GNSS Multipath Mitigation using High- Frequency Antenna Motion

Transcription

1 GNSS uliph iigio usig High- Fequey Ae oio u E. Psii By W. O'Hlo Rih A. eluzzi Seve P. Powell oell Uivesiy h NY BOGRAPHES u E is pusuig Ph.D. i he Sibley Shool of ehil Aeospe Egieeig oell Uivesiy. He eeive his B.S. i ehil Egieeig fo oell Uivesiy. His ue eseh ieess e i he es of GNSS ehologies olie esiio fileig.. Psii is Pofesso of ehil Aeospe Egieeig. He eeive B.A. i Physis.A. Ph.D. egees i ehil Aeospe Egieeig fo Pieo Uivesiy. His eseh ieess e i he es of GNSS ehology ppliios spef iue obi eeiio geel esiio fileig eeio. By W. O Hlo is gue sue i he Shool of Eleil opue Egieeig. He eeive B.S. i Eleil opue Egieeig fo oell Uivesiy. His ieess e i he es of GNSS ehology ppliios GNSS seuiy spe wehe. Rih A. eluzzi is lys he Johs Hopis Applie Physis booy woig i he fiel of bllisi issile efese. He eeive B.S. egee i Egieeig Physis se of Egieeig egee i Aeospe Egieeig fo oell Uivesiy. His ieess ilue ool heoy physis-bse syse oelig siulio wih fous ow guie vigio ool of spe vehiles. Seve P. Powell is Seio Egiee wih he GPS oosphei Suies Reseh Goup i he Depe of Eleil opue Egieeig oell Uivesiy. He hs.s. B.S. egees i Eleil Egieeig fo oell Uivesiy. He hs bee ivolve wih he esig fbiio esig luh iviies of y sieifi expeies h hve flow o high liue blloos souig oes sll sellies. He hs esige gou-bse spe-bse uso GPS eeivig syses piily fo sieifi ppliios. ABSRA A eho is evelope fo heizig opesig GNSS uliph by osieig sigl pliue phse viios i espose o e oio. his eho sees o ipove uliph ejeio pbiliies beyo hose povie by hoeig o ohe e ehologies o by uli-oelo sigl poessig. A ow e oio pofile is obie wih oifie uli-oelo isiio i oe o bee heize he uliph opoes bee isole he ie-ph opoe fo he uliph opoes. uses bh file o esie he oe phse ie phse pliue of he ie sigl he elive oe phse ie phse ie Dopple shif pliue of eh sigifi uliph opoe he ieio of ivl of he ie uliph opoes. he bh esio solves weighe olie les-sques poble h ivolves heil oels fo he i-phse quue uulios ove sp of sple ies ge of oe-phse offses. he bh esio ilues iplii high-pss fileig so h is esuls e o ffee by low-fequey phse viios h igh be use by sellie oio eeive lo if. Expeiel es esuls e pesee bse o fo e iulio syse opee ouoos i h NY. esuble ipovees e obseve i esulig esies of ie pliue i ll ses of ie-ph ie-phse i y ses of ie-ph oe phse i soe ses. Alous esuls hve lso bee obie iluig uliph opoe esies h soeies ou i oubles h hve high ie pliue ieio oe phse esies h e ely ieil ie phses h use he ely o el. hese e believe o be he esul of iffuse uliph whih is o oele i he pese suy. NRODUON Reflee uliph sigls iso he i-phse quue uulios wihi ypil GPS spewehe oio. hese isoios use eos i he esue ie phse oe phse. uh wo hs bee oe o y o iige he effes of uliph e.g. opyigh by u E. Psii By W. O Hlo Rih A. eluzzi Seve P. Powell. All ighs eseve. Pepi fo ON GNSS

2 see Refs he pese effo sees o exp o he iel of he uliph-esiig Dely- o oop ED s esibe i Refs. 6. A exple of uliph isoio of eeive's uulios is epie i Fig.. his -D figue plos i-phse uulios quue uulios o wo ohogol xes vs. oe ely offse o he hi xis. he pojee view i he figue shows wh is effeively obiio of s use by oio bou he oe offse xis vs. oe offse. As expee i loos fily siil o wh PRN oe oelio fuio woul loo lie fe ouig fo is isoio by he eeives RF fo-e file. ol Die uliph uliph uliph ie-sigl uve. he uliph iigio ehiques lie hose i Refs. 6 eogize he possibiliy of suh phse offses ou fo he i uve fiig lgoihs. he gol of suh lgoihs is o esie he os liely obiio of ue ie-sigl e uve gee blue y uliph uves h bes up o he isoe bl uve. hese ehiques lso eue he uliph isoio of ie phse whih is illuse i Fig. by he phse oio bewee he isoe bl uve he e ue ie-sigl uve bu oly if hee is suffiie oe ely bewee he ie sigl he uliph opoes. ol Die uliph uliph uliph Auulios Ely-/e PRN oe Offse hips.5 Auulios -.5 Fig.. Reeive i-phse quue uulios vs. oe-offse ely elive o pop D view. f hee wee o uliph sigls he he e iesigl uve epeses wh he eeive woul see use o eue oe phse obsevble bse o he ely/le oe offse iig of he pe. Ufouely he gee blue y uliph sigls iso he ol eeive sigl o yiel he bl sigl. he eeive is liely o esue o he bl uve bsolue oe phse h is bise.54 hips 5. se/.6 /9.7 EU le h he ue oe phse. his is elively beig siuio fo he eeive whose isoe PRN oe oelios e show i Fig.. he eo be uh lge i less beig siuios. S uliph iigio segies ep o fi he isoe bl uve o su of uves suh s he e gee blue y uves i Fig.. his fi ous fo he oplex i-phse quue ie hisoy epie i Fig.. Eh eeive opoe hs oelio pe loio log he hoizol oe-offse xis bu i lso hs phse offse i oio bou his xis. hese phse offses e o esy o visulize i Fig. ue o he hoie of viewig gle of his D plo. Fig. peses iffee viewig gle of he se seio. O his lee figue i is obvious h he bl ole uve he gee y uliph uves hve sigifily iffee phses h he e Auulios Fig..5 Auulios -.5 Ely-/e PRN oe Offse hips Re-oiee D view of eeive i-phse quue uulios vs. oe-offse. he i ew ie of he pese wo is o exploi he f h he elive phses of he ie uliph sigls i Fig. e sogly epee o he e loio. f oe oves he e jus i piul ieio whih is bou hlf wvelegh of he GPS ie sigl he he elive phsigs of he sigls i Fig. oph io he siuio show i Fig.. he i phse hges e use by he pojeio of he e ovee oo he OS ieios of he iffee sigl opoes. he elive phses hge bewee Fig. Fig. ue o he iffeig ieios of ivl of he iffee opoes. osie fo exple he phse elioship of he e ie uve he y uliph opoe. he y uve is oe ouelowise i phse elive o he e uve by bou eg. i Fig.. Fig. howeve i is oe bou 45 eg. lowise fo he e uve. Noie lso how hee is sigifi ie-phse isepy bewee he e ue sigl he bl ol eeive sigl i Fig.. Fig. his phse offse hs lgely vishe. As expeiel ofiio of his lysis osie he ul eeive sigl pliue ie hisoies -5 5

