S-CURVE SHAPING: A NEW METHOD FOR OPTIMUM DISCRIMINATOR BASED CODE MULTIPATH MITIGATION

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1 S-CURVE SHAPING: A NEW METHOD FOR OPTIMUM DISCRIMINATOR BASED CODE MULTIPATH MITIGATION Thomas Pany, Markus Irsgler and Bernd Essfeller Insttute of Geodesy and Navgaton, Unversty FAF Munch, Germany BIOGRAPHY Thomas Pany has a PhD n Geodesy from the Graz Unversty of Technology and a MS n Physcs from the Karl- Franzens Unversty of Graz. Currently he s research assocate at the Insttute of Geodesy and Navgaton at the Unversty of Federal Armed Forces Munch. Hs major areas of nterests nclude GPS/Galleo software recever desgn, Galleo sgnal structure and GPS scence. Markus Irsgler s research assocate at the Insttute of Geodesy and Navgaton at the Unversty of the Federal Armed Forces Munch. He receved hs dploma n Geodesy and Geomatcs from the Unversty of Stuttgart, Germany. Hs scentfc research work focuses - among other topcs - on GNSS recever desgn and performance wth specal focus on multpath propagaton, mtgaton and montorng. Bernd Essfeller s Full Professor and Vce-Drector of the Insttute of Geodesy and Navgaton at the Unversty of the Federal Armed Forces Munch. He s responsble for teachng and research n the feld of navgaton and sgnal processng. He receved the Habltaton (vena legend) n Navgaton and Physcal Geodesy ABSTRACT The paper characterzes the optmum code multpath mtgaton n a GNSS recever whch can be acheved by a lnear combnaton of several correlators or equvalently by correlatng the ncomng sgnal wth a code-trackngreference-functon. After a revew of currently used code multpath mtgaton technques, a new technque s developed to determne postons and weghts of the correlators (or the shape of the code-trackng-reference-functon) n order to acheve the best possble multpath mtgaton performance wthn the chosen framework. It s based on the defnton of an optmum S-curve. A non-coherent code dscrmnator s formed by dvdng the coherent dscrmnator by the punctual correlator. The technque s appled to BPSK(1) and BOC(1,1) sgnals for nfnte, 16 MHz and 8 MHz bandwdth. Outsde ts lnear regon, the S-curve s slghtly offset to avod false stable trackng ponts as they occur for BOC sgnals. It s shown that wthn ths approach hgh-bandwdth BPSK and BOC sgnals acheve vrtually dentcally multpath envelopes. The resultng dscrmnator s mplemented and tested n a software recever. INTRODUCTION The optmzaton of the shape of the code dscrmnator functon (S-curve shapng) s smply the process of placng a dstnct amount of correlators along the sgnal's correlaton functon and combnng them wth the am of optmsng the trackng performance. In the presence of multpath, the resultng multpath errors can be mnmzed by a well chosen placement and combnaton of the correlators. Ths procedure depends on the shape of the sgnal autocorrelaton functon. It also nfluences the code nose performance. As a consequence, careful correlator placng and combnaton of correlator outputs can optmse the overall trackng performance. Many multpath mtgaton technques that have been proposed and mplemented durng the past years am at enhancng the multpath performance by sutable processng of correlator outputs. Some examples are brefly descrbed n the followng paragraphs. A more detaled overvew on these mtgaton technques s provded n [3]. Soon after the SatNav communty became aware of multpath as a major error source, frst approaches to modfy the standard 1-chp early-mnus-late trackng archtecture have been proposed. The reducton of the chp spacng between early and late correlator not only led to a sgnfcant enhancement wth respect to the code multpath performance, but also code nose could be sgnfcantly reduced (Narrow Correlator TM, [12]). Code nose varance and maxmum multpath errors are proportonal to the chp spacng, so that the correspondng errors can be decreased by a factor of 1 f a chp-spacng of.1 nstead of 1 s used (at least for suffcently hgh bandwdths).

