVB].
In this paper,taking into account the quantisation,
encoding and transmitting of measurements in WSNs,we
study state estimation of dynamical stochastic processes
based on severely quantised observations.It is worth noting
that almost all of those algorithms [6,14–16] focused on
the designing of quantisation to save bandwidth and energy,
although they ignored that it can also be done in the
process of encoding and transmitting.For this reason,a
novel quantisation and transmitting strategy is proposed
here.Accordingly,the quantised ﬁltering algorithm is
discussed.In general case,since sensor observations are
vector valued,we consider the quantised ﬁltering for
general vector statevector observation case here.This paper
builds on and considerably broadens the scope of sign of
innovation Kalman ﬁltering (SOIKF) [14,15] by
16 IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26
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The Institution of Engineering and Technology 2011 doi:10.1049/ietspr.2009.0284
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addressing the middle ground between estimators based on
severely quantised (1bit) data and those based on
unquantised data for general vector statevector observation
case.The resulting method is simulated for target tracking
scenario in WSNs.The simulation results are compared
with the results of SOIKF [14,15],multiplelevel
quantised innovation Kalman ﬁlter (MLQIKF) [6,16] and
the ideal unscented Kalman ﬁltering (UKF) [17] with
measurement before quantisation.
The contribution of the present paper is precisely to address
these three issues with the goal being to construct state
estimators based on variable bytes observations so that:(i)
under the same transmission bandwidth condition,the
ﬁltering is more accuracy than those in [6,14–16];(ii) the
quantised ﬁltering problem for general vector statevector
observation case can be dealt with easily;and (iii)
quantised ﬁltering of UKF version is introduced to treat
with the highly nonlinear state estimation problems.
The paper is organised as follows.Problem statement
including the modelling assumptions and preliminaries are in
Section 2.Section 3 introduces the quantisation approach and
transmitting strategy ﬁrstly.An estimator with quantised
innovations for general vector statevector observation case is
also developed.The performance analysis is presented in
Section 4.The simulation results are presented in Section
5.Finally,conclusions are given in Section 6.
Notation:We use s{M
k
1
} to denote the sﬁeld generated by
random variables M
k
1
= {M
1
,M
2
,...,M
k
}.The probability
density function (pdf) of X conditioned on s{M
k
1
} is
represented as p(Xs{M
k
1
}).The Gaussian pdf with mean
E(X) =m and covariance matrix Cov(X) = C is
represented as p(X) = N(X;m,C).The probability mass
function for a discrete random variable m is denoted as
Pr(m).Estimators are represented using a hat,for example,
X(ks{M
k
1
}) = E[X(k)s{M
k
1
}].Finally,T stands for
transposition and
−1
stands for the matrix inverse.
2 Modelling assumptions and preliminaries
Consider the state estimation problem of general vector state
vector observation case
X(k) = F(X(k −1)) +G(k −1)w(k −1) (1)
Y(k) = H(X(k)) +v(k),k = 0,1,2,...(2)
where X(k) [ R
d
is a statevector to be estimated at time
t
k
= kDt,Dt is the time step length of sample,F(
.
) is a
continuously differentiable function of X(
.
) and G(
.
) are
timevarying matrices with suitable dimensions.Y(k) [ R
N
is the vector observation of the sensor,N is the dimension
of observed value,H(·) = {h
1
(·),h
2
(·),...,h
N
(·)}
T
is the
vector measurement function.w(k) [ R
r
and v(k) [ R
N
are
uncorrelated Gaussian noises with zero mean and
covariance matrices Q and R.The initial value X(0)
with mean m
0
and variance P
0
is independent of w(k)
and v(k).In this paper,we assume that the fusion centre
knows all the parameters of system (1),(2) and the channel
is perfect,that is,no bit error,from sensors to the fusion
centre.
In order to effect digital communication in the bandwidth
limited WSNs,the observations are quantised.The quantiser
is described mathematically by
m
i
(k) = Q
L
i
(y
i
(k)),k = 0,1,2,...;i = 1,2,...,N
(3)
where Q
L
i
(·) denotes the nonlinear quantisation mapping
with L
i
levels.Then,the individual components y
i
(k),
i = 1,2,...,N,of Y(k) be quantised,and thus Y [A
implies {a
i
≤ y
i
,b
i
,i = 1,2,...,N}.The majority of the
following results do not depend on the fact that A is a
hypercube.M(k) denotes the vector {m
1
(k),m
2
(k),...,
m
N
(k)}
T
.
Deﬁne the following events
M
k
={a
i
(k) ≤ y
i
(k),b
i
(k),i =1,2,...,N},k =1,2,...
(4)
and
M
k
1
= {M
1
,M
2
,...,M
k
},1 ≤ k
m
i
(1:k) = {m
i
(1),m
i
(2),...,m
i
(k)},1 ≤ k,1 ≤ i ≤ N
It is well known that the minimum mean square error
(MMSE) estimator is given by the conditional expectation
[18,Ch.12].Consequently,if we let
X(k) denote the
MMSE estimator of given {M
k
1
},we have
X(ks{M
k
1
}) = E[X(k)s{M
k
1
}]
=
R
d
X(k)p(X(k)s{M
k
1
})dX(k) (5)
The error covariance matrix (ECM) of the estimator is deﬁned
as
P(kk) = E{[
X(ks{M
k
1
}) −X(k)][
X(ks{M
k
1
}) −X(k)]
T
}
and the meansquare errors (MSEs) of
X(ks{M
k
1
}) are given
by the diagonal elements of the ECM.Conveniently,in the
case with no misunderstanding,we denote
X(ks{M
k
1
}) and
P(kk) as
X
kk
and P
kk
,respectively.
