A Quadratic Constraint Total Least-squares Algorithm for Hyperbolic Location

Transcription

1 I. J. Communcaton, Network and Sytem Scence, 8,, 5-6 Publhed Onlne May 8 n ScRe ( A Quadratc Contrant otal Leat-quare Algorthm for Hyperbolc Locaton Ka YANG, Janpng AN, Zhan XU Department of Electronc Engneerng, School of Informaton Scence and echnology, ejng Inttute of echnology, ejng 8, P.R. Chna E-mal: yangkbt@gmal.com Abtract A novel algorthm for ource locaton by utlzng the tme dfference of arrval (DOA) meaurement of a gnal receved at patally eparated enor propoed. he algorthm baed on quadratc contrant total leat-quare (QC-LS) method and gve an explct oluton. he total leat-quare method a generalzed data fttng method that approprate for cae when the ytem model contan error or not known exactly, and quadratc contrant, whch could be realzed va Lagrange multpler technque, could contran the oluton to the locaton equaton to mprove locaton accuracy. Comparon of performance wth ordnary leat-quare are made, and Monte Carlo mulaton are performed. Smulaton reult ndcate that the propoed algorthm ha hgh locaton accuracy and acheve accuracy cloe to the Cramer- Rao lower bound (CRL) near the mall DOA meaurement error regon. Keyword: Locaton, me Dfference of Arrval, otal Leat-quare. Introducton Determnng the locaton of a ource from t emon a crtcal requrement for the deployment of wrele enor network n a wde varety of applcaton []. Locaton fndng baed on tme dfference of arrval (DOA), whch doe not requre knowledge of the abolute tranmon tme, the mot popular method for accurate potonng ytem []. he dea of DOA to determne the ource locaton relatve to the enor by examnng the dfference n tme at whch the gnal arrve at multple meaurng enor unt, rather than the abolute arrval tme. Senor at eparate locaton meaurng the DOA of the gnal from a ource can determne the locaton of the ource a the nterecton of hyperbolae for DOA. For each DOA meaurement, the ource le on a hyperbola wth a contant range dfference between the two meaurng enor. However, fndng the oluton to the hyperbolc locaton equaton not eay a the equaton are nonlnear. Furthermore, the nonlnear hyperbolc equaton become ncontent a error occur n DOA meaurement and the hyperbolae no longer nterect at a ngle pont. In the pat, the ource locaton determnaton problem ha been mathematcally formulated a a et of lnear equaton Axb whch n matrx form and ordnary leat-quare (OLS) technque utlzed to fnd the maxmum-lkelhood oluton by aumng the ytem matrx A error-free and all error are confned to the data vector b [3 6]. However, n the DOA baed locaton problem there are error n both ytem matrx A and data vector b. Ung OLS technque for th problem wll reult n baed oluton and locaton accuracy wll decreae due to the accumulaton of the ytem matrx error. o allevate th problem, a generalzaton of the OLS oluton, called total leatquare (LS) [7 9], utlzed to remove the noe n A and b ung a perturbaton on A and b of the mallet Frobenu norm whch make the ytem of equaton content. It hown that n the DOA baed locaton problem the unknown parameter n vector x are quadratc contrant related, whch could be realzed va Lagrange multpler technque to contran the LS oluton. For practcal applcaton, the locaton etmaton algorthm hould be robut and eay to mplement, and the ource locaton etmaton hould mnmze t devaton from the true locaton. In th paper, a novel DOA baed QC-LS algorthm for ource locaton Copyrght 8 ScRe. I. J. Communcaton, Network and Sytem Scence, 8,, 5-6

