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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 trajectorys 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 receivedconsiderable attention in recent years, and havewide applications such as localization and trackingof mobile users, intrusion detection, environmentalmonitoring, climate control and disaster management,detection and identification of hazardous materials[8, 13, 15]. A common scenario for wireless sensornetworks envisions sensor nodes that are homogenousand have limited capabilities in computation, sensingrange, and communications. In this paper thescenario is considered where these sensor nodeswork collaboratively in accomplishing the goalof extracting the desired information, based onmeasurements of the mobile emitters RF signalpower recorded at different locations and possiblyat different time instants. In this paper we considerthe employment of basic receivers that measureonly the signal power from the RF emitter/targetwithin the sensors range. Oblivious of beingobserved, the target moves at constant speed andfixed direction. Localization and tracking of suchtargets is referred to as target motion analysis(TMA). TMA entails the estimation of the initialposition and the velocity of the target. TMAbased on a single sensor such as in bearings-onlymeasurements has been extensively investigated inthe 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 alsobeen investigated; and in [5] we considered theDoppler-power method. However, TMA based onpower-only measurements has not been undertakendue to the strong nonlinear dependence of the receivedsignal power on the TMA motion parameters, and,most importantly, the unknown nature of the path losscoefficient.In this paper the feasibility of employing a

wireless sensors network consisting of currentlyavailable inexpensive RF receivers with limitedsensing range which measure the received signalspower, is investigated. Such sensors are the simplestpossible for localization and tracking of RF emitters.Even though each stand-alone sensor is not capableof achieving good motion parameter estimates forTMA, collectively they may perform the TMA taskextraordinarily well. The challenge lies in the designof sensor networks and fusion algorithms. Specifically,because the sensing range is limited, the properpositioning of the sensor nodes is crucial and itstrongly impacts the estimation performancethinkof the geometric dilution of precision of themeasurement arrangement. Because sensors havelimited communication capability, a limited numberof power measurements will be transmitted to thefusion center. To avoid the difficulties associatedwith propagation effects and in order to limit theinformation flow in the sensor network, it is stipulatedthat each sensor transmits only the peak powermeasurement; the fusion center will first estimatethe targets trajectory, prior to the estimation of theTMA parameters. Due to the fact that the associatedCramer-Rao lower bound of the target motionparameters estimation error is dependent on thegeometry of the sensors network, we are able tooptimize the sensors location under the constraintof maximum allowable distance between neighboringsensors. In addition, an efficient fusion algorithm forthe estimation of the target trajectory and the TMAparameters is developed. Our results in this paperare motivated by our earlier work [4], the posteriorCramer-Rao lower bound approach to sensor networkoptimization developed in [6], the research reportedin [10] where range measurements/multilaterationis considered, and the geolocation of a stationarytarget using bearings-only measurements [16].The common thread in these papers is the questfor estimation performance improvement withapplications related to TMA. The notation in thispaper is standard and will be made clear as weproceed.

II. MAIN RESULTS

For convenience we confine our attention to TMAin the two-dimensional plane, although the results canbe easily extended to the three-dimensional space. The

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

targets trajectory is parameterized by the followingequation for a straight line in normal form:

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

where a, b are independent parameters, and c is adependent parameter used to scale a, b such thatpa2 + b2 = 1. Since the target is oblivious of being

observed, it transmits RF signals while traveling alongthe, as yet unknown, straight line trajectory (1). Thekinematic navigation assumption is invoked and it isassumed that the speed of the target is constant. Hencethe location of the target at time t is specified by(xT+ vxt,yT+ vyt) with (xT,yT) the initial location att= 0, and (vx,vy) the velocity of the target. It followsthat (xT,yT) satisfies (1), and

a2 = v2y=(v2x + v

2y ), b

2 = v2x=(v2x + v

2y ): (2)

