Jounal of Global osiioning Sysems (6) Vol. 5, No. 1-:119-16 Feasibiliy of Ai Tage Deeion Using GS as a Bisai ada E.. Glennon SigNav y Ld, Uni, 59 Tennan S, Fyshwik, ACT 69, Ausalia A. G. Dempse and C. izos Univesiy of New Souh Wales, Sydney, NSW 5, Ausalia Absa. The feasibiliy of using GS as a bisai ada illuminao fo he puposes of ai age deeion is examined. The powe budge analysis is fis pefomed assuming he use of a single saellie, bu is followed by a disussion of he expeed impovemens when muliple saellies ae employed. The analysis inludes he effe of GS signal sengh dynami ange, also known as he nea-fa poblem. The diffeene beween he ada oss-seion (CS) of a ypial ai age and goundbased lue efleions is disussed, followed by an esimaion of he effe of gound lue on he opeaion of suh a sysem. Keywods. Bisai ada, Global osiioning Sysem, ada deeion, 1 Inoduion The Global osiioning Sysem has found widespead appliaion beyond sandad posiioning and iming, wih new uses fo GS oninually being developed. One of he moe novel oneps is he seonday use of GS signals fo emoe sensing, wih he seonday appliaion onsideed in his pape onening he use of GS saellie ansmissions as he ansmie in a bisai-ada fo ai age deeion. evious publiaions on his opi inlude (Tsui and Shaw, 1993) and (Koh and Wesphal, 1995), alhough neihe of hose publiaions povides a powe budge onfiming he feasibiliy of he sysem. (Solk and Bown, 3) desibes an aibone GS emoe sensing sysem apable of deeing lage oean vessels suh as oil-ankes. Anohe pape by (Cheniakov e al., ) does povide a powe budge esimae fo ai age deeion, bu assumes he use of he highe-poweed IIDIUM saellie sysem as he ansmission soue. Alhough he mehodology of Cheniakov e-al () has been adaped o he GS ase by (Mojaabi e al., ), ha analysis onained some eos. This pape dupliaes he analysis of oiginal Mojaabi pape using a onvenional appoah and diffeen paamee seleions, esuling in less opimisi esimaes fo he maximum deeion ange. The pape also esimaes he effe of lue in a GS bisai ada using a diffeen ehnique o he mehods of (Mojaabi e al., ) and (Cheniakov e al., ). In addiion, he effeiveness of using muliple GS saellies on he powe budge is also examined and diffiulies due o he nea-fa poblem ae disussed. Bisai GS owe Budge To deemine he maximum ange of a GS-based bisai ada i is neessay o deemine he signal powe ha eahes he eeiving anenna afe being efleed fom he age. Fo any ada, he powe densiy of he age eho S a he eeive is given by (Skolnik, 1981): S G σ i Hee is he ansmied powe (W), G is he ansmie anenna gain (dimensionless fao), i is he ange fom he ansmie (saellie i) o he age (m), σ is he ada oss-seion (CS) of he age (m ) and is he ange of he age fom he eeive (m). The CS is defined as he fiional aea σ ha podues he obseved efleed powe densiy S a a eeive a a ange fom he age ha has been illuminaed wih an iniden powe densiy of S i (Skolnik, 1981). S σ lim () Si (1)
1 Jounal of Global osiioning Sysems The fis em of he podu in (1) is he powe densiy of he die (ansmied) signal a he age S die pio o being efleed (unis of W/m ). The signal powe available a he eeive anenna oupu depends on he effeive aea of he eeiving anenna A e, whih depends on is gain: λ G A e (3) whee G is he eeive anenna gain and λ is he GS aie fequeny wavelengh of.19 m. Using (1) and (3) he powe of he age efleion available a he eeive anenna oupu an be alulaed as: σ λ G S die (4) (4) pemis he aenuaion of he efleed signal measued using a high gain anenna elaive o a sandad diely eeived GS