2 Another example of how a careful selecton of correlator postons and a sutable combnaton of correlator outputs postvely affects the code multpath performance s the Double Delta Correlator ( ). It uses a lnear combnaton of 4 correlators (2 early correlators E1/E2 and 2 late correlators L1/L2) to set up the code dscrmnator functon D = a(e1-l1)+b(e2-l2). The actual multpath performance depends on the correlator locatons and on how the factors a and b are selected. If selected properly, the code multpath performance of the correlator can be sgnfcantly enhanced compared to the use of the Narrow Correlator (at the cost of ncreased thermal nose [6]). Examples for the mplementaton of the concept are Astech's Strobe CorrelatorTM [1,2], NovAtel's Pulse Aperture Correlator TM [7] and the Gated Correlator resp. the Hgh Resoluton Correlaton concept (HRC) dscussed n [6]. A thrd example of how the multpath performance depends on the selecton of a sutable correlator confguraton s the so-called Multpath Elmnaton Technque (MET, mplemented n some NovAtel recevers [1]). Ths approach makes use of two correlator pars located at both sdes of the correlaton peak. The correlator outputs are used to compute the slope of the ascendng and descendng part of the correlaton functon wth the am of detectng correlaton peak dstortons caused by multpath and determnng a pseudorange correcton by means of the two slopes. The multpath performance obtaned by means of ths technque extremely depends on how the four correlators are dstrbuted along the correlaton functon. The optmum correlator confguraton depends on the shape of the correlaton functon, whch s manly defned by the type of sgnal beng tracked (modulaton scheme) and the precorrelaton flter characterstcs (bandwdth and type of flter). As an example, a correlator confguraton, whch has been found to effectvely mtgate multpath for the GPS C/A code (BPSK modulaton scheme), wll most probably show a completely dfferent multpath performance for a BOC modulated sgnal. As a consequence, the correlator confguraton wll always have to be adjusted accordng to the underlyng sgnal/recever combnaton n order to optmse the multpath performance. Also the so-called E1/E2 Trackng [11] that makes use of two (early) correlators located at the ascendng part of the correlaton functon, results n resdual multpath errors that depend on the locaton and the spacng between both correlators. The general dea behnd that trackng scheme s to defne an early trackng pont (located as early as possble) nstead of usng the correlaton peak as the trackng pont. The man beneft of ths approach s that all multpath sgnals arrvng after ths (early) trackng pont do not cause any pseudorange errors. On the other hand, the nose performance decreases as the dstance of the trackng pont from the correlaton peak ncreases. Ths s a good example for the necessty of a trade-off between a good multpath performance and an acceptable nose performance. To summarze these ntroductory notes, t can be stated that the performance of many multpath mtgaton technques strongly depends on the amount and the exact locaton of the correlators as well as on how the correlator outputs are combned. Aganst ths background, t should be feasble to determne an optmum correlator confguraton for a gven sgnal/recever combnaton wth the am of optmsng the multpath performance. In the next secton we ntroduce two dfferent code trackng channel models, one smlar to a mult-correlator mplementaton, the other one uses reference functons nstead of multple correlators. Then we develop a scheme to optmze poston and weghts of the ndvdual correlators, resp. to optmze the waveform of the code-trackngreference-functon. Ths technque s appled to BPSK and BOC sgnals for 3 dfferent pre-correlaton bandwdths. The resultng S-curves and multpath envelopes are analyzed and a fnal concluson s drawn. Remark: Smlar results have been publshed by the authors n the proceedngs of the ENC-GNSS 25, Munch. Ths paper s a revsed and extended verson of ths prevous work. TRACKING CHANNEL, MODEL - I The trackng channel shown n Fg 1 receves as nput the dgtzed GNSS sgnal at IF-level s µ. The sgnal s correlated wth recever nternally generated replcas and the correlaton results are used to estmate the code trackng error as well as the phase or frequency trackng error. To close the trackng loops, ths data s fed back nto the code and carrer NCO. Fg 1 represents a conventonal trackng loop structure, wth one excepton, that multple correlators are used. Each correlator s shfted wth respect to the punctual correlator by a certan delay d off. Fg 1. Generc mult-correlator channel (model I)

3 The sampled GNSS sgnal at IF level s modelled as [8] ( ( ) ) s = Sc(t f τ )cos 2π f + f t ϕ + n. (1) µ µ c,sat sat IF D,sat µ sat µ µ Sample ndex, sample rate = f s s µ Receved navgaton sgnal samples at IF level t µ Tme of sample n [s], t µ=µ/ f s f c,sat Code rate of receved sgnal [chp/s] τ sat Code delay of receved sgnal [chp] f IF Intermedate frequency (IF) [Hz] f D,sat Doppler frequency of receved sgnal [Hz] ϕ sat Phase delay of navgaton sgnal [rad] S Ampltude of sgnal n µ Samples of bandlmted whte thermal nose c Bandlmted navgaton sgnal at baseband as functon of code phase The nternally generated replca of the IF plus Doppler shft s gven by pµ = exp{2π (fif + f D,rec )tµ ϕrec } (2) p µ ϕ rec f D,rec Internally generated Doppler frequency samples Phase delay of nternal sgnal [rad] Estmated Doppler frequency [Hz] and of the nternally generated code replca for the dfferent correlators by off r = c (t f τ d ). (3) µ ref µ c,rec rec r µ τ rec f c,rec c ref d off Samples of nternal correlator reference sgnal Code delay of nternal sgnal [chp] Code rate of nternally generated navgaton sgnal [chp/s] Internally generated reference PRN code (ev. ncludng BOC modulaton) Offset of th correlator n [chp] Note, the meanng of the varable s context dependent; t s used to denote the unt magnary number and to enumerate the correlators. The output of the th correlator after ntegrate and dump as a functon of the receved sgnal s defned as n R s = f s p r. (4) n f ( ) µ µ µ µ µ = 1 Number of samples wthn coherent ntegraton tme The coherent ntegraton tme n [s] s gven by the number of samples n f used dvded by the sample rate f s. The code trackng error s the dfference between estmated code delay of the nternal sgnal and the true delay of the receved satellte sgnal τ = τrec τ. (5) sat Usng ths defnton the output of the th correlator can be wrtten as shfted replca of the correlaton functon between the receved sgnal and the nternal generated sgnal multpled by the carrer phase trackng error [8] S 2 off ( τ ) = ( τ + ) ( ( ϕsat ϕrec )) R R d exp. (6) The functon R represents the (bandlmted) autocorrelaton functon shfted by d off. Accordng to Fg 1 the coherent code phase dscrmnator s defned as a lnear combnaton of the correlator outputs N ( τ ) = α ( τ) D R. (7) = 1 The dscrmnator s completely defned by choosng the postons d off and the weghts α of the ndvdual correlators. From (6) we see that the coherent dscrmnator depends on the carrer phase relatonshp between receved and nternal sgnal and on the ampltude of the receved sgnal. Ths unwanted dependency s removed by formng a non-coherent dscrmnator. Among a number of possbltes we choose a normalzed code dscrmnator whch s realzed by dvdng the output of the coherent code dscrmnator by the punctual correlator. The punctual correlator P(s µ ) s dentcal to the one correlator k whch has a vanshng offset d k off =, S P s R exp 2 k ( µ ) = ( τ) ( ( ϕsat ϕrec )) The non-coherent dscrmnator s gven by ( ) = µ ( µ ) ( µ ). (8) D s D s /P s. (9) TRACKING CHANNEL, MODEL - II It s mportant not to take Fg 1 lterally, n the sense that all correlators shown there have a one-to-one correspondence to a certan area of slcon n an ASIC or to some lnes of code n a software recever. Instead multple correlators are used to outlne the concept, but the realzaton of ths concept s a dfferent task. For example an effcent realzaton of the coherent dscrmnator can be acheved f the followng observaton s made,

