USE OF GPS MULTICORRELATOR RECEIVERS FOR MULTIPATH PARAMETERS ESTIMATION

Transcription

1 Rdha CHAGGARA, TeSA Chrstophe MACABIAU, ENAC Erc CHATRE, STNA USE OF GPS MULTICORRELATOR RECEIVERS FOR MULTIPATH PARAMETERS ESTIMATION ABSTRACT The performance of GPS may be degraded by many perturbatons such as jammng, the effects of the onosphere, and multpath. Many studes have been done to reduce the effect of multpath on the GPS measurements. Some of these are based on the radaton pattern of the recevng antenna, whle most of them concentrate on the characterstcs of the recever trackng loops. The advent of multcorrelator recevers wdens the range of methods that can be consdered to tackle ths problem as n []. In partcular, ths enables the characterzaton of multpath effects on the trackng loops through the analyss of the shape of the correlaton peak. The am of the proposed paper s to descrbe a least squares method to dentfy dfferent multpath parameters usng multcorrelator outputs, and to present results of the applcaton of ths technque. The paper starts wth a bref revew of the mpact of several reflected rays on the code and phase trackng loops. Then, the prncple of multcorrelator recevers s descrbed, and the partcular structure of the multcorrelator frmware of the NovAtel Mllenum recever s gven as an llustraton. After ths, the least squares technque used to estmate the relatve ampltude, relatve code and phase delays, code and phase trackng errors s presented. Results of the applcaton of ths technque on real data collected on a real recever connected to a Sprent GPS generator are shown. These results llustrate the overall good performance of the method and ts lmtatons. Fnally, a concluson s drawn on ths technque and ts possble capacty to mprove the performance of trackng loops by removng multpath components n correlator outputs.. MODEL AND IMPACT OF MULTIPATH. Multpath model The multpath phenomena are encountered n most rado propagaton. Ths phenomenon happens when the receved sgnal s a contrbuton of a drect ray and one or more other reflected rays, whch follow ndrect paths. The effect depends on the applcaton we deal wth. In the case of the Global Postonng System (GPS), mutpath deterorate the trackng qualty of both PLL and DLL nsde the GPS recever. Consequently, the system performance, especally ts range measurement accuracy may be dmnshed dramatcally, Ths measurement error caused by multpath ranges form centmeters to several meters, In the case of GPS applcatons t s dffcult to gve a statstcal model to descrbe the receved sgnal n presence of multpath. However, many hypotheses can be made. For nstance, the reflected sgnals are delayed wth respect to the drect one, as they travel a longer path. Furthermore, only sgnals wth a delay less than one chp are consdered. Ths latter hypothess may be justfed by the fact that sgnals wth a code delay larger than roughly one chp are uncorrelated wth the drect ray. In addton, the reflected ray s supposed to have less power than the drect one. In the presence of N- reflected rays, the receved sgnal at the nput of the trackng loops may be wrtten as follows: N s( t) = A. α. d( t τ ) c ( t τ ). cos(πf t θ ) (.) = f where α, τ and θ represent the ampltude, the tme delay and the phase delay of the th path wth respect to the drect one (the ndex zero s used for the drect path). c f s the carrer frequency c Presented at IFIS, Roma

