IEEE TRANSACTIONS ON SIGNAL PROCESSING, …broadcast their data to all other sensors, while sensors with less informative data remain inactive. This way high-quality (i.e., low-MSE)

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 8, AUGUST 2009 3193

Power-Efficient Dimensionality Reductionfor Distributed Channel-Aware Kalman

Tracking Using WSNsHao Zhu, Student Member, IEEE, Ioannis D. Schizas, Student Member, IEEE, and

Georgios B. Giannakis, Fellow, IEEE

Abstract—Estimation of nonstationary dynamical processesis of paramount importance in various applications includingtarget tracking and navigation. The goal of this paper is to per-form such tasks in a distributed fashion, using data collected atpower-limited sensors which either communicate with a fusioncenter (FC) over noisy links, or, communicate with each other overnonideal channels in an ad hoc setting. In FC-based wireless sensornetworks (WSNs) with a prescribed power budget, linear dimen-sionality reducing operators which account for the sensor-to-FCchannel are derived per sensor to minimize the mean-squareerror (MSE) of Kalman filtered state estimates formed at the FC.Using these operators and state predictions fed back from the FConline, sensors reduce the dimensionality of their local innovationsequences and communicate them to the FC where tracking esti-mates are corrected. Analytical and numerical results advocatethat the novel channel-aware distributed tracker outperformscompeting alternatives. In ad hoc WSNs deployed to perform dis-tributed tracking, one sensor broadcasts reduced-dimensionalitydata per time slot, according to a prespecified transmission order.The dimensionality reducing operators employed by the broad-casting sensor are selected to meet its transmit-power budget,while minimizing the state estimation MSE of the sensor with thelowest receiving SNR. Based on the received reduced-dimension-ality data from the broadcasting sensor, every sensor in rangeperforms the MSE optimal tracking. Corroborating distributedtarget tracking simulations based on distance-only observationsillustrate that the novel scheme provides sensors with accurateestimates at affordable communication cost.

Index Terms—Distributed tracking, wireless sensor networks(WSNs), Kalman filtering, target tracking.

I. INTRODUCTION

W ITH the advent of resource-limited wireless sensor net-works (WSNs), distributed estimation and tracking of

(non)stationary processes using fusion center (FC) based or ad

Manuscript received October 23, 2008; accepted February 16, 2009. Firstpublished April 10, 2009; current version published July 15, 2009. The asso-ciate editor coordinating the review of this manuscript and approving it forpublication was Prof. Mounir Ghogho. Work in this paper was supported bythe U.S. DoD ARO Grant W911NF-05-1-0283; and also through collaborativeparticipation in the Communications and Networks Consortium sponsored bythe U.S. Army Research Laboratory under the Collaborative Technology Al-liance Program, Cooperative Agreement DAAD19-01-2-0011. The U. S. Gov-ernment is authorized to reproduce and distribute reprints for Government pur-poses notwithstanding any copyright notation thereon. This paper was presentedin the Fourteenth Workshop on Statistical Signal Processing, Madison, WI, Au-gust 26–29, 2007.

The authors are with the Department of Electrical and Computer Engineering,University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2009.2020748

hoc WSN topologies have drawn a lot of interest recently [13],[23]. In order to save power and bandwidth, existing approachesrely on reduced-dimensionality analog-amplitude or quantizeddata to track the dynamical process of interest [6], [19]; see also[22] where compression is performed via wavelets. In additionto saving communication resources, dimensionality reductionis well motivated because the sampling rate at a sensor maybe different from the data rate supported by the sensor-to-FCor the inter-sensor communication links. In this case, dimen-sionality-reducing matrices are useful for matching the sam-pling rate, dictated by a desirable accuracy in estimation, withthe data rate that can be supported by the channel. However,transmit-power as well as the fading and receiver noise have notbeen fully accounted by existing approaches. Furthermore, ex-cept for [6], estimation based on reduced-dimensionality sensordata has dealt primarily with stationary signals [20], [23], [24].A recent alternative for data compression using WSNs relies oncompressive sampling of sparse signals. The latter is effectedvia random projections [14]; see also [17] where gossiping al-gorithms are developed for computing random projections in adistributed fashion.

In this paper, power-efficient and channel-aware Kalman fil-tering (KF)-based tracking is pursued for signals that are notnecessarily sparse based on analog-amplitude reduced-dimen-sionality multi-sensor data in both FC-based and ad hoc WSNs.In FC-based topologies, optimal linear dimensionality-reducingoperators (matrices) are derived to compress sensor data andminimize the MSE of state estimates at the FC, while adheringto power constraints prescribed at each sensor. An attractive fea-ture of this FC-based approach is the utilization of feedbackfrom the FC to the sensors which allows them to remove re-dundancy from their observations, lower the dimensionality oftheir innovation processes and thus gain in power efficiency.Without channel-aware dimensionality reduction, power sav-ings brought by FC feedback have also been studied in [12] fortracking an auto-regressive process. Our novel approach here in-cludes as a special case our earlier work in [20] for distributedestimation of stationary signals, while it differs from [6] becausebeing channel aware it accounts for fading and additive recep-tion noise.

Distributed tracking based on reduced-dimensionality datais also considered for ad hoc WSNs. In this setup, trackingschemes are developed to gain resilience to FC failures. Theresultant algorithms scale better than those developed forFC-based WSNs, as the number of sensors increases. Amongthe first distributed KF-based trackers using sensors commu-nicating any-to-any over ideal links was the one in [18]. Other

1053-587X/$25.00 © 2009 IEEE

Authorized licensed use limited to: University of Minnesota. Downloaded on July 21, 2009 at 21:21 from IEEE Xplore. Restrictions apply.

3194 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 8, AUGUST 2009

existing distributed KF approaches reduce the communicationcost either by allowing only single-hop communications amongneighboring sensors and utilizing consensus-based techniques[15], [21], or, by having sensors quantize their data before trans-mission to all other sensors [19]. Instead of quantization, theapproach here pursues channel-aware dimensionality reductionof analog-amplitude observations. Specifically, a single sensorbroadcasts per time slot its reduced-dimensionality data to allsensors in its range, which rely on the received informationto perform MSE optimal tracking. Such an approach enableseven sensors with less informative observations to acquirehigh-quality data from active sensors, and boost their trackingperformance. The dimensionality-reducing operator employedby the broadcasting sensor to compress its data is chosen tominimize the state estimation MSE at the sensor with the worstreception signal-to-noise ratio (SNR).

The order in which sensors broadcast their reduced-di-mensionality data can be selected to enhance the trackingperformance. This order can be for instance set so that sensorswith high quality observations (e.g., sensors closer to a target)broadcast their data to all other sensors, while sensors withless informative data remain inactive. This way high-quality(i.e., low-MSE) estimates can be formed across the WSN.An overview of sensor selection schemes for target trackingcan be found in [25], where the information content of eachsensor’s data is quantified using appropriate information utilityfunctions. Then, the sensor with most informative observationsis chosen to broadcast its data to all other sensors in range.

