An EM-based multipath interference mitigation in GNSS receivers

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To cite this version:

Cheng, Cheng and Tourneret, Jean-Yves

An EM-based multipath

interference mitigation in GNSS receivers. (2019) Signal Processing, 162.

141-152. ISSN 0165-1684

An EM-based multipath interference mitigation in GNSS receivers

*

Cheng Cheng

a

_,

+

_{, Jean-Yves Tourneret}

b

• Sdtool of Asrronautics, Norrltwestem Po/yrechnical University, xran 710072, China

bfNSEEIHT-IRIT-TeSA, University of Toulouse, 2 Rue Camichel Toulouse Cedex 7 31071, France

ABSTRACT

Keywords:

Global navigation satellite systems Multipath mitigation

Expectation-maximization

Particle filter Newton iteration

In multipath (MP) environments, the received signals depend on several factors related to the global

navigation satellite systems (GNSS) receiver environment and motion. Thus it is difficult to use a spe­

cific propagation model to accurately capture the dynamics of the MP signal when the GNSS receiver is

moving in urban canyons. This paper formulates the problem of MP interference mitigation in the GNSS

receiver as a joint state (containing the direct signal parameters) and time-varying model parameter (con­

taining the MP signal parameters) estimation. Accordingly, we propose to exploit the EM algorithm for

achieving the joint state and time-varying parameter estimation in the context of MP interference mit­

igation in GNSS receivers. More precisely, the proposed EM-based MP mitigation approach is decom­

posed into two iterative steps: (a) the posterior pdf of the direct signal parameters and the expected

log-likelihood function necessary in the expectation step of the EM algorithm are approximated by using

an appropriate particle filter; (b) the maximum likelihood solution for MP signal parameters is then ob­

tained using Newton's method in the maximization step. The convergence of the proposed approach is

analyzed based on the existing convergence theorem associated with the EM algorithm. Finally, a com­

prehensive simulation study is conducted to compare the performance of the proposed EM-based MP

mitigation approach with other state-of-the-art MP mitigation approaches in static and realistic scenar­

ios.

1. Introduction

With the new application requirements for global navigation

satellite systems (GNSS) in complex environments, such as in ur­

ban canyons, one important remaining challenge is to reduce the

impact of multipath (MP) on positioning methods. MP signals are

mainly due to the fact that a signal transmitted by a navigation

satellite is very likely to be reflected or diffracted and can follow

different paths before arriving at the GNSS receiver (1). In general,

there are two classes of perturbations affecting the received GNSS

signals: (a) MP interferences resulting from the sum of the direct

signal and of delayed reflections handled by the GNSS receiver (b)

Non-line of sight (NLOS) signals which result from a unique re­

flected signal received and tracked by the GNSS receiver

(2,3). In

some studies, both MP interference and NLOS signals are often

considered as MP, but they are due to different phenomena that

• Corresponding author.

E-mail addresses: cheng.cheng@nwpu.edu.cn (C Cheng),

cause different ranging errors

(4

,

5)

. Although these two phenom­

ena usually arise together in urban canyons, it is clearly interesting

to separate them in practical applications.

1.1. Previous work

The GNSS receiver has to track the signal composed of the di­

rect LOS signal possibly affected by delayed reflections in the pres­

ence of MP interferences. The correlation function of the LOS sig­

nal is distorted by the existing MP signals, and this distortion re­

sults in tracking errors leading to biases in code delay and car­

rier phase estimation

(6)

. Different MP interference mitigation ap­

proaches have been proposed in the literature for mitigating MP

interference error within the GNSS receiver. Several narrow cor­

relator delay Jock loop (DLL) methods

(7)

have been proposed in

order to eliminate MP interferences by correcting the shape of

the discriminator function or the correlation function in the DLL

such as the strobe correlator

(8)

, the early-late-slope technique

(9)

,

the double-delta correlator

(10)

. Based on the fact that the stan­

dard discriminator output, which is specificaJly normalized, has

an invariant point in the presence of MP interferences, the MP

insensitive DLL proposed in

(11)

synchronizes the late replica code

tothe direct signal code. Anothersolution is thecoupled ampli-tudeDLLconsistingofseveralparalleltrackingunits,which imple-mentsa feedback loop toseparate out the directandMP signals andthentrackeachsignalparametersinparallelunits [12] .

