An ANOVA-Based GPS Multipath Detection Algorithm Using Multi-Channel Software Receivers

Transcription

1 A ANOVA-Based GPS Multpath Detecto Algorthm Usg Mult-Chael Software Recevers M.T. Breema, Y.T. Morto, ad Q. Zhou Dept. of Electrcal ad Computer Egeerg Mam Uversty Oxford, OH 4556 Abstract: We preset a statstcal detecto test for GPS multpath based o the oe-way ANOVA method. Gve a atea array wth a GPS software recever tracg mode, the sgal from each chael s correlated wth a referece sgal blocs of oe code perod. Whe the relatve phase delay for the drect GPS sgal s strpped off from each chael, the expected values of the correlates s the same for all of the chaels oly f o multpath s preset. A oe-way ANOVA test ca the used to determe f multpath s preset. A aalyss of ths method s preseted whch shows that the parameters affectg ts detecto performace ca be grouped to three classes: the array sze, the sgal AOAs, ad the processed multpath SNR. Recever operatg characterstc curves are gve as a fucto of the processed multpath SNR for fxed array szes. They show that good detecto performace ca be acheved uder most operatg codtos wth less tha code perods of data. It s also show that the detecto performace of ths method mproves as the multpath tme delay decreases. Ths suggests ths method could be a useful tool adg multpath mtgato techques whose ablty to detect multpath typcally degrades as the multpath tme delay decreases. I. INTRODUCTION Multpath s oe of the maor sources of error precse posto determato usg GPS. A umber of methods have bee developed for multpath mtgato whch ca be grouped to two classes. The frst class of techques attempt to modfy the recever tracg loop such a way that t s ot affected by multpath. Methods that fall to ths class clude the arrow correlator [], the strobe correlator [], ad Multpath Elmato Techology [3]. The secod class of methods attempt to otly estmate the drect ad multpath sgal parameters. They clude the Multpath Estmatg Delay Loc Loop (MEDLL) [4], modfed RAKE delay loc loop [5], ad Multpath Mtgato Techology [6]. The performace for both of classes of methods, however, s degraded whe the relatve tme delay betwee the multpath ad drect le of sght (LOS) GPS sgal s short. A serous cocer s that f the multpath s very close to the LOS GPS sgal tme, ts presece ad therefore the error t produces, could potetally go udetected. I the case of the frst class of methods, for example, o detecto s performed. I the multpath-estmator based methods, detecto s mplctly corporated to the method. Ths s because the umber of multpath sources s oe of the parameters beg estmated. The estmato s ofte performed usg the maxmum lelhood (ML) method. It has bee show that the varace of Cramer-Rao lower boud (CRLB) for the ML estmated multpath parameters creases as the tme-delay decreases [7], whch could potetally lead to the msdetecto of a sgle multpath source. Notwthstadg ths problem, ML methods for multpath estmato are geerally computatoally expesve. Ths could be a lmtato for recevers o a movg platform where prompt reportg o a rapdly chagg multpath evromet s requred. Although recetly mproved versos of ML methods that reduce the complexty of the optmzato process have bee reported [8][9], a smpler ad faster method for the detecto of shorttme delay multpath would be desrable. I ths paper, we preset a method that s desged to detect the presece of multpath sgals by explotg the spatal dversty betwee the drect GPS sgal ad ts multpath. We shall show that our approach complemets the prevous methods, that ts detecto performace s optmzed exactly uder those codtos where ther performace s the worst. The structure of ths paper s as follows: Secto II descrbes the mathematcal model for the sgals. Secto III presets the theory ad algorthm for multpath detecto. The performace of ths method s aalyzed secto IV. Secto V summarzes the results ad hghlghts future mprovemets ad drectos. II. MATHEMATIL SIGNAL MODEL Fg. shows a dealzed uform lear array (ULA) cofgurato cosstg of K elemets spaced at oe-half the omal carrer wavelegth. For a collecto of sotropc array elemets that are calbrated, the agle of arrval, deoted, determes the array s respose. Uder these codtos, the array s respose, whch s ofte called the steerg vector ad wll be deoted by S, s a complex vector wth K compoets havg the parametrc form:

