Group Delay Compensation in AltBOC Receivers to Mitigate the Effect of Frequency Selective Propagation Delay Distortions

Transcription

1 Group Delay Compensaion in AlBOC Receivers o Miigae he Effec of Frequency Selecive Propagaion Delay Disorions Nagaraj C Shivaramaiah (Suden Member) and Andrew G Dempser (Senior Member) School of Surveying and Spaial Informaion Sysems, Universiy of New Souh Wales Sydney NSW 5 Ausralia {nagaraj,a.dempser}@unsw.edu.au Absrac This paper exends he Sub-carrier Phase Combinaion echnique previously presened by he auhors o miigae he effec of frequency selecive Radio Frequency (RF) propagaion delay disorions in Alernae Binary Offse (AlBOC) receivers. Two major propagaion delay disorion sources are considered: he ionosphere and mulipah. Iniial analysis shows ha he ionospheric delay errors can be reduced o being comparable o he receiver impairmens (when he error due o mulipah is low) and he maximum mulipah error (under low ionospheric disorions) can be reduced o half a meer. Under exremely high ionospheric delay disorions, he mulipah delay error envelope, wih he proposed echnique, will suffer no change in he magniude of he error excep ha he spaial characerisics are differen. The underlying idea of he proposed echnique is ha difference in he phase delays of he E5a and E5b signal racking is simply he slope of he phase response over he enire band and hence represens he group delay a he E5 cener frequency of MHz. The group delay which appears as he code phase error for he E5 signal racking, can hus be compensaed excep for he occurrence of higher order errors. I. INTRODUCTION Of he several advanages of he AlBOC signal, being able o consider he wideband signal as wo independen Quadraure Phase Shif Keying (QPSK) signals a E5a and E5b bands is he mos relevan feaure for his paper. I is discussed in [] ha AlBOC can be hough of as a frequency-diverse ransmission sysem. I is well known ha he ionospheric pah is characerized by he direcion of arrival, he wave polarizaion, he carrier frequency and he group delay. Only he las wo parameers vary for he E5a and E5b signals. The dispersion wihin he 5MHz band on eiher side of he E5 cener frequency of 9.795MHz is nearly symmeric. Even under severe condiions (or sorms) of abou m ionospheric delay, he difference due o he dispersion beween E5a and E5b signals is only.33ns []. Due o his near-symmeric naure of he ionospheric group delay dispersion, he code phase esimaes from he E5a and E5b racking loops are affeced in an opposing manner when referred o he code phase esimaes of he wideband E5 signal racking loop. The proposed echnique combines he code phase esimaes of E5a and E5b sideband racking o compensae for he error in he E5 racking loop. The effec of mulipah on he group delay has been sudied in [], [3]. Unlike he ionosphere which direcly delays he Line-Of-Sigh (LOS) signal, under mulipah condiions he non-los (NLOS) (refleced) signals are superimposed ono he LOS signal a he receiver anenna. I is shown in [] ha he phase a which he NLOS signal is superimposed is differen for E5a and E5b signals, and his difference follows a paern, and by combining he mulipah-affeced carrier phases of he E5a and E5b signals, he code phase mulipah error a E5 can be miigaed. This paper exends he work in [] o use he group delay measuremens a E5a and E5b. This paper is organized as follows. Secion II revisis he fundamenal relaion beween he group delay, phase delay and he frequency selecive delay disorions. Secion III describes he received signal model and he racking archiecure considered in his paper. Secion IV discusses he miigaion echniques. Secion V provides he simulaion and es resuls followed by conclusion in Secion VI. II. FREQUENCY SELECTIVE PROPAGATION DELAY DISTORTIONS A. Phase delay, group delay and he frequency seleciviy I is well known ha he channel hrough which he GNSS signal raverses from he saellie o he receiver acs like a filer. Le Θ(ω) be he radian phase shif experienced by each sinusoid componen of he signal when he signal