Dynamic Bandwidth Allocation for Target Tracking. Wireless Sensor Networks.

Transcription

1 4th International Conference on Information Fusion Chicago, Illinois, USA, July 5-8, Dynamic Bandwidth Allocation for Target Tracking in Wireless Sensor Networks Engin Masazade, Ruixin Niu, and Pramod K. Varshney Department of Electrical Engineering and Computer Science Syracuse University, NY {emasazad, rniu, Abstract In this paper, we study the dynamic bandwidth allocation problem for target tracking based on quantized sensor data in wireless sensor networks. At each time step, the fusion center distributes the available bandwidth among the sensors in such a way that the posterior Cramér-Rao lower bound (PCRLB) on the mean squared error (MSE) is minimized. Since the optimal solution requires a combinatorial search, we seek computationally efficient suboptimal methods for dynamic bandwidth allocation. In order to minimize the estimation error, our objective is to maximize the determinant of the Fisher information matrix (FIM) subject to the total rate constraint. To maximize the determinant of the FIM, we formulate an approximate dynamic programming (A-DP) algorithm and compare its performance with other suboptimal methods, including the generalized Breiman, Friedman, Olshen, and Stone (GBFOS) algorithm and the greedy search. A-DP is computationally more efficient than the GBFOS and simulation results show that A- DP and GBFOS algorithms yield similar tracking performance in terms of the mean squared error and outperform the greedy search. Keywords: Target Tracking, Dynamic Bandwidth Allocation, Wireless Sensor Networks. I. INTRODUCTION A wireless sensor network (WSN) consists of a large number of spatially distributed sensors which are tiny, batterypowered devices with limited on-board energy. When properly programmed and networked, a WSN performs different tasks that are useful in a wide range of applications such as battlefield surveillance, environment and health monitoring, and disaster relief operations. Dense deployment of sensors in the network introduces redundancy in coverage, so selecting a subset of sensors may still provide information with desired quality. Rather than transmitting entire sensor data to the fusion center, the sensor management policies select a subset of active sensors to meet the application requirements while minimizing the use of resources. In the literature, there exist many sensor selection algorithms (see and references therein). In, the sensor selection problem, an integer programming problem, has been relaxed and solved through convex optimization. One popular strategy for sensor selection is the information driven method, 3 where the main idea is to select the sensors that can provide the most useful information, which is quantified by entropy or mutual information. The posterior Cramér-Rao lower bound (PCRLB) is also a very important tool because it provides a theoretical performance limit for a Bayesian estimator. As we have shown in our previous paper 4, for sensor selection, the complexity to compute the mutual information increases exponentially with the number of sensors to be selected, whereas the computational complexity of Fisher information, which is the inverse of the PCRLB, increases linearly with the number of sensors to be selected. For single target tracking and in a bearing-only sensor network, a sensor selection approach which minimizes the PCRLB on the estimation error has been proposed, where the selected sensors transmit analog 5 or quantized data 6 to the fusion center. In this paper, our goal is to find the optimal bandwidth allocation for channels between the sensors and the fusion center which optimizes the target tracking performance given a total bandwidth constraint in the WSN. This problem is more general than the sensor selection problem, because in the bandwidth allocation problem, the channel corresponding to each sensor could be assigned a different number of bits, while in sensor selection problems, a sensor is either activated or not to transmit its measurement under the constraint on the total number of active sensors. Bit allocation has been previously studied for distributed sequential detection in WSNs 7. The bandwidth allocation problem can be solved by an exhaustive search which enumerates all possible bandwidth allocation solutions to minimize the PCRLB. As we show later in the paper, an exhaustive search may not be feasible for a large network. Therefore, computationally efficient suboptimal methods are required. In 8, the generalized Breiman, Friedman, Olshen, and Stone (GBFOS) algorithm has been employed to dynamically