2 S exp π cos exp K π cos () Some addtoal otato that wll be used throughout ths paper s as follows: vectors wll be deoted by boldface type, cougate trasposes by superscrpt H, trasposes by superscrpt T, ad cougates by a superscrpt *. The expected value of a radom varable wll be deoted as E{ }. III. METHODOLOGY The method we propose s a bary detecto method for multpath cast the form of a smple statstcal hypothess test. The ull hypothess s chose to be the codto that o multpath s preset ad the alteratve hypothess s the codto that multpath s preset. I the frst sub-secto, we qucly revew the aalyss of varace or ANOVA method used our method to test these two hypotheses. I the secod secto, we motvate ad expla the sgal processg steps ecessary to prepare the sgal so t ca be tested wth ANOVA. We ed ths secto wth a dscusso o how the operatg codtos whch wll satsfy a gve level of statstcal performace are determed. Fgure. Smplfed Model System A deal ULA of K elemets collects the sgals from the drect GPS ad ts multpath sgals. A ey property for each sgal s the oretato of ts wave vector relatve to the array axs, ow as ts agle of arrval ad deoted by. I a multpath evromet, the sgal s typcally modeled as the sum of a desred GPS sgal, M multpath sgals, ad radom chael ose. All of the determstc sgals wll be cosdered statoary over the tme terval of observato. The ose from each chael s dstrbuted as CWGN(, ) ad s ucorrelated both spatally ad temporally. The mathematcal model for the sgal used wll be ts complex, dgtzed form at basebad sampled wth frequecy f s. Neglectg the avgato data bt, t ca be expressed the form: M X t A C t exp t S ε t () where represet the sample dex, = correspods to the drect LOS GPS sgal, ad C t A GPS code for desred sgal Sgal ampltude Relatve tme delay Relatve carrer phase to LOS sgal to LOS sgal Itermedate frequecy wth Doppler shft S Steerg vector (3) A. ANOVA Aalyss of Varace or ANOVA s a stadard statstcal method to test whether the mea of a radom varable s the same multple populatos []. ANOVA starts wth K dstct populatos each cotag radom samples of a gve radom varable Z. I our aalyss, we wll use balaced ANOVA, whch all of the populatos cota the same umber of samples. Lettg K be the umber of samples each populato ad Z h, deote the h th sample from the th populato, f the followg codtos are satsfed for all h,h'=,..., K ad, '=,..., K:..d. h, Z ~ N, (4) Co var Z Z h, h, h,h, the ANOVA ca be used to test the followg statstcal hypotheses: H : for all,,...,k H : for some,,...,k (5). ANOVA s based o a comparso of two sample varaces. The frst varace, called the mea-square error wth populatos ad deoted by w, estmates the average sample varace tae over all the populatos: w K K Z, K K ˆ (6)

3 Because the secod codto (4) requres the varace of the radom varables to be the same for all of the populatos, the w s a ubased estmator for the true sample varace. The secod varace s the varace betwee the sample meas. It s ow as the mea-square error betwee groups ad s gve by: b K K ˆ K where s the average of the K populatos meas. Uder H, t ca be show that b s also a estmate for the true sample varace, ad by Cochra's theorem, t ca be show that b ad w are depedet chsquare radom varables havg degrees of freedom (K-) ad K( K -), respectvely. Hece ther rato follows a cetral F dstrbuto wth (K-) umerator degrees of freedom ad K( K -) deomator degrees of freedom. Uder H, eve though w ad b are stll depedet, the fact that b s a o-cetral ch-square radom varable maes ther rato a o-cetral F dstrbuto. Lettg F( (K-),K( K -)) deote the crtcal value for the hypothess