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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 22, NOVEMBER 15, 2015 6049

Compressive Sensing Based Probabilistic

Sensor Management for Target Tracking in

Wireless Sensor NetworksYujiao Zheng, Student Member, IEEE, Nianxia Cao, Student Member, IEEE, Thakshila Wimalajeewa, Member, IEEE,

and Pramod K. Varshney, Fellow, IEEE

AbstractIn this paper, we consider the problem of sensor man-agement for target tracking in a wireless sensor network (WSN).To determine the set of sensors with the most informative data, wedevelop a probabilistic sensor management scheme based on theconceptsdeveloped in compressive sensing. In the proposed scheme

where each sensor transmits its observation with a certain prob-ability via a coherent multiple access channel (MAC), the obser-vation vector received at the fusion center becomes a compressedversion of the original observations. In this framework, the sensor

management problem can be cast as the problem of finding theprobability of transmission at each node so that a given perfor-mance metric is optimized. Our goal is to determine the optimalvalues of the probabilities of transmission so that the trace of theFisher information matrix (FIM) is maximized at any given timeinstant with a constraint on the available energy. We consider twocases, where the fusion center has i) complete information and ii)

only partial information, regarding the sensor transmissions. Theexpression for FIM is derived for both casesand the optimal valuesof the probabilities of transmission are found accordingly. Withnonidentical probabilities, we obtain the results numerically while

under the assumption that each sensor transmits with equal prob-ability, we obtain the optimal values analytically. We provide nu-

merical results to illustrate the performance of the proposed prob-abilistic sensor management scheme.

Index TermsCompressive sensing, sensor management, targettracking, wireless sensor networks.

I. INTRODUCTION

A typical wireless sensor network (WSN) is composed ofa large number of densely deployed sensors, where sen-sors are assumed to be tiny, battery-powered devices with lim-

ited signal processing capabilities. When programmed and net-

worked properly, WSNs are very useful in many application

Manuscript received August 01, 2014; revised March 15, 2015 and July 13,2015; accepted July 18, 2015. Date of publication August 04, 2015; date of

current version October 06, 2015. The associate editor coordinating the reviewof this manuscript and approving it for publication was Dr. John McAllister.

This material is based upon work supported by the National Science Founda-

tion (NSF) under Grant No. 1307775 and U.S. Air Force Office of ScientificResearch (AFOSR) under Grant No. FA9550-10-1-0458. A part of this work

was presented at ICASSP, Florence, Italy, 2014.The authors are with the Department of Electrical Engineering and Computer

Science, SyracuseUniversity, Syracuse, NY 13244 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2015.2464197

areas including battlefield surveillance [1], environment moni-

toring and target tracking [2], industrial processes [3] and health

monitoringand control [4]. In this paper,we assumethat thetask

of the WSN is to track a target emitting energy in a given region

of interest (ROI). The sensors in the ROI report their measure-

ments to a central node called the fusion center which is respon-

sible for the final inference.

Typical WSNs have limited resources (energy, bandwidth),

and the sensors are deployed in the ROI densely. Thus, instead

of simply having all the sensors transmit all the time, proper

management and programming of a subset of sensors that

should transmit their observations is very important. Different

approaches have been proposed to solve the sensor manage-

ment problem in the literature for various inference tasks. To

name a few, in [5], the sensor selection problem was formulated

as an integer programming problem, which was relaxed and

solved through convex optimization. In [6], a multi-step sensor

selection strategy by reformulating the Kalman filter was pro-

posed, which was able to address different performance metrics

and constraints on available resources. In[7], a sensor selection

scheme based on an entropy-based information measure wasproposed. The recursive posterior Cramr-Rao lower bound

(PCRLB) on the mean squared error (MSE) has been explored

as the metric to select informative sensors in [8] and [9]. To

decide which sensors are important, the innovation of each

sensor which is defined as the difference between the current

measurement and the predicted measurement has been utilized

in [10]. In [11], the nondominating sorting genetic algorithm-II

method was employed for the multi-objective optimization

based sensor selection problem. In [12], the authors aimed

to find the optimal sparse collaboration topologies subject to

a certain information or energy constraint in the context of

distributed estimation. For a more complete literature review onsensor management for target tracking, see [13] and references

therein.