The parameters of the LOS and reflected signals can be esti-matedbyusingastatisticalapproachbasedonthemaximum like-lihood(ML) principle [13] , such asthe MP estimating delaylock loop(MEDLL) [14] andthevisioncorrelator [15] introducedby No-vAtel,which are two differentimplementations ofMP mitigation techniques [16] . However, these approaches need to process the receivedsignalbyusingacross-correlationfunctionwithmultiple correlatorsand algorithms that can be computationally intensive

[17] .Notethatagridsearchapproachhasbeenproposedin [18] al-lowinga betterimplementation ofthe ML estimatorvia a global maximizationapproach.Inordertoreducethecomputational com-plexityofML-based MPmitigation approaches, Newton’smethod hasbeen alsoinvestigated for iteratively computingthe ML esti-matorsofthedirectandMPsignalparameters [19,20] .

Dynamic estimation approaches assume that the time propa-gationassociatedwiththeunknown parametersofdirectandMP signalscanbemodelledbyafirst-orderMarkovmodel,which pro-videsthetime-dependentprior probability density function(pdf) for the unknown parameters [21] . The objective is then to esti-materecursivelytheposteriorpdfoftheunknownparameters as-sociatedwiththedirectandMPsignals.Consideringthatthe cor-relationfunctionis relatedtothe unknownparameters by highly nonlinear equations, the use of a Rao–Blackwellized particle fil-ter (RBPF) has been proposed in the literature to generate sam-plesdistributedaccordingto theposterior distributionofinterest andestimatethestatevectorwiththeotherunknownparameters

[22] . Some approaches have also been suggested to improvethe efficiencyofthesefilters.Forinstance,adatacompressionmethod basedonthe MLestimation wasusedtodecrease thedimension oftheobservationvectorinordertoreducethecomplexityofthe MP mitigation technique [23] . A deterministic formof a particle filterwasproposedin [24] forjointMPdetection andnavigation parameterestimationwitha low numberofparticles. Atwo-fold marginalizedBayesianfilterwasfinallyproposed in [25] allowing thenumber ofreceived MP signals andthe corresponding signal parameterstobeestimatedinatrack-before-detectfashion.

1.2.Maincontributionsandpaperorganization

A prior dynamic equation associated with the time propaga-tionofdirectsignalparameters (thispropagationisrelatedtothe vehicle motion) can be defined when the GNSS signal has been lockedinsidethereceiver [26] .An importantpropertyofMP sig-nals isthat they not only depend on the relativeposition of the receiverand GNSSsatellites, butalso on the environment where thereceiverislocated,especiallyinurbancanyons [27] .Sinceitis difficulttouseaspecificpropagationmodeltoaccurately capture thedynamicsofMPsignals,we proposein thisworkto consider theMPsignal parametersasunknown time-varyingquantities.As aconsequence,thepointofviewconsideredinthisworkisto for-mulatetheproblemofMPinterferencemitigationintheGNSS re-ceiverasa jointstate(direct signalparameters)andtime-varying modelparameter(MPsignalparameters)estimationproblemusing astate-spacemodel(i.e.,stateestimationinthepresenceofmodel uncertainty).

The expectation-maximization (EM) algorithm proposed in

[28] is an effective way to iteratively compute the ML estima-tor of unknown parameters in probabilistic models involving la-tent variables [29] . In the EM algorithm, the model parameters areestimatedusingtwoiterativestepsdenotedasexpectation(E) and maximization (M). In the E-step, the model parameters are assumedto beknown.Thereforean expectedlog-likelihood

func-tion of the complete data under the joint pdf of the state se-quence in a fixed-interval needs to be computed. In the M-step, theMLestimatesofthemodelparameters arecomputedby max-imizingthe expectedlog-likelihoodfunction. Accordingly,the EM algorithm hasbeeneffectively proposed tocompute the ML esti-matorofunknownmodelparametersforgeneralstate-space mod-elsinabatch manner [30–32] .Moreover,anonlineEMalgorithm wasinvestigatedin [33] byconstructingsufficientstatisticsforthe unknown modelparameters using thestate andobservation vec-tors.