2 c f (t) s the Pseudo Nose code waveform fltered by the RF front-end flter d(t) represent the payload data.. Multpath effect In the absence of multpath, the matched flter output s close to the symmetrc correlaton functon of the PN code. Ths symmetry s needed to obtan relable tme delay estmaton. Nevertheless, n the presence of multpath, ths symmetry s lost (as shown n fgure.), consequently, the propagaton delay becomes harder to estmate, thus, the range measurement accuracy s dmnshed MULTIPATH EFFECT Measured Lne-of-sght Multpath Fgure. Multpath effect on the normalzed correlaton functon. For the sake of smplcty, the effect of multpath on the trackng loops s studed n the case of one and two reflected rays. The trackng error of both DLL and PLL s assessed as a functon of the reflected rays parameters. In the case of one reflected ray, the phase trackng error s gven by equaton. []. ˆ α ˆ Kc ( τ + τ τ) sn( θ ) tan( θ θ ) = (.) K ( τ ˆ ) cos( ) ( ˆ c τ + α θ Kc τ + τ τ) where θ= θ θ s the phase dfference between the reflected ray and the drect one. ˆ τ s the propagaton delay estmate θ ˆ s the phase estmate of the drect sgnal. K c s the autocorrelaton functon of the fltered PN code. From. we note that the trackng error depends on the code trackng error estmate, therefore, we better have a good tme delay Presented at IFIS, Roma estmate. We note also that θ = yelds null phase trackng error. In general, only the phase trackng error envelope s gven. The latter quantty gves a better grasp of the evoluton of the phase trackng error. Furthermore, t s easer to represent. The error envelope s calculated for each multpath tme delay by maxmzng. as a functon of θ. To make ths calculus feasble we have assumed perfect tme synchronzaton.e. τ ˆ = τ and an unlmted recever flter bandwdth. Equaton.3 gves the phase trackng error envelope as a functon of the multpath tme delay and ts relatve ampltude: ˆ α ( τ ) tan( θ θ ) = ± (.3) α ( τ ) We note that that the equaton above s well defned thanks to condtons made n subsecton.. Fgure. shows the phase trackng error envelope versus the multpath tme delay for two multpath ampltudes. P H A S E E R R O R I N R A D I A N S PHASE ERROR ENVELOPE a=.5 a= Fgure.: Phase error envelope n radans TIME DELAY IN CHIPS In the stuaton of two reflected rays, the same envelope may be wrtten as below: α ( τ ) + α ( τ ) tan( θ ˆ) θ =± (.4) α α ( τ ) ( τ ) where τ s the tme delay of the th, {,}, reflected ray wth respect to the drect one. Obvously, equaton.4 may be generalzed for more than two reflected rays. In the case of the delay lock loop (DLL) t s not possble to gve the exact formula of the delay trackng error ntroduced by multpath. However, analytcal results of the error envelope may be gven. To be closer to the practcal stuaton, smulatons are done n the case of lmted bandwdth recever flter. Fgure.3 shows the delay trackng error envelope versus

3 the multpath tme delay. Results are gven for two chp spacngs, multpath ampltude s unchanged. It can be shown that the tme delay error s commensurate wth the chp spacng, therefore, operatng at low chp spacng values provdes a good help to mtgate the degradaton caused by multpath [3], [4]. We recall that ths technque s also helpful to reduce the fluctuaton of the code delay estmate caused by channel nose. 5 M 5 E T E R S CODE MULTIPATH ERROR ENVELOPE (a=.5bw=6 MHz) Fgure.3 : code error envelope. MULTICORRELATOR RECEIVERS Classcal recevers offer several trackng channels, each of them beng drven by two pars of correlator outputs. A multcorrelator recever provdes values of the correlaton of the ncomng sgnal wth several delayed replcas of the same local code n a sngle trackng channel. In that case, we get smultaneously several I and Q samples for each relatve delay d of each replca wth respect to punctual. For the experment descrbed here, we have used a Novatel Mllenum recever whose software has been modfed so as to provde trackng channel delverng 48 correlators outputs on I and 48 outputs on Q. The operatons performed n each correlaton channel are llustrated n fgure.. V V cos( π f ˆ kt s θ ) π / C ( kt s ˆ τ d) Cs=.5Tc Cs=.Tc DELAY IN METERS I&D I&D D ( n ) Fgure.: Archtecture of one correlator output. I Q The dstrbuton of the correlaton ponts wth respect to punctual can be chosen between 3 confguratons: unform, tralng edge and peak ntensve. The shapes of these dstrbutons are llustrated n fgures.,.3 and.4. CORRELATION PEAK CORRELATION PEAK CORRELATION PEAK CHIP SPACING Fgure. Samples sequence for unform dstrbuton CHIP SPACING Fgure.3 Samples sequence for peak ntensve dstrbuton CHIP SPACING Fgure.4 Samples sequence for tralng edge dstrbuton These three confguratons were used to evaluate the performance of our technque. Presented at IFIS, Roma 3