3 epie i Fig. 4. hese wee eoe fo uliple hels of GPS eeive whih ws oee o oof-oue e h uewe eyig - iesiol siusoil osillios he e of flexible ilevee be. he iiil osillio pliue ws bou pe-o-pe i ws iiie ie -9 se s esue log he figue's hoizol xis. he e osillio fequey ws bou Hz. hese osillios poue oespoig sigl pliue osillios his se fequey i vious of he hels os oble he yellow PRN 5 bl PRN e PRN 4 soli blue PRN hels. he ploe quiies i Fig. 4 e esseilly he pe pliues of he bl oelio uves i Figs. - s hey vy wih e loio. Fo soe of he sigls his pe pliue hges illy beuse he osillios use elive phse iffeees bewee he ie uliph opoes h use leely osuive esuive iefeee bewee hei oelio pes. Sigls wihou uh pliue osillio oespo o oe of wo oiios: hey y hve vey lile uliph. Aleively hei uesigl ieios of ivl hei uliph-sigl ieios y hve ely ieil o pous wih he ieio of he D e osillio i his piul expeie. Auulios Auulios -.5 Ely-/e PRN oe Offse hips -5 5 ol Die uliph uliph uliph Fig. Reeive i-phse quue uulios vs. oe-offse ely e loio isple by bou hlf wvelegh fo he loio oespoig o Figs As ofiio h he osillios i Fig. 4 wee use by uliph he se es ws u i ehoi hbe i whih GPS sigls fo oof-oue e wee e-ie. No suh pliue osillios wee obseve o he ehoi hbe. his ppe es fou oibuios o he poble of uliph iigio. Fis i peses heil oel of he uliph effes of e oio. his epeses geelizio of he oel use o esig ED lgoihs. Seo i evelops olie bh esio bse o his heil oel. his lgoih esies vious ie-sigl uliphsigl pees iluig elive elys elive ie phses he ieios of ivl of eh sigl. he bh esio osiues i effe so of pooype isiio fo obie uliphesiig D P. hi his ppe ofis he obsevbiliy of he pees of is oel by osieig he lol uiqueess of is opil soluios. Fouh i pplies he ew bh esio/isiio o expeiel i uses he esuls o evlue is effeiveess. Auulio Apliue 5.5 x PRN PRN 4 PRN 8 PRN PRN 5 PRN PRN PRN Reeive ie se Fig. 4 Osilloy pliue ie hisoies of oelio pes s eeive by spigoue e uegoig eyig - iesiol osillios. his ppe's oibuios e oie i 4 i seios plus olusios. Seio efies he uliph uulio esuee oel h ilues he effes of e oio. Seio evelops bes esio h opees o y sples of he esuee oel of Seio i oe o esie pees of he ie uliph sigls. effe his esio osiues oe-phse ie-phse isiio fo ouple uliph esiig D/P. Seio lso lyzes he syse's obsevbiliy. Seio V explis he expeiel seup olleio pig i isusses he pepoessig h h o be ie ou befoe he esio of Seio oul be pplie o he. Seio V peses lyzes he expeiel esuls. Seio V suizes he ppe's oibuios gives is olusios.. EASUREEN ODE he pese uliph iigio ppoh elies o oels of he ie uliph sigl opoes. hese oels e iiilly efie he w sigl level. hey e he popge hough he eipes fo GNSS

4 eeive's s i-phse quue uulios. he esul is se of fouls fo how he uulios epe o he pees h heize he ie-ph uliph sigls. hese fouls osiue he esuee oels use i he bh olie esio of Seio. he oel fo he eeive sigl fo give GPS sellie es he fo: yi Aos[ Fi φnb i ] { os[ i ] [ Z i f i ] si[ i ] [ Z i f i ] { i} [os φ { i } { } Z i f i δ { } si φ { } i i { } Z i f i δ ]} Asi[ Fi φnb i ] { si[ i ] [ Z i f i ] os[ i ] [ Z i f i ] { i} [ si φ { i } { } Z i f i δ { } os φ { } i i Z i f i δ { } ]} νi whee y i is he w RF fo-e oupu sple eeive sple ie i. he quiy A woul be he ie pliue he oupu of he RF fo-e if hee wee ifiie RF file bwih if hee wee o uliph. he oss F e espeively he il ieeie fequey o whih he il GPS ie fequey o ges ixe by he RF fo-e he il wvelegh of he il GPS ie fequey he spee of ligh i vuu. he ie hisoy φ NB is wh he eeive egive be ie phse of he ie sigl woul hve bee h hee bee o e oio; i is he ie iegl of he eeive ie Dopple shif. is ee "egive" beuse i hs he opposie sig o he usul be ie phse efiiio i he GPS lieue. he fuios Z f Z f e espeively he el i.e. i-phse igiy i.e. quue opoes of he sigl's PRN oe fe i psses hough he RF fo-e's bpss file. A evelope of his file's oplex ipulse espose fuio s i Ref. 7 be use o opue hese fuios fo he oigil wieb PRN oe. he fou quiies φ δ heize he h uliph opoe of he eeive sigl wih beig is elive equivle wieb ie pliue esue elive o he ie sigl A wih φ beig is phse peubio elive o he ie sigl ue he ssupio of o e oio efeee ie wih beig is elive ie Dopple shif wih δ beig is ely elive o he ie sigl ue he ssupio of o e oio. ypilly < δ e vli. he -by- ui ieio veo is he ieio of he ie-sigl lie-of-sigh y ph fo he sellie o he il e loio he -by- ui ieio veo is he ppe ieio veo of he ieio fo whih he h uliph opoe is ivig. Boh of hese ieio veos e efie i y oveie lol boy-fixe ooie syse. he -by- veo ie hisoy is he ow e oio ie hisoy i ees uis give i he se lol boy-fixe ooie syse s is use o efie eh. he e oio ie hisoy is elibeely esige o i i uliph esiio. hee ol of uliph opoes i his oel. he fil e i Eq. ν i is he eeive hel oise i he i h sple. he w RF fo-e sples e use i eeive o opue i-phse quue uulios oig o he followig eipes: i N yiz[ i D P / ] i i os[ Fi φp P i D ] i N yiz[ i D P / ] i i si[ Fi φp P i D ] b 4

5 fo he h uulio ievl whih ss eeive ie D es D. he Dopple-shife wie-b PRN oe epli is Z Z [ P / ] whee Z is he ow il PRN oe wih o oe Dopple shif P is he P's ie Dopple shif esie fo his ievl is he il bos ie fequey i.e. x575.4x 6 /se o x7.6x 6 /se. he wo ies D D us be hose by he D Z us be syhoize by he D so h Z - D is he pop PRN oe epli uig his ievl. he sple iex i is he fis sple i suh h D i. he quiy N is he ol ube of sples i he ievl so h he eil iex i N - is he lges vlue of i fo whih i < D. he phse φ P is he esie egive be ie phse he s ie D. he ipu gue is he shif of he PRN oe epli elive o he pop oe i seos. A posiive vlue of oespos o PRN oe epli i Eqs. b h lgs behi he pop epli. he uulio eipes i Eqs. b be obie wih he sigl oel i Eq. i oe o poue he followig oels of he opue uulios: γ X { os[ ] ZfZ [ ] γ si[ ] [ ZfZ ]} γ Y { si[ ] ZfZ [ ] γ os[ ] [ ZfZ ]} X {os[ γ ] [ ZfZ δ ] si[ γ ] [ ZfZ δ ]} Y { si[ γ ] [ ZfZ δ ] os[ γ ] [ ZfZ δ ]} γ X { si[ ] ZfZ [ ] γ os[ ] [ ZfZ ]} γ Y { os[ ] ZfZ [ ] γ si[ 5

6 ] [ ZfZ ]} X {si[ γ ] [ ZfZ δ ] os[ γ ] [ ZfZ δ ]} Y { os[ γ ] [ ZfZ δ ] si[ γ ] [ ZfZ δ ]} b he followig efiiios hol fo he y ew quiies use i Eqs. b: AN X os φ 4 AN Y si φ 4b e he i-phse quue phso opoes of he ie sigl wih N beig he e ube of sples i uulio i.e. he e vlue of N wih φ beig he iffeee bewee he egive be ie phse epli use i he uulios lulios he ue egive be ie phse of he ie sigl he il ipoi ie of ll he uulio ievls ue osieio : φ φp P D φnb 5 o o wih i beig he iex of he uulio ievl h epsses : D < D 6 i i he oels i Eqs. b ilue he uliph phsos AN X os φ φ fo... 7 AN Y si φ φ fo... 7b he quiies γ e espeively he ie Dopple shif esiul eo bewee he P NO he ie sigl ie he ie Dopple shif e esiul eo fo he eie ievl i quesio. hus oel of he eie esiul ie-phse iffeee bewee he P NO egive be ie phse he oe use fo bse-b ixig i he uulio eipes of Eqs. b he ue eeive ie phse of he ie sigl bse e oio effes is γ φ φp P D φnb 8 his oe oel is ssue o be esoble fo ll uulio ievls of iees... K. he ilusio of he uows X Y γ i he uliph esiio poble whe ouple wih he phse oel i Eq. 8 hs he effe of bh high-pss fileig he phse effes o he uulios pio o hig phse viios o e oios. his is esoble ppoh h els effeively wih he GNSS sellie oios he eeive lo if possible oios of he plfo o whih he e is oue. Noe h he ios of high bwih P igh e he oel i Eq. 8 ivli ue o he iouio of high-fequey viios i φ P P fo iffee vlues of i he ge o K. Suh viios oul be oise iue o hey oul be iue by he P yig o he effes of he e oio. heefoe i pplyig his oel bes-fi polyil ppoxiio of he egive be ie phse is use o ee effeive P egive be ie phse ie hisoy ove he ievl... K. he vege phse eo bewee his polyil he eeive P's ul phse NO ie hisoy fo eh uulio ievl is he use o pe-poess he oespoig [ ; ] uulio veo by oig i i he [;] ple i oe o yiel he uulios h woul hve bee opue h he NO use he egive be ie phse i he polyil oel. he polyil oel is eive by pefoig les-sques fi o he P egive be ie phse ie hisoy fo ievl h ilues sp of ie jus befoe he oio ohe sp of ie jus fe he oio. he esulig [;] uulios ei he oio effes he high-fequey oise effes. 6

