Doppler-aided positioning in GNSS receivers - A performance analysis

HAL Id: hal-02975126

https://hal.archives-ouvertes.fr/hal-02975126

Submitted on 22 Oct 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Doppler-aided positioning in GNSS receivers - A performance analysis

François Vincent, Jordi Vilà-Valls, Olivier Besson, Daniel Medina, Eric Chaumette

To cite this version:

François Vincent, Jordi Vilà-Valls, Olivier Besson, Daniel Medina, Eric Chaumette. Doppler-aided

positioning in GNSS receivers - A performance analysis. Signal Processing, Elsevier, 2020, 176,

pp.107713-107724. �10.1016/j.sigpro.2020.107713�. �hal-02975126�

an author's https://oatao.univ-toulouse.fr/26812

https://doi.org/10.1016/j.sigpro.2020.107713

Vincent, François and Vilà-Valls, Jordi and Besson, Olivier and Medina, Daniel and Chaumette, Eric Doppler-aided

positioning in GNSS receivers - A performance analysis. (2020) Signal Processing, 176. 107713-107724. ISSN

0165-1684

Doppler-aided positioning in GNSS receivers - A performance analysis

François Vincent

^a,∗

, Jordi Vilà-Valls

^a

, Olivier Besson

^a

, Daniel Medina

^b

, Eric Chaumette

^a

aUniversity of Toulouse/ISAE-SUPAERO, 10 Avenue Edouard Belin, Toulouse, France

bInstitute of Communications and Navigation, German Aerospace Center (DLR), Neustrelitz, Germany

Keywords:

GNSS Positioning Cramér-Rao bound Fisher information matrix Multilateration High-sensitivity

Harsh propagation conditions

a b s t r a c t

ThemainobjectiveofGlobalNavigationSatelliteSystems(GNSS)istopreciselylocateareceiverbasedon thereceptionofradio-frequencywaveformsbroadcastedbyasetofsatellites.GivendelayedandDoppler shiftedreplicasoftheknowntransmittedsignals,themostwidespreadapproachconsistsinatwo-step algorithm.First,thedelaysandDopplershiftsfromeachsatelliteareestimatedindependently,andsub- sequentlytheuserpositionandvelocityarecomputedasthesolutiontoaWeightedLeastSquares(WLS) problem.Thissecondstepconventionallyusesonlydelaymeasurementstodeterminetheuserposition, although Dopplerisalsoinformative. The goalofthispaperisto providesimpleand meaningfulex- pressionsofthepositioningprecision.TheseexpressionsareanalysedwithrespecttothestandardWLS algorithms,exploitingtheDopplerinformationornot.Wecanthenevaluatetheperformanceimprove- mentbroughtbyajointfrequencyand delaypositioningprocedure.Numerical simulationsassess that usingDopplerinformationisindeedeffectivewhenconsideringlongobservationtimes,andparticularly usefulinchallengingscenariossuchasurbancanyons(constrainedsatellitevisibility)ornearindoorsit- uations(weaksignalconditionswhichneedlongintegrationtimes),thusprovidingnewinsightsforthe designofrobustandhigh-sensitivityreceivers.

1. Introduction

The main objective of Global Navigation Satellite Systems (GNSS)istoprovideprecise position,velocity andtime(so-called PVT solution) to any user on Earth, thanks to the transmission of electromagnetic (EM) signalsbroadcasted from a constellation ofsatellites[1].ThePVT estimatesareobtainedby exploitingthe modifications that these EM waves undergo during their travel fromthedifferentsatellitesinviewtothereceiver.Moreprecisely foranykindofband-limitedtransmittedsignal

e

(

^t

)

=c

(

^t

)

^e2iπf0t, (1)

where c(t) represents the baseband signal and f₀ the carrierfre- quency,thereceivedsignalr(t)canbewritten,uptoascalingfac- torandunderthenarrowbandassumption[2,3],as

r

(

^t

)

⁼

α

^e

(

^t−

τ )

⁺ⁿ

(

^t

)

⁼

α

^c

(

^t−

τ )

^{e2iπf0(t−τ )+n}

(

^t

)

^{, (2)}

where

α

^{is a complex channel gain and n(t) is a thermal noise}

term,whichingeneralmayalsoincludeanyotherunmodelledef- fectsuchasmultipath.Forshorttimeperiods,giventhetransmit- ter toreceiverranged₀ andrangerated˙₀,thedelay

τ

^canbecan

∗ Corresponding author.

E-mail address: [email protected] (F. Vincent).

beapproximatedbyafirstordermodel,

τ

^=d_c^0+d_c^˙0t=

τ

⁰⁺

^{t, (3)}

sothat r

(

^t

)

c

(

^t−

τ

0

)

^{e−2iπf0τ0e−2iπf0t}

e^2iπf0t (4)

wherethedelaydrift

^{canbeneglectedinsidethebasebandsig-}

nalc(t).

