ieee transactions on signal processing, vol. 55, no. 9 ...nehorai/paper/04102539.pdf · ically,...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 9, SEPTEMBER 2007 4669

Information-Driven Distributed Maximum LikelihoodEstimation Based on Gauss-Newton Method in

Wireless Sensor NetworksTong Zhao, Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—In this paper, we develop an energy-efficient dis-tributed estimation method that can be used in applications suchas the estimation of a diffusive source and the localization andtracking of an acoustic target in wireless sensor networks. We firstpropose a statistical measurement model in which we separatethe linear and nonlinear parameters. This modeling strategyreduce the processing complexity. We then study the distributedimplementation of the Gauss-Newton method in the maximumlikelihood estimation. After that we propose a fully distributedestimation approach based on an incremental realization of theGauss-Newton method. We derive three modifications of the basicalgorithm to improve the distributed processing performancewhile still considering the energy restriction. We implement theidea of information-driven collaborative signal processing and pro-vide a sensor-node scheduling scheme in which the Cramér–Raobound (CRB) is used as the performance and information utilitymeasure to select the next sensor node. Numerical examples areused to study the performance of the distributed estimation, andwe show that of the methods considered here, the proposed mul-tiple iteration Kalman filtering method has the most advantagesfor wireless sensor networks.

Index Terms—Energy-efficient distributed estimation,Gauss–Newton method, maximum-likelihood estimation, sensornode scheduling, wireless sensor networks.

I. INTRODUCTION

RECENTLY, wireless sensor networks have attracted ex-tensive research interest as a new approach to providing

clever interaction with the physical world [1]–[4]. In thispaper, we address the issue of developing an energy-efficientdistributed parameter estimation method for applications inwireless sensor networks. The proposed methods can be ap-plied in many civilian and military applications, such as theestimation of a diffusive source [5]–[7], the localization andtracking of an acoustic target [8], [13], and most other passivesensing applications.

Manuscript received March 28, 2006; revised January 28, 2007. The asso-ciate editor coordinating the review of this manuscript and approving it forpublication was Dr. Venkatesh Saligrama. This work was supported in partby the Department of Defense under Air Force Office of Scientific ResearchMURI Grant FA9550-05-1-0443, AFOSR Grant FA9550-05-1-0018, and bythe National Science Foundation under Grant CCR-0330342.

The authors are with the Department of Electrical and Systems Engineering,Washington University, St. Louis, MO 63130 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2007.896267

In a typical wireless sensor network, each sensor node oper-ates unattended with limited battery power and communicateswith others through wireless links that consume the dominantpart of the energy in the network [4]. Therefore, a challenge inwireless sensor network research is to develop energy-efficientprocessing methods that can decrease the power usage while stillmaintaining a required processing performance, thus increasingthe lifetime of the whole sensor network. In traditional central-ized processing methods, the data measured at each sensor aretransmitted to a fusion center for processing. However, eventhough centralized methods can achieve high processing perfor-mance, some of their inherent properties, such as the large com-munication energy consumption and the lack of adaptability tolink failures and dynamical changes in the network topology,prohibit their use in a wireless sensor network. Hence, recentresearch has focused on developing energy-efficient distributedprocessing technologies suitable for the applications in wirelesssensor networks.

Most of the current distributed estimation methods fall intotwo categories. In one category, the distributed estimation isdeveloped using the sequential (incremental) Bayesian method[9]–[14]. With this method, a state belief (posterior densityfunction) is propagated in the sensor network according toan information theoretic criterion. A typical method is thedistributed particle filter [12]. A difficulty in this type ofmethods is that the convergence of the estimation is not easyto be proven. In another category, the distributed estimation isobtained by implementing the ordinary centralized estimationmethods, such as the maximum likelihood estimation and theleast-squares method in a distributed fashion [15]–[19]. Thesetype of methods maintains the optimality of the centralizedmethods, whereas the required communication amount is large.In this paper, we develop a new framework for the distributeddata exploration in wireless sensor networks, in which we jointhe advantages of the above two types of methods.

The recent relevant work is presented in [15] and [16]. In[15], a distributed Gauss-Newton method is derived. However,the difference from our work is that: i) the goal of this methodis for sensor node self-localization; and ii) all sensor nodesare involved in data communications in each iteration. In[16], a space-time diffusion scheme is derived for least-squareparameter estimation using peer-to-peer communication. Thismethod relies on the information exchange among all sensornodes during a long sampling times. The convergence is provenunder a linear measurement model assumption. However, inthis paper, the essential point is that under an framework ofmaximum likelihood estimation, we apply a information utility

1053-587X/$25.00 © 2007 IEEE

4670 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 9, SEPTEMBER 2007

measure to choose an optimal sequence of sensor node tocollect measurements such that we reduce the required datacommunications.

More specifically, we derive our new methods by combiningthe distributed implementation of a Gauss-Newton-based max-imum likelihood estimation and the information-driven sensor-node scheduling. We first modify the ordinary implementationof the Gauss-Newton method such that an update of the estimatein each Gauss-Newton iteration can be divided into a sequenceof updates at each sensor node (we do not need to wait for thecollection of data from all the sensor nodes). A performancecriterion is used to determine when the estimation process willstop. Then we implement the idea of information-driven col-laborative signal processing [20], i.e., we use the sensor-nodescheduling to optimally allocate the limited resources such asthe energy and bandwidth in a wireless sensor network. Specif-ically, according to a certain information measure, we find anoptimal subset and/or a sensor-node querying route to processthe estimation such that we can decrease energy consumptionand response time while maintaining an acceptable processingperformance.

We derive the information measure based on the Fisherinformation matrix. We find that in our distributed estimationmethod, calculation of the Fisher information matrixes is anintrinsic part of the algorithm, hence they can be obtainedwithout any extra computation and data transmission, whichis suitable for the application in wireless sensor networks.Another advantage of our new method is that we also take intoconsideration the limited processing capability of each sensornode in wireless sensor networks. For example, we propose astatistical measurement model in which we separate the linearparameters from the overall parameters such that estimationcomputation can be substantially decreased.

The paper is organized as follows. In Section II, we presentthe statistical measurement model and use the diffusivesource as an example. In Section III, we derive the standardmaximum likelihood estimation and analyze the distributedimplementation of the Gauss-Newton method. In Section IV,we propose our new fully distributed estimation method and thesensor-node scheduling scheme. Numerical examples are usedto demonstrate the performances of the proposed methods inSection V. Conclusions and ideas for future work are presentedin Section VI.

