Characterization of GPS Carrier Phase Multipath

Transcription

1 Characterzaton of GPS Carrer Phase Multpath J.K. Ray M.E. Cannon Department of Geomatcs Engneerng, Unversty of Calgary, Alberta, Canada BIOGRAPHIES Jayanta Kumar Ray s a Ph.D. student n Geomatcs Engneerng at the Unversty of Calgary. He receved a B.E. and M.Tech. n Electroncs Engneerng from the Bangalore Unversty and Indan Insttute of Scence, Inda, respectvely. He has been nvolved n GPS research snce 199 n the area of GPS recever hardware and software development, the ntegraton of GPS wth low cost sensors and the mtgaton of multpath. M. Elzabeth Cannon s Professor n Geomatcs Engneerng at the Unversty of Calgary. She has been nvolved n GPS research and development snce 1984, and has worked extensvely on the ntegraton of GPS and nertal navgaton systems for varous applcatons. Dr. Cannon s a Past Presdent of the ION. ABSTRACT GPS carrer phase measurements are affected by multpath sgnals that can sgnfcantly affect the qualty of data used for statc and knematc postonng applcatons. It s generally dffcult to characterze ths multpath usng feld data whereby the exact sources of the errors cannot be easly solated. In ths paper, carrer phase multpath parameters are dentfed and ther nfluences on measurements are nvestgated through a theoretcal analyss. Multpath effects are analyzed from a geometrcal perspectve whereby GPS sgnals are assumed to propagate through the advancement of a plane wavefront on whch the phase of the GPS sgnals s the same. The dfferental path delay of the reflected sgnal wth respect to the drect sgnal s then calculated. A multpath smulaton model s developed and descrbed wheren the multpath parameters can be vared and ther nfluences observed. These parameters nclude ) the reflecton coeffcent, ) the antenna to reflector dstance, ) the azmuth and elevaton of the reflected sgnal v) the exstence of multple reflectors, and v) satellte dynamcs. Mulpath characterstcs such as frequency, error envelope, spatal correlaton, and sgnal to nose rato are studed by varyng these parameters. Only the specular component s addressed n ths paper as the dffused component s random n nature and dffcult to model n a determnstc form. INTRODUCTION GPS carrer phase multpath errors have assumed mportance due to the hgh accuracy demands n a number of applcatons. The dfferental carrer phase measurement s nvarably used n hgh precson applcatons lke statc and knematc surveyng and atttude determnaton (e.g. Axalrad et al., 1994). Most of the errors n the carrer phase measurements, such as atmospherc delays, orbtal and clock errors are spatally correlated and generally cancel through the dfferencng process for short baselnes. However, carrer phase multpath s a hghly localzed error whch does not cancel through dfferencng and therefore has been dentfed as the maor source of error n many hgh precson applcatons (Braasch, 1996). Several technques have been developed to counter carrer phase multpath ether usng mproved recever hardware (Townsend et al., 199; Garn and Rousseau, 1997) or data processng technques (Axelrad et al., 1994; Moelker, 1997; Ray et al., 1998). Townsend et al. (199) used the MEDLL technque whereas Garn and Rousseau (1997) used the Advanced Strobe Correlator to reduce the effect of carrer phase multpath. Axalrad et al. (1994) and Sleewaegen (1997) have exploted the sgnal-to-nose (SNR) nformaton from the recever n post msson, along wth the antenna gan patterns, to estmate multpath. Moelker (1997) used a drecton fndng algorthm (e.g. MUSIC) wth a MEDLL recever to counter multpath. Ray et al. (1998) used the spatal Presented at ION NTM-99, San Dego, January -7 1

