Target Localization in Wireless Sensor Networks using Error Correcting Codes

Target Localization in Wireless Sensor etwors using Error Correcting Codes Aditya Vempaty, Student Member, IEEE, Yunghsiang S. Han, Fellow, IEEE, ramod K. Varshney, Fellow, IEEE arxiv:36.452v2 stat.a 4 ov 23 Abstract In this wor, we consider the tas of target localization using quantized data in Wireless Sensor etwors WSs). We propose a computationally efficient localization scheme by modeling it as an iterative classification problem. We design coding theory based iterative approaches for target localization where at every iteration, the Fusion Center FC) solves an M-ary hypothesis testing problem and decides the Region of Interest ROI) for the next iteration. The coding theory based iterative approach wors well even in the presence of Byzantine malicious) sensors in the networ. We further consider the effect of non-ideal channels. We suggest the use of soft-decision decoding to compensate for the loss due to the presence of fading channels between the local sensors and the FC. We evaluate the performance of the proposed schemes in terms of the Byzantine fault tolerance capability and probability of detection of the target region. We also present performance bounds which help us in designing the system. We provide asymptotic analysis of the proposed schemes and show that the schemes achieve perfect region detection irrespective of the noise variance when the number of sensors tends to infinity. Our numerical results show that the proposed schemes provide a similar performance in terms of Mean Square Error MSE) as compared to the traditional Maximum Lielihood Estimation MLE) but are computationally much more efficient and are resilient to errors due to Byzantines and non-ideal channels. Index Terms Target Localization, Wireless Sensor etwors, Error Correcting Codes, Byzantines I. ITRODUCTIO Wireless sensor networs WSs) have been extensively employed to monitor a region of interest ROI) for reliable detection/estimation/tracing of events 4. In this wor, we focus on target localization in WSs. Localization techniques proposed in the literature for sensor networs include direction of arrival DOA), time of arrival TOA) and time-difference This wor was supported in part by CASE: The Center for Advanced Systems and Engineering, a YSTAR center for advanced technology at Syracuse University; AFOSR under Grant FA955---263; the ational Science of Council SC) of Taiwan under Grants SC SC 2-222- E- -6 -MY3 and SC -222-E--69-MY3. arts of this wor were presented at the 38th IEEE International Conference on Acoustics, Speech, and Signal rocessing ICASS 23) and at the 3th International Symposium on Communications and Information Technologies ISCIT 23). art of Y. S. Han s wor was completed during his visit of Syracuse University from 22 to 23. A. Vempaty and. K. Varshney are with the Dept. of Electrical Engineering and Computer Science, Syracuse University, Syracuse, Y USA email:avempaty, varshney}@syr.edu). Y. S. Han is with the Dept. of Electrical Engineering, ational Taiwan University of Science and Technology, Taipei, Taiwan email: yshan@mail.ntust.edu.tw). Copyright c) 23 IEEE. ersonal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. of arrival TDOA) based methods 5 6. Recent research has focused on developing techniques which do not suffer from imperfect time synchronization. Received signal strength based methods, which do not suffer from imperfect synchronization and/or extensive processing, have been proposed which employ least-squares or maximum-lielihood ML) based source localization techniques 7 8. In WSs, due to power and bandwidth constraints, each sensor, instead of sending its raw data, sends quantized data to a central observer or Fusion Center FC). The FC combines these local sensors data to estimate the target location. Secure localization is very important as potential malicious sensors may attempt to disrupt the networ and diminish its capability. Only in the recent past, researchers have investigated the problem of security threats 9 on sensor networs. We focus on one particular class of security attacs, nown as the Byzantine data attac also referred to as the Data Falsification Attac). A Byzantine attac involves malicious sensors within the networ which send false information to the FC to disrupt the global inference process. In our previous wor, we have analyzed target localization in WSs in the presence of Byzantines. By considering the osterior Cramér Rao bound or osterior Fisher Information as the performance metric, we analyzed the degradation in system performance in the presence of Byzantines. We showed that the FC becomes blind to the local sensor s data when the fraction of Byzantines is greater than 5%. When the FC becomes blind, it is not able to use any information received from the local sensors and estimates the target location based only on prior information. In order to mae the networ robust to such attacs, we considered mitigation techniques. We proposed a Byzantine identification scheme which observes the sensors behavior over time and identifies the malicious sensors. We also proposed a dynamic non-identical threshold design for the networ which maes the Byzantines ineffective. An important element of WSs is the presence of nonideal wireless channels between sensors and the FC 2 3. These non-ideal channels corrupt the quantized data sent by the local sensors to the FC. This causes errors which deteriorates the inference performance at the FC. One way to handle the channel errors is to use error correcting codes 4 5. In 6, target localization based on maximum lielihood estimation at the FC was considered and coding techniques were proposed to handle the effect of imperfect channels between sensors and fusion center. In this wor, we propose the use of coding theory techniques to estimate the location of the target in WSs. In our preinary wor 7 8, we have shown the feasibility

2 of our approach by providing simulation/numerical results. In this paper, we develop the fundamental theory and derive asymptotic performance results. We first consider the code design problem in the absence of channel errors and Byzantine faults. The proposed scheme models the localization problem as an iterative classification problem. The scheme provides a coarse estimate in a computationally efficient manner as compared to the traditional ML based approach. We present performance analysis of the proposed scheme in terms of detection probability of the correct region. We show analytically that the scheme achieves perfect performance in the asymptotic regime. We address the issues of Byzantines and channel errors subsequently and modify our scheme to handle them. The error correction capability of the coding theory based approach provides Byzantine fault tolerance capability and the use of soft-decoding at the FC provides tolerance to the channel errors. In the remainder of the paper, we refer to this coding theory based localization approach as coding approach. The schemes proposed in this paper show the benefit of adopting coding theory based techniques for signal processing applications. The remainder of the paper is organized as follows: In Section II, we describe the system model used and lay out the assumptions made in the paper. We also present a brief overview of Distributed Classification Fusion using Error Correcting Codes DCFECC) 9 and Distributed Classification Fusion using Soft-decision Decoding DCSD) 2 approaches. We propose our basic coding scheme for target localization in Section III. The performance of the proposed scheme in terms of region detection probability is analyzed in this section. We extend this scheme to the exclusion method based coding scheme in Section IV to mitigate the effect of Byzantines in the networ. We present some numerical results showing the benefit of the proposed schemes compared to the traditional maximum lielihood based scheme. We also present a discussion on system design based on the performance analysis carried out in this section. We consider the presence of non-ideal channels in Section V and modify our decoding rule to mae it robust to fading channels. We conclude our paper in Section VI with some discussion on possible future wor. A. System model II. RELIMIARIES Let sensors be randomly deployed not necessarily in a regular grid) in a WS as shown in Fig. to estimate the unnown location of a target at θ x t, y t, where x t and y t denote the coordinates of the target in a 2-D Cartesian plane. We assume that the location of the sensors is nown to the Fusion Center FC). We also assume that the signal radiated from this target follows an isotropic power attenuation model 2. The signal amplitude a i received at the i th sensor is given by ) n a 2 d i, ) d i where is the power measured at the reference distance d, d i is the distance between the target and the i th sensor whose location is represented by L i x i, y i for i, 2, and n is the path loss exponent. In this wor, without loss of generality, we assume d and n 2. The signal amplitude measured at each sensor is corrupted by independent and identically distributed i.i.d.) zero-mean additive white noise with complementary cumulative distribution function given by F ; σ 2 ): s i a i + n i, 2) where s i is the corrupted signal at the i th sensor and the noise n i F ; σ 2 ) with variance σ 2. Fig.. 8 7 6 5 4 3 2 2 3 4 5 6 7 8 Wireless sensor networ layout for target localization Sensors Target Due to energy and bandwidth constraints, the local sensors quantize their observations using threshold quantizers and send binary quantized data to the FC: s i < η i D i, 3) s i > η i where D i is the quantized data at the i th sensor and η i is the threshold used by the i th sensor for quantization. The FC fuses the data received from the local sensors and estimates the target location. Traditional target localization uses MLE 2: ˆθ arg max pu θ), 4) θ where u u, u 2,, u is the vector of quantized observations received at the FC. As pointed out in the later sections, u and D can be different due to the presence of Byzantines and/or imperfect channels between local sensors and FC. B. An Overview of Distributed Classification Approaches ) DCFECC 9: In this subsection, we give a brief overview of Distributed Classification Fusion using Error Correcting Codes DCFECC) approach proposed in 9. In 9, the authors propose an approach for M-ary distributed classification using binary quantized data. After processing the observations locally, possibly in the presence of sensor faults, the local sensors transmit their local decisions to the FC. In the DCFECC approach, a code matrix C is selected to