signal as would be eeived in an omni-dieional anenna wih a gain of G dbi o be alulaed: die σ G G dbi To alulae he maximum ange of suh a sysem i is neessay o deemine he noise level N a he oupu of he F fon-end, whih an be given in ems of he equivalen noise empeaue T eff and he bandwidh BW as: (5) N k Teff BW (6) (4) and (6) allow he signal-o-noise aio of he efleed signal a he F fon-end oupu o be wien as: S die σ λ G (7) N ( ) ( k T BW ) eff If he subsequen signal poessing is hen subje o losses L sp and poessing gain G sp hen he final signal o noise aio ρ is given by: S die σ λ G Gsp ρ (8) ( ) ( k Teff BW ) Lsp Assuming a minimum value fo he deeion signal-onoise aio pemis an expession fo he maximum ange o be given as: S die σ λ G Gsp (9) ( ) ρ min ( k Teff BW ) Lsp Using values fo σ of m, a hon anenna wih a gain G of 15 db (31.6) and a ange of say 1km, he efleed signal will be aenuaed by abou 6 db, whee his ombinaion of paamees have been seleed o agee wih (Mojaabi e al., ). Hene, if he die signal has a signal-sengh of say 5 dbhz o -15 dbw, hen he efleion fom he age will have a signal level of 1 dbhz o 15. dbw. Sine he uen sae of he a in GS eeive ehnology is o dee and ak signals a aound dbhz, he deeion of suh a efleion is exemely diffiul. Howeve, sine suh weak signal GS eeives have ypially been ageed fo he E911 ellula appliaion, some of he onsains pesen in he E911 ase do no ou in he GS bisai ada appliaion heeby pemiing geae sensiiviy o be ahieved. In paiula, sine a bisai GS ada has simulaneous aess o boh he die and indie signals, he daa eeived fom he die signal an be used o sip he daa bis fom he indie signal, heeby pemiing signifianly longe oheen inegaion han he GS daa bi peiod of ms. Appendix A (Van Diggelen, 1) shows ha fo a oheen inegaion peiod of 1 ms a maximum sensiiviy of appoximaely -19 dbw o 13. dbhz is o be expeed, assuming a final SN of abou 9.3 db. This epesens an aenuaion of 38 db ompaed o he ypial maximum eeived signal of 5 dbhz, as is he ase using a ypial 3 dbi pah anenna. To alulae he ange ha his oesponds o, (4) an be used, seing / die o -38 db (158.48 1-6 ), G dbi o 3 db (), and using he pevious values fo σ and G, a value fo of 398m is obained. Table 1. Maximum ange Esimaion aamee Values aamee Unis Values S die W/m 39.81 1-15 S die dbw/m -134 σ m Bandwidh Hz.4 1 6 T eff K 344 λ m.19 G sp n/a.46 1 6 L sp n/a.11 G n/a 31.6 k 1.38 1-3 max m 39 Alenaively, using (8) wih he values given in Table 1 esuls in a maximum ange of 39m. This value is lowe han he pevious value beause i uses he minimum speified powe speal densiy fo GS of -134 dbw/m (Spilke, 1996), wheeas GS saellies ypially exeed he minimum level of pefomane. If he value fo G is ineased o 35 dbi (3481), whih is equivalen o an anenna wih effeive aea of
Glennon e al.: Feasibiliy of Ai Tage Deeion Using GS as a Bisai ada 11 3.6m 3.6m (as used in (Cheniakov e al., ), hen max ineases o.51km. 3 Bisai ada / Mulipah aadox The diffiuly in deeing an ai age efleion ous beause he powe loss following efleion fom he age is exemely high. This fa pesens an appaen onadiion; whih is how GS mulipah signals an eve be deeed sine in heoy hey should be signifianly aenuaed. The explanaion is ha hee ae essenially wo ypes of efleions fom a sufae, namely speula efleions and diffuse efleions (Kazbeg and Gaison, 1996). Mulipah efleions ae geneally speula in naue, in whih phase oheene is eained duing he efleion poess. Howeve, efleions fom an ai age ae geneally diffuse efleions in whih any iniden adiaion is saeed in muliple dieions and phase oheene is los. This means ha he CS in he dieion oesponding o a speula efleion is signifianly geae han in he ohe dieions, whih ae diffuse. This obsevaion applies o boh ages and lue efleos and is appaen on plos of CS vesus angle of inidene (Skolnik, 1981). Hene when a sandad GS eeive obseves a mulipah signal i geneally oesponds o a speula efleion. 