Research Article Fast 3D Node Localization in Multipath for UWB Wireless Sensor Networks Using Modified Propagator Method

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1 Distributed Sensor Networks, Artice ID 32535, 8 pages Research Artice Fast 3D Node ocaization in Mutipath for UWB Wireess Sensor Networks Using Modified Propagator Method Hong Jiang, Chang iu, 2 Yu Zhang, and Haijing Cui Coege of Communication Engineering, Jiin University, Changchun 32, China 2 Department of Eectrica and Computer Engineering, Kansas State University, Manhattan, KS 6652, USA Correspondence shoud be addressed to Hong Jiang; jiangh@jueducn Received 25 May 23; Revised 3 November 23; Accepted 3 December 23; Pubished 9 February 24 Academic Editor: Kapi Dandekar Copyright 24 Hong Jiang et a This is an open access artice distributed under the Creative Commons Attribution icense, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited A fast three-dimensiona (3D) node ocaization agorithm in mutipath for utra-wideband (UWB) wireess sensor networks is deveoped The agorithm empoys a modified propagator method (MPM) for time deay estimation and then uses a marriage agorithm of 3D Chan and Tayor for range-based mutiatera ocaization and node position computation In the proposed agorithm, the traditiona propagator method (PM) for direction-of-arriva (DOA) estimation is extended to frequency-domain time-of-arriva (TOA) estimation in mutipath, which can effectivey measure the distance between an unknown sensor node and an anchor node MPM agorithm requires neither spectra searching nor covariance matrix estimation and its eigenvaue decomposition, which reduces the computationa compexity The marriage ocation agorithm enhances the robustness and accuracy of node ocaization The simuations vaidate the effectiveness of the proposed agorithm in ocating mutipe unknown nodes of UWB wireess sensor networks in 3D space Introduction Ranging and ocaization of unknown sensor nodes in wireess sensor networks (WSNs) have drawn considerabe attention in many aspects such as environmenta monitoring, heath tracking, smart home, machine-to-machine (M2M), and body area networks (BAN) In actua environment, the sensor nodes are commony paced in three-dimensiona (3D) terrains, such as workshops, forests, and oceans Recenty, severa range-based and range-free 3D node ocaization agorithms in WSNs have been proposed 2 6 However, some probems shoud be investigated in node design and impementation, such as arge computationa amount, sow executing, and inaccurate ocaization in mutipath and noise Since the energy and power carried by node equipment are imited, effective methods with ow compexity, ow power consumption, and good robustness for 3D node positioning are of great significance This paper investigates fast 3D node ocaization probem based on time deay measurement in mutipath in utrawideband (UWB) wireess sensor networks As we know, time deay estimation probem has been studied with a variety of super-resoution subspace techniques 7, such as mutipe signa cassification (MUSIC), tota east square-estimation of signa parameters via rotationa invariance technique (TS-ESPRIT), and matrix penci (MP) Compared with the correator-based methods, subspace-based agorithms can increase time resoution even if the time deay is smaer than a puse width Unfortunatey, these techniques increase the compexity of WSN impementation Specificay, in MUSIC agorithm, spectra searching is needed through a the space, which increases its computationa time ESPRIT-ike agorithms require the covariance matrix estimation of the observed data