The main contribution of thispaper is to derive an EM algo-rithmforjointstateandtime-varyingparameterestimationinthe context of MP interference mitigation for GNSS receivers. Differ-entfromcurrentapplicationsoftheEMalgorithmtogeneral state-spacemodels,weproposetoimplementtheEMiterationovereach observation interval of the receiver due to the fact that the MP signal parameters areassumedto betime-varyingandthe corre-spondingsufficientstatistics cannotbe availableinpractice.More precisely,theproposedapproachisdecomposedintotwosteps:(a) the posterior pdf ofthe LOS signal parameters andthe expected log-likelihoodfunctionnecessaryintheE-stepoftheEMalgorithm are approximated by using an appropriate particle filter; (b) the MLsolution forMPsignalparameters isobtainedusingNewton’s methodintheM-step.Theconvergenceoftheproposedapproach isanalyzedbasedontheexistingconvergencetheoremassociated with the EM algorithm for general state-space models. Finally, a comprehensivesimulationstudyisconductedtocomparethe per-formanceoftheproposedapproachwithotherstate-of-the-artMP mitigationapproachesinstaticandrealisticscenarios.

This paper is organized as follows: The GNSS signal models in the presence of MP interference are presented in Section 2 .

Section 3 studies thejointestimationofdirectandMPsignal pa-rameters in the frame of the EM algorithm. The corresponding convergence properties of the proposed approach is analyzed in

Section 4 . The performance of the proposed EM-based MP mit-igation approach is evaluated in staticand dynamic scenarios in

Section 5 .Conclusionsarefinallyreportedin Section 6 .

2. GNSSsignalmodelinthepresenceofMP

Inapilotchannel,thereceivedcomplexbasebandsignal associ-atedwithGNSSsatellitesaffectedbyMMPsignalscanbewritten asfollows [1] s

(

t

)

= M  m =0 a m c

(

t −

τ

m

)

e jϕm+

ω

(

_t

)

₍₁₎ with d

ϕ

m dt =2

π

f d m (2)

where am,

τ

m,

ϕ

m and f_m d are the amplitude, code delay,carrier

phase andDoppler frequency ofthe mth received signal andthe subscript m=0 denotesthedirect LOSsignal, c(t) isthe pseudo-randomnoise(PRN)codeassociatedwiththeGNSSsignal,

ω

(t)is azeromeanadditivecomplexGaussianwhitenoisewithvariance

σ

2 _{. In the receiver, sampled data are collected over an}

observa-tion intervalT_a (thatiscomposed ofN samples oftheGNSS sig-nal,i.e.,Ta =NTs whereTs = 1 fs andfs isthesamplingfrequencyof

the GNSSreceiver). It can be assumedthat all signal parameters donot change significantly overthe observationinterval Ta .As a consequence,thelog-likelihoodfunctionofallunknownsignal pa-rameters forthe kth (k∈N) block of N collectedsamples can be

expressedasfollows L _θ∗ ∝−  kT a



s

(

t

)

− M  m =0 a m c

(

t −

τ

m

)

e jϕm



2 dt ∝−  kT a

|

s

(

t

)

|

2 dt + M  m =0 2R

{

α

∗ m R SC

(

τ

m

)

}

− M  m =0 M  n = m +1 2

(

τ

n −

τ

m

)

R

{

α

m

α

∗n

}

− M  m =0 Ta

|

α

m

|

2 (3) with

θ

∗₌



_a 0 , . . . , a M ,

τ0

, . . . ,

τM

,

ϕ

0 , . . . ,

ϕ

M , f 0 d , . . . , f m d



T

(

τ

)

=  kT a c

(

t

)

c

(

t −

τ

)

dt R SC

(

τ

)

=  kT a s

(

t

)

c

(

t −

τ

)

dt (4)

where

α

m=amejϕm (m=0,...,M)denotesthecomplexamplitude

ofthereceivedsignal,R

{

·

}

and(· )∗_{denotetherealpartand}

con-jugate of a complex number,



(· ) is the correlation function of the locally-generated PRN replicas in the receiver, R_SC (_{· )} is the crosscorrelationfunctionofthereceivedGNSSsignalwiththePRN replicasovertheobservationinterval.

