Doppler Collision and cross correlation interference in GPS receivers

Transcription

1 International Global Navigation Satellite Systems Soiety IGNSS Symposium 29 Holiday Inn Surfers Paradise, Qld, Australia 1 3 Deember, 29 Doppler Collision and ross orrelation interferene in GPS reeivers Asghar Tabatabaei Balaei University of New South Wales, Sydney, Australia Ph: , Fax: , asghart@unsw.edu.au Dennis M Akos University of Colorado Boulder Ph: (33) dma@olorado.edu ABSTRACT Global Positioning System (GPS) reeivers use Code Division Multiple Aess (CDMA) modulation. Although this modulation is immune to interferene, ross orrelation interferene is still a threat to this system. In the situation when the differene between the Doppler frequenies of the two satellite signals is within the bandwidth of the reeiver traking loop, ross orrelation has a unique behaviour. In this situation, the two signals behave like multipath with respet to eah other. Unlike normal ross orrelation interferene problems, this does not neessarily happen in the ase of strong and weak signal. Although this situation is not too frequent in GPS satellites, it happens quite frequently in SBAS satellites. In this paper this behaviour is analytially and experimentally investigated KEYWORDS: Cross Correlation, Doppler Collision, Relative Doppler Carrier, Multipath 1. INTRODUCTION Interferene has severe impats on the quality of the positioning in GNSS reeivers. One type of interferene results from the signals themselves and is referred to as ross orrelation interferene. In this type of interferene, reeived weak satellite signals are diffiult to aquire and proess as a result of simultaneously reeived strong satellite signals. Even though CDMA modulation has been used to redue the general impat, ross orrelation is an inherent interferene whih is atually aused by this modulation sheme. Different odes are

2 not ompletely orthogonal and that brings some adverse effet in the aquisition peak searh proess (Kaplan 25; Dierendonk et al. 1999). A number of different tehniques are available for mitigating ross orrelations (Dierendonk et al. 22), with most of these tehniques having been developed for CDMA ommuniation systems. The methods an be generally separated into two ategories, namely anellation (Dierendonk et al. 22; Pedersen et al. 1996) and subspae projetion (Soong, Krzymien 1996; Madhani et al. 23; Tidestav et al. 1999) tehniques. The adverse effet of ross orrelation has to do with the non-orthogonality of the odes and with the relative Doppler arrier (RDC) of the weak and strong signals. In setion 2 of this paper this relationship is derived and in setion 3 is visually analysed. The main fous of this paper however is on the ontribution of RDC in ross orrelation effet. Speifially when the Doppler frequenies of the two satellite signals are lose or equal, the ross orrelation effet is studied first in the general ase of GPS and then in more speifi ase of SBAS satellites. 2. GPS CA CODE SIGNAL ACQUISITION In order to understand and address the ross orrelation onsequene in the GNSS reeiver, this setion derives both the autoorrelation and the ross orrelation output of the mathematial proess. For this purpose the general formulation of the signal model and a typial aquisition blok in the reeiver are utilized. 2.1 Autoorrelation formulation in GNSS signal aquisition The reeived GPS or GLONASS signal an be modelled as (Kaplan 25), s(t) = AsC(t)D(t)os[2π (f )t + ϕ ] (1) where A s is the signal amplitude, C(t) is the PRN ode modulation (±1), D(t) is data modulation (±1) at a rate of 5 bps, f = arrier frequeny, and ϕ is the arrier phase. The front end bandwidth is onsidered wide enough to pass essentially all of the signal power. The interest is in evaluating the orrelation output (see Figure 1). This evaluation will help to investigate the orrelation output whih is the result of simultaneous aquisition of weak and strong signals. PRN i ( t τ) j ft e 2π Td 2 T d 2 dt ( I + Q)(.) Figure 1 Correlation struture in the reeiver Suppose the loal arrier frequeny is fˆ, and the loal ode is C(t-τ). The in-phase and quadrature part of the autoorrelation of the satellite signal after integration and dump are (Tabatabaei Balaei et al. 25):