7 he ie is he i-poi of he h uulio ievl: [ D D ] 9 he es i Eqs. b pesue h he e posiio sys fixe his vlue uig he eie h uulio ievl. heefoe he highes fequey oe i he ie hisoy us be sigifily lowe h hlf he uulio fequey. he quiy is he vege oe-phse eo bewee he pop PRN oe epli use i he uulios i Eqs. b if he ue eeive ie sigl: K [ D D ] K whee e he ul ies whe he give sege of he ue filee vesio of he PRN oe ss sops he oupu of he RF fo-e. his give sege is he se sege s he oe whose wie-b vesio exes fo Z o Z D - D osise wih he efiiio of Z i he ex h follows Eqs. b. he uulio oels i Eqs. b pesue h his oe offse woul sy os ove he K uulio ievls wee i o fo he effes of he e oio. pie his be iffiul o hieve i sevee uliph evioe. ge uliph-iue hges i he [;] uulios use lge swigs i he D's opue D D vlues. heefoe i oe o use he uulios oel i is ue fo i is eessy o use ie D ove he ievl of e oio. ie iig supply he eee isesiiviy o uliph if ipleee popely fo he ievl of oio. oe ipleeio ie-ie D hs is oe phse feeb e ue off uig he oio ievl. he bes suh ppoh woul obie zeo-oe-phse-feeb uig he oio ievl wih ie iig fo he polyil fi o he P egive be ie phse wih fi pees eeie usig oly fo ie sps jus befoe fe he oio ievl s esibe bove. his ppoh yiels D oe phse h oes o espo o he high-fequey e oio osise wih he uulios oel i Eqs. b. Noe h he speil P D equiees isusse bove pply oly beuse of he bh esiio fo of he uulios oel h hs bee eive. woul be sighfow o eive suible uliph oel of he uulios h i o ple suh uusul equiees o he P he D. h se iffee esio woul be eee h is evelope i Seio of his ppe. is liely h sequeil esio suh s Kl file woul be he ppopie hoie i suh siuio. he uliph oel fo esio use hee hve bee hose o siplify he iiil evelope esig of his uliph esiio oep usig off-lie lysis. Fuhe wo shoul be ie ou i oe o esig pil el-ie vesio of he pese uliph esiio shees. he fuios ZfZ ZfZ e espeively he el igiy opoes of he oss-oelio bewee he oplex filee PRN oe Z f jz f i he eeive sigl he wie-b PRN oe Z use i he uulio eipes. hese fuios be opue usig oifie vesio of he foul i Ref. 7: j ZfZ ZfZ j x ZfZ φ f e h whee he fuio is he ow el-vlue uooelio fuio of he wie-b PRN oe Z h is he oplex-vlue evelop ipulse espose fuio of he RF fo-e file x is he oele fiie uio of h espose φ f is he sll phse oio use he file's fequey syey Df is he vege ely hough he file. he file pees φ f Df be esie fo h by osieig is effes o sigle PRN oe hip. he file ipulse espose fuio be esie usig he ehiques of Ref. 7. he eeive hel oise es i Eqs. b e. hey boh e zeo-e Gussi o oise sples wih vie equl o σ. he quiies b e uoele fo y uulio offses b. he followig o-zeo oise oelio howeve pply o eh of hese sequees: E{ b} E{ b} σ b hus he uooelio of he wie-b PRN oe use i he uulio eipes of Eqs. b iues oise oelio bewee i-phse uulios wih iffee oe ely offses bewee quue uulios iffee offses. he uulio oels i Eqs. b be siplifie io he fos: Df 7

8 {[ X ZfZ γ ]} ][ {[ Y ZfZ γ ]} ][ ZfZ X {[ ] γ ]} [ δ {[ Y ZfZ ] γ ]} [ δ {[ X ZfZ γ ]} ][ {[ Y ZfZ γ ]} ][ ZfZ X {[ ] γ ]} [ δ {[ Y ZfZ ] γ ]} [ δ b whee he oifie -gue oelio/oio fuios use o evelop Eqs. b fo Eqs. b e efie o be si os φ φ φ ZfZ ZfZ ZfZ 4 os si φ φ φ ZfZ ZfZ ZfZ 4b A ypil eeive opues uulios se of oe-phse offses.... Fo exple uulios igh be opue fo offses -.5/f.5/f whee f f [ P / ] is he Dopple-shife PRN oe hippig e. hese woul oespo espeively o hlf-hip ely uulios pop uulios hlf-hip le uulios. op esuee oels of ll he uulios l l fo l... be evelope usig ix/veo oio. his evelope ss by sig he uulios io -iesiol uulio esuee veos s follows: z z 5 Aiiol useful veo efiiios e he -iesiol filee PRN oe i-phse quue oelio/oio fuios φ φ φ φ φ ZfZ ZfZ ZfZ ZfZ φ φ φ φ φ ZfZ ZfZ ZfZ ZfZ 6 he -iesiol oise veos 7 A useful ix is he -by--iesiol esuee oise ovie ix: } { } { E E R K O K K K σ 8 8

9 Give he foegoig veo ix efiiios he i-phse quue uulio esuees be oele by he followig wo veo equios: {[ γ z X ]} ][ {[ γ Y ]} ][ {[ ] γ X δ ]} [ {[ ] γ ]} [ Y δ x h ] ; [ 9 {[ γ z X ]} ][ {[ γ Y ]} ][ {[ ] γ X δ ]} [ {[ ] γ ]} [ Y δ x h ] ; [ 9b whee he veo of uows h is o be esie by he uliph esiig D isiio is efie o be he followig 87-iesiol veo Y X Y X Y X δ δ γ x whee Eq. 9 9b seve o efie he -by- olie esuee fuios ] ; [ x h ] ; [ x h.. UPAH GAON BAH ESAOR his seio evelops fo of uliph-isesiive D/P isiio. oes his by posig solvig opil bh esiio poble bse o he uulio esuee oel i Eqs. 9 9b. A. Exples of Siple D P Disiios Rell h D isiio povies es of foig esie of. Oe ypil D isiio is he o-ohee ely-ius-le o pou isiio. Give ely pop le uulios espeively - - his isiio yiels he esie ] {[ ]} [ }/{ ] [ f whee he ^ ovesie o iies h i is esie quiy. his esie is esoble if hee e o uliph effes if he filee PRN oe oelio fuios e ppoxiely ZfZ ZfZ i.e. if hey e ppoxiely equl o 9