After stating some preliminaries and formulating theproblem in Section II, the reduced-dimensionality KF schemefor FC-based WSNs is developed in Section III. Specifically,in Section III-A MSE optimal channel-aware linear dimen-sionality-reducing operators are designed for a single-sensorsetup. Interestingly, it turns out that these operators allocatepower optimally across the reduced-dimensionality data in awater-filling like manner. Determining the MSE optimal oper-ators in a multi-sensor setup is known to be NP-hard [13]. Forthis multi-sensor setup, a block coordinate descent algorithmis developed with guaranteed convergence to a stationary point(Section III-B), while some interesting connections are madebetween reduced-dimensionality Kalman filtering and canon-ical correlation analysis (CCA) (see, e.g., [5]) in Section III-C.Corroborating simulations in Section III-D advocate the meritsof the novel FC-based tracking scheme. Complementary dis-tributed reduced-dimensionality tracking algorithms for adhoc WSNs are developed in Section IV, where more emphasisis placed on robustness and power efficiency. Different fromthe FC-based scenario, here it becomes essential to designthe reduced-dimensionality KF separately for the cases ofideal (Section IV-A) and nonideal sensor links (Section IV-B).The reduced-dimensionality distributed KFs developed forFC-based and ad hoc topologies are compared in terms ofestimation performance in Section IV-C. Finally, concludingremarks are offered in Section V.

II. PROBLEM STATEMENT AND PRELIMINARIES

Consider a WSN with sensors deployed to esti-mate a dynamical nonstationary process . The goalis to estimate based on reduced-dimensionality distributedsensor data for two WSN configurations. The first configuration

Fig. 1. Distributed setup for estimating a dynamic stochastic process ��using FC-based WSNs.

Fig. 2. Distributed setup for estimating a dynamic stochastic process ��using ad hoc WSNs.

depicted by the topology in Fig. 1 comprises sensors that arelinked with an FC, where information gathering and trackingof takes place. The second configuration consists of sen-sors without an FC (ad hoc topology), where tracking ofis accomplished via information exchanges among neighboringsensors; see Fig. 2.

The state of interest , assumed to be obtained by sam-pling uniformly the corresponding continuous-time waveform,obeys the discrete-time difference equation

(1)

where denotes the state transition matrix anddenotes zero-mean additive white Gaussian noise

(AWGN) with covariance matrix . Sensor observes

(2)

where and the AWGN iszero-mean with (cross-) covariance matrix

, and uncorrelated with . Different from[19], the observation noise is allowed to be correlated acrosssensors. Each vector at sensor can be formedby stacking scalar observations collected by sampling fasterthan the rate varies.

A. Fusion Center-Based WSNs

In the envisioned FC-based WSN, sensor relies on amatrix with to form the reduced-dimension-

ality vector , whereis a vector subtracted from to save transmission power.(It will turn out that is in fact a predictor ofbased on data to be specified soon.) Furthermore, the followingare assumed.


ZHU et al.: POWER-EFFICIENT DIMENSIONALITY REDUCTION 3195

a1) The -to-FC link comprises a full rank channelmatrix along with zero-mean AWGN at the FC withcovariance , which is uncorrelated withand across channels. Sensors transmit over orthogonal channelsso that the FC can receive separately the vectors

(3)

a2) The covariance matrices aswell as matrices and , and channel matrices

and are known at the FC.a3) The feedback channel from the FC to the sensors is ideal.Assumption a1) is satisfied by analog or digital frequency-di-

vision multiplexing, e.g., multicarrier transmissions, whereeach entry of the reduced dimensionality vector rides on asubcarrier with nonzero channel gain. Time-invariance of thechannels is justified when the sensors are static.Matrices in a2) can be estimated at the FC usingtraining symbols as detailed in, e.g., [3, p. 383]. The matrices

and can be obtainedfrom the physics of the problem, i.e., using standard kinematicmodels with constant velocity or acceleration that are well doc-umented in the tracking literature; see e.g., [2, Ch. 6]. Finallya3), that is also assumed by [6], is reasonable since the FChas sufficient resources to mitigate errors in the FC-to-sensorchannel through powerful error control codes.

The FC concatenates , for , to form [cf. (3)]

(4)

where denotes a block diagonal matrix,, and likewise for the aggregate predictor

and the FC noise vector .The goal of the FC is to track using the received data

in (4). Specifically, the FC seeks the MMSE optimalstate estimate , recursively(via a KF) based on the observations given by (4). To this end, itis well known that the filtered estimate can be obtainedfrom the predictor aftercorrecting it using the innovations

(5)

i.e., see, e.g., [11, (13.61)]

(6)

The correction term in (6) is a function of the innovationwhich clearly depends on the selection of the local

predictor , as summarized by the following lemma(see Appendix A for the proof).

Lemma 1: If ,then ; thus .

The choice of in Lemma 1 enables each sensorto save considerable power because needs to transmit only

the nonredundant local innovation, which has much smaller variance than its raw data

. In addition, this innovation that is received at the FC, ascoincides with , is all that the FC needs to

implement the correction step (6) of the KF. Since from (2) itholds that , the FC must feedback the predictor for sensor to form[cf. a3) and Fig. 1].

To proceed with the dimensionality reduction task, note thatGaussianity and uncorrelatedness of and imply that

in (6) is a linear function of, i.e., , where

denotes the linear operator applied to the received databy the FC. Further, it follows from (4) and (6) that

depends on both and . Thus, for a prescribedaverage transmit-power per sensor the constrained opti-mization problem of interest is

(7)

where denotes the covariance matrix of .Besides transmission, sensors consume power also in the receivemode. The latter is not minimized here but if denotes av-erage receive power and the total power per sensor, theprescribed transmit power in (7) is given by .

Postponing the solution of (7) until Section III, it is usefulto outline the envisioned communication protocol that is as-sumed to rely on orthogonal access, e.g., TDMA. Supposingthat clock/timing synchronization has been established, eachTDMA slot (time index) is divided into subslots.During the first subslot, the FC broadcasts to all receiving sen-sors ; and in the next subslots the FC feeds backone after the other for , each to the corre-sponding sensor . Because each sensor’s sensing unit is de-coupled from its transceiver unit, over these first subplotsthe sensors acquire observations and each, say the th, forms

. Over the last subslots, each sensor transmits tothe FC in a round-robin fashion its reduced dimensionality data,e.g., transmits .

B. Ad Hoc WSNs

Relative to FC-based WSNs, the distinct feature of ad hocWSNs is that without a central processing unit the network it-self is in charge of tracking the state . As a result, distributedtracking algorithms can be obtained that are robust to FC fail-ures. Similar to [19], communication among sensors takes placeover a shared wireless channel that can afford transmission of asingle message per time slot. Specifically, at time slot thereis a single transmitting (broadcasting) sensor which, via a

matrix with , forms and broadcasts toall other sensors in its range the reduced-dimensionality vector

, where is a function of ’s observationsthat will be specified later in Section IV; see also Fig. 2. Themapping between time slot and sensor index is consideredgiven, and can be determined using existing sensor selection(a.k.a. scheduling) schemes [10]. Further, it is assumed that:

a4) The broadcasting channel from to , withand , is characterized by a

full rank channel matrix and a zero-mean AWGNwith covariance matrix , which is uncorrelated with



and across channels. At time slot , sensorreceives the vector

(8)

a5) Covariance matrices , andmatrices , as well as channel matrices

and are assumed available to everysensor.