4 D ( τ ) = αr ( τ) = 1 N nf off µ µ ref µ c,rec rec = 1 µ = 1 = α s p c (t f τ d ). (1) nf N off µ µ ref µ c,rec rec µ = 1 = 1 = s p α c (t f τ d ) = n f µ = 1 N s p c (t ) D µ µ µ Instead of usng multple correlators, the same result s obtaned by correlatng the ncomng sgnal wth a socalled code-trackng-reference-functon c D. It s a lnear combnaton of shfted replcas of the sampled PRN code and can be pre-computed and stored n memory [8,9]. Examnaton of (1) leads to a trackng loop mplementaton shown n Fg 2. Fg 3. Possble code-trackng reference functon DISCUSSION OF TRACKING CHANNEL MODELS Both trackng channel models can be adapted to many dfferent correlator schemes, lke early-late or double-delta trackng. On contrast, the chosen correlator structures represent a constrant n fndng an optmum correlator structure snce non-lnear operatons (apart from normalzaton) can not be modeled wth the chosen structures. For example the BOC(n,n) dscrmnators presented n [4] can nether be ftted nto the scheme of Fg 1 nor nto the scheme of Fg 2. The same s true for the early-power mnus latepower dscrmnator. Nevertheless, the used structures can be adapted to all possble coherent correlators. Fg 2. Reference functon trackng channel (model II) Wthn the context of model-ii, code multpath mtgaton s acheved by optmzng the code trackng reference functon. Ths s frst done for a sngle chp. The code-trackngreference-functon c D for a sgnal consstng of one sngle chp (and whch s otherwse zero) s approxmated by pece-wse constant parts, as shown n Fg 3. The extenson of the pece-wse constants parts s fxed and represents the resoluton of the functon. The heght s of the pece-wse constant parts s subject to the optmzaton procedure. The sngle-chp functon s convoluted wth the PRN code to obtan the code-trackng-reference-functon for the complete receved sgnal. REQUIREMENTS FOR OPTIMUM MULTIPATH PERFORMANCE The basc observaton for the presented multpath mtgaton method s that the coherent code dscrmnator s completely defned by two requrements: Lnearty around trackng pont Zero value outsde pull-n regon In the followng we wll argue why these two ponts are requred and why they are suffcent to completely defne the dscrmnator. The frst requrement, lnearty, s mathematcally expressed as D τ = τ for τ << 1. (11) ( ) Lnearty s requred to realze a lnear DLL. Although non-lnear trackng loop could be envsaged, ths would represent a rather unusual approach (at least for small trackng errors) and s not consdered here. If the coherent dscrmnator s lnear n τ also the non-coherent dscrmnator s to a good approxmaton lnear around the trackng pont of τ= as the punctual correlator s assumed to be nearly constant n ths regon. Exact lnearty of the non-coherent dscrmnator s obtaned f we would requre that the coherent dscrmnator follows (c.f. Fg 4)

5 D ( τ ) = R( τ) τ for τ << 1. (12) However, ths approach s not consdered here, as chosen values for the extenson of the lnear regon are small n the followng. Furthermore the slope of the coherent dscrmnator s rrelevant for the followng dscusson as a slope of 1, as expressed n (11), can always be obtaned by dvdng the coherent dscrmnator by ts frst dervatve at τ=, whch s smply a constant value. To ntroduce the second requrement we have to make the followng observatons. Frst of all, we recall that the code multpath trackng error τ s gven by solvng the steadystate condton D( τ)= for a gven multpath scenaro. It s mportant to note that ths s equvalent to solve the steady-state condton for the coherent dscrmnator D( τ ) = D( τ ) =. (13) Secondly, we observe that the coherent code dscrmnator s lnear wth respect to a lnear combnaton of the drect sgnal and multple multpath sgnals. Ths s expressed n the form of Fg 4. Optmum lnear To prove the optmalty property of the proposed code dscrmnator, let us consder two multpath scenaros: short range and long range multpath. The short range scenaro s shown n Fg 5. The code dscrmnator for the drect (blue) and the multpath sgnal (red) s shown. The multpath s slghtly delayed wth respect to the drect sgnal. The output of the coherent code dscrmnator s gven by the superposton of both values, c.f. (14). One sees, that the coherent code dscrmnator (of the combned sgnal) crosses the x-axes slghtly rght to the orgn causng a trackng error. k ( ) k D s = µ D s. (14) µ k k Note, that ths equaton s vald only for the coherent dscrmnator, but not for the non-coherent. The frst requrement s vsualzed n Fg 4 where the coherent code dscrmnator s shown as a functon of code trackng error τ. The red part of the S-curve represents the lnear regon around the orgn. The extenson of ths part should be suffcently large to cover the range of expected trackng errors caused by thermal nose, multpath or transent errors. Outsde ths regon, the best multpath performance s obtaned, f we requre that the S-curve vanshes, as wll be explaned below. Furthermore, t should be noted that a code dscrmnator should have just one stable trackng pont at τ=. The pull-n regon s defned as the range of code delay values, whch can be captured (and tracked) by the code trackng loop. In the case of Fg 4 t s equvalent to the lnear regon snce code trackng errors outsde the lnear regon result n a dscrmnator output of. In ths case the code trackng loop makes no attempt follow the receved sgnal. Fg 5. Multpath (short range) From Fg 5 t should be clear that the multpath trackng error for short geometrc multpath delay values s completely determned by the shape of the code dscrmnator for a sngle sgnal around the orgn. Snce ths part s determned by the requrement of lnearty, we proved (the well-known fact) that short range multpath can not be mtgated by any correlaton method whch s compatble wth Fg 1. Short range multpath can be partally mtgated f sgnal parameters of the drect and the multpath sgnal are estmated smultaneously usng multple DLLs [13]. But also n ths case, the varance of the estmated sgnal parameters goes to nfnty, when the delay of the multpath sgnal approaches zero. Thus we see that we can not do anythng wthn the dscrmnator to reduce multpath errors as long as the delay of the multpath sgnal s wthn the lnear regon. However, we are completely free to choose the shape of the S-curve outsde the lnear regon. Ths can be used to acheve the best possble multpath mtgaton performance wth a trackng channel scheme shown n Fg 1. It s real-