10 hei wie-b oueps if he oe phse offse is o oo lge. his lysis lso ssues h he wieb PRN oe uooelio fuio is ppoxiely equl o he igul pulse fuio: if f < - f if - f < f f < f ypil oe phse offses use e.5/f quehip o.5/f hlf-hip. ypil ie phse isiios fo use i P e he ge isiio: φ [ / ] he wo-gue ge isiio: φ [ ] 4 he le isiio equies opesio fo vigio bi sig hges. f hee e o uliph effes he he D isiio i Eq. he wo P isiios i Eqs. 4 will be esobly ue. he pesee of uliph howeve hey le o sysei eos. hese will effeively be bises ove he ie sle i whih he uliph pees φ δ e oss. hese bises be subsil espeilly if he RF fo-e bwih is ow o he oe of Hz 5. he gol of he pese lysis is o evelop lee isiios h eue o eve eliie hese bises. B. Opil Bh Esiio fo D/P Disiio wih uliph iigio he ew uliph isiio uses he uulios esuee oel i Eqs. 9 9b o pos he followig weighe olie les-sques bh esiio poble: fi: x 5 K o iiize: J x { z h [ x; ]} subje o: R { zh [ x; ]} { z h [ x; ]} R { zh [ x; ]} 5b δ fo fo... 5e he iequliy osis i Eq. 5 esue h he uliph sigls ive fe he ie sigl. he equliy osis i Eqs. 5 5e esue h he esie ui ieio veos obey hei lizio osis. he os fuio i Eq. 5b is he egive ul logih of he joi pobbiliy esiy fuio of he uulio esuee veos z z z z z z... z K z K oiioe o he vlues i he x veo. heefoe soluio of his poble ous o xiu lielihoo esiio. he iiizio of Jx wih espe o x is oplishe usig geel olie ueil opiizio poeue. is Newo-bse poeue h ss wih guess of he ue soluio veo γ xsh δ 6 δ his veo hs iesio 65. is foe by eleig he phso opoes X Y X Y... X Y fo x. Fo y give guess of he opil vlue of x sh he ex opil esies of hese phso opoes be opue lyilly by usig lie lgeb beuse he phso opoes ll ee he esuee oels i Eqs. 9 9b liely. he lie les-sques ix lulios h e eee o opue hese vlues e esibe i Ref. 8. he olie opiizio poeue ss wih guess of x sh. Fo eh guess i opues he opil phso opoes. Nex i opues fis seo eivives of he os fuio wih espe o x sh ue he ssupio h y viios of x sh will be opie by viios of he phso opoes i oe o ii hei opiliy. hese pil eivives be opue fo he pil eivives wih espe o he full x veo. he opiizio poeue uses he esulig qui ppoxiio o opue upes o is ue x sh guess iees uil i ehes lol iiu 89. he qui ppoxiio of he os fuio vli i he viiiy of he ue guess of x sh eploys he Hessi of he ggi fuio h is ee by joiig he osis i Eqs. 5 5e o he

11 os. he Hessi ix hs he beefiil effe of ppoxiig he os ip of bue-foe elizio of he ipove guesses of he veos he ew ipove soluio guess. he qui pog h is solve o opue ew soluio guess lso ilues fis-oe lieize ppoxiios of he osis i Eqs. 5 5e. heefoe he buefoe e-lizio ous o oly seo-oe oeio. he piul vesio of Newo's eho h hs bee use is us-egio eho 89. his eho expliily esis he giue of eh iee o x sh uig he lulio of h iee. he esiio sees o sy wihi he egio of vliiy of he lol qui os fuio ppoxiio. he esiio esues globl ovegee of he lgoih o les lol iiu of Jx.. Geeio of Nolie Opil Soluio iil Guesses fo Suessively ge Nubes of uliph opoes A esoble segy fo esiig he ie uliph opoes is o s wih he ssupio of o uliph esie he ie-sigl pees he suessively iiol uliph opoes o he poble esiig eh ew opoe's pees while efiig he esies of he pees of he peviously oele opoes. his is siil o he segy eioe i Ref. of esiig oe uliph opoe ie he subig is effes fo he uulios pio o esiig he ex opoe. h ppoh he poess of esiig he ew opoe's pees oes o ffe he esies ley obie fo ohe opoes whih is subopil. hee e lso beefis i es of geel olie opil esiio o he segy of iesig he ube of uliph opoes i iees of. Oe of he i hlleges i olie esiio is o s he olie esiio lgoih wih suffiiely goo fis guess o hieve ovegee o he globlly iiizig soluio of he os fuio. pie his be h o hieve e.g. see Ref.. he pese se oe use he followig iiil guess fo he ue soluio veo of Eq. 6 whe hee e o uliph opoes: guess γ guess x sh guess guess guess Aboy/ ef ef 7 whee ef is he ow ui ieio veo fo he sellie o he eeive i soe efeee ooie syse pehps WGS-84 ooies o lol-level ooies whee A boy/ef is esie of he -by- ieio osies ix fo sfoio fo he efeee ooie syse o he boy ooie syse i whih he e oio ie hisoy is efie. Ofe oe hs ough ie of he oe ix. hve eos s lge s egees pe xis sill povie eque fis guess fo olie lgoih ovegee. he ses osiee i Seio V he uy ppes o hve bee bou 6 eg of ol - xis oio eo i oe se 8 egees i he seo se. he fis guesses of he ohe hee pees i Eq. 7 γ e ll zeo. hese osiue goo guesses ue he esoble ssupio h he P he D ig eos e sll eough o ii lo. Sig fo he fis guess i Eq. 7 he opiizio poble Eqs. 5-5e be solve fo he uliph opoes se. Nex suppose h he opil soluio fo he se of uliph opoes is esige s x sh op. he oe geee ie soluios fo he se of uliph opoes s: xsh op guess x sh guess 8 δ guess guess he wo sl guess opoes guess δ guess he -by- ui ieio veo guess guess be geee oly. Uifo isibuios be use o sple guesses of guess δ guess. Resoble liis fo he uifo guess isibuio igh be -/ K - o / K - h is /- oe full yle of elive ie phse uig he bh esiio ievl whih ssues legh ievl o he oe of ypil uliph-iue pliue osillios fo si eeive. Resoble liis fo he uifo δ guess isibuio igh be./f o.5/f h is fo oe hueh of hip le h he ie sigl o oe que of hip le. he ui ieio veo guess guess be geee by splig fo zeoe ieiy-ovie -by- veo Gussi isibuio he lizig he esul. Suh splig poeue poues uifo isibuio of ieios o he ui sphee.

12 Oe oul siply e-s he olie solve of he opiizio poble i Eqs. 5-5e usig oize fis guess of he ew uliph opoes s i Eq. 8. A bee ppoh howeve is o geee uliple oize fis guesses s i Eq. 8 o pefo qui he of he gooess of eh guess by solvig gely siplifie opiizio poble. A suible poble is h of opiizig he weighe fieo i Eqs. 9 9b oly wih espe o he phsos of he ew uliph opoe X Y. his opiizio be ie ou i lose fo vey quily beuse hese phsos ee he esuee oel i Eqs. 9 9b liely. he esulig pilly opiize oss be logue fo y oize fis guesses of he fo i Eq. 8. he guess h gives he lowes pilly opiize os is he use s he fis guess fo full olie soluio of he opiizio poble i Eqs. 5-5e. he obie opiizio poeue wih geeio of suessive fis guesses fo pobles wih suessively lge ubes of uliph opoes is s follows:. Se geee he fis guess x sh guess usig. Eq. 7.. Solve he -uliph-opoe olie opil esiio poble i Eqs. 5-5e by usig he us-egio eho sig fo he guess x sh guess. Desige he esulig soluio s x sh op.. Deie whehe eough uliph opoes hve bee ilue. f so he eie wih he soluio x sh op. Ohewise poee o Sep Geee N oize fis guesses of he - uliph-opoe poble s efie i Eq. 8 i he ex h follows h equio. Desige he s [x sh guess ] fo... N. 5. Fo eh... N subsiue [x sh guess ] io Eqs. 9 9b fo... K log wih he opil phso esies h e ssoie wih x sh op X op Y op X op Y op... X op. iiize he weighe su-sque-eo os fuio fo he esulig equios s pe Eq. 5b i oe o eeie he ough esies [X oughop Y oughop ] he ssoie oss J ough fo... N. 6. hoose x sh guess [x sh guess ] op whee op is hose so h J ough op J ough fo... N. 7. ee go o Sep. D. os Fuio Seo Deivive Esiio Eo ovie Obsevbiliy A vge of usig xiu-lielihoo esiio o opue he esie of x is h he poeue lso yiels ppoxiio of he eo ovie i hese esies: J P xx E{ x x x } x x x 9 whee x is he esie of he ue x h solves he opil esiio poble i Eqs. 5-5e bu wih he veo of uows expe o ilue he phsos s i Eq.. he foul i Eq. 9 is i effe ppoxiio of he e-ro lowe bou fo he esiio eo ovie. he ix i Eq. 9 is ohol ix of iesio 86-by- 75. is osue so h is olus e ll of ui giue ll pepeiul o eh ohe ll pepeiul o he lieize vesios of he uilizio osis i Eqs. 5 5e s evlue he opil esie x. he ovie lulio i Eq. 9 uses i oe o popely ou fo he f h hee is o ueiy bou x log he sub-spe ieios h oespo o he leghs of he veos fo.... he obsevbiliy of he bh esiio veo i Eq. be evlue usig Eq. 9. Obsevbiliy is he oiio of hvig he veo x uiquely eeible fo se of uulio veos z z fo... s pe he oel i Eqs. 9 9b. his uiqueess is equivle o hvig well-efie uique iiu o he os fuio Jx i Eq. 5b. olly uiqueess is ssue if he pojee Hessi ix of he os fuio is posiive efiie wih pojeio beig he opeio of liiig he possible x viios o he lol ge spe of he ui-lizio osis i Eqs he equisie pojee Hessi is J H poj x f his ix is posiive efiie he he ovie ix i Eq. 9 will be posiive sei-efiie wih is zeo eigevlues log he ieios efie by fo.... heefoe he lol obsevbiliy oiio he oiio ous o he oiio h he ovie eeie i Eq. 9 be posiive sei efiie. y soluios o he olie opiizio poble i Eqs. 5-5e hve bee lule he posiive efiieess of he pojee Hessi H poj hs bee ese eh soluio. ll bu few pevese ses h file o ovege o opil soluio he posiive efiieess of H poj hs bee ofie. heefoe he uliph pee veo i Eq. is obsevble

