Target Location Estimation in Wireless Sensor Networks Using Binary Data

Transcription

1 Target Location stimation in Wireess Sensor Networks Using Binary Data Ruixin Niu and Pramod K. Varshney Department of ectrica ngineering and Computer Science Link Ha Syracuse University Syracuse, NY 344 e-mai: Abstract Based on the assumption that the received signa intensity is inversey proportiona to a poynomia function of the distance from the target, a maximum ikeihood (ML target ocation estimator that uses muti-frame binary data is proposed for wireess sensor networks (WSNs. Simuation resuts show that this estimator is much more accurate than a heuristic weighted average method, and it can reach the Cramér-Rao ower bound (CRLB even with reativey sma amount of data. As a natura extension, the estimator that uses muti-frame muti-bit discrete data is deveoped, and its corresponding CRLB is derived. Its theoretica performance bound (CRLB is compared with the CRLB of the estimator using anaog data, which is aso derived in this paper. Under a communication constraint, guideines on an optima tradeoff between number of bits and number of frames are provided for various situations. I. Introduction Many aspects of wireess sensor networks (WSNs have gained extensive attention in the past severa years. These incude network architectures, routing protocos, distributed data compression and transmission, and distributed detection, target tracking and cassification ]. A key issue for a WSN is the target ocation estimation, which is imperative for an accurate tracking of the target. Many methods have been deveoped to sove the source ocaization probem in sensor networks, 3, 4, 5, 6]. Most of them are based on time-deay of arriva (TDOA estimation or direction of arriva (DOA estimation. nergy-based methods have been deveoped in 7, 8]. In 8], a maximum-ikeihood (ML acoustic source ocaization method has been proposed, using the fact that the intensity (energy of acoustic signa attenuates as a function of distance from the source. In a WSN, usuay there are a arge number of inexpensive sensors which are densey depoyed in a region of interest. This makes accurate energy based target ocaization possibe. In addition, an advantage of the energy based methods is that they do not need a very accurate synchronization of sensors and a far-fied assumption, which are required by the TDOA and the DOA methods respectivey. However, in 7, 8], raw measurements from sensors are required to estimate the source ocation. For a typica WSN with imited resources (energy and bandwidth, This work was supported by the DoD Mutidiscipinary University Research Initiative (MURI program administered by the Army Research Office under Grant DAAD it is important to imit the communication within the network. Therefore, it is desirabe that ony binary data are transmitted from oca sensors to the processing node (fusion center. Motivated by this, we propose an energy based ML target ocation estimation method using ony binary data. As a natura extension, ML estimators using muti-bit discrete data are aso investigated. II. Probem Formuation It is iustrated in Fig. that the signa intensity attenuates as the distance from the target increases. Note that our method can hande any sensor depoyment pattern and the uniform sensor depoyment shown in Fig. is ony a specia case. Further, we make the foowing assumptions: A target has been correcty detected in a sensor fied with N sensors, whose ocations are known. We adopt an isotropic signa intensity attenuation mode: a i + αd n ( i where a i is the signa ampitude at the ith sensor, P is the signa energy at distance zero, and d i is the distance between the target and the ith sensor: P d i (x i x t + (y i y t ( in which (x i, y i and (x t, y t are the coordinates of the ith sensor and the target, respectivey. α is a constant and n is the energy decay exponent. This attenuation mode is simiar to that used in 7]. The difference is that in the denominator of quation (, instead of d n i, we use + αd n i. Therefore, our mode is vaid even if the distance d i is cose to or equa to. At each samping instance (or frame, quantized data are transmitted from sensors to the processing node. The interva between frames are sma enough such that the target is assumed static for T frames. As a resut, the signa strength at a specific sensor i is invariant to frame number, and a i a i for,, T. At each sensor, the signa ampitude a i is corrupted by an additive Gaussian noise: s i a i + ω i (3 where s i is the received signa ampitude at ith sensor during frame. The noises are independent across