2 correlaton of the multpath error between multple closely-spaced antennas that can be used n real tme at a reference staton. For successful solaton and mtgaton of ths error, t s mportant to understand the cause of the multpath and ts characterstcs. Efforts were made by several researchers to characterze code multpath effects (Hagerman, 1973; van Nee, 199; Braasch, 1996). Analyss of carrer lock loop behavor n the presence of multpath s gven n Braasch (1996) and van Nee (199). Georgadou and Kleusberg (1988) have presented some of the general characterstcs of carrer phase multpath and ts dependences on GPS L1 and L frequences. However, a comprehensve analyss on the overall behavor of carrer phase multpath under dfferent crcumstances has not been done. Ths paper dentfes parameters of concern for carrer phase multpath and analyzes ther mpact from a geometrcal perspectve through smulaton models. MULTPATH FROM A GEOMETRICAL PERSPECTIVE In Fgure 1, a typcal multpath scenaro s shown whereby A1 to A are several antennas placed close-by n a mult-antenna system and the reflecton from two sources to A1 s shown. The other four antennas wll also be affected by the reflected sgnals n a smlar way. In the dagram, θ and ϕ are the elevaton and azmuth of the drect sgnal to the antenna whle θ k and ϕ k are the elevaton and azmuth of the kth reflected sgnal to the antenna. The dstance between the antenna and the reflector n the horzontal plane s denoted by d k. Two dstnct scenaros are shown n the fgure. In the frst case (Reflector 1), the antenna (A1) s closer to the satellte compared to the reflector whereas n the second case (Reflector ), the reflector s closer to the satellte compare to the antenna (A1). These two cases are generalzed stuatons and representatve of all the possble scenaros of the antenna-reflector geometry. Snce the satellte s, km above the earth, the GPS sgnal can be assumed to travel as parallel rays on the earth s surface. A plane wavefront perpendcular to the lne of sght can be assumed to have the same carrer phase. When ths plane ntersects the phase center of Antenna 1, t has the same carrer phase at all ponts on t, ncludng pont P 1 (whch s the ntersecton of the plane and the lne of sght from Reflector 1 to the satellte). Therefore, the dfferental path delay of ths reflected sgnal wth respect to the drect sgnal s P 1 R 1 + R 1 O. The correspondng dfferental phase delay s computed by dvdng the dfferental path delay by the sgnal wavelength. Smlarly, for case, a plane perpendcular to the lne of sght from Reflector to the satellte ntersects the lne of sght from the antenna under consderaton at pont P. In ths case, the dfferental path delay s gven by R O P O. Satellte sgnal Z L Therefore, f the drect sgnal phase at the antenna s avalable, the reflected sgnal phase can be computed by addng the dfferental phase delay due to the dfferental path delay, to the drect sgnal phase. Plane wavefront perpendcular to the satellte sgnal P A A P 1 (Intersecton between the wavefront and the lne of sght) R1 (Pont of reflecton) θ A 1 θ 1 Reflected sgnal Reflector R d θ A 4 ϕ ϕ O ϕ 1 A 3 d 1 1 Reflector1 X Y Fgure 1: Drect and reflected sgnals to an antenna n a mult-antenna system Presented at ION NTM-99, San Dego, January -7

3 In order to compute the effect of multpath on the drect sgnal, the above mentoned dfferental path delays need to be determned mathematcally. By usng sold geometry t can be shown that the dfferental path delay n ether stuatons s gven by, 1 a = d tan θk sn θ cosθcos( ϕ ϕk ). (1) cosθk In general, for several satelltes n a mult-antenna system, where, 1 tan( θ k ) sn( θ ) a = cos( θ ) k d k k () cos( θ ) cos( ϕ ϕ k ) k a d θ ϕ s the satellte ndex s the antenna ndex s the reflected sgnal ndex; k = ndcates the drect sgnal s the dfferental path delay of the reflected sgnal (m) s the horzontal dstance between the antenna and the reflector (m) s the elevaton of the drect satellte sgnal or the reflected sgnals (rad), and s the azmuth of the drect satellte sgnal or the reflected sgnals (rad). The dfferental path delay expresson s a functon of the satellte elevaton and azmuth, the reflected sgnal elevaton and azmuth and the antenna-reflector dstance n the local level horzontal plane. Ths expresson s further exploted to analyze the behavor of the carrer phase multpath error. At the antenna phase center, the GPS sgnal conssts of the drect as well as the reflected sgnals such that a recever can not dstngush between them. In the recever, multpath s characterzed by four parameters, all of whch are relatve to the drect sgnal. These parameters are ampltude, path delay, phase and phase rate (Braasch, 1996). The reflected sgnal ampltude depends upon several parameters, namely the reflecton coeffcent of the reflector materal, the ncdent angle as well as the sze of the reflector wth respect to the frst