Cooperative Detection and Communication in Wireless Sensor Networks

Transcription

1 Cooperative Detectio ad Commuicatio i Wireless Sesor Networks Mohiuddi Ahmed, Yug-Szu Tu, ad Gregory Pottie Electrical Egieerig Departmet, Uiversity of Califoria, Los Ageles Box 95594, Los Ageles, Califoria 90095, USA Abstract We defie iformatio theoretic problems i the cooperative detectio of targets by a distributed wireless sesor etwork, ad i the cooperative trasmissio of the results to a remote user. The domiat costrait i both cases is the eergy cost of commuicatios, rather tha badwidth or sigal processig costs. We show that the detectio problem ca be cast as a rate distortio problem, ad boud the regio for Gaussia sources. We describe how the cooperative trasmissio problem differs from the usual cotext for space-time codes, ad preset oe practical approach. I. INTRODUCTION I wireless sesor etworks, potetially a very large umber of devices cooperate to detect a evet, ad report the results of the detectio exercise to some ed user, who may be remote from the etwork. A fudametal costrait i sesor etworks is the eergy of the odes, which may be limited i their lifetime eergy reserves by batteries, ad which likely have low peak power available for trasmissio. Questios the aturally arise as to the most efficiet meas to make reliable detectio decisios, ad i achievig loger rage commuicatios. I sectio II we set up the cooperative detectio (data fusio) problem as a rate distortio problem, ad i sectio III preset the aalytic formulatio. I sectio IV we itroduce the cooperative commuicatio problem, ad preset some simple approaches. I sectio V we preset capacity results, ad i sectio VI suggest oe simple codig approach that ca apply to a large etwork with varyig umbers of cooperatig elemets. I sectio VII we preset our coclusios. II. THE DATA FUSION PROBLEM We ow cosider the questio: what data rates ca be supported for data fusio i a sesor etwork, give a specified tolerable data distortio? Ufortuately, a exact aswer to this questio is ot available i the geeral case, ad thus we have cosidered the specific case of correlated Gaussia sources i this paper. The motivatio for this problem comes from performig distributed detectio of pheomea. It is well kow from the theory of distributed detectio that higher reliability ad lower probability of detectio error ca be achieved whe observatio data from multiple, distributed sources is itelligetly fused i a decisio makig algorithm, rather tha usig a sigle observatio data set []. This, coupled with the fact that fabricatio techological advaces have made low-cost sesors icorporatig wireless trasceivers, sigal processig ad sesig i oe itegrated package a desirable low-cost optio, it is ievitable that such devices will be widely used i detectio applicatios such as security,

2 moitorig, diagostic, remote exploratio etc. This has give rise to the developmet of wireless itegrated etworked sesors (WINS) [], Figure. However, the effective deploymet of such distributed processig systems itroduces some sigificat desig issues, most otably: etworkig ad commuicatio protocols, trasmissio chael ad power costraits, ad scalability, amog others [], [3]. It is also evidet that some fudametal limits are required to assess the optimality of ay system desig with regard to the best desig. Thus, a iformatio theoretic aalysis of the system is required. Here we assume that the primary costrait is power. A WINS system ivokes a multi-termial aalysis, as diagrammed i Figure. Satellite/remote lik to sesor etwork cotrol ceter Commuicatio liks 3 Sesor odes formig clusters of isolated ad hoc wireless etworks, liked by a commo gateway. Figure : Wireless Itegrated Network Sesor System For this type of a system, all the traditioal types of multi-termial chaels cosidered i iformatio theory appear: the multiple access chael (commuicatio pathways show i Figure umbered as ), the broadcast chael (), the relay ad iterferece chael (3), etc. [4]. Additioally, the chael may be fadig or more complex. Ufortuately, i the absece of a geeral iformatio theory of multi-termial etworks, there is, as yet, o aalytical way of evaluatig performace bouds for whole systems, for a specific type of task for which the etwork might be employed, e.g. distributed detectio. Our goal is to apply the results kow so far to obtai, if ot global optimum iformatio limits, at least optimality criteria for each of the idividual sub-blocks. I this regard, advaces have bee made with certai simplifyig cosideratios, most otably the rate-distortio bouds for multiple, correlated odes. The focus so far has bee o pathways, iside the local loop. For the idividual local etwork loops, the problem is oe of efficiet commuicatio ad data fusio for detectio. Associated problems such as etwork boot-strap, algorithms determiig the miimum umber of odes ecessary for reliable detectio of a pheomeo, etc. have bee studied ad are ot discussed here [3], [5]. Istead the codig problem is cosidered. The multi-termial codig theory problem for two correlated memoryless sources with separate ecoders has bee solved by Slepia ad Wolf [5]. The correlated sources assumptio is valid i the WINS case, sice for odes observig the same target, the data geerated for each sesor is expected to be correlated. Also, i the WINS case, it is apparet that power efficiecy ca be icorporated by allowig a distortio criteria, sice there may be several data fusio ceters, ad sice local processig provides a far higher power gai tha RF trasmissios. Thus, what becomes of iterest the is how much