test havg sze, the decso rule ANOVA s: b w b w F K,K Reect H K F K,K Fal to reect H K B. Algorthm for Multpath Detecto wth ANOVA I the cotext of multpath detecto, we use ANOVA to determe f a sgal cotas oly the drect GPS sgal, or f t also cotas ts multpath sgals. Based o the sgal model (), the statstcal test we wsh to perform ca be stated as: H : M.e. o multpath preset H : M.e. multpath s preset To ad explag how ANOVA ca be used to perform ths test, we wll troduce the two sgals for a K elemet ULA, X ad X : X t A C t exp t S ε t M X t A C t exp t S ε t (7) (8) (9) () represet the sgals uder the two hypotheses we wsh to test. No method for GPS sgal detecto ca be appled drectly to the put sgal. Ths s due to the fact that the GPS sgal s by costructo a wea drect spreadspectrum sgal whch s ot statstcally detectable. Assumg that the recever s already tracg mode, we ca correlate the sgal from each chael wth a estmate for the referece sgal of the drect GPS sgal order to crease ts effectve sgal to ose rato. Usg our prevous otato, the dscrete estmated referece sgal at tme t ca be wrtte the form: rˆ t C t exp ˆt ˆ () where s the error of the code phase tracg loop. K correlatos wll be performed usg cosecutve blocs of data, each havg a legth of oe code perod (T = msec). It s ot cocdetal that K was also used the last sub-secto to deote the umber of data pots each populato for a balaced ANOVA expermet. As we wll see later, each code perod wll cotrbute a sgle data pot to the populatos our ANOVA aalyss. To sure that we have eough data pots for relable statstcal ferece, whle also mmzg the computato tme requred for our aalyss, the umber of code perods s typcally chose to be betwee 3 ad (.e. 3 K ). The sgal s correlated over each successve code perod wth the estmated referece sgal gve (). I the correlatos, the dfferece betwee the estmated ad true carrer frequecy wll be approxmated to be zero. For a GPS recever tracg mode, the typcal error the frequeces s a few Herz, ad sce the total tegrato tme s at most msec, ths approxmato s vald. Uder ths codto, the correlato for the th source from each tme bloc wll be the same, ad s gve by []: ˆ A exp () Both test sgals () have the same ose, so we ca evaluate the ose cotrbuto to the correlator output from the th code perod. If we defe: ψˆ Sg C t T (3) the the th compoet of the post-correlated ose from the th code perod ca be expressed as: where =, N (total umber of samples) ad M>. We wll refer to these two sgals as test sgals because they

4 *N *N exp t N (4) where N deotes the total umber of data pots sampled oe code perod. Sce each compoet of the radom chael ose at each tme sample s modeled as a zero-mea complex ormal radom varable, multplcato of each elemet by a phase factor chages ether the dstrbuto or covarace propertes of the ose. From ths fact, t drectly follows that:..d. ~ CWGN, N (5) whch holds for all =,...,K ad all =,..., K. The codto that the post-correlated ose samples are depedet follows from the thrd codto (4). Combg the correlatos of the determstc ad ose sgals, the total output sgal for the two test sgals from the th code perod ca be wrtte as: Y S ε M Y S ε (6) the test sgals (whch we'll deote by Z) for the th code perod have the form: Z Z M ε exp exp K ε (9) where = [cos( -cos )]. Sce the weght s a phase factor, () has the same dstrbuto as '(). From (9) t follows that our sgal ow ot oly satsfes the assumptos ecessary to use ANOVA, but that t ca be used wth ANOVA to perform the statstcal hypothess test (9). From the frst le of (9), t follows that the expected values for the fal output sgals from each chael are the same uder H. Uder the alteratve hypothess however, the secod le of (9) shows that expected value of the fal output sgal wll be dfferet for each chael due to the cotrbuto of the multpath sgals. The smplfed bloc dagram show Fg. summarzes the overall algorthm used to detect multpath wth ANOVA for a K elemet atea array. ANOVA ca be used whe K populatos of