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Target Motion Analysis Based on Peak PowerMeasurements using Networked Sensors

Target motion analysis (TMA) using a network of wireless

sensors/receivers which measure the power from a mobile RF

emitter is considered. Due to limited communication capability

of each sensor node, only peak power measurements from

sensor nodes are transmitted to the fusion center. We present

two main results that yield the optimum sensors’ configuration

such that the asymptotically achievable error variance of the

target trajectory’s estimate is minimized, and we derive efficient

numerical algorithms for computing the optimum estimates of the

trajectory of the moving target, thus achieving the goal of TMA.

I. INTRODUCTION

Wireless sensor networks have received

considerable attention in recent years, and have

wide applications such as localization and tracking

of mobile users, intrusion detection, environmental

monitoring, climate control and disaster management,

detection and identification of hazardous materials

[8, 13, 15]. A common scenario for wireless sensor

networks envisions sensor nodes that are homogenous

and have limited capabilities in computation, sensing

range, and communications. In this paper the

scenario is considered where these sensor nodes

work collaboratively in accomplishing the goal

of extracting the desired information, based on

measurements of the mobile emitter’s RF signal

power recorded at different locations and possibly

at different time instants. In this paper we consider

the employment of basic receivers that measure

only the signal power from the RF emitter/target

within the sensor’s range. Oblivious of being

observed, the target moves at constant speed and

fixed direction. Localization and tracking of such

targets is referred to as target motion analysis

(TMA). TMA entails the estimation of the initial

position and the velocity of the target. TMA

based on a single sensor such as in bearings-only

measurements has been extensively investigated in

the literature; see [1, 3, 11] and references therein.

The employment of two sensing modalities such

Manuscript received April 21, 2009; revised September 15, 2009;

released for publication December 13, 2009.

IEEE Log No. T-AES/47/1/940057.

Refereeing of this contribution was handled by V. Krishnamurthy.

This research is supported in part by the United States Air Force.

0018-9251/11/$26.00 c° 2011 IEEE

as the Doppler-bearing method [2, 7] has also

been investigated; and in [5] we considered the

Doppler-power method. However, TMA based on

power-only measurements has not been undertaken

due to the strong nonlinear dependence of the received

signal power on the TMA motion parameters, and,

most importantly, the unknown nature of the path loss

coefficient.

In this paper the feasibility of employing a

wireless sensors network consisting of currently

available inexpensive RF receivers with limited

sensing range which measure the received signal’s

power, is investigated. Such sensors are the simplest

possible for localization and tracking of RF emitters.

Even though each stand-alone sensor is not capable

of achieving good motion parameter estimates for

TMA, collectively they may perform the TMA task

extraordinarily well. The challenge lies in the design

of sensor networks and fusion algorithms. Specifically,

because the sensing range is limited, the proper

positioning of the sensor nodes is crucial and it

strongly impacts the estimation performance–think

of the geometric dilution of precision of the

measurement arrangement. Because sensors have

limited communication capability, a limited number

of power measurements will be transmitted to the

fusion center. To avoid the difficulties associated

with propagation effects and in order to limit the

information flow in the sensor network, it is stipulated

that each sensor transmits only the peak power

measurement; the fusion center will first estimate

the target’s trajectory, prior to the estimation of the

TMA parameters. Due to the fact that the associated

Cramer-Rao lower bound of the target motion

parameters estimation error is dependent on the

geometry of the sensors network, we are able to

optimize the sensors’ location under the constraint

of maximum allowable distance between neighboring

sensors. In addition, an efficient fusion algorithm for

the estimation of the target trajectory and the TMA

parameters is developed. Our results in this paper

are motivated by our earlier work [4], the posterior

Cramer-Rao lower bound approach to sensor network

optimization developed in [6], the research reported

in [10] where range measurements/multilateration

is considered, and the geolocation of a stationary

target using bearings-only measurements [16].

The common thread in these papers is the quest

for estimation performance improvement with

applications related to TMA. The notation in this

paper is standard and will be made clear as we

proceed.