As the target passes through the area where the sensornetwork is deployed, its RF signal is sensed by someof the sensors within range. Due to affordabilityconsiderations, the deployed sensors are assumed tomeasure only the power of the received RF signalemitted by the target. In addition, due to the limitedcapability for wireless communications and tolengthen battery life, each sensor transmits to thefusion center only one measurement rather than all therecorded power measurements. A smart approach isfor each sensor to transmit only the recorded peak RFpower (the latter has the most information content).Since the RF signal travels on a straight line alongthe line of sight (LOS) to the sensor, the peak powercorresponds to the shortest distance from the sensorto the targets trajectory. This can be utilized toestimate the targets trajectory and determine the TMAparameters.Suppose that there are a total of (N +1) sensors

close to the targets trajectory, and these simplereceivers are able to measure the power of the RFsignal from the target as it traverses the sensornetwork. 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 ofgenerality, set t0 = 0. According to the the path losslaw [12], the peak power measured by each of the(N +1) sensors can be expressed as Pk = C=R

2k where

Rk is the distance between the transmitter and the kthreceiver, and is given by

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

Because Rk represents the shortest distance from thekth sensor to the targets trajectory (1), the followinggeometric relationship holds:

axk + byk + c=Rk, k = 0,1, : : : ,N: (4)The sign ambiguity is due to the relative positions ofthe targets trajectory and the kth sensor.

The difficulty of TMA based on powermeasurements lies in the unknown nature of theconstant C. The latter is a function of the transmitterspower, the propagation environment, and thetransmitter and receiver antennae gains. Note howeverthat C is invariant with respect to k due to thehomogeneity of the sensor nodes. Following themethodology developed in [5], we calculate the ratiosof peak power, to eliminate the C dependence:

k =sP0Pk=Rk

R0=axk + byk + cax0 + by0 + c

(5)

for k = 1,2, : : : ,N. In practice, k is determined fromthe ratio of peak power measurements according to

k =qP0=Pk = k + k (6)

where k = 1,2, : : : ,N and Pk is the measurement of Pk.It is assumed that the measurement errors fkg arejointly Gaussian. Although this assumption rarelyholds in practice, it is adequate if the measurementsof peak power involve Gaussian noises, and if thesignal-to-noise ratio (SNR) is relatively high. Thesign ambiguity is inherited from (4). Without lossof generality we assume that x0 = 0 and y0 = 0;this can be achieved by a suitable translation ofthe coordinate systems origin. Denote = a=c and = b=c. After the inclusion of measurement error inthe measurement equations, (5) can be rewritten as

xk+ yk = (k 1) k (7)for 1 k N. Equation (7) can be stacked into alinear regression equation of the form266664

1 12 1...

n 1

377775=266664x1 y1

x2 y2...

...

xN yN

377775

+

2666641

2...

N

377775 (8)or = A+ for short notation, where = [ ]0is the parameter vector to be determined. The TMAparameters can be recovered via 0 := [a b] = c[ ]with c= 1=

p2 +2.

LEMMA 1 Consider the target trajectory in (1) and theequations of peak power ratios in (8). Assume that theN sensors located at f(xk,yk)gNk=1 are not colinear, i.e.,the two columns of A are not linearly dependent. Underthe hypothesis that fkgNk=1 are Gaussian distributedwith mean zero and covariance , the MLE (maximumlikelihood estimate) of the parameter vector 0 of thetargets trajectory is given by

ML = (A01A)1A01c (9)where c=k(A01A)1A01k1.PROOF The measurement equations are given byA = + see, e.g., (8). Hence this is a standard

CORRESPONDENCE 713

linear regression problem with a Gaussian error. Theonly exception is the parameter c that is used to scalethe parameter a and b so that a2 + b2 = 1, or k0k= 1,which yields (9).

REMARK 1 It is well known [14] that under certainmild regularity conditions, the maximum likelihood(ML) algorithm is asymptotically unbiased, andachieves the Cramer-Rao lower bound asymptotically.Clearly, the hypotheses in Lemma 1 satisfy theregularity condition. Thus with one sensor situatedat the origin, the identifiability of the target trajectoryrequires that the remaining N sensors be not locatedcolinearly. If this is true, then for each unbiasedestimate of 0, and the ML estimate ML in (9), thereholds asymptotically, as N!1,

CovfMLg := Ef(0 ML)(0 ML)0g! c2(A01A)1 Ef(0 )(0 )0g

by (1=c)A0 = + . The matrix c2A01A is theFisher information matrix (FIM) whose inverseconstitutes the Cramer-Rao lower bound.