From (4)it is easy toidentify that the transmittedsignal un- dergo3mainmodifications,namely,apuredelayofthebaseband signal,

τ

0,aconstantphaseshift,−2i

π

^f0

τ

0,andafrequencyshift,

−f₀

^{.Thetwo firsteffects arelinked tothe distanced}0,whereas thelatter,so-calledDoppler effect,is linkedto therangerated˙0. Bothrangeandrangerateareinturnrelatedtothereceiverposi- tionandvelocity tobe inferred.The standardwayto exploitthis informationtoestimatethereceiverPVTfromthereceptionofEM wavesfrommultiplebeacons(i.e.,satellites)istofollowatwo-step approach.Inthefirststep,therangeandrangerateforeachtrans- mittertoreceiverlinkareestimated, thuscomputingasetofesti- matestoallvisiblesatellitesinparallel.Thiscanbedonethanksto theGNSSsignal design wherethedifferenttransmitted c(t) must have a very low cross-correlation among different satellites.The goalofthesecondstepistofusetheseindividualtransmittertore- ceiverestimatessolvingamultilaterationproblem,whichisusually https://doi.org/10.1016/j.sigpro.2020.107713

donethrough a WeightedLeast Square (WLS)procedure [4]. This populartwo-step approach has beenshown to be asymptotically equivalent[5]totheone-stepMaximumLikelihood(ML)solution, so-calledDirect Positioning Estimator (DPE) [6,7], which directly estimatesthereceiverpositionandvelocityfromthereceivedsig- nal.Althoughthetwo-stepprocedureisusuallysuboptimalinreal- lifenon-nominalconditions,thehighcomputationalcomplexityof theDPEpreventsitsuseinmass-marketapplications(i.e.,DPEim- pliesaMLsearchonahigh-dimensionalspace).Thatisthereason whytheconventionaltwo-stepsolutionisstillthegoldstandard.¹

As it has beenpointedout rightabove, one can exploit three differentmeasurementsfromthe receivedEMsignals toestimate thereceiverposition:

• The simplest and widespread positioning approach, beingthe state-of-the-artsolution,onlyexploits thedelaycarriedbythe baseband signal c(^t−

τ

0), conductingtothe estimationofso- called pseudoranges (i.e., pseudo because transmitters and re- ceiverare notsynchronizedandthesignal experimentsdelays duringits pass through the atmosphere).From a set of pseu- doranges a multilateration step is performed to compute the receiver position, that is, the intersection ofa set ofspheres, roughly speaking. Notice that even ifnot exploitedforthe fi- nalpositioncomputation,theDoppler shiftsmustbealsoesti- matedtoobtainacorrectdelayestimate.

• A more precise solution, consists in exploiting the additional phase informationin(4),namely, −2i

π

^f0

τ

0.Indeed,thismea- surement is linked to the wavelengthwhich is much smaller thanthebasebandsignalresolution(i.e.,foralegacyGlobalPo- sitioningSystem(GPS)L1C/Asignal,thewavelengthisapprox- imately 19cmwhilethebasebandsignalresolutionis300m).

Unfortunately, exploitingthis phase information implies solv- ingamuchmorecomplicatedproblem,mainlybecausethecar- rierphasemeasurementisambiguous(i.e.,unknownnumberof cycles inside thebaseband signal resolution), then beingsuch ambiguityresolutionthebottleneck([4],Chap21,Chap23).To thisendtwodifferentschemescanbeadvocated.Thefirstap- proach to resolve phase ambiguitiesis to turnto the class of so-calleddifferentialtechniques,wheretherelativepositionto ageo-referencedGNSSstationisobtained.Real-TimeKinemat- ics(RTK)([4],Chap26)isanexampleofsuchatechnique.Nev- ertheless,thiskindof solutionrequiresthe useofa reference stationwitha communicationlinkbetweenthetworeceivers, andisonlyvalidforshortrangesfromthebase-stationtoen- sure that thetwo receivers observethe same propagationer- rors.AnotherapproachisthefamilyofPrecisePointPositioning (PPP) techniques([4], Chap25), whichallow to getridof the referencestationbuttoreachdecimetricprecisioninturnneed (i)precisecarrierphasemeasurements,whichisnotthecasein harshpropagationconditions,(ii)highaccuracysatellite orbits, clockandpropagation(ionosphericandtropospheric)errorcor- rections,and/or(iii)multi-frequency/multi-systemarchitectures to compensate ionospheric effects. These kind of techniques received much attention in the literature (see [8] and refer- encestherein) andarestill underresearch toreachthe matu- rity neededfortheir broadreal-timeapplicability.Thepriceto bepaidistheneedtoaccessanetworkbroadcastingreal-time precisecorrections(i.e.,InternationalGNSSService(IGS) prod- ucts),andalongconvergencetimeoftensofminutes.Asstated

1Notice that DPE implies a 8-dimensional search using the baseband signal sam- ples (i.e., typically at 4, 8 or 12 Msamples/s), and the two-step approach solution is given by a 2-dimensional search per satellite (baseband) plus the PVT solution at a rate which is at most 1kHz. The 2-dimensional delay/Doppler grid search in stan- dard receivers, typically known as acquisition process, is the most time and power consuming stage of the receiver, therefore, a 8-dimensional grid search may become prohibitive in small platforms.

in[8],thesedrawbackslimittheuseofPPPformanypractical real-timeapplications.Phase-basedpositioning approachesare outofthescopeofthiscontribution.