assign bits to sensors for the tracking application in a WSN. In this paper, we treat the bandwidth allocation problem as a resource-allocation problem. Dynamic programming (DP) 9 solves the resource allocation problems by breaking them down into simpler steps. If the problem were a scalar-valued parameter estimation problem, a DP recursion can be easily formulated to find the optimal bit allocation by maximizing Fisher information due to the fact that the total Fisher information is the summation of each sensor s individual Fisher information. In target tracking, the Fisher information is in a matrix form, and we formulate an approximate DP (A-DP) recursion to maximize the determinant of the Fisher information matrix (FIM) subject to the total bandwidth constraint. We compare the performance of A-DP in terms of computation time and estimation error with other ISIF

2 suboptimal methods, including GBFOS algorithm and greedy search. Computationally, A-DP is significantly more efficient than the GBFOS for a large number of sensors. In terms of mean squared estimation errors, A-DP and GBFOS algorithms yield similar tracking performance and outperform the greedy search. The rest of the paper is organized as follows. In Section II, we introduce the problem set-up for target tracking in WSNs. In Section III, we explain the details of the proposed A-DP and other suboptimal methods. Section IV presents numerical examples and Section V concludes our work. II. TARGET AND OBSERVATION MODEL The problem we seek to solve is to track a moving target using a WSN where N sensors are grid deployed in a rectangular surveillance area of size b. The assumption of grid layout is not necessary but has been made here for convenience. Target tracking based on sensor readings can be performed for an arbitrary network layout if sensor placements are known in advance. All the sensors report to a central fusion center which estimates the target state, i.e., the position and the velocity of the target based on quantized sensor measurements. We assume that the target (e.g., an acoustic or an electromagnetic source) emits a signal from the location (,y t ) at time t. We assume that the target is based on flat ground and all the sensors and target have the same height so that a -D model is sufficient to formulate the problem. We consider a single target moving in a two-dimensional Cartesian coordinate plane. At time t, the target dynamics are defined by the 4-dimensional state vector = y t ẋ t ẏ t T where ẋ t and ẏ t are the target velocities in the horizontal and the vertical directions. The superscript T denotes the transpose operation. Target motion is defined by the following white noise acceleration model: + = F + υ t () where F models the state dynamics and υ t is the process noise which is assumed to be white, zero-mean and Gaussian with the following covariance matrix Q. F = Q = q () In (), and q denote the time interval between adjacent sensor measurements and the process noise parameter, respectively. It is assumed that the fusion center has perfect information about the target state-space model () as well as the process noise statistics. The target is assumed to be an acoustic or an electromagnetic source that follows the power attenuation model provided below. At any given time t, the signal power received at the sensor i is as follows: a i,t = P ( d d i,t ) n (3) where P denotes the target signal power at a reference distance of d from the target, n is the signal decay exponent, and d i,t is the distance between the target and the i th sensor, d i,t = (x i ) + (y i y t ), where (x i,y i ) are the coordinates of the i th sensor. At time t, the received signal at sensor i is given by z i,t = a i,t + n i,t (4) where n i,t is the noise term modeled as additive white Gaussian noise (AWGN), i.e., n i,t N(,σn), which represents the cumulative effects of sensor background noise and the modeling error of signal parameters. Without loss of generality, the reference distance d and the signal decay exponent n are assumed to be and, respectively. Rather than transmitting analog sensor observations to the fusion center, transmitting a quantized version of sensor measurements reduces the amount of communication and therefore reduces the energy consumption. A sensor measurement z i,t at sensor i is locally quantized before being sent to the fusion center using R i,t bits for R i,t = m, m {,,...,M} and M is the maximum number of bits to be transmitted to the fusion center. Then, L m = m is the number of decision intervals for transmitting m bits to the fusion center. Let D i,t be the m-bit observation of sensor i quantized with rate R