the absence of noise and fading and also show the validity of the proposed algorithm with channel constraints through simulations.

Transcription

1 1 Channel Aware Decentralzed Detecton n Wreless Sensor Networks Mlad Kharratzadeh mlad.kharratzadeh@mal.mcgll.ca Department of Electrcal & Computer Engneerng McGll Unversty Montreal, Canada Abstract We study the problem of decentralzed detecton n wreless sensor networks. Frst, we wll revew the classcal framework for decentralzed detecton. The classcal framework does not take nto account the power constrants of wreless sensor networks and also the characterstcs of wreless channels such as fadng and nose. Next, we wll revew the algorthms that consder these constrants. We assume that these algorthms know the channel fadng and nose statstcs, and hence the term channel-aware. Two man approaches are approxmatng the fuson rule based on the value of SNR, and sensor censorng, where not all the perpheral nodes send ther decsons to the central node. In the end, we wll propose a fully decentralzed channel aware algorthm for decentralzed detecton whch s based on gosspng. We wll show the valdty of the proposed algorthm by provng the convergence and also performng some smulatons. I. INTRODUCTION Wreless Sensor Networks consst of a set of sensors, whch are capable of sensng, computaton, and communcaton and are spatally dstrbuted n order to cooperatvely montor physcal or envronmental condtons. The ablty to detect events of nterest s essental to the success of emergng sensor network technology. Detecton often serves as the ntal goal of a sensng system. For example n applcatons where we want to estmate attrbutes such as poston and velocty, the frst step s to ascertan the presence of an obect. Moreover, n some applcatons such as survellance, the detecton of an ntruder s the only purpose of the sensor system. In stuatons where the perpheral nodes do some preprocessng on ther observaton before sendng data to a central node (called fuson center), the correspondng decson makng problem s termed decentralzed detecton. Assume that there are M hypotheses on the state of the envronment and each one of the sensors observes some relevant nformaton about t. In a centralzed scheme, each sensor transmts all of ts observaton (wthout any addtonal processng) to the fuson center, whch solves a classcal hypotheses testng problem and decdes on one of the M hypotheses. However, n a classcal decentralzed scheme, each sensor wll do some preprocessng and send a summary of ts observaton, whch s chosen from a fnte alphabet, to the fuson center. Then, the fuson center wll decde on one of the M hypotheses, based on the messages t has receved. The descrbed centralzed and decentralzed schemes, dffer n some aspects. Frst, t s clear that the performance of the decentralzed scheme s suboptmal n comparson wth the centralzed scheme due to loss of nformaton n the nodes local preprocesses. On the other hand the communcaton requrements of decentralzed scheme s much smaller than those of centralzed one. The reason s that nstead of sendng raw volumnous observaton, each node sends a summary of ts observaton, taken from a fnte alphabet. So, n bref, the decentralzed detecton offers a great reducton n communcaton requrements, at the expense of some performance reducton. However, t turns out that the performance reducton s often neglgble 1]. In the decentralzed scheme, whle the fuson center faces a classcal hypotheses testng problem (f we look at the messages receved from other nodes as ts observatons), the problem s more complex for the perpheral sensors. One may expect that each sensor should decde ndependently and make decson only based on ts own observaton and use ts own lkelhood rato test. Ths s not true n general. When the detectors decde n a way to acheve a system-wde optmzaton, they often use dfferent strateges than n cases where the ont costs of ther decsons separates nto a cost for each. Even under a condtonal ndependence assumpton (whch means that the observatons of dfferent sensors are ndependent from each other under the same hypothess), fndng optmal decson-makng algorthms (based on the observaton) at the sensor nodes remans, n most cases, a