using a arge number of snapshots and then performing eigenvaue decomposition (EVD) or singuar vaue decomposition (SVD) of it In 2, we deveoped a unitary matrix penci (UMP) based deay estimation agorithm for 3D ocaization in UWB wireess sensor networks, which can reduce the compexity oad However, its estimation accuracy is ower than many subspace-based methods The propagator method (PM), deveoped in 3, is a traditiona subspace method previousy used in the direction-ofarriva (DOA) estimation In 4, we extend it to the time-ofarriva (TOA) estimation PM agorithm ony requires inear

2 2 Distributed Sensor Networks operations, and thus it avoids covariance matrix estimation anditsevdorsvdwhicharethemaincomputationa burden in subspace methods However, simiar to MUSIC agorithm, a spectra peak searching through a the space is needed in traditiona PM agorithm In this paper, we deveop a fast range-based 3D node ocaization agorithm in mutipath for UWB wireess sensor networks The agorithm empoys a modified propagator method (MPM) for frequency-domain time deay estimation and then uses a marriage agorithm of 3D Chan and Tayor for range-based mutiatera ocaization and node position computation The MPM agorithm requires neither spectra searching nor covariance matrix estimation and its eigenvaue decomposition, which can reduce the computationa oad Furthermore, the marriage ocaization agorithm can enhance the robustness and accuracy of node ocaization in comparison with the previous agorithm in 5 The simuations vaidate the effectiveness of the proposed agorithm in ocating mutipe unknown nodes of UWB wireess sensor networks in 3D space This paper is organized as foows In Section 2,theUWB received signa mode for mutipath time deay estimation is given MPM-based mutipath deay estimation agorithm is proposed in Section 3 Basedonit,themutiateraocaization and node position computation using a marriage of 3D Chan and Tayor agorithms are presented in Section 4 In Section 5, the simuation resuts are given to verify the performance of the proposed method Finay, a concusion is drawn in Section 6 2 Signa Mode Assume that an utra-wideband puse is transmitted from an unknownnodetoananchornodethrough paths During the qth snapshot, q =,,Q, the received signa can be expressed as y (q) (t) = β (q) = p (t τ TOA Δτ ) +w (q) (t), () wherep(t) denotes a UWB signa having basic Gaussian puse waveform p(t) = A p e 2πt2 /T 2 p,withap and T p being the ampitude and puse width of the UWB signa, respectivey w (q) (t) denotes the compex additive Gaussian white noise τ TOA denotes the TOA of the unknown node, and Δτ and β (q) represent the reative deay and time-varying compex fading ampitude of the th path, respectivey: Δτ =Denoteτ = τ TOA +Δτ as the tota propagation time deay of the th path Then,thereceivedsignain()canberewrittenas y (q) (t) = β (q) = p (t τ ) +w (q) (t) (2) The channe impuse response of UWB signa in mutipath is a sum of components shifted from the corresponding time deays Upon discrete Fourier transform and matched fitering based on (2), the frequency-domain representation of the identified channe signa can be written as H (q) (k) = = β (q) = β (q) = e j(2πkτ /KT) +V (q) (k) z k +V (q) (k), k =,,,K,with K denoting the number of discrete Fourier transforms and T being the samping period V (q) (k) denotes the frequency-domain noise, τ is the estimated deay parameter, and z =e j(2πτ /KT) Coecting the data of a the Q snapshots from (3) yieds an observed data matrix: (3) H = Z (τ) B + V, (4) where H C K Q and V C K Q denote the observed matrix and noise matrix, respectivey Z(τ) denotes the