passes hrough a filer (ω denoing he angular velociy). The phase delay is defined by p (ω) Θ(ω) ω i.e. he phase delay expresses he phase response as ime delay. The group delay is nohing bu he ime delay of he ampliude envelope of a sinusoid a frequency ω and is defined by () g (ω) d Θ(ω) () dω If he phase response is linear over he observaion bandwidh, he group delay a a paricular frequency ω gives he slope of he phase response. The channel for he GNSS signal in pracice is no a Linear Time Invarian (LTI) filer. In addiion, for wideband signals like Galileo E5 AlBOC(5,), he frequency seleciviy wihin he signal bandwidh plays a major role in deermining he overall channel response o he signal. The wo main

2 properies of he channel ha affec he channel response are he ionosphere and mulipah propagaion. B. Group delay and Ionospheric errors The ionosphere is a dispersive medium such ha whenever a signal passes hrough he ionosphere, i speeds-up he signal (compared o ha in he free space). This resuls in a phase advance of he signal and he amoun of ha phase advance is dicaed by he elecron densiy of he ionosphere (a ha paricular pierce poin and ime). For he sake of dealing wih causal sysems, usually he ionospheric effec is specified in erms of he propagaion delay which is given by [4] b c piono (ω) = TEC ω (3) where c is he speed of he signal in free space, TEC represens he oal elecron conen and he consanb = 4.3 4π. The effec of ionospheric dispersion in he E5 Al- BOC(5,) is sudied in [5]. If I is he delay experienced by he signal a he cener of he band (9.795 MHz), hen he phase delay and he group delay a any oher frequency is given by ( ωe5 ) piono (ω) = I (4) ω ( ωe5 ) giono (ω) = I (5) ω I is very clear from (4) and (5) ha he phase delay and he group delay ionosphere errors are proporional o he square of he raio of frequencies referenced o he cener frequency. C. Group delay and Mulipah errors The phase of he refleced signal ha ges superimposed ono he direc signal a he receiver anenna depends on he addiional pah lengh raversed by he refleced signal. Equaions for phase and group delay errors in a mulipah scenario for he single refleced signal case are derived in []. The combined phase delay is given by p = p + pmuli (6) where p is he oal phase delay of he direc or he firs signal (including he ionosphere delay) and pmuli is he error in phase delay due o he refleced signals. The phase delay mulipah error is a funcion of he phase delays of he refleced signals pmuli = F( p, p3,...) and in he single refleced signal case, i is given by pmuli = ω arcan [ Asinθ +Acosθ where θ is he composie phase given by θ = (φ φ ), φ and φ being he phases of he direc signal and he refleced signal respecively, A is he raio of he ampliude of he refleced signal o he direc signal. The combined group delay of he signal is given by ] (7) g = g + gmuli (8) Phase delay (ms) Group delay (ms) p & g vs f for differen mulipah delays, SMR = 6dB x Frequency (Hz) x 9 Figure. Phase delay and Group delay vs frequency around he E5 band for differen mulipah delays; no ionospheric errors; arbirary saellie disance of 3 km; single refleced signal wih SMR = 6dB; where g is he oal group delay of he direc or he firs signal (including he ionosphere delay) and gmuli is he error in group delay due o he refleced signals. The group delay mulipah error is a funcion of he group delays of he refleced signals gmuli = H( g, g3,...) and in he single refleced signal case, i is given by [] [ gmuli = A( g g ) A+cosθ +Acosθ +A The phase of he refleced signal a he receiving anenna depends on he ype of he reflecor, he frequency of he signal and he pah lengh. Therefore boh he phase delay and he group delay mulipah errors in (7) and (9) are a funcion of ω. For a given ype of reflecor, he frequency dependency on he complex reflecion coefficien wihin he Galileo E5 AlBOC(5,) is negligible [6]. Hence he frequency dependency on he reflecor conribuion in A and θ can be negleced for all pracical purposes. Unlike he ionospheric error, he relaionship