dffcult task. Ths optmzaton problem s known to be tractable only under restrctve assumptons regardng the observaton space and the topology of the underlyng network 1]. In secton II we wll explan the problem formulaton and brefly revew the works that study the classcal framework 1], 2], 3]. Next, n secton III we wll revew 4] and 5] where some channel aware algorthms for decentralzed detecton have been ntroduced. To be specfc, the optmum lkelhood rato has been approxmated n low and hgh SNR cases there. Secton IV s devoted to the revew of sensor censorng whch s one of the man approaches n the ltearture for nteractng wth channel constrants 6], 7]. Sensor censorng s a scheme n whch, not all the nodes send ther decsons to the fuson center and only the sensors wth nformatve observatons wll communcate wth the central node. Fnally, as the orgnal part of ths work, we wll propose a channel aware algorthm for decentralzed detecton n secton V whch s based on gosspng. We wll prove the convergence of the algorthm n

2 2 the absence of nose and fadng and also show the valdty of the proposed algorthm wth channel constrants through smulatons. II. PROBLEM DEFINITION AND CLASSICAL FRAMEWORK Here, we wll focus on bnary hypotheses testng problem whch s stated as follows. There are N perpheral sensors each of whch sends ts message ( summary of ts observaton whch s chosen from a bnary alphabet) to the fuson center drectly (.e. parallel or star topology). The two hypotheses are H 0 and H 1 wth pror probabltes Pr(H 0 ) and Pr(H 1 ), respectvely. Each of these hypotheses nduces a dfferent ont probablty dstrbuton for the observatons of perpheral sensors: Under H 1 : Pr(r H 1 ) (1) Under H 0 : Pr(r H 0 ) (2) where r, = 1,..., N are the observatons of perpheral nodes. Each sensor receves an observaton r, whch s a realzaton of a random varable whose pdf s known from (1) and (2), and evaluates a message u = γ (r ), and transmts t to the fuson center. We call the functon γ, as the decson rule of the sensor. The fuson center receves all these messages and usng ts own decson rule (whch s also called fuson rule), γ 0, decdes on one of the two possble hypotheses. The set of all these decson rules s called strategy. The classcal framework, whch has been extensvely explored (e.g., see 1], 2], 3]), does not adequately take nto account mportant features of the wreless sensor networks such as nodes power constrants and lmts on the channel capacty. Also, t s assumed that the sent messages from the perpheral nodes wll be receved by the fuson center correctly, hence gnorng nose, ntereference, and fadng n communcaton channels. In ths secton we wll brefly revew the results of the classcal framework, and n the followng sectons, we wll study the effect of the mentoned constrants. Here, we wll revew the results from 1], 3]. In the Bayesan formulaton, we are gven a cost functon C(U 0, U 1,..., U N, H) (3) where C(u 0, u 1,..., u N, H ) s the cost assocated wth the event that hypothess H, = 0, 1 s true, the messages from the perpheral sensors are u 1,..., u N, and the fuson center decson s u 0. Our obectve n Bayesan framework s to fnd a strategy, γ that mnmzes the expectaton of the cost functon: J(γ) EC(U 0, U 1,..., U N, H)]. Note that the cost functon defned above s a functon of the fuson decson as well as the perpheral nodes messages. However, for dfferent purposes we can defne dfferent cost functons that only depend on some of decsons. For example, f C only depends on U 0, the performance of the system s udged on the bass of the fuson center s decson. The mnmum probablty of error crteron for the fnal decson les n ths type of cost functons. As another case, we may wsh to nterpret each sensor s message as a local decson, based on the true hypothess. Then, wth the sutable choce of C, we can penalze ncorrect decsons by the fuson center as well as the perpheral sensors. As an extreme case, the cost functon mght be ndependent of the fuson center s decson and be only a functon of the decsons of perpheral nodes and we only need to optmze wth respect to γ, = 1,..., N. Ths can happen n the case of a pror fxed fuson decson rule. The man result n 1] s the followng theorem, whch we nclude here