TRACKING PERFORMANCE OF GPS RECEIVERS WITH MORE THAN ONE MULTIPATH

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1 TRACKING ERFORANCE OF GS RECEIVERS WITH ORE THAN ONE ULTIATH ABSTRACT Chrstoph ACABIAU, Bnoît ROTURIER, Abdlahad BENHALLA CNS Rsarch Laboratory of th ENAC, ENAC, 7 avnu Edouard Bln, B 45, 3155 TOULOUSE CEDEX 4, Franc macabau@rchrch.nac.fr A wd maorty of studs on th ffct of multpath focusd on th analyss of th cod and phas trackng rrors whn only two rays ntr th trackng loops, namly th drct ray plus on dffractd ray. Unfortunatly, ths condton dos not corrspond to th most frqunt stuatons, whr th rcvd sgnal can b th dscrt sum of svral powrful rplcas of th drct sgnal. Th am of ths papr s to prsnt th thortcal rsults of a study that dntfs th prformanc of th GS rcvrs n th cas whr mor than two powrful rays ntr th trackng loops. Th thortcal analyss shows that n th cas of contnuous trackng, f th summd ampltuds of th rflctd sgnals s lowr than th ampltud of th drct sgnal, th cod rror nvlop of th compost sgnal can b approxmatd as th sum of th cod rror nvlop for ach ndvdual ray, xcpt n ts transton zons. Th dstorton n ths zons s nglgbl for th narrow corrlator DLLs and th DLLs controlld by lnar combnatons of corrlaton valus. I. INTRODUCTION Th trackng prformanc of th GS rcvrs s hghly dgradd whn th rcvd sgnal s affctd by multpath propagaton. A larg numbr of studs on th ffct of multpath on th cod and phas trackng rrors wr carrd out, as summarzd n [Braasch, 1996]. Svral studs ld to th mplmntaton of spcfc structurs dsgnd to rduc th multpath nducd rrors, lk th Narrow Corrlator rcvr [Fnton t al, 1991], [Van Drndonck t al, 199], th ultpath Estmatng Dlay Lock Loop (EDLL [Van N, 1994], [Townsnd t al., 1995], th ultpath Elmnaton Tchnology (ET [Townsnd t al., 1994], Strob and Edg corrlators [Garn t al., 1996], Early1-Early Trackr [attos, 1996], ultpath tgator Typs A and B [Hatch t al., 1997], Compnsatd Corrlators [Dors, 1997], Enhancd Strob corrlator [Garn t al., 1997] and Gatd and Hgh Rsoluton Corrlators (HRC [cgraw t al., 1999]. Th prformanc of th proposd tchnqus s publcly advrtsd by plottng th cod and phas trackng rror nvlops as a functon of th rlatv dlay of th rflctd sgnal wth rspct to th drct sgnal. Thrfor, th nd usr may only know th maxmum magntud of th synchronzaton rrors whn only two rays ntr th trackng loops, namly th drct ray plus on dffractd ray. Unfortunatly, ths condton dos not corrspond to th most frqunt stuatons, whr th rcvd sgnal can b th dscrt sum of svral powrful rplcas of th drct sgnal. For xampl, th antnna can b ht by th RHC ln-of-sght sgnal comng from th satllt, th LHC sgnal rflctd off th ground, and svral RHC sgnals scattrd by a st of buldngs and th ground. Ths papr prsnts th man thortcal rsults of a study that ams at dtrmnng th prformanc of th GS rcvrs whn th sgnal s affctd by svral multpath. It s oftn thought that th ffct of svral multpath may b xtrapolatd from th two-ray nvlop rror curv by a rsntd at GNSS 99, Gnoa 1

2 lnar suprposton. Howvr, th dscrmnators usd n practc ar non lnar n natur, hnc t s not clar f ths lnar suprposton s rally vald. Hr, w frst prsnt th thortcal analyss that was prformd, and som rsultng trackng rror nvlops obtand through analytcal dtrmnaton and by smulaton. Fnally, w dscuss th valdty of th lnar suprposton abov mntond.. II. EASUREENT ODELS Th C/A sgnal ntrng th trackng loops s th RF sgnal snsd by th antnna, fd to th RF unt whr t s fltrd, amplfd, down-convrtd to ntrmdat frquncy, sampld and quantzd. Ths ncomng compost C/A