To obtain a closedform expression for
X(ks{M
k
1
}),the
posterior distribution p(X(k)s{M
k
1
}) has to be known and
the integral in (5) needs to be computed.In principle,
p(X(k)s{M
k
1
}) can be obtained from the stateobservation
model in (1),(2) using the prediction–correction
steps,the process can be described by Algorithm 1 [A],
[B] [6].
Algorithm 1 – A Prediction step.
IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26 17
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The Institution of Engineering and Technology 2011
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Given:p(X(k −1)s{M
k−1
1
})
p(X(k)s{M
k−1
1
})
=
R
d
p(X(k)X(k −1),s{M
k−1
1
})
×p(X(k −1)s{M
k−1
1
})dX(k −1)
=
R
d
p(X(k)X(k −1))
×p(X(k −1)s{M
k−1
1
})dX(k −1) (6)
Algorithm 1 – B Correction step.
Receive:quantised observations M
k
p(X(k)s{M
k
1
}) =p(X(k)s{M
k−1
1
})
Pr(M(k)s{X(k),M
k−1
1
})
Pr(M(k)s{M
k−1
1
})
(7)
Estimator
X(ks{M
k
1
}) is then obtained by evaluating the
integral in (4).For more details,one can refer to ([6] and
references therein).
If we had inﬁnite bandwidth available,we could
communicate the observations Y(k) errorfree.This is
rightfully a clairvoyant benchmark for our bandwidth
constrained estimators and corresponds to the above
problem setup with messages M
k
= Y(k).In this case,we
have a wellknown nonlinear Gaussian state estimation
problem whose estimator can be recursively obtained by the
UKF [17] (Algorithm 2 [A],[B]).
Deﬁne the estimate
ˆ
X
kk
= E[X
k
M
k
1
] and the
corresponding ECM P
kk
= MSE[
ˆ
X
kk
M
k
1
] = E[(X
k
−
ˆ
X
kk
)(X
k
−
ˆ
X
kk
)
T
M
k
1
].
Algorithm 2 – A Prediction step.
Given:
ˆ
X
k−1k−1
and P
k−1k−1
ˆ
X
kk−1
=
2d
j=0
W
j
k−1
F(X
j
k−1
) (8)
P
kk−1
= Q+
2d
j=0
W
j
k−1
(F(X
j
k−1
) −
ˆ
X
kk−1
)(F(X
j
k−1
)
−
ˆ
X
kk−1
)
T
(9)
ˆ
Y
kk−1
=
2d
j=0
W
j
k−1
H(X
j
kk−1
),i = 1,2,...,N (10)
where {W
j
k−1
,j = 0,1,...,2d} is the weight set,
{X
j
k−1
,j = 0,1,...,2d} is the sigma point set [17,section
2.4.2],X
j
kk−1
denotes F(X
j
k−1
),and
ˆ
Y
kk−1
= (y
1
(kk −1),
y
2
(kk −1),...,y
N
(kk −1))
T
.
Algorithm 2 – B Correction step.
Receive:new observations M
k
= Y(k)
X
kk
= E[X(k)s{M
k−1
1
,Y(k)}]
=
ˆ
X
kk−1
+K
k
(Y(k) −
ˆ
Y
kk−1
) (11)
P
kk
= P
kk−1
−K
k
(R +P
yy
)K
T
k
(12)
where
K
k
= P
xy
(R +P
yy
)
−1
(13)
P
xy
=
2d
j=0
W
j
k−1
(X
j
kk−1
−
ˆ
X
kk−1
)(H(X
j
kk−1
) −
ˆ
Y
kk−1
)
T
(14)
P
yy
=
2d
j=0
W
j
k−1
(H(X
j
kk−1
) −
ˆ
Y
kk−1
)(H(X
j
kk−1
) −
ˆ
Y
kk−1
)
T
(15)
Computations for the UKF iterations in Algorithm 2 are
simpler than for the general iterations in Algorithm 1 with
quantised observations.Indeed,although Algorithm 2
requires a few algebraic operations per timestep k,
Algorithm 1 requires numerical integration in (5)–(7).
In this paper,quantisation–estimation approaches is
pursued.In this case our objectives are:(i) to give a novel
transmitting strategy so that more bandwidth can be saving
than those in [6,16],(ii) to show that the MMSE
estimation Algorithm 1 can be simpliﬁed yielding a non
linear ﬁlter of UKF version with comparable computational
cost to the UKF and (iii) to compare the performance of
resulting algorithm using quantised observations,with those
of the UKF in [17],the MLQIKF in [6,16] and the SOI
KF in [14,15].
3 State estimation using quantised
innovations
In this section,we will investigate the state estimation based
on the multiplelevel quantised innovations.Since in
general the values of innovation range are smaller than
those of observation range,quantising innovations will
bring smaller information loss than quantising observations
under the same bandwidth constraint.It is worth noting
that,in the process of data transfer,those algorithms in
[6,16] are based on a ﬁxed number of bits transmission,
which reduces the utilisation of transmission channel.In
order to further reduce the information loss and improve the
ﬁltering accuracy,we will give a novel transmitting strategy
for quantitative information.
3.1 Quantised innovations and transmitting
strategy
In this paper,we still adopt the multiplelevel quantisation
strategy,but the transmitting strategy are different from
those of Msechu’s et al.[6] and You’s et al.[16].
We consider the nonlinear system(1),(2) in a WSNs.The
activated sensor makes an observation and computes the
innovation 1(k) = Y(k) −
ˆ
Y
kk−1
,where the onestep
predictor
ˆ
Y
kk−1
of observation together with the square root
(S(:,:,k)
1/2
)
−1
= ((P
yy
+R)
1/2
)
−1
of the innovation
covariance are received by the sensor from the estimator
centre.
Denote the normalised innovation as
1(k) = (S(:,:,k)
1/2
)
−1
1(k)
18 IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26
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The Institution of Engineering and Technology 2011 doi:10.1049/ietspr.2009.0284
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Then each component
1
i
(k),i = 1,2,...,N,of the
normalised innovation
1(k) is quantised to produce a
L ¼ (2l +1)level quantised innovation J(k) = Q
L
(
1(k)).