II. STATISTICAL MEASUREMENT MODEL

In this section, we propose a statistical measurement modelbased on which we derive our distributed maximum likelihood(ML) estimation method. In this model, we assume the param-eters are stationary. The model can also be extended to the dy-namic parameter cases. This measurement model is suitable fora broad range of applications in wireless sensor networks. Wepresent an example of monitoring a diffusive source using wire-less sensor networks.

To model the measurements, we suppose that a spatiallydistributed wireless sensor network has been deployed. Eachsensor node in this network is located at a known position andcan take measurements at a sequence of time instants. We denotethe sensor node locations at the positions ;

the sensor node takes the measurements at time samples. Then we consider the following

statistical measurement model at each sensor node:

(1)

where• is the measurement of sensor node at time ;

• is a-dimensional vector, related to the underlying physical

mechanism of the source we want to estimate, where, is a real scalar function de-

pending (generally nonlinearly) on ;• is a parameter vector related to the source we

are interested in; its relation to the observation

through is nonlinear;• is a parameter vector representing the linear part

of the model (1);• represents additive noise. We assume it is a real

random variable with zero mean and Gaussian distribution.We also assume that this additive noise is correlated in thetime domain but independent in the space domain, i.e.,

In this measurement model, the term representsthe underlying physical mechanism. Both parameter vectorsand are related to the unknown source and the environmentand may include the parameters we are interested in. Note thatin our model, instead of using a general nonlinear function todescribe the relationship between the unknown parameters andthe measurements, we represent the signal into a regression formthat is a linear combination of certain regression basis functions.By providing this structure to the signal, we reduce the compu-tation complexity of the corresponding distributed estimationmethods. This is because i) the unknown parameters in can beestimated using closed-form formulas; and ii) the nuisance pa-rameters in can be removed from the unknown parameter listby applying a subspace projection method.

To simplify the presentation, we assume that each sensornode takes measurements at the same sequence of time instants

(this assumption can be relaxed withoutsignificant changes to the derived methods). We also denote

, and . Thenwe can rewrite the measurement model (1) as

(2)

When we lump all the measurements taken at sensor node intovector form, we obtain

(3)

where• , which is an vector;• , which is an

matrix;

ZHAO AND NEHORAI: INFORMATION-DRIVEN DISTRIBUTED MAXIMUM LIKELIHOOD ESTIMATION 4671

• which is an -dimension Gaussiandistributed random vector with mean 0 and covariance ma-trix . It is also independent in the space domain, i.e.,

when .This statistical measurement model can be used in many ap-

plications in wireless sensor networks, e.g., the monitoring ofa diffusive source, the localization and tracking of an acousticor electromagnetic target, and most other passive sensing appli-cations. In the following, we show an example of the proposedmodel in the application of estimating a diffusive source whichis important in the homeland security applications and environ-ment monitoring.

A. Measurement Model of a Diffusive Source

We assume that a continuous point diffusive source is locatedat position and that it releases a diffusive sub-stance at time . A wireless sensor network is deployed to mon-itor this diffusive source. The measurement taking by sensornode at time can be written as

(4)

where is the concentration of the substance of interest atposition and time . The term represents the under-lying physical mechanism of substance dispersion, and it can beobtained by solving diffusion equations under certain boundaryand initial conditions. If we assume the environment to be asemi-infinite medium with an impermeable boundary, and theconstant substance release rate from the source is , we obtainthe concentration distribution

(5)

where and is the medium diffusivity [21].The other items in (4) are defined as follows (see [22] for de-tails):

• is a bias term, representing the sensor’s response to for-eign substances, which is assumed to be an unknown con-stant;

• is the sensor’s noise, assumed to be Gaussian dis-tributed, independent in time and space.

If we denote , and, we can rewrite (4) as

(6)

which is an instance of the measurement model in the (2). Here• ;• is the linear parameter vector, in which is

the parameter of interest and is the nuisance parameter;• represents the nonlinear source and

medium parameters.

III. STANDARD MAXIMUM LIKELIHOOD ESTIMATION

In this section, we first derive the standard maximum likeli-hood estimation for the parameters and in the proposed mea-surement model (3); then we study the realization of this ML es-timation using an iterative Gauss-Newton method and therebydevelop a distributed implementation scheme motivated by theidea of distributed EM algorithms in [19]. Even though thisdistributed implementation can be used in wireless sensor net-works, some important limitations of this method restrict its ap-plications. This restriction motivates us to design a new frame-work for the distributed ML estimation.

A. Maximum Likelihood Estimation

We assume that sensor nodes are activated to process anestimation task in a wireless sensor network. At sensor node ,measurement follows the proposed measurement model (3).Here we assume the covariance matrix is known, since inpractical applications this matrix can be estimated during thecalibration step.

We can collect the measurements from allsensor nodes into vector form as

(7)

where• , which is an vector;• , which is an

matrix;• which represents the additive Gaussian

noise. According to our assumptions, it follows an-dimension multiple normal distribution with mean

0 and covariance matrix , i.e., with. Here, since the covariance

matrix is assumed to be positive definite, is alsopositive definite and can be factored aswhere is an invertible matrix. Accordingly, thecovariance matrix can also be factored aswhere .

In this measurement model, the variables and are the un-known parameters to be estimated.

According to the above measurement model (7), we find thatthe log-likelihood function of the measurements is

(8)


Maximizing this log-likelihood function, we obtain the ML es-timates of and

(9)

(10)

where . We also find that the optimization in (9)can be simplified as

(11)

where and is the projecting matrix on the columnspace of

1) Nuisance Parameters: If there exist nuisance parametersin the linear parameter vector , the ML estimates in (10)can be simplified. Assume the vector is partitioned into twoparts

where includes parameters of interest, includesnuisance parameters, and . Correspondingly,matrix can also be partitioned into two parts

where includes the first columns of whileincludes the last columns of . If we define

, and , we can ob-tain the ML estimate for the parameters of interest as

(12)

where we denote

which is the complementary projection of on thecolumn space of .

In a special case when the matrix is not dependent on, i.e., is a constant matrix as in (6), the ML estimate for

in (9) can also be simplified as

(13)

where the complexity to calculate is much less than thecalculation of in (9).