In a densely deployed WSN, since only a few nodes have

significant observations, the concatenated measurement vector

can be considered to be sparse and compressible. In [13][16],

a sparse formulation is exploited to reduce the number of se-

lected sensors. In [13], the problem of periodic sensor sched-

uling was addressed by seeking the optimal sparse estimator

gain, where a one-to-one correspondence between active sen-

sors and the nonzero columns of the estimator gain was estab-

lished. In [14], the design of the sensor selection scheme was

transformed to the recovery of a sparse matrix. In [15] and [16],

1053-587X 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

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6050 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 22, NOVEMBER 15, 2015

a sparsity-aware sensor selection problem was formulated by

minimizing the number of selected sensors subject to a certain

estimation quality. The concept of compressive sensing (CS)

has been discussed in [17] and [18]. The authors in [19] in-

troduced the application of CS to radar sensor networks. They

showed that the signal samples along the time domain could be

substantially compressed so that signals could be recovered by a

small number of measurements. Also, the maximum likelihood

(ML) algorithm was developed for parameter estimation and the

Cramer-Rao lower bound (CRLB) was provided to validate the

theoretical result. In [20], a multiple target localization approach

was proposed by formulating the multiple target locations as a

sparse matrix, and the target locations were recovered from the

noisy measurements through -minimization. The first attempt

to solve the sensor management problem by CS was presented

in [21], in which the sensor selection decision was considered as

a sparse signal, and the sensor selection problem was solved in

terms of recovering the sparsesignal by norm minimization.

However, the probabilistic sensor management problem using

the compressed data from the sparse observations of the sen-sors has not been considered in the above references.

In this paper, we propose a novel CS based sensor manage-

ment approach for target tracking in a WSN. Since not all the

sensors contain informative observations regarding the target at

a given time instant, the observation vector has only a few sig-

nificant elements. Thus, it is sufficient to forward only those sig-

nificant elements to the fusion center to perform target tracking

instead of forwarding all the measurements which consumes

a large amount of energy. To get a compressed version of the

observations at the fusion center, we employ a multiple access

channel (MAC) with probabilistic transmissions. Use of a MAC

model to get a compressed version of observations has been

discussed by several authors, for example, [22] and [23]. In

this approach, each sensor multiplies its measurement with a

random scalar drawn from a given distribution and transmits

it via a MAC. Then, the received observation vector at the fu-

sion center has an equivalent representation as with the stan-

dard CS problem. With this model, the corresponding sensing

matrix at the fusion center is completely determined by each

sensors probability of transmission. Under this framework, our

goal is to design the sensing matrix considering two cases: 1)

the fusion center has complete measurement sparsity informa-

tion (CMSI), and, 2) the fusion center has partial measurement

sparsity information (PMSI). With CMSI, the fusion center gen-

erates the sensing matrix based on the transmission probabilitiesof the sensors. With PMSI, the fusion center sends the trans-

mission probabilities to the sensors once it gets the optimal so-

lutions. Then the sensors generate the sensing matrix based on

their own transmission probabilities and decide whether or not

they should transmit their observations to the fusion center. It

is noted that, with CMSI, the fusion center has to inform which

sensor should transmit during each MAC transmission, while

with PMSI, the fusion center sends only the probability values

to each node which is the same for all MAC transmissions.Thus,

in terms of the communication burden required by the feed-

back channel (fusion center to sensor nodes), the PMSI based

method is more efficient than the CMSI method. However, sim-ulation results show that the CMSI method is capable of pro-

viding better tracking performance compared to the PMSI based

method when the processing noise is relatively large. Under

both schemes, we obtain the optimal values of transmission

probabilities that generate the measurement matrix so that a

given performance metric for target tracking is optimized.

An initial version of this work was reported in [24], where we

studied only the CMSI case. In the current work, we extend the

work reported in [24] in several directions:

We develop a probabilistic sensor management scheme for

target tracking based on compressed measurements con-

sidering PMSI in addition to the CMSI case as considered

in [24].

When the sensor nodes transmit their observations with

nonidentical probabilities, we formulate the sensor man-

agement problem as a quasi-convex optimization problem

and obtain the optimal values via a linear program for both

CMSI and PMSI cases.

With identical probabilitiesof transmission, the theoretical

results provide us with some intuitive insights into the im-

pact of the energy constraint, the number of sensors in theROI, and the number of MACs on the performance of the

CS based target tracking problem.

While a CS based sensor management approach for WSNs

has been discussed in [21], there are several major differences

between our work and the work presented in [21]: 1) The authors

in [21] considered a linear system while our model is nonlinear

and is, thus, more general. 2) In [21], a subset of sensors is se-

lected and the selected sensors send their measurements to the

fusion center over parallel channels. In this paper, a subset of

sensors is chosen probabilistically and different superpositions

of weighted measurements are sent to the fusion center over

MACs. 3) In [21], the sensor selection decision is considered

as a sparse signal and the sensor selection problem is solved by

recovering the sparse signal by norm minimization. In this

paper, the concatenated measurement vector is considered to be

sparse due to the presence of non-informative measurements,

and the sensing matrix is designed such that a desired tracking

performance is achieved with compressed measurements. Thus,

there is no recovery of signal, but the compressed signal is used

directly for state inference; 4) In [21], the sensing matrix is de-

terministic or is made semi-random by adding some random

disturbance, while in this paper, elements of the sensing matrix

are random variables whose distributions are related to sensors

probabilities of transmission.