II. MAIN RESULTS

For convenience we confine our attention to TMA

in the two-dimensional plane, although the results can

be easily extended to the three-dimensional space. The

712 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 1 JANUARY 2011

target’s trajectory is parameterized by the following

equation for a straight line in normal form:

ax+ by+ c= 0,pa2 + b2 = 1 (1)

where a, b are independent parameters, and c is a

dependent parameter used to scale a, b such thatpa2 + b2 = 1. Since the target is oblivious of being

observed, it transmits RF signals while traveling along

the, as yet unknown, straight line trajectory (1). Thekinematic navigation assumption is invoked and it is

assumed that the speed of the target is constant. Hence

the location of the target at time t is specified by

(xT+ vxt,yT+ vyt) with (xT,yT) the initial location at

t= 0, and (vx,vy) the velocity of the target. It follows

that (xT,yT) satisfies (1), and

a2 = v2y=(v2x + v

2y ), b2 = v2x=(v

2x + v

2y ): (2)

As the target passes through the area where the sensor

network is deployed, its RF signal is sensed by some

of the sensors within range. Due to affordabilityconsiderations, the deployed sensors are assumed to

measure only the power of the received RF signal

emitted by the target. In addition, due to the limited

capability for wireless communications and to

lengthen battery life, each sensor transmits to the

fusion center only one measurement rather than all the

recorded power measurements. A smart approach isfor each sensor to transmit only the recorded peak RF

power (the latter has the most information content).

Since the RF signal travels on a straight line along

the line of sight (LOS) to the sensor, the peak power

corresponds to the shortest distance from the sensor

to the target’s trajectory. This can be utilized toestimate the target’s trajectory and determine the TMA

parameters.

Suppose that there are a total of (N +1) sensors

close to the target’s trajectory, and these simple

receivers are able to measure the power of the RF

signal from the target as it traverses the sensor

network. Thus (N +1) peak power measurementsfPkgNk=0 taken at the respective time instants of ftkgNk=0by the (N +1) sensors located at fxk,ykgNk=0 aremade available to the fusion center. Without loss of

generality, set t0 = 0. According to the the path loss

law [12], the peak power measured by each of the

(N +1) sensors can be expressed as Pk = C=R2k where

Rk is the distance between the transmitter and the kth

receiver, and is given by

Rk =q(xT¡ xk + vxtk)2 + (yT¡ yk + vytk)2: (3)

Because Rk represents the shortest distance from the

kth sensor to the target’s trajectory (1), the following

geometric relationship holds:

axk + byk + c=§Rk, k = 0,1, : : : ,N: (4)

The sign ambiguity is due to the relative positions of

the target’s trajectory and the kth sensor.

The difficulty of TMA based on power

measurements lies in the unknown nature of the

constant C. The latter is a function of the transmitter’s

power, the propagation environment, and the

transmitter and receiver antennae gains. Note however

that C is invariant with respect to k due to the

homogeneity of the sensor nodes. Following the

methodology developed in [5], we calculate the ratios

of peak power, to eliminate the C dependence:

½k =§sP0Pk=§Rk

R0=axk + byk + c

ax0 + by0 + c(5)

for k = 1,2, : : : ,N. In practice, ½k is determined from

the ratio of peak power measurements according to

½k =§qP0=Pk = ½k + ´k (6)

where k = 1,2, : : : ,N and Pk is the measurement of Pk.

It is assumed that the measurement errors f´kg arejointly Gaussian. Although this assumption rarely

holds in practice, it is adequate if the measurements

of peak power involve Gaussian noises, and if the

signal-to-noise ratio (SNR) is relatively high. The

sign ambiguity is inherited from (4). Without loss

of generality we assume that x0 = 0 and y0 = 0;

this can be achieved by a suitable translation of

the coordinate system’s origin. Denote ®= a=c and

¯ = b=c. After the inclusion of measurement error in

the measurement equations, (5) can be rewritten as

xk®+ yk¯ = (½k ¡ 1)¡ ´k (7)

for 1· k ·N. Equation (7) can be stacked into alinear regression equation of the form266664

½1¡ 1½2¡ 1...