13 bse o he esuee oel i Eqs. 9 9b povie h eough sples e ilue. Noe h i is liely h he obsevbiliy will vy wih he ievl of he bh esiio poble wih he ihess of he e oio. Suies of hese elioships e beyo he sope of he pese ppe. log he veil be of he syse. he y xis lies log he hoizol hee o h is pepeiul o o i Fig. 5 is posiive ieio is ows he igh of he figue ig he ooie syse ighhe. V. DAA OEON AND PRE-PROESSNG FOR UPAH ESAON A. Hwe oios Expeies D fo uliph esiio ws ollee o Sep. 6 o he oell Uivesiy pus wo isi loios. he fis se of ws ollee i fo of oell's Rhoes Hll s see i Fig. 5. he seo se of ws ollee o he oof of Rhoes Hll s see i Fig. 6. he fis loio ws expee o hve oe sigifi uliph iefeee evioe ue o he pesee of ll builig eby. Fig. 5. D olleio syse see i fo of Rhoes Hll i h NY. A ph e oue o gou-ple ple ws use o eeive he GPS sigls. As see i Fig. 5 Fig. 6 his ouig ple ws bole o he op of he oio syse -- he gou ple ple is visible i boh figues bu he ph e is o visible beuse i is shiele fo he e's view by he ple below i. he oio syse osise of iepee seppe oos oee o hee os iiviully. he lol boy-fixe ooies syse of he oio evie he oe i whih is efie hs is x xis lige wih he hoizol hee o h is pllel o he o i Fig. 5 x poiig ows he boo lef of Fig. 5 whih is oughly oh. he z xis pois upws Fig. 6 D olleio syse see o he oof of Rhoes Hll i h NY. Oe u of he oio syse osiss of ovig suessively i x y -z -x -y z xes i he lol boy-fixe ooie syse. his se of ovees lse ppoxiely 5 seos. Fo eh of he wo expeie loios he oio syse ws u ies wih puses of o ovee i bewee while wee ollee. his se of ovees wih puses i bewee lse ppoxiely 5 iues fo eh loio. B. D D Poessig Wie-b RF wee eoe o is sve fo pos-poessig. he iig of he pue ws syhoize o he iig of he e oio o wihi bou seo. ollee GPS RF wee poesse offlie usig - lguge GPS sofwe io eeive i oe o poue pseuoge egive be ie-phse uulios Hz ee fo ll vilble sigls. Five sigls wee vilble fo he goulevel ess i fo of Rhoes Hll ie sigls wee vilble fo he oofop ess. hiy oe oe ely offses fo he D pop oe wee use o opue uulios -/f - 4/5/f - /5/f... /f. h is offses wee lule spig fo hip ely elive o

14 pop ll he wy o hip le. he ievls bewee oe offses wee /5 h of hip. wo pe-poessig lulios wee ipleee i oe o sve olie opiizio opuio ie. Boh of he eue he ube of uulio esuees i he bh os fuio i Eq. 5b. Oe lulio eues he [;] splig e fo Hz o Hz vi uulio ove suessive ses of five Hz uulios o poue Hz uulios. his opeio hs he effe of euig he ol sple ou K i he bh opiizio poble by fo of 5. he iiol ohee iegio is ie ou fe vigio bi wipe-off. oes o use sigifi loss of opil esiio uy ue o he low bwih of he oio hisoy whih is uh lowe h Hz. Aohe opuio ie svigs eues he ube of oe ely offses fo i he oigil se o. h is ppoxiely / of he oe offses e ise. he fil se osiss of offses h sp fo hip ely o hip le elive o pop wih spig of /5 h of hip bewee eighboig offses. his eiio of he oe ely offses is o expee o seiously ege he esiio uy if hee e o oo y uliph sigls beuse he o oise i he ise offses is highly oele wih oise i he eiig sples s pe Eq. 8. f howeve hee wee y uliph opoes whose piy iffeees wee hei oe elys he his eiio of oe ely offses oul hve sigifi egive ip o he fil esiio uy. Aiiol pe-poessig o he [] uulios ws ie ou i oe o eove wo effes h e o oie i he esuee oel i Eqs. 9 9b. Oe is he f h he sofwe eeive's P h high eough bwih o iely he be ie-phse hges use by he e oio. hus he effes ll show up i he be ie phse obsevble he h i he [] uulios. As isusse i Seio his poble be eeie by opuig he iffeee bewee he P be ie phse polyil fi h hes oly he be ie phse ieiely befoe ieiely fe he oio pofile h soohly siios bewee he befoe fe phse pofiles uig he oio sequee. his iffeee is he use o oe he [] uulio ie hisoy i oe o e-ise he effes. his phse iffeee is show s he sh-oe e uve i Fig. 7 fo oe of he expeies oe of he sigls. Also show i he figue e he eee P ie phse h ilues he effes soli blue uve he uve h fis his le uve oly befoe fe he oio pofile she olive gee uve. his le uve is fifh-oe polyil fi i he se osiee i Fig. 7. he effes e ppe i he soli blue uve fo 5 se o 78 se s he lge egive exusio fo he she gee uve. he she e iffeee bewee hese wo uves is zeo befoe fe he oio bu i ilues he ip bewee 5 78 se. his ip i he sh-oe e uve is use o oe he Pgeee [;] uulios i wy h sfes he effes fo he phse obsevble b io he [;] uulios. ubilly-deee Negive Be ie Phse yles P Phse w/ effes Fi o P fo pe- & pos- ies w/o effes Phse Diffeee fo use i [;] oio Rw Reeive lo ie se Fig. 7 Deee egive be ie phse wih soli blue wihou she-olive effes log wih phse iffeee sh-oe e h is use fo [;] oio. Also s isusse i Seio ele oeio us be e o he oe phses. he uliph effes use lge swigs i he D's oe phse esuees hese swigs viole he Eqs. 9 & 9b ssupio of os offse bewee he pop D oe epli he ue eeive PRN oe. his poble is illuse by osieig he iffeee bewee hges i pseuoge be ie phse uig he oio pofile s show i Fig. 8. As be see fo he blue uve i he figue he pseuoge exusios e lge up o. Obviously hese e o he ie esuls of he --pe-xis oio of he e. A siple ehique hs bee use o ppoxiely epi he oelig isepy bewee Eqs. 9 9b he D pseuoge exusios show Fig. 8. uses he iffeee bewee pseuoge be ie phse s ploe i Fig. 8 o esie whee he D woul hve ple he pop PRN oe h i use ie iig exlusively uig he oios. he opues he effeive oe phse offses of he uulios opue i Eqs. b. hese effeive vlues e he use o iepole he uulios b oo he oigil se of gi pois i oe o ppoxie he uulios h woul hve bee opue i Eqs. b h he D posiioe is pop PRN oe usig ie iig 4