3 (Ia + jqa )(m) = A s/ 2 Td sin( Δ ftd )R( τ )exp(jδφm) (2) where T d is integration time, Δ f = f - fˆ, R(τ) is the autoorrelation funtion of the PRN ode, m represents the m th integration period, and Δφm is the orresponding arrier phase. Equation (2) represents the autoorrelation output. In the following setion the fous will be on alulating the ross orrelation of the loally generated signal (k th satellite) and other reeived satellite signals (i=1 to N exept k) for both GPS and GLONASS signals. 2.2 Cross orrelation formulation in GNSS signal aquisition Considering the equation Td 1 2 T d d 2 T e m j 2π [ + Δf ] t T 1 dt = T d 1 m 2π [ + Δf T m 2sin{2π [ + Δf ] T Td m ] } = Sin{[ + Δf 2 T ] T } d The ross orrelation output for the two systems will be: N ki m ki (I + jq )(m) = Td (exp( jϕ ki) Rm ( τ )sin (( + Δf ) Td ))) (3) T i= 1 Where T is PRN ode period, R m= τ j2πl ki k i T m ( τ ) = Cm lcl e, Ck t) = l= i= k i i j 2π t T ( C e (Fourier transform of the k th PRN ode), ϕ ki is the arrier phase differene between the loally ) generated arrier for the k th signal and the reeived i th ki i k satellite signal and Δ f = f f where k f ) is the reeiver estimation of the k th satellite arrier frequeny and i f is the reeived arrier frequeny of the i th satellite.. The orrelation output is a ombination of the two outputs derived in equations (2) and (3). When the reeived signal simultaneously ontains weak and strong signals, the autoorrelation peak of the weak signal not only ompetes with noise but also it needs to be distinguished from all the ross orrelation peaks as well. In (Lestarquit et al. 29), the effet of Doppler frequeny ollision investigates the ross orrelation effet between two signals of equal frequeny, with frequeny being one of the two parameters in the analysis done in this paper. In setion 3, these two parameters are visually analysed. 3. GPS CROSS CORRELATION PROBLEM Equations (2) and (3) are a generalized formulation of the autoorrelation and ross orrelation of the GNSS orrelation output. As far as the signals are onerned, the

4 autoorrelation depends on the R (τ ) whih is the orresponding ode autoorrelation. Also as far as the orrelation struture is onerned, it depends only on the integration duration time (T d ). Therefore the ase of GPS, the autoorrelation peak looks like a triangle with base of two hips period if and it looks like a sin funtion if the loal and the reeived odes are perfetly aligned. This is shown in Figure 2 in whih for the GPS signal PRN 1, the Doppler frequeny is hosen to be at 1 khz and T d =1 ms. GPS ACF 6 5 x 1 4 Correlation output ode offset (hip) freq offset (Hz) 11 Figure 2 Autoorrelation peaks for GPS 12 In the ase of GPS, in whih the signal modulation is CDMA, the magnitude of eah ross orrelation terms in (3) depends on the power level of the satellite Pi as well as the ross orrelation between the odes for satellite k (the loal PRN ode) and satellite i (the reeived odes). It will also be observed that the ross orrelation terms are modulated by a relative ki Doppler arrier (RDC = Δ f ) whih is the differene between the Doppler frequeny of the two signals. For GPS signals, the Doppler differene between two satellites an be as muh as 8 khz (see Figure 4) and several yles of RDC an our within a given integrate & dump sample period, whih for a typial GPS reeiver will be 1 ms orresponding to the C/A ode period. In Figure 3, the ross orrelation peaks are shown. In this experiment, for GPS, the strong satellite signal is hosen to be PRN 2 and its Doppler frequeny is 14 Hz higher than that of PRN 1 whih is hosen to be the weak signal. GPS CCF 4 x 1 4 Correlation output ode offset (hip) freq offset (Hz) Figure 3 Cross orrelation peaks for GPS 3. DOPPLER COLLISION EFFECT When the Doppler of two satellites are equal, it is possible to introdue a multipath-like error into the traking of the orrelation peak as a result of potential ross orrelation sidelobes. In Figure 4 the Doppler frequeny of all the visible satellites are plotted over the period of 24