sgnal s modld as follows: N V k = A D kt τ C kt τ cos πf kt θ n ( k (1 ( ( ( ( f + = whr k s th dscrt tm ndx and T s th ntrnal samplng prod of th rcvr. f s th ntrmdat frquncy of th rcvr. N+1 s th total numbr of rplcas ntrng th loops. Th ln-of-sght sgnal s dnotd wth subscrpt. A s th ampltud of ach rplca, and D s th /NRZ/L navgaton mssag. C f s th C/A cod as fltrd by th RF unt, and n f s th fltrd addtv nos. τ and θ ar rspctvly th total group and phas propagaton dlays of ach ray. Th opratons prformd by th trackng loops calld th has Lock Loop (LL and th Dlay Lock Loop (DLL ar llustratd n fgur 1 for a classcal rcvr. Th am of ths trackng loops s to gnrat local rplcas of th cod and carrr that ar synchronzd wth th ln-of-sght sgnal. To achv ths obctvs, th loops control local oscllators whos outputs ar contnuously compard wth th ncomng sgnal. Th sgnals usd to control th oscllators ar fltrd vrsons of th dscrmnaton sgnals. Ths dscrmnaton sgnals ar non lnar functons of th dgtal I and Q data dlvrd by th Intgrat and Dump fltrs that cumulat nput sampls. V cos DCO sn E-L Cod gn DCO I D Q D I Q Cod and has Dscrmnators Loop Fltr + f 154 Loop Fltr V DLL V LL E-L ASIC (Hardwar CU (Softwar To othr channls... Fgur 1: Archtctur of a dgtal GS rcvr (courtsy SEXTANT AVIONIQUE. For xampl, th dscrmnaton functon of th classcal dot-product DLL s proportonal to: I D I + QD Q E L E L VDLL = ( I + Q BUS rsntd at GNSS 99, Gnoa

3 whr ID = V( k mcos(π f k m T k C k m T k C k m T k E L ( θˆ( [ (( ˆ( τ (( τ(ˆ QD = V( k msn(π f k m T k C k m T k C k m T k E L ( θˆ( [ (( ˆ( τ (( τ(ˆ and I = V ( k mcos(πf ( k m T θˆ( k C( ( k m T ˆ( τ k Q = V ( k msn(πf ( k m T θˆ( k C( ( k m T ˆ( τ k In ths xprssons, τˆ and θˆ ar th cod and carrr phas dlay stmats that hav to b appld to th local oscllators n ordr to cancl th dscrmnaton functon. It can b shown that ths dscrmnaton functon has th followng xprsson whn th ncomng sgnal s th nos fr sgnal modld as n (1: N N ] ] ( τ D( kt τ R( ε [ R( ε + τ R( ε + τ + ]cos( θ V ( k, = A A D kt θ DLL = = D whr ε D = τ ˆ τ s th dvaton btwn th group dlay of ray and th cod dlay stmat. s th arly-lat chp spacng. Anothr ntrstng xampl s a DLL controlld by a lnar combnaton of svral corrlaton ponts, as n th Strob corrlator, th ET, or th HRC. For xampl, f th rfrnc wavform of th DLL s th wghtd sum of two sts of Early mnus Lat corrlators spacd 1 and apart, th dscrmnaton functon s: VDLL = VDLL( k, 1 + avdll ( k, (6 Th cod and phas trackng rrors, ε and ε, ar th dvatons btwn th tru group and D phas propagaton dlays of th drct sgnal, τ and θ, and thr corrspondng stmats τˆ and θˆ. III. CODE TRACKING ERRORS If svral rplcas of th drct sgnal ntr th trackng loops, th qualty of th cod and phas trackng opratons can b svrly dgradd, and n som cass of dstructv collaboraton btwn all th multpath, th loops may vn los lock for som nstants. For a classcal Early mnus Lat DLL ncludng narrow corrlator DLLs, f th rcvr mantans contnuous trackng, f th drct sgnal s nvr blockd, and f th sum of th ampltuds of ach rflctd sgnal nsd th DLL s lowr than th ampltud of th drct sgnal, thn th cod trackng rror s boundd by half th Early mnus Lat chp spacng. Ths rsult s xpland by th fact that th Early mnus Lat loop s not snstv to any dformaton of th corrlaton pak outsd of ts xploraton ntrval. If th sum of th ampltuds of ach rflctd sgnal s largr than th ampltud of th drct sgnal, thn th cod trackng rror may b much largr than half th Early mnus Lat chp spacng, as th multpath sgnals may cooprat to buld a corrlaton pak far away from th drct sgnal corrlaton pak. Ths cas can not b analyzd n dtal n ths papr and t wll b prsntd latr. Th xtrm valus of th cod trackng rrors ar obtand n two dstnct cass: whn th drct sgnal and all th rflctd sgnals ar n phas whn ach of th rflctd sgnals s out of phas wth rspct to th drct sgnal. For short dlays and powrful multpath, th absolut maxmum magntud of th phas trackng rror s th product of λ/4 by th numbr of rays ntrng th LL. Howvr, th phas D D (3 (4 (5 rsntd at GNSS 99, Gnoa 3