wthout proof: Theorem 1. (a) Fx some 0 and suppose that γ has been fxed for all. Then γ mnmzes J(γ) f and only f where γ (r ) = argmn d=0,1 1 Pr(H r ) a (H, d) w.p. 1 (4) =0 a (H, d) = EC(γ 0 (U 1,..., U 1, d, U +1,..., U N ), U 1,..., U 1, d, U +1,..., U N, H ) H ] (5) (b) Suppose that the decson rules for perpheral nodes have been fxed, then γ 0 mnmzes J(γ) f and only f γ 0 (U 1,..., U N ) = argmn d=0,1 1 Pr(H U 1,..., U N ) =0 C(d, U 1,..., U N, H ) w.p. 1 (6) Ths means that wth person-by-person optmzaton (fxng all the sensors decson rules except one, and try to mnmze J wth respect to that sensor decson rule) we wll get a set of lkelhood rato tests whose thresholds depend on each other. Thus, n order to fnd the person-by-person optmal strategy, we need to solve a system of 4N nonlnear equatons n as many unknowns. Therefore, the acheved decson rule s equvalent to dvdng the observatons space nto two regons (because we are consderng bnary hypotheses testng), and decdng to send u = d, d = 0 or 1 to the fuson center f the vector of lkelhood belongs to the correspondng regon. Each regon s specfed by a set of lnear equaltes and therefore s a polyhedral. Note that ths structure s the same as the structure of the optmal decson rule for the classcal bnary hypotheses testng. The Neyman-Pearson formulaton of the problem has been studed n 3] and smlar results have been derved, that we do not menton them here. In the next secton we wll revew the extenson of the classcal parallel fuson structure, by ncorporatng the fadng channel layer that s present n wreless sensor networks. III. DECISION FUSION UNDER FADING CHANNEL ASSUMPTION In ths secton, we wll revew 4] and 5] whch explore decson fuson algorthms that take nto account channel fadng effects. Here we assume that we have flat fadng channels and the fuson center knows the fadng coeffcent (envelope), and hence t s channel-aware. Although t s always possble to reduce the effects of nose and fadng by ncreasng SNR n channel aware systems, n wreless

3 3 sensor networks t s desrable to reduce the communcaton power (because WSNs are energy constraned). The problem formulaton s almost smlar to the prevous secton, except for the fact that the messages sent from the perpheral sensors to the fuson center pass through channels wrh flat fadng and are corrupted by nose (see Fg. 1). Here, we do not look at the perpheral sensors decson rules and only modfy the fuson center decson to cope wth nose and fadng. Fg. 1. H 0, H 1 Sensor 1 Sensor 2 u u 1 2 h1 h2 n n 1 2 y1 y 2... Fuson Center hn n N yn Sensor N Prallel fuson model n the presence of fadng and nosy channel Assume that the kth local sensor makes a bnary decson u {+1, 1} wth P f = Pr(u = 1 H 0 ), P d = Pr(u = 1 H 1 ), called false alarm and detecton probabltes respectvely. If u s transmtted from sensor through a fadng channel, then the output of the channel (nput to the fuson center) would be: y = h u + n (7) where h s the real valued (postve) fadng coeffcent (envelope) of the channel between sensor and the fuson center. n s zero mean Gaussan nose wth varance. In the followng, we try to derve a fuson rule based on y, = 1,..., N, that can determne whch hypothess s true wth the best achevable performance. Assumng full knowledge of the fadng channels, h, and the local sensors performance, P f and P d, we can wrte the lkelhood rato at the fuson center as: Λ(y) = f(y H 1) f(y H 0 ) P d e (y h ) 2 2σ = 2 + (1 P d )e (y +h )2 2 (8) =1 P f e (y h )2 2 + (1 P f )e (y +h )2 2 where y s the vector of receved messages from all the sensors at the fuson center. Now, we try to smplfy the above expresson n two specal cases: hgh SNR (.e., 0) and low SNR (.e., ). 