i,t = m at time step t, then < z i,t < η m η m < z i,t < η m D i,t = (5). L m ηl m < z m i,t < where η m = η m η m... ηl m m with η m = and ηl m m =. The quantization thresholds are assumed to be identical at each sensor for simplicity. We explain the selection of the quantization thresholds for each data rate R i,t = m in the next section. Given and m, it is easy to show that the probability of a particular quantization output l is, ( ) ( η m P(D i,t = l,r i,t = m) = Q l a i,t η m ) Q l+ a i,t σ n σ n (6) where Q(.) is the complementary distribution function of the standard Gaussian distribution, ( ) Q(x) = exp t dt (7) π x At time t, let fusion center receive the data vector D t = D,t,...,D N,t from N sensors with a corresponding rate vector R t = R,t,...,R N,t, then N p(d t,r t ) = p(d i,t,r i,t ) (8) and we assume p(d i,t,r i,t = ) =. At time step t, we assume that the network of N sensors can transmit reliably at a maximum rate of M bits per sampling time, R i,t M (9)

3 where M is an integer. A. PCRLB with quantized data Let p(d t, ) be the joint probability density of D t and, and ˆ be an estimate of at time step t. Based on the received data D t quantized with rate vector R t, and the prior probability distribution function of, p( ), the PCRLB on the mean squared estimation error has the form, E { ˆ ˆ T R t } J t (R t ) () where J t (R t ) is the 4 4 Fisher information matrix (FIM) with the elements J t (R t )(i,j) = E log p(d t, R t ) i, j {,...,4} (i) (j) () where J t (R t )(i,j) denotes the i th row, j th column element of the matrix J t (R t ) and (i) denotes the i th element of vector. Let xt = xt T denote the operator of the second order partial derivative with respect to. Using this notation () can be rewritten as, J t (R t ) = E xt log p(d t, R t ) () Since p(d t, R t ) = p(d t,r t )p( ), J t can be decomposed into two parts as, and J t (R t ) = J D t (R t ) + J P t (3) J D t (R t ) E xt log p(d t,r t ) J P t E xt log p( ) Note that J D t (R t ) represents the Fisher information obtained from the data and J P t represents the a priori Fisher information. B. Optimization of Quantization Thresholds As we show later in the paper, the Fisher information and hence the PCRLB are the functions of the quantization thresholds corresponding to each data rate R i,t = m. Thus, the quantization thresholds should be designed to achieve better estimation accuracy. An algorithm to obtain the optimal quantization thresholds that minimize the variance of the estimation error has been proposed in. If we assume that (x i,y i ) and (,y t ) are uniformly distributed in a region, we can define the average cost function by performing a multiple fold integration over the random parameter which may result in a large computational load. To alleviate this problem, in some alternative methods to design the quantization thresholds have also been developed. In this paper, we use the Fisher information based heuristic quantization method which maximizes the Fisher information about the signal amplitude a i,t contained in the quantized data D i,t. Given the vector of quantization rates R,t,...,R N,t and using (8) in (3), the data part of the Fisher information can be derived as, J D t (R,t,...,R N,t ) = J D i,t(r i,t ) (4) where J D i,t(r i,t ) = J S i,t(r i,t )p( )d Note that given state and quantization rate R i,t, J S i,t (R i,t ) is the standard Fisher information of the unknown parameter, J S i,t(r i,t ) (5) E p(di,t,r i,t) log p(d i,t,r i,t ) For quantized data based target localization problem, J S i,t (R i,t ) has been derived in and provided as follows, where J S i,t(r i,t = m) = n κ i,t (R i,t )a i,td 4 i,t (6) (x i ) (x i )(y i y t ) (x i )(y i y t ) (y i y t ) κ i,t (R i,t = m) = (7) L m e (η m l a i,t ) σ e (η m l+ a i,t ) σ 8πσ p(d i = l ) l= Note that all the information about, y t T is contained in sensors signal amplitudes (a i,t ) s. If all the signal amplitudes can be recovered from their quantized data D i,t, an accurate estimate of,y t T can be obtained. Let F a (η R i,t = m) be the Fisher information of the signal amplitude contained in quantized data D i,t using a threshold η. Then given R i,t = m, sensor location (x i,y i ) and source location (,y t ), it has been derived in that F a (η R i,t = m) = 4κ i,t (R i,t = m). The Fisher information based heuristic quantization method finds the decision thresholds maximizing F a (η R i,t = m) averaged over the probability distribution of d i,t, where (x i,y i ) and (,y t ) are assumed to be independent and identically distributed and follow a uniform distribution U b/, b/. The details of this quantizer design approach can be found in. C. Particle