ormally dstrbuted radom varables are depedet ad have the same varace. From (5), t follows that our sgal model uder both hypotheses satsfes the assumptos ecessary to use ANOVA. However, for a ANOVA test to be successful, the data must also satsfy the codto that the mea of each populato wll be the same whe H holds. If we cosder each of the chaels as beg a populato (so our K populatos are represeted by the K chaels), we see from (6) that the expected values of our K populatos uder H are: E Y exp π cos exp K π cos (7). Equato (7) shows us that uder H the meas for the dfferet chaels are geerally ot all equal. To mae them equal uder H, the th chael sgal eeds to be multpled by the weght: w exp - π cos (8) whch ca be costructed, assumg that the drect GPS sgal s AOA s ow. After the applcato of the weght,

5 Fg.. Flowchart for ANOVA wth GPS Software Recever C. ANOVA Expermetal Desg The algorthm ust descrbed omts a mportat prelmary step whch s cetral to all ANOVA expermets: sample sze determato. I ths secto, we wll cosder ths smple desg ssue ad show how the problem of sample sze determato s solved for multpath detecto. I sgal detecto problems, we wat to choose our system parameters so that our detecto method wll perform accordg to some pre-determed performace codtos. The two measures of detecto performace are the false alarm rate ad the mssed detecto probablty. Oe typcally sets predetermed tolerace levels for both ad that we wat the detecto method to satsfy. Sample sze s a mportat parameter that affects both ad. Although sample sze determato for ANOVA s geeral rather complcated, we ca evertheless llustrate the basc uderlyg cocept of how t wors. Fg. 3a shows a hypothetcal probablty desty for the samplg dstrbuto of the mea a sgle populato uder H ad H based o some fxed sample sze. The predetermed codto o determes the crtcal value, Z C, upo whch the decso rule s based: Z Z Reect H C Z Z Fal to reect H C () Based o ths decso rule, t follows that s the probablty that the mea wll be less tha Z C whe H holds. We see from Fg. 3a that ths s the shaded area uder the probablty desty for H to the left of Z C. Although the decso rule sures that the false alarm rate s satsfed, what f the area uder the curve s greater tha the value of desred? The soluto to ths problem les the fact that the varace of both dstrbutos ANOVA s versely proportoal to the sample sze. Thus, creasg the sample sze maes both dstrbutos more cocetrated about ther meas ad reduces the probablty desty the tals of the dstrbuto. Fg. 3b shows the same dstrbuto after the sample sze has bee creased by a factor of four. We see qute clearly that, the probablty the mea wll fall below Z C whe H s true s substatally decreased. Fg. 3 If we cotue to crease the sample sze, the probablty of both ad would go to zero. Whle that may seem very desrable, practce t s ofte more useful to try to collect as few samples as possble. I the case of multpath detecto for example, collectg more samples requres loger tmes. I a evromet wth a rapdly chagg multpath evromet, prompt reportg of the multpath codtos s a mportat cosderato. The goal the s to fd the mmum sample sze that wll satsfy both ad. There are a varety of statstcal methods desged for determg the sample sze wth ANOVA []. To mplemet such methods however, oe eeds a estmate of how close the mea values of the statstc uder H ad H are. Referrg to Fg. 3, oe could mage that as the meas of the two dstrbutos approached oe aother (whle the varace of the dstrbutos remaed the same), the umber of samples requred to mae the dstrbutos suffcetly well separated would have to crease. Therefore, to determe the sample sze for a gve performace specfcato, a estmate of the mmum dfferece betwee ad oe wshes to detect, must be ow a pror. For the problem of multpath detecto, a estmate of the mmum dfferece the sample meas s dffcult to determe. For that reaso, we stead use Mote-Carlo smulatos to estmate the sample sze requred to meet a specfed performace crtera. The parameters our smulatos are those whch affect detecto performace: the umber of array elemets K, the data legth, the drect sgal AOA, ad the multpath sgal parameters cludg the AOA, relatve tme delay, ad sgal stregth. Based o our aalyss, we ca defe a parameter whch essetally represets the effectve multpath sgal stregth at the output of our correlato ad weghtg algorthm: SNR Log N () K T

