Least Squares Algorithms for Time-of-Arrival Based Mobile Location

Transcription

1 Leat Square Algorthm for me-of-arrval Baed oble Loaton K. W. Cheung H. C. So W.-K. a Y.. Chan Dept. of Computer Engneerng & Informaton ehnology Cty Unverty of Hong Kong at Chee Avenue Kowloon Hong Kong Dept. of Eletron Engneerng he Chnee Unverty of Hong Kong Shatn N.. Hong Kong gary.heung@tudent.tyu.edu.hk tho@tyu.edu.hk wkma@ee.uhk.edu.hk ythan@ee.uhk.edu.hk ABSRAC Aurate loalzaton of moble phone of onderable nteret n wrele ommunaton. In th paper two algorthm are developed for aurate moble loaton ung the tme-of-arrval meaurement of the gnal from the moble taton at three or more bae taton. he frt algorthm an unontraned leat quare (LS) etmator whh enjoy t mplementaton mplty. he eond algorthm onder olvng a nononvex ontraned LS problem for mprovng etmaton auray. Smulaton reult how that the ontraned LS etmator yeld better performane than t unontraned ounterpart and aheve both the Cramer-Rao lower bound and the optmal rular error probablty. Keyword me-of-arrval potonng algorthm moble termnal. INRODUCION oble loaton ha reeved a lot of nteret ne the frt rulng of the Federal Communaton Common for deteton of emergeny all n the Unted State n 996 []. he apablty of aurate potonng of a moble phone one of the eental feature that wll help thrd generaton (3G) wrele ytem n reahng a wde ue and trggerng a large number of nnovatve applaton. In addton to emergeny management moble poton nformaton wll alo be ueful n ntellgent tranport ytem loaton bllng nteratve map onultaton and montorng the mentally mpared []-[6]. Wrele loaton ytem uually requre two or more bae taton (BS) to nterept a moble taton (S) gnal. Common loaton approahe are baed on tme-of-arrval (OA) reeved gnal trength (RSS) tme-dfferene-of-arrval (DOA) or angle-of-arrval (AOA) meaurement determned from the S gnal reeved at the BS [6]-[8]. In the OA method the dtane between the S and BS meaured by fndng the one-way propagaton tme of a gnal travellng between them. Geometrally the S poton gven by the ntereton of rle aoated wth at leat three BS n order to reolve ambgute arng from multple rong of the lne of poton aumng perfet OA meaurement are employed. In pratal tuaton where the OA meaurement are ubjet to error nonlnear leat quare (NLS) an approprate but omputatonally demandng approah to etmate the S poton [4]. he RSS meaurement are obtaned by ung the ame trlateraton onept where the path loe of the rado propagaton from the S to the BS are meaured to gve ther dtane. In the DOA method the dfferene n OA of the S gnal at multple par of BS are meaured. Eah DOA meaurement defne a hyperbol lou on whh the S mut le and the loaton etmate gven by the ntereton of two or more hyperbolod. Fnally the AOA method neetate the BS to have mult-element antenna array for meaurng the arrval angle of a gnal from the S at the BS. From the AOA etmate a lne of bearng (LOB) from the BS to the S an be drawn and the poton of the S alulated from the ntereton of a mnmum of two LOB. he prnple of thee method an be vualzed n Fgure. In the general ae wth extra meaurement the OA RSS DOA or AOA etmate are onverted nto a et of nonlnear equaton from whh the S poton an be determned wth the knowledge of the BS geometry. Copyrght held by the author. PGDay 3 Jan. 5 3 he Hong Kong Polytehn Unverty Hong Kong. Fgure. OA DOA RSS & AOA meaurement