time deay matrix: Z (τ) = z z d z K z K K and B C Q denotes the fading ampitude matrix: B = β (), (5) β (Q) (6) β (Q) β () The probem interest is to estimate the parameter τ based ontheobservedmatrixin(4), =,, 3 Mutipath Deay Estimation Using Modified Propagator Method 3 Review of Traditiona Propagator Method et us first review the traditiona propagator method The propagator method is previousy presented in DOA estimation 3 We introduce its appication in TOA estimation 4 From (5), it is noted that Z(τ), as a Vandermonde matrix, is of fu coumn rank with ineary independent rows The other K rows can be expressed as a inear combination of these rows Therefore, Z(τ) canbepartitionedinto Z (τ) = Z (τ) Z 2 (τ) } (7) } K, where Z (τ) C and Z 2 (τ) C (K ) Thepropagator is the unique inear operator P of C K into C such that or P H Z (τ) = Z 2 (τ) (8) P H, I K Z (τ) = Q H Z (τ) =, (9)

3 Distributed Sensor Networks 3 where H denotes the Hermitian transpose I K R (K ) (K ) and R (K ) denote the identity matrix andnumatrix,respectivey,andthematrixq is given by Q P () I K Equation (9)reveasthatZ(τ) is orthogona to the coumns of Q Therefore, the foowing TOA estimator can find the peaks of the spectrum: F PM (τ) = Z H (τ) QQ H Z (τ) () In (), Q can be effectivey estimated from the inear operations of the observed data We simiary partition the observed matrix H in (4)into H = H H 2 } } K, (2) where H C Q and H 2 C (K ) Q Inthenoise-freecase, from (4), (8), and (2), it yieds H 2 = P H H (3) In the presence of noise, a east squares (S) estimate of the propagator operator P is P =(H 2 H + ) H, (4) where Η + is the Moore-Penrosepseudo inverse of Η, Η + =(Η H Η ) Η H ThustheestimateofQ can be cacuated from () 32 The Modified Propagator Method The traditiona propagator method can reduce computationa oad compared with the common subspace techniques such as MUSIC and ESPRIT since it uses inear operations on the observed data, avoiding the EVD or SVD of covariance matrix However, it reveas from () that ong time is needed for spectra peak searching through a the scope of time deays To avoid it, we combine the idea of shift invariance with propagator method in the paper, presenting a modified propagator method (MPM) for mutipath deay estimation et H and H 2 be two (K ) Q submatrices of the matrix H consisting of the first and ast K rows of H;then, H = Z (τ) B + V, H 2 = Z 2 (τ) B + V 2, (5) where Z (τ) and Z 2 (τ) are the first and ast K rows of the matrix Z(τ),respectiveyThus,we have Z 2 (τ) = Z (τ) Φ, (6) where Φ = diag{e j(2πτ/kt),,e j(2πτ/kt) } is a diagona matrix, which contains the estimated time deay parameters τ,,τ Compose the two matrices H and H 2 to form a 2(K ) Q matrix: X = H = Z (τ) H 2 Z (τ) Φ B +V =D (τ) B + N, (7) V 2 B A C Figure : The geometric reationship of 3D mutiatera ocaization with five anchor nodes where D(τ) is a 2(K ) deay matrix and N is the noise matrix upon composition From the idea of propagator method, if D(τ) is partitioned into D (τ) = D (τ) D 2 (τ) E D } } 2 (K ), then there exists a unique propagator operator P satisfying (8) P H D (τ) = D 2 (τ) (9) Simiary, the matrix X can be partitioned into two matrices X and X 2 inthesamewayasd(τ) Ifthenoise matrix N is ignored, then X = D (τ) B, X 2 = D 2 (τ) B (2) Premutipy P H in X yieding P H X = P H D (τ) B = D 2 (τ) B = X 2 (2) In the noisy environment, P canbeachievedbythes soution; that is, P =(X 2 X H (X X H ) ) H (22) et P = I P H,whereI is a identity matrix; then PD (τ) = I P H D (τ) = D (τ) =D(τ) (23) D 2 (τ) From (23), we observe that P spans signa subspace Eveny dividing P into two (K ) matrices P and P 2 and based on (7), we have Therefore, P D P (τ) = Z (τ) (24) 2 Z (τ) Φ Z (τ) Φ = ΨZ (τ), (25)