½n¡ 1

377775=266664x1 y1

x2 y2

......

xN yN

377775·®

¯

¸+

266664´1

´2

...

Ń

377775 (8)

or Ã = Aμ+ ´ for short notation, where μ = [® ¯]0

is the parameter vector to be determined. The TMA

parameters can be recovered via μ0 := [a b] = c[® ¯]

with c= 1=p®2 +¯2.

LEMMA 1 Consider the target trajectory in (1) and the

equations of peak power ratios in (8). Assume that the

N sensors located at f(xk,yk)gNk=1 are not colinear, i.e.,the two columns of A are not linearly dependent. Under

the hypothesis that f´kgNk=1 are Gaussian distributedwith mean zero and covariance §, the MLE (maximum

likelihood estimate) of the parameter vector μ0 of the

target’s trajectory is given by

μML = (A0§¡1A)¡1A0§¡1Ãc (9)

where c=§k(A0§¡1A)¡1A0§¡1Ãk¡1.PROOF The measurement equations are given by

Aμ = Ã+ ´–see, e.g., (8). Hence this is a standard

CORRESPONDENCE 713

linear regression problem with a Gaussian error. The

only exception is the parameter c that is used to scale

the parameter a and b so that a2 + b2 = 1, or kμ0k= 1,which yields (9).

REMARK 1 It is well known [14] that under certain

mild regularity conditions, the maximum likelihood

(ML) algorithm is asymptotically unbiased, and

achieves the Cramer-Rao lower bound asymptotically.

Clearly, the hypotheses in Lemma 1 satisfy the

regularity condition. Thus with one sensor situated

at the origin, the identifiability of the target trajectory

requires that the remaining N sensors be not located

colinearly. If this is true, then for each unbiased

estimate μ of μ0, and the ML estimate μML in (9), there

holds asymptotically, as N!1,

CovfμMLg := Ef(μ0¡ μML)(μ0¡ μML)0g

! c2(A0§¡1A)¡1 · Ef(μ0¡ μ)(μ0¡ μ)0gby (1=c)Aμ0 = Ã+ ´. The matrix c

¡2A0§¡1A is theFisher information matrix (FIM) whose inverse

constitutes the Cramer-Rao lower bound.

Because the FIM is dependent on the location of

the N sensors, a meaningful optimization problem

arises naturally: with one sensor situated at the origin,

how is one to place the remaining N sensors so that

the error variance associated with the ML algorithm

is minimized? This is referred to as the optimum

sensors’ location problem. However, this problem is

not well posed in the sense that these (N +1) sensors

discussed thus far are assumed to be near the target’s

trajectory, and may constitute only a subset of all

the sensors in the network. Moreover some of these

(N +1) sensors can be far away from each other. For

this reason we focus on the position assignment of

(n+1) local sensors: with one sensor situated at the

center, the remaining n surrounding sensors are no

more than a distance rmax away from it. The value of

rmax is chosen to be the maximum distance between

neighboring sensors that is determined by the effective

range of the receivers and the sparsity of the network.

Once the local position assignment problem is solved,

the result can be extended to the global assignment

of all the sensors, thereby solving the problem of

an optimum network configuration. We therefore set

¢xi = xi¡ x0 and ¢yi = yi¡ y0, and modify the linearregression equation Aμ = cÃ+ c´ according to266664

¢x1 ¢y1

¢x2 ¢y2

......

¢xn ¢yn

377775·a

b

¸=

266664½1¡ 1½2¡ 1...

½n¡ 1

377775c+266664´1

´2

...

ń

377775c (10)

where ¢x2i +¢y2i = (xi¡ x0)2 + (yi¡ y0)2 · r2max for

i= 1,2, : : : ,n.

THEOREM 1 Consider the linear regression (10)

in which ¢x2i +¢y2i · r2max, and the equation errors

fígni=1 are Gaussian with mean zero and covariance§ = diag(¾21,¾

22, : : : ,¾

2n). Then the optimum locations of

the n local sensors for minimizing the corresponding

Cramer-Rao lower bound are given by xi = x0 +

rmax cos(®i) and yi = y0 + rmax sin(®i) satisfying

nXi=1

cos(2®i)

¾2i= 0,

nXi=1

sin(2®i)

¾2i= 0 (11)

where ®i is the angle between the positive x-axis to the

ray from the center to the ith sensor.