The rest of the paper is organized as follows. In Section II,we introduce the formulation of our problem. In Section III, par-

ticle filtering based Fisher information matrix (FIM) for CMSI

and PMSI are derived. We present the optimization problem for

probabilistic sensor management in Section IV and the numer-

ical experiments in Section V, respectively. Our work is con-

cluded in Section VI.

II. PROBLEM FORMULATION

A. System Model

We consider a WSN consisting of sensors which are de-ployed uniformly in a square region of interest (ROI) of size

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. Note that our approach can handle any sensor deploy-

ment pattern as long as the sensor locations for all

are known in advance. We assume that the target and

all the sensors are based on a flat ground, so that we can formu-

late the problem with a 2-D model. We focus on a target tracking

problem, where a moving acoustic or electromagnetic target is

tracked by the WSN. The dynamics of the target is defined by a

4-dimensional state vector where

is the location of the target at time instant and , are the

velocities in the and directions. The model of the target mo-

tion is assumed to be

(1)

where is the state transition model and is the process noise

which is assumed to be Gaussianwith mean zero and covariance

matrix .

At time , the measurement model at each sensor is

(2)

where , is the signal power of the

source, is the signal decay parameter, denotes the dis-

tance between the target and the th sensor at time , i.e.,

, where is the loca-

tion of the th sensor, and is the measurement noise, which

is assumed to be Gaussian with mean zero and variance and

mutually independent over for .

Let the measurement vector be at time

, where denotes the matrix or vector transpose. We con-

sider a relatively large distributed network. Based on the obser-

vation model (2), it is seen that the signal amplitude received

at a given node at a given time becomes smaller and eventu-

ally negligible as the distance between that particular node and

the true target location increases. Therefore, at time ,

contains only a few significant values. This

motivates us to consider a scheme where only a compressed

version of is transmitted to the fusion center instead of the

complete observation vector .

B. Compressive Sensing (CS)

For a densely deployed WSN, the signal measurements are

considered to be sparse and compressible. CS is a recently de-

veloped signal processing technique for acquiring and recon-

structing a sparse signal with a small number of measurements

compared to the original signal dimension. Consider a signal, that can be expressed in an orthonormal basis

as

where is the coefficient of the signal projected on to and

. The signal is said to be -sparse, if only

coefficients in are significant and all the others are zeros or

negligible.

To get a compressed signal, the sparse signal is projected

to a lower dimension via a sensing matrix with dimension

, where , i.e.,

Then, the standard CS problem is to recover from only

measurements . The reconstruction capability is deter-

mined by the properties of the sensing matrix in addition to

the sparsity index and the number of compressed measurements

[17]. Several such properties including restricted isometry prop-

erty (RIP) and mutual coherence of the sensing matrix, and re-

covery algorithms are discussed in [25][27].

C. Sparsity Formulation of Sensor Management Problem

To obtain a compressed version of observations at the fusion

center, we consider the following transmission scheme as con-

sidered in [23]. In particular, to get a compressed version of

the observations at the fusion center, we employ a multiple ac-

cess channel (MAC) based communication scheme with prob-

abilistic transmissions. In this approach, each sensor multiplies

its measurement with a random scalar, which denotes whether

or not to transmit its measurement to the fusion center, drawn

from a given distribution and transmits it via a MAC. Then, the

received observation vector at the fusion center is a compressed

version of the original observation vector and has an equiva-lent representation as in the standard CS problem. Let the th

sensor transmit its measurement after multiplying it by (to

be defined later) via a MAC, so that after transmissions, the

received signal at the fusion center is given by

(3)

where is the receiver noise, which is assumed to be white

and Gaussian with mean zero and variance . Note that (3) can

be written in a vector form as

(4)

where , the -th element of is

given by for and , and is

the receiver noise, which is assumed to be white Gaussian with

mean zero and covariance matrix , where

is an identity matrix of size .

We consider each to be a random variable so that

with prob.

with prob.

with prob.

(5)

where is the probability of transmission of th sensor attime instant .

Based on how is constructed, it is obvious that, though the

elements in a given column in are independent and identi-

cally distributed (i.i.d.), elements in different columns are inde-

pendent but not identically distributed. It is noted that, the ma-

trix can be very sparse when only a small number of sensors

decide to transmit with a high probability. With this sensing ma-

trix, we show numerically in Section V that compressed obser-

vations in (4) provide us with tracking performance comparable

to that with (2) with relatively small .