3 distortio ca be tolerated if the sesor etwork is to achieve some measure of efficiecy i distributed detectio i other words, the rate-distortio boud. Previous work i this area by Wyer ad Ziv [7], Ha ad Kobayashi [9], [0] ad Csizar [] have all focused o special extesios of Slepia ad Wolf, but the geeral ratedistortio regios characterizatio problem has remaied usolved. We have exteded the special case for two correlated memoryless Gaussia sources (Oohama, [7]) to the - sources case (with partial side iformatio). I what follows, the otatio is adopted from Oohama, [7]. The ext sectio presets the aalytic formulatio of the problem ad the mai result. III. ANALYTIC FORMULATION Cosider the multi-sesor system as show below (Figure ). Satellite or some remote lik betwee gateway ad sesor etwork ifrastructure Y Gateway/Fusio Ceter Y... Y A cluster of sesor odes with a mai source () ad -helpers (Y i ) Figure : Data Fusio for a Wireless Networked Sesor System A portio of a distributed cluster of sesor odes (perhaps mobile) is observig a pheomeo ad geeratig source data. Algorithms exist which ca determie which odes i the proximity of the pheomeo eed to be activated ad which ca remai dormat [4]. Oce this boot-up process is completed, the ode observatio data is assumed to be Gaussia (for aalytical simplicity), with oe data ode actig as the mai data source (e.g. that which is closest to the pheomeo), ad the remaiig odes geeratig correlated data. The codig challege is the to determie appropriate codes ad data rates such that the gateway/data-fusio ceter ca reproduce the data from the mai ode usig the remaiig odes as sources of partial side iformatio, subject to some distortio criteria. Thus for a mai source,, ad correlated sources, Y i, with {, Y, L Y } beig t t t t = statioary Gaussia memoryless sources, for each observatio time, t=,, 3,..., we let the radom (+)-tuplet (, Y, L Y ) take values i L. The joit probability desity t t t Y Y fuctio is give by the usual expressio for the multi-dimesioal Gaussia probability desity fuctio, where the covariace matrix ca be deoted as: σ ρ σ σ Y Y L ρyσ σ Y ρ σ σ σ L ρ σ σ Y Y Y Y Y Y Y Λ= M M O M ρ σ σ ρ σ σ L σ Y Y YY Y Y () We ca write idepedet copies of { t} as m t = =,, L, ad similarly for { } m kt t m m k=,,,. Next, we cosider a codig system where data sequeces, L are Y k Y =,