As explained in Section II-A, the matrices in a5) can beobtained from the physics of the problem, or during a trainingphase. These matrices are acquired offline to enable onlinetracking of . For simplicity in exposition and since thereis a prescribed mapping between time slot index and broad-casting sensor index , in the subsequent analysis we omit thesuperscript index in .

Each sensor wishes to track by forming the MMSEestimate of given . This will be accomplishedthrough Kalman filtering at each sensor. The state estimate of

will be obtained at sensor using the predictionand the received data as

(9)

where . Unlike (6), thepredictor-corrector (9) for ad hoc WSNs is sensor specific.

Note that [and thus ] is a function ofwhile is again, due to Gaussianity, a linearfunction of the form . One way to determine

at the broadcasting sensor could be to solve per slotthe minimization problem

(10)

However, (10)wouldthenyieldacompressionmatrix thatdependsonthereceivingsensor with , thoughthebroad-casting sensor can only afford broadcasting with a commondimensionality-reducingmatrixoverall channel linkswith there-ceiving sensors . Note that determining ina closed form even for a fixed , appears to be impossible.

In order to bypass these difficulties, will be deter-mined in order to minimize the MSE between and its re-construction obtained at sensor , where

is the sensor whose reception link from is the worst inthe sense that it has the weakest receiving SNR; i.e.,

(11)

Thus, for a prescribed power at broadcasting sensor ,matrix will be determined as the optimal solution of

(12)

where denotes the covariance matrix of atthe broadcasting sensor .

Remark 1: In both WSN settings KF-based tracking al-gorithms will be developed based on reduced-dimensionalitysensor data. In FC-based WSNs, the FC must feed back tothe sensors and per time slot, thatis scalars. The extra parameters required

relative to [6] is a relatively small increase in the feedbackoverhead especially in target tracking applications where typ-ically . Thus, [6] and the approaches considered hereessentially incur comparable feedback overhead. On the otherhand, both approaches in the present paper remain applicableeven if the observation and channel noise vectors do not adhereto a Gaussian distribution-case in which the resultant KF yieldsthe linear (L-) MMSE optimal estimates of .

Remark 2: Although exposition here deals with dimension-ality reduction of analog-amplitude data, the model considereddoes not exclude digital communications. In fact, dimension-ality reduction of analog-amplitude samples is the stage pre-ceding rate-optimal bit loading for a quadratic distortion metricof a source coder, e.g., [7, p. 345]. From another perspective,digital transmissions can be viewed as analog ones with quan-tization errors lumped in the additive (generally non) Gaussiannoise terms. To ensure that the noise terms are temporarilyuncorrelated, the number of quantization bits per sensor andtime slot must be sufficiently large.

Remark 3: The minimization problems in (7) and (12), arerelated to those in [20]. However, the system and data settingshere are different from those in [20]. Specifically, [20] deals withdistributed parameter estimation of stationary signals whereasthe setup here entails distributed state tracking for nonstationarysignals.Fromthispointofview, theapproachheresubsumes[20].Finally, the present setup addresses both FC-based and ad hoctopologies, whereas only FC-based WSNs are considered in [20].

III. REDUCED-DIMENSIONALITY KF IN FC-BASED WSNS

State estimation is dealt with in this section using reduced-dimensionality data transmitted from distributed sensors to anFC. The operation of the reduced-dimensionality tracker will bedescribed first, followed by the development of optimal matrices

and .Supposing that all quantities at step are available, the

FC relies on to obtain the state predictor

(13)

and on the estimation error covariance matrix

to find the covariance matrix of the state innovation (predictionerror) as

(14)

With MSE optimal and assumed determinedas described in the ensuing section, recall that the FC feeds backto sensor the matrix and the vector .

Using the feedback and its local observation, sensor formsthe innovation

and transmits to the FC .The FC receives each separately as in (3) and relies on

to obtain the filtered estimate (corrector)

(15)

where represents the columns of matrix withindexes through and

. (Recall also that .)Matrix represents the KF gain. Further, as ischosen so that is the (L)MMSE estimator of based



on , the orthogonality principle and the linearity of theexpected value operator imply that the filtered error covariancematrix (ECM) can be updated as

(16)

where the second equality comes from the fact that the estima-tion error is uncorrelated with the received data

.

A. Dimensionality-Reducing Matrices for a Single-SensorSetup

Before tackling the multi-sensor setup, consider first the de-sign of MSE optimal dimensionality reducing matrix fora WSN with a single sensor, which turns out to be a basic stepfor the multi-sensor problem. Using (4) with the opti-mization problem in (7) can be rewritten as follows [cf. (15)]

(17)

After straightforward algebraic manipulations, the Lagrangianfunction associated with (17) can be expressed as

(18)

where is the corresponding Lagrange multiplier andis the minimum

MSE achieved when estimating based onwithout channel distortion and additive noise at the FC; and

for any

(19)

(20)

The Lagrangian function in (18) resembles that in [20, eq.(24)], which deals with distributed estimation of stationary sig-nals via reduced-dimensionality observations. However, it is im-portant to stress that here distributed state estimation pertains tononstationary processes.

In order to obtain and , we will first sim-plify the Lagrangian in (18) using appropriate matrix de-compositions. To this end, consider the SVD (see e.g.,[9]) , as well as theeigen-decompositions and

, wherewith and

with

. Notice that captures the SNR of the thentry in the received vector at the FC. Further, de-fineand let , while ma-trix can be factorizedas , with

and. Moreover, let

denote the invertible matrix which simultaneously diagonalizesthe matrices and .

Relying on these matrices and their corresponding decompo-sitions it is possible to establish the following proposition (theproof follows using the arguments in [20, App. G and H]).

Proposition 1: Under a1), a2) with , and for, the optimal matrices and

that minimize (7) are

(21)

where the matrix is diagonal with diag-onal entries given by

(22)

Index is the maximum integer in for whichare strictly positive; and is the Lagrange multiplier, associ-ated with and , given by

(23)

Although and are all functions of , the timeindex is dropped for notational brevity.

The optimal matrices and obtained in Propo-sition 1 are given in closed form as a function of the stateand observation model parameters and

, as well as the channel matrix , the received noisecovariance and the transmission power .

Remark 4: Intuitively, the optimal in Proposition 1selects the entries of in which is strongestand the channel imperfections weakest, and allocates poweramong them in a water-filling like manner. It is also worthmentioning that (22) dictates a minimum power per sensor.Indeed, in order to ensure that , it must hold that

, which implies that the power must satisfy

(24)

Notice that when the sensor power is relatively low, willbe equal to one no matter how many orthogonal channels areavailable. This way transmits the most informative entry of

(associated with ), using the most reliablechannel (associated with ). On the contrary, if the sensorhas much more available power, is allocated judiciously toall the available channels so that the error between and



is as small as possible. Besides, when there isno improvement in the MSE obtained in (17), i.e., forany . This is exactly the reason why dimensionality re-duction is considered within the interval in Proposition 1.

The computations involved in obtaining andper time slot , include those needed for: i) evaluating the SVDof and , [cf. (21)] and ii) performingthe necessary matrix multiplications. Thus, the complexity forevaluating is per time step [9, Sec. 5.4].