6 To obta a quattatve uderstadg of the depedece has o ts parameters, Fg. 4 shows a cotour plot of as a fucto of the multpath put SNR ad tme delay. The thrd parameter o whch depeds, total umber of data pots (N= K N ) was ept fxed at 5, (5 code perods sampled at a rate of 5 MHz). Fg.4 Effect of Multpath Parameters o Net Multpath SNR Usg () wth N = 5,, s plotted as a fucto of the multpath parameters. Cotour plot values are uts of db. For multpath whose AOA s well separated from that of ts drect sgal, prcpally determes the detecto algorthm s performace for a fxed array sze. Ths ca be demostrated by the smulato results show Fg. 5 where both ad the mea F statstc values are plotted as cotours fuctos of the umber of code perods K, ad the multpath SNR. The smulato s performed wth a sgal model cotag a drect sgal, a sgle multpath, ad ose receved by a three elemet ULA wth fxed at.*t. The samplg frequecy s aga 5MHz ad the multpath ad drect sgal AOAs are 45 o ad 86 o respectvely. For each par of multpath SNR ad umber of codes sampled,,5 Mote-Carlo smulatos were performed ad the average F-statstc value computed. We see that the cotours of ad the F-statstc correlate very well, dcatg that the detecto performace ca be completely accouted for by. Ths agreemet s tutvely soud because the et multpath SNR efectvely specfes the smallest sgfcat sgal that ca be detected. Based o ths result, t s possble to determe the sample sze for a gve par of (, ). Fg. 6 s a cotour plot of for a three elemet ULA usg smulato. The values are estmated from 5 Mote-Carlo smulatos for gve pars of ad values, by determg the percetage of F statstc values that fell below the crtcal value for. We see from the plot for example, that for the multpath detecto to have a false alarm rate of o more tha.5 ad a mssed detecto probablty of o greater tha.5, a mmum value of = 3.66 db would be requred. If the mmum multpath SNR to be detected s specfed alog wth the maxmum tme delay ad samplg rate, t s possble, usg equato () to compute the mmum umber of samples requred. For example, substtutg the values =3.3 db, alog wth a multpath SNR of -34 db, a samplg rate of 5 MHz, ad the maxmum tme delay of.5 T to equato (), we fd that a sample sze of 9, or 9 code perods, would be requred. Fg. 6 Determato of sample sze for 3 elemet ULA The cotour for the desred value of s foud. The pot where ths cotour tersects the desred value of s the foud. The value of at whch they tersect ca the be used wth () to determe the sample sze. IV. RESULTS Fg. 5 Correlato betwee ad the Detecto Statstc The F-statstc values estmated from,5 Mote Carlo smulatos show the same fuctoal tred as the values. We wll preset results frommat LAB smulatos to evaluate the detecto performace of ANOVA. The relevat parameters o whch the performace depeds are (whch cotas the total umber of codes used alog wth the multpath SNR ad tme delay), the umber of array elemets, ad the AOAs of multpath ad drect GPS sgal. Fg. 7 llustrates how the detecto performace s affected by ad K, the umber of array elemets. Two sets of levels curves for are plotted as a fucto of the false