We consider a symmetric quantiser J
i
(k) = Q
L
(
1
i
(k)) for
1
i
(k) given by
J
i
(k) =
j
l
if
1
i
(k) [ (h
l
,+1)
˙ ˙
˙ ˙
˙ ˙
j
2
if
1
i
(k) [ (h
2
,h
3
]
j
1
if
1
i
(k) [ (h
1
,h
2
]
0 if
1
i
(k) [ (−h
1
,h
1
]
−j
1
if
1
i
(k) [ (−h
2
,−h
1
]
−j
2
if
1
i
(k) [ (−h
3
,−h
2
]
˙ ˙
˙ ˙
˙ ˙
−j
l
if
1
i
(k) [ (−1,−h
l
]
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
(16)
Thus,the normalised innovation
1(k) is quantised to
produce a quantised innovation J(k) = {J
1
(k),J
2
(k),...,
J
N
(k)}
T
.
Remark 1
1.Comparing the deﬁnition of M(k) in (4) with
J(k) = Q
L
(
1(k)),the difference is that the former gives
only where the range of
1(k),whereas the latter gives not
only a range information of
1(k),but also a speciﬁc value.
Clearly,the deﬁnition of J
i
(k) gives a oneone mapping
from M
i
(k) to R.Hence in following text we will not
distinguish M
i
(k) and J
i
(k).If we take the value of
j
j
= E[
1
i
(k))h
j
,
1
i
(k)) ≤h
j+1
],j = 1,2,...,l
i
,that is,
J
i
(k) = E[
1
i
(k))h
j
,
1
i
(k)) ≤h
j+1
],then the numerical
integration cannot be executed in the ﬁltering algorithms.
Thus,the online computation cost can be reduced.More
details will be given in Section 3.3.
2.It should be noted that,in practical applications,we often
require the measurement information from multiple sensors
for target tracking.In this case,an approximate quantisation
algorithm can be used.Speciﬁcally,the fusion centre only
send the predicted measurements and the corresponding
diagonal elements of prediction error covariance to the
sensor nodes,respectively.Then,the sensor nodes
approximately calculate the quantised innovations according
to the quantisation algorithmand return themto fusion centre.
After quantisation,the quantised innovation J(k) =
{J
1
(k),J
2
(k),...,J
N
(k)}
T
is transmitted to the fusion
centre by a dynamic transfer strategy,which is different
from the ﬁxed bit transmission in [6,16].The dynamic
transfer strategy is described mathematically by
u
i
(k) =
0 if −h
1
,
1
i
(k) ≤h
1
⌈log
2
( j +1)⌉ if h
j
,
1
i
(k) ≤h
j+1
or −h
j+1
,
1
i
(k) ≤ −h
j
,
j = 1,2,...,l
⎧
⎪
⎪
⎨
⎪
⎪
⎩
where h
l+1
= +1,⌈ ⌉ operator rounds up to the
nearest integer.Speciﬁcally,the dynamic transmission
strategy is given by:when quantised innovation m
i
(k) = 0,
it will not be transmitted to the fusion centre;when
J
i
(k) [ {j
1
,−j
1
},it will be transmitted to the fusion
center by u
i
(k) = 1 bit;when J
i
(k) [ {j
2
,−j
2
,j
3
,−j
3
},
it will be transmitted to the fusion centre by u
i
(k) = 2 bits;
when J
i
(k) [ {j
4
,−j
4
,j
5
,−j
5
,j
6
,−j
6
,j
7
,−j
7
},it will
be transmitted to the fusion centre by u
i
(k) = 3 bits and so
on.Hence,for example,when L
i
= 7 the quantised
innovation J
i
(k) can be transmitted by not more than
u
i
(k) = 2 bits,although there needs u
i
(k) = 3 bits for the
ﬁxed bit transmission in [6,16].Obviously,the dynamic
transfer strategy can effectively save bandwidth and
transmitting power.On the other hand,under the same
bandwidth constraint condition,our approach should
perform better than those in [6,16] because more quantised
messages are added.
After receiving the quantised measurement messages,the
fusion centre will combine the received messages J(k) to
estimate the state X(k).Similar as in [6,14,15],we assume
that the fusion centre has enough bandwidth and energy to
transmit the information and ignore the receiving constraints
of sensors.
3.2 State estimation using quantised innovations
In this subsection,following the idea of Curry et al.[1],we
introduce a nonlinear approximate MMSE ﬁltering
algorithm with quantised innovations.
By (3),we have that
s{M
k−1
1
,Y(k)}.s{M
k
1
},k = 1,2,...(17)
By the property of iterated conditional expectation [19,p.37],
for the state variable X(k) (k = 1,2,...),it holds that
E[X(k)s{M
k
1
}]
= E[E[X(k)s{M
k−1
1
,Y(k)}]s{M
k
1
}] (18)
Hence,the expectation of X(k) conditioned on quantised
measurements {M
k
1
} can be performed in two steps:
1.Find E[X(k)s{M
k−1
1
,Y(k)}],this is the usual goal of
estimation with unquantised measurements;
2.Find the expectation of E[X(k)s{M
k−1
1
,Y(k)}]
conditioned on the quantised measurements {M
k
1
}.
The ﬁrst step,under the Gaussian assumption of prior pdf
p(X(k)s{M
k−1
1
}),can be performed (similar to the
Algorithm 2 in a Gauss–Markov model) as
X
∗
kk
= E[X(k)s{M
k−1
1
,Y(k)}]
=
ˆ
X
kk−1
+K
k
(Y(k) −
ˆ
Y
kk−1
)
=
ˆ
X
kk−1
+K
k
(1(k)) (19)
P
∗
kk
= P
kk−1
−K
k
(R +P
yy
)K
T
k
(20)
where,
ˆ
X
kk−1
,P
kk−1
,K
k
and P
yy
can be obtained by (8),(9),
(13) and (15),respectively.