1) Hgh SNR ( 0): In ths case the authors proposed a two-stage approxmaton n 4]. The lkelhood rato descrbed n (8) consders the channel effects and the sensors specfcatons smultaneously. An alternatve s to separate ths nto a u N two stage procedure. Frst, nfer about u usng y, and then apply the optmum fuson rule based on the acheved value of u. Gven the model ntroduced n (7), the maxmum lkelhood estmaton of u s: ũ = sgn(y ) Now, f we rewrte (8), we have: Λ = sgn(y )=1 sgn(y )= 1 P d e 2y h + (1 P d ) P f e 2y h + (1 P f ) P d + (1 P d )e 2y h P f + (1 P f )e 2y h Because of the hgh SNR assumpton ( 0), we have: lm Λ = 0 sgn(y )=1 P d P f sgn(y )= 1 1 P d 1 P f In above dervaton we used the fact that as goes to zero, the exponental terms wth a postve power over wll go to nfnty and we can gnore other terms that are added to them. Takng logarthms from both sdes, we wll acheve the followng smplfed log-lkelhood rato: Λ 1 = lm log Λ = log 0 sgn(y )=1 Pd P f ] + sgn(y )= 1 ] 1 Pd log 1 P f Note that Λ 1 does not depend on the channel statstcs at all and s only a functon of P d and P f for = 1,..., N. 2) Low SNR ( ): The hgh SNR assumpton s not realstc, because of the power constrant of the wreless sensor networks. So, n ths case we consder the more realstc case of low SNR. Rewrtng (8), we have: =1 P d + (1 P d )e 2y h P f + (1 P f )e 2y h For we have e (2yh)/σ2 1 and thus we can approxmate t by the frst-order Taylor seres expanson: Therefore: lm Λ = lm 0 0 e (2yh)/σ2 = lm =1 0 =1 Takng logarthm from both sdes: lm log Λ = lm 0 0 lm 1 2y h P d + (1 P d )(1 2yh ) P f + (1 P f )(1 2yh ) 1 (1 P d ) 2yh 1 (1 P f ) 2yh N =1 N 0 =1 log 1 (1 P d ) 2y ] h log 1 (1 P f ) 2y h ]

4 4 Usng the approxmaton log(1 + x) x for x 0, and smplfyng, we have: lm log Λ = N (P d P f ) 2y h 0 =1 If we assume that P d P f s constant for all sensors, and elmnate t from the expresson as well as the constant term, and also multplyng by a constant term 1/K we wll acheve a lkelhood rato at the fuson center whch s analogous to maxmum rato combner (MRC): Λ 2 = 1 N h y K =1 In fact, Λ 2 s the frst order approxmaton of the log-lkelhood rato based fuson rule and s asymptotcally accurate for low SNR ( ). Note that the acheved expresson for Λ 2 does not depend on the sensor specfcatons and s only a functon of the channel statstcs. The valdty of the above approxmatons and performance evaluaton have been studed n 4], whch we skp here. The same authors, also study the same problem for mult-hop wreless sensor networks n 5] and found smlar approxmatons for the fuson rule. The dervatons of formulas are smlar to the ones revewed above and thus we do not repeat them here. In the setup we revewed n ths secton, we assumed that all the perpheral nodes send ther decsons to the fuson center. In the next secton we wll revew another approach used n lterature for nteractng wth channel constrants whch s called sensor censorng. IV. SENSOR CENSORING Here we brefly revew the sensor censorng approach. We ust revew the general concepts and do not go nto detals. The dea of censorng sensors was ntroduced by Rago, et al. n 6]. In ths scheme, sensors are assumed to censor ther observaton so that each sensor sends to the fuson center only nformatve observatons. How should we defne nformatve to mnmze the probablty of error was studed n 6] and t was shown that wth condtonally ndependent observatons, the sensors should send ther decsons to fuson center f and only f ther local lkelhood rato do not fall nto a certan sngle nterval. The exact expressons for the ntervals for dfferent frameworks such as Neyman-Pearson and Bayesan are derved n 6]. The mportant fact s that these ntervals depend on the avalable communcaton rates, so, when the avalable rate s low, the ntervals that defnes the no-send regons are larger, and only a few sensors wll transmt to the fuson center. Also, t was shown, through experments, that the performance of censorng scheme s very close to optmal even wth qute severe communcaton rates. The drect consequence of censorng s the savng n communcaton. Also, t wll help the sensors save energy by not sendng unnformatve messages. Fuson of censored decsons transmtted over fadng channels n wreless sensor networks s studed n 7], 8]. In these works t s assumed that the local sensors employ a sensor censorng scheme wth known thresholds and the man focus s on developement of fuson algorthms whle the channels have flat fadng. Both cases, one assumng the knowledge of channel fadng envelope and the other the fadng statstcs, are studed and the optmal lkelhood rato test s derved under each scenaro. In summary, to save power and also decrease communcaton rate, perpheral sensors can employ censorng schemes so that they do not