4 m m separately ecoded to ϕ ( ), ϕ ( Y k ) ad set to the iformatio processig / data fusio ceter. The decoder fuctio, Ψ= ( ψ, ψ, L, ψ ), observes the (+)-tuplet m m m { ϕ( ), ϕ( Y ), L, ϕ( Y )} ad estimates ( ˆ ˆm, ˆm Y,, Y ) L. We let I, δ ( R, R, L, R) deote the set of all such codig ad decodig schemes, ( ϕ, ϕ, L, ϕ, Ψ), which ca exist with the properties metioed above. We take : dx : [ 0, ), d: Y [ 0, ), L, d : Y [ 0, ) as the distortio measures, which, i our case, is the squared distortio measure, ad we let m the average distortios be ˆ = E d ( t, t) (similar expressios for the other sources). m t = The for give positive umbers D, D,,D, a rate (+)-tuplet R, R,,R,, is admissible if for ay δ > 0, 0 ( δ), there exists a (+)-tuplet ( ϕ, ϕ, L, ϕ, ψ) I, δ ( Rx, R, L, R), such that i Di + δ. I our specific case, we do ot care about the reproductio of the Y i s, so the D i s ca be large. Rather, the Y i s act as helpers to reproduce by providig side iformatio at the data fusio ode. This is the so called -helper case. The for a ecodig system usig the Y i, s as -helpers, the rate-distortio regio give by: R ( D, D, LD ) = {( R, R, LR )( : R, R, L R ) is admissible} for a give set of rates ad distortio measures, is desired. Mai Result: For the special case of the correlated Gaussia source, we ca ow state the mai result that we have obtaied. We cosider the ecodig fuctios: m m ϕ : ℵ = {, L, C}, L, ϕ : Y ℵ = {, L, C i i i i} to be such that the rate costraits beig satisfied are: log C m i R i + δ, i=,,,... Extedig previous results [4]-[0], we show that for a admissible rate ( R, R, R, L R ), ad for some D i s > 0, the -helper system data rates for correlated Gaussia sources ca be fused to yield a effective date rate (with respect to source ) satisfyig the followig lower boud: R Proof: σ log Y Y D + k = Rk ( ρ ρ ) k k This is the desired rate distortio regio. I the iterests of brevity, a outlie of the mai steps of the proof is give. The method of employig joit weakly δ-typical tuplet, based o typical sequeces, is used i the proofs of the characterizatio, rather tha a measure-theoretic approach []. The: We assume that a admissible set of rates exists, ad we let Wx ϕ ( ), Wi ϕi( Y ) = =. ( ) ( ) ( ) ( ˆ x + δ log ; ) ( ; k ) ad ( R ) log ( ) ( ) ( i δ Ci H Wi I Yi ; Wi) R C H W I I W Note that W Y forms a Markov Chai, thus defiig: i i k = +.

5 ( ) ( ˆ ) ( ) ( ) F D = if I ;, Gi R = sup I ; Wi we get: ˆ : D : Wi I( Yi ; Wi) R k k k= ( ) ( ) R + δ F D + δ G R + δ Wi Yi By the Gaussia property of the sources, ad cocavity of the logarithm fuctio, we get: ( ˆ σ σ ; ) log( ) ( ) log( ) I F D D D Fially, a upper boud of Gi ( R ) is obtaied by a extesio of the techique for the two ode case (etropy of the power iequality, mootoicity, Jese's iequality): I( ; Wi ) log R ρy ρ + i Y i Substitutig the bouds i the expressio for F ( D ), we obtai: + δ ( + δ) ( + δ) R F D G R R k k k = σ ( Rk + δ ) log + log ( ρy + ρ k Y k ) D + δ k = Lettig δ 0 σ, we obtai the fial result: log Yk Yk D + k = Rk ( ρ ρ ). IV. THE COOPERATIVE COMMUNICATION PROBLEM With may low-power ad low-cost sesors available, the goal is to achieve more robust ad higher rate commuicatios. For example, we may agai cosider the situatio i Figure, where ow commuicatio to the satellite may be accomplished by a cooperatig group of odes to overcome their peak power limitatios. This problem is of iterest for example i remote plaetary exploratio, where the cost of providig power geeratio o the surface of a plaet is orders of magitude greater tha for the dowlik. The commuicatio schemes to accomplish this goal must be isesitive to the umber of trasmitters ad their locatio, ad robust to the trasceiver failure or motio. We are also iterested i problems where for example clusters of odes o the surface must cooperate to overcome gaps i the multi-hop etwork caused by ifelicitous ode placemet, or ode failures. Thus multiple trasmitters ad receivers may cooperate. The simplest way to achieve cooperative commuicatio is by multiplexig. Frequecy, time, code divisio multiple access (FDMA, TDMA, CDMA), ad multiuser orthogoal frequecy divisio multiplex (multiuser OFDM) are all feasible approaches. Usig these methods, chaels startig from differet trasmitters ca work i parallel ad idepedetly. Differet trasmitters could cooperate with each other or process iformatio idepedetly. For FDMA, TDMA, if they process iformatio idepedetly, obviously, the improvemet o the performace over the sigle trasmitter system is that iformatio data rate icreases by a factor of the umber of trasmitters. Meawhile, the badwidth expads by the same factor. Eve if codig is used across trasmitters, the data rate icreases by the same factor because the received symbol rate icreases by the same factor. Although the aalyses for CDMA ad multiuser OFDM are ot so straightforward, there is also a