Filtering with Quantized Data It is known that Kalman Filter provides the optimal solution to the Bayesian sequential estimation problem for linear and Gaussian systems. In nonlinear systems, the extended Kalman filter (EKF) can be used to provide a suboptimal solution by linearizing the nonlinear state dynamics and/or nonlinear measurement equations locally. However, it has been shown that, even for linear and Gaussian systems, when the sensor measurements are quantized, the EKF fails to provide an acceptable performance especially when the number of quantization levels is small. For our tracking problem, in addition to the nonlinear mapping from the target state to the sensor observations, the final measurement model at the fusion

4 center consists of quantization of sensor observations resulting in a highly nonlinear and non-gaussian system. Therefore, we propose to employ a particle filter to solve the Bayesian sequential estimation problem. Let D :t = D,...,D t be the received sensor data up to time t which are obtained according to the data rates R :t = R,...,R t. In particle filtering, the main idea is to find a discrete representation of the posterior distribution p( D :t ) by using a set of particles x s t with associated weights w s t. The posterior density at t can be approximated as, N s p( D :t ) wtδ(x s t x s t) (8) s= where N s denotes the total number of particles. In this paper, we employ sequential importance resampling (SIR) particle filtering algorithm to solve the nonlinear Bayesian filtering problem. Here, we provide a summary of the algorithm rather than discussing the details. Note that T S in Algorithm denotes the number of time steps over which the target is tracked. A more detailed treatment of particle filtering can be found in a wide variety of publications such as. Algorithm SIR Particle Filter with Dynamic Bandwidth Allocation Set t =. Generate initial particles x s p(x ) with s, w s = Ns. while t T S do x s t+ = Fx s t + υ t (Propagating particles) p(+ D :t ) = Ns N s s= δ(+ x s t+) Decide R t+ and obtain sensor data D t+ wt+ s p(d t+ x s t+,r t+ ) (Updating weights) wt+ s w = s t+ (Normalizing weights) Ns ˆ+ = N s {x s t+,n s t = t + end while s= ws t+ s= ws t+x s t+ } = Resampling(x s t+,w s t+) In Algorithm, p(d t+ x s t+,r t+ ) has been obtained according to (6) and (8). Resampling step avoids the situation that all but one of the importance weights are close to zero. At time t, we find the data rate vector R t+ that maximizes the Fisher information about + conditioned on the data that have been received so far, i.e. D :t. In other words we maximize, max J t+ (R t+ ) = (9) R t+ E p(dt+,+ D :t,r t+) xt+ + log p(d t+,+ D :t,r t+ )) where using the property p(d t+,+ D :t,r t+ ) = p(d t+ +,R t+ )p(+ D :t ) it is straightforward to show that the Fisher information defined in (9) can be decomposed into data part and prior part as in (3). Given the target state + and the quantization rate vector R t+ = R,t+,...,R N,t+, the standard Fisher information has the form J S t+(r,t+,r,t+,...,r N,t+ + ) = () E p(di,t+ +,R i,t+) = J S t+(r i,t+ + ) xt+ + log p(d i,t+ +,R i,t+ ) The computation of J S t+(r i,t+ + ) has been given previously in (6). The data part of the Fisher information matrix, J D t+, can then be obtained by averaging J S t+(+ ) over the particle approximation of the posterior distribution p(+ D :t ) as, J D t+(r i,t+ ) N s J S N t+(r i,t+ x s t+) () s s= J P t+ = E p(xt+ D :t) xt+ + log p(+ D :t ) has been defined as the prior Fisher information of p(+ D :t ) as in (3). According to (8), p(+ D :t ) has a non-parametric representation by a set of random samples with associated weights, so it is very difficult to calculate the exact J P t+ 3. Instead, we use a Gaussian approximation such that p(+ D :t ) N(µ t+,σ t+ ), where and µ t+ = N s N s x s t+ s= Σ t+ = N s (x s t+ µ N t+ )(x s t+ µ t+ ) T s s= Given the Gaussian approximation, it is easy to show that J P t+ = Σ t+. The Fisher information introduced in (3) and (9) can be found as, J t+ (R t+ ) = J D t+(r i,t+ ) + Σ t+ () If one is more concerned with position estimates, at time step t, the best strategy to determine the bandwidth allocation for the next time step t + is to minimize the sum of two diagonal elements of the PCRLB matrix which correspond to the bounds on the MSE estimates for the position subject to the rate constraint. min J t+ (R t+)(, ) + J t+ (R t+)(,) R,t+,...,R N,t+ s.t. R i,t+ = M (3) 3