transmt to fuson center f ther lkelhood rato s n a certan nterval 6]. Then, knowng these ntervals and also the nformaton about the channel, the fuson center can use ts modfed lkelhood rato test 7], 8] to decde on the true hypothess. It s shown that the performance of ths scheme s close to optmal. V. GOSSIP-BASED ALGORITHM In ths secton, we are gong to propose a gossp-based fully decentralzed detecton algorthm. To the best of our knowledge, the proposed methods s novel. The algorthms that we studed untl now, were mostly sem-decentralzed, n the sense that all the perpheral sensors transmt a quantzed functon of ther observatons to a central node (fuson center). So, t s clear that we stll have the problems of centralzed systems such as sngle pont of falure, data management (floodng more data to fuson center than t can process), and securty. Also, n practcal wreless sensor networks, because of the power constrants, nodes are only capable of short-range communcatons. Thus, each node can communcate wth only a few other nodes that are close to t. Our proposed gossp-based algorthm, tres to reach a consensus among all the nodes by only local communcatons. Frst, we wll propose our method for the deal channels between the nodes (.e., no fadng or nose) and show that t converges and the soluton s exactly the same as the soluton of the centralzed scheme, and hence globally optmum. Then, we wll modfy t for the case where there s nose and channels between nodes have flat fadng and show by smulaton that our algorthm stll converges and fnd the optmum soluton. Assume that we have a wreless sensor network, wth N sensors, and wthout any central node. Also, assume that we have a bnary hypotheses testng problem. Sensor S can communcates only wth a few other nodes that are n ts communcaton range. We call these nodes the neghbors of sensor S and denote them by the set V. The notaton ntroduced n secton II stll holds. Our proposed algorthm for fully decentralzed detecton s brefly as follows: 1- Each sensor receves an observaton and based on that, computes ts lkelhood rato. Let us denote ths lkelhood rato by : = f(r H 1 ) f(r H 0 ), = 1,..., N 2- The sensors make ntal decsons based on ther observatons. Note that they make ther decsons under ether Bayesan or Neyman-Pearson crteron. Also, note that, snce we have a bnary hypothess testng, under ether of the frameworks, all the nodes mplement a lkelhood rato test. 3- Gosspng: After calculatng the lkelhood ratos and makng ntal decsons, the nodes perform several

5 5 rounds of gosspng. In each round, the nodes communcate two-by-two and update ther lkelhood ratos. If the nodes and communcate wth each other (or gossp ) at teraton k, then, the update procedure s as follows: = = (9) (10) where denotes the value of the lkelhood rato of sensor S at teraton k of the algorthm (0 k). 4- Based on ther new lkelhood ratos, the sensors update ther decsons. As mentoned n step 2, all the nodes mplement a lkelhood rato test wth fxed thresholds. In our algorthm, the lkelhood ratos are updated and the thresholds of the tests are remaned fxed and also are the same for all the sensors (λ 1/N ). 5- Steps 3 and 4 are repeated untl all the sensors decde on the same hypothess (denoted as convergence ). In the followng, frst, we wll show that the mentoned algorthm wll converge to the same decson as the centralzed scheme n the case where there s no nose or fadng. Then we wll modfy our algorthm for the case where there exsts nose and channels have flat fadng. A. Analyss of Proposed Algorthm Havng N condtonally ndependent observatons (r 1,..., r N ), and assumng a Bayes cost or Neyman- Pearson performance crteron, the optmum (centralzed) test for the bnary hypotheses testng s well known to be: =1 Λ(r ) H1 λ (11) H 0 where n NP framework, λ can be found based on the probablty of error crteron, and n Bayesan framework t can be found based on the pror probabltes of the two hypotheses and the costs. By optmum decson, we mean a decson whch mnmzes the probablty of the error. In a fully decentralzed sensor network, each node s only aware of ts own lkelhood rato and can perform a test lke: Λ (r ) H1 H 0 In our method we have equal thresholds for all the