6 zed by requrng that the code dscrmnator vanshes outsde left to the lnear regon. Ths s vsualzed n Fg 6. Here the delay of the multpath sgnal s just slghtly larger than the lnear regon, but the dscrmnator output does not produce any trackng error. The second modfcaton results n a dscrmnator of Fg 8 whch lmts the mnmum and maxmum values of the S- curve. Fg 6. Multpath (long range) To put t n other words: the optmum code dscrmnator s multpath-error-free as soon as the delay of the multpath sgnal s larger then the extenson of the lnear regon. Furthermore, t should be noted that the shape of the S-curve rght to the lnear regon s rrelevant for multpath mtgaton, as multpath delays are always postve. Modfcatons to the optmum S-curve As a sde remark we present two slghtly modfed versons of the optmum code dscrmnator. The frst modfcaton results n a dscrmnator of Fg 7 whch does not vansh outsde the lnear regon, but retans a small non-zero value whose sgn s dfferent on the left and rght sde. Fg 8. Modfcaton to lmt max. multpath error Ths modfcaton reduces the maxmum multpath errors, as wll be shown n Fg 29, at the cost of a decreased lnear regon. MULTIPATH OPTIMIZATION FOR MODEL I After defnng the shape of an deal code dscrmnator D deal of Fg 5 or Fg 7, we are now n the poston to realze t by choosng proper postons d off and weghts α of the correlators. To do ths, we present two dfferent approaches: the dscrete and the contnuous approach. For the dscrete approach we defne a set of control ponts τ l. We assume those control ponts are equally dstrbuted wthn a chosen fttng range wth a gven resoluton. Values for the fttng range and the resoluton can be found n Table 1 and Table 2. The poston d off of one ndvdual correlator s assumed to be equal to the value of one these control ponts, whch means that the ndvdual correlators are also equally dstrbuted. The weghts ˆα of the correlators are calculated by mnmzng Fg 7. Modfcaton to ncrease pull-n regon By mposng ths addtonal requrement, we ncrease the pull-n regon of the dscrmnator. The pull-n regon s larger than the lnear regon and the dscrmnator s capable of followng sgnals even f they have large code trackng errors. In that case the trackng loop s non-lnear, untl trackng errors fall wthn the lnear regon. Ths modfed S-curve s especally mportant f the lnear regon s kept small to lmt maxmum multpath errors. By mposng ths condton we can also acheve for BOC sgnal trackng that the code dscrmnator has only one stable trackng pont (see below). ( ( l ) deal ( l )) τ 2 D D τ mn. (15) l Ths equaton can be rewrtten as α ˆ = 2. (16) N off = mn α ( τ + ) R l d Ddeal ( τl ) { α } l = 1 We essentally try to ft the deal S-curve by a lnear combnaton of shfted autocorrelaton functons. The estmated weghts ˆα are obtaned by solvng ths equaton usng standard methods of least-squares-adjustment. Thus the

7 dscrete approach tres to ft the deal S-curve by a fnte amount of coeffcents. The result of ths ft (the ftted S- curve) s n general not n exact agreement wth the deal S-curve. The agreement s better when more control ponts are used and f the shape of the correlaton functon R s sutable to reproduce the deal S-curve. Ths last statement leads us to the second approach, the contnuous approach. In the second (contnuous) approach, we generalze (7) and allow an nfnte number of correlators accordng to D d R d d d. (17) off off off ( τ ) = α( ) ( τ + ) ( ) off d = For all possble delay values d off rangng from mnus nfnty to plus nfnty, there exsts one correlator. The ndvdual weght of ths correlator s gven by α(d off ). Note, that the correlator weght s now a real-valued functon of one argument. We now ntroduce the Fourer Transform of real-valued functons of one argument by { D }( k) F =. (18) 1 = ( τ) { τ} ( τ) π D exp k d 2 τ= The Fourer Transform s appled on the deal dscrmnator, on the correlaton functon and on the weght functon. After usng the convolutonal theorem we can wrte (17) as F F{ α}( k) = conj { D deal }( k) F{ R}( k), (19) whch completely defnes the weght of the ndvdual correlators. It needs however to be clear, that the above expresson s not always well defned, as dvson by zero may occur. Ths s especally mportant f bandlmted sgnals are consdered. The Fourer Transform of the correlaton functon R equals the power spectral densty of the receved bandlmted sgnal (Ths statement s strctly vald, f the receved sgnal s correlated wth an nfnte bandwdth replca and f an deal rectangular precorrelaton flter s used). In ths case F{R}( k) vanshes f k s outsde the passband regon of the pre-correlaton flter and dvson by zero occurs. For those value of k, the Fourer Transform of the deal F{ D deal }( k) s n general non-zero (dependng on the actual extenson of the lnear regon) and F{α}( k) s not defned. As a consequence, no weght functon exsts and (17) has no soluton. Thus one sees that hgh sgnal bandwdth s a crucal factor to acheve good multpath mtgaton potental wth a sngle dscrmnator based approach. For the remander of ths paper, we wll not consder the contnuous approach further. All further nvestgatons are based on the dscrete approach, manly for ts computatonal smplcty. Result for BPSK Sgnals To evaluate the postons and weghts for an optmum BPSK code dscrmnator we consder a BPSK(1) sgnal. The obtaned results can be also appled to BPSK(n) sgnals provded that the rato between chp rate (here 1.23 Mchp/s) and sgnal bandwdth s constant. For example, ths s fulflled for the nfnte bandwdth case for all values of n. The receved sgnal s assumed to be of lmted bandwdth and three dfferent pre-correlaton bandwdth values are consdered (nfnte bandwdth, MHz and MHz). An deal (rectangular) low-pass flter s used. We choose the extenson of the lnear regon accordng to Table 1. Dscrete correlators were placed wthn the fttng range wth the gven resoluton. It should be noted that the fttng regon also nclude parts of the multpath-rrelevant regon of Fg 4. Ths was done to get symmetrc weght coeffcents α. Below another case wll be presented, where the fttng regon excludes ths multpath rrelevant part of the S-curve. The least-squares algorthm fnds for each correlator a weghtng factor n order to ft the optmum S-curve wthn the fttng range. A very small offset (c.f. Fg 7) was ntroduced for the two fnte bandwdth cases n order to extend the pull-n regon of those dscrmnators. Table 1. Settngs for BPSK sgnals Scenaro Inf. BW 16 MHz 8 MHz Lnear regon [+/- chp] Ft range [+/- chp] Resoluton [chp] Bandwdth [MHz] Infnte 16 8 Offset [chp].5.2 Fg 9 shows postons and weghts of the ndvdual correlators for the nfnte bandwdth case. Frst of all, one sees that the correlators are regularly dstrbuted wthn the fttng range (+/- 1.5 chps) wth a resoluton of.5 chp. Most of the correlators have a weght of and are thus rrelevant to compute the coherent code dscrmnator. One of them s the punctual correlator. The remanng correlators are ant-symmetrcal around the orgn. The 4 correlators nearest to the orgn form a lke correlator. They