4 Clue owe Esimae The lue powe ha is obseved in a GS bisai ada sysem is dependen on he opeaing envionmen and he sysem haaeisis. One fao unde onol of he designe is he eeiving anenna. Fo his ase, a suiable anenna is one in whih hee is high dieiviy and gain in a paiula dieion, bu has small anenna sidelobes and gain in all ohe dieions. The polaizaion of he anenna should be uned o bes mah he polaizaion of he efleions and as suh, will pobably be lef hand iula polaized (LHC) whih is opposie o he HC of he die GS signal. Clue powe is assumed o ene he sysem via he anenna sidelobes only, an assumpion ha is also made in (Cheniakov e al., ). The effe of obseving he die signal in he anenna main lobe is a moe diffiul issue ha is sepaaely onsideed in a subsequen seion. This analysis employs a simple anenna model in whih he boesigh gain G and beamwidh θ bw ae onsan and he sidelobe gain G sl is onsan and omni-dieional. The model does no inlude ealisially-shaped anenna sidelobes and is only inended fo a fis ode analysis. When alulaing he lue fo a GS bisai ada, eall ha GS is a spead speum sysem wih a hipping ae of f (1.3 MHz) and hene has a ange esoluion d of / f, whee is he speed of ligh. Fo his eason, he lue will geneally be ange limied by he GS ange esoluion of 93m. Assuming ha all lue is due o diffuse gound efleion, he powe eeived a he anenna wihin a paiula ange ell an be alulaed as he powe efleed by he aea beween he isoange onous on he gound saing a a ange and ending a d and adjused fo he fee spae loss of 1/( ). Using his poedue, he lue powe densiy a he eeiving anenna an be given as σ d SC S die log e 1 (1) whee σ is he CS of he lue pe uni aea and whih is ypially a funion of he gazing angle, alhough fo his analysis is aken o be onsan. A deivaion of his esul is given in Appendix B. The powe of he lue a he eeive anenna oupu an now be given as: σ d Gsl λ C S die log e 1 (11) This pemis he lue-o-age powe aio a he eeive anenna oupu o be given as: C π σ σ π σ σ G G sl G G sl d d loge 1 Table. Clue-o-Tage owe aio σ (α)(db) G sl (db) (m) C / (db) - 1, -.36 - -1 1, -1.36 - -1 5, -3.37 - -1 1, -.36 - -1 1, 7.64-1, 17.64 (1) Table gives he lue-o-age powe aio fo vaious values of σ, and G sl. The σ values of - db and - db ae aken fom (Willis, 1995) and oespond o daa fo ou-of-plane, hoizonally polaized, σ B daa fo all weeds and sub ees measued a a fequeny of 1.3 GHz, a value ha is lose o he GS aie fequeny of 1.5754 GHz. The maximum value fo his paamee is abou - db oesponding o a speula efleion, while he ypial value is abou - db. The minimum value is abou -3 db. All daa is fo bisai mode of opeaion. The sidelobe gain values fo G sl ae