3 perform both local decision and fault-tolerant fusion at the FC. The code matrix is an M matrix with elements c j+)i, }, j,,, M and i,,. Each hypothesis H j is associated with a row in the code matrix C and each column represents a binary decision rule at the local sensor. The optimal code matrix is designed off-line using techniques such as simulated annealing or cyclic column replacement 9. After receiving the binary decisions u from local sensors, the FC performs minimum Hamming distance based fusion and decides on the hypothesis H j for which the Hamming distance between row of C corresponding to H j for j,, M and the received vector u is minimum. It is important to note that the above scheme is under the assumption that > M and the performance of the scheme depends on the minimum Hamming distance d min of the code matrix C. 2) DCSD 2: In this subsection, we present a brief overview of Distributed Classification using Soft-decision Decoding DCSD) approach proposed in 2. This approach uses a soft-decision decoding rule as opposed to the hard-decision decoding rule used in DCFECC approach. The use of softdecision decoding maes the system robust to fading channels between the sensors and the FC. The basic difference between the two approaches DCFECC and DCSD) is the decoding rule. In DCFECC, the minimum Hamming distance rule is used. In the presence of fading channels, the received data at the FC is analog although the local sensors transmit quantized data based on the code matrix C as described before. Then, the FC can use hard-decision decoding to determine the quantized data sent by the local sensors and use minimum Hamming distance rule to mae a decision regarding the class. However, in 2, the authors show that the performance can deteriorate when hard-decision decoding is used. Instead, they propose a soft-decision decoding rule based on the channel statistics to mae a decision regarding the class. We sip the derivation of the soft-decision decoding rule but present the decoding rule here for the case when binary quantizers are used at the local sensors, i.e., the elements of the code matrix are or. Let the analog data received at the FC from the local sensors be v v,, v when the local sensors transmit u u,, u, where u i / is decided by the code matrix C. For fading channels between the local sensors and the FC, v i and u i are related as follows v i h i ) ui Eb + n i, 5) where h i is the channel gain that models the fading channel, E b is the energy per bit and n i is the zero mean additive white Gaussian noise. Define the reliability of the received data v i as ψ i ln vi ui ) ui ) + vi ui ) ui ) v i u i ) u i ) + v i u i ) u i ) for i,, }. Here v i u i ) can be obtained from the statistical model of the fading channel considered and u i d s) for s, d, } is the probability that the decision is d given s is present at the bit i before local decision maing and is given as follows 6) u i d s) M j u i d H j ) i H j s). 7) u i d H j ) depends on the code matrix while i H j s) is the probability that the hypothesis H j is true given s is present at the bit i column i of the code matrix) before local decision maing, and can be expressed as where i H j s) i s H l ) i s H j ) M l i s H l ), if c l+)i s, if c l+)i s 8). 9) Then the decoding rule is to decide the hypothesis H j where j arg min j M d F ψ, c j+ ). Here d F ψ, c j+ ) i ψ i ) c j+)i ) 2 is the distance between ψ ψ,, ψ and j + ) th row of C. III. LOCALIZATIO USIG ITERATIVE CLASSIFICATIO In this section, we propose the localization scheme using iterative classification. Our algorithm is iterative in which at every iteration, the ROI is split into M regions and an M-ary hypothesis test is performed at the FC to determine the ROI for the next iteration. The FC, through feedbac, declares this region as the ROI for the next iteration. The M-ary hypothesis test solves a classification problem where each sensor sends binary quantized data based on a code matrix C. The code matrix is of size M with elements c j+)i, }, j,,, M and i,,, where each row represents a possible region and each column i represents i th sensor s binary decision rule. After receiving the binary decisions u u, u 2,, u from local sensors, the FC performs minimum Hamming distance based fusion. In this way, the search space for target location is reduced at every iteration and we the search based on a pre-determined ping criterion. The optimal splitting of the ROI at every iteration depends on the topology of the networ and the distribution of sensors in the networ. For a given networ topology, the optimal region split can be determined offline using -means clustering 2 which yields Voronoi regions 22 containing equal number of sensors in every region. For instance, when the sensors are deployed in a regular grid, the optimal splitting is uniform as shown in Fig. 2. In the remainder of the paper, we consider a symmetric sensor deployment such as a grid. Such a deployment results in a oneto-one correspondence between sensors across regions which is required in our derivations. Further discussion is provided in the later part of this section. In this section, the sensors are assumed to be benign and the channels between the local sensors and the FC are assumed to be ideal. Therefore, in this section, the binary decisions received at the FC are the same as the binary decisions made by the local sensors, i.e., u i D i, for i,,. We relax these assumptions in the later sections. The FC estimates the target location using the received data u.