8 are equvalent to two narrow correlators, one wth a correlator spacng of.1, the other one wth a correlator spacng of.2. The more nner correlator par has twce the ampltude as the outer correlator par. Furthermore we observe two lke correlator structures around +/- 1 chp. They ensure, that the resultng S-curve vanshes wthn the fttng regon of (+/- 1.5 chp). Ths s not the case for the standard correlator, where non-vanshng S-curve values are located around +/- 1 chp. Correlator Weght Correlator Dagram Fg 9. Correlator dagram for nfnte bandwdth BPSK sgnal The resultng S-curve for the nfnte bandwdth case s shown Fg 1. Here we show the deal S-curve (blue) and the S-curve of the coherent dscrmnator (green, nearly completely below the red curve), whch s the result of the dscrete fttng procedure. We also show the S-curve of the non-coherent dscrmnator (red). Note, that the latter one s not nvolved n the fttng procedure. Thus devatons of the red curve from the deal curve are expected whch ncrease wth ncreasng code phase values Obtaned Ft Normalzed In Fg 1 we observe that the deal S-curve s well reproduced wthn +/- 1 chp. The deal S-curve returns faster to zero than the ftted S-curve. Ths s a drect consequence of the used resoluton (here.5 chp). The correlator dagram for the 16 MHz bandlmted BPSK(1) sgnal s vrtually dentcal to Fg 9. By vsual nspecton no dfference can be observed and thus t s not shown here. The resultng S-curve for the 16 MHz case s shown Fg Obtaned Ft Normalzed Fg 11. S-curve for 16 MHz bandwdth BPSK sgnal Compared to Fg 1 the ftted S-curve (green) looks more smooth, but otherwse essentally dentcal. Addtonally, the small offset s vsble, whch fnally ncreases the pulln regon. By mposng ths offset, we also avod spurous trackng ponts n the normalzed code dscrmnator, whch are caused by the band lmtaton and normalzaton. Due to the normalzaton, the S-curve approaches +/- nfnty outsde +/- 1 chp. In a recever, ths does not cause any problems, snce for trackng errors larger than +/- 1 chp, the loss-of-lock detector wll be actvated and reacquston of the satellte sgnal wll be ntated. The results for the 8 MHz case look dfferent to the frst two cases, because we had to ncrease the sze of the lnear regon. We were not able to reproduce the deal S-curve of the frst two cases usng a 8 MHz correlaton functon. The resultng correlator dagram for the 8 MHz case s shown n Fg 12. Agan we observe lke correlator structure around Fg 1. S-curve for nfnte bandwdth BPSK sgnal