In the context of sensor management, plays the role of

a sensor management entity that divides the sensors into

subsets. Sensors in the same subset send their superimposed

measurements over the same MAC (there are a total of

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MACs) if FDMA is used or in the same time slot (there are a

total of time slots) if TDMA is used. Note that, the weight

could be 0, which means that the associated sensor does not

send its measurement. In this scenario, the sensor observations

are compressed from dimension to , and the compression

is achieved as a result of coherent transmission over MAC chan-

nels (either with FDMA or TDMA). Therefore, the problem of

managing sensors is equivalent to the design of the sensing ma-

trix or the probability vector ,

such that a certain objective function is optimized. In our paper,

we choose the Fisher information, which is the inverse of

the bound on the mean squared error of the estimator namely

PCRLB, as our optimization criterion, i.e., we address this

probabilistic sensor management problem so that the trace of

the FIM is maximized at each time step .

III. FISHERINFORMATION MATRIX FORCS BASED

TARGET TRACKING

In our framework, the fusion center decides the transmission

probabilities of the sensors, and then communicates with the

sensors to get the compressed data through the MACs. Thus,

regarding the communication scheme between the fusion center

and the sensors, the natural question is, who decides the sensing

matrix in (4)? We consider two different cases: 1) the fusion

center generates the sensing matrix and informs the sensors as

to which sensors are to transmit during each MAC transmission,

in which case the fusion center has CMSI; and, 2) the fusion

center sends only the transmission probabilities of the sensors,

in this case, the sensors generate the sensing matrix and decide

how they send their observations to the fusion center, i.e., the

fusion center has PMSI. Therefore, before proceeding with the

optimization problem, in this section, we find the FIM as a func-tion of the transmission probabilities of the sensors for the two

different transmission schemes.

For the target tracking problem under consideration, a nice

recursive computation of the FIM at time , , is pro-

posed in [28], which is given as follows

(6)

where is the FIM at time , and

In the above expressions for , , , , denotes

the probability density function, is the state of the target

at time , is the measurement vector at time ,

denotesthe second derivative operator, namely ,

and isthe gradient operator, and denotes the mathemat-

ical expectation.

For the problem considered in this paper, we have

, , . To compute

, it is required to compute . It is noted that

depends on the random matrix . Here we consider two

different cases: 1) the fusion center generates the sensing matrix

and decides the selection state of each sensor,so that it has

complete information about the sensing matrix . Then the

conditional probability can be applied to

obtain ; 2) the fusion center sends the transmission prob-

ability to sensor , and sensor generates

corresponding to its selection prob-

ability. In this case, the fusion center does not have the exact

values of . Thus, is computed by averaging

over the sensing matrix .

A. Fisher Information Matrix With Complete Measurement

Sparsity Information (CMSI)

With complete sparsity information, is calculated from

the conditional probability ,

(7)

where the expectation is with respect to the joint distribution

. Hence,

(8)

where

(9)

Notice that the second expectation in (9) is with respect to

. Recall (4), where we have

(10)

where , i.e., it is the concatenation

of themeasurement noises of sensors. Since themeasurement

noises are mutually independent, , where

represents the multivariate Gaussian distribution withmean vector and the covariance matrix .

Given and , based on (10), one can get

, where . Then

Therefore, in (9) reduces to

(11)

where

and

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for . Note that is the location of the th

sensor and represents the location of the target at

time .

Letting , we observe that it is

difficult to obtain directly. Thus, in the following propo-

sition, we employ some approximations when is large.

Proposition 1: If the number of sensors in the WSN is

large, then, at any given time , we may approximate

where denotes a diagonal matrix, with on the

main diagonal.

Proof: See Appendix A.

We see that Proposition 1 yields a diagonal formulation of

, which makes the relationship between and

much more clear. Thus, we have

(12)

(13)

where

Note that in (13) we take the expectation of because

is the only element that is related to the target state

. Therefore, the FIM for the CMSI case is computed by

substituting in (13) back to (6).

B. Fisher Information Matrix With Partial Measurement

Sparsity Information (PMSI)

In this subsection, we consider the second case where the sen-

sors generate sensing matrix based on by them-

selves and the fusion center does not have the exact information

regarding . In this case, to compute ,

is computed as

With similar approximations as considered in Section III-A, we

have the following proposition.

Proposition 2: If the number of sensors is large, then, at

any given time , we may approximate

(14)

Proof: See Appendix B.

Hence,

(15)

(16)

where is a 4 4 matrix with

and

All the other elements in are zeros. Also note that in (16) we

take the expectation of because is the only

element thatis related tothetarget state . Thus, the FIM for

the PMSI case is computed by substituting in (16) back

to (6).