B. Implementation With Gauss-Newton Method

For a centralized ML estimation, the measurements from allsensors are transmitted to a fusion center, and the fusion centerapplies (9) and (10) to calculate the ML estimates directly. How-ever, the severe energy restriction in wireless sensor networksprohibits the transmission of raw data from all sensor nodesto a fusion center. Therefore, the estimation results in (9) and(10) are not enough to derive a distributed estimation method. Inthis section, we analyze the realization of ML estimation usinga Gauss-Newton method and develop a distributed implemen-tation scheme. In this distributed implementation, in stead oftransmitting all the measurements to a center for processing,only several sufficient statistics are transmitted between nearbysensor nodes; hence, we lower the communication cost.

1) Gauss-Newton Method: To obtain the ML estimates, weobserve that the nonlinear optimization problem in (9) to calcu-late is a nonlinear weighted least squares (LS) problem (see[23] and [24]) under the assumption that the covariance matrix

is known. Then we can solve it using the effective and effi-cient Gauss-Newton method. Note that an iterative method suchas Gauss-Newton method can only converge to a local max-imum value. In general this method does not yield the max-imum-likelihood estimate unless the optimization problem canbe formulated as, e.g., a convex optimization problem. There-fore, the choice of the initial value for this iterative method isvery important.

When employing the Gauss-Newton method to solve the non-linear weighted LS problem in (9), we obtain an iterative processto update the estimate of as

(14)

where

(15)

is an Jacobian matrix. In order to obtain a distributedestimation, we study the details of the implementation in (14).First, we can partition as

(16)

where is the th column of and can be derived as

(17)


Hence, the second term on the right-hand side (RHS) of (14) canbe rewritten as

.... . .

...

... (18)

In this formula, the term can be fur-ther derived as

(19)

where .From the (17) and (19), we make the following two

observations.1) In order to update using (14), we need only

statistics. Each statistic is a vector, denoted by, as follows:

(20)

(21)

Therefore, for the estimation we do not need to collect theactual raw data but only these statistics.

2) We can rewrite and as

(22)

(23)

which means that each of these statistics is a summation ofthe local statistics at each sensor node.

From these two observations, we obtain a distributed implemen-tation of the ML estimation of , motivated by the idea of dis-tributed EM algorithms in [19], as follows.

2) Distributed Implementation: An initial value of , denotedas , is provided at the beginning. Assume that at iterationstep , node has the current parameter estimate .

The next ML estimate can be computed by performingone message cycle through all the nodes. To begin, initializethe sufficient statistic to zero. When themessages are transmitted to the sensor node , the node performsthe following two steps.Step 1: It computes the local sufficient statistics using

(24)

(25)

Note that the statistics are computed using onlyand the local measurements ; hence, these calcu-lations can be implemented completely locally.

Step 2: It then increments the global sufficient statistics ac-cording to

(26)

(27)

After that the current node passesand to the node

.At the last sensor node , the node obtains the complete

sufficient statistics and , and usesthem to calculate by applying (14). We observe that foreach iteration, real valued numbers are transmitted be-tween two sensor nodes. Hence, the transmission volume is in-dependent of the number of measurements taken by each sensornode.

3) Discussion: Compared with the centralized ML estima-tion method, this distributed implementation algorithm avoidsthe necessity of transmitting raw data to the fusion center (onlyseveral sufficient statistics are transmitted between the nearbysensor nodes); hence, the algorithm decreases the required com-munication burden. However, several inherent properties of thisalgorithm limit its application in wireless sensor networks.

• Since the Gauss-Newton is an iterative algorithm, in orderto obtain the ML estimate, the messages should cyclethough all the sensor nodes several times (each cyclecorresponding to one iteration). Even though the volumeof communication for one cycle is not large, the potentialtotal communication burden is still heavy, especially incases where the algorithm converges very slowly.

• In order to update the estimate once, the messages need tocycle through all the sensor nodes, which is time costly andnot suitable for some applications where a rapid responseis required.

• A predetermined message transmission route is needed toguarantee that the messages pass through all the sensornodes. This requirement is not suitable for applications inwireless sensor networks, because the network topologychanges dynamically due to the failure of some sensornodes or communication links.

• Not all the sensor nodes provide useful information to im-prove the estimation; furthermore, some information maybe redundant. Therefore, collecting information from allavailable sensor nodes is not an optimal strategy for thedistributed processing due to the power restriction.


In the following, we propose a new and fully distributed estima-tion method motivated by the above distributed implementationand the analysis. The new method addresses these limitationsand is suitable for a broader range of applications in wirelesssensor networks.

IV. INFORMATION-DRIVEN DISTRIBUTED MAXIMUM

LIKELIHOOD ESTIMATION

In this section, we develop an information-driven distributedmaximum likelihood estimation method based on the aforemen-tioned Gauss-Newton implementation. We first derive the basicdistributed estimation algorithm where we assume the linearparameter is known; then we propose several modificationsof the basic algorithm to improve the estimation performance.Finally, we extend the algorithm to the case when is un-known. We introduce information-driven collaborative signalprocessing techniques and propose a sensor-node schedulingscheme.

A. Information-Driven Distributed Estimation Algorithm

For the case in which linear parameter is known, the ML es-timation of parameter in (9) becomes the following nonlinearweighted least-squares problem:

(28)

which is equivalent to (9) except that is replaced by a con-stant vector . Then, the corresponding Gauss-Newton iterationis

(29)

where

In order to develop our distributed estimation method, weintroduce the following theorem, in which we prove that anupdate of the estimate in each iteration of the Gauss-Newtonmethod can be divided into a sequence of updates at each sensornode. This theorem implements the Proposition 1 in [25] into thederivation of distributed estimation methods.

Theorem 4.1: We define

and assume that the matrix is posi-tive definite. Then, to solve the nonlinear LS problem (28), the

th iteration of the Gauss-Newton method in (29) can be gener-ated by the algorithm

(30)

where we let and we obtain . Thepositive definite matrixes are generated by

(31)

with

Proof: See the Appendix.In the above theorem, we replace by in (30). Then,

the updating algorithm is modified as

(32)

where

(33)

After we make this modification, in the implementation of theGauss-Newton method we linearize the function at thecurrent estimate rather than at the estimate availableat the start of the iteration. Therefore, the resulting update ofthe estimate at the end of each Gauss-Newton iteration is moreclose to the true value than the result without this modification.