B. Dimensionality-Reducing Matrices for the MultisensorSetup

In the multi-sensor scenario it turns out that the minimizationproblem in (7) does not lead to a closed-form solution, and ingeneral incurs complexity that grows exponentially with [13],[20]. For this reason, we resort to a block coordinate descent al-gorithm which converges at least to a stationary point of the costin (7). For the -sensor setup, the cost function in (7) becomes[cf. (15)]

(25)

Specifically, suppose temporarily that matricesand are fixed and satisfy

the power constraints , forand . Upon defining

the cost in (25) can be written as

(26)

which is a function of and . Interestingly, (26)falls under the realm of Proposition 1. This means that when

and are given, the matricesand minimizing (26) under the power constraint

can be directly obtainedfrom Proposition 1, after setting and

in (17).In order to apply the corresponding matrix decomposition, it

is also necessary to update the covariance matrices as follows

(27)

(28)

In summary, the following proposition has been established:Proposition 2: Under a1) and a2) and for given

matrices and satis-fying , the optimal

and matrices minimizing (26) subject toare provided by Propo-

sition 1, after setting and, and using the (cross-)

covariances in (27) and (28).Based on Proposition 2, a block coordinate descent algorithm

follows whereby the FC determines in an alternating fashion(successively for ) the matrices and ,which are guaranteed to converge at least to a stationary pointof (7). This round-robin scheme is tabulated as Algorithm 1.

Algorithm 1 Fusion Center: Solving for Optimal Matrices

1: Initialize randomly and sothat .

2: for do3: for do4: Given ,

5: ,

determine using Proposition 2.6: end for7: if

for aprescribed

8: break9: end if

10: end for

Remark 5: The KF in this subsection relies on reduced-di-mensionality sensor data and is both channel- and power-aware.The MMSE optimal dimensionality-reducing matrices in thesingle-sensor setup as well as those in the multi-sensor case se-lect the most ‘informative’ entries of and com-municate them to the FC through the most reliable subchannelsof . On the other hand, the channel-unaware approach in [6]is challenged by error propagation when AWGN is present inthe sensor-to-FC channels. Further, the feedback link from theFC to sensors enables forming and reducing the dimensionalityof the innovation that has much smaller dynamicrange than , and thus effects transmit-power savings. Suchsavings are not available in [6] or [20] where raw sensor obser-vations are compressed directly without preprocessing. It is alsoworth mentioning that the novel reduced-dimensionality KF of-fers a neat generalization of the stationary results in [20] to non-stationary Markov processes.

The reduced-dimensionality KF scheme is run at the FCto track the dynamical process , and is summarized asAlgorithm 2-A. The associated computational complexity forimplementing Algorithm 1 is per iteration. Thedata acquisition and dimensionality reduction carried out atthe sensors is summarized under Algorithm 2-B, where each



sensor only needs to perform a matrix-vector multiplicationoperation that incurs computational complexity . Notethat at each time-step sensors have to receive from the FC ,which certainly consumes power. However, matricesdepend only on the covariance of the sensor data and thatof . With matrices and

available, can thus be determined and stored atthe sensors offline pretty much as KF covariance iterations canbe run offline. In this case, transmission of the dimensionality-reducing matrices from the FC to the sensors is not an issue.

Algorithm 2-A Fusion Center: Reception and Estimation

1: Initialize prior estimate and ECM2: for do3: Compute and using (13)

and (14)4: Compute optimal matrices and

using Algorithm 15: Transmit and to sensor for

6: Receive from sensor transmissions7: Compute and using (15) and (16)8: end for

Algorithm 2-B Sensor : Observation and Transmission

1: Observes and receives the feedbackand from FC

2: Forms usedto construct the innovations sequencethereafter.

3: Forms and transmits this to the FC

C. Reduced-Dimensionality KF With Ideal Links

If sensors utilize powerful error control codes, then theadverse effects of nonideal links can be mitigated and thesensor-to-FC channel links can be considered essentially ideal.In this case, it is interesting to examine how the reduced-di-mensionality KF schemes developed in Sections III-A andIII-B specialize to ideal channels. To this end, let us replaceassumption a1) with:a1’) The link from sensor to the FC is ideal, so that the FCreceives separately the vectors

Notice that a1’) boils down to a1) if the multiplicative fadingmatrix is set equal to the identity matrix , andthe noise covariance matrices are set equal to . The op-timization problem in (7) then becomes

(29)

The power constraint per sensor in (7) is not necessary forideal sensor-to-FC links. This holds because the requiredtransmission power can be made negligible by scaling theoptimal in (29) with an arbitrarily small factor andcorrespondingly multiplying by without inducingloss in performance.

The optimal solution for the minimization problem in (29),obtained from (7) under the assumption of ideal sensor-to-FClinks and after setting , can be easily found if (29) isviewed as a canonical correlation analysis (CCA) problem.From this perspective, it follows readily from ([5], p. 368) that:

Proposition 3: For and , the optimalmatrices at the sensor and at the FC that minimize(29) under a1’) and a2), are

(30)

where is any invertible matrix andis the orthonormal matrix formed by the eigenvectors cor-responding to the largest eigenvalues of

.Interestingly, the results of Proposition 1 generalize

the noise-free CCA analysis in Proposition 3 to nonidealsensor-to-FC links. Recall that in the classical CCA setup,estimation of is performed using noiseless reduced-di-mensionality data which is not the case inour setup. Specifically, it is possible to establish that (seeAppendix B for the proof):

Corollary 1: Proposition 1 generalizes the CCA analysissummarized in Proposition 3 to nonideal channel links; i.e., theoptimal matrices in (21) coincide with (30) if and

.Clearly, Proposition 1 allows determination of the MSE op-

timal dimensionality-reducing operator and the Kalmangain , even when fading and noise are present. Thesecannot be handled by the classical CCA approach, especiallyfor the nonstationary time-varying setup considered here. Fur-ther, notice that if fading and reception noise are not taken intoaccount, estimation errors can accumulate. This is the case withthe channel-unaware approach in [6], as will be confirmed withsimulated tests presented next. Also, if and the dimen-sionality of is reduced down to , the reduced-dimensionalityKF coincides with the classical one and there is no loss in MSEperformance.

D. Simulations

Here we test the performance of our channel-aware re-duced-dimensionality KF and compare it with [6]. Considera WSN deployed for measuring e.g., room temperature. Acommon state zero-acceleration propagation model is adoptedfrom [19], where the temperature contained in the first entry of

evolves according to

with and covariance matrix



Fig. 3. Trace of the correction ECM at the FC versus time � for (a) � � ��,and (b) � � �.

The WSN comprises sensors, where each sensor ac-quires temperature measurements. The observation matrix

has dimensionality with its first column equalto the all-one vector and the second one , where ischosen randomly within the range ; andfor . Each sensor reduces the dimensionalityof to . Furthermore, sensors here transmitonly scalar data over scalar channels with , while the re-ception noise variance is set so that the

is equal to 20 dB for . Whenthe reduced dimension is , it turns out that the ma-trices obtained by either Proposition 1 for , or, Algo-rithm 1 have the form , where is chosen tosatisfy the power constraint in (7). This happens because thedominant eigenvector of and in this case, in-troduced before Proposition 1, is the scaled all-one vector, i.e.,

. Thus, for this special case is a scaled all-onevector with the scaling constant determined by the prescribedtransmit-power. Intuitively, this is expected since the sample av-erage of the entries in is a sufficient statistic forestimating in the presence of Gaussian noise.