4 8 7 6 H H Sensors Target are given by c j+)i if i S j otherwise, 2) Fig. 2. 5 4 3 2 H 2 H 3 2 3 4 5 6 7 8 Equal region splitting of the ROI for the M-hypothesis test for j,,, M and i ROI. The above construction can also be viewed as each sensor i using a threshold ηi for quantization as described in 3)). Let each region Rj correspond to a location θj for j,,, M, which in our case is the center of the region Rj. Each sensor i decides on a if and only if the target lies in the region R r i). Every sensor i, therefore, performs a binary hypothesis test described as follows: H : θ R r i) H : θ / R r i). 3) A. Basic Coding Based Scheme In this subsection, we present the basic coding based scheme for target localization. Since there are sensors which are split into M regions, the number of sensors in the new ROI after every iteration is reduced by a factor of M. After M iterations, the number of sensors in the ROI are and, therefore, the code matrix at the + ) th iteration would be of size M. Since the code matrix should always M have more columns than rows, < log M, where is the number of iterations after which the scheme terminates. M After iterations, there are only sensors present in the ROI and a coarse estimate ˆθ ˆθ x, ˆθ y of the target s location can be obtained by taing an average of locations of the sensors present in the ROI: M and ˆθ x M ˆθy M i ROI i ROI x i ) y i, ) where ROI is the ROI at the last step. Since the scheme is iterative, the code matrix needs to be designed at every iteration. Observing the structure of our problem, we can design the code matrix in a simple and efficient way as described below. As pointed out before, the size of the code matrix C at the +) th iteration is M, M where. Each row of this code matrix C represents a possible hypothesis described by a region in the ROI. Let Rj denote the region represented by the hypothesis H j for j,,, M and let Sj represent the set of sensors that lie in the region Rj. Also, for every sensor i, there is a unique corresponding region in which the sensor lies and the hypothesis of the region is represented as r i). It is easy to see that Sj i ROI r i) j}. The code matrix is designed in such a way that for the j th row, only those sensors that are in Rj have as their elements in the code matrix. In other words, the elements of the code matrix We assume that is divisible by M for,,..., log M. If d i,θ j represents the Euclidean distance between the i th sensor and θj for i, 2,, and j,,, M, then r i) arg min d i,θ l. Therefore, the condition θ l R r i) can be abstracted as a threshold η i on the local sensor signal amplitude given by η i d i,θ. 4) r i) This ensures that if the signal amplitude at the i th sensor is above the threshold η i, then θ lies in region R r i) leading to minimum distance decoding. B. erformance Analysis In this subsection, we present the performance analysis of the proposed scheme. Although the performance metric in this framewor is the Mean Square Error MSE), it is difficult to obtain a closed form representation for MSE. Therefore, typically, one uses the bounds on MSE to characterize the performance of the estimator. In our previous wors 2,, we analytically derived the expressions of MSE bound osterior Cramér Rao Lower Bound) on target localization under both non-adversarial 2 and adversarial scenarios. An analytically tractable metric to analyze the performance of the proposed scheme is the probability of detection of the target region. It is an important metric when the final goal of the target localization tas is to find the approximate region or neighborhood where the target lies rather than the true location itself. Since the final ROI could be one of the M regions, a metric of interest is the probability of zooming into the correct region. In other words, it is the probability that the true location and the estimated location lie in the same region. The final region of the estimated target location is the same as the true target location, if and only if we zoom into the correct region at every iteration of the proposed scheme. If d denotes the detection probability probability of correct classification) at the + ) th iteration, the overall detection probability is given by

5 D d. 5) Exact Analysis: Let us consider the + ) th iteration and define the received vector at the FC as u u, u 2,, u, where are the number of local sensors reporting their data to FC at +) th iteration. Let Dj be the decision region of j th hypothesis defined as follows: D j u d H u, c j+) d H u, c l+) for l M }, where d H, ) is the Hamming distance between two vectors, and c j+ is the codeword corresponding to hypothesis j in code matrix C. Then define the reward ru j, associated with the hypothesis j as ru j, q when u D u j, 6) otherwise where qu is the number of decision regions to whom u belongs to. ote that qu can be greater than one when there is a tie at the FC. Under such scenarios when qu >, we brea the tie using random decision. Since the tie-breaing rule is to choose one of them randomly, which is successful with probability q, the reward is given by 6). According u to 6), the detection probability at the + ) th iteration is given by d M j M H j ) M j u Dj u,} u H j )r j, u ) u i Hj ) i qu, 7) where u i H j ) denotes the probability that the sensor i sends the bit u i, }, i, 2,,, when the true target is in the region Rj corresponding to H j at the +)th iteration. From the system model described before, we get u i H j ) E θ H j u i θ, H j ). 8) Since 8) is complicated, it can be approximated using θj which is the center of the region Rj. 8) now simplifies to u i H j ) F η i a ij; σ 2), 9) where ηi is the threshold used by the i th sensor at th iteration, σ 2 is the noise variance, a ij is the signal amplitude received at the i th sensor when the target is at θj and F x; σ 2 ) is the complementary cumulative distribution function of noise at the local sensors. Using 5), the probability of detection of the target region can be found as the product of detection probabilities at every iteration. It is clear from the derived expressions that the exact analysis of the detection probability is complicated and, therefore, we derive some analytical bounds on the performance of the proposed scheme. erformance bounds: In this section, we present the performance bounds on our proposed coding based localization scheme. For our analysis, we will use the lemmas in 23, which are stated here for the sae of completeness. Lemma 3. 23): Let Z j } j be independent antipodal random variables with rz j q j and rz j q j. If λ mez + + Z m /m <, then rz + + Z m } λ 2 m) m/2. 2) Using this lemma, we now present the performance bounds on our proposed scheme. Lemma 3.2: Let θ Rj be the fixed target location. Let e θ) be the misclassification probability of the target region given θ at the + ) th iteration. For the received vector of /M observations at the + ) th iteration, u u,, u, assume that for every j, l M and l j, qi,j < M, 2) M + i S j S l where qi,j z i,j θ}, z i,j 2u i c j+)i ), and C c j+)i} is the code matrix used at the + )th iteration. Then ) 2 e i S 2q j θ) S i,j ) l l M,l j M ) d 2 m, d m, /2 22) λ j,maxθ) ) 2 ) d m, /2, 23) where d m, is the minimum Hamming distance of the code matrix C given by d m, 2 due to the structure of our M + code matrix and λ j,maxθ) max l M,l j d m, i Sj S l 2q i,j ). 24) roof: Let d H, ) be the Hamming distance between two vectors, for fixed θ R j, e θ) detected region Rj θ } } d H u, c j+) min d Hu, c l+) θ l M,l j d H u, c j+) d H u, c l+) θ } l M,l j l M,l j i,, :c l+)i c j+)i } zi,j θ. 25) Using the fact that c l+)i c j+)i for all i S j S l, l j, we can simplify the above equation. Also, observe that z i,j } i are independent across the sensors given θ.