2 he fou of th paper to develop aurate and omputatonally effent moble potonng algorthm ung the OA meaurement. Our prevou work [9] on OA-RSS hybrd potonng wa baed on the maxmum lkelhood (L) approah. Although the performane of the L-baed method an attan the Cramer-Rao lower bound (CRLB) ther omputatonal omplexty extremely demandng. oreover the L ot funton are multmodal and thu ntal guee loe to the true poton whh may be dffult to obtan n prate are requred for global onvergene. o allow real-tme mplementaton and enure global optmzaton we adopt the dea of the pheral nterpolaton (SI) n DOA-baed loaton [] that reorganze the nonlnear hyperbol equaton nto a et of lnear pheral equaton by ntrodung an ntermedate varable whh a funton of the oure poton. However the SI etmator olve the lnear equaton dretly va the tandard leat quare (LS) wthout ung the known relaton between the ntermedate varable and the poton oordnate. o mprove the loaton auray of the SI approah Chan and Ho had propoed [] to ue two weghted lnear quare (WLS) to etmate the oure poton by explotng th relaton mpltly whle [] norporate t expltly by mnmng a ontraned LS funton baed on the tehnque of Lagrange multpler. Aordng to [] thee two modfed algorthm are referred to a the quadrat-orreton leatquare (QCLS) and lnear-orreton leat-quare (LCLS) repetvely. Reently the QCLS method ha been modfed for OA-baed loaton n the preene of non-lne-of ght (NLOS) propagaton [3] that when at leat one of the dret or lne-ofght (LOS) path between the S and BS bloked. he ret of the paper organzed a follow. In Seton the model for the OA meaurement derbed. In Seton 3 we frt derve a mple LS OA-loaton algorthm baed on the SI. An mproved algorthm whh wegh the LS funton and explot the relaton between the range and the poton oordnate then deved. wo mportant performane meaure of loaton auray namely the CRLB [6] and rular error probablty (CEP) [4] are revewed n Seton 4. Smulaton reult are preented n Seton 5 to evaluate the loaton etmaton performane of the two method. Fnally onluon are drawn n Seton 6.. OA EASUREEN ODEL It aumed that a relable NLOS deteton algorthm ha frt been employed to elmnate the meaurement wth large error. A a reult all meaurement we utlze for the S loaton ome from LOS propagaton. Let the true loaton of the S be u [ x ] and the oordnate of the th BS be [ x ] y L where the total number of reevng LOS BS. he dtane between the S and the th BS denoted by gven by d ( x x ) + ( y y ) K y () In the abene of meaurement error the one-way propagaton tme taken for gnal to travel from the S to the th BS t d t K () d 8 where 3 m the peed of lght. he range meaurement baed on t n the preene of dturbane denoted by r modeled a r d + n ( x x ) + ( y y ) + n K where n the meaurement error n (3) at the th BS. For eae of analy we aume that eah meaurement error n a zero- mean whte proe wth varane. 3. OBILE LOCAION ALGORIHS 3. Leat Square In th eton we frt apply the SI tehnque to develop an LS moble loaton etmator ung the OA meaurement. Wthout meaurement error (3) beome: ( x x ) + ( y y ) L r (4) Squarng both de of (4) yeld r σ r ( x + y ) x x y y + ( x + y ) x x + y y.5r ( x + y r ) L where R x + y the range varable ntrodued n order to reorganze (4) nto a et of lnear equaton n x y and. Expreng (5) nto matrx form we have where (5) Aθ b (6) x A x x θ y R y y x + y b x + y.5.5 r r In the preene of meaurement error θ an be etmated ung unontraned LS: θˆ arg mn θ ( A A) A b ( Aθ b) ( Aθ b) 3. Weghted Leat Square wth Contrant For better performane we an add a weghtng matrx W to (7) and retrt θ to atfy the ba relatonhp R x + y (8) h lead to a ontraned optmzaton problem a follow: (7) R

