IJCSS Internatonal Journal of Computer Scence and etwork Securty, VOL.7 o., January 7 59 Weghted Leat-Square Soluton for nergy-baed Collaboratve Source Localzaton Ung Acoutc Array Kebo Deng and Zhong Lu Department of lectronc ngneerng, anng Unverty of Scence and echnology anng, Jangu 94, Chna Summary he Leat-Square (LS) acoutc ource locaton etmaton technque reported for e applcaton n a wrele enor network. he technque ue acoutc gnal energy meaurement taken at ndvdual enor of a wrele enor network to etmate an acoutc ource locaton. In paper, an mproved formulaton of localzaton problem, whch clarfe e LS etmaton error, frtly preented. hen two weghted oluton, weghted nonlnear LS and weghted lnear LS, are gven. he weghtng coeffcent are derved from e energy meaurement, whch drectly relate e energy treng w e etmaton error. Compared w extng LS meod, e weghted LS oluton delver more accurate reult and offer flexble mplementaton to reduce computatonal load. xtenve mulaton are conducted to confrm e performance advantage. Key word: Acoutc enor, Collaboratve gnal proceng, Leat quare, Senor network, Source localzaton. Introducton ffcent collaboratve gnal proceng algorm at conume le energy for computaton and le communcaton bandwd are hghly mportant for e applcaton of e wrele enor network [], []. Source localzaton one of e mportant collaboratve gnal proceng tak. It obectve to etmate e poton of one or more target wn a enor feld montored by e enor network. he extng acoutc ource localzaton technque are typcally baed on ree type of enor meaurement from phycal varable: tme delay of arrval [3]-[6], drecton of arrval [7]-[9] and receved enor gnal treng or energy []-[]. It found at e energy-baed meod derved from e receved gnal energy are much approprate for e applcaton to enor network []. In paper, we focu on collaboratve ource localzaton of a ngle target w acoutc enor. Let ere be enor deployed randomly but w known poton n a enor feld n whch a target emt omndrectonal acoutc gnal from a pont ource. It ha been hown at e acoutc energy n ground urface wll attenuate at a rate at nverely proportonal to e quare of e dtance from e ource []. he energy meaurement y at e r r enor can be modeled a y g + n,,, L () where g e gan factor of e enor, e gnal energy radated by e acoutc ource, r and r are e p coordnate of e ource and e enor (p or 3), n (,, L ) are meaurement noe approxmated well a Gauan noe w n ~ ( μ, σ ) and ndependent at dfferent enor. he problem to etmate e ource coordnate from e meaurement. In [], tartng from e acoutc energy decay model, e auor formulate e ource localzaton a a maxmum-lkelhood (ML) etmaton problem. By ntroducng e concept of energy rato and approxmatng e meaurement noe a t mean value, e etmaton problem furer formulated repectvely a oluton of nonlnear and lnear leat-quare (LS) one (n [], ey are named a R-LS and R-LS meod, repectvely). Smulaton and feld experment how at ee LS meod yeld promng reult. Fundamental e concept of e energy rato, whch elmnate e etmaton of ource energy n e ML etmaton. In addton, e energy rato alo gve u elegant geometrc explanaton of e ource localzaton. A hown n [], e ource locaton can be retrcted to a hyperphere (a crcle n -D coordnate) whoe center and radu are functon of e energy rato and e two enor locaton. If more enor are ued, more hyperphere can be determned. If all enor meaurement contan no noe, e correpondng hyperphere wll nterect at a partcular pont at correpond to e ource locaton. However, t hould be noted at e LS etmaton formulaton of ource locaton aume addtve whte Gauan noe (AWG). In practce, alough t reaonable to aume at e enor have AWG, e LS model error may not be AWG. In paper, we reformulate e LS etmaton problem by ncorporatng e meaurement noe nto etmaton model and fnd at weghted LS approache are more approprate for etmatng e ource locaton. Keepng n mnd at e Manucrpt receved January 5, 7. Manucrpt reved January 5, 7.