6 icrease for the data rate. Thus, i order to deal with a peak power costrait, icreased commuicatios rage or rate ca be achieved if the trasmitters first share the message to trasmit, ad the all sed it. Some weighted combiig scheme ca be used i the receiver array. Usig more badwidth as the umber of trasmitters is icreased is thus a simple way to improve eergy efficiecy, takig advatage of the absece of competig uses. For fadig chaels, optimal combiig is simple as well. Because each chael is ucorrelated, the umber of received sigals are the umber of trasmitters times the umber of receivers. The same optimal combiig formula applies to these received sigals. I additio to the badwidth expasio, it is oted that cotrol sigalig overhead is ecessary. Its fuctio is to allocate ad maage chael resources as well as esure the messages of the trasmitters are coordiated. For FDMA ad CDMA, the receiver complexity grows with the umbber of trasmitters liearly. For TDMA, the receiver speed goes up with the umber of trasmitters. For multiuser OFDM, the umber of trasmitters does't affect the receiver complexity so dramatically. However, time ad frequecy sychroizatio [][3] are critical issues that must be dealt with carefully. Multiplexig affects the trasmitter desig as well. For FDMA ad CDMA, each trasmitter uses a uique frequecy bad ad spreadig code, respectively. This uiqueess could be implemeted by hardware or by software. TDMA ad multiuser OFDM trasmitters are assiged chael resources dyamically, ad this results i more complicated hardware desig. No matter whether multiplexig is adopted, it is valuable to ivestigate commuicatios with multiple trasmitters over the same chael. They are orthogoal commuicatio schemes i the sese that multiplexig just deals with how chaels are allocated, ad does't care how each chael is used. So, there could be differet codig, modulatio, etc. at differet multiplex chaels. I our study, we address the commuicatios over the same chael. Oe advatage of this kid of commuicatios is the simplicity of trasmitter desig. Trasmitters usig the same chael ca be almost idetical. We cosider two kids of chaels that may be ivolved. They are the AWGN chael ad the fadig chael with uequal path loss. The availability of chael state iformatio affects RF sychroizatio ad the choice of modulatio ad demodulatio schemes. It is easy to appreciate the importace of estimatig the chael state iformatio for fadig chaels. It icludes the magitude ad phase of the chael respose, both critical quatities i optimal combiig. For AWGN chaels, there is o path loss. However, due to the differet path legths, trasmitted sigals arrive at the trasmitter with differet RF phases. This phase relatioship is part of the chael state iformatio. Here, ackowledge of chael state iformatio is characterized ito three types, as follows. We assume chaels are ot reciprocal. Thus, chael state iformatio must be estimated at the receiver side. Oe approach is to isert a pilot sequece [4][5] periodically i each trasmitted sigal sequece. Because more tha oe trasmitter uses the same chael ad there is iterferece betwee pilot sequeces from differet trasmitters, some properties must be satisfied amog these pilot sequeces so that the chael state iformatio for each chael ca be estimated. The, the chael state iformatio is feed back to the trasmitters explicitly or implicitly. From this chael state iformatio, trasmitters ca