1 Jounal of Global osiioning Sysems esimaes only. Values fo G and σ have been hosen as 3 db and m espeively o mah he pevious seion and d has been se a 93m. This daa shows ha even fo elaively sho anges, he lue onains almos as muh powe as he age ada eun. The siuaion an be impoved by eduing he magniude of he anenna sidelobe gain. Howeve, oasionally i will be possible ha speula ype lue will ou ha will ompleely dominae ove he age efleion. One fao ha has no been aken ino aoun is he lue Dopple fequeny, whee he analysis assumes ha he lue will have he same Dopple offse as he age. This is no ealisi fo ai ages heeby enabling he opion of using Moving Tage Indiao (MTI) o sepaae he age fom he lue. The use of MTI is onsisen wih he use of long oheen inegaion and is assoiaed naow bandwidhs equied fo he deeion of vey weak signals. ope analysis of he MTI impovemen is beyond he sope of his pape and in he ase of GS, ae mus be aken sine he Dopple fequeny eun vaies wih disane fom he eeive. 5 GS Nea-Fa oblem Coss-oelaion o he nea-fa poblem ous when ying o dee weak GS signals in he pesene of ohe song GS signals. Due o he use of 1-bi Gold odes, he dynami ange of he GS C/A ode is nomally limied o signal levels ha ae no moe han 1.6 db weake han he songes signal pesen (Spilke, 1996). Sine he age signal efleions ha need o be deeed by a GS-based bisai-ada ae signifianly weake han he songes signals pesen, his pesens a lea limiaion ha mus be addessed. Thee ae seveal mehods ha ould be used o suppess he song GS signals. One mehod is o employ a highly dieional high-gain anenna ha has vey low-gain sidelobes, alhough his will pobably esul in he song signal suppession of beween 3 o 4 db, assuming he anenna gain is abou 3 o 4 db above he sidelobes. Howeve, i has aleady been shown ha age ehoes ould easily be aenuaed by appoximaely 6 db ompaed o he main signal, so unless addiional ossoelaion miigaion ehniques ae employed eliable deeion of he efleions is unlikely. Suh ehniques ange fom anellaion o subspae pojeion ehniques (Madhani e al., 3; Glennon and Dempse, 4; Glennon and Dempse, 5), alhough no ommeialoff-he-shelf GS hipses uenly implemen ossoelaion miigaion heefoe leaving sofwae oelaion as he only viable alenaive. Coss-oelaion miigaion epesens a hallenging poblem and has appliaions fo weak signal GS eeives in geneal, alhough sine hese ehniques ae no in widespead use i is diffiul o esimae hei effeiveness. One ineesing ase ha an aise is when he age happens o lie in he line-of-sigh of a GS saellie. Clealy he eeiving anenna offes no poeion agains oss-oelaion sine he boh efleion and die signal ae amplified equally. Howeve, fom a bisai ada poin of view his ase is signifian fo anohe eason, namely he fowad sae CS of he age expeienes signifian enhanemen ompaed o he ypial value. The fowad-sae CS σ F fo a age is given by (Willis, 1995): A σ F (13) λ whee A is he age shadow aea. If (13) is subsiued ino (4), hen he powe efleed by he age and eeived by he eeive an be expessed as: S die 1 A G (14) Assuming ha he die signal is no signifianly aenuaed due o blokage by he age, hen he die eeived signal powe a he anenna oupu is: λ G die S die (15) and he aio of he efleed o he die signal is: die A (16) λ I should also be noed ha aoding o (Koh and Wesphal, 1995) his assumpion abou he die signal no being aenuaed may no be eniely ue. This is beause he age size ould be ompaable o he size of he fis Fesnel zone, his being he size a whih signifian F blokage beomes appaen. If his is he ase hen i means ha bee esuls han he above analysis pedis may be expeed. The ohe diffiuly wih his paiula senaio is ha he pah diffeene beween he die pah and he efleion will be smalle and unless i exeeds one GS hip (93m), sepaaion of he wo signals will be diffiul. The geomey of his senaio also means ha he veloiy of he age anno be deemined sine he Dopple diffeene beween he die and indie pah is small. Using a value fo A of 5 m and a value fo of say 1km, he aenuaion of he efleion ompaed o he die signal omes ou a abou 51.6 db. As a esul, alhough he fowad sae enhanemen of he CS is