4 Example of the effect of multpath on the correlator outputs s shown n fgure.5. OUTPUT Fgure.5 Example of I and Q correlator outputs varaton as a functon of reflected ray phase shft (α =.5, τ =.5 chp). 3. USE OF THE LEAST SQUARE METHOD 3. Prncple In the prevous secton, we have seen that the multcorrelator recever we are usng provdes 48 samples of the correlaton functon for both n-phase and quadrature components of the Integrate & Dump flter outputs. Those 96 samples form the set of observatons whch s updated every second. The goal of the least squares method s to mnmze the Eucldan dstance between the observaton and the mathematcal model, then, we chose the MSE soluton as an estmaton of the multpath parameters and both phase and code trackng errors. We note that we deal wth a non-lnear model. Therefore, an teratve least squares method s used. In addton, n our algorthm, we have taken nto account the channel nose contrbuton, thus, the teratve generalzed least squares method has been adopted. 3. The Least Squares Algorthm Observaton model may be mathematcally expressed as follows: Y Y obs obs ( k) = N = = α K ( d k = 49,5, c c k k ε τ )cos( ε + θ ) + n( k) τ τ θ = π θ = CHIP SPACING k =, (3.) N ( k) = α K ( d ε τ )sn( ε + θ ) + n( k) n (k) refers to the nose terms θ θ = ( d ) 48 d s the samplng nstants vector normalzed wth respect to chp tme duraton. Let X be the set of parameters we want to estmate. Then, observaton Y may be related to X by a non-lnear equaton as follows: Y obs = h( X ) + n (3.) The nverse functon of h can t be analytcally calculated. Furthermore the nose term makes the calculaton of the exact soluton mpossble. Consequently, we can only approxmate the exact value. The estmator s the soluton of the least square equaton: Xˆ = mn Y h( X ) (3.3) X obs As we are faced wth a non-lnear equaton, an teratve method s the only possble way to estmate multpath parameters. The n+ order estmator s deduced form the nth order one by a lnear functon: Xˆ n+ = Xˆ n+ δx n (3.4) Here, δ X n stands for the adjustment of the nth order estmator, t s gven by: T T δ X n = [ H ( Xˆ n) W. H ( Xˆ n) ] H ( Xˆ n). W. δy (3.5) where W s the nverse matrx of the nose covarance matrx H X ˆ ) s the gradent vector of h n vcnty ( n of Xˆ n. δ Y = Yobs h( Xˆ n ) s the measurement predcton error. The teraton process s stopped when the Eucldan dstance between observaton and model s smaller than a fxed threshold. 3.3 Theoretcal performance The theoretcal performance of a least squares algorthm depends on the condton number of the gradent matrx of the observaton model. Let H be a matrx, the condton number of H s defned by µ + µ =, where µ µ + (respectvely µ ) denotes the largest (respectvely the smallest) egenvalue of the matrx H T. H and H T s the hermtan transpose matrx of H. Large condton numbers ndcate a nearly sngular matrx. The effect of the multpath parameters, namely the multpath tme and phase delays on the condton number were assessed. The results Presented at IFIS, Roma 4