beween he mulipah phase delay and group delay errors a any wo frequency componens is quie complicaed o visualize. θ(ω) will have a (modulo π) linear relaionship wih he frequency, bu pmuli and gmuli will experience non-linearly damped oscillaions according o (7) and (9). Fig. shows he variaions of he phase delay and he group delay versus frequency for differen mulipah delays. Observe ha he variaions follow a paern which we will furher explore laer in he paper. D. Relaion beween he phase/group delay and he phase measuremen errors in GNSS From (4) and (7) i can be inferred ha he phase delay resuls in carrier phase measuremen error. The relaion is sraighforward; if he phase delay is p radians, hen he carrier phase measuremen will incur a phase error of he same amoun. On he oher hand, he relaionship beween he group delay and he code phase measuremen error is no easy o ] (9)

3 rif() ˆ ˆ e j x () e j Complex y () y() * se5 ˆ T c y() * se5 ˆ AlBOC Ref Signal Generaor NCO y() * se5 ˆ ( n )TT d ( n )T T T c d ( n )TT Code NCO Loop Filer T T d y l T y m y m Code Loop Filer discrimin aor Code discrimin aor ˆ E5a/b band ranslaor s * s * E5a ˆ y a() E5a and E5b Code Generaor y b() E5b ˆ E5b NCO E5a NCO ( n )TT ( n )TT Loop Filer (E5b) Loop Filer (E5a) d T d T y a y b discrimin aor(e5b) discrimin aor(e5a) c Combiner ca cb Code phase measuremen Figure. The racking archiecure visualize. The opimal racking circuiry for synchronizing he spreading code involves a delay locked loop [7]. In an aemp o measure he group delay in spread specrum sysems like GNSS, [8] shows ha he iming error of an early/lae code correlaor indicaes he group delay experienced by he signal. III. THE RECEIVED SIGNAL AND THE TRACKING A. The Received Signal ARCHITECTURE The received signal along wih N- refleced signals can be expressed as r() = R{P s( giono goh ) N i= exp(jω eff (+ piono + poh ))+ P i s( giono gimuli goh ) exp(jω eff (+ piono + pimuli + poh ))} where poh and goh represen he phase delay and group delay respecively, due o all oher sources han he ionosphere and he mulipah, P denoes he received signal power, ω eff denoes he effecive frequency (including Doppler). These sources mainly include he signal ransi ime, he roposphere error, anenna induced errors and he errors in he receiver due o he RF downconverer and filer. Depending on he mehod of downconversion, r() could be eiher complex or real. However, a bandpass sampling o a moderae IF is assumed in his paper and as a resul r() is real. B. The Tracking Archiecure An innovaive archiecure for mulipah miigaion has been presened in [], [9]. This paper uilizes a similar archiecure and is shown in Fig.. Observe ha he code phase esimaes of he Wideband E5 AlBOC(5,) are provided o he code generaion modules of he sideband racking. IV. MITIGATING THE IONOSPHERIC AND MULTIPATH ERRORS A. Miigaing he Effecs of Ionosphere (in he absence of mulipah errors) Ionospheric effecs on he wideband signals are discussed in []. There exiss he effec of ionosphere dispersion wihin he useful bandwidh of 5MHz of he Galileo E5 AlBOC(5,) signal [5]. This effec is in erms of he difference in group delay and phase delay beween he frequency componens. As a resul he code and carrier phase oupus of he wideband E5 racking will be erroneous. The effec due o he dispersion can be negleced for all pracical purposes as deailed in [5]. One of he main feaures of Galileo E5 AlBOC modulaion is ha he E5a and E5b sidebands can be processed independenly of each oher. Hence, i is possible o obain hree code and carrier phase measuremens from E5a, E5b and he wideband E5 ha can be used o obain he esimaes of ionosphere errors. Figures 3 and 4 show he difference in he phase delay and he correlaion values compared o ha of a dispersionless case. The corresponding parameers a he cener frequency are used as references o obain he differences. Observe ha he difference is only up o.33 ns which shows ha we can use he combinaion of E5a and E5b measuremens o obain a good esimae of he ionospheric delay a he cener frequency. Neverheless, he qualiy of such esimae depends on he conribuion of he receiver noise. B. Miigaing he Effecs of Mulipah (in he absence of ionospheric errors) The effec of he phase delay and he group delay for differen mulipah delays and differen frequency componens around he E5 band was shown in he previous secion. I is imporan o observe he behavior of he phase and group delays a he E5a, E5b and E5 cener frequencies. Fig. 5 shows