Because the FIM is dependent on the location ofthe N sensors, a meaningful optimization problemarises naturally: with one sensor situated at the origin,how is one to place the remaining N sensors so thatthe error variance associated with the ML algorithmis minimized? This is referred to as the optimumsensors location problem. However, this problem isnot well posed in the sense that these (N +1) sensorsdiscussed thus far are assumed to be near the targetstrajectory, and may constitute only a subset of allthe sensors in the network. Moreover some of these(N +1) sensors can be far away from each other. Forthis reason we focus on the position assignment of(n+1) local sensors: with one sensor situated at thecenter, the remaining n surrounding sensors are nomore than a distance rmax away from it. The value ofrmax is chosen to be the maximum distance betweenneighboring sensors that is determined by the effectiverange 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 assignmentof all the sensors, thereby solving the problem ofan optimum network configuration. We therefore setxi = xi x0 and yi = yi y0, and modify the linearregression equation A = c+ c according to266664

x1 y1

x2 y2...

...

xn yn

377775a

b

=

2666641 12 1...

n 1

377775c+2666641

2...

n

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 +y

2i r2max, and the equation errors

figni=1 are Gaussian with mean zero and covariance = diag(21,

22, : : : ,

2n). Then the optimum locations of

the n local sensors for minimizing the correspondingCramer-Rao lower bound are given by xi = x0 +rmax cos(i) and yi = y0 + rmax sin(i) satisfying

nXi=1

cos(2i)2i

= 0,nXi=1

sin(2i)2i

= 0 (11)

where i is the angle between the positive x-axis to theray from the center to the ith sensor.

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

Efk0 MLk2g= Tr[Ef(0 ML)(0 ML)0g] c2Trf(A01A)1g: (12)

The optimum sensors location for positionassignment of n local sensors minimizes the righthand side of the inequality (12) which correspondsto the Cramer-Rao lower bound. Recall that for eachi > 0, xi = xi x0, yi = yi y0, and (xi,yi) =(xi=i,yi=i). Using polar coordinates,

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

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

A01A=nXi=1

12i

x2i xiyi

yixi y2i

=nXi=1

x2i xiyi

yixi y2i

: (14)

For convenience, use the notation ci = cos(i) andsi = sin(i) for 1 i n. It follows from the abovetwo equations that

det(A01A) =

nXi=1

r2i c2i

!nXk=1

r2k s2k

!

nXi=1

r2i cisi

!nXk=1

r2k cksk

!

=nXi=1

nXk=1

r2i r2k

c2i 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)

=nX

k>i=1

r2i r2k (skci cksi) sin(k i)

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

=nX

k>i=1

r2i r2k sin

2(k i)

=nXk=2

k1Xi=1

r2i r2k sin

2(k i)

=12

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 twosums with one for k < i and the other for k > i sincethe terms for i= k are zero, and then each termindexed with (i,k) for i < k is replaced by the negativeterm indexed with (k, i), that leads to the sum forall terms such that indexes satisfy k > i. The samereasoning is used again to obtain the final expression.Denote Trfg as trace operation. We can now obtainthe expression for the right hand side of (12) as

Trf(A01A)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=iand ri, subject to the constraint 0< ri rmax for1 i n. Alternatively, minimization of (15) can beconverted to a maximization of

J = [Trf(A01A)1g]1

=12

nXi=1

nXk=1

krk22i 2k sin2(k i) (16)

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

max

(nXi=1

nXk=1

2i 2k sin

2(k i) :nX`=1

2` = 1

)(17)

plus the maximization of krk2 subject to 0 iri rmax 8i. Consequently, the optimum local sensorslocation is achieved by taking ri = rmax=i or xi =rmax cos(i) and yi = rmax sin(i). Thus the optimumlocal sensors location is determin...