5 An exhaustive search can be employed to find the optimal bandwidth distribution which minimizes the cost defined in (3). For a network( of N sensors) and a total of M bits, M + N there are a total of = (N+M )! N (N )!M! possible bandwidth distribution solutions. For large N and M, such an exhaustive search may not be feasible in real time. Therefore suboptimal but computationally more efficient algorithms are required. III. APPROXIMATE DYNAMIC PROGRAMMING BASED BANDWIDTH DISTRIBUTION In this section, we present the bandwidth allocation algorithm based on A-DP which will be shown to provide near optimal solution but require much less computation time than the GBFOS approach. If the problem were a scalarvalued parameter estimation problem, minimizing the PCRLB would be equivalent to maximizing the Fisher information and a suitable DP trellis would yield the optimal bandwidth allocation since the Fisher information can be expressed as the summation of each sensor s individual Fisher information as defined in (). In this section, we try to formulate an approximate DP recursion in tracking applications where the Fisher information is a matrix rather than a scalar entity and we can maximize the Fisher information by maximizing its determinant subject to the total bandwidth constraint as max det(j t+ (R t+ )) (4) R,t+,...,R N,t+ s.t. R i,t = M Since A-DP is performed at each time step for resource management, for simplicity, the time inde + for Fisher information matrix is dropped. Instead an index for the stages in DP is adopted. Let J N = J t+ and A i (R i ) = J D t+(r i ) be the reward in terms of Fisher information when sensor i quantizes its measurement in R i bits (R i {,,...,M}). While constructing the DP trellis, the bandwidth allocation problem is first divided into N + stages which correspond to N sensors and a termination stage. We define the state of a stage as the remaining bandwidth for the usage of sensor i. So each stage has M + states associated with it. The bandwidth allocation chosen at any sensor (stage) determines the feasible states at the next sensor. An example DP trellis is shown in Fig. with N = 6 and M = 3 which implies a total of 7 stages and 4 states in the DP trellis. As an example, sensor is at state r = means bits have already been used by N sensors and bit is available for sensor. Then, sensor can only take the action A () and the DP goes to the termination stage (stage ) which has only the bit available state. For such a DP trellis, we have, J N = A N (R N ) + {A N (R N ) A (R ) + J } = A N (R N ) + J N : J = A (R ) + J (5) where J = Σ t+ and R +...+R N = M. According to the matrix determinant lemma det(x + A) = det(x + AI) = det(x)det(i + X A) With X = J i, A = A i (R i ), and I being the identity matrix, we have { } log det(j N ) = (6) { } { log det(j N ) + log det I + J N A N(R N ) } : { } log det(j ) = { } { } log det(j ) + log det(i + J A (R )) Figure : Trellis of the DP for tracking time step t. (N = 6, M = 3). We { can } maximize det(j N ), by maximizing log det(j N ). The DP recursion at each stage is formulated as follows: the trellis starts from J () and for the first stage (i = ), log detj (r) = (7) log deti + J ()A (r) for all r {,,...,M} + log detj () Then for all the intermediate stages i {,...,N }, log detj i (r) (8) { max log deti + J i (r k)a i(k) + k=,,...,r log detj i (r k) } ; for all r {,,...,M} 4

6 Finally for the last stage i = N, log detj N (M) (9) { max log deti + J N (M k)a i(k) + k=,,...,m } log detj N (M k) In (7), (8), and (9), the reward of sensor i s transmission in R i bits depends not only on A i (R i ) but also on the FIM of the previous stage J i. So at each stage i, the FIM, J i(r), which has the maximum determinant should be stored in a memory for its usage at the next recursion. All J i (r) terms are updated at each state of the next stage. For Eqs. (8) and (9), we choose the sign, since the proposed A-DP may not yield the maximum matrix determinant at the final stage. The suboptimality of the A-DP recursions is discussed in the next subsection. A. Suboptimality of the A-DP recursion For a given state of a stage, we choose the path with the maximum determinant of the FIM and dismiss all the other paths arriving at this state. The proposed DP recursions would yield the optimal solution to maximize the determinant of the FIM, if the following property were satisfied, if det{j } det{j } (3) then det{a + J } det{a + J } Unfortunately, the above property is( not necessarily ) true. Consider the simple example, J = and J = ( ). where det{j } det{j }. Let A = (. ).. Then det{a + J } < det{a + J }.. At each stage of the DP, we only store the FIM with the maximum determinant. Therefore the final solution obtained by the DP recursions becomes suboptimal, since not all the feasible solutions are enumerated. B. Existing Suboptimal Bandwidth