4 4 Distributed Sensor Networks Unknown node (x, y, z) (X,Y,Z ) (X,Y,Z ) (X 2,Y 2,Z 2 ) (X 3,Y 3,Z 3 ) Anchor nodes (X 4,Y 4,Z 4 ) τ TOA τ TOA τ TOA2 τ TOA3 τ TOA4 MPM-based TOA τ TDOA τ TDOA2 τ TDOA3 τ TDOA4 TDOA Position computation of unknown node ( x, ŷ, ẑ) 3D ocation agorithm Figure 2: Procedure of ocating an unknown node with five anchor nodes where Ψ = P 2 ( P H P ) P H Itisshownfrom(25) thatφ is theeigenvauematrixofψ Therefore, the time deays of paths can be cacuated from main diagona eements of Φ; that is, τ = KT 2π ange (λ ), (26) where λ is the th eigenvaue of Φ,for=,, The procedure of MPM-based time deay estimation agorithm can be summarized as foows (i) Divide the observed matrix H into H and H 2 consisting of the first and ast K rows of H,respectivey (ii) Compose H and H 2 to form a 2(K ) Qmatrix X Partition X into X =X T, XT 2 T such that X C Q and X 2 C (2(K ) ) Q (iii) Obtain the propagator P =(X 2 X H (X X H ) ) H et P = I, and eveny divide P into two (K ) P H matrices P and P 2 (iv) Cacuate the eigenvaue matrix Φ of the matrix Ψ = P 2 ( P H P ) P H Then, τ can be obtained from the diagona eements of Φ 4 Node Position Computation Agorithm 4 Mutiatera ocaization From the resuts of MPMbased mutipath deay estimation given in (26), the τ TOA and distance r fromanunknownnodetoanchornodescanbe determined In the foowing procedure of 3D node position computation, we consider mutiatera ocaization other than triatera ocaization to improve the accuracy The number of anchor nodes is no ess than five in our agorithm A geometry reationship of the cross point of spheres with five anchor nodes(a,b,c,d,e)isshowninfigure Assume that (x,y,z) denotes the coordinate of an unknown node (X,Y,Z ) represents the coordinate of a main anchor node, and (X i,y i,z i ) represents the coordinate of the other anchor nodes, i =,2,,M,withM denoting the number of anchor nodes et t be the time of the transmitted UWB signa by the unknown node and et t i be the time of the received first-path signa by the ith anchor node Then, the distance between the unknown node and the ith anchor node is r i =c(t i t)=cτ TOAi Theequationof3D node position computation can be written by r 2 =(x X ) 2 +(y Y ) 2 +(z Z ) 2, r 2 i =(x X i ) 2 +(y Y i ) 2 +(z Z i ) 2, Δr i =r i r =cτ TDOAi, (27) where c denotes the veocity of wave, Δr i and τ TDOAi denote the distance difference and time-difference-of-arriva (TDOA) from the unknown node to the ith anchor node and to the main anchor node, and τ TDOAi =τ TOAi τ TOA, i=,2,,m Chan agorithm 6 andtayoragorithm7 aretwo traditiona methods to sove noninear positioning equations for radio ocation in a 2D space Chan agorithm has ow computationa compexity and high accuracy in high signanoise-ratio (SNR) and Gaussian noise environment However, with the increase of TOA measurement error, the performance of Chan agorithm in ow SNR environment wi degrade Tayor agorithm has higher accuracy and good robustness in noise However, the performance of Tayor agorithm is highy dependent on the initia estimate of iterative computation An improper initia estimate wi ead to no convergence Inthepaper,wepresentamarriageagorithmof3DChan andtayorocationtocacuatethe3dpositionofunknown