Fig. 3(a) and (b) depict the MSE of the novel tracker alongwith that of [6], and the one corresponding to the clairvoyant

Fig. 4. Error in temperature and the �� (3-� bounds) versustime �.

KF which does not involve dimensionality reduction. TheMSE is obtained from the trace of (theoretical), aswell as through Monte Carlo simulations (empirical) for both

and and . The comparison isdone so that sensors utilize the same sampling rate both for theclairvoyant KF and the reduced-dimensionality KF. Under thisequal footing, each sensor acquires and transmits scalarsin the clairvoyant setup; but when dimensionality-reduction isapplied, each sensor acquires observations and compressesthem down to a scalar. Thus, the required bandwidth in theclairvoyant KF is times larger that the one needed whendimensionality reduction is applied. Fig. 3(a) and (b) demon-strates that increasing from 5 to 10 reduces the trace of ECMby approximately 13% (from 0.76 to 0.66), which quantifiesthe improvement in state estimation performance. However,the computational burden required per sensor increases sincecomplexity is in the order of .

Moreover, the MSE in the novel channel-aware approachreaches steady-state, while in the channel-unaware approach[6] the MSE eventually diverges. This is expected since thechannel-unaware approach does not account for noise whendesigning the dimensionality-reducing operators in (7). Inter-estingly, the tracking performance achieved by the proposedreduced-dimensionality KF is close to the one enjoyed by theclairvoyant KF even when the compression ratio is 10. Despitethe fact that the clairvoyant KF needs times more bandwidththan the reduced-dimensionality KF, the MSE amplificationfactor penalizing for latter scheme due to compression is muchsmaller than the compression ratio . This advocates thatsensor observations are “judiciously squeezed” without muchinformation lost.

Similar conclusions can be drawn from Fig. 4 that depicts theerror in temperature, namely , compared withthe 3- bounds (see, e.g., [2, Sec. 5.4]) for both channel-awareand channel-unaware approaches with . Notice that theestimation error in the channel-aware approach falls within the3- bounds obtained by the reduced-dimensionality KF, thuscorroborating its MSE optimality. However, the error associ-ated with [6] does not comply with the 3- bounds obtained



Fig. 5. MSE performance of reduced-dimensionality KF in the presence ofmodel mismatch.

by the corresponding KF since it does not account for the pres-ence of noise in the sensor-to-FC channels. Fig. 5 displays theempirically computed MSE achieved by our approach when thechannel matrices are not perfectly known at the FC. Here thereduced-dimensionality KF uses matrices , where the pertur-bation is upper bounded as for .Clearly, the proposed tracking scheme exhibits robustness tomodel mismatch.

IV. REDUCED-DIMENSIONALITY KF IN AD HOC WSNS

In this section, distributed state tracking algorithms are de-veloped based on reduced-dimensionality data collected by sen-sors configured in an ad hoc topology. The motivation behind adhoc configurations is that information processing is performedacross sensors and not at a central location. As a consequence,the resultant tracking schemes exhibit resilience to FC failures.On the other hand, the reduced-dimensionality tracking algo-rithms developed for FC-based topologies offer better MSE per-formance since all the available data per time slot are collectedand processed at the FC. However, their ad hoc counterparts areflexible to trade-off estimation accuracy for robustness. In addi-tion, in the ad hoc setup only one sensor transmits per time slot,which further improves power efficiency.

A possible approach to reduce the communication cost couldbe to have each sensor perform tracking based on its own (local)observations. However, some sensors may not attain adequatetracking performance simply because they acquire poor obser-vations. This motivates a communication scheme in which sen-sors with high-quality observations broadcast their data to allother sensors in range. In target tracking for example, sensorsthat are closer to the target acquire more accurate observationsthan sensors located further away. Thus, remote sensors can per-form better if they receive and utilize information from sensorsin the vicinity of the target. Next, we first outline the steps of thereduced-dimensionality KF running locally at each sensor, andthen elaborate on the selection of the dimensionality reducingoperator as well as the data vector whose dimen-sionality is reduced at sensor .

Suppose that at time slot , each sensor has availablea local state estimate and the corresponding

covariance matrix . Then, each sensor carriesout the prediction phase, which relies on toobtain the state predictor as

(31)

and also on to update the prediction ECM via

(32)

Consider temporarily that and are given (meansof selecting them optimally will follow in the next subsections).At time slot , sensor broadcasts the reduced-dimension-ality vector to all other sensors in range. Each ofthese sensors, say the th, receives in (8) and corrects itscorresponding state estimate as

(33)

where is the Kalman gain that depends on the statistics of. The gain will be chosen to obtain

as the MMSE estimate of state based on the data sequence. The associated ECM can be written as

(34)

where is the cross-covariance of with, and the Kalman gain is

(35)

The issue of which sensor broadcasts at slot depends onthe underlying scheduling algorithm and is assumed given.One possibility is to have only neighboring sensors with highquality observations broadcast their reduced-dimensionalitydata allowing the rest, even those sensors with less informativedata, to form accurate state estimates.

Further, recall that in FC-based WSNs, state tracking whenlinks are ideal can be viewed as a special case of the one fornonideal links. However, this is not the case in ad hoc WSNs,where reduced-dimensionality KF for ideal and nonideal inter-sensor links must be treated separately.

A. Ideal Sensor Links

Under operational conditions elaborated in Section III-C heretoo we assume ideal links between sensors. Thus, it follows froma4) that at time slot sensor receives the vector

(36)

Since the received sequence is the same for all, every sensor at time slot computes the same state es-

timate given by . There-fore, the sensor index in (31)–(36) can be dropped for sim-plicity in exposition. Similar to Section III-A, let denotethe observation innovation , broadcasted by sensor

to gain in power efficiency. Since, it follows that

, which implies that the received data across sensors con-tain the minimal information required to perform the correctionstep in (33).



Because all sensor-to-sensor links are ideal, the power con-straint in (12) can be dropped as discussed in Section III-C. Fur-ther, as the received sequence is the same for allsensors it follows that the optimal in (12) is the same forall . Keeping these properties in mind, it is pos-sible to reformulate the optimization problem in (12) as (detailsin Appendix C)

(37)

Similar to the optimization problem in (29) with , (37)can also be solved using the CCA approach. Thus, the optimalmatrices and are provided by Proposition 3 as

(38)

From (37) it follows that is the MMSE op-timal estimate of the innovation based on

. Thus, plays the role of the Kalman gain asso-ciated with the KF steps in (31)–(34). Further, the ECM in (34)can be updated readily if one takes into account that

(39)

B. Nonideal Sensor Links

When sensor links are nonideal, the received datadiffer from sensor to sensor, due to the presence of additivereception noise as well as multiplicative fading. The fact thatdata received by different sensors are different, renders the cor-responding filtered and predicted state estimates also different.Unfortunately, this unavoidable fact does not allow KF viareduced-dimensionality innovations, as explained next.

Suppose that the broadcasting sensor at time slot formsthe reduced-dimensionality vector ; i.e.,

. Then, sensor receives the vector

Clearly, sensor can perform the correction step (33) so long asit has available the innovation

, where

Since links are nonideal, for , fromwhich we infer that

. In fact, computingat sensor requires that

transmits to the cross-covariance between andfor , with denoting the

broadcasting sensor at time slot . However, this requirementrenders inter-sensor communications quite demanding in termsof power and complexity.