6 According to 2) in 23, λ m d m, i c l+)i c j+)i )2q i,j ) d m, i Sj S l d m, i S j S l 2q i,j ) 2q i,j 2 M 26) since c l+)i c j+)i for all i S j S l, l j. Here, we have used the fact that cardinality of Sj /M for all j, and Sj and S l are disjoint sets for all l j. Condition λ m < of Lemma 3. is then equivalent to condition 2). Therefore, using Lemma 3. and 26), we have i,, :c l+)i c j+)i } i S j S l ) 2 2qi,j ) d 2 m, zi,j θ d m, /2. 27) Substituting 27) into 25), we have 22). ote that condition 2) λ m < ) implies λ j,max θ) < by definition. Hence, 23) is a direct consequence from 22). The probabilities qi,j u i c j+)i θ} can be easily computed as below. For j M and i, if i Sj, q i,j u i θ} F η i a i ; σ 2), 28) where ηi is the threshold used by the i th sensor at + ) th iteration, σ 2 is the noise variance, a i is the amplitude received at the i th sensor given by ) when the target is at θ. If i / Sj, qi,j u i θ}. Before we present our main theorem, for ease of analysis, we give an assumption that will be used in the theorem. ote that, our proposed scheme can still be applied to those WSs where the assumption does not hold. Assumption 3.3: For any target location θ Rj and any, there exists a bijection function f from Sj to Sl, where l M and l j, such that and fi j ) i l, η i j η i l, d ij < d il, where i j Sj, i l Sl, and d i j d il ) is the distance between θ and sensor i j i l ). One example of WSs that satisfies this assumption is given in Fig. 3. For every sensor i j Sj, due to symmetric region splitting, there exists a corresponding sensor i l Sl which is symmetrically located as described in the following: Join the centers of the two regions and draw a perpendicular bisector to this line as shown in Fig. 3. The sensor i l Sl is the sensor located symmetrically to sensor i j on the other side of the line L. These are the sensors for which the thresholds are the same. In other words, due to the symmetric placement of the sensors, ηi j ηi l c.f. 4)). Clearly, when θ Rj, d ij < d il. Fig. 3. 8 7 6 5 4 3 2 H j Line L H l 2 3 4 5 6 7 8 ROI with an example set of paired sensors Theorem 3.4: Let D be the probability of detection of the target region given by 5), where d is the detection probability at the + ) th iteration. Under Assumption 3.3, where and Sensors Target Centers of Regions aired sensors d M ) λ max) 2) d m, /2, 29) λ max max j M λ j,max λ j,max max θ R j λ j,maxθ). roof: First we prove that condition 2) is satisfied by the proposed scheme for all θ when the noise variance, σ 2 <. Hence, the inequality 23) can be applied to the proposed scheme. The probabilities qi,j given by 28) are qi,j F ηi a i; σ 2), for i Sj F ηi a i; σ 2), for i Sl. 3) By Assumption 3.3, there exists a bijection function f from Sj to S l. The sum i S q j S i,j of 2) can be evaluated by l considering pairwise summations as follows. Let us consider one such pair i j Sj, fi j) i l Sl ). Hence, their thresholds are ηi j ηi l η. Then, from 3), q i j,j + q i l,j F η a ij ; σ 2) + F η a il ; σ 2) 3) F η aij ; σ 2) F η a il ; σ 2). ow observe that, by the assumption, a ij > a il d ij d il 32)

7 and, therefore, F η aij ; σ 2) > F η a il ; σ 2) for all finite values of noise variance σ 2. From 32), the sum qi j,j +q i l,j is strictly less than. Therefore, the sum i S q j S i,j < l M d m, M + 2. Therefore, the condition in 2) is satisfied for the code matrix used in this scheme. Hence, e θ) can always be bounded by 23). By using 23), d d can be bounded as follows: M θ Rj } detected region Rj θ Rj } j M M j θ θ θ Rj } detected region Rj θ, θ Rj } dθ M θ θ Rj }e θ) dθ M M j M j θ R j θ R j θ θ R j }M ) λ j,maxθ) ) 2 ) d m, /2 dθ M M λ ) 2 ) dm, /2 j,max M j θ θ Rj } dθ 33) θ R j M M M ) M j λ ) 2 ) dm, /2 max 34) λ max) 2 ) dm, /2. 35) Both 33) and 34) are true since λ j,max < and λ max <. ext we analyze the asymptotic performance of the scheme, i.e., we examine D when approaches infinity. Theorem 3.5: Under Assumption 3.3), roof: We have λ j,max max l M,l j > M + 2 i S j S l d m, i Sj S l D. 2q i,j ) ) 36) for all j M since not all qi,j. Hence, by definition, λ max is also greater than. Since < λ max <, we have < λ max) 2 <. Under the assumption that the number of iterations are finite, for a fixed number of regions M, we can analyze the performance of the proposed scheme under asymptotic regime. Under this assumption, d m, grows linearly with the number of sensors 2 M + for. Then D d M ) λ max) 2 ) d m,/2 M ) ) λ max) 2 ) d m,/2 M ). Hence, the overall detection probability becomes as the number of sensors goes to infinity. This shows that the proposed scheme asymptotically attains perfect region detection probability irrespective of the value of finite noise variance. ote that the above result also holds when M increases with as long as d m, 2 grows with the number of sensors M + for. In other words, our theory can be extended to scenarios when M increases with as long as as for. M + C. umerical Results We now present some numerical results which justify the analytical results presented in the previous subsection and provide some insights. In the previous subsection, we have observed that the performance of the basic coding scheme quantified by the probability of region detection asymptotically approaches irrespective of the finite noise variance. Fig. 4 shows that the region detection probability approaches uniformly as the number of sensors approaches infinity for Gaussian sensor observation noise with variance σ 2. Observe that for a fixed noise variance, the region detection probability increases with increase in the number of sensors. This can also be observed from Table I. Also, for a fixed number of sensors, the region detection probability decreases with σ when the number of sensors is small. But when the number of sensors is large, the reduction in region detection probability with σ is negligible and as, the region detection probability converges to. TABLE I TARGET REGIO DETECTIO ROBABILITY FOR FIXED OISE VARIACE σ 4) WITH VARYIG M 4) Target Region Detection probability 64.6753 52.7982 496.8433

8 Region detection probability.9.8.7.6.5.4.3.2. 64 52 496.5 2 2.5 3 3.5 4 4.5 5 Standard Deviation of noise σ) Fig. 4. Region detection probability versus the standard deviation of noise with varying number of sensors IV. LOCALIZATIO I THE RESECE OF BYZATIES Let us now consider the case when there are Byzantines in the networ. As discussed before, Byzantines are local sensors which send false information to the FC to deteriorate the networ s performance. We assume the presence of B α number of Byzantines in the networ. In this paper, we assume that the Byzantines attac the networ independently where the Byzantines flip their data with probability before sending it to the FC. ote that the Byzantines can flip with any probability ɛ. However, since it has been shown in that the optimal independent attac strategy for the Byzantines is to flip their data with probability, we focus on the optimal attac case which is ɛ. In other words, the data sent by the i th sensor is given by: D i if i th sensor is honest u i D i if i th sensor is Byzantine. 37) For such a system, it has been shown in that the FC becomes blind to the networ s information for α.5. Therefore, for the remainder of the paper, we analyze the system when α <.5. For the basic coding scheme described in Section III-A, each column in C contains only one and every row of C contains exactly s. Therefore, M + the minimum Hamming distance of C 2 is and, at M + the + ) th iteration, it can tolerate a total of at most faults data falsification attacs) due to the presence M + of Byzantines in the networ. This value is not very high and we would lie to extend the basic scheme to a scheme which can handle more Byzantine faults. A. Exclusion Method with Weighted Average As shown above, the scheme proposed in Section III-A has a Byzantine fault tolerance capability which is not very high. The performance can be improved by using an exclusion method for decoding where the two best regions are ept for next iteration and a weighted average is used to estimate the target location at the final step. This scheme builds on the basic coding scheme proposed in Section III-A with the following improvements: Since after every iteration two regions are ept, the code matrix after the th iteration is of size M 2 and the M number of iterations needed to the localization tas needs to satisfy < log M/2. At the final step, instead of taing an average of the sensor locations of the sensors present in the ROI at the final step, we tae a weighted average of the sensor locations where the weights are the -bit decisions sent by these sensors. Since a decision u i would imply that the target is closer to the sensor i, a weighted average ensures that the average is taen only over the sensors for which the target is reported to be close. Therefore, the target location estimate is given by i ROI ˆθ x u ix i i ROI u i 38) and ˆθy i ROI u iy i i ROI u. i 39) One can extend this scheme to consider other weights such as based on Euclidean distance which can be determined after processing the initial data to derive a coarse estimate of the target location. However, further processing is required for this and, therefore, we have not used such a scheme. The exclusion method results in a better performance compared to the basic coding scheme since it eeps the two best regions after every iteration. This observation is also evident in the numerical results presented in Section IV-C. B. erformance analysis Byzantine Fault Tolerance Capability: When the exclusion based scheme described in Section IV-A is used, since the two best regions are considered after every iteration, the fault tolerance performance improves and we can tolerate a total of at most 2+ faults. This improvement in the fault M + tolerance capability can be observed in the simulation results presented in Section IV-C. roposition 4.: The maximum fraction of Byzantines that can be handled at the + ) th iteration by the proposed exclusion method based coding scheme is ited by αf 2 M M 2. roof: The proof is straight forward and follows from the fact that the error correcting capability of the code matrix C at +) th iteration is at most 2+. Since there are 2 M + M sensors present during this iteration, the fraction of Byzantine sensors that can be handled is given by α f 2 M M 2. The performance bounds on the basic coding scheme presented in Section III-B can be extended to the exclusion based coding scheme presented in Section IV-A. We sip the details for the sae of brevity of the paper. When there are Byzantines in the networ, the probabilities qi,j of 28) become q i,j α) F η i a i ; σ 2) + α F η i a i ; σ 2)).