6 IJCSS Internatonal Journal of Computer Scence and etwork Securty, VOL.7 o., January 7 le energy for computaton needed n enor network applcaton, we derve mple weghtng matrce from e enor meaurement. he reultng weghted LS approache (denoted a R-WLS and R-WLS) take a bt more computatonal load an e LS one n [], however, e etmaton accuracy greatly mproved. Anoer advantage of e weghted LS meod able to chooe oe hyperphere or hyperplane, whch have great contrbuton to e accurate etmaton of e ource locaton, accordng to e weghtng coeffcent. In ene, e R-WLS and R-WLS can be mplemented n maller ze of LS etmator an e R-LS and R-LS can do. h paper organzed a follow. In Secton, an mproved etmaton model of target locaton formulated. he weghted oluton (R-WLS and R-WLS) are preented n Secton 3. In Secton 4, extenve mulaton are performed to how uperor performance of e weghted oluton over R-LS and R-LS meod. Secton 5 e concluon.. tmaton Model Intead of ung model () drectly, we conder a mean-removed one. Defne h ( y μ) g a normalzed and mean-removed enor meaurement. hen () can be expreed a h +ε,,, L () r r where ε ~ (, σ g ). From (), e ource energy related w any enor meaurement h a r r h d ε,,, L (3) where d r r e dtance between e ource and e enor. For any two enor, we have ε r r h d r r h d ε,, L and +, L, (4) Defne e energy rato [] κ of e and enor a For < κ, (4) can be reexpreed a r c ρ + ζ,,, L and +, L, (5) where c and ρ are e center and e radu of e hyperphere defned n [], and c r κ r κ κ r r, ρ κ d ε d ε ζ h h a compote noe twtng e hyperphere. he noe a zero-mean varable dependng on meaurement noe, enor readng and dtance between ource and enor. It varance dfferent from one (,) to anoer (,). quaton (5) e etmaton model of e ource locaton we wll ue n e followng dcuon. A great dfference between (5) and (8) n [] at (5) clearly defne e etmaton error, whch depend on e enor meaurement noe, enor meaurement and dtance from ource to enor. For acoutc enor, ere wll be C equaton mlar to (5). For e brevty of notaton, uppoe at M C hyperphere are ued for locatng e ource and let u denote ee hyperphere equaton a (6) r c ρ + ζ,,, L, M (7) 3. nergy Rato-Baed Weghted Leat Square Soluton Baed on (7), two leat quare formulaton can be defned. In [], e noe term ζ are aumed to be lnear, ndependent Gauan random varable w zero mean and dentcal varance. Obvouly, uch an aumpton may not be true a revealed n (6) and hence may caue ome performance degradaton. In ene, a well-known, weghted leat-quare formulaton [3] wll gve mproved oluton. Parallel to e development n [], we derve two weghted leat-quare oluton. κ h h / 3. nergy rato-baed weghted nonlnear leat For κ, q. (5) reduced to e twted hyperplane equaton between r and r. For mplcty of preentaton, we wll only dcu e cae of < n e followng development. κ
IJCSS Internatonal Journal of Computer Scence and etwork Securty, VOL.7 o., January 7 6 quare oluton (R-WLS) Defne ree M vector ar (), b and n w r c, ρ and ζ a er element, repectvely. hen (7) gven n matrx form a ar () b+ n (8) he weghted nonlnear leat-quare oluton to (8) gven by mnmzng ( ) ( ) J () r a() r b W a() r b (9) where W an arbtrary M M potve defnte weghtng matrx. he R-LS meod [] aume at e noe term ζ (,, L, M ) n e noe vector n are ndependent and dentcally dtrbuted, and erefore e unt weghtng matrx ued. A expected, e optmal weghtng e nvere of e noe covarance matrx Q { nn } f e matrx nonngular [4]. However, a ndcated n (6), e noe term ζ n e noe vector n due to e compoton of e two enor noe. Any enor noe may be ued to formulate everal noe element n e vector n. herefore, e noe term ζ are generally not ndependent and u reult n ngular covarance matrx Q. o overcome, we tll aume at e noe term are ndependent, but not dentcally dtrbuted. hen e noe covarance matrx wll be dagonal one w e varance of ζ a t element. h aumpton wll acrfce ome etmaton performance but can ave computatonal reource, whch hghly dered n energy-avng enor network. From (6) and (8), we ee at e dagonal element of ζζ, whch concern two Q wll be gven by { } enor meaurement. ote from () at e can be approxmated a h. hen hu d ε d ε ε ε ζ h h h h h h d r r () where σ σ ω +. W e double hg hg ( h h) ndce replaced by a ngle ndex m for e brevty of notaton, e weghtng matrx W can be elected a W dag. L,, L () ωm where ωm ω. Dcardng wll not affect e mnmzaton of (9). If e noe varance at dfferent enor are equal, e noe varance term can alo be dcarded from ω m. After fndng e weghtng matrx, e ource locaton can be etmated from (9) by ome nonlnear optmzaton meod, uch a exhautve earch, multreoluton earch, and gradent-baed teepet decent earch meod. In Secton IV, we wll ue multreoluton meod [] to conduct mulaton experment. Fnally, we note at e (9) can be tranformed nto a lnear LS problem w quadratc contrant, a dcued n [6]. 