4 x 8 x p x E5 E5a E5 E5b pe5b pe5a p 3 p E5 E5a p E5 E5b Ionospheric error a E5 (m) ge5b ge5a x 7 Figure 3. Difference in he phase delays in E5a and E5b w.r.. E5 (op); Difference of he wo curves in he op figure (boom) Mulipah delay (m) Correlaion loss (db) E5a E5b Figure 5. Difference of E5a and E5b phase and group delays for differen mulipah delays (Analyical); Single refleced signal case; A=.5; 6 4 E5 LB E5 UB E5a UB E5a LB correlaion losses (db) E5a E5b Ionospheric error a E5 (m) Group Delay (m) 4 Figure 4. Difference in he correlaion values in E5a and E5b signal componens w.r.. Ionosphere free siuaion (op); Difference of he wo curves in he op figure (boom) he difference of phase delay and group delay beween E5a and E5b frequencies. This plo is generaed using (7) and (9) I should be noed ha he equaion only involves he carrier frequencies and does no include he effecs of he spreading code. Wih he spreading code in place, he errors a larger mulipah delays are aenuaed, following he shape of he correlaion funcion. The composie phase delay and he composie group delay for a single reflecion case a he oupu of he correlaor are as follows (see appendix for he derivaion) pcomposie = p [ ] ω arcan AR(ε+δ)sinθ () R(ε)+AR(ε+δ)cosθ gcomposie = g +AR(ε+δ)( g g ) [ ] AR(ε+δ)+R(ε)cosθ R (ε)+ar(ε)r(ε+δ)cosθ +A R (ε+δ) () where R(.) is he auo-correlaion funcion of he underlying spreading code, ε is he code phase error (in chips), δ is Mulipah Delay (m) Figure 6. Envelope of he group delay error due o mulipah he pah delay difference beween he refleced signal and he direc signal (in chips). The envelope of he group delay error is shown in Fig. 6. (3) and () are ploed in Fig. 7. The effec of he correlaion shape can be observed in boh he phase delay difference and he group delay difference responses. I is shown in [] ha a combinaion of phase delays in E5a and E5b can be used o miigae he effec of code phase mulipah in E5 wideband racking. Wih he proposed archiecure, he local baseband reference signals in he cases of E5a and E5b componens are generaed no a he peak of he corresponding correlaion riangle, bu a an offse equal o he difference in group delay w.r.. he E5 (cener frequency) wideband racking. C. Group delay compensaion when boh Ionospheric and Mulipah errors are presen Previous secions discussed he ionosphere and mulipah errors wihou considering he dependency on each oher. The individual analysis provided a good insigh ino he errors.

5 pe5b pe5a ge5b ge5a x x 7 Wihou he spreading code Wih he spreading code Mulipah delay (m) Wihou he spreading code Wih he spreading code Figure 7. Difference of E5a and E5b phase and group delays for differen mulipah delays (Analyical); Single refleced signal; A=.5 g (chips) p (rad) Mulipah delay (m) E5a, m E5a, 5m E5a, m E5b, m E5b, 5m E5b, m E5, m E5, 5m E5, m Figure 8. Phase delay and group delay for E5a, E5b and E5 frequencies under mulipah condiion for differen ionospheric delay; nominal saellie disance of 3km; Single refleced signal; A=.5 Siuaions wih only ionosphere errors or only he mulipah error may occur in some applicaions. However, a more pracical siuaion is he case when boh he ionsphere and mulipah errors co-exis. In addiion, he receiver may experience range esimaion errors from sources such as roposphere errors, esimaion of he clock error (boh saellie and he receiver), saellie posiion and velociy esimaion (due o orbi parameers / ephemeris), previous epoch pseudorange esimaion error and oher secondary effecs. I is assumed in his paper ha all errors from he oher sources are frequency independen. Fig. 