9 Correlator Weght Correlator Dagram Fg 12. Correlator dagram for 8 MHz bandwdth BPSK sgnal The ftted S-curve s shown n Fg 13. Agan, the deal S- curve s reproduced, but we would lke to stress agan, that ths was only possble snce the extenson of the lnear regon was ncreased compared to the frst two cases. For mplementaton n a real recever, a good compromse between sgnal bandwdth, extenson of the lnear regon and eventually some further small modfcatons to the deal S-curve (lke the offset value) must be found. Ths s however beyond the scope of ths paper, where the theoretcal outlne of the concept s presented Obtaned Ft Normalzed Fg 13. S-curve for 8 MHz bandwdth BPSK sgnal In Fg 13, we also see substantal devatons of the normalzed (non-coherent) S-curve from the coherent S-curve. However, f we assume small to medum code trackng errors of about a few meters (lets say +/- 5 m), ths corresponds to a code phase value of +/-.2 chps n case of a BPSK(1) sgnal. Wthn ths area the normalzed dscrmnator ncely follows the deal S-curve. Result for BOC Sgnals The same procedure as for BPSK sgnals s repeated for BOC(n,n) sgnals. Agan we present results for a BOC(1,1) sgnal keepng n mnd that the results can be transferred to BOC(n,n) sgnals f the rato between the code rate and the bandwdth s constant. The settngs for BOC sgnals are lsted n Table 2. Table 2. Settngs for BOC sgnals Scenaro Lnear regon [+/- chp] Inf. BW 16 MHz 8 MHz non-sym. 8 MHz Ft range [+/- chp] [-2,.2] Resoluton [chp] Bandwdth [MHz] Infnte Offset [chp] The correlator dagram for the nfnte bandwdth case s shown n Fg 14. Ths dagram s more dffcult to nterpret than n the BPSK case, but a correlator structure s located around and +/-.5 chps. At +/- 1 chp there s another correlator structure. Altogether they ensure, that the S-curve vanshes wthn the fttng regon of +/- 1.5 chp (but outsde the lnear regon). The correlator dagram for the 16 MHz sgnal s shown n Fg 16. A correlator structure can be seen around chp, but otherwse no nterpretaton can be gven. The resultng S-curves for the nfnte bandwdth case and for the 16 MHz case are shown n Fg 15 and Fg 17. Both look smlar, but the band lmted curve s smoother. The normalzed S-curve shows n both cases a sngularty at +/- 1/3 chp. Ths s the pont where the BOC(n,n) autocorrelaton functon changes ts sgn. Thus the pull-n regon n both cases s lmted to +/- 1/3 chp. If the code trackng error due to some reasons becomes larger than +/- 1/3 chp, the DLL wll not follow the sgnal anymore. Instead t wll further ncrease the code trackng error untl a loss-of-lock event. Thus no stable false lock ponts occur. The sze of the pull-n regon corresponds to trackng errors of about +/- 1 m n case of a BOC(1,1) sgnal and trackng errors of ths sze are very unlkely to occur. If they occur, the value of the offset determnes the tme untl

10 DLL drves the channel to a loss-of-lock event and to reacquston. The hgher the offset value, the shorter ths tme s. Durng ths tme however, code pseudorange measurements of ths channel wll contan rather bg errors. 6 4 Correlator Dagram 6 4 Correlator Dagram Correlator Weght 2-2 Correlator Weght Fg 16. Correlator dagram for 16 MHz bandwdth BOC sgnal Fg 14. Correlator dagram for nfnte bandwdth BOC sgnal Obtaned Ft Normalzed Obtaned Ft Normalzed Fg 15. S-curve for nfnte bandwdth BOC sgnal Fg 17. S-curve for 16 MHz bandwdth BOC sgnal The correlator dagram for the 8 MHz BOC sgnal s shown n Fg 18. An nterpretaton s hard to gve, but one should note, that t s symmetrc wth respect to the orgn. Ths symmetry requrement was not ntroduced a pror nto the least-squares adjustment procedure; nstead t s a result of t.

11 .5 Correlator Dagram.3 Correlator Dagram Correlator Weght Correlator Weght Fg 18. Correlator dagram for 8 MHz bandwdth BOC sgnal The correspondng S-curve s shown n Fg 19. The ftted coherent S-curve follows generally the deal S-curve, but s not able to reproduce the sharp edges of the deal S-curve. The pull-n regon of the normalzed non-coherent S-curve s lmted to +/- 1/3 chp and a bgger offset had to be ntroduced n order to avod stable trackng ponts outsde the pull-n regon Obtaned Ft Normalzed Fg 19. S-curve for 8 MHz bandwdth BOC sgnal Another ft for the 8 MHz BOC sgnal was performed, where the fttng regon was chosen to range from -2 to.2 (.e. not to be symmetrc wth respect to the orgn). Accordng to Fg 4 t was chosen to cover only those parts of the S-functon whch are actually relevant for trackng. The resultng correlator dagram s shown n Fg 2. Correlators left to -1 chp get a weght of. Fg 2. Correlator dagram for 8 MHz bandwdth BOC sgnal (non-symmetrc) The correspondng S-curve s shown n Fg 21. It can be seen, that the deal S-functon s reasonably well reproduced wthn the fttng regon (especally wthn the lnear regon). Outsde ths regon (>.2 chp) the S-curve s completely unconstraned. The normalzed S-curve shows sngulartes at +/- 1/3 chp. A stable trackng pont occurs at about.75 chp. Comparng the symmetrc case (Fg 19) to the non-symmetrc case (Fg 21), we see that the deal S- curve s slghtly better reproduced n the non-symmetrc case. However, the occurrence of a second stable trackng represents a major dsadvantage Obtaned Ft Normalzed Fg 21. S-curve for 8 MHz bandwdth BOC sgnal (nonsymmetrc)

12 MULTIPATH OPTIMIZATION FOR MODEL II To demonstrate the optmzaton procedure for the second trackng channel model we consder nfnte bandwdth sgnals only and apply the two modfcatons to the optmum S-curve (offset to ncrease pull-n regon and clppng to reduce maxmum multpath errors). The results apply to arbtrary code rates. Table 3. Model II settng for BPSK and BOC(n,n) sgnals Name Value Lnear regon [+/- chp].1 Ft range [+/- chp] [-1,+1] Resoluton [chp].5 Bandwdth [MHz] nfnte Offset [chp].1 Clp.4 The presented settngs represent a reasonable compromse, between sze of pull-n regon, sze of lnear regon and maxmum multpath errors. Those settngs have been mplemented n a software recever (see later) and proved to work stable. Result for BPSK Sgnals Fg 22 shows the modfed optmum S-curve a BPSK sgnal. The deal and the ftted S-curve nearly concde, only the slope at the outer border of the lnear regon s steeper n case of the deal case. Ths dscrepancy s a drect consequence of the fnte resoluton of the code-trackngreference-functon Obtaned Ft Normalzed Fg 22. S-curve for nfnte bandwdth BPSK sgnal (model-ii) Fg 23 shows the correspondng code-trackng reference functon. When code trackng reference functon and the receved (nfnte bandwdth) sgnal are algned, then the sngle chp falls exactly wthn the [-.5, +.5] chp regon. As soon as a code trackng error occurs, the receved chp correlates wth one of the two nner bars of Fg 23. Functon Code Trackng Reference Functon Fg 23. Code trackng reference functon for BPSK sgnal Result for BOC Sgnals Fg 25 shows the modfed optmum S-curve for a BOC(n,n) sgnal. Apart from the sngularty, Fg 25 looks dentcal to Fg Obtaned Ft Normalzed Fg 24. S-curve for nfnte bandwdth BOC sgnal (model- II) Fg 25 shows the code-trackng-reference-functon for the BOC(n,n) case. Compared to the BPSK case, t looks more complex. Ths s caused by the sgn change of the BOC(n,n) sgnal wthn one chp.