5 Distributed Sensor Networks 5 Ampitude TOA Time (ns) Transmitted signa Received mutipath signa Figure 3: The transmitted UWB signa and the received mutipath superposition signa Tabe : MPM-based deay estimation resuts when SNR = db Path deays True (ns) Estimate (ns) Path Path Path Path Path node Figure 2 shows the procedure of ocating an unknown node with M=5anchor nodes 42 3D Chan Agorithm The traditiona Chan agorithm can be extended from 2D space to 3D space From the positioning equation in (27), we have (X X i ) x+(y Y i ) y+(z Z i ) z=k i +r Δr i, (28) where k i = (/2)Δr 2 i (X 2 i +Y2 i +Z 2 i )+(X2 +Y2 +Z2 ) for i=,2,,m Thecoordinate(x,y,z)of unknown node canbeobtainedusingtheweighteds(ws)soution8 for in (28), and thus, X, Y, Z, x y X 2, Y 2, Z 2, = z X M, Y M, Z M, Δr 2 K +K Δr { Δr 2 2 K 2 +K Δr 2 } 2 + { r, { Δr 2 M K } M +K Δr M } (29) where X i, =X i X, Y i, =Y i Y, Z i, =Z i Z, K i =X 2 i + Y 2 i +Z 2 i, K =X 2 +Y2 +Z2, and + denotes pseudoinverse of a matrix In (29), since r contains unknown coordinate (x,y,z), substituting (29) into(27) yieds a second-order equation with respect to r Obtaining the WS soution of r and substituting its positive root into (29),wecanattainanaccurate positionestimate ( x, y, z) of unknown node 433DTayorAgorithm Simiary, Tayor agorithm can aso be extended to 3D space Starting with an initia estimate (x (),y (),z () ),the3dtayoragorithmiterativeycacuates the position error Δ using WS: Δx Δ = Δy =(G T Q G) G T Q h, (3) Δz where Q isthecovariancematrixoftdoameasurement Consider the foowing: + (X x) (X x) r r (X x) (X 2 x) G = r r 2 (Z z) (Z 2 z) r r 2 (Y y) r (Y y) r (Y y) r (Y y) r (Y 2 y) r 2 (Y M y) r M (Z z) r (Z z) r (Z z) r (Z z) r (Z 2 z) r 2 (Z M z) r M, (3) Δr (r r ) Δr 2 (r 2 r ) h = Δr M (r M r )

6 6 Distributed Sensor Networks 4 RMSE (ns) 5 Run time (s) SNR (db) snapshots MPM TS-ESPRIT MPM TS-ESPRIT (a) RMSE versus SNR (Q =5) (b) Run time versus Snapshots (SNR = db) Figure 4: Performance comparison between MPM and TS-ESPRIT 44 Marriage of 3D Chan/Tayor ocation Based on the two agorithms mentioned above, a marriage agorithm of 3D Chan and Tayor ocation is presented to cacuate the 3D position of the unknown node The procedure is as foows In the first step, 3D Chan agorithm is empoyed to cacuate the estimated coordinate of an unknown node based on (29), denoted by ( x, y, z) In the foowing step, taking the resut of 3D Chan agorithm as an initia vaue of Tayor series expansion, 3D Tayor agorithm is used to iterativey cacuate the ocation error Δ in (3) The iterative procedure has ended untiδ is smaer than a threshod Then, an accurate estimate ofthenodepositioncanbecacuatedby x x Δx y = y + Δy (32) z z Chan ΔzTayor Therefore, the 3D ocation method inherits the advantages of Chan and Tayor agorithms, having sma computationa compexity, high accuracy, and good robustness in noise 5 Simuations In the simuations, the performance improvement achieved by the MPM is presented Then, node ocaization accuracy using MPM-based mutipath deay estimation and the marriage of 3D Chan/Tayor ocation is verified 5 MPM-Based Mutipath Deay Estimation Resuts The transmitted UWB signa (T p = ns, A p = )and the received five-path superposition signa are shown in Figure 3 Tabe iustrates the resuts of time deay estimation using the proposed MPM agorithm SNR =db =5 Q= K = T = 4 ns The simuations confirm highresoution performance of the proposed MPM-based deay estimation agorithm in mutipath environment, even at ower SNRs Figure 4 presents the comparison of root mean square error (RMSE) and run time between the MPM agorithm and TS-ESPRIT agorithm There are =3paths with time deays 6 ns, 7 ns, and 8 ns K=4 T = ns, with Monte Caro trias In Figure 4(b),we evauate the computationa compexity using TIC and TOC instructions in MAT- AB Figure 4 demonstrates that MPM agorithm and TS- ESPRIT agorithm have simiar RMSE performance