8 shows he phase delay and group delay for E5a, E5b and E5 frequencies vs. mulipah delay for hree ionospheric delays viz. m, 5m and m experienced a he cener frequency. The error shape due o he mulipah is no visible because he magniude of he mulipah error is small compared o he scale of he plos. Fig. 9 shows he phase delay and group delay differences a differen ionospheric delays and mulipah delays. Observe ha he difference in phase delay shows an offse depending on he ionospheric delay. The error due o mulipah is around his offse. The group delay difference also shows a similar behavior. However, he error due o mulipah is in he order comparable o he ionosphere errors for he delays shown in Fig. 9. The phase delay differences pbc = pe5b pe5 pca = pe5 pe5a pba = pe5b pe5a () provide an easy compuaion of he ionosphere-affeced phase a he E5 cener frequency due o he similar differences wih respec o boh he side bands. This is expeced as he ionospheric delay is almos symmerical around he E5 cener frequency. I should be noed ha he knowledge of ineger number of cycles is no required due o wo reasons. p diff (rad) g diff (chips) Mulipah delay (m) E5b E5a, m E5b E5a, 5m E5b E5a, m E5b E5, m E5b E5, 5m E5b E5, m E5 E5a, m E5 E5a, 5m E5 E5a, m Figure 9. Phase delay and group delay differences a differen ionospheric delays and mulipah delays ) Because he phase difference mehod experiences he oscillaions a he frequency difference (3.69 MHz for E5b-E5a and MHz for E5b-E5c or E5b-E5) here can be a maximum of 3 cycles difference beween E5a and E5b for ionospheric delays of up o m (.5 cycles in he oher wo cases). ) he availabiliy of hree phase measuremens helps isolae he ionosphere error a he cener frequency. To miigae he effecs of mulipah, a unique combinaion of he carrier phases of he E5a and E5b sidebands has been used in []. In [] he firs auhor proposed a mehod called Sideband Phase Combinaion (SCPC) mehod o miigae he code phase mulipah error produced in he E5 AlBOC(5,) racking. In he SCPC mehod, carrier phases of he E5a and E5b are appropriaely combined and he resulan follows he shape of he code phase mulipah error a he cener frequency. Using his informaion, he insananeous

6 mulipah error has been reduced by up o four imes. Referring back o Fig. 9, a wo sep approach is followed here, firs o resolve he ionospheric error and second o apply he SCPC algorihm o miigae he mulipah error. The difference in he phase delays of he E5a and E5b signal componen racking is nohing bu he slope of he phase response over he enire band and hence represens he group delay a he cener frequency E5=9.795 MHz. ) Effec of previous epoch pseudorange errors on he ionosphere and mulipah miigaion process: As menioned earlier, apar from he wo major errors under consideraion, he receiver may experience oher frequency-independen errors. As an example assume ha he pseudorange is in error. The firs consequence of his erroneous pseudorange is ha he phase and he chip shif (fracional) of he incoming signal differ from he acual values. However, his error will be common o all he hree componens E5, E5a and E5b of he signal and he mehod of obaining he difference in phase delay and he difference in group delays nullifies his common error. The second consequence is he effec of his pseudorange error on he code mulipah error. In (3) and (), he ε parameer which indicaes he error in he pseudorange alers he mulipah error characerisics. Thanks o he SCPC mehod he code delay esimaes from he E5 AlBOC(5,) racking loop are provided o he code delay esimaes of he wo sidebands. Wih his sor of aiding, all he hree componens of he signal ge he pseudorange esimae from a single source (of he previous insan) and keeping he noise characerisics undisurbed. Hence he effec of pseudorange error on he mulipah error is removed by he racking loop archiecure. ) Effec of Doppler frequency on he ionosphere and mulipah miigaion process: In moderae o high dynamics applicaions, each frequency componen of wideband signal experiences differen Doppler shifs. In