5 hours. This data is olleted by Novatel reeiver on 27-April-29 in Boulder, Colorado, USA. As it an be seen, there are many oasions in whih the Doppler lines interset, thus Doppler ollision do take plae for all GPS satellites. From equation (3), it is possible to highlight a phenomenon whih happens as a result of the Doppler frequeny ollision for two satellite signals Doppler frequeny (Hz) Time (se) x 1 5 Figure 4 Doppler frequeny hange for all the visible GPS satellites in 24 hours If during the period in whih two signals have a ommon Doppler frequeny, there is a orresponding ross orrelation sidelobe, then this sidelobe impats the proessing in the same way as a multipath refletion. The reason is that in equation (3), Δf ki = and for m= equation (3) beomes similar to equation (2) whih shows the autoorrelation output. This is formulated here. ( I 21 + jq )( m) T exp( jϕ ) R ( ) (4) = d 21 τ Where R 21 ( τ ) is the ross orrelation between the two satellite signals and ϕ 21 is the arrier phase differene between the loally generated arrier for the k th signal and the reeived i th satellite signal. This is exatly similar to equation (2) with the differene that instead of the autoorrelation, here we have the ross orrelation. This term behaves exatly the same as a multipath with power equivalent to the ross orrelation. The autoorrelation funtion will be summed with any ross orrelation sideslobes whih result from the Doppler ollision. In (Tabatabaei Balaei, Akos 29) it is shown that for every interval of 1 KHz from the weak signal arrier frequeny, the ross orrelation peaks shown in Figure 3 are visible. In the next setion, the partiular ase in whih the Doppler frequeny of the two signal are lose or equal is further investigated first for the ase of GPS in general and then speifially for SBAS satellite signals. 4. EXPERIMENTS AND RESULTS Among the two parameter affeting the ross orrelation interferene was the ross orrelation between the odes of the two reeived signals. In Figure 5, the ross orrelation of

6 two PRN odes is shown. This subset plot of the ode period is representative of the ross orrelation between all GPS PRN odes. Any level of sidelobe will result in a distortion of the autoorrelation funtion, but one ould set a threshold suh that only when the value of the sidelobe is 3% or greater of its maximum value it would be of interest/suffiient degradation to the autoorrelation peak, illustrated in blue on the plot. Examining this plot, whih again is representative of all PRN odes should that this 3% threshold is exeeded 3% of the time. Thus although the potential impat an be determined by the differential range at the time of the Doppler ollision, these approximations provide insight into the general problem for all GPS satellites. The impat of this ross orrelation effet an again be paralleled to multipath. When the signals are of equal power, the error whih results from this ross orrelation is presented infigure 8. It is important to reognize that if the signals are not at equal power levels, the ross orrelation impat on traking error an be even more severe Normalized ross orrelation Time (hips) Figure 5 Cross orrelation between two PRNs More interestingly, and the fous of (Lestarquit et al. 29), was the ross orrelation impat result on SBAS ranging. The reason is simple: Doppler ollisions take plae for SBAS satellite muh more frequently given their apparent stationary nature. Leveraging the earlier derived formulas this phenomenon is further studied for this speial ase of SBAS satellites using olleted data/analysis, extending that in (Lestarquit et al. 29). In the US urrently, there are two SBAS satellites, PRNs 135 and 138, visible to most of the ountry (see FAA referene). The geostationary satellites tend to have Doppler ollisions a majority of the time (Figure 6). Thus when used for ranging the ross orrelation sidelobes, when present, an introdue a multipath like error (equation (4)). Negleting the speifi Doppler frequenies for now, it is possible to examine the differential range to the two SBAS satellites. Although they are geostationary satellites, there still exists onsiderable movement, on the order of 5-15 km, relative to a stationary observer in the US. Sine these are used for ranging it is possible to extrat the differential range between two satellites and ompute the sidelobes from the ross orrelation funtion assuming a frequeny offset of zero.