• Finally,referring to (4), thelast measurement that can beex- ploited to infer information on the receiver position is the Doppler effect. Indeed, since this termis linked to the range rate,andbecausethepositionandvelocityvectorsofthetrans- mitterareknowna-priori,itbringsinformationonboththere- ceivervelocityandits positionthroughtheLine-of-Sight(LOS) direction.Itisimportanttonoticethatthislastinformationon theposition isan angularinformation andnot aranging one, thus supplementing the two other measurements. Although thisinformationhashistoricallybeenatthecoreoftheformer GPS,namelyTransit,itisseldomusedinmodernGPSreceivers.

ThereasonsofthislackofinterestinDopplerpositioningiscer- tainlyduetoitspoorprecision comparedtorangingmeasure- ments,atleastforshortobservationtimes.Nevertheless,itsuse isvery simple andknown tobe more robust to harshpropa- gationenvironmentssuch asurbancanyons affected by dense multipathorinindoorconditions.Oneoftherareusageofthis information in GPS receivers is an improved coarse position- ing acquisition technique where Dopplersare exploited inde- pendentlyfromthetwoother rangingmeasurementsto speed uptheinitializationofthetrackingprocess[9].

Inthis contributionwe focus onthe solutions exploitingboth delayandDopplermeasurementswiththeaimtoprovideafunda- mentalanalysisanddetermineifitisworthconsidering,andunder which conditions,Doppler informationin GNSS positioning algo- rithms. Tothisend, weprovideasimpleandstrikingformulation ofthecovariancematrixonthepositionestimationbasedonboth delayandDoppler measurements. Inthisformulation,we do not take into account the carrier phase information mentioned right above, mainlybecause thisleadsto veryspecific solutions which do not apply forstandard standalone receivers.Nevertheless, the Cramér-RaoBound(CRB)fora mixtureofrealandinteger-valued parameters, and its usefor carrierphase-based positioning tech- niquesperformancecharacterization,hasbeenderivedin[10].

When dealing withprecision, a popular way to proceed is to determine theFisher Information Matrix(FIM) orits inverse,the CRB,whichgives alower bound forthecovariancematrix ofthe estimates. In the case of GNSS receivers, the FIM associated to the one-step ML solution (i.e., DPE) has been calculated in [11], butnoinsightsontheperformanceassociatedtothedelay/Doppler two-stepapproachwereprovided.EvenifDPEhasbeenshownto be asymptotically efficient, it suffers from a huge computational burden which prevents its use in real-time applications. On the other hand, the two-step approach is suboptimal because it re- laxes the linksexisting among all delaysand Dopplers andsim- plyconsidersthemasindependentmeasurements.However,ithas beenrecentlyshowntobeasymptotically efficientwhenusingan appropriatedweighting[5].Such optimalweights areobtainedby resorting tothe EXtended Invariance Principle (EXIP) [12], which statesthat usinga re-parametrizationoftheproblemcanlead to asimpler solutionwhilepreservingtheasymptoticperformances.

Moreprecisely,theintermediateestimatesobtainedfromasimpler firststepcanberefinedtoasymptoticallyachievetheperformance oftheinitialmodelusinganappropriateWLSminimization.Obvi- ously,thisoptimalsolutionmustexploitnotonlypseudorangesto eachsatelliteinview(i.e.,delays)butalsoDopplermeasurements.

Although it is widely used in all GNSS receivers, the perfor- mance analysis of thistwo-step procedure through the determi- nation of the corresponding receiver position covariance matrix (i.e.,CRB)hasnotbeenproperlyhandledintheliterature.Indeed, [11]showstheperformancedifferencebetweenDPEandthetwo- stepprocedure,butthelatteronlyconsidersdelaymeasurements,

then missing all the information broughtby Dopplers.Moreover, to the bestof the author’sknowledge, thereis no complete(de- lay/Doppler) closed-form expression of the covariance matrixfor the positionestimatesoftheWLS two-stepprocedure.Ofcourse, the concept of Geometric Dilution Of Precision (GDOP) hasbeen introduced for a long time [1], butit only describes the second stepoftheprocessinganddoesnottakeintoaccounttheinforma- tionbroughtbyDopplers.SeveralpapersdealwiththeCRBinthe contextofradiolocation.Forinstance,inthereversecaseofsource localizationthankstosynchronizedsensors,theCRBhasbeencal- culated[13–17],butseveraldifferencespreventusingtheseresults intheGNSScase, namelythefactthat thesignal isunknown and consideredasrandominthecaseofpassivelocalization.Otherau- thors havecalculated theCRB inthe caseof GNSSReflectometry (GNSS-R)altimeters[18],butonceagain,manyintrinsicdifferences about the processing prevent fromadapting these results to the caseofGNSSpositioning.

Inthiscontributionwederivethecovariancematrixofthepo- sitionestimationforanyWLSprocedurebasedonbothdelaysand Dopplers.Thisresultisvalidforanykindofweighting, andespe- ciallyfortheoptimalWLSscheme[5]conductingtothebestpre- cision.Weobtainasimpleandmeaningfulformulationofthepre- cision one can obtainusing a GNSSreceiver,that clearlyexhibits the improvement provided by the useof Dopplers whenconsid- eringlongintegrationtimesorinconstrainedsatellitevisibility.Of course,thisformulationcanalsobeexploitedinthemorestandard way, whereonly thedelays aretaken intoaccount. Theseresults provide newinsightsto beexploitedinharsh propagationcondi- tionsandespeciallymeaningfulforhigh-sensitivityGNSSreceivers [19] (i.e.,indoor GNSS), which are expected to be at the core of precise time synchronization for next generation 5G small cells.