9 We have shown in Section III-B that the detection probability at every iteration approaches as the number of sensors goes to infinity. However, this result only holds when the condition in 2) is satisfied. otice that, in the presence of Byzantines, we have qi,j α) F ηi a i; σ 2)) + α F ηi a i; σ 2), for i Sj α) F ηi a i; σ 2) + α F ηi a i; σ 2)), for i Sl which can be simplified as qi,j α) 2α) F ηi a i; σ 2), for i Sj α + 2α) F ηi a i; σ 2), for i Sl. 4) ow using the pairwise sum approach discussed in Section III-B, we can re-write 32) as follows: q i j,j + q i l,j 2α) F η aij ; σ 2) F η a il ; σ 2),4) which is an increasing function of α since F η a ij ; σ 2) > F η a il ; σ 2) for all finite σ as discussed before. Therefore, when α <.5, the pairwise sum in 4) is strictly less than and the condition 2) is satisfied. However, when α.5, i S j S l qi,j M. Therefore, the condition fails when α.5. It has been shown in that the FC becomes blind to the local sensor s information when α.5. ext we state the theorem when there are Byzantines in the networ. Theorem 4.2: Let α be the fraction of Byzantines in the networs. Under Assumption 3.3), when α <.5, D. ote that the performance bounds derived can be used for system design. Let us consider sensors uniformly deployed in a square region. Let this region be split into M equal regions. From roposition 4., we now that αf is a function of M and. Also, the detection probability equations and bounds derived in Section III-B are functions of M and. Hence, for given fault tolerance capability and region detection probability requirements, we can find the corresponding number of sensors req ) to be used and the number of regions to be considered at each iteration M req ). We now present guidelines for system design of a networ which adopts the proposed approach. Let us suppose that we need to design a system such that we split into M 4 regions after every iteration. How should a system designer decide the number of sensors in order to meet the target region detection probability and Byzantine fault tolerance capability requirements? Table II shows the performance of the system in terms of the target region detection probability and Byzantine fault tolerance capability with varying number of sensors found using the expressions derived in roposition 4. and in Section III-B. From Table II, we can observe that the performance improves with increasing number of sensors. However, as a, TABLE II TARGET REGIO DETECTIO ROBABILITY AD BYZATIE FAULT TOLERACE CAABILITY WITH VARYIG M 4) Target Region Detection probability 32.4253.4688 28.687.4844 52.6994.4922 Byzantine fault tolerance capability system designer, we would lie to minimize the number of sensors that need to be deployed while assuring a minimum performance guarantee. In this example, if we are interested in achieving a region detection probability of approximately.7 and a Byzantine fault tolerance capability close to.5, we get 52 sensors to be sufficient. C. Simulation Results In this section, we present the simulation results to evaluate the performance of the proposed schemes in the presence of Byzantine faults. We analyze the performance using two performance metrics: mean square error MSE) of the estimated location and probability of detection D ) of the target region. We use a networ of 52 sensors deployed in a regular 8 8 grid as shown in Fig. 2. Let α denote the fraction of Byzantines in the networ that are randomly distributed over the networ. The received signal amplitude at the local sensors is corrupted by AWG noise with standard deviation σ 3. The power at the reference distance is 2. At every iteration, the ROI is split into M 4 equal regions as shown in Fig. 2. We the iterations for the basic coding scheme after 2 iterations. The number of sensors in the ROI at the final step are, therefore, 32. In order to have a fair comparison, we the exclusion method after 4 iterations, so that there are again 32 sensors in the ROI at the final step. Fig. 5 shows the performance of the proposed schemes in terms of the MSE of the estimated target location when compared with the traditional maximum lielihood estimation described by 4). The MSE has been found by performing 3 Monte Carlo runs with the true target location randomly chosen in the 8 8 grid. As can be seen from Fig. 5, the performance of the exclusion method based coding scheme is better than the basic coding scheme and outperforms the traditional MLE based scheme when α.375. When α >.375 the traditional MLE based scheme has the best performance. However, it is important to note that the proposed schemes provide a coarse estimate as against the traditional MLE based scheme which optimizes over the entire ROI. Also, the traditional scheme is computationally much more expensive than the proposed coding based schemes. In the simulations performed, the proposed schemes are around 5 times faster than the conventional scheme when the global optimization toolbox in MATLAB was used for the optimization in ML based scheme. The computation time is