7 adjust their costellatio phase with respect to each other so that their sigals arrive at the receiver i phase. This is called RF sychroizatio. It is also possible that the chael state iformatio is ot fed back to the trasmitter side. Thus, the costellatio phase caot be cotrolled, ad RF sychroizatio caot be achieved. Sigals may arrive at the receivers costructively or destructively. Idividual trasmitted sigals must be determied at the receiver with the help of chael state iformatio. Sice the costellatio ad chael state iformatio are kow, the possible combied receive sigal ca be computed, ad coheret commuicatio schemes ca be used if the phase of the combied receive sigal ca be tracked as well. Whe chael state iformatio caot be estimated, obviously, there is o way to achieve RF sychroizatio. Sice the possible clea received sigal caot be determied, it would be more feasible to adopt ocoheret commuicatio schemes. V. CAPACITY RESULTS The iformatio theoretic capacity for multiple ateas commuicatios has bee ivestigated to by Foschii[6], ad Telatar [7] idepedetly. Let us assume there are T trasmitters ad R receivers, ad there is o feedback lik from trasmitter to receivers. The T trasmitted sigal compoets are take to be statistically idepedet Gaussias. The virtue of a Gaussia distributio for the trasmitted sigal is well established, see [8]. Uder the assumptio of ucorrelated oise o each chael, the capacity takes the form [8-0], H C = log det( I + GA r sg ) () No where A s is the trasmitted sigal covariace matrix, G is the chael respose matrix that icludes path loss ad fadig, ad No is the oise variace. It is observed that capacity is a fuctio of the chael state iformatio G ad the trasmitted sigal covariace matrix A s. Furthermore, from Hadamard's iequality [], i order to maximize the capacity, GA s G H must be a diagoal matrix. Whe chaels are idepedet from each other, G ad G H are diagoal, ad A s has to be diagoal as well. For Gaussia chaels, it is obvious to see the best power allocatio policy is the equal power distributio. Several combiatios of the umbers of trasmitters ad receivers are briefly discussed here.. Receive Diversity: maximum ratio combiig, T =, R = For Gaussia chaels, C = log [ + SNR* ] (3) For uequal path loss fadig chaels, C = log [ + SNR* G ] (4) where G is a x vector, ad SNR is defied as the ratio of trasmit power to the receiver oise power.. Trasmit Diversity T =, R =. The capacity ca be derived from (). For Gaussia chaels, C log [ SNR i ] (5) = + i where SNR i is defied as the ratio of the i-th trasmit power to the receive oise power, assumig the oise power is equal at all receivers. For uequal path loss chaels,