From Theorem 4.1, we observe that when applying theGauss-Newton method to the ML estimation, the update of theestimate in each iteration can be divided into sequentialupdates . In each of these updates, only thelocal measurements and the variable and matrixpassed from the previous sensor nodes are needed; hence, theseupdates can be processed completely locally. This theoremmotivates us to develop a new distributed estimation method, inwhich the messages are transmitted through the chosen sensornodes, and the estimate of is updated when the new data ateach sensor node are obtained. Then we do not need to waitfor the accumulation of the measurements from all the sensornodes in order to obtain a new estimate.

Another advantage of the new method is that it is convenientfor us to implement information-driven dynamic collaborativeinformation processing, which is a very useful technique to beapplied in wireless sensor networks. Wireless sensor networksare characterized by limited battery power, frequent node attri-tion, and varying data and communication quality. Hence, goodcollaboration among distributed sensor nodes is necessary to im-prove estimation performance and reduce power consumption.

In our case, it is critical to select an optimal subset and an op-timal order of incorporating these measurements into our esti-mate update. This process is called the sensor-node scheduling.Usually this scheduling process provides a faster reduction inestimation uncertainty compared with blind or nearest neighborsensor node selection schemes, and it incurs a lower commu-nication burden for meeting a required estimation performancethreshold. By applying this idea in our method, instead of trans-mitting the messages through all the available sensor nodes insequence, at each sensor node we use some information mea-sures to determine to which sensor node in the neighbor of thecurrent node we should transmit the current messages. This esti-mation process will continue until the final performance fits ourrequirements.


According to Theorem 4.1 and the above discussion, wepropose a new information-driven distributed ML estimationmethod as follows.

Distributed estimation algorithm:• The distributed estimation algorithm is initialized by

sensor node . We assume an initial value of theparameter is available at this sensor node, denoted by

; we let .• At sensor node , we perform the following steps:Step 1. Data receiving: the sensor node is activated by the

previous node and receives the transmitted data

and . Note: when , this step isomitted.

Step 2. New variable calculation: Calculate the new ma-

trix as

(34)

then, update the matrix as

(35)

Step 3. Estimate update: Update and obtain the current es-

timate as

(36)

Step 4. Estimation quality test: Test the quality of the up-dated estimate according to some performancemeasure, e.g., the trace or determinant of the co-variance matrix. If the estimate is “good enough,”the estimation process is terminated; otherwise,the algorithm continues with the following steps.

Step 5. Sensor node selection: Select a sensor node fromits neighbor according to certain informationutility criteria (the details are discussed in Sec-tion IV-D).

Step 6. Data transmission: The current sensor node wakes

up and transmits the current estimate andto the selected sensor node, and then the currentnode returns to sleeping status.

When comparing this new distributed ML estimation algo-rithm with the ordinary Gauss-Newton method, we observe twocharacteristics: i) the new algorithm corresponds to just one iter-ation of the ordinary Gauss-Newton method; therefore, we de-crease the required communications and shorten the responsetime; and ii) we linearize the nonlinear function at thecurrent estimate rather than at the estimate available at the be-ginning of the iteration; hence, we speed the convergence of thealgorithm.

It can be verified that after we make these changes, the newmethod has the same update steps as the extended Kalman filter(EKF) specialized to the case where the state of the underlyingdynamic system stays constant [25]. Because of this relation-ship, we can obtain a new explanation of the proposed dis-

tributed estimation method under the framework of the sequen-tial Bayesian estimation. We linearize each new measurementaround the current estimate and treat the measurement as ifit were linear. At that point the belief is approximated by aGaussian density function, and it is updated when the new mea-surement is available, until it is “good enough.” The mean ofthe final obtained belief is calculated as the estimate of the un-known parameters.

B. Modifications to Improve Performance

This algorithm provides a basic framework for the dis-tributed estimation. However, its estimation performance islimited under the restriction of the available energy. In thissection, we propose three modifications of the above basicmethod to improve estimation performance and reduce pro-cessing time while still maintaining a low communicationenergy consumption.

1) Levenberg-Marquardt Method: One important problem ofthe ordinary Gauss-Newton method is that in each iteration toupdate the estimate, even though the update is in the correct di-rection, the Gauss-Newton method may take steps that are toolong, thereby slowing the converge speed. An efficient modifi-cation is to apply a trust region strategy [26], in which when weupdate the estimate, we limit the distance between the new andold estimates below a bound, i.e., . The so-lution to this problem when applied to our measurement modelis

(37)

where is a positive scalar. This formulation is known asthe Levenberg-Marquardt method. It can be verified thatwe can implement the Levenberg-Marquardt method toour new distributed estimation method by setting the ini-tial condition instead of , so that matrix

is positive definite even if

matrix is not.2) Multiple Local Iteration Implementation: As we dis-

cussed before, the new distributed estimation method can beimplemented as an extended Kalman filter. The basic idea ofthe extended Kalman filter is to linearize the measurementfunction at the current estimate so it can be treated as a linearfunction. In the new distributed estimation algorithm, after weupdate the estimate using (36) at sensor node , the new estimate

is probably closer to the true value of than was .Therefore, if we relinearize function at the new estimate

and perform the update step in (36) at the same sensor nodeagain, we would on average reduce the approximation error andincrease the estimation accuracy [27]. Hence, instead of using(36) in Step 3. Estimate update to obtain the new estimate, weapply the following multiple local iteration updating step:

Step 3. Estimate update: We denote and. At the sensor node , we update the estimate using

the following multiple iteration scheme

(38)


where

(39)

When this iteration converges, the values and are

used as new estimate and , respectively.Another consideration underlying the above modification isthat in wireless sensor networks, communications are the majorsource of power consumption, rather than data processing.Therefore, in this modification we employ more effective andintensive calculations at each sensor node and thus potentiallyreduce the total communication burden.

3) Iterated Kalman Filtering Implementation: In our pro-posed distributed estimation method, we implement a single it-eration of the Gauss-Newton method, which becomes an ex-tended Kalman filter. The consideration for this single iterationis to decrease the required communications and speed the taskresponse. However, the estimation accuracy is degraded underthis scenario. Since our new method is not absolutely a ML es-timation, even though the variance of the estimation error con-verges to zero, the estimate itself may converge to a biased value[28].

An improvement designed to address this problem extendsthe proposed single iteration method to a multiple global iter-ation method, which becomes an iterated Kalman filter (IKF)[29]. To apply this IKF implementation into our distributedestimation method, we first carry out the single iteration dis-tributed algorithm and select a subset of the sensor nodes; thenthe obtained estimate is transmitted back and cycles through thechosen sensor nodes and is updated again, until it converges.