4 trackng rror nvlops wll b prsntd latr, and w only focus hr on th cod trackng rror nvlops. Fgur 3 shows th nvlop of th cod trackng rror for a.1 T c narrow corrlator, n th cas whr th sgnal s th sum of th ln-of-sght sgnal plus two rflctd sgnals. Ths fgur was plottd usng thortcal xprssons obtand usng (5, assumng th C/A cod autocorrlaton functon s an dal trangl functon, and assumng th pr-corrlaton bandwdth s nfnt. As w can s n ths fgur, th shap of th cod trackng rror nvlop can b dvdd n four man rgons. Th frst rgon corrsponds to th cas whr both rays combn to nduc non-zro trackng rrors, and th rsultng nvlop s th box vsbl n th forground. Its hght s th sum of th maxmum trackng rrors nducd by ach ray. Th scond rgon s th doman whr both rays hav a rlatv dlay largr than Tc+/, thrfor not nfluncng th DLL any mor. Th thrd and fourth rgons ar th rgons whr on of th ray has gon out of th ara of nflunc of th DLL, and only rmans th trackng rror du to th closst ray. Fgur 4 shows th nvlop of th cod trackng rror for a.1 T c.5 Tc lnar combnaton DLL obtand usng (6, n th sam cas as prvously. As ths typ of loop s solly nfluncd by multpath havng short and chp rang dlays, only thr small non-zro rgons xst. Th frst non-zro rgon corrsponds to th cas whr at last on of th rays has a short dlay. In that cas, th maxmum trackng rror s approxmatly th sum of th maxmum trackng rror nducd by ach ray. Th scond and thrd non-zro rgons ar th rgons whr at last on of th rays has a dlay clos to on chp, and th maxmum trackng rror of ach ndvdual ray can b summd, xcpt f on of th rays has a short dlay. Fgur 3: Cod trackng rror nvlop for rflctd rays wth a total ampltud < LOS ampltud (.1 T c narrow corrlator DLL. Fgur 4: Cod trackng rror nvlop for rflctd rays wth a total ampltud < LOS ampltud (.1 T c -.5 T c DLL. As ntroducd prvoulsy, n most of th stuatons, ths nvlops concd wth th sum of th sngl nvlops obtand for ach ndvdual ray, xcpt n thr transton zons. For a narrow corrlator rcvr, ths happns whn th rlatv dlays ar clos to or 1 chp, as w can s n th dagonal cross scton prsntd n fgur 5. For a standard 1 chp spacng corrlator, th dvaton wth rspct to th lnar addton assumpton s sgnfcant, as llustratd n fgur 6. For a DLL controlld by a rfrnc sgnal that s a lnar combnaton of two Early-Lat sgnals, th dvaton btwn th tru thortcal 3D functon and th lnar approxmaton s lss svr, as llustratd n fgurs 7 and 8. rsntd at GNSS 99, Gnoa 4

5 15 CODE TRACKING ERROR ENVELOE (A=1,A1=.5,A=.4, =.1 Tc 15 CODE TRACKING ERROR ENVELOE (A=1,A1=.5,A=.4, =1 Tc 1 LINEAR AROX 1 LINEAR AROX τ 1 =τ DELAY IN CHIS Fgur 5: Comparson of th tru nvlop and th nvlop obtand usng th lnar approxmaton assumpton for a.1 T c narrow corrlator (dagonal cross-scton τ 1 =τ DELAY IN CHIS Fgur 6: Comparson of th tru nvlop and th nvlop obtand usng th lnar approxmaton assumpton for a.1 T c standard corrlator (dagonal cross-scton. 