under medium and high SNR However, MPM agorithm has ess computationa compexity compared with TS-ESPRIT agorithm The computation oad of TS-ESPRIT mainy reies on estimating the covariance matrix R XX C 2(K ) 2(K ) and performing eigenvaue decomposition for three times, whie MPM agorithm avoids the covariance matrix estimation, whose computation oad mainy reies on the cacuation of propagator P and an eigenvaue decomposition 52 3D Position Computation Resuts In Figure 5, we assume that M=7anchor nodes are arranged in a 2 m 2m 2m space Among them, a main anchor node is in the center of thecube,thecoordinateofwhichis(,,)thecoordinates of the other six anchor nodes are (,, ), (,, ), (,, ), (,, ), (,, ), and(,, ), respectivey Mutipe unknown nodes are randomy generated within the space Assume that UWB signa is sent from an unknown node to an anchor node through =2paths SNR = db At first, we can obtain τ TOA τ TOA6 using MPM-based estimation agorithm, in which Q = K=2 T=ns Then, the

7 Distributed Sensor Networks z (m) z (m) y (m) 5 5 x (m) 5 5 y (m) 5 5 x (m) 5 Anchor node Unknown node Estimate of unknown node (a) unknown nodes, 7 anchor nodes Anchor node Unknown node Estimate of unknown node (b) unknown nodes, 7 anchor nodes Figure 5: Node position computation resut positions of mutipe unknown nodes can be cacuated using the marriage agorithm of 3D Chan/Tayor ocation The resuts of 3D position computation of or unknown nodes are demonstrated in Figures 5(a) and 5(b), respectivey The resuts show that the coordinates of mutipe unknown nodes can be effectivey estimated with high accuracy, even with ow density of anchor nodes 53 Accuracy of the Proposed Agorithm We investigate the ocaization accuracy and robustness of the proposed agorithm by simuation The node position error can be cacuated by Position error (cm) Position Error N = (( x 3N n x) 2 +( y n y) 2 +( z n z) 2 ), n= (33) The number of anchor nodes Figure 6: Position error versus the number of anchor nodes (SNR =db) where N isthenumberofunknownnodesand( x n, y n, z n )is the coordinate estimate of (x n,y n,z n )forthenth unknown node unknown nodes are randomy generated in the space When SNR = db, the performance of node position error versus the number of anchor nodes is shown in Figure 6, in which the number of anchor nodes varies from 5 to 8 The simuation resut from Figure 6 indicates that the accuracy is promoted with the increase of the number of anchor nodes Furthermore, the performance of the proposed agorithm is compared with the matrix penci based ocaization agorithm in 2 and the previousy proposed ocaization agorithm in 5, as shown in Figure 7 The simuation resuts revea that the performance of the proposed agorithm outperforms the other agorithms By empoying the MPMbased time deay estimation agorithm, the TOA measurement accuracy of our method can be greaty improved compared with matrix penci based TOA measurement agorithm in 2 In addition, by the use of the marriage of 3D Chan/Tayor ocation, the proposed method has better robustness in ow SNR compared with the previous agorithm in 5 6 Concusion In this paper, we propose a fast 3D node ocaization agorithm in mutipath for UWB wireess sensor networks It