PROOF Denote by Tr(¢) the trace operator. The errorvariance for the ML algorithm is given by

Efkμ0¡ μMLk2g= Tr[Ef(μ0¡ μML)(μ0¡ μML)0g]

¸ c2Trf(A0§¡1A)¡1g: (12)

The optimum sensors’ location for position

assignment of n local sensors minimizes the right

hand side of the inequality (12) which corresponds

to the Cramer-Rao lower bound. Recall that for each

i > 0, ¢xi = xi¡ x0, ¢yi = yi¡ y0, and (¢xi,¢yi) =(¢xi=¾i,¢yi=¾i). Using polar coordinates,

¢xi = ri cos(®i), ¢yi = ri sin(®i) (13)

for i= 1,2, : : : ,n. Then, by straightforward calculation,

A0§¡1A=nXi=1

1

¾2i

·¢x2i ¢xi¢yi

¢yi¢xi ¢y2i

¸

=

nXi=1

·¢x2i ¢xiyi

¢yi¢xi ¢y2i

¸: (14)

For convenience, use the notation ci = cos(®i) and

si = sin(®i) for 1· i· n. It follows from the above

two equations that

det(A0§¡1A) =

ÃnXi=1

r2i c2i

!ÃnXk=1

r2k s2k

!

¡Ã

nXi=1

r2i cisi

!ÃnXk=1

r2k cksk

!

=

nXi=1

nXk=1

r2i r2k

³c2i s

2k ¡ siciskck

´

=

nXi=1

nXk=1

r2i r2k cisk(cisk ¡ sick)

=

nXi=1

nXk=1

r2i r2k cos(®i)sin(®k) sin(®k ¡®i)

=

nXk>i=1

r2i r2k (skci¡ cksi) sin(®k ¡®i)


=

nXk>i=1

r2i r2k sin

2(®k ¡®i)

=

nXk=2

k¡1Xi=1

r2i r2k sin

2(®k ¡®i)

=1

2

nXi=1

nXk=1

r2i r2k sin

2(®k ¡®i):

In obtaining the fourth line of the above derivation,

the double sum from 1 to n is first replaced by two

sums with one for k < i and the other for k > i since

the terms for i= k are zero, and then each term

indexed with (i,k) for i < k is replaced by the negative

term indexed with (k, i), that leads to the sum for

all terms such that indexes satisfy k > i. The same

reasoning is used again to obtain the final expression.

Denote Trf¢g as trace operation. We can now obtainthe expression for the right hand side of (12) as

Trf(A0§¡1A)¡1g= 2Pn

`=1 r2`Pn

i=1

Pnk=1 r

2i r2k sin

2(®k ¡®i):

(15)

For optimal position assignment of n local sensors,

minimize the above expression by choosing ri = ri=¾iand ri, subject to the constraint 0< ri · rmax for1· i· n. Alternatively, minimization of (15) can beconverted to a maximization of

J = [Trf(A0§¡1A)¡1g]¡1

=1

2

nXi=1

nXk=1

krk2¹2i ¹2k sin2(®k ¡®i) (16)

under the same constraint where ¹= r=krk and r =[r1 r2 ¢ ¢ ¢ rn]0. Since ¹ has a unit norm, the constrainedmaximization of J is equivalent to

max

(nXi=1

nXk=1

¹2i ¹2k sin

2(®k ¡®i) :nX`=1

¹2` = 1

)(17)

plus the maximization of krk2 subject to 0· ¾iri ·rmax 8i. Consequently, the optimum local sensors’

location is achieved by taking ri = rmax=¾i or xi =

rmax cos(®i) and yi = rmax sin(®i). Thus the optimum

local sensors’ location is determined by the optimality

of the performance function

Jopt =

"2

nX`=1

1

¾2`

#¡1max

®`2[0,2¼)

nXi=1

nXk=1

sin2(®k ¡®i)¾2i ¾

2k

(18)

in which maximization is now over the angular

positions of the n local sensors. Let ¯` = 2®` for `=

1,2, : : : ,n. Using the trigonometric identity sin2(x) =

12[1¡ cos(2x)],

nXi=1

nXk=1

sin2(®k ¡®i)¾2i ¾

2k

=

nXi=1

nXk=1

0:5

¾2i ¾2k

¡nXi=1

nXk=1

0:5cos(¯k ¡¯i)¾2i ¾

2k

:

(19)

It follows that Jopt is achieved by minimizing

nXi=1

nXk=1

cos(¯k ¡¯i)¾2i ¾

2k

=Re

(nXi=1

nXk=1

ej(¯k¡¯i)

¾2i ¾2k

)

=

ÃnXi=1

e¡j¯i

¾2i

!ÃnXk=1

ej¯k

¾2k

!¸ 0:

Hence the optimum angular position of the n local

sensors are determined by the optimality condition in

(11), which concludes the proof.

REMARK 2 It is important to note that Theorem 1 is

very general. In the case of a homoscedastic equation

error covariance § = ¾2I, the optimum local sensors’

configuration includes the case of n equally distributed

sensors on the circumference of a semicircle of radius

rmax with center (x0,y0). It also includes the case of

even n in which each pair of sensors is situated on the

circumference of a circle of radius rmax with center

(x0,y0) and that are ¼=2 apart. Both cases ensure

condition (11). The former implies that f¯ig areequally distributed over the circle of radius rmax by

¯i = 2®i for each i, thereby satisfying (11), and the

latter implies that

sin(2®2k¡1) =¡sin(2®2k)cos(2®2k¡1) =¡cos(2®2k)

(20)

if ®2k¡1 and ®2k are ¼=2 apart, thus leading to (11).The most important case is n > 2 with § = ¾2I. By

taking ®i = ®0 +2i¼=n and, recall, ¯i = 2®i,

nXi=1

ej¯i = ej2®0nXi=1

ej4i¼=n = 0: (21)

Hence the optimum local sensors’ location includes

the case of equally distributed n sensors on the

circumference of the circle of radius rmax with center

at (x0,y0), provided that n > 2. In order to extend

this result to optimum global sensors’ location,

an additional condition is imposed by requiring

symmetry. It can be verified that for n= 4, the 4 local

sensors on the circle form a square, and for n= 6,

the 6 local sensors form a hexagon which is not only

symmetric, but can also be replicated so that all the

sensors are used (recall that the plane can be tiled

with squares and hexagons). The reader is referred

to Fig. 1 for illustration. If the measurement errors are

uncorrelated and have unequal variances, then the n

CORRESPONDENCE 715

Fig. 1. Optimum sensors’ location with (a) n= 4 and (b) n= 6

where § = ¾2I.

sensors are still on the same circle, but they are not

equally distributed anymore. Basically, sensors with

large measurement error variances are required to be

densely spaced, and sensors with small measurement

error variances can be sparsely distributed.

After the problem of optimum sensors’ location

is solved, we obtain the ML estimate for the target

trajectory’s parameters, as presented in Lemma 1.

The sign ambiguity poses a difficulty when N, the

number of sensors near the trajectory, is large. We

do not favor solving for the parameter vector μ as

in Lemma 1 for every possible sign combination,

as that would require computing a total of 2N least

squares (LS) solutions. On the other hand, with μML in

Lemma 1, the estimation error is given by

kAμML¡Ãck= k[A(A0§¡1A)¡1A0 ¡§]§¡1Ãk£ jcj:(22)

Let S = sign(§1, : : : ,§1) be a diagonal sign matrix, ½be a column vector of the power ratios f½ig, and letª = [A(A0§¡1A)¡1A0 ¡§]§¡1. Then the estimationerror in (22) can be written as

kAμML¡Ãck2=jcj2 = (S½¡ 1)0ª 0ª(S½¡ 1): (23)

Our goal is to choose the signs of S such that the

expression (23) is minimized, yielding the correct

ML solution. This minimization process can be

accomplished by using the linear matrix inequality

(LMI) toolbox from Mathworks. Let the minimum

achievable estimation error be ² > 0, which can be

estimated using the Cramer-Rao lower bound. Then

(S½¡ 1)0ª 0ª (S½¡ 1)¡ ² < 0 (24)

if, and only if the following LMI holds:· ¡² (S½¡ 1)0ª 0

ª (S½¡ 1) ¡I

¸< 0: (25)

Thus efficient LMI tools can be employed to search

for the correct sign matrix in the above LMI.