Glennon e al.: Feasibiliy of Ai Tage Deeion Using GS as a Bisai ada 13 quie signifian (σ F /σ is 6 db), he benefi is paially anelled by he loss of he anenna dieiviy and gain in eduing he die signal. This analysis shows ha he fowad sae CS enhanemen advanage may no be as benefiial as expeed and is sill vey muh dependen on he popeies of he eeiving anenna sysem. This is beause he fowad sae CS effe ypially akes plae when he bisai angle is wihin 1 of he 18 ideal angle. This means ha fo an exemely naow anenna beam-widh i would sill be possible o gain some of he advanage of he enhaned CS while sill gaining he benefi of some eduion in he die signal. Howeve ying o analyze a ase suh as his using he simple anenna model employed in his pape lealy epesens a limiaion of his appoah, wih a pope analysis of his paiula ase equiing a paiula anenna and age CS pofile be employed. The GS fon-end hip may also fail o opeae oely if he inpu signal levels exeed is design paamees. 6 Use of Muliple Saellies Up o his poin he analysis has assumed he use of a single GS saellie and a single GS eeive. Howeve, he numbe of visible saellies vaies fom a minimum of abou fou o a maximum of abou welve saellies and his pesens an oppouniy o impove pefomane. GS SV i Die ah si si sj eeive sj β i GS SV j β j sj Indie ah Tage Coelaion Die Coelaion Tage Coelaion Fig. 1. Sysem Geomey & Coelaion Cuves Delay (Chips) Figue 1 shows he sysem geomey fo a GS bisai ada as well as an illusaion of one of he oelaions ha ould be expeed (neihe o sale). The diffiuly in using muliple saellies aises beause alhough he ange fom he age o he eeive is he same fo all saellies, he mulipah delay beween he die pah and he efleed pah is no. Fo he example, he mulipah delay fo SV i would be expeed o be lage han fo SV j whih is lose o he line-of- sigh (LOS). If he ange veos fom he eeive o saellie i, he age o saellie i and he eeive o he age ae denoed by si, si and espeively and he bisai angle is given by β i, hen he mulipah delay beween he die pah and he efleed pah is given by: d Die ah si 1 si eeive si i os( β ) os ( β ) i si Fig. Delay ah Lengh Tage Indie ah (17) (17) makes i possible o pedi he mulipah delay fo a given age ange and bisai angle (he angle beween he ansmie and eeive measued a he age). Sine he ange fom he age o he eeive is fixed aoss all saellies and he bisai angle is onsan fo a given saellie on a paiula age LOS veo, i is possible o sale he oelaion oupus fo eah saellie so ha he age delays ae he same fo all saellies. ovided his is done afe emoval of he die signal (o emove he effe of loal oelaion peaks and ossoelaions), i should be possible o aumulae he saled oelaion oupus fom all saellies ogehe. The mos saighfowad mehod of aumulaing oelaions aoss muliple saellies is o do so nonoheenly by fis aking magniudes of he omplex oelaion oupus. The impovemen ha is o be expeed an be appoximaed by assuming ha use of N sv saellies eah wih powe speal densiy S die is equivalen o he use of a single saellie wih powe speal densiy N sv S die. Unfounaely his appoah suffes fom he poblem of non-linea inegaion loss (Baon, 1969; Lin e al., ), whee he magniude of he loss depends on he signal deeabiliy-fao (signalo-noise aio) as well as he numbe of non-oheen inegaions pefomed. A plo of inegaion loss vesus numbe of inegaions fo diffeen oupu single-pulse deeabiliy faos D (1) an be found in (Baon, 1969) and is epodued below in Figue 3. β i