5 gve us an dea about the sngularty of the gradent matrx. The smallest the condton number s, the easer the MSE equaton to solve s. Fgures 3. and fgure 3. show the condton number varaton as a functon of multpath ampltude, delay and phase. CONDITION NUMBER CONDITION NUMBER (n db) CODE MULTIPATH DELAY IN CHIPS AMPLITUDE Fgure 3. Condton number versus multpath tme delay CONDITION NUMBER CODE PHASE DELAY Fgure 3. Condton number versus multpath phase delay As we see from fgure 3., the multpath parameters are harder to estmate for small multpath tme delays. Ths concluson s obvous because reflected rays wth small delays are very close to the drect one. Moreover, we have found that small multpath parameters yeld large condton number, consequently, the estmaton of those multpath s more dffcult. In addton, as shown n fgure 3., parameters are harder to estmate when the reflected ray s n quadrature wth respect to the drect one. In the case of two reflected rays, the condton number s calculated as a functon of the dfference tme delay between those two rays. We have concluded that the algorthm Presented at IFIS, Roma delay=.5 delay=.5 delay=.5 delay=.35 delay=.45 delay=.55 delay=.65 delay=.75 delay=.85 5 effcency s dmnshed for small tme dfferences. In fact, such a stuaton ntroduces an ambguty when we try to separate the two sgnals smply because the observaton functon s not njectve (.e. two dfferents sets of multpath parameters can yeld to the same vector of I and Q correlaton samples). Fgure 3.3 shows the evaluaton of the condton number versus the frst tme delay, the second tme delay s taken as constant and equal to.45. CONDITION NUMBER (n db) Phase shft= Phase shft =.346 Phase shft =.683 Phase shft =.9448 Phase shft =.566 Phase shft =.578 Phase shft =.885 Phase shft =.99 Phase shft =.533 Phase shft =.874 Phase shft = MULTIPATH CODE DELAY Fgure 3.3 Condton number versus the frst multpath tme delay. 4. RESULTS WITH A REAL RECEIVER 4. Sgnal generaton The GPS sgnal we process s provded by a SPIRENT GPS sgnal generator GSS 76. Then, t s fed to a Novatel Multcorrelator GPS recever. Subsequently, the correlator outputs are stored nto a computer. Our MSE algorthm wll process obtaned raw data n order to estmate multpath parameters. We note that those parameters are defned n the scenaro nserted n the GSS Scenaro wth one reflected ray In the case of only one reflected ray, the multpath sgnal s charactersed by a fxed ampltude throughout the scenaro. The tme delay wth respect to the drect ray ranges from to. Tc. That tme delay vares by slces of 3 s: t s constant durng 3 seconds, then ncreased by. Tc, then agan kept constant durng 3s, etc... The phase shft of the reflected ray has a lnear varaton versus tme. The slope s equal to π n s. The latter scenaro was run for 3 ampltude values (.5,. and.5) and for the three dfferent recever confguratons. Major results are llustrated by the followng fgures.

6 Fgures 4., 4., 4.3 and 4.4 show estmates of multpath code, ampltude, phase, code and phase trackng errors n the case of the unform dsrtbuton recever confguraton wth a multpath ampltude set to...5 CODE DELAY ESTIMATE Code delay estmate True code delay PHASE DELAY IN RADIANS PHASE DELAY ESTIMATE CODE DELAY IN CHIPS Fgure 4. Multpath tme delay estmaton performance wth unform recever confguraton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s). MULTIPATH AMPLITUDE MULTIPATH AMPLITUDE ESTIMATE Fgure 4. Multpath ampltude estmaton performance wth unform recever confguraton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s) Fgure 4.3 Zoom of multpath phase delay estmaton performance wth unform recever confguraton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s). Those fgures show the qualty of the multpath parameters estmaton (code delay, ampltude, phase). As we can see n fgure 4., the multpath tme delay s estmated wth an accuracy close to.5 chp (4 m) when that delay s larger than. chp. When the multpath tme delay s lower than. chp, the estmaton s not robust. Smlarly, we can see n fgure 4. that the ampltude estmate s very good (accuracy better than.5) when the multpath delay s larger than. chp. When the delay s smaller than chp, the ampltude estmaton s not robust. As shown n fgure 4.3, the phase estmate s also very good, dsplayng a lnear evoluton wth sudden phase shfts every 3 s. Fgure 4.4 shows the evoluton of the code and phase trackng error estmates. We can see that, n lne wth theory, these two estmates are n quadrature []. In addton, the phase trackng error changes suddenly every 3 s. The code trackng error estmate s more affected by nose than the phase trackng error estmate. Presented at IFIS, Roma 6