2 8 6 4 Basic Coding Approach Coding with Exclusion Localization using MLE scheme improves with increase in the number of sensors and approaches αf.5 asymptotically. Table III shows the reduction of MSE with increasing for a fixed fraction of Byzantines, α. MSE m 2 ) 2 8 6 8 6 4 2 64 52 496 4 2..2.3.4.5 Fraction of Byzantines MSE m 2 ) 8 6 4 Fig. 5. MSE comparison of the three localization schemes 2 very important in a scenario when the target is moving and a coarse location estimate is needed in a timely manner. Fig. 6 shows the performance of the proposed schemes in terms of the detection probability of the target region. The detection probability has been found by performing 4 Monte Carlo runs with the true target randomly chosen in the ROI. Fig. 6 shows the reduction in the detection probability with increase in α when more sensors are Byzantines sending false information to the FC. Region detection probability Fig. 6..9.8.7.6.5.4.3.2. Basic Coding approach Coding with Exclusion..2.3.4.5 Fraction of Byzantines robability of detection of target region as a function of α In order to analyze the effect of the number of sensors on the performance, we perform simulations by changing the number of sensors and eeping the number of iterations the same as before. According to roposition 4., when M 4, the proposed scheme can asymptotically handle up to 5% of the sensors being Byzantines. Figs. 7 and 8 show the effect of number of sensors on MSE and detection probability of the target region respectively when the exclusion method based coding scheme is used. As can be seen from both figures Figs. 7 and 8), the fault-tolerance capability of the proposed Fig. 7. Detection robability of target region Fig. 8...2.3.4.5 Fraction of Byzantines α).9.8.7.6.5.4.3.2. MSE of the target location estimate with varying 64 52 496..2.3.4.5 Fraction of Byzantines α) robability of detection of target region with varying TABLE III MSE OF THE TARGET LOCATIO ESTIMATE FOR FIXED UMBER OF BYZATIES α.25) WITH VARYIG MSE m 2 ) 64 7.79 52.24 496.55 V. SOFT-DECISIO DECODIG FOR O-IDEAL CHAELS In this section, we extend our scheme to counter the effect of non-ideal channels on system performance. Besides the faults due to the Byzantines in the networ, the presence of nonideal channels further degrades the localization performance.

To combat the channel effects, we propose the use of a soft-decision decoding rule, at every iteration, instead of the minimum Hamming distance decoding rule. ote that the code design is independent of the hard-decoding or soft-decoding since according to the code-design, a sensor sends a when the sensor decides that the target is in the same region as the sensor. A. Decoding rule At each iteration, the local sensors transmit their local decisions u which are possibly corrupted due to the presence of Byzantines. Let the received analog data at the FC be represented as v v, v2,, v, where the received observations are related to the transmitted decisions as follows: v i h i ) u i Eb + n i, i,, }, 42) where h i is the fading channel coefficient, E b is the energy per channel bit and n i is the additive white Gaussian noise with variance σf 2. In this paper, we assume the channel coefficients to be Rayleigh distributed with variance σh 2. We assume that the FC does not have nowledge of the fraction of Byzantines α. Hence, instead of adopting the reliability given in 6), we propose to use a simpler reliability measure ψi in our decoding rule that is not related to local decisions of sensors. It will be shown that this reliability measure performs well when there are Byzantines in the networ. We define the reliability measure for each of the received bits as follows: ψ i ln v i u i ) v i u i ) 43) for i,, }. Here vi u i ) can be obtained from the statistical model of the Rayleigh fading channel considered in this paper. Define F -distance as d F ψ, c j+) ψi ) c j+)i ) 2, i where ψ ψ,, ψ and c j+ is the jth row of the code matrix C. Then, the fusion rule is to decide the region Rj for which the F -distance between ψ and the row of C corresponding to Rj is minimized. B. erformance Analysis In this section, we present some bounds on the performance of the soft-decision decoding scheme in terms of the detection probability. Without loss of generality, we assume E b. As mentioned before in 5), the overall detection probability is the product of the probability of detection at each iteration, d. We first present the following lemma without proof which is used to prove the theorem stated later in this section. Lemma 5. 2): Let ψ i ψ i Eψ i θ, then E ψ i ) 2 θ 8 Eh σ 4 i ) 4 + Eh i ) 2 σf 2 }, 44) where σ 2 is the variance of the noise at the local sensors whose observations follow 2). For the Rayleigh fading channel considered in this paper, both Eh i )4 and Eh i )2 are bounded and, therefore, the LHS of 44) is also bounded. Lemma 5.2: Let θ Rj be the fixed target location. Let e,j θ) be the misclassification probability of the target region given θ Rj at the +)th iteration. For the reliability vector ψ ψ,, ψ of the /M observations and code matrix C used at the + ) th iteration, e,jθ) l M,l j i S j S l i ψ i i S j S l i Eψ i θ θ, 45) where i 2 )c j+)i ) c l+)i ). roof: e,jθ) detected region Rj θ} } d F ψ, c j+) min d F ψ, c l+) θ l M,l j } d F ψ, c j+) d F ψ, c l+) θ l M,l j l M,l j } ψi ) c j+)i ) 2 ψi ) c l+)i ) 2 θ i l M,l j l M,l j i S j S l i S j S l i ψ i θ i where 46) comes from the fact that ψ i i S j S l ψ i ) c j+)i ) 2 ψ i ) c l+)i ) 2 2 ) c j+)i ) c l+)i )ψ i i ψ i Let σ 2 ψθ) i S E j S i ψ i )2 θ l i S E ψ j S i )2 θ, then the above result can be l re-written as e,jθ) σ ψθ) l M,l j i S j S l i ψ i < σ ψθ) i S j S l 46) i Eψ i θ θ, i Eψ i θ θ. 47)

2 Under the assumption that as for M +,,, we have the following result for asymptotic performance of the proposed soft-decision rule decoding based scheme. Theorem 5.3: Under Assumption 3.3), when α <.5, D. roof: First we prove that when α <.5, then i Eψ i θ, 48) since the thresholds corresponding to sensors i j and i l are same due to Assumption 3.3. Therefore, u i l θ) u i j θ) 2α) F η ail ) F η a ij )). 57) ote that, since θ R j, F η ail ) < F η a ij ). ext we prove that Eψ i u i Eψ i u i < 58) for all finite noise variance of the fading channel σ 2 f ). i S j S l where i 2 )c j+)i ) c l+)i ). Based on our code matrix design, i for i Sj S l is given as, for i Sj i +, for i Sl. 49) By using the pairwise summation approach discussed in Section III-B, we notice that, for every sensor i j Sj and its corresponding sensor i l Sl, when θ R j, i j Eψ i j θ + i l Eψ i l θ Eψ i l ψ i j ) θ. 5) ow, for a given sensor i, we have the following, Eψ i θ u i θ)eψ i θ, u i + u i θ)eψ i θ, u i 5) u i θ))eψ i u i + u i θ)eψ i u i 52) Eψ i u i + u i θ) Eψ i u i Eψ i u i,53) where we used the facts that u i θ) + u i θ) and that the value of ψi depends only on u i. ote that the channel statistics are the same for both the sensors. Therefore, Eψi u i d for d, } given by Eψ i u i d E ln v i u i ) v i u i u i d is the same for both the sensors. The pairwise sum Eψi l ψi j ) θ now simplifies to the following, Eψ i l ψ i j ) θ Eψ i u i + u i l θ) Eψ i u i Eψ i u i Eψi u i u i j θ) Eψi u i Eψi u i ) u i l θ) u i j θ) Eψ i u i Eψi u i. 54) When θ Rj, we have u i j θ) α + 2α) F ) η a ij 55) u i l θ) α + 2α) F η a il ) 56) Eψi u i Eψi u i E ln v i u i ) vi u i ) u i E ln v i u i ) vi u i ) u i v i u i ) ln v i u i ) v i u i ) dv i v i u i ) ln v i u i ) v i u i ) dv i D v i u i ) v i u i )) D v i u i ) v i u i )), 59) where Dp q) is the Kullbac-Leiber distance between probability distributions p and q. Since vi u i ) v i u i ) for all finite σf 2, we have D v i u i ) vi u i )) > and D vi u i ) vi u i )) >. This concludes that Eψi u i Eψi u i <. Hence, when α < /2, from 54), 57), and 58), Eψi l ψi j ) θ > and the condition i S j S i Eψ i θ is satisfied. l We now show that when the condition 48) is satisfied, the proposed scheme asymptotically attains perfect detection probability. D Define M j θ d θ Rj } detected region R j θ Rj } M M j θ θ Rj } detected region R j θ, θ Rj } dθ. e,j,max max e,jθ) 6) θ R j 6)