Fresnel zone (Beckmann and Spzzchno, 1963; Braasch, 1998). The dfferental path delay of the reflected sgnal depends upon several parameters ncludng the antenna reflector dstance and geometry as shown n equaton (). The other two multpath parameters essentally depend upon the dfferental path delay and can be easly computed from t. The relatve phase s obtaned drectly by dvdng the dfferental path delay by the sgnal wavelength and the phase rate s obtaned by dfferentatng the phase. Therefore, equaton () forms an mportant relatonshp that can be used to characterze the carrer phase multpath. The composte sgnal at the antenna phase center, whch conssts of the drect and reflected sgnals s then gven by, where, ω t + φ = n s + τ α (t) D (t)c (t k ) k S cos = π a (3) k k + λl s s the composte sgnal to the antenna D s the satellte data bt C s the Pseudo Random Nose (PRN) code,.e. ether C/A or P code S s the drect sgnal ampltude (v) τ s the dfferental tme delay of the reflected sgnal (s) α s the reflecton coeffcent, defned by the rato of drect sgnal ampltude to the reflected sgnal ampltude ω s the nomnal carrer frequency (Hz) t s the tme (s) φ s the ntal carrer phase (rad), and λ L s the carrer wavelength (m). For smplcty, only one of the PRN code sequences and a nose-less stuaton s assumed n equaton (3). Each of the drect and reflected sgnals conssts of a carrer modulated by the code as well as a navgaton data bt. The data bt s extracted n the recever at a later stage and s of no concern as long as the correlaton ntegraton tme n the recever trackng loops s from data bt boundary to boundary. The PRN code s despreaded n the recever by beatng the ncomng sgnal wth a local replca of the PRN code n a Delay Lock Loop. The carrer s tracked n a Phase Lock Loop, generally a Costas loop (Ward, 1996), where the ncomng sgnal s mxed wth the nphase and quadrature phase components of a local carrer generated from a Numercally Controlled Oscllator (NCO). The carrer lock loop phase dscrmnator expresson s gven by, ( τ k ) α k sn ( ψ k + γ k ) ( ) ( ) τ k α k cos ψ k + γ k n A Ψ = k = arctan (4) n A k = Presented at ION NTM-99, San Dego, January -7 3

4 where, A where, T Ψ ψ γ s the correlaton functon. For a PRN code wthout band lmtaton t s defned as, τ A( τ) = 1, τ T T () =, τ > T s the PRN code bt perod s the measured phase dfference between the composte sgnal carrer and the locally generated carrer (rad) s the true phase dfference between the drect sgnal carrer and the locally generated carrer (rad), and s the dfferental phase delay due to the dfferental path delay of the reflected sgnal and π a equal to (rad). λl In a recever, the phase measurement s generated by accumulatng the phase of the NCO output. In a bengn envronment where there are no reflected sgnals, the ncomng sgnal carrer s the same as the drect sgnal carrer. The NCO-generated local carrer locks onto the drect carrer very accurately, and as a result, the true phase dfference between the ncomng sgnal carrer and the locally generated carrer s nearly zero (actually zero mean) and the resultng phase measurement s very accurate. In the presence of multpath, the composte sgnal phase shfts from the drect sgnal phase and the NCO-generated local carrer locks onto the composte carrer phase resultng n an error n the phase measurement. Ths error s equal to the dfference between the composte sgnal carrer phase and the drect sgnal carrer phase. It can be easly seen from the equaton that when multpath s absent, α k =, for k = 1,.; then, Ψ = ψ. Under these crcumstances, the measured phase s the correct phase when assumng no phase nose. The error n the measured carrer phase s then calculated by, ψ = Ψ ψ. Usng equaton (4) t can be easly shown that, ( γ ) n A( τ ) α sn k k k k = ( ) ψ = arctan. (6) n 1+ A( τ k ) α k cos γ k k= If there s only one domnant reflector, the above equaton reduces to, A( τ 1 ) α 1 sn γ 1 ψ = arctan. (7) 1+ A( τ α γ 1 ) 1 cos 1 From equaton (7), t can be observed that the multpath error ampltude (n radans) s a functon of the PRN code correlaton functon and the reflecton coeffcent, and ndependent of the carrer wavelength. Ths means that the L1 and L carrers wll have the same ampltude of multpath error (n radans). The ampltude s also a functon of the