13 Functon Code Trackng Reference Functon Fg 25. Code trackng reference functon for BOC(n,n) sgnal MULTIPATH ENVELOPES The last step of our evaluaton procedure conssts of analyzng and comparng the obtaned multpath envelopes. They are defned as the mnmum and maxmum code trackng errors plotted as a functon of the geometrc delay of one multpath sgnal. The ampltude of the multpath sgnal s assumed to be half of the ampltude of the drect sgnal (-6 db). pull-n regon and by doublng the extenson of the lnear regon. Ths leads us to the conjecture, that the same optmal (sngle coherent dscrmnator based) multpath performance can be acheved wth all possble navgaton sgnals, provded the sgnal bandwdth s hgh enough. In further nvestgaton we wll try to prove ths statement analytcally. Mn./Max. Multpath Error [chp] Multpath Envelope BPSK BOC Geometrc Delay [chp] Fg 26. Multpath envelopes for nfnte bandwdth BPSK(1) and BOC(n,n) sgnals (model I) Trackng Channel Model I In the followng we always compare the BPSK wth the BOC case for all three dfferent bandwdth values. We show the error envelope only wthn -1 chp, because multpath delays larger than 1 chp (.e. larger than 293 m for 1.23 Mchp/s) are extremely rare to occur [5]. It should however be noted, that the multpath error envelopes can be non-zero for delay values outsde the fttng regon. If ths demonstrates to be problematc for a certan applcaton, the fttng regon needs to be ncreased. Here we stck to +/- 1.5 resp. +/- 2 chp, whch we thnk s already a reasonable hgh value. Fg 26 shows the envelope for both the nfnte bandwdth BOC and the nfnte bandwdth BPSK sgnal. It s somehow surprsng to observe that both look vrtually dentcal. Ths curve represents the optmum multpath performance, whch s achevable wth a sngle dscrmnator and the assumed requrements (especally lnearty). One sees, that for nfnte bandwdth, the multpath performance s manly determned by the extenson of the lnear regon and not by the modulaton scheme of the satellte sgnal. The same s to a good approxmaton also true for the 16 MHz case, whch s shown n Fg 27. The dfference of Fg 27 to Fg 26 s caused by mposng an offset to ncrease the Mn./Max. Multpath Error [chp] Multpath Envelope BPSK BOC Geometrc Delay [chp] Fg 27. Multpath envelopes for 16 MHz bandwdth BPSK(1) and BOC(1,1) sgnals (model I) Fg 28 shows the multpath envelope for the 8 MHz case. We observe for the BOC sgnal a larger multpath envelope, whch s caused by the undulated structure, especally the reduced slope at the orgn, of the BOC S-curve wthn the lnear regon.

14 Mn./Max. Multpath Error [chp] Multpath Envelope BPSK BOC BOC (asymmetrc) THE NO-MULTIPATH-ERROR-S-CURVE A coherent code dscrmnator whch s completely free of any multpath errors can be (theoretcally) obtaned f the S-curve of the coherent code dscrmnator takes the shape shown n Fg Geometrc Delay [chp] Fg 28. Multpath envelopes for 8 MHz bandwdth BPSK(1) and BOC(1,1) sgnals (model I) Trackng Channel Model II In case of the second trackng channel model, we dd only consder nfnte bandwdth sgnals, and n ths case the deal S-curve vrtually perfectly reproduced (c.f. Fg 26) by the fttng procedure. As the assumed deal S-curve s dentcal for BPSK and BOC(n,n) sgnals, the resultng multpath envelopes are also dentcal and are shown n Fg 29. Mn./Max. Multpath Error [chp] Multpath Envelope BPSK and BOC(n,n) Geometrc Delay [chp] Fg 29. Multpath envelopes for nfnte bandwdth BPSK(1) and BOC(n,n) sgnals (model II) Fg 3. Perfect coherent S-curve As can be seen from Fg 3 a multpath sgnal does not cause a code trackng error. Unfortunately t seems that such a coherent dscrmnator can not be realzed. The correspondng code-trackng-reference-functon s composed of Drac delta-functons, as shown n the case of an nfnte bandwdth BPSK sgnal n Fg 31. Fg 31. Perfect code-trackng reference functon for BPSK sgnals Snce t s mpossble to generate a Drac delta-functon, as well as t s mpossble to transmt and receve an nfnte bandwdth BPSK sgnal, ths coherent dscrmnator remans fcton. Furthermore t would result n a hghly nonlnear code trackng loop. In addton t should be noted, that an S-curve wth a shape of Fg 3 can be reproduced by non-lnear operatons, but n ths case the whole dscusson presented here does not apply and the perfect multpath mtgaton performance wll not be obtaned ether. Compared to Fg 26, Fg 29 shows the effect of the ncreased pull-n regon and of clppng the S-curve values. The frst effect shows up n non-zero multpath errors for delay values larger than.16 chp. The second modfcaton results n flat top parts of the envelope.