2 frames and across sensors and foow the standard Gaussian distribution ω i N(, for i, N and, T. The ML estimation is now the foowing optimization probem: max n p(i θ θ The Cramér-Rao ower bound (CRLB for this probem has been derived and stated in the foowing theorem 3 Theorem Assuming the existence of an unbiased estimator ˆθ(I, the CRLB is given by { ] ] } T ˆθ(I θ ˆθ(I θ J (8 3 Sensors Target 3 3 in which the Fisher information matrix J is where β i J i β ib i e (η i a i 8π( + αd n i Q(η i a i Q(η i a i] and the eements of matrix B i are Fig. : The signa intensity contour of a target ocated in a sensor fied. III. Location stimation using Binary Data In this section, we assume that during each frame, each sensor sends either or to the processing node, depending on whether its received signa exceeds a specific threshod: { si < η i I i (4 s i η i where η i is ith sensor s threshod during frame, and I i is the binary data from ith sensor during frame. We denote the muti-frame muti-sensor data as I {I i : i, N,, T }. After coecting data I, the processing node wi estimate the parameter: θ P x t y t]. Based on the above assumptions and notations, it is easy to derive the ikeihood function at the processing node p(i θ N i T Q(η i a i(θ] I i Q(η i a i(θ] I i (5 where Q( is the compementary distribution function of the standard Gaussian distribution: Q(x The og-ikeihood function is therefore n p(i θ i x π e t dt. (6 I inq(η i a i(θ] +( I in Q(η i a i(θ] (7 b (i /a i b (i b (i αnd n i (x i x t b 3(i b 3(i αnd n i (y i y t b (i α n a id n4 i (x i x t b 3(i b 3(i α n a id n4 i (x i x t(y i y t b 33(i α n a id n4 i (y i y t Proof: See Appendix A. In Fig., the performance of the ML estimator is compared with the CRLB and that of a heuristic weighted average (WA method. In the WA method, the estimated target ocation is obtained by taking a weighted average of a the sensors ocations, where each sensor s weight is proportiona to the tota number of s it sent during the T frames. In this exampe, the threshods are the same for a the sensors during a specific frame, i.e. η i η. But they are inear functions of time: η P (9 T + for,, T. The performances are based on 5 simuation runs. It is cear that the ML method has a much better performance than the WA method. As the number of frames T increases, the ML estimator s performance quicky converges to the CRLB. The ML estimator s performance is very cose to the CRLB even when the number of frames is reativey sma (T 5. Therefore, the ML estimator can be efficient even with reativey sma amount of data. IV. Location stimation using Muti-Bit Data A Muti-Bit Data In this section, we study the ocation estimation probem using muti-bit data. Suppose that during each frame, each sensor can transmit discrete M-bit data. We denote them as D {D i : i, N,, T }, where D i can take any discrete vaue from to M. To simpify the

3 RMS error (P RMS error (x t RMS error (y t Tota number of frames (T Fig. : Root mean square (RMS errors of different estimation methods (N 49, n, α, P 64, x t y t.5. Sensors are uniformy depoyed as shown in Fig.. Soid ine: CRLB, dashed ine: ML, dashdot ine: WA. notation, we define L M. Further, we assume that the set of quantization threshods for ith sensor during frame is η i η i, η i,, η i(l ]. The quantization process for ith sensor during frame is such that s i < η i η i s i < η i D i.. L η i(l3 s i < η i(l L s i η i(l The probabiity that D i takes a specific vaue is ( p i ( η i, θ ( Q(η i a i Q(η i( a i Q(η a i L Q(η i( a i L Therefore, the ikeihood function of D is p(d θ N T L i where δ( is defined as foows { x δ(x x The og-ikeihood function of D is n p(d θ i p i ( η i, θ δ(d i ( (3 L δ(d i np i ( η i, θ] (4 The ML estimation is now the foowing optimization probem: max n p(d θ (5 θ The Fisher information matrix for this estimation probem is provided in the foowing theorem: Theorem The Fisher information matrix J for estimators using muti-frame muti-bit data is J i κ ib i where B i has been defined in Theorem, and κ i L γ i p i ( η i, θ 8π( + αd n i γ i e (η ia i ] e (η i( a i e (η i a i L e (η i( a i L Proof: See Appendix A. B Anaog Data For the purpose of comparison, we have derived the CRLB of the ocation estimator using anaog data. We denote the muti-frame anaog data as S {s i : i, N,, T }. Based on the additive Gaussian noise assumption, it is easy to show that the ikeihood function is f(s θ N i The og-ikeihood function is n f(s θ i T e (s i a i (6 π (si ai n ] π (7 The Fisher information matrix for this estimation probem is given in the foowing theorem: Theorem 3 The Fisher information matrix J for estimators using muti-frame anaog data is J T i where the eements of matrix C i are c (i 4P ( + αd n i C i c (i c (i αndn i (x i x t 4( + αd n i c 3(i c 3(i αndn i (y i y t 4( + αd n i c (i Pα n d n4 i (x i x t 4( + αd n i 3 c 3(i c 3(i Pα n d n4 i (x i x t(y i y t 4( + αd n i 3 c 33(i Pα n d n4 i (y i y t 4( + αd n i 3 Proof: See Appendix B.