7 alarm rate ad mssed detecto rate. The frst group correspods to a array wth 3 elemets, whle the secod group s for a array wth 7 elemets. Fg. 8 Depedece of G o multpath tme delay, the put multpath SNR ad umber of samples collected Fg. 7. Performace Curves for Sgle Multpath Sgal The relatoshp betwee,, ad for a 3 ad 7 elemet array s show. Gve ay of these 3 quattes, the thrd ca be estmated from the plot. Fg. 7 shows that for a fxed array dmeso, a larger value correspods to lower mss detecto rate ad false alarm rate. Ths s tutvely expected sce a larger value represets stroger processed multpath sgals. For the same value, a larger array also reduces the false alarm ad mss detecto rate. For example, f acceptable false alarm ad mss detecto rates of 5% ad 7.5% respectvely are chose, the the multpath that ca meet ths crtero should have a mmum value of.97 f the array has 3 elemets. For a 7 elemet array, the correspodg value s aroud.5. What are the multpath sgal parameters for the above metoed values? Smlar to Fg. 4, s plotted as a fucto of the multpath parameters, but ow two sets of cotours for are show. The frst set of cotours are the values whch would be obtaed usg the lower boud for the umber of samples, K =3. The secod set of cotours are the values obtaed usg the upper boud, K =. Fg. 8 provdes a quattatve descrpto of the value s depedecy o the basc multpath sgal parameters, ad SNR. Two sets of curves are plotted Fg. 8. The sold les are geerated for = code perods ad the dashed les are for 3 code perods. Based o ths fgure, we see that for =.97, the multpath SNR has to be larger tha -4 db order to meet the detecto crtera, f =. For =3, the mmum multpath SNR s db. For a gve multpath SNR, the value sets the upper lmt of the multpath delay tme that ca meet the prevously stated detecto crtera. For example, f the multpath SNR s -35dB, the the maxmum multpath delays are.5t ad.t for = ad 3 respectvely. Fg. 8 shows that f all of the system parameters are fxed except the multpath tme delay, creases as the tme delay decreases. From Fg. 7, t was see that both detecto errors decrease as creases, hece we ca coclude that there s a uform mprovemet detecto performace as the multpath tme delay decreases. Ths s a terestg result because t s the exact opposte of typcal MLE methods such as the MEDLL, whose performace becomes worse as the tme delay decreases. The bass for ths result les the fact that the ANOVA method maes use of dffereces the AOAs betwee the drect ad multpath sgal. By usg the sgal's spatal, rather tha ts temporal dversty, ANOVA does ot ecouter ths lmtato due to the tme delay. Fg. 9 shows how sgfcat the mprovemet detecto s as the multpath tme delay decreases. Usg a desred drect GPS sgal wth a put SNR of -8 db ad a multpath sgal wth a put SNR of -8 db, the tme delay s chaged cremets of.*t. For each tme delay, 5, Mote-Carlo smulatos are ru ad the crtcal value s chose at whch =. We see that at a tme delay of.7* T, a probablty of false alarm ad mssed detecto of. ca be acheved, but at a.3* T, the probablty of false alarm ad mssed detecto drop dramatcally to less tha.. Fg. 9 Effect of multpath tme delay o detecto

8 I Fg., we demostrate the sestvty of the multpath detecto algorthm to the spatal separato betwee the multpath ad drect sgal AOAs. Fg. was geerated from a Mote-Carlo smulato volvg two sgals: oe wth the drect GPS ad a sgle multpath sgal ad the other wth ust the drect GPS sgal. These two sgals represet the two sgals uder our two competg hypothess. A grd search s performed over the plot Fg. : the drect sgal s AOA s vared from zero to ety degrees fve degree cremets ad at drect sgal AOA, the multpath AOA s vared from the drect sgal AOA by zero to 5 degrees ( degree cremets). For each effectve (, ) par,,5 Mote-Carlo smulatos are ru. The emprcal dstrbuto fuctos for both sgals are computed ad the crtcal value at whch the estmated type I ad type II errors are the same s foud. Ths approach gves us a estmate for the detecto errors wthout weghtg a specfc detecto error over the other. The value of (or equvaletly ), s the chose as our detecto performace metrc ad are the cotour values plotted Fg.. Fg. shows that the spatal proxmty of the multpath sgal has a great effect o the detecto performace whe drect sgal AOA s relatvely small. For example, whe <, the multpath AOA has to be at least greater tha the drect sgal s to esure that both ad are less tha %. As the desred LOS AOA