Distribution Algorithms In this section, we review some existing suboptimal algorithms that are suitable for solving the bandwidth allocation problem in target tracking applications. ) GBFOS Algorithm: This algorithm has been proposed in 8 for dynamic bandwidth allocation in target tracking. The GBFOS algorithm starts by assigning the maximum number of bits, M to each sensor in the network and then reduces the number of bits one bit at a time until the sum rate constraint N is satisfied, i.e., R i = M, in (N )M iterations. Reduction of the bits is carried out to ensure the minimum reduction of the determinant of the FIM at each iteration. The GBFOS algorithm can be stated as in Algorithm. Algorithm GBFOS - Bandwidth Distribution Algorithm Set R = R,...,R N for i =,...,N with R i = M. FOR c = : (N )M () FOR j = : N, Define R j c = R,...,R j,...,r N and calculate det(j(r j c )). ENDFOR () Find the sensor p for which det(j(r j c )) is the maximum: p = arg max j det(j(r j c )). (3) Decrement R p = R p and update R c = R,...,R p,...,r N. ENDFOR ) Greedy Algorithm: The greedy algorithm is basically the reverse of the GBFOS algorithm which makes the algorithm much more faster. The greedy algorithm starts by assigning bits to each sensor in the network and then increases the number of bits one bit at a time until the sum rate constraint is satisfied in M iterations. At each iteration a single bit is added to maximize the determinant of the resulting FIM. The greedy algorithm can be stated as in Algorithm 3. Algorithm 3 Greedy Bandwidth Distribution Algorithm Set R = R,...,R N for i =,...,N with R i =. FOR c = : M () FOR j = : N, Define R j c = R,...,R j +,...,R N and calculate det(j(r j c )). ENDFOR () Find the sensor p for which det(j(r j c )) is the maximum: p = arg max j det(j(r j c )). (3) Increment R p = R p + and update R c = R,...,R p,...,r N. ENDFOR Let us compare the computational complexity in terms of search space of each scheme. For the A-DP, the first stage needs M + matrix summations to compute the FIM at all the states. For all the intermediate stages, at each state r, (r {,,...,M}), (r + ) different matrix summations are required to find the FIM with the maximum determinant. Finally at stage N, A-DP again needs M + matrix summations in order to maximize the determinant of J N. So the A-DP totally searches over, M (M + ) + (N ) (j + ) + (M + ) = r= (N )(M + )(M + ) + (M + ) matrix summations which is linear in N. Because of the inner FOR loop (Step ()) of Algorithms and 3, the GBFOS and greedy algorithms search for N different bit decrement or increment strategies to maximize the determinant of the FIM. The GBFOS and Greedy search then meet the bandwidth constraint after (N )M and M iterations respectively. We evaluate the computation time of each bandwidth allocation approach by using the etime function of MATLAB and averaging it over trials. In Fig., the mean computation times 5

7 of the considered suboptimal bandwidth allocation schemes are compared. Since the search space of A-DP increases linearly with N, for large number of sensors, the computation time of A-DP is significantly less than the computation time of GBFOS algorithm whose search space increases quadratically as N Actual Track Approx. DP Greedy Sensor Locations Approx. DP GBFOS Greedy Y 4 6 Computation Time X (a) Number of Sensors Figure : Computation time for A-DP, GBFOS, and greedy search (M = 5). IV. SIMULATION RESULTS In this section, we provide some numerical results where we compare the MSE performance of the approximate DP based bandwidth allocation method with the GBFOS approach and greedy search. We assume that N sensors are grid deployed in a b = m m surveillance area as shown in Fig. 3- (a). We select P = 3 and sensor observation noise σn =. The probability distribution function of the target s initial state, p(x ), is assumed to be Gaussian with mean µ = 8 8 and covariance Σ = diag... The target motion follows a near constant velocity model with a process noise parameter q =.5 3. Measurements are assumed to be taken at regular intervals of =.5 seconds and the observation length is s. Namely, we perform target tracking over T S = time steps for each Monte-Carlo trial. The number of particles used in the particle filter is N s = 5. We assume M = 5 bits of bandwidth is available at each time step for data transmission. The MSE at each time step is averaged over T trials = 5 trials as, T trials MSE(t) = (3) T trials u= (xt,u () ˆ,u ()) + (,u () ˆ,u ()) where in the u th trial,u and ˆ,u are the actual and estimated target state at time t respectively. Fig. 3-(b) shows the average number of sensors activated at each