he case of Galileo E5 AlBOC(5,), he Doppler observed on he E5a and E5b componens differ from ha of he wideband AlBOC racking ha experiences a Doppler corresponding o he cener frequency E5. However, aiding he E5a and E5b carrier racking loops wih he frequency esimae of he E5 AlBOC racking loop eliminaes he effec of any difference in Doppler frequency esimaion in he sideband racking loops. Secondly, he effec of he rae of change of he user dynamics on he mulipah is discussed in []. In he case of E5a and E5b, hese errors will be opposing each oher when referenced o he E5 cener frequency. Hence, he difference of E5a and E5b carrier phase measuremens is void of he range rae effecs. 3) Effec of anenna induced errors on he ionosphere and mulipah miigaion process: Due o pracical limiaions in he anenna design, some properies of he anenna depend on he frequency []; of ineres o his paper are he phase cener and he axial raio. The phase cener of he anenna varies wih frequency and boresigh angle. The axial raio of he anenna which is an indicaor of he amoun of he rejecion of a LHCP (refleced) signal varies wih he frequency and he inciden angle [3], [4]. Knowing he frequency dependan variaion will help in calibraing for ha effec. However, he variaion caused due o he inciden angle canno be calibraed apriori. A possible soluion is o employ muliple closely spaced anennas [5]. The deailed analysis of he anenna-induced errors are no addressed in his paper. V. SIMULATION RESULTS AND DISCUSSION To es he mulipah miigaion echnique in he presence of ionosphere, he GIOVE-A E5 AlBOC(5,) signal srucure is used as a reference. Two ypes of verificaion of he echnique menioned in he previous secion are performed. In he firs case, he signal wih he mulipah and he ionosphere effecs is generaed in a Malab environmen. Second, an IF signal is colleced from he GIOVE-A saellie and while doing so i is ensured ha here are no reflecors wihin a disance of 3 meers ha can cause mulipah errors. Then, ionosphere and mulipah errors are added in Malab (a he IF sage) o his signal. This signal is ermed as pseudo-real signal. The IF samples of he GIOVE-A saellie signal are colleced from he Sepenrio GeNeRx receiver which has he capabiliy o oupu 5 ms of he IF signal sampled a MHz. The bandwidh of he receiver is 55 MHz around he E5 cener frequency. The simulaed delay disorions are added o he signal from he 6h ms. Figs.-3 show he resuls of he sideband carrier phase combinaion mehod. Fig. shows he applicaion of SCPC mehod under differen mulipah delays for a simulaed signal. I can be observed ha he code minus he scaled difference of he carrier phases a E5a and E5b successfully reduces he code phase mulipah error a E5. Similar resuls are observed when he real signal is used insead of he simulaed signal as shown in Fig.. Fig. shows he performance of he group delay compensaion echnique a a paricular mulipah delay for wo ionospheric delay errors of 5 m and m. Again he reducion in he code phase error is clear from he simulaed signal. Wih he real signal, he error iself is small as shown in he op lef porion of Fig. 3. However, he group delay compensaion brings down a significan par of his error. VI. CONCLUSION AND FURTHER WORK The proposed echnique compensaes for he propagaion delay disorion wihou he need of esimaing he absolue value of eiher ionospheric or mulipah delay. In summary, he proposed group delay compensaion echnique explois he muli-frequency, wideband feaure of he AlBOC modulaion o produce an accurae range measuremen. Presence of he oher frequency-selecive channel impairmens affec he performance of he proposed mehod and have o be sudied furher. REFERENCES [] N. C. Shivaramaiah, Code phase mulipah miigaion by exploiing he frequency diversiy in galileo e5 alboc, in nd In. Tech. Meeing of he Saellie Division of he U.S. Ins. of Navigaion ION GNSS, Savannah, Georgia, Sepember 9. [] T. Ooshi, The effecs of mulipah on he measuremen of anenna ime delays, Anennas and Propagaion Magazine, IEEE, vol. 35, no. 5, pp. 8 36, Oc 993.