C. Particle Filter Based Fisher Information Matrix

After computing the FIM for CMSI and PMSI cases, in this

subsection, we briefly introduce particle filters and give simpler

expressions of the FIM with the help of particle filters. Since

particle filtering is a general filtering procedure that can be ap-

plied to any state-space model and thus is more general than

the Kalman filter [29], we apply particle filtering for our non-

linear model to track the target. In particle filtering, the main

idea is to find a discrete representation of the posterior distri-

bution by using a set of particles with as-

sociated weights . The posterior density at can be

approximated as,

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where is the Dirac delta measure and denotes the total

number of particles. We employ the sequential importance re-

sampling (SIR) particle filtering algorithm [29] to solve the non-

linear Bayesian filtering problem. Algorithm 1 provides a sum-

mary of SIR particle filtering for sensor selection. A more de-

tailed treatment of particle filtering can be found in a wide va-

riety of publications such as [29].

Algorithm 1:SIR Particle Filter with Probabilistic Sensor

Management for Target Tracking

Set . Generate initial particles with.

while do

a) Propagating particles:

b) Obtain sensor data based on the sensormeasurements through the MAC of transmissions.

c) Updating weights:

(for CMSI)

(for PMSI)

d) Normalizing weights:

e) Target Estimation:

f) Resample:

end while

In Algorithm 1, denotes the number of time steps over

which the target is tracked. For the first case, i.e., CMSI,

is according to (10),

and for the second case, i.e., PMSI, is approxi-

mated by (14). The resampling step avoids the situation that all

but one of the importance weights are close to zero [29].

With particle filtering, expectation of in (13) and

expectation of in (16) can be written as

(17)

and

(18)

Thus, we provide the FIM of our system in the following

proposition.

Proposition 3: The FIM of our system for both the CMSI and

PMSI can be written as

(19)

where is different for the two cases,

for CMSI

for PMSI.

Proof: We get (19) by substituting (17) and (18) back into

(13) and (16).

IV. THESENSORMANAGEMENT PROBLEM

Having computed the FIM in Section III for both CMSI and

PMSI cases, in this section, we focus on our CS based sensor

management problem in the WSN. We observe that for our

model, the problem of maximizing the determinant of the FIMin

(19) subject to the resource constraint is not convex, thus global

optimal solutions are not guaranteed, and the computation costs

will be high. Therefore, we solve the CS based resource man-

agement problem in a WSN by maximizing the trace of theFIM.

For simplicity, we assume that each transmission from a local

sensor to the fusion center consumes unit power. Finding the op-

timal values for transmit power at sensor nodes while achievinga desired performance is an interesting aspect which will be

studied in the future. We aim to solve the following optimiza-

tion problem:

(20a)

(20b)

(20c)

where denotes the trace of a matrix, is the total energy

constraint, and is the lower bound of the selection probability.

Here we impose a lower bound on the probability to avoid pro-ducing columns with all zeros in the measurement matrix .

In other words, we allow each node to transmit with a certain

nonzero probability.

At time step , the fusion center first solves the optimiza-

tion problem in (20) to obtain the optimal before measure-

ments at this time are available. Then, 1) For the CMSI case

in III-A, the fusion center generates the sensing matrix

using , and, according to which, sends control messages

to local sensors. Based on these control messages, local sen-

sors send their measurements over assigned MACs to the fu-

sion center. For the PMSI case in III-B, the fusion center sends

the transmission probability to sensor .Based on its selection probability , sensor generates

, according to which it sends its mea-

surements to the fusion center.

A. With Identical Probabilities

We first consider a simple scenario in which each sensor has

the same probability to be selected to transmit their measure-

ments. Since from (19) we know that does not depend on

the transmission probability of the sensors, the FIM for both the

CMSI and PMSI cases with the assumption of identical trans-

mission probability is

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The objective function of the optimization problem (20a) is

Thus, no matter whether or not the fusion center has completeinformation about the sensing matrix , the resource manage-

ment problem of (20) is equivalent to

(21)

Since is an increasing function of , the

optimal solution is simply the largest value that can achieve

under the constraints

(22)

We observe that the optimal solution for such a simple sce-

nario is no longer dependent on the FIM of each sensor, i.e.,

this formulation treats all the sensors to be equally important,

even though they make different contributions for tracking the

target. When we do not consider the case that every sensor has

probability 1 to transmit its measurements, we get some insights

from (22): 1) A larger energy constraint results in a higher

transmission probability for the sensors. 2) In the identical prob-

ability case, having more sensors in the WSN yields a lower

transmission probability for the sensors. It is because all the sen-

sors are forced to transmit their measurements with the sameprobability. 3) In the context of CS, is the number of com-

pressed measurements. A larger is supposed to yield a higher

probability to recover the original signal, so that better perfor-

mance is expected. However, on the other hand, in (20b), the

value of also decides the upper bound of the total selection

probability of the sensors, thus increasing decreases the sen-

sors transmission probability as shown in (22), which may yield

worse tracking performance. Therefore, there exists a tradeoff

between the accuracy of the recovered signal and tracking per-

formance.