2 A QUADRAIC CONSRAIN OAL LEAS-SQUARES ALGORIHM 3 FOR HYPEROLIC LOCAION problem preented. he ret of the paper organzed a follow. he formulaton of DOA baed locaton etmaton problem decrbed n Secton. In Secton 3, th problem hown to be n the form of an overdetermned et of lnear equaton wth error n both A and b and the QC-LS algorthm appled to the DOA meaurement data for ource locaton etmaton. Computer mulaton are ued to verfy the valdty of the algorthm and mulaton reult are preented n Secton, whch followed by concluon.. Problem Formulaton here are two bac way to meaure the DOA n wrele enor network: the drect way and the ndrect way. In the drect way, we could obtan the DOA through the ue of cro-correlaton technque, n whch the receved gnal at one enor correlated wth the receved gnal at another enor. he tmng requrement for th method the ynchronzaton of all the recever partcpatng n the DOA meaurement, whch more avalable n locaton applcaton. However, n multpath channel there ambguty n detectng the DOA ung cro-correlaton technque nce the correlaton peak to be detected may not be caued by the two drect lne of ght (LOS) path []. In the ndrect way, we frt obtan the arrval tme of receved gnal tranmtted from the ource at patally eparated enor and then ubtract the meaurement of arrval tme at two enor to produce a relatve DOA. DOA alo could be obtaned by ubtractng abolute OA meaurement, but th requre the ynchronzaton of ource and enor. In th paper, becaue of thee factor above we obtan the DOA by ubtractng the meaurement of arrval tme at two enor. When DOA baed locaton method adopted to gve ource locaton etmaton n wrele network, accordng to DOA meaurement a et of hyperbolc equaton gven by r cτ r r,,3, L, M + r x x y y () where c the peed of gnal propagaton, τ the true value of DOA meaurement between enor and, r the dtance between the ource and enor, M the number of enor, and (x, y ) and (x, y ) are the coordnate of the ource and enor repectvely. Solvng thoe nonlnear equaton dffcult. Lnearzng them and then olvng one poble way. From (), we have r + r r () Subttutng the expreon of r and r nto () and then quarng both de of () produce (( x x ) ( y ) ) y r x x x x + y y y y + r r + Formulaton t n matrx form we have where A (3) Ax b () x x y y r x x y y r M xm x ym y rm [ x x y y r] x x x + y y r x3 x + y3 y r 3 b M ( xm x) + ( ym y) rm and the upercrpt denote the matrx tranpoe operaton. In the abence of noe and nterference, hyperbolae from two or more enor wll nterect to determne a unque locaton whch mean that the overdetermned et of lnear equaton () content. In the preence of noe, more than two hyperbolae wll not nterect at a ngle pont whch mean that () ncontent. In the followng ecton, we wll develop the oluton to the overdetermned equaton. 3. Hyperbolc Locaton Soluton In th ecton we frt analyze the OLS oluton to () and then gve the QC-LS oluton. 3.. OLS Approach In our applcaton, the uual aumpton for OLS approach that the matrx A cont of M- obervaton on each of the 3 ndependent varable. he dependent varable repreented by vector b, n whch we try keepng the correcton term b a mall a poble whle multaneouly compenatng for the noe preent n b by forcng Axb+ b [7]. Under thee aumpton the leat-quare etmator x A A A b (5) Copyrght 8 ScRe. I. J. Communcaton, Network and Sytem Scence, 8,, 5-6

3 3 K. YANG E AL. n whch t mplct that A known exactly. Suppoe now that the element of the A matrx contan error. he ytem matrx A reultng from the calculaton of () by ung the meaured value of DOA can be gven by AA +E, where A and E repreent the true value and error repectvely. Under the aumpton that the error E reaonably mall, retanng only the lnear error term, the OLS etmator gven by [] + x E x A A E η AA AEx (6) where η the vector of redual b-ax whch would be obtaned f A were known accurately. he entvty of each etmated parameter x k to each element A j of ytem matrx A come mmedately from (6) by takng the dervatve of x k wth repect to A j and th yeld x AA AA (7) 3 ( ) η ( ) k j A k kl jl x A j l he entvte are dfferent, both n value and rankng, from the entvte to change n the dependent varable. It would n partcular gve detal of thoe obervaton mot lable to caue etmaton error. 3.. QC-LS Approach In hyperbolc locaton problem, t can be argued that both the ytem matrx A and vector b are ubject to error, whch out of accord wth the aumpton of OLS. o b b xa ( a, b ) ( a, b ) 3 3 ( a, b) a geometrc nterpretaton, t hown that the LS method better than the LS method wth repect to the redual error n the curve fttng. When error ext n DOA meaurement, () can be repreented by A + E x b + w (8) where E and w are the perturbaton of A and b, repectvely. And (8) alo can be put nto the followng form or Where b M A + w M E (9a) x ([ ] [ ]) ( +D) α (9b) [ M ] [ M ],, b A D w E α x he rank of matrx three, becaue t equal to the rank of matrx A. he LS oluton to th problem to look for the mnmal (n the Frobenu norm ene) perturbaton matrx E and perturbaton vector w o that ubject to { } xew ˆ,, argmn D () xew,, F A + E x b + w where the ubcrpt F denote the Frobenu norm. he LS optmzaton nvolve to fnd the optmum f α ˆ, ˆα for α that mnmze the cot functon F f αˆ αˆ αˆ α ˆ () In practcal applcaton, the value of matrx unknown. We now make a ngular value decompoton (SVD) of the matrx [-b A], that, U Σ V () H Fgure. Leat-quare veru total leat-quare allevate the effect of thee error, we propoe to ue the LS oluton for the ource locaton problem. he total leat-quare method [8,9], whch a natural extenon of LS when error occur n all data, deved a a more global fttng technque than the ordnary leat-quare technque for olvng overdetermned et of lnear equaton by tryng to remove the data error. he LS and LS meaure of goodne are hown n Fgure. In the LS approach, t the vertcal dtance that are mportant; wherea n the LS problem, t the perpendcular dtance that are crtcal. So, from th where Σ compoed of the ngular value of. o olve (), matrx can be replaced by a rank three optmum approxmaton of, that, % U Σ V (3) 3 where Σ 3 compoed of the maxmum three ngular value of. Now we have the cot functon ( α) H f ˆ αˆ α % % ˆ () he parameter n vector ˆα are ubject to a quadratc contrant Copyrght 8 ScRe. I. J. Communcaton, Network and Sytem Scence, 8,, 5-6