15 VERIFICATION WITH A SOFTWARE RECEIVER The code-trackng-reference-functon wth parameters lsted n Table 3 has been mplemented n the GNSS software recever pexsr [9]. The man objectve of ths verfcaton was to assess the thermal nose behavor of the new dscrmnator and to verfy proper operaton. A GPS C/A code sgnal has been tracked, whch was generated by a Sprent STR476 GPS sgnal smulator. 1 satelltes were tracked smultaneously, wth a hgh sgnal-to-nose rato. Code Trackng loop bandwdth was 1 Hz. The same sgnal was tracked by two recevers, whose only dfference was the used code dscrmnator. A conventonal early-late dscrmnator (d=.1) was compared to the BPSK S-curve shapng dscrmnator of Table 3. The resultng scatter-plot of the obtaned postons s shown n Fg 32. Note that here no multpath was present, as well as all other errors (orbt, clock and atmosphere) were absent. From Fg 32 one sees the proper operaton for the S-curve shapng dscrmnator, wth a postonng accuracy n the 1-meter regon. It s a well-known fact, that multpath mtgatng dscrmnators (lke the double-delta dscrmnator) have a slghtly reduced thermal nose performance. Multpath causes postonng errors at a few meter level and thus good multpath mtgaton can n general be seen to be more mportant than good thermal nose performance. Fg 32. Verfcaton (thermal nose assessment) of code-trackng-reference-functon, for GPS C/A code

16 CONCLUSIONS Based on two generc trackng channel structures we presented a method to calculate weghts of ndvdual correlators to obtan an optmum S-curve or to buld an optmum code-trackng-reference-functon. The method works very well wth BPSK sgnals and the deal S-curve can be reproduced. The resultng dscrmnator s very smlar to a double-delta dscrmnator. One reason for ths good fttng performance les n the fact that the BPSK correlaton functon (a trangle) s somehow smlar to the deal S-curve. When applyng the method to BOC sgnals, we obtan the same optmum S-curve as long as the sgnal bandwdth s hgh enough. However, more correlators are needed compared to BPSK sgnals. Ths les n the fact, that the shape of BOC correlaton functon s more complex. Furthermore, for 8 MHz bandwdth, the obtaned S-curve shows more devatons from the optmum S-curve compared to the 8 MHz BPSK sgnal. Our nvestgaton leads us to the conjecture, that the best possble multpath mtgaton performance wth a sngle coherent dscrmnator s a feature of the sgnal bandwdth and the sze of lnear regon of the correspondng code dscrmnator and not whch modulaton scheme s used. We wll try to prove ths statement n future work analytcally. When comparng the two trackng channel models (model I: correlator based / model II: trackng-functon based) no dfference can be found as long as nfnte bandwdth sgnals are consdered. Model II has not been appled to fnte bandwdth sgnals yet. Model II represents a trackng channel n a software recever [9] and an optmzed codetrackng-reference-functon for the GPS C/A code has been successfully verfed and compared to an early-late dscrmnator. [5] E. Lutz, M. Werner, and A. Jahn. Satellte Systems for Personal and Broadband Communcatons, Berln: Sprnger, 2. [6] McGraw, G. A. and Braasch, M. S., "GNSS Multpath Mtgaton Usng Gated and Hgh Resoluton Correlator Concepts," Proc. ION-NTM 1999, San Dego. [7] NOVATEL INC. OEM4 Famly of Recevers, User Manual Vol. 1: Installaton and Operaton. OM-246, pp [8] Pany, T. and Essfeller, B., Code and Phase Trackng of Generc PRN Sgnals wth Sub-Nyqust Sample Rates NAVIGATION, vol. 51, no. 2, pp , 24. [9] Pany, T., Irsgler, M., Essfeller, B., and Fürlnger, K., "Performance Assessment of an Under Samplng SWC Recever for Smulated Hgh-Bandwdth GPS/Galleo Sgnals and Real Sgnals," Proc. ION-GPS 23, Portland, pp [1] Townsend, B. and Fenton, P., "A Practcal Approach to the Reducton of Pseudorange Multpath Errors n a L1 GPS Recever," Proc. ION-GPS 1994, Salt Lake Cty. [11] van Derendonck, A. J. and Braasch, M. S., "Evaluaton of GNSS Recever Correlaton Processng Technques for Multpath and Nose Mtgaton," Proc. ION-NTM 1997, Santa Monca. [12] van Derendonck, A. J., Fenton, P., and Ford, T., Theory and Performance of Narrow Correlator Spacng n a GPS Recever NAVIGATION, vol. 39, no. 3, pp , [13] van Nee, R. and et al., "The Multpath Estmatng Delay Lock Loop: Approachng Theoretcal Accuracy Lmts," Proceedngs of the IEEE Poston, Locaton and Navgaton Symposum, Las Vegas. REFERENCES [1] Garn, L. and Rousseau, J., "Enhanced Strobe Correlator Multpath Rejecton for Code & Carrer," Proc. ION- GPS 1997, Kansas Cty. [2] Garn, L., van Dggelen, F., and Rousseau, J., "Strobe & Edge Correlator Multpath Mtgaton for Code," Proc. ION-GPS 1996, Kansas Cty. [3] Irsgler, M. and Essfeller, B., " Comparson of Multpath Mtgaton Technques wth Consderaton of Future Sgnal Structures," Proc. ION-GPS/GNSS 23, Portland. [4] Julen, O., Cannon, M. E., Lachapelle, G., and Mongréden, C., "A New Unambguous BOC(n,n) Sgnal Trackng Technque," Proc. ENC-GNSS 24, Rotterdam.