nodes, and choose them n a way such that ther product s equal to the approprate λ n (11). Therefore, λ = λ 1/N, = 1,..., N. These thresholds reman fxed through the algorthm. However, each node updates ts lkelhood rato n each teraton of the algorthm. The procedure s as follows: n each teraton, nodes gossp two-by-two wth each other and update ther lkelhood ratos as descrbed n (9) and (10). Note that although the ndvdual lkelhood ratos change, the product of the lkelhood ratos of all nodes reman fxed through the algorthm. In other words: =1 = =1 λ, 1 k (12) It s clear that only has the classcal defnton of the lkelhood rato for sensor S, and hence equals to f(y H1) f(y H 0). However, we stll call the updated versons, for 1 k, lkelhood ratos for ease of notaton. Then, based on the updated versons of lkelhood ratos, the nodes revse ther decsons. The algorthm wll converge f all the nodes have the same decsons. Assume that at stage k all the nodes have the same decsons. Wthout loss of generalty, assume that they all decde on H 1. Thus: > λ = λ 1/N, = 1,..., N Multplyng all the nequaltes, we have: =1 =1 > λ > λ (usng (12)) Whch shows that H 1 s also the soluton of the centralzed test, and hence optmum (mnmzes the probablty of error). Thus, f all the nodes agree wth each other and reach a consensus decson, then ther decson matches the centralzed decson exactly. However, a bg queston remans; Does ths algorthm converge or not? Note that, n order to prove that the mentoned algorthm converges, t s enough to show that as k : =1 1/N, = 1,..., N (13) The reason s that n ths case, all the lkelhood ratos become equal. Also, we know that the thresholds are equal by defnton. Therefore, all the nodes perform the same lkelhood rato test and thus get the same result (convergence). The type of convergence that we are concerned about n (13), s convergence n expectaton. In the followng theorem, we prove that (13) s true and hence, our proposed algorthm converges. Theorem 2. Assume that we have N nodes wth ntal values, = 1,..., N. Consder the followng gossp algorthm: At each teraton, nodes choose one of ther neghbors at random and gossp wth each other. When two nodes gossp wth each other, they update ther values accordng to the followng procedure: = (14) = (15) where denotes the value of the lkelhood rato of sensor S at teraton k of the algorthm (0 k). Ths gossp algorthm converges n expectaton and we have: lm E k ] = =1 1/N, = 1,..., N (16)

6 6 The proof can be found n Appendx A. Another mportant queston s: How fast does the algorthm converge? or How many messages do we need to send among sensors n order to converge? Ths can hghly affect the energy consumpton of the sensors (the more transmtted messages, the more battery consumpton), and also the delay of the system. Actually there are many fast gossp algorthms that we can employ. For example, the algorthm proposed n 9], acheves ɛ-accuracy wth hgh probablty after O ( ) n log log n log kn ɛ messages. B. Consderng Channel Constrants Untl now, we dd not consder the effects of nose and fadng n our algorthm. Now, we are gong to consder these effects. Snce, the nodes are communcatng only two-by-two, we only need to consder the followng case: Assume that nodes and communcate wth each other at teraton k over a channel wth flat fadng and addtve whte Gaussan nose (see Fg. 2). Also assume that node sends ts lkelhood rato,, to node. If we name the message receved at sensor from sensor as y (k), then we can wrte: C. Performance Evaluaton For evaluatng the performance of the proposed algorthm, we performed t on random geometrc graph wth 25 nodes. In our experment the two hyotheses have equal pror probabltes. At the begnng, each node takes an observaton whch s corrupted by nose and computse ts log-lkelhood rato. Then all nodes try to reach a consus by gosspng ther lkelhood ratos n several rounds (n our program, we performed teratons). We consdered two measure of performance: probablty of detecton, Pr(H 1 H 1 ), and probablty of false alarm, Pr(H 1 H 0 ). The former shows the probablty of detectng the H 1 hypotheses and we want t to be as hgh as possble. The latter s the probablty of detctng H 1 whle H 0 s true, and clearly we