antenna-reflector dstance through the correlaton functon. If the reflector s far away from the antenna (.e. τ s large), the correlaton value decreases and so does the multpath error ampltude. As the dstance approaches the PRN code chp, the correlaton value as well as the multpath error dmnsh. The multpath error phase s drectly related to the relatve phase of the reflected sgnal. Multpath error phase varaton s due to the varaton of the reflected sgnal relatve phase or dfferental path delay. For the same dfferental path delay, GPS L1 and L sgnals wll have dfferent relatve phase delays and correspondngly dfferent phases of the multpath error. The rate of change of the phase or frequency of the multpath error may also be determned from ths equaton. Ths s also obtaned by takng the tme dervatve of the dfferental path delay expresson. Expressng the dervatve n terms of phase rate rather than dstant rate from equaton (), we get, sn θ cos( ϕ ϕ ) 1 δθ δγ πd 1 1 δ cosθ θ t = tan 1. (8) δt λl δϕ + { cosθ sn( ϕ ϕ 1 )} δt The above expresson s obtaned under the assumpton that the antenna-reflector geometry does not change sgnfcantly over the perod under consderaton. Ths assumpton does not generally hold for knematc recevers, where the antenna-reflector geometry may change rapdly. In statonary stuatons, f the geometry changes sgnfcantly, the partal dervatves wth respect to the reflected sgnal elevaton and azmuth are to be added n the above equaton. Presented at ION NTM-99, San Dego, January -7 4

5 It s evdent from equaton (8) that the multpath error frequency s, - drectly proportonal to the dstance between the antenna and the reflector - nversely proportonal to the wavelength of the carrer sgnal - drectly proportonal to the rate of change of elevaton of the satellte - drectly proportonal to the rate of change of azmuth of the satellte, and - dependent upon the antenna-reflector and the lne-of-sght vectors. The above statements allow an analyss of the carrer phase multpath characterstcs as follows, - reflectors whch are far away from an antenna cause hgh frequency or fast-changng multpath and close-by reflectors cause low frequency or slowly changng multpath - GPS L1 and L carrers wll have the same multpath ampltude but dfferent nstantaneous phases. The L1 carrer has hgher frequency multpath compared to the L carrer - reflectors whch are far away from an antenna cause a weak multpath error compared to ther close-by counterparts, and - a low elevaton satellte s more lkely to cause carrer phase multpath (due to more potental reflectors) but requres a larger surface (due to the large Fresnel zone) for strong multpath. On the other hand, a hgh elevaton satellte s less lkely to cause carrer phase multpath but requres a smaller surface for strong multpath (Braasch, 1998). MULTIPATH AND SNR GPS sgnal power or SNR s related to the carrer phase multpath parameters. It s to be emphaszed that the code and data bt n the GPS sgnal do not contrbute to the sgnal power as they merely change the phase of the carrer dependng upon the modulaton technque employed. The carrer or sgnal power wth or wthout the data and code bts remans the same. Therefore, the recever determnes the power of the carrer, not code and data, and generally expresses t as the rato of average sgnal power to nose power spectral densty or C/N (Splker Jr, 1996). GPS sgnal power can therefore be determned from the composte sgnal n equaton (3) by gnorng the code and data bt as follows, ( ω t + φ + γ k ) n s = S α k cos. (9) k= For a sngle domnant reflector, the equaton smplfes to, 1 s = S cos( ωt + φ ) + S α cos( ω t + φ + γ ). (1) 1 1 Assumng a unform antenna gan pattern, the average receved sgnal power can be easly calculated from the above expresson and s gven by (Close, 1966), P ( 1+ ( α ) + α cos( γ )) = P (11) where, P s the average power of the drect carrer sgnal (S ) and s equal to. From equaton (11), the total sgnal power n the recever s a functon of the reflecton coeffcent and relatve phase of the reflected sgnal. The maxmum and mnmum sgnal power may be obtaned from equaton (11) as, P P ( + α ) 1 ( α ) = P max 1, and (1) = P mn 1 1. (13) Ther rato s P P max mn Therefore, = ( 1+ α 1 ) ( 1 α ) 1 1 ( C/ N ) max 1 = ( C / N ) mn = R. (14) R 1 α 1 =. (1) R + 1 Equaton (1) relates the reflecton coeffcent to the SNR, whch means that the reflecton coeffcent may be estmated from the maxmum and mnmum SNR of the GPS sgnal n the recever. These relatonshps may be used to estmate the reflecton parameters from the SNR. SIMULATION DESCRIPTION A carrer phase Multpath Smulaton and Mtgaton software (MultSM) for GPS system was developed on a PC platform. The maor nputs to the smulator are, Presented at ION NTM-99, San Dego, January -7