7 CODE AND PHASE TRACKING ERRORS ESTIMATES PHASE TRACKING ERROR ENVELOPE TARCKING ERROR Code trackng error(n m) Phase trackng error n cm Fgure 4.4 Code and phase trackng error estmates wth unform recever confguraton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s). Fgures 4.5 and 4.6 show the code and phase trackng error estmates as a functon of the multpath delay. These estmates are compared wth the exact theoretcal trackng error envelope. As we can see, the code trackgn error estmate s very close to the theoretcal envelope when the delay s lower than. chp (3 m). As already seen n fgure 4.4, the phase trackng error estmate s very close to theory whatever the multpath tme delay. PHASE TRACKING ERROR IN RADIANS CODE DELAY IN CHIPS Fgure 4.6 Tme trackng error estmate envelope wth unform recever confguraton for α =. as a functon multpath delay. The performance of the estmaton s degraded when the relatve multpath ampltude s low. Fgure 4.7 (to be compared wth fgure 4.) shows the tme delay estmate when the multpath relatve ampltude s equal to.5. As we can see, that estmate s noser than when α =...5 CODE DELAY ESTIMATE Code delay estmate True code delay CODE TRACKING ERROR CODE ERROR ENVELOPE MULTIPATH DELAY IN METERS Fgure 4.5 Tme trackng error estmate envelope wth unform recever confguraton for α =. as a functon multpath delay. CODE DELAY IN CHIPS Fgure 4.7 Multpath tme delay estmaton performance wth unform recever confguraton for α =.5 as a functon of tme n the run (multpath delay vares by. chp every 3 s). The results obtaned wth other recever confguratons have slghtly the same behavour. Fgures 4.8 and 4.9 show the tme delay and ampltude estmate wth the tralng edge recever confguraton. Presented at IFIS, Roma 7

8 CODE DELAY IN CHIPS CODE DELAY ESTIMATE Code delay estmate True code delay confguraton, the tme delay and ampltude estmate are robust when the multpath delay s larger than.6 chp, whch s a very good performance. CODE DELAY IN CHIPS.5.5 CODE DELAY ESTIMATE Code delay estmate true code delay Fgure 4.8 Multpath tme delay estmaton performance wth tralng edge recever confguraton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s). MULTIPATH AMPLITUDE ESTIMATE Fgure 4. Multpath tme delay estmaton performance wth peak ntensve dstrbuton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s) MULTIPATH AMPLITUDE ESTIMATE MULTIPATH AMPLITUDE Fgure 4.9 Multpath ampltude estmaton performance wth tralng edge recever confguraton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s). As we can see n fgure 4.8, the tme delay estmaton s not very robust untl the multpath delay s larger than. chp. In addton, when the multpath delay s larger than chp, the estmate s nosy. As shown n fgure 4.9, the multpath ampltude estmate s not robust untl the tme delay s larger than. chp. Fgures 4. and 4. show the tme delay and ampltude estmaton performance usng the peak ntensve recever confguraton. We see that wth that recever MULTIPATH AMPLITUDE Fgure 4. Multpath ampltude estmaton performance wth peak ntensve dstrbuton for α =. as a functon of tme n the run (multpath delay vares by. chp every 3 s). Therefore, the proposed technque provdes a relable estmate of the multpath tme delay, ampltude and phase when the multpath tme delay s larger than.6 chp usng the peak ntensve recever confguraton. In addton, we have shown that the code and phase trackng errors have a consstent behavour (quadrature) and do ft closely to ther theoretcal envelope. Presented at IFIS, Roma 8