want t to be as low as possble. We plotted these two measures for dfferent SNRs. Actually, we computed the average of these measures over all the nodes and for several repettons of the algorthm (5000 tmes). It s clear from Fg. 3 and Fg. 4 that as SNR ncreases, we have the hgher probablty of detecton and lower probablty of false alarm. And, of course, f the SNR s large enough, the algorthm works perfectly (Pr(detecton) = 1 and Pr(false alarm) = 0). y (k) = h + n where h s the fadng coeffcent (envelope) of the channel Sensor (k ) h n (k) y Sensor Probablty of Detecton Fg. 2. Gosspng wth channel constrants (AWGN and flat fadng channel) between sensors and, and n s Gaussan nose wth zero mean and varance. Upon recevng y (k), sensor tres to estmate. Snce h s known to sensor (cahnnel aware), the maxmum lkelhood of s smply: = y(k) (17) h SNR Fg. 3. Probablty of detecton versus SNR. As SNR ncreases, the probablty of detecton approaches So, the algorthm s exactly as before, except for the update procedure, whch now changes to: = = (y (k 1) (y (k 1) /h ) (18) /h ) (19) Probablty of False Alarm Note that the only dfference from the smple case s that and are replaced by ther ML estmates for the other sensor, whch s derved as n (17). Provng theoretcally that ths algorthm converges to the same decson as the centralzed scheme s more complcated n ths case. However, n the next subsecton we wll show ths by some smulatons SNR Fg. 4. Probablty of false alarm versus SNR. As SNR ncreases, the probablty of false alarm approaches 0

7 7 The result s that for large enough SNR (about 10 here), the algorthm works well and the event of nterest s detected wth probablty one. Also, the probablty of wrong detecton goes to zero as we ncrease the SNR. So, we showed that even wth channel constrants our proposed algorthm converges to the same decson as the centralzed scheme. D. Comparson Wth Other Algorthms The man dfference of our proposed algorthm wth the prevous ones s that t s fully decentralzed. In other words, there s no central node (fuson center) n our algorthm, whle n the other ones, all the nodes send ther messages to the fuson center and the fnal detecton process s performed centrally. Also, after performng the gossp-based algorthms, all the nodes are aware of the fnal decson whle n the other algorthms only central node knows about the detected hypothess. However, n our algorthm all the nodes should be aware of the channel fadng coeffcents, whle n the other ones only the fuson center needs to have such nformaton about the channel. In case of performance, our algorthm performs slghtly better than the prevous ones. For example the algorthms proposed n 4] acheve same probabltes of detecton as our proposed algorthm, wth hgher SNRs. VI. CONCLUSION In ths proect, we revewed the problem of channel aware decentralzed detecton n wreless sensor networks. Frst, we revewed the classcal framework, where we do not consder power and channel constrants and showed that wth personby-person optmzaton we wll get a set of lkelhood rato tests whose thresholds depend on each other. Then, we revewed some channel aware algorthms that take nto account the constrants of the wreless sensor networks n real world, such as power constrants, nose, and fadng. Frst, we derved some approxmatons for the fuson center lkelhood rato n the presence of nose and flat fadng channels n the cases of large and low SNR. Then, we brefly revewed the sensor censorng approach n whch nodes only transmt to the fuson center f ther observatons are nformatve (ther lkelhood rato do not belong to a certan nterval). Fnally, we proposed a totally decentralzed and channel aware algorthm for decentralzed detecton, whch s based on gosspng. We proved the optmalty and convergence of the algorthm n the case where there s no nose or fadng. The performance of the proposed algorthm for the case of AWGN and flat fadng channels have been evaluated through smulatons and t was shown that for large enough SNRs, the algorthm performs perfectly (Pr(detecton)=1 and Pr(false alarm)=0). REFERENCES 1] J. N. Tstskls, Decentralzed Detecton, Advances n Sgnal Processng, vol. 2, pp , ] R. Tenney and N. Sandell, Detecton wth dstrbuted sensors, Aerospace