6 - reflector parameters, and - antenna parameters whle the maor outputs from the smulator are, - true carrer phase - measured carrer phase contamnated wth multpath and phase nose, and - estmated carrer phase. The software conssts of two man modules namely, Smulaton and Mtgaton. The frst module allows the user to defne the multpath envronment and the antenna setup through the nput parameters. The user can nput the number of reflectors per satellte and ther locatons wth respect to the reference antenna poston n order to smulate a controlled multpath envronment. The user can also confgure the antenna setup,.e. the number of antennas and ther placement. The carrer phase of the drect and reflected sgnals at each antenna may be determned by computng the dstance traveled by the sgnal up to the antenna. For the drect sgnal, t s the dstance between the satellte and the antenna; whle for the reflected sgnals t s the total dstance from the satellte to the reflector plus the reflector to the antenna. The satellte poston s determned from stored ephemers data. The measured carrer phase wthout nose contans two parts the nteger and fractonal cycle components. Assumng that the drect sgnal s stronger than the ndrect sgnal, the nteger cycles n the measured carrer phase s same as the drect sgnal s nteger cycles. The phase of the fractonal cycle of the reflected sgnal s what actually corrupts the phase of the fractonal cycle of the drect sgnal, dependng upon ts relatve strength and phase. A sngle observaton from the drect and all the reflected sgnals are generated per satellte-antenna combnaton. Gaussan nose wth selected characterstcs s added to the measurement. SIMULATION RESULTS Fgures (a) to (e) show the effect of a reflected sgnal on a drect sgnal for three dfferent stuatons. In Fgure (a), the drect sgnal modulated by the code and data s shown. There are many L1 carrer cycles wthn a code bt, and only a small fracton of t s llustrated to demonstrate the behavor. Fgure (b) s the reflected sgnal delayed by two nteger cycles. It s also 9 degrees out of phase wth respect to the drect sgnal and one half ts ampltude. Fgures (c), (d) and (e) show the composte sgnals consstng of a drect and reflected sgnal for a relatve reflected sgnal phase of 9, and 18 degrees, respectvely. It s observed that n the frst case, the composte sgnal has a phase error, but no change n ampltude. In contrast, n the second and thrd cases, the composte sgnals do not exhbt phase error, but the sgnal ampltude s ncreased and decreased respectvely. Ths wll affect the SNR or the more wdely used C/N of the carrer. For a large out of phase reflected sgnal, the recever may lose lock of the ncomng sgnal. Dr Sg Refl Sg Comp Sg Comp Sg Comp Sg Tme Tme Tme(s) Fgure (a-e): Waveforms of the drect, reflected and composte sgnals for 9, and 18 degrees relatve phase of the reflected sgnal from a reflector wth a reflecton coeffcent of.. Fgure 3(a) plots the multpath error aganst the dfferental path delay for a reflector wth a reflecton coeffcent.9. It can be seen from equaton (6) that the multpath error s a functon of the correlaton functon. As the dfferental path delay ncreases, the correlaton value of the reflected sgnal code wth the locally generated replca decreases, thereby reducng multpath error. For a reflected sgnal delayed by more than one PRN code chp, the correlaton value s zero (gnorng the correlaton sdelobes) and so s the multpath error. Therefore for C/A and P code recevers, a dfferental path delay of more than 93.6 m and 9.33 m respectvely do not contrbute to multpath errors n the carrer phase measurements. Also, t can be noted that carrer phase multpath has a zero mean. Fgure 3(b) plots the multpath error envelope aganst the dfferental path delay for varous reflecton coeffcents of the reflector. Multpath error s plotted n unts of cycle length and dstance n unts of code chps. Therefore, ths fgure s representatve for multpath on an L1 as well as L carrer n a C/A or P code recever. As expected, the multpath error envelope reduces for weaker reflected sgnals. Presented at ION NTM-99, San Dego, January -7 6