Noticethatwedonotconsiderspecificmultipathorindoorpropa- gationconditions,butrathershowanalyticallyandthroughsimu- lationsthatwhenusinglongintegrationtimes(i.e.,asinHS-GNSS) orunderconstrainedsatellitegeometries(i.e.,beingthecaseinur- banconditions)theperformancecanbeimprovedifproperlyusing Dopplers.

The paper is organizedas follows.First, the problem athand andthetwo-stepWLSproceduretoestimatetheuserpositionare describedinSection 2.Then, thecovariancematrix oftheseesti- mates isderivedin Section3,andsome insightsforthestandard weighting procedures are provided inSection 4.Section 5 allows to analyseinwhichconfiguration itisworth usingDoppler mea- surementsin additiontodelaymeasurements, throughnumerical simulations.ConcludingremarksareprovidedinSection6.

2. Problemstatement 2.1. Signalmodel

We assume that K scaled, delayed and Doppler-shifted front waves, transmitted by the set of satellites in view impinge on a GNSS receiverantenna.Under thenarrowband assumption,the complexbasebandmodelcanbewrittenasfollows,

y

(

^t

)

= K

k=1

α

kck

(

^t−

τ

k

)

^{e−2iπf0bkt}+n

(

^t

)

, (5)

where the phase termin (4) isabsorbed by the complex ampli- tudes

α

k.Thiscanberewritteninamorecompactformas,

y=A

α

^{+n, (6)}

where

• y=[y(⁰)...y((^N−1)^Ts)^]T, Ts being the samplingperiod and Nthenumberofcoherentavailablesamples,

• A=[a₁...a_K] is the manifold corresponding to all in- view satellite signals, with a_k=c_ke_k, where c_k= [c_k(−

τ

k)· · ·c_k((^N−1)^Ts−

τ

k)^{]T is the sampled} transmitted codeforsatellitekaffectedbythecorrespondingdelay

τ

k,and e_k=[1· · ·e^{−2iπf0bk(N−1)Ts}]^T its frequency signature due to the

f_k=−f₀b_kDoppler shift,being the element-wise product,

•

α

=[

α

1...

α

K]^T thecorrespondingcomplexamplitudes,

• n=[n(⁰)...n((^N−1)^Ts)^{]T the complexnoise, assumedto be} circularlywhiteandGaussian,withnoisepower

σ

^2.

Theobserved delay,

τ

k,anddelaydrift,b_k, dependon theac- tualrelative distanceandvelocity fromsatellitek tothe receiver, aswell as secondary propagation effects(ionospheric and tropo- spheric additional delays,...) and receiver ortransmitter defaults (clockbiasanddrift).Theycanbeexpressedasfollows,

τ

k

| |

^pk−p

| |

c +

τ

⁰⁺

δτ

k, (7)

b_k

(

^vk−v

)

^T.uk

c +b0+

δ

^bk, where

• p,v,p_kandv_k ∈R³ are,respectively,thepositionandvelocity vectorsofbothreceiverandk-thsatellite,

• u_k=

| |

^ppkk−−pp

| |

is the unitary steering vector toward the k-th satellite,

•

τ

0 andb₀ arethereceiverclockdelayanddelaydriftwithre- spectto(w.r.t)theGNSStimereference,

•

δτ

k and

δ

^bk include all secondary biases (satellites clock de- faults,propagation,...)andaresupposed tobeknownfromthe navigationmessage,

• cthecelerityofEMwaves.

The unknownsofthe positioning problemcan be gathered in vector

ζ

=[

α

^Tr

θ

^{T]T where}

α

^r=[Re

{ α

1

}

^Im

{ α

1

}

^{· · ·Re}

{ α

K

}

^Im

{ α

K

}

^]T

isthevectorofthesignalamplitudesand

θ

=[p^Tc

τ

0v^Tcb₀]^Tisthe 8-dimensionalvector correspondingto theuserposition,velocity, clockdelayanddrift.

Noticethat we can makeexplicitin (6)thedependence on

θ

^,

y=A(

θ

)

α

+n. Ifwe assume thecomplex amplitudes

α

^asdeter-

ministicandunknown,itisstraightforwardto showthat theDPE ML-basedsolutionoftheproblemisgivenbymaximizingthenon- linearfollowingcriterion[6],

θ

ˆML=argmax θ

y^HPA

( θ )

^y

(8)

where ( · )^H stands for the Hermitian transpose operation and the projection matrix onto the signal subspace, spanned by the K received signals, is P_A=A(^AHA)^{−1AH. We can observe that} A^HAࣃNI,asthe GNSSpseudo-randomcodesbroadcastedby the satellitesarealmostorthogonalandtheDopplershiftmodulations arerelativelyslowcomparedtothesignalvariations.Thisnearor- thogonality ofthe columns of A isthe assumption that allows a separate processing foreach satellite signal in all standard GNSS receiver.Therefore,wecansimplywritethat

θ

ˆMLargmax θ

A

( θ )

^Hy

²

=argmax θ

_K

−1

k=0

|

^ak

( θ )

^Hy

|

^{2 , (9)}

which is a nonlinear 8-dimensional optimization problem.