Even though we process multiple iterations with this method,it still has advantages compared with the distributed implemen-tation of the ordinary Gauss-Newton method in Section III-B–2.

• In the first iteration, we choose a subset of the sensor nodesthat contains information closely related to our estimationtask. Then the messages need to cycle through only the se-lected sensor nodes. Therefore, we still save much com-munication energy compared with the ordinary distributedimplementation based on Gauss-Newton.

• Even though we use multiple iterations, an estimate updateis still obtained at each sensor node, rather than at the endof the current iteration. Therefore, an estimation result canbe provided at any time according to the tradeoff betweenthe required estimation accuracy and the processing time.

C. Distributed Estimation With Unknown Linear Parameters

The derivation of the above new distributed estimationmethod is based on the fact that the cost function we wantto optimize can be written as a summation of several localcomponent functions; each component function is related onlyto the measurements at the current sensor node. However,when the linear parameter is unknown, its ML estimate isobtained using the measurements fromall the sensor nodes, as in (10). Hence, the cost function, asgiven in (9), cannot be formulated as a summation of severallocal component functions. Therefore, the above proposed

distributed estimation method cannot be directly applied to thecases where is unknown.

A straightforward scheme to implement the proposed dis-tributed estimation method under such a situation is not to sepa-rate the linear parameters from the total parameters in the mea-surement model (7), i.e., we include into the unknown param-eter . Under this situation, the above proposed distributed esti-mation method can be directly applied when we replacewith another function, e.g., . However, in this scheme, wecannot benefit from the separation of linear and nonlinear pa-rameters in the measurement model, so the resulting computa-tion complexity and the volume of transmitted data are increasedsignificantly. Therefore, we propose the following more effi-cient estimation scheme for the case in which is unknown.

In this new scheme, we still separate the linear parameterfrom the total parameters. However, we reformulate the ML es-timation of and in (9) and (10) as the following iterativeprocess:

(40)

(41)

This iterative process generates a sequence of estimates

which converges to the ML estimates of and [24].According to this iterative process, the proposed new dis-

tributed estimation algorithm in Section IV–A can still be ap-plied after we introduce two new variables, a vector and amatrix , and make the following two modifications of the dis-tributed estimation algorithm in Section IV-A at sensor node .

1. Modify Step 2. New variable calculation as follows.

Calculate the new matrix as

(42)

then, update the matrix as

(43)

update the matrix as

(44)

update the vector as

(45)

2. Modify Step 3. Estimate update as follows:


Update and obtain the current estimate and as

(46)

(47)

D. Sensor-Node Scheduling

Sensor-node scheduling is an important strategy used in wire-less sensor networks. Through this scheduling, we can deter-mine the optimal scheme to allocate the limited resources avail-able in the wireless sensor network and potentially obtain highprocessing performance. In our proposed distributed estimationmethod, sensor-node scheduling involves two tasks:

1. Sensor node selection. We use an information utilitymeasure to determine which sensor node in the neighborof the current node we should select next and include itsmeasurements into our estimation task. This sensor nodeshould provide the information that has the most potentialto decrease the estimation uncertainty and increase theperformance.

2. Termination strategy. We need to find a proper estima-tion performance criterion to determine when we shouldstop including the new sensor nodes into our estimationprocess, such that we can obtain an optimal subset fromall available sensor nodes. The nodes in this subset shouldcarry the necessary and sufficient information for our esti-mation task.

The neighbor of the current sensor node is defined as a subsetof the sensor nodes that are located within the communicationrange of the current node. Note that in our sensor node selectionscheme we want to choose the next node that is the most infor-mative. However, consider a tradeoff between the communica-tion cost and the processing performance, we can only choosethe sensor nodes that are within certain radio range of the cur-rent node.

The measures for the estimation performance and the in-formation utility include: covariance-based measures, Fisherinformation matrix-based measures, entropy of estimationuncertainty, volume of high probability region, sensor ge-ometry-based measures, mutual information-based measures,and so on [4]. Different measures will be used according tothe sensor measurement models, the available computationcapabilities and energy resources, and the required estimationaccurate and processing time. In our method, we propose to usethe Cramér–Rao bound (CRB) as the estimation performanceand information utility measures. The reasons are: 1) Theproposed distributed estimation is based on the ML estimationwhich is asymptotically efficient, i.e., its variance attains theCRB asymptotically [30]. The CRB is the lower bound onthe variance of any unbiased estimators; it equals the inverseof the Fisher information matrix (FIM). Hence, we calculateour measures directly from the FIM. 2) We will find that thecalculation of the FIM as a performance measure is an intrinsicpart of the algorithm, i.e., they can be obtained without anyextra computation and data transmission. This is suitable for the

applications in wireless sensor networks. 3) When we use theFIM as an information measure to select the next sensor node,the FIM is computed by averaging over all possible values ofthe measurements from the next sensor node; therefore, thecurrent sensor node does not require the transmission of mea-surements from its neighbor and hence avoids communicatinguseless information. The similar idea has been used in [31] foradaptive electromagnetic-induction sensing of buried targets.

In order to compute the measures, we first derive a recur-sive process to update the FIM when information from the newsensor node is obtained. We denote the matrix in the se-quence as the FIM of the nonlinear param-eter when we collect the measurements from the first sensornodes, i.e., ; we have a similar notation for theFIM sequence for the linear parameter .According to the definition of FIM, we have

(48)

The second term on the RHS of (48) can be calculated as

(49)

where

(50)

Hence, we obtain the recursive equation for the FIM sequence

(51)

with . Similarly, we can obtain the recursive equationfor as

(52)

with . By using the recursive equations in (51) and(52), we can obtain measures for estimation performance andinformation utility. Here, we observe an interesting result thatwhen we compare (51) and (52) with (43) and (44) in the pro-posed distributed estimation algorithm, we find that the updatingformulas for the matrixes and are the same as the FIM

and , respectively. These equivalences represent theessential significance of the matrixes and used in ourmethod. It also shows that using FIM as the performance mea-sure will not increase the computation complexity and the re-quired transmission.


We can directly calculate the estimation performance mea-sure using the determinant or the trace of the FIM and

. At sensor node , we first update the FIMs according to(51) and (52); then we compare their determinants or traces witha predetermined threshold: if their determinants or traces equalor exceed the threshold, we will not include further measure-ments from the new sensor nodes into our estimation task.