B. With Nonidentical Probabilities

Without the simplifying assumption that each sensor employs

an equal transmission probability, our optimization problem is,

(23)

Note that

for CMSI

for PMSI.

By denoting , the objective func-

tion is further expressed as

(24)

where is a unit vector, . The

objective function in (24) is a linear-fractional function, so that

the optimization problem is a quasilinear optimization problem

[30]. It has been shown in [30] that, as the feasible set

is not nonempty, the linear-fractional problem (24) can be trans-

formed to an equivalent linear program. We first define the fol-

lowing two variables

such that once we get the optimal solutions for and , we can

also easily get the optimal values for the original variables .

The equivalent linear program optimization problem is

(25)

where

......

. . ....

......

. . ....

We apply the MATLAB function linprog to solve the con-strained linear optimization problem (25). Having the optimal

solutions and , the transmission probabilities of the sensors

are .

V. SIMULATION RESULTS

In this section, we illustrate the performance of the proposed

sensor management algorithm by numerical examples. We com-

pare the MSE of CMSI and PMSI presented in Section III-A and

Section III-B, respectively, with different process noise param-

eters. The MSE is computed through

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Fig. 1. Target trajectory and the estimated track with different sensor manage-

ment schemes.

where is the number of Monte Carlo trials. Also, the effect

of the parameters in the model, i.e., the energy constraint ,

the number of sensors and the number of MACs ( of the

sensing matrix ) is evaluated.

We consider a WSN, consisting of sensors grid deployed

in a surveillance area. The dynamical

model of the target is given by (1). The state transition model

and the covariance of the process noise are given as follows:

where is the sampling time interval and isthe process noise parameter. The parameters of the observation

model (2) are set as and . The initial state

of the target is assumed to be Gaussian with mean

and covariance matrix .

We perform target tracking over time steps for each

Monte-Carlo trial, and set particles for the particle

filter. The MSE of the estimation at each time is averaged over

Monte-Carlo trials.

In Fig. 1, a WSN is illustrated where sensors track

a target with MACs under the energy constraint

. We plot the true target trajectory, the estimated target

trajectory when all the sensors send their measurements to thefusion center, the estimated target trajectory with the proposed

CS based sensor management method when the FC has CMSI

and the transmission probabilities of the sensors are noniden-

tical, and the estimated target trajectory under the random selec-

tion scheme. For the random selection scheme, the sensors are

randomly selected by the fusion center with probability .

Thus, the random selection method is expected to give worse

tracking performance compared to the CS based sensor man-

agement method and the all-send case. It is also observed that

the proposed CMSI based scheme shows similar performance

as that of the all-send case.

In Fig. 2, we study the tracking performance with CMSI

and PMSI respectively. With sensors in the ROI, we

compare the MSE under different process noise parameters.

Fig. 2. Comparing CMSI and PMSI via MSE with different noise parameters.

Note that under the identical probability condition, sensors

are equally important, so that whether the fusion center has

CMSI or PMSI does not affect the tracking performance. We

observe in Fig. 2(a) that when the process noise is small

, PMSI outperforms CMSI a little bit, and in

Fig. 2(b) when the noise is relatively large , CMSI

performs much better than PMSI. The reason is that when the

process noise is small, the target motion model has relatively

smaller uncertainty about the predicted target location. Thus,

averaging the sensing matrix helps us get even better tracking

performance. However, when the process noise is relatively

large, the fusion center has higher uncertainty about the target

trajectory. Moreover, without having complete information

about the sensing matrix, the fusion center calculates the

weights of the particles in particle filtering with errors. There-fore, the fusion center is able to achieve better performance

with CMSI compared to PMSI.

Now we study the effect of the parameters in the WSN (the

energy constraint , the number of sensors and the numberof

MACs) on the CS based sensor management problem. The per-

formance is evaluated through the tracking performance under

the assumption that the process noise is , and that the

fusion center has CMSI to track the target.

In Fig. 3, the tracking performance is illustrated when

sensors track a target with MACs under

different energy constraints . In Fig. 3, we give the MSE

for and , to show the efficiency of our model,where Fig. 3(a) shows the MSEs when the sensors have non-

identical probabilities, and Fig. 3(b) shows the MSEs when

the sensors have identical probabilities. We also present the

MSE for the random selection method for both cases. Note

that for the random selection scheme, the sensors are randomly

selected with probability in Fig. 3(a) and with probability

in Fig. 3(b) corresponding to the optimal solution in

(22). From both figures, we observe that when increases from

6 to 8, the MSE decreases for both the random selection and

the CS based approach, and the latter one always outperforms

the former under the same energy constraint. Both methods are

compared to the all-send case where all sensor measurements

are available at the fusion center via a set of parallel channels.