Then, because of the near orthogonality condition, the origi- nal ML position estimation can be decoupled into K individual (reparametrized) ML delay/Doppler estimation problems, which areinturnfusedtoobtaintheMLpositionsolution[5].

2.2.Standardandoptimaltwo-stepsolution

Giventhat the direct solution of(8) is difficult to implement in practice, as already stated, the classical way to estimate the

receiver position and velocity consists in a two-step procedure:

i)first, thedelays andDoppler shifts foreach satellite signal are estimated, and then ii) a WLS procedure allows to estimate the receiverpositionandvelocity.Thefirststepofthisalgorithmcor- respondstoa MLprocedureandisalsoperformedintwo stages.

Indeed, as the electronic noise is assumed to be Gaussian and white,the MLis shown tobe a 2D correlation maximization for each couple of unknowns

τ

k and b_k. Conventionally, this maxi- mizationis first performed using a loose grid (acquisition stage) andthen alocal andtight smallergrid isusedto trackthe max- imum(tracking loops)inordertoreduce thecomputationalcom- plexity.The output of the trackingstage is fed into the datade- modulationblock whichallowsto computethe so-calledpseudo- ranges.Noticethat time-delays(i.e.,pseudoranges), Dopplers(i.e., pseudorangerates)andcarrierphaseobservablesarethemainout- putsofthesynchronizationstageofanyreceiver,typicallyobtained viadelay/frequency/phaselockedlooparchitectures.Therefore,de- layandDopplermeasurementsarereadilyavailableinalmostany commercialreceiver.Thesecondstepoftheproceduretendstoes- timate

θ

^{fromthenonlinearproblemin(7).Asthereceiverusually}

getsan approximate initial solution(from theBancroft algorithm [20], for instance), the standard wayto solve this problemis to linearize(7)nearaninitialguess,

θ

0=[p₀^Tc

τ

00v^T₀cb₀₀]^T,

η

k

=

τ

k−^||pk−p_c ^0||−

δτ

k

b_k−^(vk−v_c^0)T.uk0−

δ

^bk

=1

cHk

( θ

−

θ

0

)

, (10) with

H_k=

−u^T_k0 1 0^T 0

−

ν

^Tk0 0 −u^T_k0 1

, (11)

whereu_k0=

| |

^{ppkk−−pp00}

| |

^{isthedirectionvectortowardthek-thsatel-}

litefrom the supposed position p₀ and

ν

k0=

| |

^{pPk⊥k−0vpk0}

| |

^{, withP⊥k0}

the projection matrix on the subspace orthogonal to u_k0, corre- spondstotheangularvelocityvector.Observing(11),itisnotewor- thythattheDoppler (ordelaydrift)dependsonthevelocity, but alsoon the position through thisangular velocity vector. Hence, Dopplersbringadirectpieceofinformationontheuserposition.

It is important to mention that standard GNSS receivers that useDopplerstoestimatethevelocity,usuallyassumethatthean- gular velocity vectors in the linerized matrix (11) are null, i.e.,

ν

k0=0,andthen donotexploitthetotalityoftheDopplerinfor- mation[21,Chap7].

Using the linearized observationmodel, thead-hoc procedure isaWLSclosed-formsolution,

θ

ˆ⁻

θ

0=argmin θ

c

η

^−H

( θ

⁻

θ

0

)

T

W

c

η

^−H

( θ

⁻

θ

0

)

=c

(

^HTWH

)

^−1HTW

η

⁽¹²⁾

where

η

=

η

^T1...

η

^TK

T

, H=

H^T₁...H^T_K

T

and W is the diagonal weightingmatrix,dependingonthechosenWLSscheme.Asstated before,thestandardwaytoproceed,istoconsideronlythedelay measurementsinthisWLSstep,that simplyconsistsinremoving thecorrespondinglinesinmatrixHandvector

η

^{.Twoweightsare}

conventionallyused: (i) W=I,leadingto a standard LS, or(ii)a weight related to the inverse of the measurement noise covari- ance, which is typically approximated as a function of the esti- matedsignal-to-noiseratio(SNR)and/orthedifferentsatellites’el- evation.In short,thisisrelatedto thereceived signal powerand then,uptoascalefactor,W=diag(

α α

^∗)^[4].

The optimalwayto proceedwouldbetoconsider alsothein- formationcontainedinthe Doppler.ConsideringDopplersornot,

theweightingmatrixcanbewrittenas W=

W_τ O O W_b

. (13)

where W_τ=diag(^wτ) ^{are the delays weighting and} W_b=diag

w^b

the delay drifts weighting, leaved identically nullifnotconsideredintheWLSminimization.In[5],theoptimal weightingisshowntobe

W_τ=

β

^PαandWb=

δ

^Pα (14) with

β

⁼²

π

^2NB2

3

σ

^{2 , (15)}

δ

=2

π

^2N

(

^N−1

)(

^N+1

)

^f0²T_s²

3

σ

^{2 , (16)}

and

P_α=diag

( α α

^∗

)

^{, P}α

(

^k,k

)

⁼

| α

k

|

^{2, (17)}

Bbeingthesignalbandwidth.