4 C Performance Comparison and Tradeoff between T and M In this section, the CRLB on the estimation error for the estimator using muti-bit data is studied and compared with that for the estimator using anaog data. In Fig. 3, the theoretica vaues of CRLB on RMS errors are potted as a function of the number of bits (M. In this exampe, data from ony one frame (T are used. Here, for simpicity, the threshods are identica for a the sensors during a the frames, i.e. η i η, for,, L. The threshods are set as foows: η L P (8 for,, L. As we can see, as M increases, the CRLB on RMS error for muti-bit data converges to that for anaog data very fast. The performance in terms of CRLB for the estimation method that uses 6 bit data is indistinguishabe from that uses anaog data. CRLB RMS error (P CRLB RMS error (x t CRLB RMS error (y t Number of Bits M Fig. 3: CRLB RMS errors of estimators using muti-bit data as a function of number of bits M (N 49, n, α, P 64, x t y t.5, T. Sensors are uniformy depoyed as shown in Fig.. Soid ine + star: muti-bit discrete data, dashed ine: anaog data. It is obvious that by increasing the number of frames T or number of bits M, the estimation accuracy can be improved significanty. But what is the best choice of T and M, especiay when communication is very expansive? Here, we investigate the optima tradeoff between T and M under a communication constraint. We assume that to estimate the ocation of a target, for each sensor bits of data are aowed to be transmitted from sensors to the processing node, or equivaenty MT. In Tabe, for certain parameters (P 64, x t y t.5, the RMS errors are isted for different combinations of (M, T under the constraint MT. Here, to get best position estimation, the optima soution is to transmit frames of binary data. However, to obtain best estimation of P, the optima soution is to transmit 4 frames of 3-bit data. In Tabe, for different situations, the optima combinations of (M, T for the best position estimation are isted. From this tabe, it is cear that for a sma P and a target cose to a particuar sensor, it is better to use sma vaue of Tab. : CRLB RMS errors of estimators using different combinations of bit-frame numbers under a communication constraint (MT, N 49, n, α, P 64, x t y t.5. (M, T P error x t error y t error (, (, (3, (4, (6, (, Tab. : Optima bit-frame tradeoff (M, T for different situations under a communication constraint (MT, N 49, n, α. (x t, y t P P P (, (, (, (4, 3 (, 4 (, (, (4, 3 ( 4, 4 (, (, (4, 3 (, (, (, 6 (4, 3 ( 4, (, (3, 4 (4, 3 (, (, (3, 4 (4, 3 M and arge vaue of T. On the other hand, for a arge P, and a target that is not cose to any particuar sensor and has simiar distances to severa neighboring sensors, it is better to use arge vaue of M and sma vaue of T. This is because with a sma P and a target very cose to a particuar sensor, ony the signa received at that sensor is strong. The benefit of quantizing weak signas at other sensors with arge number of quantization eves is very imited. However, in this situation, by using mutipe frames of independent data, the noise is smoothed out even though the quantization is coarse. V. Concusions In this paper, based on an isotropic signa attenuation mode, we presented a ML target ocation estimator that uses mutiframe binary data for WSNs. Simuation resuts show that this estimator is much more accurate than a heuristic weighted average method and it is efficient even with reativey sma number of frames. In addition, the estimator that uses muti-frame muti-bit discrete data is deveoped, and its corresponding CRLB is derived. Its performance bound (CRLB is compared with that of the estimator using anaog data. Under a communication constraint, guideines on how to choose the best combination of M and T are provided for various situations. In this paper, the quantization threshods are designed in a heuristic manner. In the future, the optima design of threshods wi be investigated. Appendix A. Proof of Theorems and Because binary data are a specia form of muti-bit data when the bit number M, Theorem is more genera than Theorem, and it is sufficient to ust prove Theorem.