9 4.3 Scenaro wth two reflected rays In ths case, two reflected rays are consdered: the frst one has a fxed tme delay equal to.5 chp when the tme delay of the second ray vares every 3 s and s equal to.5,.75,.,.5,.,.5,.3,.35,.4,.6,.8,.9,.,.,. chp. The multpath ampltude of the frst fxed ray s equal to., and the ampltude of the varyng ray s set to.3. Tme delay estmates and ampltude estmates are shown n fgures 4. and 4.3 for the unform recever confguraton. CODES DELAYS IN CHIPS.5.5 Delay estmate Delay estmate Exact delay Exact delay MULTIPATHS DELAYS ESTIMATES Fgures 4. Multpath tme delay estmaton performance wth unform dstrbuton for α =., α =.3 as a functon of tme n the run (ray delay s constant equal to.5 chp, ray delay s as presented above). MULTIPATHS AMPLITUDE MULTIPATHS AMPLITUDES ESTIMATE Fgures 4.3 Multpath ampltude estmaton performance wth unform dstrbuton for α =., α =.3 as a functon of tme n the run (ray delay s constant equal to.5 chp, ray delay s as presented above). It s clear that even f the algorthm s able to estmate reflected rays parameters wth the same accuracy as n the case of one Presented at IFIS, Roma 9 reflected ray, ts performance collapse when the two reflected rays have slghtly the same tme delay due to the fact that the observaton functon s not njectve. We can also see by the color change that the parameters estmates relatve to each are often nterchanged by the algorthm. Fgures 4.4 and 4.5 show tme delay estmates and ampltude estmates for the peak ntensve recever confguraton. As we can see, the estmate s less affected by nose and s more robust for a short delay of ray and when both delays are dentcal. Note agan that the parameters estmates relatve to each are often nterchanged by the algorthm. CODES DELAYS IN CHIPS Delay estmate Delay estmate Exact delay Exact delay MULTIPATHS DELAYS ESTIMATES Fgures 4.4 Multpath tme delay estmaton performance wth peak ntensve dstrbuton for α =., α =.3 as a functon of tme n the run (ray delay s constant equal to.5 chp, ray delay s as presented above). MULTIPATHS AMPLITUDE MULTIPATHS AMPLITUDES ESTIMATE Fgures 4.5 Multpath ampltude estmaton performance wth peak ntensve dstrbuton for α =., α =.3 as

10 a functon of tme n the run (ray delay s constant equal to.5 chp, ray delay s as presented above). 5. CONCLUSION Ths paper has presented a least squares technque for multpath parameters dentfcaton and code and phase trackng error estmaton usng a multcorrelator recever. In the case of one reflected ray, the technque s able to estmate the multpath parameters (tme delay, relatve ampltude, phase shft) wth good accuracy (better than.5 chp for code delay, better than.5 for relatve ampltude) when the multpath delay s strctly larger than.6 chp. The technque also provdes estmates of the code and phase trackng errors that seem to have a consstent behavour and ft perfectly n ther theoretcal envelope. In the case of reflected rays, the multpath parameters are estmated wth the same accuracy, although the technque s not robust when both multpath delays are dentcal because the observaton functon s not njectve. Potental applcatons are twofold: stng and more generally channel characterzaton through dentfcaton of multpath parameters, trackng performance mprovement through removal of multpath components from trackng loops dscrmnaton functons. Note that the current lmtaton here s on the relatve delay of all rays (strctly larger than.6 chp) and on the multpath relatve ampltude (larger than.5). Further work am at testng ths technque on lve sgnals and at refnng the estmaton technque to reduce those lmts. [] Braasch M. (996) «Global Postonng System: Theory and Applcatons», volume, chapter Multpath Effects, pages , AIAA. [3] Fenton P., Falkenberg B., Ford T., Ng K. and Van Derendonck A.J. (99) «NovAtel s GPS Recever the Hgh Performance OEM Sensor of the Future», proceedngs of ION GPS-9, Albuquerque, Sept 9-3. [4] Van Derendonck A.J., Fenton P. and Ford T. (99) «Theory and Performance of Narrow Correlator Spacng n a GPS Recever», proceedngs of ION Natonal Techncal Meetng, San Dego, January 7-9. ACKNOWLEDGMENTS The authors wsh to thank Dane Rambach for her help durng the evaluaton of the performance of the proposed algorthm on lve measurements. REFERENCES [] Van Nee R., Sereveld J., Fenton P. and Townsend B. (994) «The Multpath Estmatng Delay Lock Loop : Approachng Theoretcal Lmts», proceedngs of IEEE PLANS 94, Las Vegas, Aprl -5. Presented at IFIS, Roma