3. Closed-formpositioncovariancematrix(CRB)expression

The precision performance of thistwo-step procedure is con- tainedinthecovariancematrixoftheestimate

θ

ˆ^{in(12).Assuming} that the first step procedurereaches its asymptotic performance, thiscovariancematrixisobtainedas[11]

cov

( θ

^ˆ

)

=c²

(

^HTWH

)

^−1HTWF−_η^1WTH

(

^HTWH

)

⁻¹, (18) where F_η is the right-lower block of the complete FIM F_γ on theintermediateparameters

γ

=[

α

^Tr

η

^T]T,whose(k,)elementis givenby

F^k,_γ = 2

σ

^2Re

∂ (

^A

α )

^H

∂ γ

k

∂ (

^A

α )

∂ γ

, (19)

Inordertoobtainaclosed-formexpressionofthecovariancema- trix (18), we have first to compute the FIM on the intermediate parametersgivenin(19).

3.1. FIMontheintermediateparameters

As shown in Appendix A,the intermediate parameters of the FIMcanbewritteninthefollowingblockform,

F_γ =

A B B^T D

, (20)

withA= ²_σ^N2I_2K, B=4

π

^f0TsN−1n=0n

σ

^{2 diag}

0 Im

{ α

k

}

0 Re

{ α

k

}

k=1:K

,

D=8

π

²

σ

^{2 diag}

NB²|αk|²

12 0

0 f₀²T_s²

| α

k

|

^2Nn=⁻0¹n²

k=1:K

.

Usingtheblockmatrixinversionformula,itisreadilyseenthat F⁻_η¹=

D−B^TA⁻¹B

−1

(21)

=diag

σ

²

8

π

²

| α

k

|

²

₁₂

NB² 0

0

(

^f0²T_s²Var

{

ⁿ

} )

⁻¹

k=1:K

,

withVar

{

ⁿ

}

=^N(N−1₁₂^)(N+1).

Itcan benoticed thatthanks tothediagonalstructure ofF⁻_η¹, theCRBonthedelaysissimply F⁻_τ¹= _β¹P⁻_α¹,andtheCRBonthe delay drifts is F⁻_b¹= ¹_δP_α⁻¹, with P_α defined in from Eq. (17). It

is noteworthy that theoptimal weights introduced atthe endof Section 2.2,W_τ andW_b,correspond tothisFIM.Althoughthisis only an intermediate result, it isinteresting, as it givesprecisely theasymptoticprecisiononecanobtainonthedelayandDoppler measurementsincaseofGNSSsignals.

3.2. Covariancematrix(CRB)onthepositionestimation

The covariancematrix on the 8-D vector

θ

^{can be computed}

from(18)and(21).Because weare interestedinthereceiverpo- sition we focuson thefirst 4 parameters of

θ

^,pos=[p^T(^c

τ

0)^]T, correspondingtotheposition.Forthatpurposeweconductablock matrixinversionofH^TWH,whichcanbewritten

H^TWH=

(

^UWτU^T+VW_bV^T

)

^VWbU^T UW_bV^T UW_bU^T

, (22)

with U=

[−u^T₁₀ 1]^T...[−u^T_K0 1]^T

and V=

[−

ν

^T10 0]^T...[−

ν

^TK0 0]^T

.Hence,wehave

(

^HTWH

)

⁻¹=

^{−1 B} B^T −

,

where

=UW_τU^T+VW_b^1/2P_⊥W¹_b^/2V^T, B=−

^−1VWbU^T

(

^UWbU^T

)

⁻¹, with

P_⊥=I−W¹_b^/2U^T

(

^UWbU^T

)

^−1UW1_b^/2.

Then, usingthis block decomposition in (18), we can obtain the followingcovariancematrixforthepositionparametersonly, cov

(

^pos

)

=c²

^−1[

(

^UW_τ^1/2F_τ⁻¹W¹_τ^/2U^T

)

⁽²³⁾

+

(

^VW1_b^/2P⊥F_b⁻¹P_⊥W¹_b^/2V^T

)

^]

⁻¹,

where F_τ⁻¹=W_τ^1/2F⁻¹_τ W¹_τ^/2 andF_b⁻¹=W¹_b^/2F⁻¹_b W¹_b^/2 are the nor- malizedFIM onthedelayandDopplers.It hastobe noticedthat we simply have F_τ⁻¹=F_b⁻¹=I when one chooses the optimal weights(14)forW_τ andW_b.

Theresultin(23)givesthepositionprecision(i.e.,CRB)associ- atedtoanyGNSSWLSmultilaterationprocedurewhetherDopplers areused(W_b=0)ornot(W_b=0).Obviouslythisperformancede- pendsonthenumberofsatellitesandtheirpositionsthroughthe directionvectorsU,butalsoontheir velocitythroughtheangular velocity vectors V.In the caseof an optimalweighting (14), this lastexpressionsimplifiesaswehaveF_τ⁻¹=F_b⁻¹=I.Inthefollow- ingSection4weprovidetheperformancecomparisonfordifferent WLSprocedures.

4. InsightsonthestandardandoptimalWLSposition estimation

InthisSection,weaimtocomputethepositioncovariancema- trix for standard weighting matrices W andcompare the results to assess the benefits ofusing the optimalweighting, exploiting notonlydelays(^Wτ=

β

^Pα)^{butalsoDopplers}(^Wb=

δ

^Pα)^.Asthe conventional processingonly exploitthe delays,we firstconsider thecaseofpseudorangesonlymultilateration.