7 . Code phase error. Difference in carrier phases Code minus scaled cp diff.. No mulipah..7m 5.4m 8m. 5 5 Time (ms) Figure. Mulipah miigaion wih he simulaed signal a hree differen mulipah delays.6 Code phase (chips).3 Difference in cp Code phase and scaled cp diff No Mulipah.7m 5.4m 8m Time (ms) Figure. Mulipah miigaion wih he pseudo-real signal a hree differen mulipah delays

8 Code phase x 3 Difference in cp 5.5. Code minus scaled cp diff 5m iono m iono Time (ms) Figure. Mulipah miigaion wih he simulaed signal wih mulipah a 5.4m and ionospheric delays of 5m and m a E5.6 Code phase (chips).5 Difference in carrier phases Code minus scaled cp diff... 5m m Time (ms) Figure 3. Mulipah miigaion wih he pseudo-real signal wih mulipah a 5.4m and ionospheric delays of 5m and m a E5

9 [3] G. J. Bishop, J. A. Klobuchar, and P. H. Dohery, Mulipah effecs on he deerminaion of absolue ionospheric ime delay from gps signals, Radio Science, vol., no. 3, pp , 985. [4] J.-H. Won and J.-S. Lee, A noe on he group de and phase advance phenomenon associaed wih gps signal progpagaion hrough he ionosphere, Navigaion: Journal of he Insiue of Navigaion, vol. 5, pp , 5. [5] Sleewaegen, Galileo alboc receiver, in ENC GNSS, 4. [6] Seybold, Inroducion o RF Propagaion. Wiley Inerscience, 5. [7] B. Parkinson and J. Spilker Jr, Eds., Global Posiioning Sysem: Theory and Applicaions. American Insiue of Aeronauics and Asronauics, 995. [8] P. T. J. S. Ascarrunz, F.G., Group-delay errors due o coheren inerference, vol., 999, pp. 98 vol.. [9] N. C. Shivaramaiah and A. G. Dempser, Processing complexmodulaed signals involving spreading code and subcarrier in ranging sysems, Ausralian Provisional Paen , 3, 9. [] G. Gao, S. Daa-Barua, T. Waler, and P. Enge, Ionosphere effecs for wideband gnss signals, in ION NTM, 7. [] S. Nedic, On gps signal mulipah modeling in dynamic environmens, in Aerospace Conference, 9. [] D. Orban and G. Moernau, Gnss anennas: An inroducion o bandwidh, gain paern, polarizaion and all ha, GPS World, Feb 9. [3] I. Y. Sohyeun Yun, Dongpil Chang, Wideband receive anennas of sensor saions for gps/galileo saellie, Aug. 8, pp.. [4] Zhuang and Tranquilla, Effecs of mulipah and anenna on gps observables, IEE Radar Sonar and Navigaion, vol. 4, no. 5, pp , 995. [5] J. K. Ray, Miigaion of gps code and carrier phase mulipah effecs using a muli-anenna sysem, Ph.D. disseraion, Universiy of Calgary,. gmuli = AβR(ε)R(ε+δ)cos(βω) A βr (ε+δ) R (ε)+a R (ε+δ)+r(ε)ar(ε+δ)cos(βω) Subsiuing β = ( g g ) and revering back o he phase difference represenaion θ we ge. gmuli = AR(ε+δ)( g g )[AR(ε+δ)+R(ε)cosθ] R (ε)+a R (ε+δ)+r(ε)ar(ε+δ)cosθ APPENDIX Group delay error caused by mulipah The mulipah phase error observed a he oupu of he correlaor when he direc signal is affeced by a single refleced signal is given by [] pmuli = [ ] ω arcan AR(ε+δ)sinθ R(ε)+AR(ε+δ)cosθ (3) where ε is he code phase error caused by he delay locked loop, and δ is he ime difference beween he direc and he refleced signal, θ is he phase difference beween he direc and he refleced signal and A is he ampliude raio of he refleced signal o he direc signal. Using he definiion of group delay, he error due o mulipah is gmuli gmuli = d dω [ p muli ] = d [ ( )] AR(ε+δ)sinθ arcan dω R(ε)+AR(ε+δ)cosθ The phase difference θ can be wrien as θ = βω where β = δt c C, C being he speed of ligh. Using he differeniaion rules for he nesed funcions, gmuli = ( + ( AR(ε+δ) sin(βω) R(ε)+AR(ε+δ) cos(βω) d dω ( ) ). AR(ε+δ)sin(βω) R(ε)+AR(ε+δ)cos(βω) )