This challenge can be bypassed by having perform di-mensionality reduction on its raw observation vector ;

i.e., . In this case, the innovationin (33) takes the form

which can be computed easily at sensor for and. This task requires only the compression matrix

that can also be found locally at [cf. a5)], and matrixwhich is available at sensor .

To proceed, consider writing explicitly the (cross-) covari-ance matrices

(40)

(41)

(42)

which are required to obtain . Based on these, the opti-mization in (12) can be rewritten as

(43)

The optimal solution of (43) is provided by Proposition 1, sincethe minimization problem in (43) resembles the one in (17).Thus, the optimal can be obtained using the (cross-)covariance matrices of and in (40)–(42), the cor-responding channel matrices and the reception noise co-variance . Due to a5), the quantities that are necessary forsolving (43) are all available per sensor, which implies that eachsensor can locally obtain the optimal matrix . Uponreceiving , sensor is then able to perform the steps ofthe reduced-dimensionality KF as outlined next.

At time slot , the broadcasting sensor computes the in-novation of its reduced-dimensionality observation, namely

(44)

and the Kalman gain with

(45)

(46)

Then, sensor performs the correction step in (33) and updatesits filtered state ECM as specified in (34). Notice that dimension-ality reduction is not necessary at the broadcasting sensor, sinceit can readily utilize and perform tracking through theordinary KF recursions. Though, for symmetry in exposition di-mensionality reduction is performed across all sensors.

Similarly, after receiving sensor , with ,computes the innovation

, as well as the Kalmangain where

(47)



(48)

and forms through (33). Using (47), (48), and (33) thecorrection step can be performed locally once sensor receivesthe reduced-dimensionality data from the broadcasting sensor

, without any extra communications.The reduced-dimensionality KF schemes in this section (for

either ideal or nonideal links) entail two phases. In the first phasethe broadcasting sensor forms from , performs di-mensionality reduction of and broadcaststo all other sensors. This phase is tabulated as Algorithm 3–A.The second phase involves reception of at all sensors inrange, through (non)ideal links and implementation of prop-erly designed KF recursions. This phase is tabulated as Algo-rithm 3–B.

Algorithm 3-A Broadcasting Sensor : Observation andtransmission

1: Compute matrix after updating the corresponding(cross-) covariance matrices

2: Form

3: Construct and transmit

Algorithm 3—B Sensor : Reception and estimation

1: Initialize prior estimate and ECM

2: for do

3: Compute and using (31) and(32)

4: Compute via (43).

5: Receive

6: Compute and using (33) and (34).

7: end for

Algorithm 3–A is run at the broadcasting sensor and providesreduced-dimensionality data to the rest of the neighboring (lis-tening) sensors; Algorithm 3–B is run by all sensors in rangeto keep track of the state via the filtered estimate . Thecomputational complexity per sensor is determined by the stepsinvolved in determining the optimal in Proposition 1,which is per time slot .

Remark 6: The tracking schemes summarized as Algorithm3 offer resilience to FC failures as well as savings in transmit-power. The price paid is increase in the estimation MSE as willbecome more apparent in the next subsection. However, at anytime slot only one sensor is broadcasting while the rest are justlistening. This way the lifetime of the ad hoc network can beextended relative to the FC-based approach. Additionally, thedistributed tracking schemes developed for ad hoc WSNs scalebetter than Algorithm 2 as the number of sensors increases,and offer operational robustness (to FC failures). On the other

hand, the distributed tracking schemes for ad hoc and FC-basedWSNs have complementary strengths. Indeed, if estimation ac-curacy is at a premium, the FC-based tracker can be applied,whereas when robustness, scalability and extended WSN life-time are more important the ad hoc distributed schemes shouldbe preferred.

C. Simulations

Target tracking based on distance-only measurements is acommon problem in power-limited distributed estimation usingWSNs (see e.g., [8]) for which estimation with low communi-cation cost is particularly attractive. Consider sensorsrandomly deployed uniformly in a square region ofsquare meters, where Km, and suppose that using any ofthe existing localization algorithms, e.g., [16], every sensor hasavailable position vectors of all sensors in range.

The sensors are deployed to track the positionof a target. The initial position is

randomly placed within 10 meters from the reference point. The associated state model used in the tracking process

accounts for both the target position and its velocitywhich is initialized as ,

but not for the acceleration which is lumped into the randomnoise term. The maneuvering target’s position and velocityevolve according to

(49)

where is the sampling period and the covariance matrix ofis given by

(50)

Consider a radar located far from the WSN, tracking a targetmoving in the direction of the field where sensors are deployed.Sensors observe the power of the signal return off the target thatis modeled as

(51)

where and depends on the transmissionpower of the radar , the distance between the radar and theWSN , as well as the path-loss propagation law. Also,

denotes the distance between the target and thebroadcasting sensor , while in (51) stands for zero-mean observation noise with variance .

Mimicking the steps applied to an extended (E)KF (see, e.g.,[11, Sec. 13.7]), sensor linearizes (51) in the neighborhoodof to obtain

(52)

where and. Note that (52) is a linear ob-

servation model with all the parameters locally available at .With the linearized model in (49) and (52) obeying (1)–(2), it

is possible apply the KF scheme outlined in Algorithms 3–A and



Fig. 6. Target tracking with EKF and reduced-dimensionality EKF under:(a) ideal sensor links, and (b) nonideal links. The parameters are set to � � �

km, � � �� m/s � � � �, and �� dB.

3–B to track the target’s position . This amounts to an ex-tended KF based on reduced-dimensionality data (abbreviatedas RDim-EKF in the plots).

Similar to Section III-D, the broadcasting sensor collectssamples successively in time and reduces their

dimensionality to . With a sufficiently small samplingperiod, we can assume that the target position remainsinvariant during these successive sampling periods. Further,the order in which sensors broadcast is specified by how closethey are to the target. Specifically, the broadcasting sensorevaluates the Euclidean distance of each sensor from thetarget’s estimated location, namely , andselects the next one which is closer to the estimated sensorlocation. An overview of information-driven sensor selectionschemes tailored for target tracking applications can be foundin [25]. However, different from the signal model establishedin Section II, the observation model (51) is a nonlinear oneand therefore the broadcasting sensor must rely on numericalmethods to find and that are necessaryfor obtaining when links are nonideal. The receptionquality at sensor from sensor is characterized by the

Fig. 7. Standard deviation of the reduced-dimensionality EKF estimate is inthe order of 2 m–15 m for (a) ideal links, and 5 m–20 m for (b) nonideal links,both of which are much smaller than those of the clairvoyant EKF, and almostthe same to the EKF without the communication rate constraint.

signal-to-noise ratio , wherei) accounts for the path-loss since and

have inter-sensor distance ; and ii) is thereception SNR without path-loss that is set to 20 dB.

Next, the tracking performance of RDim-EKF is comparedagainst the clairvoyant EKF. Note that the clairvoyant EKF en-tails no dimensionality reduction but for fairness the communi-cation cost as well as the communication rate are kept equal asin Algorithm 3. To ensure identical communication cost eachbroadcasting sensor in EKF acquires one sample and broad-casts it to the other sensors, while during the same interval inRDim-EKF each sensor acquires samples and implementsAlgorithm 3 to only broadcast a scalar sample to the rest of thesensors.