time step using A-DP based bandwidth distribution. Average number of active sensors N = 5 N = Time Steps (b) Figure 3: (a) A WSN with N = 9 sensors tracking a target. (b) Number of active sensors averaged over 5 trials based on A-DP. For N = 9, between the time steps 8 and, the target is relatively close to the sensor located at (, ). Hence almost all the bits are allocated to this sensor. When the target is not relatively close to any of the sensors, multiple sensors are activated with relatively coarse information and this increases the estimation error as shown in Fig. 4-(a). For the case of N = 5 sensors, since the sensor density is increased, the A- DP based bandwidth allocation scheme often assigns all the resources to a single sensor which is more informative about the target. This improves the tracking performance at each time step as clearly shown in Fig. 4-(b). In Fig. 4-(a), for N = 9, we also compare the MSE of the suboptimal bandwidth distribution schemes with the exhaustive search which minimizes the sum of the PCRLBs on MSEs for position estimates as defined in (3). Simula- 6

8 tion results show that the exhaustive search based bandwidth distribution is slightly better than the suboptimal bandwidth allocation approaches based on A-DP and GBFOS. For N = 9 and N = 5 sensors, A-DP and GBFOS yield similar estimation performances and they significantly outperform the greedy search in terms of the MSE. On the other hand, for N = 5 sensors, A-DP requires approximately 5 times less computation time as compared to GBFOS. MSE MSE Exhaustive Search Approx. DP GBFOS Greedy 5 5 Time Step (a) Approx. DP GBFOS Greedy 5 5 Time Step (b) Figure 4: MSE at time instant t (M = 5). (a) N = 9, (b) N = 5. the greedy search. On the other hand, the proposed A-DP is computationally more efficient than the GBFOS based bandwidth allocation scheme especially for a large sensor network. ACKNOWLEDGMENT This work was supported by U.S. Air Force Office of Scientific Research (AFOSR) under Grant FA REFERENCES S. Joshi and S. Boyd, Sensor selection via convex optimization, IEEE Transactions on Signal Processing, vol. 57, no., pp , 9. F. Zhao, J. Shin, and J. Reich, Information-driven dynamic sensor collaboration, IEEE Signal Processing Magazine, vol. 9, no., pp. 6 7, Mar. 3 G. M. Hoffmann and C. J. Tomlin, Mobile sensor network control using mutual information methods and particle filters, IEEE Transactions on Automatic Control, vol. 55, no., pp. 3 47, Jan.. 4 E. Masazade, R. Niu, P. K. Varshney, and M. Keskinoz, Energy aware iterative source localization schemes for wireless sensor networks, IEEE Transactions on Signal Processing, vol. 58, no. 9, pp , Sept.. 5 L. Zuo, R. Niu, and P. K. Varshney, Posterior CRLB based sensor selection for target tracking in sensor networks, in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 7, vol., 7, pp. II 4 II 44. 6, A sensor selection approach for target tracking in sensor networks with quantized measurements, in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 8, April 8, pp Q. Cheng, P. K. Varshney, K. G. Mehrotra, and C. K. Mohan, Bandwidth management in distributed sequential detection, IEEE Transactions on Information Theory, vol. 5, no. 8, pp , 5. 8 O. Ozdemir, R. Niu, and P. K. Varshney, Dynamic bit allocation for target tracking in sensor networks with quantized measurements, in Proc. IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP,, pp D. P. Bertsekas, Dynamic programming and optimal control, vol. I and II. Athena Scientific, 7. R. Niu and P. K. Varshney, Target location estimation in sensor networks with quantized data, IEEE Transactions on Signal Processing, vol. 54, no., pp , Dec. 6. Y. Ruan, P. Willett, A. Marrs, F. Palmieri, and S. Marano, Practical fusion of quantized measurements via particle filtering, IEEE Transactions on Aerospace and Electronic Systems, vol. 44, no., pp. 5 9, January 8. M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracking, IEEE Transactions on Signal Processing, vol. 5, no., pp , Feb. 3 O. Ozdemir, R. Niu, P. K. Varshney, and A. Drozd, Modified Bayesian Cramér-Rao lower bound for nonlinear tracking, in accepted to be presented at IEEE International Conference on Acoustics Speech and Signal Processing, ICASSP. V. CONCLUSION In this paper, we studied the dynamic bandwidth allocation problem for target tracking in a WSN with quantized measurements. Under the sum rate constraint, we proposed an approximate DP algorithm which maximizes the Fisher information by maximizing its determinant. Simulation results show that A-DP and GBFOS algorithms yield similar tracking performance in terms of mean squared errors and outperform 7