3 ubjet to where P ( Aθ b) W( Aθ b θˆ arg mn ) (9) q θ + θ Pθ θ and q Here () a matrx repreentaton of (8). () Let u tudy the dturbane n b whh wll lead to a uggeton on the hoe of W. For uffently mall meaurement error or hgh gnal-to-noe rato (SNR) ondton the quared value of an be approxmated a r ( d + n ) d + d n L () A a reult the dturbane between the true and meaured quared dtane ε r d d n L () In vetor form { ε } expreed a [ d n d n L d n ε ] he ovarane matrx of the dturbane thu of the form { εε } BQB ψ E ( ) (3) (4) ( ) where B dag d d K d and Q dag σ σ K σ. Under the above aumpton the optmum weghtng matrx for (9) W ψ whh depend on the unknown {. For th reaon d } { } we approxmate ψ BQB ˆ ˆ where B ˆ dag r r K r. We now go bak to olve (9) ubjet to () whh equvalent to mnmzng the Lagrangan [8] Γ ( θ ) ( Aθ b) ψ ( Aθ b) + ( q θ + θ Pθ) (5) where the Lagrange multpler. he mnmum of (5) obtaned by dfferentatng Γ ( θ) wth repet to ( θ ) and then equatng the reultant expreon to zero: ( θ ) Γ θ ( θ ) Γ ( A ψ A + P) θ A ψ b + q q θ + θ Pθ Gven the oluton to (6) θ ( ) ( A ψ A + P) A ψ b q ˆ o fnd the that atfe (7) we ubttute (8) nto (7) q A ψ A q ( A ψ A + P) + b ψ he matrx ( A ψ b q (6) (7) (8) (9) ( A ψ A + P) P( A ψ A + P) A ψ b q A) P an be dagonalzed a r ( ψ A) P UΛU A () where dag( γ γ γ ) 3 of the matrx ( ψ A) ( A ) ψ A + P gve Λ and γ ( A ψ A + P) U( I + Λ) U ( Puttng () nto (9) we get 3 are the egenvalue A P. Subttutng () nto A A ψ ) () ( I + Λ) f ( I + Λ) g + e ( I + Λ) Λ( I + Λ) e + 4 where ( I + Λ) Λ( I + Λ) g ( I + Λ) Λ( I + Λ) ( I + Λ) Λ( I + Λ) g q U g U e b ψ f U [ ] 3 ( ) [ ] A ψ A q g g AU g 3 [ e e e ] 3 ( A ψ A) A ψ b [ f f ] f 3 f f () Sne the matrx ( A ψ A) P of rank one of t egenvalue ay γ mut be zero. After expanon of equaton () and 3 puttng γ 3 f g e g γ () mplfed to: f + γ g + + γ + ( + γ ) ( + γ ) ( + γ ) 4 ( + γ ) f γ g γ e f γ (3) whh an equaton of 5 root. he dered found by the followng proedure.. Obtan the 5 root ung a root fndng algorthm. Dard any omplex root beaue the Lagrange multpler alway real for real optmzaton problem.. Put the real ' bak to (8) and obtan ub-etmate of θ.. he ub-etmate that yeld the mallet objetve value of ( Aθ b) W( Aθ b) taken a the globally optmal ontraned WLS oluton. 4. CRLB and CEP he CRLB gve a lower bound on varane attanable by any unbaed etmator and thu t an be erved a a benhmark to ontrat wth the mean quare error of the OA potonng algorthm. he CRLB for the kth parameter etmate of u denoted by CRLB uˆ ) where u [ u ˆ uˆ ] [ xˆ yˆ ] an be omputed from [5]: where ( k CRLB û ˆ ( ) [ ( )] u k k I (4) kk

4 [ I( u) ] j E ln p u u ( r u) j (5) the orrepondng Fher nformaton matrx (FI). he quantty p( r u) the probablty denty funton of the meaurement r r [] [ r K r ] of the S and E. the expetaton operator. From (5) the FI an be alulated a ( x x) ( x + ( y y) ( x x)( y y) ( x x) + ( y y) ondtoned on the loaton ( x x)( y y) ( x x) + ( y y) ( y y) ( x x) + ( y y) ( ) [ ] σ [ ] σ I u (6) [ ] [ ] σ σ hen the orrepondng CRLB an be alulated by ung (4) and (6). It noteworthy the CRLB hould reman unhanged when the NLOS meaurement f any are alo nluded n the omputaton [6]. Apart from the CRLB there another approxmate but mple meaure of auray for loaton etmate whh alled the CEP. It defned a the radu of the rle that ha t entre at the mean and ontan half the realzaton of the loaton etmate. If the loaton etmator unbaed the CEP a meaure of the unertanty n the loaton etmate u ˆ relatve to the atual S loaton. Partular Etmate h rle ontan half of the loaton etmate Fgure 3 how that the geometry of the BS. hey were tuated at [ ]m [ 3 33]m [ 6]m [ 3 33]m and [ 3 3 3]m. Fgure 4 how the mean quare range error (SRE) defned a E ( ) [ x of the LS ˆ x + ( y yˆ ) ] and ontraned WLS method a well a the CRLB [8][5] veru the average SNR gven by d σ. In th mulaton d the SNR were dental for all OA meaurement. It an σ be een that the performane of the ontraned WLS wa very loe to optmum and t wa uperor to the LS by approxmately 5 db for the whole range of SNR. Fgure 5 and 6 how the dtrbuton of loaton etmate obtaned by the LS and ontraned WLS tehnque repetvely. he SNR wa at 5dB. he rle n the fgure were entred at the true loaton of the S whh nluded half of the loaton etmate. A hown n the fgure the rad or CEP of the LS and ontraned WLS were 9.m and 9.97m repetvely. hu the ontraned