creases however, the two sgals ca get relatvely close to each other before performace s sgfcatly degraded. Whe =6, the multpath sgal AOA ca be about 5 o away from the drect sgal AOA ad stll be detected wth less tha 5% false alarm ad mss detecto rate. V. CONCLUSIONS Multpath s oe of the maor error sources hgh accuracy GPS applcatos. The most dffcult type of multpath are those whose tme delay relatve to the drect GPS sgal, s short. For these types of multpath, exstg methods to may ot be able to detect the presece of multpath ad therefore ot recogze the error produced by the multpath. The ANOVA algorthm preseted paper taes advatage of the spatal dversty betwee the multpath ad the drect GPS sgal to detect the presece of such multpath. We have show that the ANOVA-based algorthm ca detect the presece of multpath usg 3- code perods wth modest computatoal cost. The algorthm requres the costructo of a sgle weght vector based o ow drect sgal AOA ad multplcato of the weght vector to the correlator outputs. The ANOVA-based algorthm complmets prevous methods, that ts performace mproves as the multpath tme delay decreases. As the agle of arrval for the multpath ad the drect sgal becomes close to each other, the detecto performace s degraded as expected. Smulatos suggest however that ths reducto performace s lmted to a relatvely small rego ad that by creasg the umber of array elemets, performace ca be mproved to the desred level. The goal of ths paper was to show that multpath sgal detecto for short tme delay multpath ca be performed usg the spatal dversty betwee the desred ad udesred sgal a way that was geerally applcable, smple to mplemet, ad yet had good performace. Improvemets of ths method are plaed cosderg the optmzato of the detecto method by comparg the performace of the ANOVA method wth varous egestructure techques. The corporato of the temporal ad spatal dversty should help also mprove detecto ad mae the method more robust. VI. BIBLIOGRAPGHY Fg. Effect of relatve multpath ad drect sgal AOA o detecto. [] Deredoc, A. J., P. Feto, ad T. Ford, ``Theory ad Performace of Narrow Correlator Spacg a GPS Recever'', NAVIGATION, Joural of the Isttute of Navgato, vol. 39, o. 3, pp , 99. [] Gar, L. ad J.M. Rousseau, ``Ehaced Strobe Correlator Multpath Mtgato for Carrer Code'', Proc th Iteratoal Techcal Meetg of the Satellte Dvso of the Isttute of Navgato, ION-GPS, vol., pp , 997. [3] Towsed, B. ad P. Feto, ``A Practcal Approach to the Reducto of Pseudorage Multpath Errors a L GPS Recever'', Proceedgs of the 7 th Iteratoal Techcal Meetg of the Satellte Dvso of the Isttute of Navgato, Part (of ), Proceedgs of ION GPS, vol., pp , 994.

9 [4] R.D.J. va Nee, ``The Multpath Estmatg Delay Loc Loop'', IEEE d Iteratoal Symposum o Spread Spectrum Techques ad Applcatos, Yoohama, Japa, pp. 39-4, 99. [5] Cah, C. ad M. M. Chasarar, ``Multpath Correctos for a GPS Recever'', Proceedgs of the th Iteratoal Techcal Meetg of the Satellte Dvso of the Isttute of Navgato, ION-GPS, vol., pp , 997. [6] L.R. Wel, ``Multpath Mtgato usg Moderzed GPS Sgals: How Good Ca t Get?'', Proceedgs of the 5$^{th}$ Iteratoal Techcal Meetg of the Satellte Dvso of the Isttute of Navgato ION GPS, pp ,. [7] Selva, J., ``Effcet Multpath Mtgato Navgato Systsems'', Ph.D. Thess, Uverstat Poltècca de Cataluya, 3. [8] Sahmoud, M. ad M.G. Am, ``Fast Iteratve Maxmum- Lelhood Algorthm (FILMA) for Multpath Mtgato Next Geerato of GNSS Recevers'', 4 th Aslomar Coferece o Sgals, Systems ad Computers, 6. ACSSC '6, pp , 6. [9] Soubelle, J., I. Falow, P. Duvaut, ad A. Bbaut, ``GPS Postog a Multpath Evromet'', IEEE Tras Sgal Proc., vol. 5, o., pp. 4-5,. [] Casella, G. ad R.L. Berger, ``Statstcal Iferece'', Duxbury Thomso Learg, d edto,, Chapter. [] Msra, P. ad P. Ege, ``Global Postog System: Sgals, Measuremets, ad Performace '', Gaga-Jamua Press, d edto, 6, Chapter. [] Cohe, J. ``Statstcal Power Aalyss for the Behavoral Sceces'', Lawrece Erlbaum, d edto, 988, Chapters ad 3.