We conclude this section by providing an initial

value for the sign matrix S prior to applying the LMI

Fig. 2. Sensors’ location and TMA trajectory.

tool. Let L0L=ª 0ª be the Cholesky factorization

with L lower triangular, and let Li,k be the (i,k)th

element of L. Then

kAμML¡Ãck2=jcj2 = kL(S½¡ 1)k2

and by direct calculation,

kL(S½¡ 1)k2 = L21,1(s1j½1j ¡ 1)2 + [L2,1(s1j½1j ¡ 1)

+L2,2(s2j½2j ¡ 1)]2 + ¢ ¢ ¢+[LN,1(s1j½1j ¡ 1)+LN,2(s2j½2j ¡ 1)

+ ¢ ¢ ¢+LN,N(sN j½N j ¡ 1)]2:Hence we can minimize L21,1(s1j½1j ¡ 1)2 over s1 =§1.Once s1 is obtained, we then minimize

[L2,1(s1j½1j ¡ 1)+L2,2(s2j½2j ¡ 1)]2 (26)

over s2 =§1. This can be continued until theminimizer sN =§1 is obtained. The procedure issimple but does not guarantee that the optimum

sign matrix S is obtained. On the other hand, it does

provide a small value of the error functional (23),

thereby providing a good starting sign matrix S prior

to applying the LMI algorithm to (25).

III. AN ILLUSTRATIVE EXAMPLE

This section considers assignment of RF power

sensors over a near square area with unit length

of 500 ft along the x axis, and 500p3 ft along the

y axis. For simplicity the case of homoscedastic noise

error with ¾ = 0:1 is studied. The value of rmax is

chosen to be 1000 ft. In light of Remark 2, a total

of 33 identical RF power sensors can be assigned

(in hexagon geometry) covering approximately one

square mile as shown in Fig. 2.

The TMA trajectory is assumed to be governed by

x+ y = 1 =) a= b =¡c= 2=p2 (27)

represented by the straight line in the figure. Those

sensors within the sensing range are marked with

circles, and those outside the range are marked with

squares. So there are a total of N = 17 sensors marked

with circles. The ML estimate in Lemma 2.1 is


employed to compute the optimum estimate of (a,b,c).

The MSE (mean-squared error) value associated

with such an estimate based on 2000 simulations

fluctuates around 1:427£ 10¡4 (by the Cramer-Raolower bound).

For performance comparison with other

assignments of RF sensors, we consider the case

of assigning 17 sensors within the sensing range

along with the TMA trajectory with one at the

origin in order to have a fair comparison. For the

16 sensors not at the origin, their x-coordinates are

uniformly distributed over an interval of [¡3,5], andthe y-coordinates are normal distributed with mean

zero and variance 1 which are then scaled such that

the maximum distance to the trajectory does not

exceedp3 units or 500

p3 ft. A total of 100 different

realizations are tested. The associated MSE values

fluctuate around their respective Cramer-Rao lower

bounds with

MSEmin = 3:024£10¡4

MSEmax = 1:108£10¡3

MSEmean = 5:702£10¡4:

(28)

The above MSEs are all significantly greater than the

one from the sensor assignment in Fig. 2.