7 Observed Doppler Freq (Hz) Colorado University PRN138 PRN GPS TOW (seond) x 1 5 Stanford University Observed Doppler Freq (Hz) 5-5 PRN138 PRN x 1 5 GPS TOW (seond) Figure 6 Doppler frequeny of the two PRNs in Colorado (CO) and Stanford (CA) University Live signals have been olleted for 24 hours using a NovAtel OEM5 reeiver on 2 th of April 29 in Boulder, Colorado. In Figure 7, range differene of the two SBAS satellites is shown. This range differene varies between 62 km to 66 km. Considering that every milliseond orresponds to 3 km, then it is possible to utilize this differential range to exatly determine the ross orrelation of the two PRN odes. This is also shown in Figure 7. Range differene {meter) -6.2 x time (seond) x ross orrelation time (seond) x 1 4 Figure 7 Range differene and orresponding ross orrelation between two SBAS satellite signals in the period of 24 hours

8 In the nominal situation where the Doppler differene between the two satellites is zero, we should be able to see the effet of all of the ross orrelation peaks in Figure 7 on the pseudoranges of the two satellites. In reality however, this is not the ase. As shown in Figure 6, the Doppler frequenies of the two satellites are not equal during the whole period of the data olletion. They do ollide around 15 seond into the data and also around 6 seond into the data. However, the Doppler frequeny differene between these two satellites is quite small, atually within the Delay Lok Loop (DLL) bandwidth, thus the ross orrelation effet should be observable over the entire 24 hour period..3 ranging error (hips) EL=.1 EL=.2 EL=.3 EL=.1 EL=.2 EL= multipath delay (hips) Figure 8 Ranging error in the presene of multipath with different delay and different earlylate spaing The effet of SBAS satellite ross orrelation has to be observed onsidering both ross orrelation pattern and also the Doppler ollision window between the two signals. However, reognize that there are two other parameters whih play an important role given this is an examination into the impat on traking performane. These two additional parameters are: (1) the loop bandwidth of the DLL inside the reeiver; and (2) the early-late spaing of the disriminator in the DLL. First, given the integration period and equation (3), it is possible to detet ross orrelation energy at non zero frequeny offsets as the ross orrelation power rolls off as a sin funtion of frequeny. If Doppler ollision frequeny is exat, then narrowing of the DLL bandwidth an minimize this impat. However, even if the frequenies are merely lose, within the bandwidth of the DLL, then the impat will be seen. Further, if it is outside that bandwidth, but within the integration period, then there an still be a multipath like effet where the non-zero average of the error urves in Figure 8 would show the resulting impat. In Figure 8, the ranging error is shown in the presene of multipath. This error depends on the multipath delay and early late spaing in the DLL. It is important to note that this multipath error envelope is shown for the ase where the two signals are reeived in equal power and the ross orrelation is at its peak. If for any reason one signal is reeived stronger than the other signal, then the multipath effet will be even more severe. It is also worth noting that for the ase when RDC= then the range error is as shown in Figure 8. If RDC is not zero, then the upper side and the lower side of the envelope average out in time but due to the speifi shape of the multipath error envelopes, this average is not equal to zero (Phelts 21). This is the reason why there is still some range error in Figure 9 and Figure 1 even at times when the RDC is not zero. Also from Figure 8 it is possible to see that the range error is zero if the multipath delay is more than 1.5 hips. In the ase of WAAS satellite signals, as shown in Figure 7 the multipath envelope repeats as there are ross orrelation all aross the period of 24 hours. It is due to the Doppler ollision, that the ross orrelation between the two signals plays like