15 exlusively. Oe his hs bee oe he uulios oels i Eqs. 9 9b e fily well sisfie. Noe howeve h his poeue leves he ie effes i he effeive D eeiio of he pop oe phse whih ehilly violes ssupio of Eqs. 9 9b. Fouely give he oio level of his oelig isepy will hve oly egligible ip i he uliph esiio esuls fo /A oe h e pesee i Seio V. Deee Pseuoge ius ie Phse Fig. 8 Pseuoge/Be-ie-Phse ie Sp of pofile Rw Reeive lo ie se Exusios of D pseuoge uig oio pofile s esue elive o be ie phse.. Ae oio Deeiio Ufouely he oio syse i o ilue posiio feeb seso. hee ws ow pespeifie oio pofile bu hee ws quesio bou how uely he evie followe he ple pofile. hee ws lso uow iig bis bewee he oio pofile he GPS pue evie of bou seo. iio he ll be o whih he e soo ws poe o osillios log is os flexible xis i.e. log he hoizol y xis. hese osillios ee o ou he s sop of he y-xis oio ievls hei giue fequey wee uesue uoele. Give he ipoe of owig give he l of peisio wih whih his pofile oul be ow pioi he l of ie esuig syse ie-phse Diffeeil GPS DGPS ehiques wee use o esie he pofile. A speil RK-lie DGPS lgoih ws evelope pplie o he oofop whee fo 9 sellies wee vilble. his speil lgoih i o use eeive-o-eeive iffeeig. se i use sellie-o-sellie iffeeig polyil fi o esiul o-oioiue ie-phse viios he ifoio h ei pois he s sop of give oio ievl oespoe o he se -es hie posiio. he DGPS lgoih solve fo he oio pofile elive o his -es posiio. his efiiio of he elive oio whe ouple o he f h he evie se is es posiio eue o i he e of is oio pofile oue o so of seo iffeeig i ie of he be ie phses i he DGPS soluio lgoih. he esul of he DGPS lulio ws fily osise se of pofiles fo he oof-op ses. hese pofiles wee vege i oe o eeie pesue hie oio pofile. he oigil pofiles wee esie i lol veil-es-oh ooies. Nex hey wee vege usig lgoih h lso fou hei opil elive iig liges. Filly ohol ooie sfoio fo veil-esoh ooies o hie ooies ws esie by exploiig he pesupio h he i feues of he oio pofile i hie ooies wee suessive ps log x y -z -x -y z. his ooie sfoio ws use o sfo he vege oio pofile io hie ooies. he esulig pofile ppes i Fig. 9 log wih he pese pofiles h wee vege i oe o eeie he fil pofile. Ae Defleio Fig. 9 hie X vg hie Y vg hie Z vg hie X oio se ie se Reeive e oio pofile s eeie fo DGPS lulios pplie o oof-op oio ses followe by vegig ooie sfoio. is ppe fo Fig. 9 h he pe-poge sequee of oio log he suessive xis x y -z -x -y z is osise wih he DGPS esuls. Also he iffeees bewee he iiviul ses he vege esul is o gee h.5 usully i is uh slle. Also sll osillios i he gee y-xis uves e ppe fe he e of is iiil p 9 se espeilly fe he e if is fil p se. heefoe hese solve-fo oio pofiles see esoble fo use i he uliph esiio lgoih. Of ouse hee is ope quesio bou he effes of uliph o his DGPS-bse esie of he oio pofile. Give h ees o be ow i 5

16 oe o esie uliph i sees quesioble o use eho of eeiig h is susepible o uliph eos. uh i woul be bes o hve iepee o-gps eho of eeiig uely. he oly eso fo esoig o DGPS o eeie ws ue o he l of seso. he fuue i will be uos ipoe o o iiol expeies wih suh seso. he e ie i is hope h he eos i his DGPS soluio fo e sll ue o sevel fos: he lielihoo of lowe uliph o he Rhoes Hll oofop he use of 9 sellie sigls o opue he DGPS soluio he vegig of iffee us ie ou ove 5 iue ievl log eough ievl fo uliph eos o hge fo u o u. he vege DGPS pofile oes h he pepoge pofile fily losely eve hough he DSGPS lulios h o ifoio bou he pofile ohe h he f h i se soppe i he se loio. heefoe oe igh suise h he uliph effes wee o ge eough o use seious pobles wih his ehique of eeiig. Fuhe eviee suppoig his ojeue is fou i he ep o h pofiles fo he ses h wee u i he high-uliph evioe i fo of Rhoes Hll. he hig ws uh pooe h i Fig. 7. V. EXPERENA UPAH ESAON RESUS he olie opil esiio lgoih of Seio hs bee pplie o he expeiel esibe i Seio V he esuls hve bee lyze o ssess he effeiveess of he uliph esiio lgoihs. Assesse of expeiel esuls howeve is hllege. hee is o ow "uh" vlue fo he uliph opoes. Eve he ie-ph sigl's uh pees e uei ue o he ueiy of higs suh s he uow iue of he hie ooie syse esue elive o efeee ooies. heefoe i is hllege o fi pply effeive eho of ssessig pefoe. Fou ehos hve bee use o ssess pefoe. he fis is esiuls ssesse. osies he opil fi eos he fil vlue of he opil os i Eq. 5b whehe hey e sisilly esoble. he seo ssesse eho osies he esobleess of he opue esiio pees h is of he iiviul elees of he x op veo. he hi ssesse ei osies he esie ieios of ivl of he ie sigls he op elees of x op. hei esobleess be hee by opig he gles bewee esies fo uliple sigls wih he oespoig ow gles i efeee ooies. he fouh eho of ssesse is o opue he wieb pseuoge fo he [;] uulio fe subio of he opilly esie uliph opoes. he esulig oee uulios be use i D-isiio-ype lulio. A goo uliph soluio shoul eue he ifluee of uliph o he esulig pseuoge esies. Ohe ehos of evluio igh hve bee pplie h he expeiel se-up bee iffee. Oe woul be o opee he ovig-e syse e ow sog efleo suh s l boy of we o builig wih lge fl elli sufe. he esie elive ely ieio of ivl of he uliph sigl oul he be ope wih he expee vlues bse o he geoey of he efleive sufe elive o he sellie ieio he eeive loio. Aleively oe oul opee he syse o wo suessive ys he ex se loio exly 4 sieel hous p. he GPS sellies' epee gou s ypilly use he uliph o epe. he syse's esie uliph pees oul be ope ove he wo suessive ys' us o see if he uliph pees epe. Ufouely he expeiel olleio pig epoe hee ws o suue o exploi eihe of hese possibiliies. A. Opiize os Fuio Resiuls Sisil lysis peis h he opiize vlue of he opil os i Eq. 5b if ouble shoul be ppoxiely sple fo hi-sque isibuio wih egee K his is ue beuse he esiul fi eo shoul be use piily by Gussi o oise beuse R - ix i Eq. 5b lizes he esiul Gussi oise o hve ieiy ovie ix beuse hee e iepee K uulio esuees i he eie bh 78 esie quiies i he esiio poble. heefoe he expee e vlue of he opil os is.5k-7-8. f he esuee oel i Eqs. 9 9b pues os of wh is hppeig he he ul vlue of Jx op opue usig will be e he expee vlue. heefoe ssesse of he esiuls povies he o he uliph oel's esobleess fo ul expeie. Visully oe ssess fi eos bewee esue oele by osieig plos of uulios. osie Fig. Fig.. hey plo he pop uulio pliue ie hisoies / [ ] vs. fo wo ses. Fig. is fo PRN whe opeig he ovig-e syse i fo of Rhoes Hll whih is osiee o be high uliph evioe. Fig. pplies o PRN 8 whe he syse is opeig o he oof of Rhoes Hll 6