3 and Then, D M j θ R j e,max max e,j,max. 62) j M M M M e,j,max M j Since E M j θ θ R j } e,j,max dθ θ R j e,max M e,max ψ i )2 θ θ θ θ R j } e,j θ)dθ θ θ Rj } dθ M j. 63) is bounded as shown by Lemma 5.2, Lindeberg condition 24 holds and σ ψθ) i S j S l i tends to a standard Gaussian random variable by Lindeberg central it theorem 24. Therefore, from 47), we have e,jθ) i S j S l l M,l j σ ψθ) i S j S l σ ψθ) } i Eψ i θ θ Q l M,l j σ ψθ) i S j S l i S j S l i ψ i < ψ i 64) i Eψ i Hj. Since, for a fixed θ, σ ψθ) will grow slower than i Eψ i θ when i Sj S l i Eψ i θ, e,j θ) for all θ. Hence, e,max and from 63), D for all finite noise variance. ote that the detection probability of the proposed scheme can approach even for extremely bad channels with very low channel capacity. This is true because, when M increases M + sub-linearly with, i.e., when as for,,, as approaches infinity, the code rate of the code matrix approaches zero. Hence, even for extremely bad channels, the code rate is still less than the channel capacity. C. umerical Results In this section, we present some numerical results which show the improvement in the system performance when softdecision decoding rule is used instead of the hard-decision decoding rule in the presence of Byzantines and non-ideal channels. As defined before, α represents the fraction of Byzantines and we evaluate the performance of the basic coding approach with soft-decision decoding at the FC. We simulate the scenario with following system parameters: 52, M 4, A 8 2 64 sq. units, 2, local sensor observations are corrupted with Gaussian noise with σ 3, E b, σ f 3 and Eh i )2 which corresponds to σh 2 π 4. The basic coding approach is ped after 2 iterations. ote that in the presence of non-ideal channels, α blind is less than.5 since the non-ideal channels add to the errors at the FC. The number of Byzantine faults which the networ can handle reduces and is now less than.5. In our simulations, we observe that the performance of the schemes completely deteriorates when α.4 as opposed to.5 observed before) and, therefore, we plot the results for the case when α.4 Fig. 9 shows the reduction in mean square error when the soft-decision decoding rule is used instead of the hard-decision decoding rule. Similarly, Fig. shows the improvement in target region detection probability when the soft-decision decoding rule is used. The plots are for 5 3 Monte-Carlo simulations. MSE 6 4 2 8 6 4 Hard decision decoding Soft decision decoding 2.5..5.2.25.3.35.4 Fraction of Byzantines Fig. 9. MSE comparison of the basic coding scheme using soft- and harddecision decoding As the figures suggest, the performance deteriorates in the presence of non-ideal channels. Also, the performance worsens with an increase in the number of Byzantines. The performance can be improved by using the exclusion method based coding approach as discussed in Section IV in which two regions are stored after every iteration. Figs. and 2 show this improved performance as compared to the basic coding approach. ote that the exclusion method based coding approach also follows the same trend as the basic coding approach with soft-decision decoding performing better than hard-decision decoding.

4.55.5 Hard decision decoding Soft decision decoding.55.5 Hard decision decoding Soft decision decoding.45.45 Detection robability of target region.4.35.3.25.2 Detection robability of target region.4.35.3.25.2.5.5...5.5..5.2.25.3.35.4.45.5 Fraction of Byzantines α).5.5..5.2.25.3.35.4.45.5 Fraction of Byzantines α) Fig.. robability of detection of target region comparison of the basic coding scheme using soft- and hard- decision decoding Fig. 2. robability of detection of target region comparison of the exclusion coding scheme using soft- and hard- decision decoding 4 2 Hard decision decoding Soft decision decoding.9.8.7 64 52 496 MSE 8 6 4 Region detection probability.6.5.4.3 2.2.5..5.2.25.3.35.4 Fraction of Byzantines α) Fig.. MSE comparison of the exclusion coding scheme using soft- and hard- decision decoding In our theoretical analysis, we have shown that the probability of region detection asymptotically approaches irrespective of the finite noise variance. Fig. 3 presents this result that the region detection probability approaches as the number of sensors approach infinity. Observe that for a fixed noise variance, the region detection probability increases with increase in the number of sensors and approaches as. However, as σ f increases, the convergence rate decreases. For example, when σ f.5, 496 is large enough to have D close to.9. However, for σ f 4, 496 results in D.65 which is not very large. It is expected that D much later for σ f 4 and, therefore, the convergence rate is less compared to when σ f.5. VI. COCLUSIO In this paper, we considered the problem of target localization in wireless sensor networs. Traditionally, research has focused on conventional maximum lielihood approaches for estimating the target location. However, maximum lielihood..5 2 2.5 3 3.5 4 oise standard deviation of fading channel σ ) f Fig. 3. robability of detection of target region of the exclusive coding scheme using soft- decision decoding with varying number of sensors ) based approaches are computationally very expensive. To reduce the computational complexity, we proposed a novel coding theory based technique for target localization. Modeling the estimation problem as an iterative classification problem, we can determine a coarse estimate of the target location in a computationally efficient manner. This efficiency in terms of computation becomes important in a scenario when the target is not stationary. The proposed scheme estimates the target location iteratively using M-ary classification at each iteration. We provided the theoretical analysis of the proposed scheme in terms of the detection probability of the target region. Considering the presence of Byzantines malicious sensors) in the networ, we modified our approach to increase the faulttolerance capability of the coding scheme used. This approach, called the exclusion method based approach, is more tolerant to the presence of Byzantines than the basic coding scheme. We showed with simulations that the exclusion method based scheme provides an accurate estimate of the target location in a very efficient manner than the traditional MLE based scheme