The second step is to take the expectation of (19) conditioned
on {M
k
1
} to ﬁnd the mean of X(k) conditioned on quantised
IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26 19
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The Institution of Engineering and Technology 2011
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measurements
ˆ
X
kk
= E[X(k)s{M
k
1
}]
= E[E[X(k)s{M
k−1
1
,Y(k)}]s{M
k
1
}]
=
ˆ
X
kk−1
+K
k
E[1(k)s{M
k
1
}] (21)
Correspondingly
P
kk
= MSE[
ˆ
X
kk
s{M
k
1
]
= E[(X
k
−
ˆ
X
kk
)(X
k
−
ˆ
X
kk
)
T
s{M
k
1
}]
where,by (19) and (21)
X
k
−
ˆ
X
kk
= X
k
−
ˆ
X
∗
kk
+(
ˆ
X
∗
kk
−
ˆ
X
kk
)
= X
k
−
ˆ
X
∗
kk
+K
k
(1(k) −E[1(k)s{M
k
1
}])
Then
E[(X
k
−
ˆ
X
kk
)(X
k
−
ˆ
X
kk
)
T
s{M
k
1
}]
= P
∗
kk
+K
k
(1(k) −E[1(k)s{M
k
1
}])(1(k)
−E[1(k)s{M
k
1
}])
T
K
T
k
and ﬁnally
P
kk
= P
∗
kk
+K
k
Cov[1(k)s{M
k
1
}]K
T
k
= P
kk−1
−K
k
(R +P
yy
)K
T
k
+K
k
Cov[1(k)s{M
k
1
}]K
T
k
(22)
Thus,the approximate MMSE ﬁlter with quantised
measurements is given by (19)–(22).The ﬁrst step is directly
obtained from the Kalmantype ﬁlter solution because of the
Gauss–Markov property of system (1) and (2).In the second
step,an important practical problem is to compute efﬁciently
E[1(k)s{M
k
1
}] in (21) and Cov[1(k)s{M
k
1
}] in (22).Despite
this difﬁculty,this approach does provide one common point of
departure for designing purposes and approximations can be
made that depend on the speciﬁc method of quantisation.
3.3 Algorithm
In this subsection,we will discuss how to calculate
E[1(k)s{M
k
1
}] and Cov[1(k)s{M
k
1
}].
First,given
ˆ
X
k−1k−1
and
ˆ
P
k−1k−1
,by unscented transform
(UT),we have
ˆ
Y
kk−1
=
2d
j=0
W
j
k−1
H(X
j
kk−1
),i = 1,2,...,N (23)
P
yy
=
2d
j=0
W
j
k−1
(H(X
j
kk−1
) −
ˆ
Y
kk−1
)(H(X
j
kk−1
) −
ˆ
Y
kk−1
)
T
(24)
Then we can obtain the covariance of innovation
S(:,:,k) = MSE[1(k)M
k−1
1
] = E[1(k)1(k)
T
M
k−1
1
]
= E[(Y
k
−
ˆ
Y
kk−1
)(Y
k
−
ˆ
Y
T
kk−1
M
k−1
1
]
= P
yy
+R (25)
Further,under the Gaussian assumption of the prior
pdf p(X(k)s{M
k−1
1
}) = N(X(k);X
kk−1
,P
kk−1
),for the
innovation 1(k),we obtain
p(1(k)M
k−1
1
) =
p(1(k),X(k)M
k−1
1
)dX(k)
=
p(1(k)X(k),M
k−1
1
)p(X(k)M
k−1
1
)dX(k)
=
p(1(k)X(k))p(X(k)M
k−1
1
)dX(k)
=
N(1(k);H(X(k)) −H(X
kk−1
),R)
×N(X(k);X
kk−1
,P
kk−1
)dX(k)
≃N(1(k);0,P
yy
+R) = N(1(k);0,S(:,:,k))
Thus,for the normalised innovation
1(k),we have
p(
1(k)M
k−1
1
) = N(
1(k);0,I
N×N
) (26)
where I
N×N
is an Norder identity matrix.Further
p(
1(k)M
k−1
1
,M
k
) =
p(
1(k)M
k−1
1
)
Pr(M
k
M
k−1
1
)
if Q
L
(
1(k)) = M
k
0 else
⎧
⎨
⎩
=
e
−(1/2)
1(k)
T
1(k)
(2p)
(N/2)
Pr(M
k
M
k−1
1
)
I
[a
k
,b
k
)
(
1(k))
By the property of conditional probability,we have
p(
1(k)M
k−1
1
,M
k
) =
p(
1(k)M
k−1
1
)
Pr(M
k
M
k−1
1
)
if Q
L
(
1(k)) = M
k
0 else
⎧
⎨
⎩
=
e
−(1/2)
1(k)
T
1(k)
(2p)
(N/2)
Pr(M
k
M
k−1
1
)
I
[a
k
,b
k
)
(
1(k))
(27)
where [a
k
,b
k
) is the hypercube corresponded to M
k
Pr(M
k
M
k−1
1
) =
b
k
a
k
p(
1(k)M
k−1
1
)d
1(k)
=
b
k
a
k
N(
1(k);0,I
N×N
)d
1(k) (28)
and
I
[a
k
,b
k
)
(
1(k)) =
1 if
1(k) [ [a
k
,b
k
)
0 else
(29)
20 IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26
&
The Institution of Engineering and Technology 2011 doi:10.1049/ietspr.2009.0284
www.ietdl.org
Then
E(
1(k)M
k−1
1
,M
k
) =
b
k
a
k
1(k)p(
1(k)M
k−1
1
,M
k
)d
1(k)
=
b
k
a
k
1(k)p(
1(k)M
k−1
1
)d
1(k)
b
k
a
k
p(
1(k)M
k−1
1
)d
1(k)
=
b
k
a
k
1(k)N(
1(k);0,I
N×N
)d
1(k)
b
k
a
k
N(
1(k);0,I
N×N
)d
1(k)
(30)
and
Cov(
1(k)M
k−1
1
,M
k
)
= E(
1(k)1(k)
T
M
k−1
1
,M
k
)
−E(
1(k)M
k−1
1
,M
k
)E
T
(
1(k)M
k−1
1
,M
k
)
=
b
k
a
k
1(k)
1(k)
T
p(
1(k)M
k−1
1
,M
k
)d
1(k)
−E(
1(k)M
k−1
1
,M
k
)E
T
(
1(k)M
k−1
1
,M
k
)
=
b
k
a
k
1(k)