17 lowe uliph evioe. Eh of he figues plos 6 pliue ie hisoies. he e os e he ul pop uulio pliue. he soli uquoise uve is he opil pliue esie ue he ssupio of o uliph. he ohe 4 uves oespo o 4 iffee ubes of esie uliph opoes she gee uves shoe blue uves oe bl uves 4 she ge uves. Auulio Apliue sq pop pop.5 x D oel w/o P oel w/ P oel w/ Ps.5 oel w/ Ps oel w/4 Ps Reeive lo ie se Fig. esue opil-fi pop uulio pliue ies hisoies fo PRN i fo of Rhoes Hll. Auulio Apliue sq pop pop 5.5 x D oel w/o P oel w/ P oel w/ Ps oel w/ Ps oel w/4 Ps Rw Reeive lo ie se Fig.. esue opil-fi pop uulio pliue ies hisoies fo PRN 8 o oof of Rhoes Hll. boh of hese ses he oels h ssue uliph fi he pop uulio bee h he oel h ssues oly ie opoe. he ul hve ie-vyig pliue i boh ses liely ue o uliph eo bu he oel wih o uliph o ou fo suh viios. heefoe is vlue is jus os h ies o fi he e vlue of he fluuig pliue. he suessive uliph esies fi he pliue bee s he ssue ube of uliph opoes ieses. he high uliph evioe of Fig. he fi eo es oieble ipovees s is iese fo o 4. Noe how he she ge uve fis he vey well fe 66 se. Oly he she ge uve he oe bl uve hve eoely esoble fis o he fis o pliue osillios befoe 66 se he she ge uve fis he bes fe 8 se. Fig. he fi eos e sll fo ll. Noe how he sh-oe blue uve he oe bl uve he she ge uve ll fi he e os well fo he eie bh. boh of hese ses he visul fis iie h he uliph oel suessfully ous fo he obseve povie i ilues eough uliph opoes. he opil esiuls io γ Jx op /K-.5-4 ells siil soy. Fo Fig. he espeive vlues of γ o γ 4 e his lely iies iesig gooess of fi s ieses. Ufouely hi-sque pobbiliy heoy iies h γ 4 shoul be below.99 i 999 ou of sples. heefoe he fi eo io.68 is sill uesobly lge. his is illuse visully i Fig. by he f h he exusios of he she ge uve fo he e os is soeie sigifily lge h he o sisilly spe of he pois. he uliph oel wih 4 uliph opoes oes o full ou fo he ollee i his se. Fig. ppes o be bee i his espe bu is oespoig γ o γ 4 vlues e Agi fi eo ipoves wih iesig bu i is o ely s goo s i shoul be bse o sisil lysis; γ 4 shoul be less h.46 i 99.9% of ses. Auulio hve bee opilly poesse esiuls hve bee osiee fo evey e sellie fo he ile oio pofiles i eh expeie loio i fo of Rhoes Hll he Rhoes Hll oof. he 5 sellies e i fo of Rhoes Hll wee PRNs All of he expeiee ipove fis/eese esiuls wih ieses of up o 4 bu oe of he poue γ 4 vlue h ws sisilly esoble. he lowes hieve vlue ws γ 4.54 fo PRN 4 bu i h siuio wih he 99.9% hi-sque heshol ws uh slle.4. hus he uliph oel i Eqs. 9 9b oes o pue ll h is ouig i hese siuios. Fuheoe he olie opil esiio lgoih expeiee ovegee iffiulies i few ses: i he 4 ses fo PRN 7 i he 4 ses fo PRNs 4 8. hei ovegee ws vey slow. his issue will be isusse oe i he ex subseio. he 9 sellies e fo he oof of Rhoes Hll wee PRNs ie he fo i fo of Rhoes Hll hese ll expeiee eue fi esiuls wih ieses i he ube of uliph 7

18 opoes. Ulie he fo he fo of Rhoes 6 of hese 9 sellies ehe sisilly esoble γ vlues fo 4. PRNs 7 hieve sisilly esoble fis fo uliph opoes PRNs 4 hieve esoble fis fo opoes PRN 4 hieve esoble fi 4. hus he oel i Eqs. 9 9b ppes o h he oe losely i he elively beig uliph evioe o he Rhoes Hll oof. shoul lso be oe h few of he uulio pliue fis ppee uh pooe visully h hose show i Fig. Fig.. piul PRN 's pliue fis e piul poo fo ll ses o 4 fo he ile oio pofile i fo of Rhoes Hll. As ye hee is o explio fo why his ous. B. Resobleess of Esies i x op he pese seio osies he esobleess of he opil esies of he vious elees of he x veo h is efie i Eq.. Fo puposes of his lysis hese elees be osiee i goups. he fis goup is h of Dopple shif Dopple-shif e elees. his goup osiss of γ.... he seo goup osiss of he oe elys i δ δ... δ. he hi goup osiss of he ieio-of-ivl ui veos i.... he fouh goup osiss of he phsos [X ;Y ] [X ;Y ] [X ;Y ]... [X ;Y ]. hee e o piul heisis o expe fo he Dopple-shif Dopple-shif-e pees γ... exep h hey be esobly sll. Sll is efie i es of he lisig h ous hlf he uulio fequey i.e. / -. All of he esie Dopple pees... hve bsolue vlues less h % of his lisig fequey. f ll hve bsolue vlues less h 4/ K - so h oe of poue oe h full osillios ove he eie esiio bh. he esie γ Dopple shif e vlues espe oespoig lii o hei bsolue vlues equl 8/ K -. Neihe will hey poue oe h full oios ove he bh. pie hese esies lly poue less h oe full phse oio ove he bh ievl of he ie sigl elive o he P phse o of he uliph sigls elive o he ie sigl. he esie bsolue ie-sigl oe elys i he esie uliph oe elys i δ δ... δ e o esoble vlues fo ll ses osiee. he esies of he uliph-iue D oe eos epesee by e ypilly o he oe of. hips o less. he uliph oe phse ely esies i δ δ... δ sp he ge fo hips o.5 hips. Evluio of he esobleess of he esie ieph ieio veos is eseve fo lysis i he ex subseio. he esie uliph ui ieio veos... o be evlue fo hei esobleess beuse hee is o wy o ow hei ue ieios i he usuue evioe h hs bee use o ou he expeies. is ieesig o oe howeve h y of he esie ieios h egive elevios. Elevio oul be eue esily beuse i ws ow h he z xis of he e oio evie poie ely veil. heefoe posiive vlue of he hi opoe of y give veo iie is ivl fo ieio of egive elevio. hese egive elevios e esoble beuse he gou use uliph efleios. Aohe ipo pheeo oe fo soe of he esies is h hey soeies oue i losely spe pis. his oue fo PRNs i he high uliph ses i fo of Rhoes Hll. he Rhoes oof his siuio oue oly fo PRN 8. hese ouees e ouple o eve sge pheeo i he uliph phso esies isussio of his pheeo is lef o he isussio of he esobleess of he phso esies. he esies of he ie uliph phsos [X ;Y ] [X ;Y ] [X ;Y ]... [X ;Y ] give ifoio bou he ie phses pliues of he vious eeive sigl opoes. hee e o obviously uesoble vlues fo hei YX phses bu hei X Y / pliues be lyze fo esobleess. Oe woul expe he ie sigl pliue X Y / o be sigifily lge h y uliph pliue X Y /. he joiy of ses his is ue. soe ses howeve hee exiss uliph opoe wih lge pliue. Soe of hese ses e beig. he elive oe ely of he offeig uliph opoe is δ. his ou whe uliph opoe he ie sigl ive viully he se ie. hee is o wy fo he bh opiizio poble solve ssoie wih Eqs. 5-5e o isiguish whih sigl is he uliph sigl whih is he ie sigl. A siple opiso of pliues howeve be use o isiguish he ie-ph sigl whe δ. he ie-ph sigl is lge. evey se whee his oue he ojeue h he sigl wih he lges pliue ws he ie-ph sigl ws ofie by opig is ieio of ivl wih he expee ieio of ivl of he ie-ph sigl. 8