After the target’s trajectory is determined, the

point (xk, yk) on the trajectory which is closest to the

kth sensor at location (xk,yk) can be computed in the

fusion center via

xk = b2xk ¡ abyk ¡ ac, yk =¡abxk + a2yk ¡ bc

(29)

for 1· k ·N where a, b, and c are estimated

according to Lemma 1. Recall the constraint a2 +

b2 = 1. Since ftkgNk=0 and the sensors’ locations areboth known, the velocity of the target is estimated

according to

vx =1

N

NXk=1

xk+1¡ xktk+1¡ tk

, vy =1

N

NXk=1

yk+1¡ yktk+1¡ tk

:

(30)

Together with the information on fxk, ykg, this solvesthe TMA problem.

In our example, it is assumed that v = 0:05p2 or

35.35 ft/s with vx =¡vy = 0:05 or 25 ft/s. Under theoptimum sensors’ configuration in Fig. 2, (xT,yT) =

(¡4:598,5:598) on the trajectory is chosen as theinitial position. The RMSE (root MSE) values are

16.292 ft for the position RMSE, and 0.160 ft/s for

the velocity RMSE. For randomly distributed sensors

as was previously discussed, we use as the initial

position the most up-left point on the target trajectory

fxk, ykgNk=0. A total of 100 different distributions of17 sensors are simulated. The corresponding RMSE

values (in [ft]) are summarized below:

Position:

8><>:RMSEmin = 18:341

RMSEmax = 49:516

RMSEmean = 30:787

Velocity:

8><>:RMSEmin = 0:3699

RMSEmax = 0:9599

RMSEmean = 0:6151

:

The above RMSEs are all greater than those of therespective RMSE values achieved in the optimumcase.

IV. CONCLUSION

This paper addresses the TMA of an RF emittingobject of interest using a wireless sensor network,where the received peak power is measured. Twoproblems are investigated. First, the optimumsensors’ placement in order to minimize theachievable estimation error of the target’s trajectoryis investigated. The Gaussian assumption is usedfor the measurement error of the ratio of peakpower measurements. Next, attention is given tothe estimation of the trajectory’s parameters. Anumerically efficient algorithm using the LMItoolbox is developed. This addresses the complexityissue involved in the estimation of the target’strajectory/TMA.Sensor placement and sensor fusion have been

an active research field. Past work has been reportedin [4, 6, 10, 16], to name a few. Our contributionin this paper differs from the past work in that theoptimum sensors’ location are obtained for RF powersensors, and in that each sensor transmits only thepeak value of its measurements, which takes limitedsensing range and limited communication capabilityinto consideration. It is our hope that the results ofthis paper complement the existing work in the area ofwireless sensor networks and TMA.

YICHAO CAO

College of Information Science and Technology

Donghua University

Shanghai 201620

P.R. China

GUOXIANG GU

Dept. of Electrical and Computer Engineering

Louisiana State University

Baton Rouge, LA 70803-5901

E-mail: ([email protected])

MEIR PACHTER

Dept. of Electrical Engineering

Air Force Institute of Technology

Dayton, OH 45433-7531

REFERENCES

[1] Aidala, V. J. and Hammel, S. E.

Utilization of modified polar coordinates for bearing-only

tracking.

IEEE Transactions on Automatic Control, 28 (Mar. 1983),

283—294.

CORRESPONDENCE 717

[2] Chan, Y. T. and Rudnicki, S. W.

Bearing-only and Doppler-bearing tracking using

instrumental variables.

IEEE Transactions on Aerospace and Electronic Systems,

28 (Oct. 1992), 1076—1083.

[3] Gavish, M. and Weiss, A. J.

Performance analysis of bearing-only target location

algorithm.


28 (Oct. 1992), 22—26.

[4] Gu, G., Chandler, P. R., Schumacher, C. J., Sparks, A., and

Pachter, M.

Optimal cooperative sensing using a team of UAVs.


42 (Oct. 2006), 1446—1458.

[5] Gu, G., Pachter, M., Chandler, P. R., and Schumacher, C. J.

Target motion analysis based on RF power and Doppler

measurements.

In Proceeding of the American Control Conference, Seattle,

WA, June 2008.

[6] Hernandez, M. L., Kirubarajan, T., and Bar-Shalom, Y.

Multisensor resource deployment using posterior

Cramer-Rao bounds.


40 (Apr. 2004), 399—416.

[7] Ho, K. C. and Chan, Y. T.