3. nergy rato-baed weghted lnear leat quare oluton (R-WLS) he optmzaton obectve J () r a 4- order nonlnear equaton, caued by quadratc term r. he R-WLS formulaton elmnate e quadratc term and ha a cloed form oluton. For any two equaton (7), ubtractng each de and arrangng er term, we get hyperplane equaton ( c ) ( ) ( ) c r c ρ c ρ + ( ζ ζ),, L M and +, L, M (3) Defne a matrx C w ( c c ) obervaton vector a each row, an b w ( c ) ( ) ρ c ρ a t element, and a noe vector ζ ζ a t element. In matrx form, (3) rewrtten a n w ( ) Cr b + n (4) σ σ + ω h h hg hg { ζζ} () hen e R-WLS oluton read a mnmzng ( ) ( ) J () r Cr b W Cr b (5)
6 IJCSS Internatonal Journal of Computer Scence and etwork Securty, VOL.7 o., January 7 where W a potve defnte weghtng matrx. he optmal weghtng e nvere of e noe covarance Q n n. matrx, { } A n lat ubecton, we only conder e dagonal element. Referrng to (4), (5), and (6), any dagonal element of Q wll concern four enor meaurement noe. here are two poblte for e generaton of (3): four dfferent enor and ree enor w one common. Let u denote e dagonal element a ζ ζ ζ ζ, whch can be expanded a {( kl )( kl )} {( ζ ζkl )( ζ ζkl )} { ζζ } { ζζ kl kl} { ζζ kl} + he lat term gven by { ζζ kl}, ( h h)( hk h) k l σ, k, l hhg l hu, ung () and (7), we have where τ kl {( )( )} kl kl kl (6) (7) ζ ζ ζ ζ τ (8) ω + ωkl, k l (9) σ ω + ωkl +, k, l ( h h)( hk h) hhg l 4 For e frt poblty, ere are 3 C hyperplane. For e econd poblty, ere need ree enor to determne a hyperplane, o ere are C hyperplane. W e four ndce kl replaced by a ngle ndex p for e brevty of notaton, e weghtng matrx W gven by. where τ p τ kl W dag. L,, L () τ p W e weghtng matrx defned by (), e etmated ource locaton gven by ( ) r C W C C W b () 3 4. Performance Smulaton We have conducted extenve mulaton experment to ae e performance of e R-WLS and R-WLS meod. wo mulaton reult are preented here. he frt experment compare e performance of e R-WLS and R-WLS meod to at of e R-LS and R-LS meod. he mproved performance of e R-WLS and R-WLS meod n term of locaton etmaton error and range etmaton error apparent n e mulaton reult. he econd experment how e performance varaton a e weghtng number. It found at e R-WLS and R-WLS meod have e advantage to chooe mportant hyperphere and/or hyperplane for etmaton. herefore, mall number of hyperphere and/or hyperplane can be ued to mplement e ource locaton etmaton w undeterorated performance. We aume ere are enor node, whch are randomly cattered n a -D (p) enor feld of ze m by m. All e enor gan calbraton et at, and e meaurement noe at dfferent enor aumed..d w e varance σ. A ngle ource locaton alo choen randomly from wn e enor feld. quaton () ued to generate e acoutc energy readng and. For e R-LS and R-WLS meod, we ue multreoluton (MR) earch oluton w ree level of grd ze at meter, meter, and.4 meter, repectvely. A. Performance Comparon for dfferent enor number and noe level In tudy, we compare e ource locaton etmaton error and e range etmaton error for dfferent enor number and noe level w four meod of R-LS, R-LS, R-WLS and R-WLS. Frt, at e noe level σ., we calculate e etmaton error (recorded n x- and y-coordnate and range, repectvely) w enor number 6,, and 5. hen, w, we calculate e range etmaton error at dfferent noe level. For each σ and ettng, we conduct repeated tral whch are averaged to obtan e etmaton error. able how e mean and covarance matrce of e locaton etmaton error. From e table, t een at e mean value of ee four meod do not how any tattcally gnfcant ba and, hence, e four etmate are unbaed. Furermore, e locaton error n dfferent dmenon are uncorrelated and e related varance are approxmately equal. It alo noted at e varance n bo x- and y-coordnate of all e meod decreae a e enor number ncreae. he R-WLS and R-WLS meod contently outperform e