8 8 Distributed Sensor Networks Position error (cm) SNR (db) Agorithm in 2 Agorithm in 5 Proposed agorithm Figure 7: The comparison of ocaization accuracy (7 anchor nodes) utiizes MPM-based mutipath time deay estimation and the marriage of 3D Chan/Tayor ocation Compared with the traditiona range-based methods, our method has greaty reduced the computationa compexity and enhanced the robustness and ocaization accuracy in mutipath and noise environment The proposed agorithm has key superiorities in fast executing, high accuracy in mutipath, and ow SNR, which is appicabe for node ocaization in 3D space Confict of Interests The authors decare that there is no confict of interests regarding the pubication of this paper Acknowedgments The research is supported by the Nationa Natura Science Foundations of China (674 and 63758) and the Jiin Province Natura Science Foundations (2254) References Cheng,CWu,YZhang,HWu,Mi,andCMape, A survey of ocaization in wireess sensor network, Internationa Distributed Sensor Networks, vo22,articeid , 2 pages, 22 2 H Qu, G Hou, Y Guo, N Wang, and Z Guo, ocaization with singe stationary anchor for mobie node in wireess sensor networks, Distributed Sensor Networks, vo23,articeid2286,pages,23 3 S-H Hong, B-K Kim, and D-S Eom, ocaization agorithm in wireess sensor networks with network mobiity, IEEE Transactions on Consumer Eectronics, vo55,no4,pp92 928, 29 4 Y-Y Cheng and Y-Y in, A new received signa strength based ocation estimation scheme for wireess sensor network, IEEE Transactions on Consumer Eectronics, vo55,no3,pp , 29 5 E Kim, S ee, C Kim, and K Kim, Mobie beacon-based 3Docaization with mutidimensiona scaing in arge sensor networks, IEEE Communications etters, vo 4, no 7, pp , 2 6 M T Isik and O B Akan, A three dimensiona ocaization agorithm for underwater acoustic sensor networks, IEEE Transactions on Wireess Communications, vo8,no9,pp , 29 7 C-D Wann and S-H Hsu, Estimation and anaysis of signa arriva time for UWB systems, in Proceedings of the IEEE 6th Vehicuar Technoogy Conference,vo5,pp ,September 24 8 M-A Paas and G Jourdain, Active high resoution time deay estimation for arge BT signas, IEEE Transactions on Signa Processing,vo39,no4,pp78 788,99 9 Dumont, M Fattouche, and G Morrison, Super-resoution of mutipath channes in a spread spectrum ocation system, Eectronics etters,vo3,no9,pp ,994 H Saarnisaari, TS-ESPRIT in a time deay estimation, in Proceedings of the 47th IEEE Vehicuar Technoogy Conference, vo 3, pp , May 997 N Dharamdia, R S Adve, and R Farha, Mutipath Deay Estimation using Matrix Penci, in Proceedings of the IEEE Wireess Communications and Networking Conference, vo,pp , 23 2 C iu, H Jiang, and D Zeng, Unitary matrix penci agorithm for range-based 3d ocaization of wireess sensor network nodes, Networks,vo7,no9,pp384 39,22 3 S Marcos, A Marsa, and M Benidir, The propagator method for source bearing estimation, Signa Processing, vo42,no2, pp 2 38, H Jiang, F Cao, and R Ding, Propagator method-based TOA estimation for UWB indoor environment in the presence of correated fading ampitudes, in Proceedings of the 4th IEEE Internationa Conference on Circuits and Systems for Communications (ICCSC 8), pp , May 28 5 HJiang,YZhang,HCui,andCiu, Fastthree-dimensiona node ocaization in UWB wireess sensor network using propagator method (digest of technica papers), in Proceedings of the IEEE Internationa Conference on Consumer Eectronics (ICCE 3), pp , January 23 6 Y T Chan and K C Ho, Simpe and efficient estimator for hyperboic ocation, IEEE Transactions on Signa Processing, vo42,no8,pp95 95,994 7 W H Foy, Position-ocation soutions by tayor-series estimation, IEEE Transactions on Aerospace and Eectronic Systems, vo2,no2,pp87 94,976 8 Z Jian-Wu, Y Cheng-ei, T Bin, and J Ying-Ying, Chan ocation agorithm appication in 3-dimension space ocation, in Proceedings of the ISECS Internationa Cooquium on Computing, Communication, Contro, and Management (CCCM 8), pp , August 28

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