19 his opiso use lyses siil hose esibe i he ex subseio. os of ses h hve bee lyze iluig he beig ses of evesible is-ieifiio of he ue ie sigl he uliph sigl pliues ge fo less h 5% of he ue ie-ph pliue o s uh 75%. os ses he uliph opoes ll hve pliues h e less h 5% of he ie ph pliues. hese ppe o be esoble esuls. Aohe ype of lge uliph pliue lso ous oe h is o beig. his se wo uliph opoes hve lge pliues h he ie ph uh lge pehps sevel oes of giue lge. hese uliph sigls hve o-egligible elys δ ps he ie-ph sigl. evey suh se he wo lge-pliue uliph sigls lso hve ely ieil ieios of ivl s ley oe fo PRNs i fo of Rhoes Hll fo PRN 8 o he oof of Rhoes Hll. hey lso hve ely opposie ie phses. h is he [X ;Y ] phso of oe of hese opoes is los exly equl o he egive of he phso of he ohe opoe. hese sge uliph opoes osiue so of uliph ouble. hey los el bu hey o o quie el so h he poue uusul sigl suue h bee fis he. is ojeue h hese e he esul of ul sigl pheeo: iffuse uliph. hey y be he esio's bes wy o ope wih his pheeo h oes o fi io is speul uliph oel fo. Regless of whehe o o his pheeo is iffuse uliph i peses sigifi poble fo he esio. his siuio uses vey slow pogess of he us-egio olie esiio lgoih ows soluio i is foe o qui befoe ehig ue lol opiu of he os fuio. his poble of oubles iplies h bee uesig of his pheeo is eee o opee he pese lgoihs i high-uliph evioe. is hope h bee uesig woul le o pbiliy of ig ew es o he oel i Eqs. 9 9b. hese ew es woul oel i bee wy he pheeo h is uely beig ppoxie by uliph oubles. Pesubly he ew es woul oi uow pees h oul be esie i oe o pefo uliph iigio.. Alysis of Esie Die-Sigl Aivl Dieios he esie ivl ieios of he ie-ph sigls e bes evlue siuleously fo ll vilble sellie sigls uig give e oio pofile. Suppose h hese esies e op fo... N whee is he sellie iex whee N is he ol ube of sellies whose sigls hve bee e lyze o esie hei ie-ph uliph opoes. Kowlege of he eeive loio he sellie epheeies he ie be use o esie hese se ie-ph ieio-of-ivl veos i soe ow efeee ooie syse. Suppose h he hose efeee ooie syse is he lol-level veil/es/oh ooie syse suppose h he ow ieio-of-ivl veos i his syse e ve fo... N. he wo ses of ieio-of-ivl veos e give i wo iffee ooie syses hey be use o esiio he ohogol sfoio bewee hese syses. Suppose h A boy/ve is he uow x ohol ooie sfoio ix fo veil/es/oh efeee ooies o he e iulio ooie syse i whih is efie i whih op fo... N hve bee esie by he lgoih of Seio. he goo esie of A boy/ve be opue by solvig Whb's poble : fi: A boy/ve o iiize: J subje o: A boy /ve N w [ op A boyve / ve ] [ / op Aboyveve] A boy / veaboy/ ve b whee he quiies w fo... N e posiive weighs. he pese lysis hese weighs e se equl o he eeive ie-o-oise ios of he ieph sigls. hus soge sigls e eee o hve oe uely eeie op veos. his iue esiio poble be solve i lose fo usig he q-eho. he q-eho solves fo he opil queio epeseio of A boy/ve by solvig ele eigevlue/eigeveo poble fo syei 4x4 ix. he effiy of he uliph esiio lgoih of Seio be evlue by opig wo soluios o he iue eeiio poble i Eqs. -. Oe soluio uses h op esies h ssue he pesee of uliph sigls i he oel i Eqs. 9 9b. he ohe soluio uses op esies h ilue eough uliph opoes o hieve esobly sll opil os oespoigly esoble sll esuee eo esiuls s isusse i Subseio V.A. he bee soluio is he 9

20 oe h poues he lowes opil os vlue fo he os fuio i Eq. b. he lii of sll gul eos his os is popoiol o he e sque of he gul eos bewee he esie ui ieio veos op he oespoig sfoe efeee ooies veos boy Aboyve / ve. A sll os vlue lso iies h he elive gles bewee he esie veos op fo... N h he elive gles bewee he oigil efeee veos ve fo... N. he biliy o h hese elive gles iies he esobleess of he op esies. hese lulios hve bee pefoe fo wo ses. Oe se is fo he seo oio pofile o he oof of Rhoes Hll ilues N 9 sellies i he iue eeiio poble of Eqs. -. he hoie of uliph soluios fo he se h uses > uliph opoes o fo he op esies ws se fo ll sellies exep PRN 4. PRN 4 use is op soluio fo is uliph soluio h use ws se 4 uliph opoes beuse 4 opoes seee eessy o ge esoble low esiuls fi. his se yiele RS gul eo bewee is esie op veos is sfoe efeee veos boy of 4.8 eg whe usig he op h ssue o uliph opoes i poble 5-5e. Whe usig op h wee esie log wih uliph opoe pees howeve he RS gul eos bewee he op boy ieio veos ws eue o.9 eg. hus he uliph esiio poble ppes o yiel ipove ieio esiios fo is ie-sigl opoes. As fuhe ofiio of he effiy of he uliph esiio he pe gul eo bewee op veo is oespoig boy veo ws. eg whe uliph ws igoe i he esiio of op bu i oppe o 5.5 eg whe uliph ws osiee. he se es ws pefoe fo he fo he fo of Rhoes Hll. his se hee wee oly N 5 sellies vilble. he RS eos bewee he op esies he sfoe efeee veos boy ws 8. eg fo he se h igoe uliph i he esiio of op bu i ose o 8.9 eg whe uliph opoes wee osiee uig he esiio of he op veos. he pe gul eos we he opposie wy. he pe ws 4.7 egees fo he op veos h wee esie igoig uliph bu oly. eg whe usig he op h wee foe while osieig uliph effes i he Eqs. 5-5e opil esiio poble. he biliy o esie op uely by solvig he esiio poble i Eqs. 5-5e seves s poxy fo he biliy o eove uliph effes fo esue be ie phse. ll siuios he biliy o esolve op is bse piily o is ip o he ie phse h is oie ipliily i he [;] uulios. Give he elioship bewee op esiio uy he biliy o eove he ie-phse effes of uliph he esuls of his subseio pese ixe essge bou he effiy of his ppe's syse o esie eove he effes of uliph. he elively beig uliph evioe of he Rhoes Hll oof soluio of he poble i Eqs. 5-5e wih uliph ilue es o ipove he effeive esuee of he ie-ph ie phse. he high uliph evioe i fo of Rhoes Hll howeve he lgoih ppes o hve file o ipove is iplii esies of he ie-ph ie phse. D. Alysis of Deee Wieb Pseuoge Afe Reovl of uliph Effes he pese subseio lyzes wieb pseuoge s eeie usig ype of D isiio he esue [;] uulios se of oe offses. oe seio he w [;] esuees e use. seo seio uliph-oee vesios of he [;] esuees e use. f he uliph esiio lgoih of Seio is effeive he hese oee esuees shoul poue uh soohe wieb pseuoge esuee uig he lge uliph viios h ou uig he e oio pofiles. he oeio of he w [;] uulio esuees fo uliph effes poees s follows: Fis he uliph esiio lgoih of Seio is use o solve he poble i Eqs. 5-5b. Seo he X op Y op ie-ph phso opoes i he esulig x op soluio veo e eple by zeo vlues o poue he veo x opxy. sly his veo is subsiue io he uulios esuee fuios i Eqs. 9 9b he esuls e sube fo he w uulio esuee veos z z i oe o poue he uliph-oee vesios of hese veos: zpo z h [ xopx Y ; ] z po z h x ; ] b [ opx Y he isiio use i his lysis is bse o solvig gely siplifie vesio of he opiizio poble