7 MP Error(Cycle) MP Error(Cycle) Fgure 3(a-b): Multpath error vs. reflected sgnal path delay for a reflecton coeffcent of.9 and multpath error envelopes for reflecton coeffcents.1,.3,.,.7 and.9. Fgures 4(a) to 4(c) demonstrate the varaton of the multpath error as a functon of satellte elevaton and azmuth for satellte 4. Fgure 4(c) s plotted usng equaton (7) for a reflector wth reflecton coeffcent of. at a dstance of m from the antenna. A nomnal phase nose of 3 mm (1σ) s added. Though n practce, t s unlkely to have reflecton from the same pont for a long perod, n ths case t serves the purpose of understandng the general behavor of multpath error over tme. In the fgure, the multpath phase rate changes substantally dependng upon the change n satellte azmuth and elevaton. Elv(deg) Az(deg) Delay(Chp)..1.1 R.C. =.9. R.C. = Delay(Chp) GPS Tme(s) Fgure 4(a-c): Multpath error varaton as a functon of elevaton and azmuth for satellte 4 due to a reflector wth reflecton coeffcent of. at a dstance of m from the antenna. Fgures (a) to (d) are generated under smlar crcumstances as n Fgure 4(c) except that the reflector s now placed m away from the antenna. In fgures (b) and (c), the reflecton coeffcent s changed to.9 and.3, respectvely. In fgure (d), the reflector s placed n a dfferent locaton but at the same dstance wth respect to the antenna. These fgures show several mportant characterstcs of multpath. It s clear from the fgures that n a weak multpath stuaton, the error tends to be snusodal whereby the maxma and the mnma are unformly spaced at 9 and 7 degrees relatve phase of the reflected sgnal. In a strong multpath stuaton however, the error tends to be an nverted sawtooth wth sharp transtons n the vcnty of 18 degrees relatve phase of the reflected sgnal. Also, the multpath phase and frequency s hghly dependent on the locaton of the reflector. In fact, a small change n locaton, n the order of several cm, may change the dfferental path delay thereby causng a change n the reflected sgnal relatve phase and multpath error. Ths makes the day to day predcton of carrer phase multpath hghly vulnerable unless the envronment s exactly the same. Furthermore, code multpath phase depends upon reflected sgnal relatve phase and that makes code multpath error day to day predcton vulnerable as well GPS Tme(s) Fgure (a-d): Multpath error for satellte 4 due to a sngle reflector wth reflecton coeffcents of.,.9,.3 and. respectvely at a dstance of m from the antenna. For (d), the reflector s at dfferent locaton. Fgures 6(a) to 6(d) dsplay multpath errors from a large reflector at varous closely-spaced antennas. It s clear from the fgure that the multpath error s hghly correlated between antennas. They have dfferent phases due to dfferent dfferental path delays of the reflected sgnal. Ther correlaton can be exploted to estmate the multpath error at ndvdual antennas from sngle dfference carrer phase measurements between antennas as demonstrated by Ray et al. (1998). Presented at ION NTM-99, San Dego, January -7 7

8 Ant1 MP(cm) Ant MP(cm) Ant3 MP(cm) Ant4 MP(cm) GPS Tme(s) Fgure 6(a-d): Multpath error at multple antennas separated by to 1 cm. Fgure 7(a) dsplays the multpath error due to an addtonal reflector compared to the set up used to generate data for Fgures (a) and 6(a) to 6(d). It demonstrates the error behavor for both L1 and L carrers for satellte 4. Fgure 7(b) shows the error behavor for satellte 16 due to the same reflectors. In these fgures, the darker shaded error corresponds to the L1 carrer. Several mportant observatons can be made from them: ) the multpath error may change substantally due to the addton or subtracton of another reflector, ) the same set of reflectors have a dfferent effect for varous satelltes dependng upon the lne-ofsght vector, antenna-reflector vector, elevaton and azmuth of the satellte, ) the multpath error has the same ampltude (n radans) for the L1 and L carrer, v) the multpath