For the sensor-node selection, among the neighbors of thecurrent sensor node, we want to choose a node that will pro-vide the most important information to reduce the estimationuncertainty. For this purpose, we use the following recursiveequations to predict the FIM when we include the measurementsfrom the next sensor node:

(53)

(54)

Since and are always the same for any sensor nodesin the neighbor of the current node , we can use only the ma-trixes and to calculatethe information utility measure. Here, and is a col-lection of the sensor nodes that are in the neighborhood of thecurrent sensor node . For example, if we use the trace of thematrix (denoted as ) as the information measure, we cancreate an information utility function:

(55)

where the parameter determines the tradeoff between the con-tributions from the FIM of and . Then, we will select thesensor node that maximizes this information utility functionas the next sensor node

(56)

Since we need the values of and to compute this information

utility, we can use the current estimates and to replacetheir true values.

V. NUMERICAL EXAMPLES

In this section we use numerical examples of localizing a dif-fusive source to analyze and demonstrate the performance of theproposed distributed estimation methods. The proposed basicdistributed estimation and its modifications can be summarizedas the following four methods.

1. Single local iteration implementation. This method is theproposed basic Gauss-Newton-based distributed max-imum likelihood estimation method in Section IV-A. Itprocesses a single iteration at each sensor node. Sincethis method is equivalent to an extended Kalman filter, wedenote it as the single iteration Kalman filtering (SIKF).

2. Multiple local iterations implementation. This is a modi-fication of the above basic distributed estimation method,as we presented in Section IV-B-2. In this algorithm, weprocess multiple iterations at each sensor node to improve

the estimation performance. We denote this method as themultiple iteration Kalman filtering (MIKF).

3. Iterated Kalman filter implementation with single localiteration. This is another modification of the basicdistributed estimation method, as we presented in Sec-tion IV-B-3. In this method, we process several iterationsover all the selected sensor nodes to increase the estimationaccuracy. At each sensor node, we apply a single iterationGauss-Newton step. Hence, we denote this method as theiterated Kalman filtering with single iteration (IKFSI).

4. Iterated Kalman filtering implementation with multiplelocal iterations. This method is similar to the IKFSI,except that at each sensor node, we process multiple localGauss-Newton iterations. Hence, we denote it as the iter-ated Kalman filtering with multiple iterations (IKFMI).

Now we use numerical examples to compare the performance ofthese four methods. Through these comparisons, we investigatethe properties of the proposed distributed estimation methods,and draw a conclusion concerning which method best fits theapplications in wireless sensor networks when considering bothestimation performance and consumption of the network re-sources, particularly energy.

In these numerical examples, we use the measurement modelfor a diffusive source in (6). Here, source position is theunknown we need to estimate. We apply the Levenberg-Mar-quardt method, i.e., instead of setting the initial value forto be 0, we let where is chosen according to variousscenarios to obtain a possible best performance. For the setup ofthe sensor network, we use 100 wireless sensor nodes randomlydeployed in a square area of 100 100 m. We define a pair ofneighboring sensor nodes whose distance is less than 20 m. Forthe diffusion model, we consider the environment as a homo-geneous semiinfinite medium with an impermeable boundary,which can represent dispersion in air above the ground. We usea scenario of a stationary impulse source located at position

meter. The bias term in (4) is g/m ,and the noise standard deviation is g/m . Wetake 10 temporal samples at each sensor node with a samplinginterval of 5 s. The other parameters , and are taken to be1 g/s, 20 m /s, and 0 s, respectively.

In the first example, we compare the estimation performanceof the SIKF method and the MIKF method. In Fig. 1 we showthe message transmission route (sensor-node selection route)when applying the MIKF method and the proposed sensor-nodescheduling scheme. We observe that even though the initialsensor node, located at meter, is relativelyfar from the source, the messages are still transmitted to theneighbor of the source. This demonstrates the optimality of thesensor node selection algorithm: according to the dispersionmechanism, the concentration near the diffusive source isrelatively high; hence, the data measured by the nodes close thesource have a high signal-to-noise ratio (SNR) and provide sig-nificant contributions to improve the accuracy of the distributedestimation method.

In Figs. 2 and 3 we compare the estimation biasand the log-determinant of the CRB matrix of with respect tothe number of the chosen sensor nodes for the methods SIKFand MIKF. We find that the estimation performance of MIKF is


Fig. 1. Message transmission routing for localizing a diffusion source in asquare area with 100 sensor nodes randomly placed. “Star” denotes the sourceposition; “square” denotes the initial sensor node; “circle” denotes the selectedsensor nodes.

Fig. 2. Estimation bias versus number of selected sensor nodes for methodsSIKF and MIKF.

much better than the SIKF, which means that multiple iterationsat each sensor node can greatly improve the accuracy of theestimation without increasing the communication burden.

In the second example, we compare the performance of theSIKF with the IKFSI. Here, the initial sensor node is located at

meter. In the IKFSI method, the first iteration isused to select a subset of the sensor nodes that carries the mostinformation. We set the threshold of the log-determinant of theCRB to be , and 8 sensor nodes around the source arechosen. The comparison results are shown in Figs. 4 and 5. Wefind that for the SIKF, the bias and the log-determinant of theCRB nearly do not decrease after 10 sensor nodes are included.This means the chosen sensor nodes provide little in-formation to improve the estimation performance. We also findthat the performance of the IKFSI is substantially better thanthe SIKF even though these two methods use the same trans-mission, which demonstrates that global multiple iterations overall the selected sensor nodes can decrease estimation bias andimprove performance. However, the cost is that the IKFSI con-sumes more communication energy to obtain this improvement.

Fig. 3. Log determinant of the CRB versus number of selected sensor nodesfor methods SIKF and MIKF.

Fig. 4. Estimation bias versus number of selected sensor nodes for methodsSIKF and IKFSI.

Fig. 5. Log determinant of the CRB versus number of selected sensor nodesfor methods SIKF and IKFSI.

In the last example, we compare the performance of theMIKF, the IKFSI, and the IKFMI methods. The comparisonresults are shown in Figs. 6 and 7. We observe that the estima-tion accuracy and performance of the MIKF is better than the


Fig. 6. Estimation bias versus number of selected sensor nodes for methodsMIKF, IKFSI, and IKFMI.