14 Jounal of Global osiioning Sysems Inegaion Loss Li (db) To obain he inegaion loss L i, he deeabiliy fao D (1) ha povides equied pobabiliy of false-alam fa and pobabiliy of deeion d fo a single pulse deeo is deemined using ayleigh-ie pobabiliy disibuion uves. The loss aused by obaining his quaniy hough inegaion of muliple pulses an hen be deemined by inepolaing he uves given in Figue 4 fo he given value of D (1) and hen eading he loss oesponding o he numbe of non-oheen inegaions n. Fo he muliple GS saellie ase, he value of n will ange fom beween 4 and 1, while fo a fa of less han 1-6 and a d of geae han.9 a D (1) value of geae hen 13 db is equied (Baon, 1969). These ombinaions of paamees imply an inegaion loss of less han db. Depending on he signal sengh of he age efleion, his aumulaion aoss muliple saellies should esul in impoved deeabiliy. Moe impoanly howeve, wih he use of muliple saellies he bisai angle fo one of he saellies may be moe favouable, leading o a bee CS, also impoving he pobabiliy of deeion. I is pobably no possible o pefom a fully oheen inegaion aoss muliple saellies sine he phase oheene of he signal is unlikely o be eained duing he diffuse efleion poess. As a esul, he inegaion losses of he non-oheen poess pobably anno be eliminaed. 7 Conlusions Inegaion Loss Li (db) vs No of Inegaed ulses n (given D(1)) 1 db 18 db 1 16 db 14 db 1 db 8 1 db 8 db 6 db 6 4 db db 4 1 1 1 1 No Of Inegaed ulses n Fig 3. Non-oheen inegaion loss fo a given numbe of inegaion peiods. This analysis has shown ha due o he exeme weakness of he ansmied GS signal, deeion of ages wih small CS using eeive anennas of gain 15 db is pobably no feasible. Ohe poblems ha need o be oveome involve he nea-fa poblem aused by he dynami ange of he eeived GS signals and he poblem of he eeived gound lue powe having a geae powe level han he age efleion. Given hese diffiulies, he following eommendaions ae suggesed if onsuion of suh a sysem is o be undeaken. Fisly, he gain of he eeive anenna should be as lage as possible. The woked example used a hon anenna wih a gain of 15 db, howeve i should be possible o ahieve a gain of say 5 db whih would inease he maximum ange by a fao of 3. Seondly exemely long oheen inegaion peiods of appoximaely 1 o seonds using full daa-wiping should be employed. Thidly, he mehod of non-oheenly ombining he oupu fom all he available saellies should be implemened. Sine hee ae geneally a leas 6 saellies visible and someimes as many as 1, his would pobably inease he maximum ange by a fuhe fao of 1.5 o 3, depending on he squaing loss. GS signal anellaion ehniques need o be developed in ode of miigae agains he nea-fa poblem. Use of he sysem ould also be limied o deeion of lage ages (wih lage CS) heeby also ineasing he deeion ange, alhough if a suffiien numbe of saellies ae pesen hen some may have a moe favouable geomey han ohes esuling in a bee CS in hese iumsanes. In onlusion, i has been shown ha implemening a GS-based bisai ada fo he puposes of age deeion is signifianly onsained by he available powe budge. These onsains explain why he paen (Tsui and Shaw, 1993) and ealy-published papes (Koh and Wesphal, 1995) do no appea o have esuled in any follow-on wok. Howeve, i is possible ha use of he suggesions oulined above and he availabiliy and use of new GNSS signals ould pemi onsuion of a wokable sysem, albei one wih limied maximum ange. Appendix A: Maximum GS Sensiiviy To esimae he maximum sensiiviy of a GS eeive i is fis neessay o esimae he noise