5 As we know, the Fisher information matrix can be obtained as foows J { θ n p(d θ] θ n p(d θ] T} ] θ T θ n p(d θ (9 A (, ement of J and From (4, we have n p(d θ n p(d θ P i + Now P L i δ(d i p i ( η i, θ ( p i ( η i, θ δ(di pi ( η i, θ ( ηi, θ p i δ(di p i ( η i, θ p i ( η i, θ P i + i δ(di ] p i δ(di ] p i p i p i P pi ] ( ] pi ] + p i P ( Note the fact that δ(d i ] p i ( η i, θ has been used in the derivation of (. The second part in ( is Therefore, i i P p i ( η i, θ P Q(η i a i +Q(η i a i Q(η i a i + +Q(η i(l3 a i Q(η i(l a i +Q(η i(l a i ] (3 P i It is easy to show that For, p i pi ] (4 Q(η i a i e (η i a i πa i( + αd n i (5 ] pi( η i, θ { } Q(η i a i] (6 Substituting (5 into the above equation, we have ] pi( η i, θ 8πa i ( + e(η ia i αdn i 8πa i ( + αdn i Simiary for L, we have ] pi η i, θ and for L, we have ] pi(l ( η i, θ 8πa i ( + αdn i e (η i( a i γi (7 e (η i a i ] 8πa i ( + αdn i γ i (8 8πa i ( + e(η i(l a i αdn i 8πa i ( + γ αdn i i(l (9 Substituting (7, (8 and (9 into (4, we finay have P L i i κi a i B (, and (3, 3 ements of J γ i p i ( η i, θ8πa i ( + αdn i Foowing a simiar procedure in Section A, we have x t It is easy to show that Q(η i a i x t i Substituting (3 into (3, we have x t p i pi x t (3 ] (3 αnaidn i (x i x t π( + αd n i e (η i a i (3 L i i γ iαna id n i (x i x t] p i 8π( + αd n i κ iαna id n i (x i x t] (33 Due to the symmetry between eements x t and y t, κ iα n a id n4 i (y i y t (34 y t i

6 C (, and (, 3 ements of J Foowing a simiar procedure in Section A, we have ] pi p i x t p i x t i Substituting (5 and (3 into (35, we have x t i i (35 L γ iαnd n i (x i x t p i ( η i, θ8π( + αd n i κ iαnd n i (x i x t (36 Due to the symmetry between eements x t and y t, we aso have y t D (, 3 ement of J i κ iαnd n i (y i y t (37 Foowing a simiar procedure in Section A, we have ] pi p i x t y t p i x t y t i We aso have Q(η i a i y t (38 αnaidn i (y i y t (η i a i π( + αd n i e (39 Substituting (3 and (39 into (38, we have and y t From (7, we have n f(s θ P L γ i p i 8π( + αd n i i αna id n i ] (x i x t(y i y t i κ i αna id n i ] (x i x t(y i y t (4 B. Proof of Theorem 3 n f(s θ i i (s i a i ai (4 (s i a i a i P ] ai (4 We aso have a i P ( + αd n i (44 Substituting (44 into (45, we finay have ] n f(s θ P T i 4P ( + αd n i (45 The other eements of J can be derived simiary. The detaied derivations are omitted due to page imitations. References ] S. Kumar, F. Zhao, and D. Shepherd edts., Specia issue on coaborative signa and information processing in microsensor networks, I Signa Processing Magazine, vo. 9, Mar.. ] K. Yao, R.. Hudson, C.W. Reed, D. Chen, and F. Lorenzei, Bind Beamforming on a Randomy Distributed Sensor Array System, I Jouran on Seected Areas in Communications, vo. 6, no. 8, pp , October ] C.W. Reed, R.. Hudson, and K. Yao, Direct Joint Source Locaization and Propagation Speed stimation, in Proc. ICASSP99, Phoenix, AZ, March 999, vo. 3, pp ] L.M. Kapan, Q. Le, and P. Monar, Maximum Likeihood Methods for Bearings-Ony Target Locaization, in Proc. ICASSP, Sat Lake City, UT, May, vo. 5, pp ] J.C. Chen, R.. Hudson, and K. Yao, A Maximum Likeihood Parametric Approach to Source Locaization, in Proc. ICASSP, Sat Lake City, UT, May, vo. 5, pp ] J.C. Chen, R.. Hudson, and K.Yao, Maximum- Likeihood Source Locaization and Unknown Sensor Location stimation for Wideband Signas in the Near-Fied, I Transactions on Signa Processing, vo. 5, no. 8, pp , August. 7] D. Li, K.D. Wong, Y.H. Hu, and A.M Sayeed, Detection, Cassification, and Tracking of Targets, I Signa Processing Magazine, vo. 9, no. 3, pp. 7 9, Mar.. 8] X. Sheng and Y.H. Hu, nergy Based Acoustic Source Locaization, in The nd Internationa Workshop on Information Processing in Sensor Networks, Pao Ato, CA, Apri 3, pp Therefore, ] n f(s θ P T i ] ai ] ai (43 i