An asymptotically unbiased estimator for bearing-only

and Doppler-bearing target motion analysis.

IEEE Transactions on Signal Processing, 54 (Mar. 2006),

809—822.

[8] Li, D., Wong, K. D., Yu, H-H., and Sayeed, A. M.

Detection, classification, and tracking of targets.

IEEE Signal Processing Magazine, 19 (Mar. 22, 2002),

17—29.

[9] Lindgren, A. G. and Gong, K. F.

Position and velocity estimation via bearing-only

observations.


14 (July 1978), 564—577.

[10] Martinez, S. and Bullo, F.

Optimal sensor placement and motion coordination for

target trackingstar.

Automatica, 42 (Apr. 2006), 661—668.

[11] Nardone, S. C., Lindgren, A. G., and Gong, K. F.

Fundamental properties and performance of conventional

bearing-only target motion analysis.

IEEE Transactions on Automatic Control, 29 (Sept. 1984),

775—787.

[12] Rappaport, T. S.

Wireless Communications: Principles and Practice (2nd

ed.).

Upper Saddle River, NJ: Prentice-Hall, 2003.

[13] Sayed, A. H., Tarighat, A., and Khajehnouri, N.

Network-based wireless location: challenges faced in

developing techniques for accurate wireless location

information.

IEEE Signal Processing Magazine, 22 (July 2005), 24—40.

[14] Stoica, P. and Moses, R.

Introduction to Spectral Analysis.

Upper Saddle River, NJ: Prentice-Hall, 1997.

[15] Wang, A. and Chandrakasan, A.

Energy-efficient DSPs for wireless sensor networks.

IEEE Signal Processing Magazine, 19 (July 2002), 68—78.

[16] Oshman, Y. and Davidson, P.

Optimization of observer trajectories for bearings-only

target localization.


35 (July 1999), 892—902.

Multi-Sensor Centralized Fusion withoutMeasurement Noise Covariance by VariationalBayesian Approximation

The work presented here solves the multi-sensor centralized

fusion problem in the linear Gaussian model without the

measurement noise variance. We generalize the variational

Bayesian approximation based adaptive Kalman filter (VB AKF)

from the single sensor filtering to a multi-sensor fusion system,

and propose two new centralized fusion algorithms, i.e.,

VB AKF-based augmented centralized fusion algorithm and

VB AKF-based sequential centralized fusion algorithm, to deal

with the case that the measurement noise variance is unknown.

The simulation results show the effectiveness of the proposed

algorithms.

I. INTRODUCTION

Multi-sensor data fusion has been widely used

in many fields, such as aerospace, defense, robotics,

and automation systems. In general, multi-sensor

data fusion architectures can be divided into three

categories, i.e., centralized, decentralized (or

distributed), and hybrid types. Centralized fusion is

used in the sensor level, and decentralized fusion is

used in the estimation level. The hybrid architecture

involves both centralized and decentralized fusion.

The results of centralized fusion methods are more

accurate than those of the decentralized fusion

methods, because the data used in the former are the

original sensor measurement information, while the

data employed in the latter are the information already

handled by the local sensor processing unit.

In the past thirty years, many algorithms have been

proposed for multi-sensor data fusion [1—3], and most

of them belong to decentralized fusion algorithms. In

[4] and [5], the centralized, decentralized, and hybrid

architecture fusion are implemented in a uniform

framework, and optimal fusion rules are presented in

Manuscript received April 5, 2009; revised November 15, 2009;

released for publication April 13, 2010.

IEEE Log No. T-AES/47/1/940059.

Refereeing of this contribution was handled by T. Luginbuhl.

This research was supported by the National Basic Research

Program of China (973 Program) (Grant No. 2011CB707000),

the National Natural Science Foundation of China (Grant Nos.

60832005, 41031064, 60702061, and 61072093), the Ph.D.

Programs Foundation of Ministry of Education of China under

Grant 20090203110002, and the Natural Science Basic Research

Plan in Shaanxi Province of China under Grant 2009JM8004.

0018-9251/11/$26.00 c° 2011 IEEE