figue of he F fon-end (Figue 4) using he Fiss fomula (Van Dieendonk, 1996). Table 3 shows he esuling noise figue and effeive empeaue a eah sage of he poess wih ypial values fo he gain, loss and noise figues. The end esul is a oal noise figue NF T of.4 db and effeive empeaue T eff of 344 K assuming an anenna soue empeaue T A of 13 K. Using he effeive empeaue of he fon-end (FE) and assuming availabiliy of he full GS navigaion message o enable daa wiping on he deeed signal and hene enabling long oheen inegaion, he abiliy o dee vaious signal levels an be esablished. Table 4 (Van Diggelen, 1) shows ha wih a vey weak inpu signal of -16 dbm a he anenna oupu and using a 1 seond
Glennon e al.: Feasibiliy of Ai Tage Deeion Using GS as a Bisai ada 15 oheen inegaion peiod wih full daa wipe, he oupu signal o noise aio is 9.3 db and heefoe a he limi of deeabiliy. T a Loss L 1 G 1 NF 1 Loss L G NF Fig. 4 F Fon-End Table 3. Fon-End Noise Figue F NF 3 bi digial L1 G1 La Lb G FE Gain (db).1 19. 6. 3. 19. Gain.98 79.4.3.5 79.4 Toal Gain.98 77.6 19.5 9.8 776 NF (db).1 1.9 6. 3. 1.9 9. F 1 NF (db) / 1 1. 1.6 4.. 1.6 7.9 Toal F T 1. 1.6 1.6 1.7 1.7 1.7 Toal NF T (db).1..1.3.4.4 T eff T A (F T -1) T 137 3 311 36 34 344 aamee Table 4. Maximum GS Sensiiviy Value Noes Anenna O (dbm) -16. Signal a Anenna Oupu Anenna O (dbhz) 13.3 AnO 31 log 1 (kt eff ) IF Bandwidh (MHz).4 Typial F FE is.4 MHz wihin 3dB. T eff (K) 344.4 FE effeive empeaue Noise powe (dbm) -19.43 1 log 1 (k T eff BW) 3 IF SN (db) -5.57 An O - Noise powe Inpu Bandwidh (MHz).46 Coheen eiod (ms) 1. Coheen inegaion Oupu Bandwidh (Hz) 1. 1/Coheen Inegaion Coheen Gain 143.4 (Inpu BW/Oupu BW) Coheen Gain (db) 63.1 log 1 (Coheen Gain). Misuning Loss (db). Coelaion&Misuning loss Quanizaion Loss (db) 1.5 -bi quanizaion Loss Aual Gain (db) 59.9 Coheen Gain Losses Final SN (db) 9.3 IF SN Aual Gain Final SN ( aio).91 eak/sigma aio 1 (db/) Appendix B: Clue owe Densiy Esimaion The lue powe densiy affeing a age a ange an be esimaed by inegaing he powe efleed fom he gound beween he isoange onous, saing fom and ending a d wih a fee-spae loss-fao weighing of 1/( ). This isoange onou an be obained fo a given mulipah delay using he elaionship (Solk and Bown, 3) k i os β i whee is he ange veo fom he eeive o he age (lue), k i is he uni line-of-sigh veo o he saellie and β i is he (bisai) angle beween and k i. Assuming a fla eah model wih he eeive a he oigin, he mulipah delay an be wien as: ( os( e)os( a) os( e)sin( a) sin( e) ) ( osθ sin θ ) ( x y ) ki osθos( e)os( a) sin θos( e)sin( a) e a ( 1 os( )os( θ )) whee e is he saellie elevaion, a is he saellie azimuh and and θ ae he ange and angle fom he eeiving anenna o he lue espeively. Hene he isoange lue onou Iso( ) fo a age a a ange ous a a lue ange is defined by: ( ) ( 1 os( e) os( a) ) Iso θ A lue elemen a onibues ds C o he oal lue powe densiy, wheeσ is he lue CS pe uni aea. ds S die σ C d dθ Applying a hange of vaiables fom lue ange o age ange and hen inegaing ds ove he full θ ange and a age ange wih he appliable ange esoluion yields he oal lue powe densiy as equied: S C π Iso( Iso( ) Sdieσ S S d ) π d d σ π d d θ σ d log 1 die die Sdie σ 4 d dθ π ( 1 os( e)os( θ a) ) ( 1 os( e)os( θ a) ) d dθ Noe ha he oal lue powe densiy is independen of saellie posiion despie he fa ha his is no he ase fo he isoange onous. A oss hek fo he ase whee he bisai illuminao is diely ovehead an be pefomed sine in his ase he isoange onous ae
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