4.1. Multilaterationwithpseudorangesonly

Inthiscase,W_b=0,sothat=UW_τU^T,and(23)becomes cov

(

^pos

)

^=c2

(

^UWτU^T

)

⁻¹

(

^UWτF⁻_τ¹W_τU^T

)(

^UWτU^T

)

^{−1. (24)}

As statedbefore,twoprocedures areconventionallyused tocom- putethereceiverposition,namelytheLSprocedure,whereW_τ=I

andthe WLSone,wheretheoptimalweightis W_τ=

β

^Pα.Inthe LScase,wehave

cov_LS

(

^pos

)

^{= c}

β

²

⁽

^UUT

⁾

⁻¹

⁽

^UP−α1UT

⁾⁽

^UUT

⁾

^{−1. (25)}

IntheWLScase,wehave covWLS

(

^pos

)

=c²

β ⁽

^UPαUT

⁾

^{−1, (26)}

where we recall that P_α=diag(

α α

^∗) ^{(i.e., P}α(^k,k)=

| α

k

|

^{2) is}

simplythematrixofthepowersreceivedoneachsatellitechannel (conventionallymeasuredbymeansofthecarrier-to-noisedensity ratioC/N₀inGNSSreceivers).Hence,introducinganormalizeddi- rectionvectormanifoldmatrixU^T=P_α^1/2U^T,(26)reducesto covWLS

(

^pos

)

=c²

β ⁽

^UUT

⁾

^{−1. (27)}

When computing the square root of the trace of this co- variance matrix we recognize the so-called GDOP [1], through (^Tr

{

(UU^T)⁻¹

}

)^{1/2. In order to get rid of the unknown clock bias} andfocusingonthe3-Dpositionparametersonly,itisconvenient toconductablockinversionofUU^T.Itisstraightforwardtoobtain thecovariancematrixonthepositionvectoronly,p,as

covWLS

(

^p

)

=c²

β ⁽

^UcUcT

⁾

^{−1 (28)}

where Uc=[(^u10− −u₀),. . .,(^uK0− −u₀)^]P1α^/2, with −u₀=

|

^αk

|

^2uk0

|

^αk

|

^{2 the} power-weighted mean direction vector. Hence, the position precision when using a WLS procedure is linked to the inverse of the covariance matrix driven by the weighted and centred unit vectors towards the visible satellites. In the specialcasewhere all thereceived signalshave thesame power, UcUc^T is simply the covariance matrix of these unit vectors. This interpretationhasalreadybeennoticedin[22],forinstance.

4.2.Multilaterationwithbothpseudorangesanddopplers

Now, we compute the position covariance matrix in the case whereweusethecompleteinformationbroughtbytheintermedi- ateparameters(delaysandDopplers),withtheaimtodrawacom- parisonwiththeprevioussimplified,butwidelyusedcase.When usingtheoptimalweighting matrices(^Wτ=

β

^Pα,W_b=

δ

^Pα),the positioncovariancematrix(23)becomes,

covWLSopt

(

^pos

)

=c²

UF_τU^T+VF¹_b^/2P_⊥F¹_b^/2V^T

−1

. (29)

Again,introducing the power-normalizedmatrix V^T=P¹_α^/2V^T,we canrewrite(29)as

covWLSopt

(

^pos

)

=c²

β

^UUT+

δ

^VP⊥V^T

−1

, (30)

or

cov_WLSopt

(

^pos

)

=c²

β

UU^T+

δ

V⊥V_⊥^T

−1

, (31)

whereV_⊥^T=P_⊥V^T.This last expression has to be compared with (27).Wecanseethatthispositioncovariancematrix,whenusing bothdelaysandDopplers,iscomposedoftwoterms.Thefirstone isthesameasinthedelaysonlycase,andislinkedtothesignal bandwidth, through

β

^{, and the GDOP, through} UU^T. The second one,thatwillhaveatendencytoreduce thecovariancematrix,is linked to the observation time, through

δ

^{anda kindof angular}

velocityGDOP,through V⊥V_⊥^T.Thislast matrixisalsosimilarto a covariancematrixdrivenbytheangularvelocityvectorscontained inV,afteraprojectionontothesubspaceorthogonaltoU^T.Hence, themoresatelliteswe have,thesmallerV⊥V_⊥^T is,asonlythepart

Fig. 1. Skyplot of the complete satellite configuration.

inthesubspaceorthogonaltothe4DsubspacespannedbyU^T re- mains.Togoastepfurther,usingthematrixinversionlemma,we have

1

c²covWLSopt

(

^pos

)

=

(

UU^T

)

⁻¹

β

−

(

^UUT

)

⁻¹

β

^V⊥

I

δ

^+V⊥T

⁽

^UU

T

)

⁻¹

β

^V⊥

−1

V_⊥^T

(

^UUT

)

⁻¹

β

^{. (32)}

As noticed in [5], for a small integration time, V_⊥^T(UU_β^T)−1V⊥ is muchsmaller than _δ^I so that wecan drawthe followingapprox- imation

1

c²covWLSopt

(

^pos

) (

UU^T

)

⁻¹

β

⁻

⁽

^UU

T

)

⁻¹

β

^r

⁽

^VP⊥UVT

⁾⁽

^UUT

⁾

^{−1, (33)}

wherer=_β^δ = ^f02Ts2(N−_B2^1)(N+1).