4 A QUADRAIC CONSRAIN OAL LEAS-SQUARES ALGORIHM 33 FOR HYPEROLIC LOCAION or, equvalently, a ˆ ˆ ˆ + x x + y y r (5) ασα ˆ ˆ (6) where Σdag(,,, ) a dagonal and orthonormal matrx. he technque of Lagrange multpler ued and the modfed cot functon of the form f ( αˆ) αˆ α % % ˆ + λ { αˆ Σαˆ } αˆ ( % % + λσα ) ˆ λ (7) where λ the Lagrange multpler. he requred optmum parameter vector ˆα found by olvng the followng lnear ytem of equaton [] ( + λσα ) ˆ ke % % (8) where, e [ ], and normalzng contant k elected o that the frt element of oluton vector ˆα equal to one. Solvng (8) very eay, and t ndependent of normalzaton contant k, whch unknown. Calculatng the nvere of matrx ( % % + λσ) and then normalzng the frt element of ˆα, we can obtan the oluton vector. he QC-LS oluton vector to (9) % % % % αˆ e Σ e e Σ e + Σ ke + Σ e % % + k % % + he etmated x and y coordnate of the ource are where xˆ ˆ α e + x yˆ αˆ e + y e e 3 3 [ ] [ ] (9) () Lagrange Multpler λ n the expreon of oluton vector ˆα unknown. In order to fnd λ, we can mpoe the quadratc contrant drectly by ubttutng (9) nto (6), o that % % + Σ e % % + Σ e Σ % % + % % + e Σ e e Σ e () y ung an egenvalue factorzaton, the matrx Σ % % can be dagonalzed a % % () Σ UΛU where Λdag(γ, γ, γ 3, γ ), γ,,, are the egenvalue of the matrx Σ % %, and U the correpondng egenmatrx of Σ % %. Subttutng () nto () produce ( + λ ) ( + λ ) e ΣU Λ I U e e ΣU Λ I U e For notatonal convenence, we defne 3 ΣU p p p p [ p p p 3 p ] [ 3 ] [ q q q q ] p U q q q q q 3 (3) () Subttutng () nto (3), then the quadratc contrant n the form of λ become pq f λ γ + λ ( ) ( γ + λ) pq ( γ + λ) γ + λ (5) Lagrange Multpler λ can be olved from (5) by ung Newton method wth zero a the ntal gue. o um up, a bref decrpton of the propoed DOA baed locaton algorthm a follow: ) Calculate the optmum approxmaton of augmented matrx ung () and (3). ) Solve λ by fndng the root of (5) ung Newton method. 3) Obtan the QC-LS oluton to (9) ung (9) and determne x and y coordnate of the ource ung ().. Smulaton We have propoed a QC-LS algorthm for DOA baed locaton problem. In th ecton, we evaluate t performance at practcal meaurement error level ung Monte-Carlo mulaton. Source and enor were located n random poton n a quare of area m a hown n Fg.. We aumed that the DOA meaurement error were whte random procee wth zero mean and varance σ, DOA and the DOA varance of all enor nput were dentcal. For mplcty, the DOA varance wa tranlated nto correpondng dtance varance σ (cσ DOA ). Smulaton reult provded were average of ndependent run. We compare the propoed QC-LS approach wth OLS approach and CRL. Copyrght 8 ScRe. I. J. Communcaton, Network and Sytem Scence, 8,, 5-6