1(k)
T
N(
1(k);0,I
N×N
)d
1(k)
b
k
a
k
N(
1(k);0,I
N×N
)d
1(k)
−E(
1(k)M
k−1
1
,M
k
)E
T
(
1(k)M
k−1
1
,M
k
) (31)
Remark 2:Numerical computation of the above integrals has a
heavy computational burden,but it is worth noting that the
integrals are uniquely determined by its integral intervals,
whereas the integral intervals are only related to the
quantitative strategies.Therefore these integrals can be
calculated ofﬂine.As described in Remark.1,we can take
j
j
= E[
1
i
(k))h
j
,
1
i
(k)) ≤h
j+1
],j = 1,2,...,l
i
,that is,
J
i
(k) = E[
1
i
(k))h
j
,
1
i
(k)) ≤h
j+1
].Furthermore
J(k) = {J
1
(k),J
2
(k),...,J
N
(k)}
T
= E(
1(k)M
k
)
Similarly,the covariance matrix of quantisation error can also
be calculated ofﬂine for different quantitative results M
k
of
normalised innovation,denoting it as S(k).
Finally,by property of expectation and covariance,we
have
E(1(k)M
k−1
1
,M
k
) = S(:,:,k)
1/2
E(
1(k)M
k−1
1
,M
k
)
= S(:,:,k)
1/2
J(k) (32)
Cov(1(k)M
k−1
1
,M
k
) = S(:,:,k)
1/2
Cov(
1(k)M
k−1
1
,M
k
)
×(S(:,:,k)
1/2
)
T
= S(:,:,k)
1/2
S(k)(S(:,:,k)
1/2
)
T
(33)
Thus,equations (8)–(15) and (21) –(33) constitute the UKF
based on multilevel quantised innovations (QIUKF)
algorithm.
QIUKF is implemented in three stages:Given
ˆ
X
00
and
ˆ
P
00
;Compute the values of E(
1(k)M
k−1
1
,M
k
) and
Cov(
1(k)M
k−1
1
,M
k
) using (30) and (31) for all possible
values of M
k
ofﬂine.
Algorithm 3 – A Prediction step.
Given:
ˆ
X
k−1k−1
and P
k−1k−1
ˆ
X
kk−1
=
2d
j=0
W
j
k−1
F(X
j
k−1
) (34)
P
kk−1
= Q+
2d
j=0
W
j
k−1
(F(X
j
k−1
) −
ˆ
X
kk−1
)(F(X
j
k−1
)
−
ˆ
X
kk−1
)
T
(35)
ˆ
Y
kk−1
=
2d
j=0
W
j
k−1
H(X
j
kk−1
) (36)
(S(:,:,k)
1/2
)
−1
= (P
yy
+R
1/2
)
−1
(37)
where {W
j
k−1
,j = 0,1,...,2d} is the weight set,
{X
j
k−1
,j = 0,1,...,2d} is the sigma point set [17,section
2.4.2],X
j
kk−1
denotes F(X
j
k−1
),
ˆ
Y
kk−1
= (y
1
(kk −1),
y
2
(kk −1),...,y
N
(kk −1))
T
and
K
k
= P
xy
(R +P
yy
)
−1
(38)
P
xy
=
2d
j=0
W
j
k−1
(X
j
kk−1
−
ˆ
X
kk−1
)(H(X
j
kk−1
) −
ˆ
Y
kk−1
)
T
(39)
P
yy
=
2d
j=0
W
j
k−1
(H(X
j
kk−1
) −
ˆ
Y
kk−1
)(H(X
j
kk−1
) −
ˆ
Y
kk−1
)
T
(40)
Algorithm 3 – B Measurement and quantisation
Given:
ˆ
Y
kk−1
and (S(:,:,k)
1/2
)
−1
= ((P
yy
+R)
1/2
)
−1
.
Measure:Y(k).
Construct quantised innovations M
k
= Q
L
(
1(k)) as in
Section 3.1.
Transmit quantised innovations M
k
.
Algorithm 3 – C Correction step.
Receive:Quantised observations M
k
According to M
k
,take the corresponding values of J(k)
and S(k) as in Remark 2.
Compute E(1(k)M
k−1
1
,M
k
) and Cov(1(k)M
k−1
1
,M
k
)
using (32) and (33)
ˆ
X
kk
=
ˆ
X
kk−1
+K
k
E[1(k)s{M
k
1
}] (41)
P
kk
= P
kk−1
−K
k
(R +P
yy
)K
T
k
+K
k
Cov[1(k)s{M
k
1
}]K
T
k
(42)
Remark 3:When the stateobservation model (1),(2) are
linear and L ¼ 3,it is obvious that the QIUKF of this paper
can vary to quantised KF based on 1bit quantised
innovation as h
1
0.Consequently,one may wonder
whether there is a relation between the QIUKF of this paper
and the vectorobservation case SOIKF [14,15].Apart
from the using of UT and multiplelevel quantisation,the
IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26 21
doi:10.1049/ietspr.2009.0284
&
The Institution of Engineering and Technology 2011
www.ietdl.org
two schemes are fundamentally different.In the SOIKF,the
computing process of gain matrix,state estimation and
estimation covariance matrix need to be carried out N times
repeatedly in the correctional step.In the QIUKF algorithm,
however,the all computing processes are only performed
one time.In the work of You’s et al.[16] and Msechu’s et
al.[6],only the multiplelevel quantisation ﬁlter for scalar
observation case is considered.