5 and also has a better Byzantine fault tolerance capability. We also considered the effect of non-ideal channels between local sensors and the fusion center. To minimize the effects of these non-ideal channels, we proposed soft-decision decoding at the fusion center. We showed with simulations, the improvement in performance of soft-decision decoding rule based scheme over hard-decision decoding rule based scheme in the presence of non-ideal channels. In the future, we plan to extend our wor by relaxing Assumption 3.3 and to also derive the convergence rates using Berry-Essen inequalities. One can also extend this wor to the case of target tracing when the target s location is changing with time and the sensor networ s aim is to trac the target s motion. The proposed schemes provide an insight on M -ary search trees and show that the idea of coding based schemes can also be used for other signal processing applications. For example, the application involving search such as rumor source localization in social networs. R EFERECES I. F. Ayildiz, W. Su, Y. Sanarasubramaniam, and E. Cayirci, A survey on sensor networs, IEEE Commun. Mag., vol. 4, no. 8, pp. 2 4, Aug. 22. 2 R. iu and. K. Varshney, Target location estimation in sensor networs with quantized data, IEEE Trans. Signal rocess., vol. 54, no. 2, pp. 459 4528, Dec. 26. 3 X. Wang, M. Fu, and H. Zhang, Target tracing in wireless sensor networs based on the combination of KF and MLE using distance measurements, IEEE Trans. Mobile Comp.,, vol., no. 4, pp. 567 576, April 22. 4 V. Veeravalli and. K. Varshney, Distributed inference in wireless sensor networs, hilosophical Transactions of the Royal Society A: Mathematical, hysical and Engineering Sciences, vol. 37, no. 958, pp. 7, Jan 22. 5 L. M. Kaplan, L. Qiang, and. Molnar, Maximum lielihood methods for bearings-only target localization, in roc. IEEE Int. Conf. Acoust., Speech, Signal rocess. 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Liu, Target localization in wireless sensor networs for industrial control with selected sensors, International Journal of Distributed Sensor etwors, vol. 23, 23. 7 A. Vempaty, Y. S. Han, and. K. Varshney., Target localization in wireless sensor networs using error correcting codes in the presence of Byzantines, in roc. IEEE Int. Conf. Acoust., Speech, Signal rocess. ICASS 2), Vancouver, Canada, May 23, pp. 595 599. 8, Byzantine tolerant target localization in wireless sensor networs over non-ideal channels, in roc. 3th Int. Symp. Commun. Inf. Technologies ISCIT 23), Samui Island, Thailand, Sep. 23. 9 T.-Y. Wang, Y. S. Han,. K. Varshney, and.-. Chen, Distributed fault-tolerant classification in wireless sensor networs, IEEE J Sel. Areas Comm., vol. 23, no. 4, pp. 724 734, April 25. 2 T.-Y. Wang, Y. S. Han, B. Chen, and. K. Varshney, A combined decision fusion and channel coding scheme for distributed fault-tolerant classification in wireless sensor networs, IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 695 75, 26. 2. Berhin, Survey of clustering data mining techniques, Tech. Rep., 22. 22 F. Aurenhammer, Voronoi diagram- A survey of a fundamental geometric data structure, ACM Comput. Surv., vol. 23, no. 3, pp. 345 45, Sep. 99. Online. Available: http://doi.acm.org/.45/ 6873.688 23 C. Yao,.-. Chen, T.-Y. Wang, Y. S. Han, and. K. Varshney, erformance analysis and code design for minimum Hamming distance fusion in wireless sensor networs, IEEE Trans. Inf. Theory, vol. 53, no. 5, pp. 76 734, May 27. 24 W. Feller, An Introduction to robability Theory and Its Applications. ew Yor: Wiley, 966. Aditya Vempaty S 2) was born in Hyderabad, India, on August 3, 989. He received the B. Tech degree in electrical engineering from Indian Institute of Technology IIT), Kanpur, India, in 2 with academic excellence awards for consecutive years. Since 2, he has been woring towards his h.d. degree in electrical engineering at Syracuse University. He is a Graduate Research Assistant at the Sensor Fusion Laboratory where he was also an Undergraduate Research Intern during summer of 2. He was a Graduate Research Intern in the Data Systems Group at Intel Corporation in Santa Clara, CA in summer 23. His research interests include statistical signal processing, human decision maing, security in sensor networs, and data fusion. He has been awarded the Syracuse University Graduate Fellowship award for the academic year 23-4. He is a member of hi Kappa hi and Golden Key International Honor Society.

6 Yunghsiang S. Han S 9 M 93 SM 8 F ) was born in Taipei, Taiwan, 962. He received B.Sc. and M.Sc. degrees in electrical engineering from the ational Tsing Hua University, Hsinchu, Taiwan, in 984 and 986, respectively, and a h.d. degree from the School of Computer and Information Science, Syracuse University, Syracuse, Y, in 993. He was from 986 to 988 a lecturer at Ming-Hsin Engineering College, Hsinchu, Taiwan. He was a teaching assistant from 989 to 992, and a research associate in the School of Computer and Information Science, Syracuse University from 992 to 993. He was, from 993 to 997, an Associate rofessor in the Department of Electronic Engineering at Hua Fan College of Humanities and Technology, Taipei Hsien, Taiwan. He was with the Department of Computer Science and Information Engineering at ational Chi an University, antou, Taiwan from 997 to 24. He was promoted to rofessor in 998. He was a visiting scholar in the Department of Electrical Engineering at University of Hawaii at Manoa, HI from June to October 2, the SURIA visiting research scholar in the Department of Electrical Engineering and Computer Science and CASE center at Syracuse University, Y from September 22 to January 24 and July 22 to June 23, and the visiting scholar in the Department of Electrical and Computer Engineering at University of Texas at Austin, TX from August 28 to June 29. He was with the Graduate Institute of Communication Engineering at ational Taipei University, Taipei, Taiwan from August 24 to July 2. From August 2, he is with the Department of Electrical Engineering at ational Taiwan University of Science and Technology as Chair professor. His research interests are in error-control coding, wireless networs, and security. Dr. Han was a winner of the 994 Syracuse University Doctoral rize and a Fellow of IEEE. ramod K. Varshney S 72-M 77-SM 82-F 97) was born in Allahabad, India, on July, 952. He received the B.S. degree in electrical engineering and computer science with highest honors), and the M.S. and h.d. degrees in electrical engineering from the University of Illinois at Urbana-Champaign in 972, 974, and 976 respectively. From 972 to 976, he held teaching and research assistantships with the University of Illinois. Since 976, he has been with Syracuse University, Syracuse, Y, where he is currently a Distinguished rofessor of Electrical Engineering and Computer Science and the Director of CASE: Center for Advanced Systems and Engineering. He served as the Associate Chair of the department from 993 to 996. He is also an Adjunct rofessor of Radiology at Upstate Medical University, Syracuse. His current research interests are in distributed sensor networs and data fusion, detection and estimation theory, wireless communications, image processing, radar signal processing, and remote sensing. He has published extensively. He is the author of Distributed Detection and Data Fusion ew Yor: Springer- Verlag, 997). He has served as a consultant to several major companies. Dr. Varshney was a James Scholar, a Bronze Tablet Senior, and a Fellow while at the University of Illinois. He is a member of Tau Beta i and is the recipient of the 98 ASEE Dow Outstanding Young Faculty Award. He was elected to the grade of Fellow of the IEEE in 997 for his contributions in the area of distributed detection and data fusion. He was the Guest Editor of the Special Issue on Data Fusion of the IEEE ROCEEDIGS January 997. In 2, he received the Third Millennium Medal from the IEEE and Chancellors Citation for exceptional academic achievement at Syracuse University. He is the recipient of the IEEE 22 Judith A. Resni Award. He is on the Editorial Board of the Journal on Advances in Information Fusion. He was the resident of International Society of Information Fusion during 2.