WLS etmator outperformed the LS etmator by approxmately m n CEP. oreover the CEP for the L etmator wa alulated to be 9.95m [9] and a a reult the optmalty of the ontraned WLS method agan demontrated. ( )m ean Etmate CEP Fgure 3. Loaton of the BS Fgure. Illutraton of CEP herefore the maller the CEP the more relable the etmator hould be. Note that an ellpe whh haraterzed by t angle of rotaton from x-ax major and mnor axe an generally derbe the ontour that ontan half the realzaton of etmate muh better than the CEP rle. he omplete proedure for omputng th ellpe a well a the CEP ung the L loaton etmate n Gauan noe an be found n [4]. Sne the L method an alway gve optmum loaton etmate the CEP ung the L loaton etmate the optmal CEP. 5. SIULAION RESULS Computer mulaton were performed to evaluate the performane of the propoed OA-baed loaton algorthm. he evaluated performane wa alo ompared wth the CRLB and CEP baed on L etmaton. here were 5 BS nvolved n the mulaton and the loaton of S wa at [ ]m. All reult were average of ndependent run. (db) a n q u are range erro r me No. of BS5 S at [m m] LS ontraned WLS CRLB Sgnal-to-Noe Rato (db) Fgure 4. ean quare range error of propoed method

5 6 No. of BS5 S at [m m] radu9.m SNR5dB 4 ()m y oordnate 98 Fgure 7. Loaton of the BS 96 No. of BS5 S at []m x oordnate 4 Fgure 5. CEP for LS OA-baed loaton algorthm 6 No. of BS5 S at [m m] radu9.97m SNR5dB 4 LS ontraned WLS CRLB 35 6 mean quare range error (db) Sgnal-to-Noe Rato (db) 6 Fgure 8. ean quare range error of propoed method 98 6 No. of BS5 S at [m m] radu9.8m SNR5dB x oordnate Fgure 6. CEP for ontraned WLS OAbaed loaton algorthm he above experment wa repeated for the followng BS geometry: []m [6]m [6]m [6]m and [6]m whh hown n Fgure 7. In Fgure 8 we ee that the SRE of the ontraned WLS algorthm remaned loe to the CRLB and outperformed the LS by about 5dB. In Fgure 9 and whh how the dtrbuton of loaton etmate for the LS and ontraned WLS tehnque the rad of the rle were 9.8m and.69m repetvely. he optmal CEP wa alulated to be.6m whh llutrate that the ontraned WLS algorthm optmal. y-oordnate y oordnate x-oordnate 4 Fgure 9. CEP for LS OA-baed loaton algorthm 6

6 y-oordnate No. of BS5 S at [m m] radu.69m SNR5dB [5] D.Porno Performane of a ODOA-IPDL potonng reever for 3G-FDD mode Pro. Int. Conf. 3G oble Communaton ehnologe pp x-oordnate 6. CONCLUSIONS wo OA-baed loaton algorthm are developed baed on the SI approah ung DOA meaurement. he frt LS algorthm dretly extend the SI ung the OA meaurement. he eond method an mproved veron of the frt algorthm va employng WLS and ontrant at the expene of nreang the omputatonal omplexty. It hown that the ontraned WLS approah an attan the CRLB and the optmal CEP. 7. ACKNOWLEDGENS h work partally upported by a grant from the Reearh Grant Counl of the Hong Kong Speal Admntratve Regon Chna (Projet No. CtyU 9/E). 8. REFERENCES [] [] C.Drane.anaughtan and C.Sott "Potonng GS telephone" IEEE Communaton agazne pp Aprl 998 [3] H.Kohma and J.Hoen Peronal loator erve emerge IEEE Spetrum pp.4-48 Feb. [4] Y.Zhao "oble phone loaton determnaton and t mpat on ntellgent tranport ytem" IEEE ran. Intellgent ranportaton Sytem vol. pp arh [6] J.J.Caffery Jr. Wrele Loaton n CDA Cellular Rado Sytem Kluwer Aadem Publher [7] J.C.Lbert and.s.rappaport Smart Antenna for Wrele Communaton: IS-95 and hrd Generaton CDA Applaton Upper Saddle Rver: Prente- Hall 999 [8].Gure K.N.Platanot A omparon of radoloaton for moble termnal by dtane meaurement Pro. Int. Conf. Wrele Communaton pp [9] K.W.Cheung H.C.So and Y..Chan oble loaton ung tme-of-arrval and reeved gnal trength meaurement Pro. of the 4th Int. Conf. Wrele Communaton vol. pp July Fgure. CEP for ontraned WLS OAbaed loaton algorthm [] J.O.Smth J.S.Abel Cloed-form leat-quare oure loaton etmaton from