4 Performance analysis
In this section,we analyse the performance of the QIUKF.
Comparison of the ECM corrections for the UKF in (12)
with those for the Algorithm 3 (QIUKF) in (42) reveal that
they are identical except for the third term in (42).The
similarity is quantiﬁed by deﬁning the ECM reduction per
correction step [cf.(42)]
DP(k):= P
kk−1
−P
kk
= K
k
(R +P
yy
)K
T
k
−K
k
Cov[1(k)s{M
k
1
}]K
T
k
(43)
If we use Y
k
instead of M
k
in the correction step,the ECM
reduction will be [cf.(12)]
DP(k):= P
kk−1
−P
kk
= K
k
(R +P
yy
)K
T
k
(44)
Comparing (43) with (44),we see that the ECM reduction
achieved by the Algorithm 3 is less than that of the UKF.
Similar to [6],the optimal quantiser is deﬁned as the one
that maximises the average variance reduction,that is
{h
∗
i
(k)}
L
i=1
:= arg max
{h
i
(k)}
L
i=1
E
M
k
(DP(k)M
k−1
1
)
= arg min
{h
i
(k)}
L
i=1
E
M
k
(Cov(1(k)M
k
1
)M
k−1
1
) (45)
An MSE distortion conditioned on M
k−1
1
is adopted and the
optimal quantiser of
1(k) is deﬁned as
{h
†
i
(k)}
L
i=1
:= arg min
{h
i
(k)}
L
i=1
E
M
k
(Cov(
1(k)M
k
1
)M
k−1
1
) (46)
It is easy to obtain [6] that the corresponding optimal
thresholds are uniform,that is
{h
∗
i
(k)}
L
i=1
= {h
†
i
(k)}
L
i=1
From (26),we know that the components
1
i
(k)(i = 1,
2,...,N) of
1(k) are independent identically distributed
(IID).Obviously,the quantisation does not undermine this
independent.In other words,the components m
i
(k)
(i = 1,2,...,N) of M
k
are still IID.Hence,the covariance
Cov(
1(k)M
k
1
) is a diagonal matrix,and
Cov(
1(k)M
k
1
)
=
var(
1
1
(k)M
k
1
) 0 · · · 0
0 var(
1
2
(k)M
k
1
) · · · 0
˙ ˙ ˙
˙˙ ˙ ˙
˙˙ ˙ ˙
0 0 · · · var(
1
N
(k)M
k
1
)
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(47)
Furthermore,noting that E
M
k
(var(
1
i
(k)M
k
1
)) (i =1,2,...,N)
are equal,we have,without loss of generality
E
M
k
(Cov(
1(k)M
k
1
)M
k−1
1
)
=E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
) ×I
N×N
(48)
Hence,the optimisation problem in (46) is equivalent to
{h
†
i
(k)}
L
i=1
:=arg min
{h
i
(k)}
L
i=1
E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
) (49)
and for the covariance of 1(k) conditioned on M
k
1
,we have
E
M
k
(Cov(1(k)M
k
1
)M
k−1
1
)
=E
M
k
(S(:,:,k)
1/2
Cov(
1(k)M
k
1
)(S(:,:,k)
1/2
)
T
M
k−1
1
)
=S(:,:,k)
1/2
E
M
k
(Cov(
1(k)M
k
1
)M
k−1
1
)(S(:,:,k)
1/2
)
T
=S(:,:,k)
1/2
×E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
)
×I
N×N
×(S(:,:,k)
1/2
)
T
=E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
) ×I
N×N
×S(:,:,k) (50)
Combining (50) with (43),we have
E
M
k
(DP(k)M
k−1
1
)
:= E
M
k
(P
kk−1
−P
kk
M
k−1
1
)
= K
k
(R +P
yy
)K
T
k
×−K
k
E
M
k
(Cov[1(k)s{M
k
1
}]M
k−1
1
)K
T
k
= K
k
(R +P
yy
)K
T
k
−E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
)
×I
N×N
×K
k
(P
yy
+R)K
T
k
= (1 −E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
))
×I
N×N
×K
k
(P
yy
+R)K
T
k
(51)
It means that QIUKF exhibits MSE performance identical to a
UKF based on analogueamplitude observations applied to
the same observation model with MSE reduction per
correction step multiply by a factor of
a= 1 −E
M
k
(var(
1
1
(k)M
k
1
)M
k−1
1
) (52)
The optimisation problem in (49) has a wellknown solution
22 IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26
&
The Institution of Engineering and Technology 2011 doi:10.1049/ietspr.2009.0284
www.ietdl.org
given by the Lloyd–Max quantiser [20].The optimal
normalised thresholds values for L ¼ 3,7 and 15,from
[20],are given in Table 1.The corresponding values of J
and S are also given in Table 1.The corresponding
a
obtained by using (52) and the maximal number u of
transmitting bytes (Max u) are summarised in Table 2.
Now,we will calculate the transmission bandwidth of
quantised information in the sense of probability average.
By the transmitting strategy stated in Section 3.1,for the
number u of transmitting bytes,it is easy to obtain the
average transmission bandwidth of quantised information in
the sense of probability
E(u
i
(k)M
k−1
1
) = 2
l
j=1
h
j+1
h
j
u
i
(k)
1
2p
√
e
−t
2
/2
dt (53)
where h
l+1
= +1.