and Electronc Systems, IEEE Transactons on, vol. AES- 17, no. 4, pp , ] Q. Yan and R. Blum, Dstrbuted sgnal detecton under the Neyman- Pearson crteron, Informaton Theory, IEEE Transactons on, vol. 47, pp , May ] B. Chen, R. Jang, T. Kasetkasem, and P. Varshney, Channel aware decson fuson n wreless sensor networks, Sgnal Processng, IEEE Transactons on, vol. 52, no. 12, pp , ] Y. Ln, B. Chen, and P. Varshney, Decson fuson rules n multhop wreless sensor networks, Aerospace and Electronc Systems, IEEE Transactons on, vol. 41, no. 2, pp , ] C. Rago, P. Wllett, and Y. Bar-Shalom, Censorng sensors: a lowcommuncaton-rate scheme for dstrbuted detecton, Aerospace and Electronc Systems, IEEE Transactons on, vol. 32, no. 2, pp , ] R. Jang and B. Chen, Fuson of censored decsons n wreless sensor networks, Wreless Communcatons, IEEE Transactons on, vol. 4, no. 6, pp , ] R. Jang and B. Chen, Decson fuson wth censored sensors, n Acoustcs, Speech, and Sgnal Processng, Proceedngs. (ICASSP 04). IEEE Internatonal Conference on, vol. 2, pp vol.2, May ] K. Tsanos and M. Rabbat, Fast decentralzed averagng va multscale gossp, Proc. IEEE Dstrbuted Computng n Sensor Systems, June ] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, Randomzed gossp algorthms, IEEE/ACM Trans. Netw., vol. 14, pp , June APPENDIX A PROOF OF THEOREM 2 We use the result of 10] to prove Theorem 2. It was shown n 10] that: Lemma 1. Assume that we have N nodes wth ntal values x (0), = 1,..., N. Consder the followng gossp algorthm: At each teraton, nodes choose one of ther neghbors at random and gossp wth each other. When two nodes ( and ) gossp wth each other, they update ther values accordng to the followng procedure: x (k) = x (k) = x(k 1) + x (k 1) 2 (20) where x (k) denotes the value of the sensor at teraton k of the algorthm (0 k). Ths gossp algorthm converges n expectaton and we have: lm E k x (k) ] = N =1 x(0) ], = 1,..., N (21) N The proof of the lemma can be found n 10]. Now, f we do the followng replacement: x (k) log( ) for = 1,..., N and 0 k (22) then Lemma 1 transforms to Theorem 2, because the update procedure n Theorem 2: = (23) s equvalent to ( ) ( ) ( ) log log + log = 2 whch s the update rule of Lemma 1. Also, we have: 1/N ) N log =1 (Λ log (0) = N =1 (24) (25) whch shows that Theorem 2 s equvalent to Lemma 1 n the log doman and thus the proof s complete.

8 8 APPENDIX B MATLAB CODES 1 % Wreless Proect 2 % Gossp-Based Decentralzed Detecton 3 % Mlad Kharratzadeh 4 % Aprl, % Intalzatons 7 n = 25; 8 SNR = logspace(-1,2,200); 9 K=5000; 10 A = RGG(n,0.4); % Adacency Matrx 11 H = abs(raylrnd(ones(n,n))).* A; 12 LLR = zeros(n,1); % Local Log-Lkelhood Rato 13 U = zeros(n,1); % Local Sensors Decsons 14 d = sum(a,2)'; 15 sz = sze(a); 16 edges = fnd(a == 1); 17 nel = numel(edges); 18 l=1; 19 P_Detecton = zeros(200,1); 20 P_FalseAlarm = zeros(200,1); % Performng Algorthm 23 for snr=snr 24 detected = 0; 25 falsealarm = 0; 26 hyp1 = 0; 27 hyp0 = 0; 28 sgma = 1/sqrt(snr); 29 for =1:K 30 hyp = rand>0.5; 31 r = hyp*ones(n,1) + sgma*randn(n,1); % Observaton at sensors 32 LLR = (2*r-1)/(2*sgmaˆ2); % Calculatng local Log-LRs 33 for =1: rand_, rand_] = nd2sub(sz, edges(rand(nel))); 35 temp = LLR(rand_)+LLR(rand_); 36 % Updatng log-lkelhood ratos 37 LLR(rand_)=(temp+(sgma*randn)/H(rand_,rand_))/2; 38 LLR(rand_)=(temp+(sgma*randn)/H(rand_,rand_))/2; 39 end 40 U = (LLR>0); % Makng fnal decsons 41 f hyp==1 42 detected = detected + sum(u==1); 43 hyp1 = hyp1 + 1; 44 else 45 falsealarm = falsealarm + sum(u==1); 46 hyp0 = hyp0 + 1; 47 end 48 end P_Detecton(l) = detected / (hyp1*n); 51 P_FalseAlarm(l) = falsealarm / (hyp0*n); 52 l = l+1; 53 end semlogx(snr,p_detecton); 56 hold on; 57 semlogx(snr,p_falsealarm); 58 hold off; 59 fgure; 60 plot(sort(p_falsealarm),sort(p_detecton));

9 9 1 % Producng a random geometrc graph and returnng ts adacency matrx 2 functon A = RGG(n,r) 3 4 A = zeros(n,n); 5 X = rand(n,1); 6 Y = rand(n,1); 7 8 % Plottng the graph: 9 10 % scatter(x,y,'red','flled'); 11 % hold on; for =1:n 14 for =:n 15 f (((X()-X())ˆ2 + (Y()-Y())ˆ2) rˆ2) && ( ) 16 A(,) = 1; 17 % lne(x() X()], Y() Y()],'Color','k','LneWdth',1.5); 18 % hold on; 19 end 20 end 21 end % hold off; 24 A = A + A';