range-dfferene meaurement IEEE ran. Aout. Speeh Sgnal Proeng vol. ASSP-35 pp De. 987 [] Y..Chan and K.C.Ho A mple and effent etmator for hyperbol loaton IEEE ran. Sgnal Proeng vol. 4 pp Aug.994 [] Y.Huang J.Benety G.W.Elko and R..erereau Real-tme pave oure loalzaton: a pratal lnear-orreton leat-quare approah IEEE ran. Speeh Audo Proeng vol.9 pp Nov. [3] X.Wang and Z.Wang A OA-baed loaton algorthm redung the error due to non-lne-of-ght (NLOS) propagaton Pro. VC Fall vol. pp.97- [4] D.J.orrer Stattal theory of pave loaton ytem IEEE ran. on Aeropae and Eletron Sytem vol. pp arh 984 [5] S..Kay Fundamental of Stattal Sgnal Proeng: Etmaton heory Prente- Hall 993 [6] Y.Q and H.Kobayah Cramer-Rao lower bound for geoloaton n non-lne-of-ght envronment Pro. ICASSP vol.3 pp

7 Bography K. W. Cheung wa born n Hong Kong. He reeved the B.Eng. degree wth Frt Cla Honour n Eletral & Eletron Engneerng from Imperal College of Sene ehnology & edne Unverty of London n. From Otober to November he wa a Reearh Atant n the Department of Computer Engneerng & Informaton ehnology at the Cty Unverty of Hong Kong. He urrently purung an.phl. degree n the ame department. H reearh nteret are n array gnal proeng and developng effent method n radoloaton for moble termnal. r. Cheung now an Aoate ember of Inttuton of Eletral Engneer n U.K. and the Hong Kong Inttuton of Engneer. H. C. So wa born n Hong Kong. He obtaned the B.Eng. degree n Eletron Engneerng from Cty Polytehn of Hong Kong n 99. From 99 to 99 he wa an Eletron Engneer at the Reearh & Development Dvon of Everex Sytem Engneerng Ltd. In 995 he reeved the Ph.D. degree n Eletron Engneerng from he Chnee Unverty of Hong Kong. He then worked a a Pot-Dotoral Fellow at he Chnee Unverty of Hong Kong and wa reponble for devng and analyzng effent algorthm for geoloaton. From 996 to 999 he wa a Reearh Atant Profeor at the Department of Eletron Engneerng Cty Unverty of Hong Kong. Currently he an Atant Profeor n the Department of Computer Engneerng & Informaton ehnology at Cty Unverty. H reearh nteret nlude adaptve flter theory deteton and etmaton wavelet tranform and gnal proeng for ommunaton and multmeda. Wng-Kn a reeved the B.Eng. (wth frt la honour) degree n eletral and eletron engneerng from the Unverty of Portmouth Portmouth U.K. n 995. He obtaned the.phl. and Ph.D. degree both n eletron engneerng from the Chnee Unverty of Hong Kong n 997 and repetvely. In he wa wth the Department of Eletral and Computer Engneerng ater Unverty Hamlton ON Canada a a Vtng Sholar. Preently he a Reearh Aoate wth the Department of Eletron Engneerng the Chnee Unverty of Hong Kong. H reearh nteret le n ommunaton and gnal proeng wth reent fou on multuer deteton and advaned reever tehnque for ommunaton. Dr. a' Ph.D. the wa ommended to be "of very hgh qualty and well deerved honorary mentonng" by the Faulty of Engneerng the Chnee Unverty of Hong Kong n. Y.. Chan wa born n Hong Kong. He reeved the B.S. and.s. degree from Queen Unverty Kngton Ontaro Canada n 963 and 967 and the Ph.D. degree from the Unverty of New Brunwk Frederton Canada n 973 all n eletral engneerng. He ha worked wth Northern eleom Ltd. and Bell-Northern Reearh. From 973 to he wa at the Department of Eletral and Computer Engneerng of Royal ltary College of Canada. Currently he a Vtng Profeor at the Department of Eletron Engneerng of he Chnee Unverty of Hong Kong. H reearh nteret are n onar gnal proeng and pave loalzaton and trakng tehnque. He ha erved a a onultant on onar ytem. He wa an Aoate Edtor (98-98) of the IEEE ranaton on Sgnal Proeng and wa the ehnal Program Charman of the 984 Internatonal Conferene on Aout Speeh and Sgnal Proeng (ICASSP-84). He dreted a NAO Advaned Study Inttute on Underwater Aout Data Proeng n 988 and wa the General Charman of ICASSP-9 held n oronto Canada.