Fig. 6(a) and (b) depicts the true and estimated target trajec-tories obtained by Algorithm 3 and the EKF at a specific sensor.Fig. 7(a) and (b) shows the standard deviation of the estimationerror associated with Algorithm 3 and EKF, confirming that theformer outperforms the latter for the same communication cost.



Fig. 8. Position error � �� and the�� curves (3-�bounds) versus time � under (a) ideal, and (b) nonideal channel links.

This should be expected because the information broadcasted inAlgorithm 3 is more informative about the target position thanin EKF, since it is constructed by judicious dimensionality re-duction of multiple, namely , observations. Similar conclu-sions can be drawn from Fig. 8(a) and (b), where clearly Algo-rithm 3 outperforms the EKF, while the corresponding estima-tion errors of both fall within the corresponding bounds.Notice that in Fig. 7(a) and (b) the estimation error’s standarddeviation for the EKF with no communication rate constraint isalso plotted as a baseline. In this case, each sensor transmits 10scalars per time slot , instead of 1 as in RD-EKF. It can be seenthat the RDim-EKF has almost the same performance with theEKF when no communication rate constraint is imposed. Intu-itively, this is expected since for the Gaussian observation noisesetup here, a sufficient statistic for estimating will be thesample average of each sensor’s observations. Thus, dimen-sionality reduction is expected to introduce a negligible perfor-mance loss in MSE.

Fig. 9. (a) Trace of the correction ECM for both FC-based and ad hoc WSNsversus time �; (b) Temperature error and the corresponding �� (3-� bounds) versus time �.

Next, the tracking performance of Algorithms 2 and 3 is com-pared in order to gauge the loss in estimation MSE when givingmore emphasis to robustness. The system model is similar to theone in Section III-D and both the FC-based and ad hoc WSNshave sensors. Each sensor acquires data vectors of size

and reduces them to . Note that within eachiteration of Algorithm 2 the FC receives data from all five sen-sors. For fairness in comparison (same communication rate persensor) all sensors in the ad hoc WSN broadcast in a round-robinfashion during one iteration of Algorithm 3. The available trans-mission power per sensor is for the FC-based and for thead hoc WSN. As a result, Algorithm 2 consumes 5 times morepower than Algorithm 3 per iteration. Fading is not incorporatedand reception noise variance is set so that dB for thesensor-to-FC links. Fig. 9(a) depicts the estimation MSE (boththeoretical and empirical) corresponding to Algorithms 2 and 3.The tracking performance of Algorithm 2 in FC-based WSNsas expected outperforms Algorithm 3 in the ad hoc setup. The



performance loss experienced by Algorithm 3 is roughly 50%.However, recall that in every step of Algorithm 2 the power con-sumption is 5P, while in Algorithm 3 it is over the same band-width. This implies that Algorithm 3 consumes 80% less totalpower than Algorithm 2, with the first leading to improved esti-mation performance and the second exhibiting improved robust-ness and lower power consumption. Similar conclusions can bedrawn from Fig. 8(b).

V. CONCLUDING REMARKS

We developed channel-aware algorithms for tracking nonsta-tionary state processes based on reduced-dimensionality datacollected by power-limited wireless sensors. This distributedtracking problem was considered in both FC-based WSNs,where sensors are linked with an FC, as well as in ad hocWSNs, where each sensor is responsible for both data acquisi-tion and processing.

Linear dimensionality-reducing matrices were derived at theFC to account for the sensors’ limited power and noisy links,as well as to minimize the estimation MSE. In the single sensorcase, mean-square error optimal schemes were found in closedform, while an algorithm relying on block coordinate descentiterations was developed for multi-sensor tracking. Power sav-ings result when allowing the FC to feed back state predic-tions to the sensors. The resultant KF tracker is channel-awareand optimally allocates the available transmission power amongthe reduced-dimensionality components in a water-filling likemanner.

Further, distributed KF schemes using ad hoc WSNs weredeveloped to gain robustness to FC failures and save transmit-power. Both ideal links (feasible when powerful error controlcodes are employed) and nonideal inter-sensor links were con-sidered. When sensor links were assumed ideal, each broad-casting sensor compresses the innovation sequence of its ob-servations, thus transmitting the minimal required informationto perform tracking across sensors. This was not the case, how-ever, for the nonideal links where sensors must reduce the di-mensionality of their raw observations.

In a nutshell, the tracking schemes for FC-based and ad hocWSN topologies offer complementary strengths. The FC-basedones yield higher estimation accuracy, while their ad hoc coun-terparts gain in robustness and power savings. Simulations cor-roborated these relative merits.

APPENDIX

A. Proof of Lemma 1

The predictor of based on past observations can bewritten as [cf. a1)]

where for the last equality we used that the predictor of isgiven by .

Q.E.D.

B. Proof of Corollary 1

Upon setting the channel matrices and ,it follows readily that and ; thus,for . The fact that the power inequality in (24)holds for any , implies that(full rank). An inspection of the Lagrange function reveals thatthe diagonal entries of the MSE optimal operator are

(53)

Next, it is shown that the matrices obtained in Proposition 1 co-incide with these in Proposition 3. To this end, letbe the diagonal matrix whose th diagonal entry is equal toin (53), while denotes the matrix obtained fromthe identity matrix after removing its last rows.Thus, matrix can be rewritten as

(54)

Substituting the matrices and in (21) as specified ear-lier we obtain that

(55)

Notice that the right and left eigenvector matrices ofcoincide with the unitary ma-

trices and , respectively. Further, letdenote the diagonal matrix containing the singular values of

. Using these properties it followsthat

(56)

where denotes the upper left part of .From (56) it follows readily that obtained from Propo-sition 1 coincides with the one in Proposition 3 after setting

. Next, it is shown that the optimal compres-sion matrix provided by CCA coincides with the one inProposition 3 when sensor links are ideal. To this end, in(21) can be re-expressed as

(57)

where . Now, recall: i) that

; and ii) that. Using the latter, (57) can be

re-written as

(58)

Q.E.D.



C. Proof of Equation (37)

Recall from (36) that. Adding and subtracting inside the Euclidean

norm of the cost in (12) yields

(59)

where the second equality stems from the fact that the pre-dictor is uncorrelated with the innovation signals

and . Since doesnot depend on and , the first term in (59) can bedropped leading to the MSE cost given in (37). Q.E.D.

REFERENCES

[1] B. D. O. Anderson and J. B. Moore, Optimal Filtering. EnglewoodCliffs, NJ: Prentice-Hall, 1979.

[2] Y. Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation With Appli-cations to Tracking and Navigation. New York: Wiley, 2001.

[3] S. Benedetto and E. Biglieri, Principles of Digital Transmission WithWireless Application. Boston, MA: Kluwer/Plenum, 1999.

[4] D. P. Bertsekas, Nonlinear Programming, Second ed. Belmont, MA:Athena Scientific, 2003, 2nd Printing.

[5] D. R. Brillinger, Time Series: Data Analysis and Theory, 2nd ed. SanFrancisco, CA: Holden-Day, 1981.

[6] H. Chen, K. Zhang, and X. R. Li, “Optimal data compression for mul-tisensor target tracking with communication constraints,” in Proc. 43rdConf. Dec. Control, Atlantis, Bahamas, Dec. 2004, pp. 2650–2655.