Fig. 7. Log determinant of the CRB versus number of selected sensor nodesfor methods MIKF, IKFSI, and IKFMI.

IKFSI, even though in the MIKF we use only a single iterationover the selected sensor nodes. And the performance of theMIKF is worse than the IKFMI; however, the difference is notvery large.

Note that the results from the above examples show not onlythe comparison between the performance but the comparisonbetween the required communications to reach a performancethreshold. According to our methods, the data transmissionbetween neighbored sensor nodes is the same. Therefore, thenumber of data transmission hops approximates the requiredcommunications (we assume the sensor nodes are uniformlydistributed in an area). For example, in Fig. 7, if we set thethreshold of the log-determinant of the CRB to be , thenthe MIKF and IKFMI methods need four transmission hops toreach this performance threshold, but the IKFSI method needs25 hops. This means the MIKF and IKFMI are more energyefficient than IKFSI.

To summarize the above results, when comparing the estima-tion performance of these four methods, we obtain the followingorder:

That is, the performance of the iterative Kalman filtering isbetter than the extended Kalman filtering implementation,and the multiple local iteration is better than the single localiteration implementation. When comparing the required com-munications to reach their performance limits, we obtain

Therefore, when considering both estimation performance andenergy consumption, the MIKF method has the most advantagesfor applications in wireless sensor networks, since it can obtaina high processing performance using relatively low communi-cation energy. By fully taking advantage of the local operationat each sensor node, we can obtain a substantial improvementin processing performance while still keeping energy consump-tion low.

VI. CONCLUSION

In this paper, we addressed the problem of developing anenergy-efficient distributed estimation method for applicationsin wireless sensor networks. We first proposed a statisticalmeasurement model in which we separated the linear andnonlinear parameters. This modelling strategy can reduce thecorresponding processing complexity. Based on this model,we analyzed the implementation of the maximum likelihoodestimation using the Gauss-Newton method and derived abasic framework of a fully distributed estimation method. Thismethod is motivated by the analysis of incremental least squaresmethods by Bertsekas [25]. In our method, we decreased theenergy usage while still maintaining an acceptable estimationperformance. We also proposed three modifications of the basicdistributed estimation method to improve processing perfor-mance. We introduced information-driven collaborative signalprocessing and proposed a sensor-node scheduling scheme toreduce further the communication energy consumption. Finally,we used numerical examples to study the performance of theproposed methods and determined that of the methods consid-ered here the multiple iterations Kalman filtering method isthe most suitable approach to be applied in the wireless sensornetwork, considering both the estimation accuracy and requiredcommunication burden. In future work, we will extend theproposed distributed estimation method to the dynamic signalmodel. We will study the distributed implementation of othernumerical methods, such as the full Newton-type methods. Wewill also investigate the application of the proposed distributedmethod to solve additional practical problems.

APPENDIX

PROOF OF THEOREM 4.1

Recall that the recursive equation (29) in the th iteration ofthe Gauss-Newton method is obtained by linearizing the non-linear signal model at the current estimate and applying thelinear least squares procedure to solve it. We denote the errorfunction as ,


where represents the Euclidean norm of a vector. Then wecan linearize this error function at as

(57)

Then, by applying the linear least squares procedure, we have

(58)

which is equivalent to (29).Next, according to Proposition 1 proved in [25], we have the

following results.Assuming that matrix is positive definite, the linear

least squares estimates

(59)

can be generated by the algorithm

(60)

where is an arbitrary vector, and the positive definite ma-trixes are generated by

(61)

with .Comparing the linear least squares problem (59) with the lin-

earized error function (57), we find that when we apply theidentifications

(62)

(63)

and let , we can generate using the incrementalalgorithm in Theorem 4.1.

REFERENCES

[1] S. Kumar, F. Zhao, and D. Shepherd, Eds., “Special issue on collabora-tive signal and information processing in microsensor networks,” IEEESignal Process. Mag., vol. 19, no. 2, Mar. 2002.

[2] I. F. Alyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Asurvey on sensor networks,” IEEE Commun. Mag., vol. 40, no. 8, pp.102–114, Aug. 2002.

[3] C.-Y. Chong and S. P. Kumar, “Sensor networks: Evolution, oppor-tunities, and challenges,” Proc. IEEE, vol. 91, no. 8, pp. 1247–1256,Aug. 2003.

[4] F. Zhao and L. J. Guibas, Wireless Sensor Networks: An InformationProcessing Approach. San Francisco, CA: Morgan Kaufmann, 2004.

[5] T. Zhao and A. Nehorai, “Localization of diffusive sources using dis-tributed sequential Bayesian methods in wireless sensor networks,”in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP),Toulouse, France, May 2006, vol. 4, pp. 985–988.

[6] L. Rossi, B. Krishnamachari, and C.C.J. Kuo, “Distributed param-eter estimation for monitoring diffusion phenomena using physicalmodels,” in Proc. 1st IEEE Int. Conf. Sensor and Ad hoc Commun.Networks (SECON), Santa Clara, CA, Oct. 2004, pp. 460–469.

[7] Y. Levinbook and T. F. Wong, “Maximum likelihood diffusive sourcelocalization based on binary observations,” in Proc. 38th AsilomarConf. Signals, Syst. Comput., Nov. 2004, vol. 1, pp. 1008–1012.

[8] R. R. Brooks, P. Ramanathan, and A. M. Sayeed, “Distributed targetclassification and tracking in sensor networks,” Proc. IEEE, vol. 91, no.8, pp. 1163–1171, Aug. 2003.

[9] A. T. Ihler, J. W. Fisher, R. L. Moses, and A. S. Willsky, “Nonpara-metric belief propagation for self-localization of sensor networks,”IEEE J. Sel. Areas Commun., vol. 23, no. 4, pp. 809–819, Apr. 2005.

[10] A. T. Ihler, J. W. Fisher, and A. S. Willsky, “Particle filtering undercommunications constraints,” in Proc. IEEE/SP 13th Wkshp. Statist.Signal Process., Jul. 2005, pp. 89–94.

[11] J. L. Williams, J. W. Fisher, and A. S. Willsky, “Optimization ap-proaches to dynamic routing of measurements and models in a sensornetwork object tracking problem,” in Proc. IEEE Int. Conf. Acoust.,Speech, Signal Process. (ICASSP), Mar. 2005, vol. 5, pp. 1061–1064.