Compared to the all-send case where a total of 25 units of

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Fig. 3. MSE for different energy constraints, , 8 (a) Nonidentical probabilities. (b) Identical probabilities.

Fig. 4. MSE for different number of sensors, , 25, 36, 49 (a) Nonidentical probabilities. (b) Identical probability.

energy are consumed at each time since parallel

transmissions are necessary, the proposed approach loses only

a little performance for both and cases.

In Fig. 4, we show the tracking performance for different

number ofsensors , 25,36, 49withthe energy constraint

and the number of MACs , where Fig. 4(a) shows

the MSEs when the sensors have nonidentical probabilities, and

Fig. 4(b) shows the MSEs when the sensors have identical prob-

abilities.In Fig. 4(a),the MSEs show that theproposed approach

achieves a better performance as increases. This is because,in target tracking without the strict constraint that every sensor

is equally important, the sensors which can obtain more infor-

mative observations in the ROI will be assigned higher proba-

bilities. On the other hand, as the number of sensors in the ROI

increases, the density of the sensors increases, so that the fu-

sion center has a better chance to activate even more informative

sensors. When the density is large enough, the tracking perfor-

mance tends to saturate as shown in Fig. 4(a). However, from

Section IV-A, we know that under the assumption that all the

sensors have identical transmission probabilities, the transmis-

sion probability of the sensors becomes smaller as the number

of sensors in the ROI increases, so that the tracking performance

worsens as shown in Fig. 4(b). Note that in Fig. 4, in order to

clearly see that the number of sensors in the WSN affects the

tracking performance, we have set the number of MACs to

be relative small, which results in divergence of the MSE espe-

cially when is relatively small. We shows the effect of the

number of MACs in Fig. 5.

In Fig. 5, the MSEs of the CS based approach with different

values are presented, where Fig. 5(a) shows the MSEs when

the sensors have nonidentical probabilities, and Fig. 5(b) shows

the MSEs when the sensors have identical probabilities. In

Fig. 5, we let and . As observed before, it is

seen from Fig. 5(a) and Fig. 5(b), that the tracking performanceis improved when nonidentical probabilities are assigned to

sensors compared to assigning identical probabilities under

the same energy constraints and the same number of MAC

transmissions. In both cases (identical and nonidentical proba-

bilities) the performance is improved as M increases. However,

it is observed that the rate of performance improvement reduces

as M increases. It is noted that, in Section IV-A, we indicated

that with the identical transmission probability assumption, the

transmission probability of the sensors decreases as M increases

due to the energy constraint. However, the following reasons

justify the improvement of the tracking performance as M

increases (even though the transmission probability decreases):

1) As M, the number of MAC transmissions, increases the

fusion center receives different superpositions of the observa-

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Fig. 5. MSEs for different number of MACs, , 3, 4 (a) Nonidentical probabilities. (b) Identical probability.

tions from the sensors with good observations. This enables

the fusion center to better extract the most informative data

residing at distributed nodes. 2). M is the dimension of the

compressed observation vector received at the fusion center

in (4). In the context of CS with respect to complete signal

recovery, it has been shown in [31] that reliable recovery is

possible with a small number of compressive measurements

(compared to original signal dimension) even if the projection

matrix contains sparse random elements as long as the matrix

satisfies some required properties. Then, increasing M beyond

this particular value is not necessary. While these analyses

(e.g., [31]) may not be directly applicable to the scenario

considered in this paper, (where we consider tracking instead of

complete recovery when the random projection matrix contains

sparse nonidentical elements), it is intuitive that when M isincreased beyond a certain value, a significant improvement in

performance is not observed. The performance of the proposed

approach is quite close to the all send case, especially when

, 4, but is energy efficient in the sense that it consumes

only units of energy at any given time on an average.

Note that the compressed measurements are used directly for

state estimation in the considered target tracking problem,

which is different from the traditional CS problems where the

goal is to recover a sparse signal.

VI. CONCLUSION

In this paper, we proposed a novel probabilistic sensor man-

agement approach for target tracking in sensor networks based

on compressed observations. With the proposed approach, the

sensor management problem becomes a constrained optimiza-

tion problem, where the goal is to determine the optimal values

of probabilities that each sensor should transmit with such that

the trace of the FIM at any given time step is maximized. We de-

rived the formulations for the FIM when the fusion center has

complete measurement sparsity information (CMSI) and partial

measurement sparsity information (PMSI). Theoretical results

for the case when each sensor employs identical transmission

probability give us useful insights into the impact of the three

important parameters of our model. Numerical results for both

the identical probability case and the more general nonidentical

transmission probabilities case show that the proposed approach

saves a lot of energy with a little performance loss compared to

the optimal scenario in which all sensor observations are trans-

mitted to the fusion center via parallel channels. Under the same

energy constraint, the proposed scheme outperforms the random

selection approach significantly. An interesting future work is to

take the channel statistics into consideration.