The maximal number u of transmitting bytes (Max u) and
the average transmitting bandwidth (Ave u) in the sense of
probability are given in Table 3.In order to compare with
the transmitting bandwidth needed by the ﬁxedbits
transmitting strategy in Msechu et al.[6] and You et al.
[16],the byte numbers (Msechu u and You u) needed under
the same output level condition are also listed in Table 3.
Remark 4:On the one hand,in scalarobservation case,with
the same quantisation level,the performance of Algorithm 3
and is similar as that of the algorithms in [6,16].However,
it should be noted that the required transmitting bandwidth
is different,the bandwidth required of Algorithm 3
proposed in this paper is the minimum.Then,Algorithm 3
can effectively save the transmission bandwidth.At the
same time,the communication cost is also reduced.On the
other hand,under the same transmission bandwidth
condition,innovation can be quantiﬁed into more levels in
Algorithm 3 than in Msechu’s and You’s.For example,
under the 2bit bandwidth constraining condition,for
Msechu’s algorithm [6],fourlevel quantisation is allowed;
for You’s [16],ﬁfthlevel is allowed,but for Algorithm 3,
seventhlevel is allowed.Thus,Algorithm 3 will result in
the least loss of measurement information.Then,the novel
transmission strategy can effectively improve the utilisation
of communication bandwidth.Furthermore,the state
estimation will be more accuracy than those in [6,16].
5 Simulation results
In this section,we consider a target tracking systemin WSNs.
There are N ¼ 81 sensors,which are uniformly located in a
square region of about 400 × 400 m
2
and the position
(x
i
,y
i
) of the ith sensor is known.It is assumed that there is
no communication loss and that the sensors are synchronised.
The target (e.g.handcart etc.) is considered as a point
object moving in a twodimensional plane.Here,we
consider a quite general,nonlinear motion model,that is,
the coordinated turn rate model [21].This model assumes
that the target moves with a nearly constant speed and
unknown turn rate.We denote X(k) as the state of the
target.X(k) represents the coordinates X
1
,X
2
,the velocities
˙
X
1
,
˙
X
2
and the turn rate fas
X(k) = {X
1
(k),
˙
X
1
(k),X
2
(k),
˙
X
2
(k),f(k)}
T
(54)
The target motion in the Cartesian coordination system is
modelled as
X(k) = F(k −1)X(k −1) +G(k −1)w(k −1) (55)
where the w(k 21) is the process noise with covariance
matrices of Q(k −1).The system noise is a Gaussian noise
w(k) N(0,diag[@
2
1
,@
2
2
,@
2
f
]),where @
1
=@
2
=@
f
=
0.001.The state transition matrix F(k) and process noise
coefﬁcient matrices G(k) can be written as,respectively
F(k) =
1
sin f(k)Dt
f(k)
0
1 −cos f(k)Dt
f(k)
0
0 cos f(k)Dt 0 −sin f(k)Dt 0
0 −
1 −cos f(k)Dt
f(k)
1
sin f(k)Dt
f(k)
0
0 sin f(k)Dt 0 cos f(k)Dt 0
0 0 0 0 1
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(56)
Table 1
Optimal normalised thresholds values,the
corresponding values of J and S
h
j
J
j
S
j
L ¼ 3
j ¼ 1 0.6120 1.2241 0.2508
L ¼ 7
j ¼ 1 0.2803 0.5607 0.0287
2 0.8744 1.1883 0.0425
3 1.611 2.0338 0.1401
L ¼ 15
j ¼ 1 0.1369 0.2739 0.0060
2 0.4143 0.5550 0.0065
3 0.7030 0.8514 0.0074
4 1.013 1.1752 0.0098
5 1.361 1.5464 0.0137
6 1.776 2.0068 0.0240
7 2.344 2.6814 0.0947
Table 2
avalues
L ¼ 3 L ¼ 7 L ¼ 15
Max u (bits) 1 2 3
a 0.8098 0.9560 0.9893
Table 3
Transmitting bandwidth
L ¼ 3 L ¼ 7 L ¼ 15
Max u (bits) 1 2 3
Ave u (bits) 1 1.1611 1.8806
Msechu u (bits) 2 3 4
You u (bits) 1 3 4
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The Institution of Engineering and Technology 2011
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and
G(k) =
Dt
2
2
0 0
Dt 0 0
0
Dt
2
2
0
0 Dt 0
0 0 1
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(57)
For all simulations,we take the following parameters.The
total time T is 120 s and the time step is Dt ¼ 1 s.The turn
rate f takes the value of 20.05 (1 ≤ k ≤ 70),0.15
(71 ≤ k ≤ 100) or 0.25 rad (101 ≤ k ≤ 120).The target
initial state is
X
0
= [90,3,300,4,−0.05]
T
Here,we employ the tracking algorithm starting from
X
00
= [100,2,280,5,0.08]
T
P
00
= 30 ×Q
Example 1:In order to compare the performance of QIUKF
with the performance of MLQIKF in [6,16] and the ideal
UKF [17] in target tracking,we consider the WSN with
scaler measurements [6] under the same bandwidth
constraint conditions.In this example,we consider the
target tracking problem with 2bit bandwidth restriction.
The measurement is the noisy relative distance between
the sensors and the target (e.g.radar,acoustic sensors,
sonar etc.).Sensor S
i
,located at position X
i
= (x
i
,y
i
)
measures [6]
y
i
(k) = h
i
(X(k)) +v
i
(k)
=
X(k) −X
i
+v
i
(k)
=
(X
1
(k) −x
i
)
2
+(X
2
(k) −y
i
)
2
+v
i
(k) (58)
The measurement noise is a Gaussian noise v
i
(k) N(0,9).
Linearising (58) about a generic state prediction
X
kk−1
in
similar fashion to the extended Kalman ﬁlter (E)KF,one
can obtain
y
i
(k) ≃
∂h
i
(X(k))
∂X(k)

X
kk−1
X(k) +y
0
i
(k) +v
i
(k) (59)
where y
0
i
(k) is a function of
X
kk−1
and X
i
.The linearised
observations (59) together with (55)–(57) are amenable to
the MLQIKF algorithms in [6,16].