error has a dfferent phase for the L1 and L carrer. At a partcular nstant the multpath error for the L1 and L carrers look arbtrary, but over tme the error sgnals have the same shape. The multpath error dependency on frequency s exploted by Georgadou and Kleusberg (1988). Fgures 8(a) and 8(c) show the multpath error for satellte 4 and 16, respectvely whle Fgures 8(b) and 8(d) are ther estmated perods. Perods are estmated not from the error themselves, but from the antenna-reflector geometry usng equaton (8). Comparng the errors wth ther estmated perods, t s seen that the estmaton s approxmately correct for the entre duraton. In the fgures, estmaton s based on the known poston of the reflector, whch s not avalable n practcal applcatons. However, ths relatonshp may be used to estmate the multpath error from the avalable measurements. For example, one can assume the reflector poston and reflecton coeffcent to be the unknown state varables and then estmate them usng measurements from the closely-spaced antennas. SV4 MP(cm) SV16 MP(cm) GPS Tme(s) Fgure 7(a-b): L1 and L multpath errors due to two reflectors for satelltes 4 and 16, respectvely. SV4 MP(cm) - 1 Perod(s) SV16 MP(cm) Perod(s) - L1 MP error L MP error L MP error L1 MP error GPS Tme(s) Fgure 8(a-d): Multpath errors and ther estmated perods for satelltes 4 and 16. Fgure 9(a) shows the multpath error for the same setup used to generate the data for Fugure 8(a) except that the reflecton coeffcent s changed to.3. Fgure 9(b) s the rato of the average composte sgnal power to the nose spectral densty generated usng equaton (11). Smlarly Fgures 9(c) and 9(d) are the multapth error and the SNR due to a reflector wth reflecton coeffcent.9. The nomnal value of C/N due to the drect sgnal alone s 4 db-hz. It s clear from the plots that multpath error and SNR have dstnct relatonshps. Ths s because the composte sgnal power and multpath error depend upon the relatve phase of the reflected sgnal. For a small reflecton coeffcent, the multpath error s at an absolute maxmum at 9 and 7 degrees and s zero at and 18 degrees of the reflected sgnal relatve phase (Fgure 9(a)). However, for a hgh reflecton coeffcent, multpath error s at absolute maxmum value at the vcnty of 18 degrees of the reflected sgnal relatve Presented at ION NTM-99, San Dego, January -7 8

9 phase (Fgure 9(c)). In contrast, the carrer power s maxmum and mnmum at and 18 degrees respectvely of the reflected sgnal relatve phase for all reflecton coeffcents. Also from fgure 9(b), the maxmum and mnmum power s approxmately 49. and 39 db-hz. Usng equatons (14) and (1) the estmated reflecton coeffcent s approxmately.3, whch s correct. The relatonshp between the SNR and multpath error s exploted by researchers to estmate carrer phase multpath (Axelrad et al., 1994; Sleewaegen, 1997). C/No(dB-Hz) C/No(dB-Hz) Fgure 9(a-d): Multpath error and SNR for satellte 4 due to a reflector wth reflecton coeffcent of.3 and.9 at a dstance of m from the antenna. CONCLUSIONS GPS Tme(s) GPS Tme(s) Carrer phase multpath s a maor source of error for hgh accuracy dfferental carrer phase postonng. Effectve multpath mtgaton technques or multpath avodance requres a sound understandng of ts characterstcs. In ths paper, varous parameters of the carrer phase multpath are analyzed from theoretcal and smulaton models. The problem s approached from a geometrcal perspectve and explots the antenna-reflector geometry to characterze multpath. Maor fndngs of ths work are the computaton of the varous relatonshps between such parameters as the multpath ampltude, phase and frequency wth the satellte dynamcs, antenna-reflector dstance, antennareflector geometry, sgnal frequency, and SNR. The analyss s also extended for multple reflectors and multple closely-spaced antennas. Ths analyss may be further extended usng mage theory of electromagnetc sgnals. The extent of the change n sgnal polarzaton due to reflecton, and ts effect on varous antennas, requres further research. A comprehensve comparson of code multpath and carrer phase multpath s yet to be compled. Furthermore, smulatons need to be substantated through real data analyss to confrm the fndngs. REFERENCES Axelrad, P., C. Comp, and P. MacDoran, (1994), Use of Sgnal-To-Nose Rato for Multpath