8 C = + log [ SNR i * G i ] (6) i where G i is the chael respose from the i-th trasmitter to the sigle receiver. It has a chisquare distributio with degrees of freedom ad a variace that depeds o the path loss. 3. Combied Trasmit-Receive Diversity. The capacity ca be computed umerically by Mote Carlo simulatio. For T R, equal trasmit power ad equal path loss Rayleigh fadig chaels, the capacity lower boud was derived i [7] T C> log [ + χ ksnr] (7) k= T ( R ) where χ k is a chi-square distributio with k degrees of freedom. The above discussio for trasmit diversity is based o the situatio whe RF phases are ot sychroized at the receivers. Whe their relative phases are adjusted so that they are aliged at the receiver, the amplitude of the combied sigal is the sum of the costituet trasmit amplitudes. Therefore, for uequal path loss fadig chaels, the capacity is L NM F T = + H G I K J C log SNR i * G i (8) k= O QP The capacity for Gaussia chaels is obtaied by replacig G i with. For receiver diversity, RF sychroizatio is ot a problem because there is oly oe trasmit sigal at ay istat of time. For combied trasmit-receive diversity, RF sychroizatio is almost impossible to be achieved simultaeously at all receivers uless trasmitters ad receivers are deliberately located i the fixed chael eviromet, ad ca move to ay locatio i the varyig chael eviromet. The capacity for RF sychroizatio is much higher tha that for RF o-sychroizatio. VI. A PRACTICAL APPROACH I wireless sesor etworks, the umber of available trasmitters may vary with time. Sesor failures or additios are expected to take place. Thus, the commuicatio scheme used i the eviromet must be adaptive with respect to the umber of trasmitters. Spacetime trellis codes [] are desiged for fixed umbers of trasmitters. We propose a approach which is easier to implemet ad more isesitive to the chage of the umber of trasmitters. I the approach, the relative costellatio phases of trasmitters chage every symbol iterval so that sigals combied destructively may combie costructively i other symbol itervals. Cosequetly, the chael behaves like a fast fadig chael. We have coducted a computer search for the best relative phase rotatio. There are two trasmit ateas usig BPSK sigal costellatios ad the rate / repetitio code. The chaels are AWGN chaels. The costellatio phase of the first trasmit atea is fixed, while the secod trasmit atea icreases its costellatio phase by the value of relative phase rotatio at the secod symbol iterval. It turs out that the best phase is 80 degrees. I other words, the secod trasmit atea flips its costellatio phase every symbol iterval. It is iterestig that the BER at this poit is equal to the BER for oe trasmit atea with the same total trasmit power. If the costrait is o the idividual trasmit power, which is the usual case i wireless sesor etwork, we will obtai 3dB gai i performace. For three trasmit ateas, the miimum appears at three poits, (0,π), (π,π) ad (π,0). These three poits ca be obtaied with the same operatio. Two costellatios chage by the same

9 phase, while the other chages 80 degree more. Whe ad 3 trasmit atea cases are cosidered together, we ca observe the robustess to the umber of trasmit ateas. Whe oe trasmit atea is added to the trasmit atea sceario, the origial two ateas work i the same way, while the third atea ca choose to flip every symbol iterval, as the secod atea, or ot, as the first atea. Therefore, the system scales up without chagig the operatio of the existig trasmitters. VII. CONCLUSION I this work, we have cosidered the rate distortio problem for a sesor etwork employig data fusio at a ode. The mai assumptio has bee that the -helper odes are all producig correlated Gaussia data, which has the eabled us to obtai a aalytic form for the rate distortio boud. The primary utility of this result is to compare practical data fusio schemes with the predicted bouds i particular, to determie which are the most critical ad sesitive parameters affectig rate ad performace of a data fusio scheme i a etwork sesor system. The mai limitatio of the work is that curretly, extesios to the more real-world sceario of o-gaussia sources ad chaels are ot obvious. However, efforts are uderway to formulate umerically solvable versios for the more realistic scearios. Our ultimate goal is, i the absece of ay tractable aalytical expressio, to obtai at least a iterative algorithm, or a covex optimizatio form. Ideally, this would allow us to accurately predict maximum rate/ miimum distortio pairs for a wide variety of chaels ad sources for radom arragemets of sesors. We would also like to be able to ivestigate various what-if scearios i simulatio set-ups for particular types of cofiguratios ad codig implemetatios. Thus, we hope evetually to be able to use these bouds to defiitively compare various data fusio ad etwork commuicatio schemes for wireless sesor etworks, with regards to their performace ad efficiecy. We have also preseted a simple phase rotatio scheme that eables a variable umber of trasmitters without a commo phase referece to achieve gais over simple o-coheret combiig, assumig the same chael must be used. However, much work remais to be doe i devisig schemes with superior performace. Simple capacity calculatios idicate that substatial gais ca be obtaied by providig a coheret phase referece, ad thus it is likely that approaches which devote a sigificat fractio of the eergy to beacos may be fruitful. ACKNOWLEDGEMENTS This work was supported by JPL uder cotract G3378, ad HRL Laboratories i associatio with the UC MICRO program. REFERENCES [] P.K. Varshey, Distributed Detectio ad Data Fusio, New York: Spriger-Verlag, 997. [] G. J. Pottie, et. al., Wireless Sesor Networks, Iformatio Theory Workshop Proceedigs, 998, Killamey, Irelad, Jue -6, 998.

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