[12] M. Coates, “Distributed particle filters for sensor networks,” in 3rd Int.Symp. Inf. Process. Sensor Netw. 26–27 Apr. 2004, 2004, pp. 93–107.

[13] J. Liu, J. Reich, and F. Zhao, “Collaborative in-network processingfor target tracking,” EURASIP J. Appl. Signal Process., vol. 4, pp.378–391, Mar. 2003.

[14] M. Chu, H. Haussecker, and F. Zhao, “Scalable information-drivensensor querying and routing for ad hoc heterogeneous sensor net-works,” Int. J. High-Performance Computing Applicat., vol. 16, no. 3,pp. 90–110, 2002.

[15] B. H. Cheng, R. E. Hudson, F. Lorenzelli, L. Vandenberghe, and K.Yao, “Distributed Gauss-Newton method for node localization in wire-less sensor networks,” in Proc. IEEE 6th Wkshp. Signal Process. Adv.Wireless Commun., Jun. 2005, pp. 915–919.

[16] L. Xiao, S. Boyd, and S. Lall, “A space-time diffusion scheme forpeer-to-peer least-squares estimation,” in Proc. 5th Int. Conf. Inf.Process. Sensor Netw. (IPSN), Apr. 2006, pp. 168–176.

[17] M. Rabbat and R. Nowak, “Distributed optimization in sensor net-works,” in Proc. 3rd Int. Symp. Inf. Process. Sensor Netw. (IPSN), Apr.2004, pp. 20–27.

[18] D. Blatt and A. Hero, “Distributed maximum likelihood estimationfor sensor networks,” in Proc. IEEE Int. Conf. Acoust., Speech, SignalProcess. (ICASSP), May 2004, vol. 3, pp. 929–932.

[19] R. D. Nowak, “Distributed EM algorithm for density estimation andclustering in sensor networks,” IEEE Trans. Signal Process., vol. 51,no. 8, pp. 2245–2253, Aug. 2003.

[20] F. Zhao, J. Shin, and J. Reich, “Information-driven dynamic sensorcollaboration,” IEEE Signal Process. Mag., vol. 19, no. 2, pp. 61–72,Mar. 2002.

[21] T. Zhao and A. Nehorai, “Detecting and estimating biochemical dis-persion of a moving source in a semi-infinite medium,” IEEE Trans.Signal Process., vol. 54, no. 6, pp. 2213–2225, Jun. 2006.

[22] A. Nehorai, B. Porat, and E. Paldi, “Detection and localization ofvapor-emitting sources,” IEEE Trans. Signal Process., vol. SP-43, pp.243–253, Jan. 1995.

[23] A. R. Gallant, Nonlinear Statistical Models. New York: Wiley, 1987.[24] G. A. F. Seber and C. J. Wild, Nonlinear Regression. New York:

Wiley, 1989.[25] D. P. Bertsekas, “Incremental least squares methods and the extended

Kalman filter,” SIAM J. Optimiz., vol. 6, no. 3, pp. 807–822, 1996.[26] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained

Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Pren-tice-Hall, 1983.


[27] R. P. Wishner, J. A. Tabaczynski, and M. Athans, “A comparison ofthree non-linear filters,” Automatica, vol. 5, pp. 487–496, 1969.

[28] B. M. Bell and F. W. Cathey, “The iterated Kalman filter update as aGauss-Newton Method,” IEEE Trans. Autom. Control, vol. 38, no. 2,pp. 294–297, Feb. 1993.

[29] B. M. Bell, “The iterated Kalman smoother as a Gauss-Newtonmethod,” SIAM J. Optimiz., vol. 4, no. 3, pp. 626–636, Aug. 1994.

[30] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[31] X. Liao and L. Carin, “Application of the theory of optimal experimentsto adaptive electromagnetic-induction sensing of buried targets,” IEEETrans. Pattern Anal. Machine Intell., vol. 26, no. 8, pp. 961–972, Aug.2004.

Tong Zhao (S’02–M’07) received the B.Eng. andM.Sc. degrees in electrical engineering from theUniversity of Science and Technology of China(USTC), Hefei, in 1997 and 2000, respectively. Hereceived the Ph.D. degree in electrical engineeringfrom Washington University, Saint Louis, MO, in2006.

Currently, he is a Postdoctoral Research Associatewith the Department of Electrical and ComputerEngineering, University of California, Davis. Hisresearch interests are adaptive and statistical signal

processing, including detection and estimation theory, optimization, distributedsignal processing, sequential Bayesian inference, Monte Carlo methods, etc.,and their applications in sensor arrays, communications, wireless sensornetworks, and biochemical sensors.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom the Technion, Israel, and the Ph.D. degreein electrical engineering from Stanford University,Stanford, CA.

From 1985 to 1995, he was a faculty memberwith the Department of Electrical Engineering, YaleUniversity, New Haven, CT. In 1995, he joined theDepartment of Electrical Engineering and ComputerScience, University of Illinois at Chicago (UIC), asFull Professor. From 2000 to 2001, he was Chair of

the Department of Electrical and Computer Engineering (ECE) Division, whichthen became a new department. In 2001, he was named University Scholar ofthe University of Illinois. In 2006, he became Chairman of the Department ofElectrical and Systems Engineering at Washington University, St. Louis, MO.He is the inaugural holder of the Eugene and Martha Lohman Professorshipand the Director of the Center for Sensor Signal and Information Processing(CSSIP) at WUSTL since 2006.

Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING during 2000 to 2002. From 2003 to 2005, he was Vice President(Publications) of the IEEE Signal Processing Society, Chair of the PublicationsBoard, member of the Board of Governors, and member of the ExecutiveCommittee of this Society. From 2003 to 2006, he was the founding editor ofthe special columns on Leadership Reflections in the IEEE Signal ProcessingMagazine. He was corecipient of the IEEE SPS 1989 Senior Award for BestPaper with P. Stoica, coauthor of the 2003 Young Author Best Paper Award,and corecipient of the 2004 Magazine Paper Award with A. Dogandzic. He waselected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005 andreceived the 2006 IEEE SPS Technical Achievement Award. He is the PrincipalInvestigator of the new multidisciplinary university research initiative (MURI)project entitled Adaptive Waveform Diversity for Full Spectral Dominance. Hehas been a Fellow of the Royal Statistical Society since 1996.

ieee transactions on signal processing, vol. 55, no. 9 ...nehorai/paper/04102539.pdf · ically,...

Documents