9 multipath for both signals. In other word it has similar effet on the two psuedoranges over the ourse of 24 hours of olleting data. The reason for this similarity is that if x and y are the two signals and is ross orrelation between x and y then we have. On 15-May-29 two parallel data olletion were done in Boulder, CO, USA and Stanford, CA, USA. to investigate this phenomena in more detail. Of interest are the observed Doppler frequeny of the two visible SBAS satellites, whih is shown to be quite similar, within the expeted bandwidth of the DLL ode traking loop in the Novatel GPS reeiver. Also reognize that the Novatel GPS reeiver utilizes a 1 hip orrelator spaing for the SBAS traking (onformed in orrespondene with Novatel engineers). Next it is possible to ompute the differential range between the two SBAS satellites at eah loation and this is shown in the seond subplot in both Figure 9 and Figure 1. This an be used to determine the ross orrelation sidelobes between these two speifi PRN odes. These ross orrelation sidelobes are shown in the third subplot. Finally, in an attempt to assess if these periods of Doppler ollision and the presene of ross orrelation sidelobes do indeed impat the ode traking auray, a plot of the differential of the lok orreted pseudorange, or range, on a sample by sample basis is shown. Sine the reeived signal power level of the SBAS satellite does not hange, as it remains fixed in the same part of the antenna pattern, one would expet this to have a onstant value. However, this is not the ase as is shown in the forth subplot in Figure 9 and Figure 1. It is lear that the noise on the measurement inreases during periods of the ross orrelation sidelobes as is shown in these plots.

10 Observed Doppler Freq (Hz) 1-1 Colorado University PRN138 PRN GPS TOW x 1 5 prange differne (m) time (seond) x 1 4 ross orrelation differene of prange (m) time (seond) x x time (seond) x 1 4 Figure 9 Cross orrelation effet on the pseudorange aused by Doppler frequeny ollision Colorado University As shown in Figure 9 and Figure 1, the ross orrelation patterns for the two ases are totally different as expeted. This is due to the fat that the pseudoranges are different at the two points. However, the orrespondene between the ross orrelation peaks and the pseudorange differene for eah ase is easily observable.

11 Observed Doppler Freq (Hz) 5-5 Stanford University PRN138 PRN GPS TOW x 1 5 prange differne (m) ross orrelation differene of prange (m) time (seond) x time (seond) x x time (seond) x 1 4 Figure 1 Cross orrelation effet on the pseudorange aused by Doppler frequeny ollision Stanford University One point whih is worthy of note here is that there are two peaks in the pseudorange differene for eah peak in the ode ross orrelation. To explain this phenomenon it is important to note that multipath error has both bias and noise nature. The bias nature is for the ases where the ross orrelation is fixed and does not hange. The noise nature on the other hand omes from the hanging in ross orrelation. To demonstrate this in Figure 11, the differential of pseudorange and the differential of ross orrelation is shown simultaneously. It is possible to see that the pseudorange differene inreases with the inrease in the ross orrelation differene and this explains the reason why there are two peaks in pseudorange differene for eah peak in the ross orrelation.

12 Overlay Doppler Collision Problem Differential Range to PRN Differential of Cross Correlation Sidelobes Sample Number x 1 4 Figure 11 Differential of ross orrelation sidelobes and differential of pseudorange for SBAS satellite PRN 138 Using this haraterization and noting the fat that the ross orrelation effet in this ase behaves like multipath, and also onsidering the fat that arrier smoothing helps signifiantly in averaging the multipath effet, it is possible to solve this problem. Also sine the range differential is known as well as the ross orrelation sidelobes, it is even possible to predit and anel this ross orrelation effet. 5. CONCLUSIONS A general formula is derived for ross orrelation output in the aquisition stage of the reeiver. Using the generalized formula, also the speial ase of ross orrelation interferene and Doppler frequeny ollision has been studied. This phenomenon speifially is demonstrated for the ase of SBAS satellite signals. It is shown why this effet has a multipath harater. These analyses have been validated using real experimental data. 6. REFERENCES Behrens R. T, Sharf L. L, "Signal proessing appliations of oblique projetion operators," Signal Proessing, IEEE Transations on [see also Aoustis, Speeh, and Signal Proessing, IEEE Transations on], vol. 42, pp , FAA Glennon, EP and Dempster, AG. (25), A novel ross orrelation mitigation tehnique. 18th Int. Teh. Meeting of the Satellite Division of the U.S. Institute of Navigation, Long Beah, California, September, Jade-Morton Y, Tsui Y. B. J, Lin D. M., Liou L. L, Miller M, Zhou Q, Frenh M, Shamus J, "Assessment and Handling of CA Code Self Interferene during Weak GPS Signal Aquisition," presented at Proeedings of the 16th International Tehnial Meeting of the Satellite Division of the U.S. Inst. Of Navigation, Portland, OR, 23.

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