IJCSS Internatonal Journal of Computer Scence and etwork Securty, VOL.7 o., January 7 63 R-LS and R-LS meod. he uperorty of e R-WLS meod to e R-LS meod obvou, and a enor number ncreae, e R-WLS meod perform bet. he range etmaton error can be furer analyzed by calculatng t probablty denty dtrbuton. We ue e htogram of e error a approxmaton of e dtrbuton. he reult are hown n Fg., w 5-m ncrement bn. In fgure, each row repreent reult obtaned from a partcular meod. ach column repreent reult from a partcular enor number. he mean and e tandard devaton of e error are alo calculated and lted n each ubfgure. It agan een at R-WLS meod perform bet among all ee meod. LS WLS LS WLS able. Mean covarance matrce of locaton etmaton error.5 Fg.. LS WLS LS WLS 6 enor mean5.4 tdev8.5 6 enor enor 5 enor [.5.4] 7 94 [..33] 47 5 5 65 [.3.7] 35 33 33 378 [..] 36 54 54 369 4 6 8 mean9.96.5 tdev4.59 4 6 8 mean5.87.5 tdev.83 4 6 8 mean4.65.5 tdev.93 4 6 8.5 [.6.38] 45 59 [..] 84 87 [..] 4 3 3 47 [.8. ] 4 5 5 6 enor mean3.97 tdev7.6 4 6 8 mean9.39.5 tdev4..5 4 6 8 mean6.4.5 tdev.53.5 4 6 8 mean7.5.5 tdev.57.5 4 6 8 [..5] 5 6 [.7.4] 3 9 [.3.8 ] 44 4 [.7. ] 5 5 5 enor mean3.5 tdev6.4 4 6 8 mean.65 tdev5.9 4 6 8 mean4.46.5 tdev8.5 4 6 8 mean.9 tdev6.5 4 6 8 Dtrbuton of e range etmaton error of e four meod ext, e root-mean-quared error (RMS) of e range etmaton of e four meod are mulated at dfferent noe level and. Fg. gve e varaton of e RMS veru noe level σ. From fgure, we can ee at e RMS ncreae a e noe level ncreae. he R-WLS and R-WLS meod yeld maller error an e R-LS and R-LS meod can do. Agan, e R-WLS meod outperform oer meod. RMS (n meter) 4 35 3 5 5 5 LS LS WLS WLS...3.4.5.6.7.8.9 σ Fg.. RMS of e range etmaton at dfferent noe level σ () B. Localzaton Accuracy w Reduced Hyperphere and Hyperplane If enor are ued to detect e ource, a analyzed before, ere wll be C hyperphere for e nonlnear LS oluton and C 3 3 4 + C hyperplane for e lnear LS oluton. However, n eory, e nonlnear LS meod only need 3 hyperphere, and e lnear LS meod only need hyperplane. In anoer world, ere wll be large redundant hyperphere/hyperplane n LS etmator. he weghted LS oluton propoed n paper enable u to chooe e mportant (large weghtng) hyperphere or hyperplane for locaton etmaton. In mulaton, we tudy e effect of weghtng number on etmaton accuracy. Let. hen ere are all C 45 hyperphere for e R-WLS meod 3 4 and C + 3 C 75 hyperplane for e R-WLS meod,.e., ere are equal number of weght to be computed. We conduct tral, and n each tral, all e weght of e hyperphere or hyperplane are computed ahead and arranged n decendng order. Fg.3 and 4 how e varaton of e mean value of e weght for σ.. A een, ere are only mall number of weght w large weghtng coeffcent. he etmaton accuracy of e ource locaton largely due to ee weght. Fg. 5 and 6 gve e RMS varaton of e locaton etmaton error w R-WLS and R-WLS meod a e number of weght. he etmated RMS decreae
64 IJCSS Internatonal Journal of Computer Scence and etwork Securty, VOL.7 o., January 7 a e number of weghtng coeffcent ncreae. When e number ncreae to certan value, e RMS do not decreae gnfcantly and even acend lghtly for e R-WLS meod. For e mulaton example, hyperphere for e R-WLS meod or hyperplane for e R-WLS meod are enough to enure e etmaton accuracy. he number of e hyperphere/hyperplane are greatly le an oe requred n R-LS and R-LS etmator. he advantage of e weghted meod n applcaton qute obvou. A rehold et to elect e weghtng coeffcent and en e computaton ource can be aved wout gnfcantly acrfcng e localzaton accuracy. 4 3 RMS (n meter) 4.5 4 3.5 3.5.5.5 5 5 5 3 35 4 45 o. of hyperphere Fg.5. RMS of e R-WLS meod veru e number of hyperphere Mean value (log) - - -3-4 5 5 5 3 35 4 45 Fg.3. Mean value ( log 3 Index of orted weghtng coeffcent ) of e orted weghtng coeffcent for e hyperphere RMS (n meter) 4 35 3 5 5 3 4 5 6 7 8 o. of hyperplane Fg.6. RMS of e R-WLS meod veru e number of hyperplane Mean value (log) - - -3-4 3 4 5 6 7 8 Index of orted weghtng coeffcent Fg.4. Mean value ( log ) of e orted weghtng coeffcent for e hyperplane 5. Concluon In paper, an mproved energy baed acoutc ource localzaton etmaton model propoed and two weghted leat quare oluton are preented. xtenve mulaton how at ee weghted oluton yeld performance uperor to at of e extng leat quare oluton. he weghted formulaton enable u to remove redundant hyperphere and/or hyperplane n orgnal formulaton and ave computatonal ource. Acknowledgment he work wa partally upported by e atonal atural Scence Foundaton of Chna rough e grant number 64759.
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