5 3 K. YANG E AL. able and compare the mean abolute locaton error (MALE) of OLS and QC-LS approach for varou DOA error varance, and MALE wa defned a E ( x ˆ ) ( ˆ ) x + y y. he number of enor wa et to 7 and for able and able, repectvely. In able and, CRL, whch a fundamental lower bound on the varance of any unbaed etmator, calculated ung the functon n []. he dtance unt meter. We oberve that ncreang the number of enor can ncreae the locaton accuracy, becaue ncreang the number of enor can provde more redundant nformaton whch could be helpful for olvng the overdetermned lnear equaton. Y /m enor ource X /m Fgure. Locaton n -D plane able. comparon of MALE for QC-LS and OLS method for varou varance: even enor σ.... OLS QC-LS CRL able. comparon of MALE for QC-LS and OLS method for varou varance: ten enor σ.... OLS QC-LS CRL he performance of QC-LS approach much better than OLS approach epecally when the DOA varance large, and t cloer to CRL. hey verfy that the propoed approach could nhbt the nfluence of DOA meaurement error on the etmaton reult better than OLS. When DOA varance large, the noe n matrx A, whch would be allevated by QC-LS approach, could gnfcantly reduce the performance of OLS approach. Mean abolute locaton error one of the metrc to evaluate the performance of QC-LS algorthm, and computatonal complexty another mportant metrc to evaluate the performance of algorthm. he prce of the better performance for QC-LS that t computatonal complexty larger than OLS. In the ame mulaton condton, the tme of ndependent run for QC- LS and OLS are approxmately m and 6.3 m, repectvely. It obvouly that QC-LS algorthm requre matrx ngular value decompoton once, egenvalue decompoton once, Newton teraton once, nvere operaton once and matrx multply operaton everal tme, wherea OLS algorthm only requre matrx peudonvere operaton once and matrx multply operaton everal tme. It noteworthy that QC-LS algorthm perform better at a cot of larger computatonal complexty. 5. Concluon A novel DOA baed quadratc contrant total leatquare locaton algorthm propoed n th paper. he propoed algorthm utlze LS to nhbt the nfluence of DOA meaurement error. And the technque of Lagrange multpler utlzed to explot the quadratc contrant relaton between the ntermedate varable to contran the oluton to the locaton equaton and mprove the locaton accuracy. Smulaton reult ndcate that the propoed QC-LS algorthm gve better reult than the OLS oluton. 6. Reference [] N. Patwar, J. N. Ah, S. Kyperounta, A. O. Hero III, R. L. Moe, and N. S. Correal, Locatng the Node: Cooperatve Localzaton n Wrele Senor Network, IEEE Sgnal Proceng Magazne, vol., no., pp. 5 68, July 5. [] X. Jun, L. R. Ren, and J. D. an, Reearch of DOA baed elf-localzaton approach n wrele enor network, n proceedng of IEEE Internatonal Conference on Intellgent Robot and Sytem, ejng, pp. 35, October 6. [3] J. Smth and J. Abel, Cloed-form leat-quare ource locaton etmaton from range-dfference meaurement, IEEE ranacton on Acoutc, Speech, and Sgnal Proceng, vol. 35, pp , December 987. [] Y.. Chan and K. C. Ho, A mple and effcent etmaton for hyperbolc locaton, IEEE ranacton on Sgnal proceng, vol., no. 8, pp , Augut 99. [5] R. Schmdt, Leat quare range dfference locaton, IEEE ranacton on Aeropace and Electronc Sytem, vol. 3, no., pp. 3, January 996. [6] Y. Huang, J. enety, G. W. Elko, and R. M. Merereau, Real-tme pave ource localzaton: a practcal lnearcorrecton leat-quare approach, IEEE ranacton Copyrght 8 ScRe. I. J. Communcaton, Network and Sytem Scence, 8,, 5-6

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