Error Correcton n GPS Dfferental Phase Measurements: Methodology and Expermental Results, Proceedngs of ION GPS-94, Salt Lake Cty, September -3, pp Backer, D., K.H.Thel, and P.Hartl (1994), A specal Method of Managng Multpath Effects, Proceedngs of ION GPS-94, Salt Lake Cty, September -3, pp Beckmann, P. and A. Spzzchno (1963), The scatterng of Electromagnetc Waves from Rough Surfaces, Pergamon Press, 3 pp. Braasch, M.S. and F. van Graas (1991), Gudance Accuracy Consderatons for Realtme GPS Interferometry, Proceedngs of ION GPS-91, Albuquerque, September 9-13, pp Braasch, M.S. (1996), Multpath Effects, Global Postonng Systems: Theory and Applcatons, Amercan Insttute of Aeronautcs and Astronautcs, Vol. 1, Chapter 14, pp Braasch M. S. (1998), Vewgraphs of the lecture presented at the Unversty of Calgary, July 8, 73 pp. Breeuwer, E. (199), Modelng and Measurng GPS Multpath Effects, Master s Thess, Faculty of Electrcal Engneerng, Delft Unversty of Technology, Delft, The Netherlands, January pp. Cannon, M.E. and G. Lachapelle (199), Analyss of a Hgh-Performance C/A-Code GPS Recever n Knematc Mode, NAVIGATION: Journal of The Insttute of Navgaton, Vol. 39, No. 3, Fall, pp Cannon, M.E. and J. Sh (199), Precse Arborne Carrer Phase-based GPS Postonng wth a Mult- Recever Confguraton: Data Processng and Accuracy Evaluaton, Canadan Aeronautcs and Space Journal, Vol. 41, No. 1, March, pp Presented at ION NTM-99, San Dego, January -7 9

10 Close, C.M. (1966), The Analyss of Lnear Crcuts, Harcourt, Brace & World, Inc., 716 pp. Dxon, R.C. (1976), Spread Spectrum Technques, IEEE Press, New York, 4 pp. Garn L and J. Rousseau (1997), Enhanced Strobe Correlator Multpath Reecton for Code & Carrer, Proceedngs of ION GPS-97, September 16-19, Kansas Cty, pp GP1 Data Sheet (1996), GPS 1 Channel Correlator wth Mcroprocessor Support Functons, GEC Plessey Semconductors, July, 64 pp. Georgadou, Y and A. Kleusberg (1988), On Carrer Sgnal Multpath Effects n relatve GPS Postonng, manuscrpta geodatca, Sprnger- Verlag, Vol. 13, No. 3, pp Hagerman, L.L. (1973), Effects of Multpath on Coherent and Non-coherent PRN Rangng Recever, Aerospace Report No. TOR-73 (3 3) 3, Development Plannng Dvson, The Aerospace Corporaton, 39 pp. Jordan, E.C. and K.G. Balman (197), Electromagnetc Waves and Radatng Systems, Second Edton, Prentce-Hall, Inc. 73 pp. Lachapelle, G., W. Falkenberg, D. Neufeldt and P. Kelland (1989), Marne DGPS Usng Code and Carrer n Multpath Envronment, Proceedngs of ION GPS-89, Colorado Sprngs, September 7-9, pp Splker Jr., J. J. (198), GPS Sgnal Structure and Performance Characterstcs, Global Postonng Systems (Red book), The Insttute of Navgaton, Vol. 1, pp Splker Jr., J. J. (1996), GPS Sgnal Structure and Theoretcal Performance, Global Postonng Systems: Theory and Applcatons, Amercan Insttute of Aeronautcs and Astronautcs, Vol. 1, Chapter 3, pp Townsend, B., P. Fenton, K.van Derendonck and R.D.J. van Nee (199), L1 Carrer Phase Multpath Error Reducton Usng MEDLL Technology, Proceedngs of ION GPS-9, Palm Sprng, September 1-1, pp Tranqulla, J. and J. Carr (199-91), GPS Multpath Feld Observatons at Land and Water Stes, NAVIGATION: Journal of the Insttute of Navgaton, Vol. 37, No. 4, Wnter, pp van Nee, R.D.J. (199), Multpath and Mult-Transmtter Interference n Spread-Spectrum Communcaton and Navgaton Systems, Delft Unversty Press, Delft, The Netherlands, 8 pp. Ward P. (1996), Satellte Sgnal Acquston and Trackng, Understandng GPS Prncples and Applcatons, Artech House, Chapter, pp Well, L.R. (1997), Conquerng Multpath: The GPS Accuracy Battle, Innovaton, GPS World, Vol. 8, No. 4, Aprl 1997, pp Lachapelle, G. (1997), Lecture notes of GPS Theory and applcatons, The Unversty of Calgary, Fall 1997, 444 pp. Moelker, D. (1997), Multple Antennas for Advanced GNSS Multpath Mtgaton and Multpath Drecton Fndng, Proceedngs of ION GPS-97, September 16-19, Kansas Cty, pp Ray, J.K., M.E. Cannon and P. Fenton (1998), Mtgaton of Statc Carrer Phase Multpath Effects Usng Multple Closely-Spaced Antennas, Proceedngs of ION GPS-98, Nashvlle, September 1-18 (n press). Sleewaegen, J. (1997), Multpath Mtgaton, Benefts from usng the Sgnal-to-Nose Rato, Proceedngs of ION GPS-97, Kansas Cty, September 16-19, pp Presented at ION NTM-99, San Dego, January -7 1