8 CODE TRACKING ERROR ENVELOE (A=1,A1=.5,A=.4, 1=.1 Tc 6 CODE TRACKING ERROR ENVELOE (A=1,A1=.5,A=.4, 1=.1 Tc 6 LINEAR AROX 4 LINEAR AROX τ 1 DELAY IN CHIS (τ = Tc Fgur 7: Comparson of th tru nvlop and th nvlop obtand usng th lnar approxmaton assumpton for a.1 T c -.5T c DLL (dagonal cross-scton. IV. CONCLUSION τ 1 DELAY IN CHIS (τ =.5 Tc Fgur 8: Comparson of th tru nvlop and th nvlop obtand usng th lnar approxmaton assumpton for a.1 T c -.5T c DLL (prpndcular cross-scton. Ths papr shows th thortcal rsults of an analyss that ams at dtrmnng th prformanc of GS rcvrs trackng a sgnal dgradd by mor than on multpath. Th rsults prsntd hr focusd on th cod trackng rror nvlops whn th sum of th ampltuds of th rflctd sgnal s lowr than th ampltud of th ln-of-sght sgnal. W show that whn th ncomng sgnal s composd of th drct sgnal and svral rplcas of ths sgnal, th cod trackng rror nvlop can b approxmatd as th sum of th cod trackng rror nvlops xpctd for ach ndvdual ray, xcpt n th transton zons of ths nvlops. rsntd at GNSS 99, Gnoa 5

6 In ths transton zons, th dvaton s small as a frst approxmaton n th cas of a narrow corrlator rcvr for th maxmum xpctd cod trackng rrors, but th approxmaton rror s hghly dgradd whn th corrlator spacng s wdnd. In th cas of a DLL controlld by a lnar combnaton of corrlaton valus, th approxmaton rror n th transton zons s also small, f th chp spacngs usd ar narrow. Th nxt rsults to b prsntd ar th phas trackng rror nvlops for ths rcvr structurs, and th cod and phas trackng rrors for othr structurs. In addton, w wll prsnt th charactrstcs of th cod trackng rror whn th sum of th ampltuds of th rflctd sgnals s largr than th ampltud of th drct sgnal, and th masurd cod and phas trackng rror nvlops usng a sgnal gnrator. ACKNOWLEDGENTS Th authors wsh to thank th STNA for supportng ths rsarch, and SEXTANT AVIONIQUE for havng provdd tchncal assstanc. In addton, Rdouan Yazd dsrvs our thanks for hlpng us prformng ths study. REFERENCES Braasch. (1996 «Global ostonng Systm: Thory and Applcatons», volum 1, chaptr ultpath Effcts, pags , AIAA. Dors D. (1997 odélsaton d récpturs GS. Applcaton à l étud d l nflunc ds multtrats sur ls prformancs du récptur L1 GS, Thès INT, Sptmbr. Fnton., Falknbrg B., Ford T., Ng K. and Van Drndonck A.J. (1991 «NovAtl s GS Rcvr th Hgh rformanc OE Snsor of th Futur», procdngs of ION GS-91, Albuqurqu, Spt Garn L., Van Dggln F. and Roussau J.. (1996 Strob and Edg Corrlator ultpath tgaton for Cod, procdngs of ION GS-96, Kansas Cty, Sptmbr 17-. Garn L. and Roussau J.. (1996 Enhancd Strob Corrlator ultpath Rcton for Cod and Carrr, procdngs of ION GS-97, Kansas Cty, Sptmbr Hatch R., Kgan R. and Stansll T. (1997 Lca s Cod and has ultpath tgaton Tchnqus, procdngs of ION Natonal Tchncal tng, Santa onca, January attos. (1996 ultpath Elmnaton for th Low-Cost Consumr GS, procdngs of ION GS-96, Kansas Cty, Sptmbr 17-. cgraw G. and Braasch. (1999 GNSS ultpath tgaton Usng Gatd and Hgh Rsoluton Corrlator Concpts, procdngs of ION Natonal Tchncal tng, San Dgo, January 5-7. Townsnd B. and Fnton. (1994 A ractcal Approach to th Rducton of sudorang ultpath Errors n a L1 GS Rcvr, procdngs of ION GS-94, Salt Lak Cty, Sptmbr -3. Townsnd B., Van N R., Fnton. and Van Drndonck K. (1995 rformanc Evaluaton of th ultpath Estmatng Dlay Lock Loop, procdngs of ION Natonal Tchncal tng, Anahm, January 18-. Van Drndonck A.J., Fnton. and Ford T. (199 «Thory and rformanc of Narrow Corrlator Spacng n a GS Rcvr», procdngs of ION Natonal Tchncal tng, San Dgo, January 7-9. Van N R., Srvld J., Fnton. and Townsnd B. (1994 «Th ultpath Estmatng Dlay Lock Loop : Approachng Thortcal Lmts», procdngs of IEEE LANS 94, Las Vgas, Aprl rsntd at GNSS 99, Gnoa 6