[7] T. M. Cover and J. A. Thomas, Elements of Information Theory. NewYork: Wiley-Interscience, 1991.

[8] P. M. Djuric, M. Vemula, and M. F. Bugallo, “Tracking with particlefiltering in tertiary wireless sensor networks,” in Proc. IEEE Int. Conf.Acoustics, Speech, Signal Processing, Philadelphia, PA, Mar. 2005,vol. 4, pp. 757–760.

[9] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Bal-timore, MD: The John Hopkins Univ. Press, 1993.

[10] V. Gupta, T. Chung, B. Hassibi, and R. M. Murray, “On a stochasticsensor selection algorithm with applications in sensor scheduling andsensor coverage,” Automatica, vol. 42, pp. 251–260, Feb. 2006.

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

[12] M. Kisialiou and Z.-Q. Luo, “Reducing power consumption in a sensornetwork by information feedback,” presented at the Eur. Signal Pro-cessing Conf., Florence, Italy, Sep. 2006.

[13] Z.-Q. Luo, G. B. Giannakis, and S. Zhang, “Optimal linear decentral-ized estimation in a bandwidth constrained sensor network,” in Proc.Int. Symp. Info. Theory, Adelaide, Australia, Sep. 2005, pp. 1441–1445.

[14] W. U. Bajwa, J. D. Haupt, A. M. Sayeed, and R. D. Nowak, “Jointsource-channel communication for distributed estimation in sensor net-works,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp. 3629–3653, Oct.2007.

[15] R. Olfati-Saber, “Distributed Kalman filter with embedded consensusfilters,” in Proc. 44th Conf. Dec., the Eur. Control Conf., Seville, Spain,Dec. 2005, pp. 8179–8184.

[16] N. Patwari, J. N. Ash, S. Kyperountas, A. O. Hero III, R. L. Moses, andN. S. Correal, “Locating the nodes: Cooperative localization in wirelesssensor networks,” IEEE Signal Process. Mag., vol. 22, pp. 54–69, Jul.2005.

[17] M. Rabbat, J. Haupt, A. Singh, and R. Nowak, “Decentralized com-pression and predistribution via randomized gossiping,” in Proc. IPSN,Nashville, TN, Apr. 2006, pp. 51–59.

[18] B. Rao and H. Durrant-Whyte, “Fully decentralized algorithm for mul-tisensor Kalman filtering,” Proc. Inst. Electr. Eng.—Control TheoryAppl., vol. 138, pp. 413–420, Sep. 1991.

[19] A. Ribeiro, G. B. Giannakis, and S. I. Roumeliotis, “SOI-KF: Dis-tributed kalman filtering with low-cost communications using thesign of innovations,” IEEE Trans. Signal Process., vol. 54, no. 12, pp.4782–4795, Dec. 2006.

[20] I. D. Schizas, G. B. Giannakis, and Z.-Q. Luo, “Distributed estima-tion using reduced-dimensionality sensor observations,” IEEE Trans.Signal Process., vol. 55, no. 8, pp. 4284–4299, Aug. 2007.

[21] I. D. Schizas, G. B. Giannakis, S. I. Roumeliotis, and A. Ribeiro, “Con-sensus in Ad Hoc WSNs with noisy links—Part II: Distributed estima-tion and smoothing or random signals,” IEEE Trans. Signal Process.,vol. 56, no. 4, pp. 1650–1666, Apr. 2008.

[22] K. M. Wong, Z.-Q. Luo, Q. Jin, and E. Bosse, “Data compression, datafusion and kalman filtering in wavelet packet sub-bands of a multi-sensor tracking system,” Proc. Inst. Electr. Eng.—Radar, Sonar, Nav-igat., vol. 145, pp. 100–108, Apr. 1998.

[23] J.-J. Xiao, A. Ribeiro, Z.-Q. Luo, and G. B. Giannakis, “Distributedcompression-estimation using wireless sensor networks,” IEEE SignalProcess. Mag., vol. 23, pp. 27–41, Jul. 2006.

[24] K. Zhang, X. R. Li, P. Zhang, and H. Li, “Optimal linear estimationfusion—Part vi: Sensor data compression,” in Proc. Int. Conf. Info.Fusion, Queensland, Australia, 2003, vol. 3, pp. 221–228.

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

Hao Zhu (S’07) received the Bachelor’s degree in in-formation electronics and engineering from the De-partment of Electronic Engineering, Tsinghua Uni-versity, Beijing, China, in 2006.

From 2005 to 2006, she was a Research Assistant,working on radar waveform design, with the Intelli-gent Transportation information Systems (ITiS) Lab-oratory, Tsinghua University. Since September 2006,she has been working towards the Ph.D. degree as aResearch Assistant with the Department of Electricaland Computer Engineering, University of Minnesota,

Minneapolis. Her research interests lie in the areas of communication theory,and signal processing. Her current research focuses on distributed signal pro-cessing for wireless sensor networks and sparsity-exploiting detection with fi-nite-alphabet constraints.

Ioannis D. Schizas (S’06) received the Diploma de-gree in computer engineering and informatics (withHons.) from the University of Patras, Patras, Greece,in 2004 and the M.Sc. degree in electrical and com-puter engineering from the University of Minnesota,Minneapolis in 2007.

Since August 2004, he has been working towardthe Ph.D. degree with the Department of Electricaland Computer Engineering, University of Minnesota,Minneapolis. His research interests lie in the areasof communication theory, signal processing, and net-

working. His current research focuses on distributed signal processing with ap-plications in wireless ad hoc and sensor networks, as well as distributed com-pression and source coding.

Georgios B. Giannakis (F’97) received the Diplomadegree in electrical engineering from the NationalTechnical University of Athens, Greece, in 1981,the M.Sc. degree in electrical engineering, the M.Sc.in mathematics, and the Ph.D. degree in electricalengineering, all from the University of SouthernCalifornia (USC), Los Angeles, in 1983, 1986 and1986, respectively.

Since 1999 he has been a Professor with theECE Department at the University of Minnesota,Minneapolis, where he now holds an ADC Chair in

Wireless Telecommunications. His general interests span the areas of com-munications, networking and statistical signal processing-subjects on whichhe has published more than 250 journal papers, 450 conference papers, twoedited books and two upcoming research monographs on Space-Time Codingfor Broadband Wireless Communications (Wiley, 2006) and Ultra-WidebandWireless Communications (Cambridge Press, 2007). Current research focuseson diversity techniques, complex-field and space-time coding, multicarrier,cooperative wireless communications, cognitive radios, cross-layer designs,mobile ad hoc networks, and wireless sensor networks.

Dr. Giannakis is the (co-)recipient of six paper awards from the IEEE SignalProcessing (SP) and Communications societies, including the G. Marconi PrizePaper Award in Wireless Communications. He also received Technical Achieve-ment Awards from the IEE Signal Processing Society (2000), from EURASIP(2005), a Young Faculty Teaching Award, and the G. W. Taylor Award for Dis-tinguished Research from the University of Minnesota. He has served the IEEEin a number of posts.


Documents

IEEE TRANSACTIONS ON SIGNAL PROCESSING, …broadcast their data to all other sensors, while sensors with less informative data remain inactive. This way high-quality (i.e., low-MSE)