Remark 5:In the algorithm running,the choice of sensor
data uses the nearest neighbour principle.Speciﬁcally,
every ﬁve seconds,the fusion centre will activate the three
sensors that are nearest to the predicted location of target.
The predicted measurements and the corresponding
diagonal elements of prediction error covariance are sent
to them.Then,the sensor nodes approximatively calculate
the quantised innovations according to the quantisation
algorithm in Section 3,and return them to fusion centre.
Thus,the fusion centre can use the quantised innovations
returned by the three sensor nodes for target state estimation.
In Figs.1–3,the simulation results of QIUKF the
MLQIKF [6,16]),and the ideal UKF [17] in target tracking
are shown.The simulation results are based on 100 Monte
Carlo runs.The criterion for comparison is the mean
squared error and root meansquare error (RMSE) on the
position of the target.
Fig.1 Target position estimation of Algorithm3(QIUKFtwo bits),
the MLQIKF (Msechutwo bits,Youtwo bits) and the UKF
Fig.2 MSE
a MSE of position along the Xaxis
b MSE of position along the Yaxis
24 IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26
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The Institution of Engineering and Technology 2011 doi:10.1049/ietspr.2009.0284
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In Fig.1,we show the simulation result of last Monte Carlo
for position estimation.In Figs.2a and b,all plots generated
illustrate the evolution of the MSEs,obtained from the
diagonal elements of the respective ECMs,against the time
index k.These simulation results conﬁrm the correctness of
the theoretical analysis in Section 4.In Fig.3,the RMSEs of
position along the Xaxis and Yaxis are given,respectively.
Seen from Figs.3a and b,the estimation performance of
Algorithm 3 is obviously better than those of Msechu’s et al.
[6] and You’s et al.[16].Furthermore,the position RMSEs
of Algorithm 3 and those of UKF are very close.The
performance of Algorithm 3 is better than those of Msechu’s
et al.[6] and You’s et al.[16],is because Algorithm 3 have
a fuller use of communications bandwidth than Msechu’s
et al.[6] and You’s et al.[16].
Example 2:In this example,the performance of SOIKF is
compared with that of QIUKF for vector observation WSN.
In this WSN,each sensor node can simultaneously measure
relative distance the target to their and the azimuth.The
quantisation bandwidth for each measurement innovation is
only 1bit.Then,for SOIKF,only twolevel quantisation is
allowed for each measurement innovation;for Algorithm 3,
threelevel is allowed.
Sensor S
i
,located at position X
i
= (x
i
,y
i
) measures
y
1
i
(k) =
X(k) −X
i
+v
1
i
(k) (60)
y
2
i
(k) = arctan
X
2
(k) −y
i
X
1
(k) −x
i
+v
2
i
(k) (61)
The measurement noise is a Gaussian noise v
i
(k)
N(0,diag[(3 m)
2
,(0.003 rad)
2
]).In the algorithm running,
the choice of sensor data is similar as in Example 1.The
Fig.3 RMSE
a RMSE of position along the Xaxis
b RMSE of position along the Yaxis
Fig.4 Target position estimation of Algorithm 3 (QIUKF one
bit),the SOIKF and the UKF
Fig.5 RMSE
a RMSE of position along the Xaxis
b RMSE of position along the Yaxis
IET Signal Process.,2011,Vol.5,Iss.1,pp.16–26 25
doi:10.1049/ietspr.2009.0284
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The Institution of Engineering and Technology 2011
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difference is that here only two sensors measurement
information are used for each estimating cycle.
In Figs.4 and 5,the comparing of SOIKF with Algorithm
3 (QIUKF) in target tracking are shown.The simulation
results are based on 100 MonteCarlo runs.In Fig.4,we
show the simulation result of last Monte Carlo for position
estimation.In Figs.5a and b,the RMSE of position along
Xaxis and Yaxis are given,respectively.Seen from Figs.4
and 5,SOIKF diverge,whereas Algorithm 3 still works
well.Similar as in Example 1,the performance of
Algorithm 3 is better than that of SOIKF [14,15],is
because QIUKF have a fuller use of communications
bandwidth.
6 Conclusion
This paper widens the scope of SOIKF [14,15] and
MLQIKF [16] by addressing the middle ground between
estimators based on severely quantised (1bit) data and
those based on unquantised data for general vector state
vector observation case.The problem of state estimation
with quantised measurements is considered in WSNs.An
novel transport strategy is given for quantised innovations,
which can effectively improve the utilisation of
transmission channel.An efﬁcient solution for state
estimation with quantised innovations has been proposed
for general vector statevector observations case.The
performance of resulting ﬁlter is discussed.Performance
analysis shows that,under the same bandwidth constraint
condition.The performance of Algorithm 3 is better than
those in [6,14–16].The novel method is simulated for
target tracking scenario in WSNs.Its results are compared
with the results of the ideal UKF and those of the algorithm
in [6,14–16].The performance analysis and numerical
results show that the MSEs and RMSEs of the QIUKF
tracker is better than those of SOIKF [14,15] and of
MLQIKF [6,16].
7 Acknowledgments
The authors thank the anonymous referees and the associate
editor for valuable comments that helped to improve the
presentation of the paper.This work was jointly supported
by National Natural Science Foundation (60935001,
60874104);973Project (2009CB824900,2010CB734103);
Shanghai Key Basic Research Foundation:08JC1411800;
the Space Foundation of SupportingTechnology No.2008
HTSHJD003 and cultivation Fund of the Key Scientiﬁc
Technical Project Ministry of Education of China(706022).
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&
The Institution of Engineering and Technology 2011 doi:10.1049/ietspr.2009.0284
www.ietdl.org
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