Thisapproximationshowsthat thepositioncovariancematrix, when using the appropriate delay and Doppler WLS scheme, is the one we obtained when using the delays only, but reduced bya correctionmatrix.Thisimprovementcorrectionmatrixisin- verselyproportionaltothecovariancematrixonthedirectionvec- tors,UU^T,whichshowsthattheimprovementwhenincludingthe Doppler informationwill belargerincaseofbadgeometries(i.e., bad GDOP).Inother words,we canexpect a betterimprovement incaseofchallengingenvironments,suchasurbancanyons,forin- stance.

5. Numericalsimulations

To evaluate the gain provided by the optimal use of Doppler information,through the matched WLS procedure,we assess the positionig performance in3 representative scenarios:(i) first, we consideranopen-skyconfigurationwith12satellitesandanomi- nalC/N₀=45dB-Hz;then,(ii)inordertodiscusstheimpactofa constrainedsatellite visibilitywelimit thestudyto aconstrained setof6satellitesinabadGDOPscenario;andfinally,(iii)forcom- pletnesswe simulatea near indoor scenario with multipathand randomsignalattenuations.

5.1. Open-skyscenario

Inthisfirstsimulation, weconsideran idealscenario wherea GNSS receiver exploits GPS L1 C/A signals (B=1 MHz) from 12 satellites.Weconsiderthecasewhereallthesignalshavethesame nominal strength, C/N₀=45 dB-Hz. The satellite configuration is drawnfromarealGPSconstellation,throughsp3files,andthecor- respondingskyplot ispresentedinFig.1.The receiverhasacon- stantspeed of15 m/s, forinstance beingthe caseofa carin an urban environment.Fig.2 representsthesquare rootofthe trace ofthepositioncovariancematrices, limitedtoitsfirst3elements, namelythepositionvectorp.AdoptingthestandardGNSSnomen- clature,thisso-calledPositionDilutionOfPrecision(PDOP)simply

Fig. 2. Delay only vs. delay and Doppler CRB for position estimation (open-sky configuration). The CRB approximation is given in (33) .

Fig. 3. Delay only vs. delay and Doppler RMSE and CRB for position estimation (open-sky configuration).

Fig. 4. Skyplot of a constrained satellite configuration.

representsthestandarddeviationofthepositionerror,thatis,the square-root of the trace of the CRB derived in this contribution.

We comparethe PDOPobtainedwithboth(27),wherethepseu- doranges only are exploited, and (31), where pseudoranges and Dopplers areused. Wehave alsoplottedthe approximation from (33).Wecanfirstnoticethat theproposedapproximated formula isvalidupto2sofintegrationtime.

Moreinteresting isthegain providedbytheDoppler exploita- tion.Inthisopen-skyconfiguration,theimprovementseemsweak, even ifthere is onlya marginal additional computationalcost in

addingtheDopplerinformationintheWLSprocedure.Forashort integrationtimeinanopen-skyscenariothereisnoapparentgain.

But,although the majorityof nowadays applicationsdo not con- siderlongintegrationtimes,thereisarising demandforimprov- ing theperformance ofGNSS systemsin harshenvironments. In- deed,underfoliagecanopy,urbancanyonsorindoorenvironment, conventionalprocessingdoesnotallowtorecoverthesignalswith C/N₀ upto 20dB lower thenominaloutdoor level.Themain so- lutiontocompensate forthesestrongattenuations consistsinin- creasingtheintegrationtime[19,23].Thisso-calledHighSensitiv- ityGNSS(HS-GNSS)has attractedmuch attentionduringthelast decadeandsome experimentstendtoprovethepracticalbenefits of such receivers for indoor pedestrianapplications, for example [24]. But,while the main effort to improvethe precision perfor- mancehasbeenfocused on theelectronic sensitivityandthe in- creaseof theintegration time, the second step ofthe processing usually remains the sub-optimal delay-based only WLS process- ing. This articleproposes a complementary wayof improvement insuch long integrationapplications, usingthe Doppler informa- tiondirectlyintheWLSpositionformulation.Italsohastobeno- ticedthat inpractical situations,the integrationtime islinked to theFrequencyLockLoop(FLL) filterbandwidth.Inthiscasefrac- tionsofHzofprecisionontheDopplerestimationcanbeachieved, correspondingtosomesecondsofintegrationtimeofthissimula- tion.

To go a step further, and in order to assess the validity of the asymptotic covariance (CRB) derivation of this paper, we compare this results with estimated covariance errors obtained fromMonte-Carlo simulations. To this end, Fig. 3 represents the samePDOP asFig.2, butwith longerintegration times,for both theasymptoticformulationgiveninthisarticleandtheestimated errors from 1000 Monte-Carlo simulations. First of all, we can observea perfect matchbetweenthe closed-formformulation of

Fig. 5. Delay only vs. delay and Doppler RMSE and CRB for position estimation (constrained satellite visibility, i.e., urban canyon).

Fig. 6. Delay only vs. delay and Doppler RMSE and CRB for position estimation (near indoor (i.e, multipath and strong fadings) scenario).