APPENDIX A

PROOF OF PROPOSITION 1

Let . Note that in this proof the time index is

omitted for the sake of simplicity. Diagonal elements of are

given by

and off-diagonal elements are

It is straightforward to obtain ,

, , and

.

Therefore, according to the law of large number (LLN) for

independent and nonidentical random variables, we get

Hence, and

(26)

Then,

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Diagonal elements of are given by

and off-diagonal elements are

Therefore,

completing the proof.

APPENDIX B

PROOF OF

PROPOSITION

2Based on Eq. (10), one can get

, where .

Then we have

With LLN, we have the approximation for as in (26).

Then

and

(27)

According to LLN, we can get approximations

(28)

and

(29)

Therefore, the proof is completed by substituting (28) and (29)

into (27).

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Yujiao Zheng (S13) received her B.S. degree

in electronic engineering and information sciencefrom University of Science and Technology of

China (USTC) in 2008, M.S. degree in electricalengineering and Ph.D. degree in electrical and

computer engineering from Syracuse University,

Syracuse, NY, in 2011 and 2014 respectively. Herresearch interests are in the areas of statistical signal

processing with its application in target tracking,sensor management, and compressive sensing.

She received the Best Student Paper Award at theThirteenth International Conference on Information Fusion in 2010.

Nianxia Cao (S12) received the B.S. and M.S.degrees in control theory and control engineering

from Northwestern Polytechnical University (NPU),Xian, China, in 2009 and 2012. She has been

pursuing the Ph.D. deg ree in the Department of Elec-trical Engineering and Computer Science, Syracuse

University since 2012. Her research interests include

statistical signal processing, sensor managementin the wireless sensor networks, game theory, and

mechanism design.

Thakshila Wimalajeewa(M15) received the B.Sc.degree in electronic and telecommunication engi-

neering with First Class Honors from the University

of Moratuwa, Sri Lanka, in 2004 and the M.S., andPh.D. degrees in electrical and computer engineering

from the University of New Mexico, Albuquerque,NM in 20 07 and 2009, respectively.

She spent 20102012 as a postdoctoral researchassociate in theDepartmentof Electrical Engineering

and Computer Science, Syracuse University (SU),Syracuse, NY. She currently holds a research faculty

position at SU. Her research interests lie in the broad areas of communication

theory, signal processing and information theory. Her current research focuseson compressive sensing, low dimensional signal processing for communication

systems, blind signal recognition, and resource optimization in distributedsensor networks.

Pramod K. Varshney (S72M77SM82F97)

was born in Allahabad, India, on July 1, 1952. He

received the B.S. degree in electrical engineeringand computer science (with highest honors), and

the M.S. and Ph.D. degrees in electrical engineeringfrom the University of Illinois at Urbana-Champaign

in 1972, 1974, and 1976 respectively.From 1972 to 1976, he held teaching and research

assistantships with the University of Illinois. Since1976, he has been with Syracuse University, Syra-

cuse, NY, where he is currently a Distinguished

Professor of Electrical Engineering and Computer Science and the Directorof CASE: Center for Advanced Systems and Engineering. He served as the

Associate Chair of the department from 1993 to 1996. He is also an AdjunctProfessor of Radiology at Upstate Medical University, Syracuse. His current

research interests are in distributed sensor networks and data fusion, detectionand estimation theory, wireless communications, image processing, radar

signal processing, and remote sensing. He has published extensively. He is the

author of Distributed Detection and Data Fusion (New York: Springer-Verlag,1997). He has served as a consultant to several major companies.

Dr. Varshney was a James Scholar, a Bronze Tablet Senior, and a Fellowwhile at the University of Illinois. He is a member of Tau Beta Pi and is the

recipient of the 1981 ASEE Dow Outstanding Young Faculty Award. He waselected to the grade of Fellow of the IEEE in 1997 for his contributions in the

area ofdistributed detectionand data fusion. Hewas theGuest Editor of the Spe-

cial Issue on Data Fusion of the IEEE PROCEEDINGS January 1997. In 2000, he

received the Third Millennium Medal from the IEEE and Chancellors Citation

for exceptional academic achievement at Syracuse University. He is the recip-ient of the IEEE 2012 Judith A. Resnik Award. He is on the Editorial Boards of

theJournal on Advances in Information Fusionand IEEE SIGNALPROCESSINGMAGAZINE. He was the President of International Society of Information Fusion

during 2001.

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