In practice, thetime propagationof the direct LOS signal pa-rameters x=



a₀ ,

τ

0 ,

ϕ

0 ,f0 d



T

(this propagation is related to the dynamicsofthereceiver)canbedescribedbyaconditional proba-bilitydensityfunction(pdf)ofastate-spacemodelwhentheGNSS signalhasbeenlockedinsidethereceiver [34] ,i.e.,

x _k ∼ f

(

x _k

|

x _{k −1}

)

=N



F_{k |k −1} x _{k −1} , G _{k |k −1}



_{x,k −1} G _k T _{|k −1}



(5)

where k=1,...,∞ is the kth observation interval, f

(

x_k

|

x_{k −1}

)

is thepdfassociatedwiththedirectLOSsignalparameterdynamics,

N

(

μ

,

)

denotes themultivariateGaussian pdfwithmeanvector

μ

and covariancematrix



,F_{k |k −1} and G_{k |k −1} arethe transition ma-trices ofthe direct LOSsignal parameters andthe process noises and



x,k −1 is the covariancematrix of theprocess noise (azero

mean Gaussian white noise). More precisely, the matrices F_{k |k −1 ,} Gk|k−1 and



x,k −1 canbedefinedasfollows [34]

F _{k |k −1} =

⎛

⎜

⎜

⎝

1 0 0 0 0 1 0

λ

Ta 0 0 1 Ta 0 0 0 1

⎞

⎟

⎟

⎠

, G _{k |k −1} =

⎛

⎜

⎜

⎝

T a 0 0 0 Ta 0 0 0 Ta2 2 0 0 T a

⎞

⎟

⎟

⎠

and



x,k −1 =

⎛

⎜

⎝

σ

2 a 0 0 0

σ

_τ2 0 0 0

σ

_f 2

⎞

⎟

⎠

(6)

where

λ

= f_co/ fca isascalefactorconvertingthecarrierDoppler frequencyto thecode Doppler frequency,fco andfca are thePRN codeandtheGNSSsignalcarrierfrequencies,respectively.

An importantproperty ofMPsignalsisthat theynot only de-pend onthe relative positionof thereceiver andGNSS satellites, but also on the environment where the receiver is located, es-pecially in urban canyons. Since it is difficult to use a specific propagationmodeltoaccurately capturethedynamicsofMP sig-nals, we propose inthis work to consider the MP signal param-eters as an unknown time-varying model parameter vector, i.e.,

θ

k =



θ

1 ,k ,. . .,

θ

M,k



T

where

θ

_m,k =



a_m,k ,

τ

m,k ,

ϕ

m,k ,fm,kd



T

and m= 1_{,. . .,}M.Accordingly,thelikelihoodfunctionin (3) canbedefined usingthefollowingconditionalpdf

y _k ∼ g_θk

(

y k

|

x k

)

(7)

wherey_k =



s_{(k −1 )T a+ T s},...,s_{(k −1 )T a+ NT s}



TisthesampledGNSS sig-nal vector over the kth observation interval in GNSS receiver,

g_θ

k

(

yk

|

xk

)

is the pdf associated with the observed measurements

dependingon the unknown model parameter vector

θ

_k over the

kth observation interval. Note that g_θ

k

(

yk

|

xk

)

can be easily

ob-tainedbyconsideringthedirectLOSandMPsignal parametersin

(3)asthestatevectorx_k andunknowntime-varyingmodel param-etervector

θ

_k respectively.

3. TheEM-basedMPinterferencemitigationintheGNSS receiver

Accordingto (5) and (7) ,theestimationofthestatevector (con-tainingthedirectLOSsignalparameters)resultsinanonlinear fil-teringproblemthatcanbesolvedusing,e.g.,theextendedKalman filter(KF)ortheparticlefilter(PF)intheabsenceofMP interfer-ences.IthasbeenrecognizedthattheKFandPF-basedGNSSsignal trackingloopscanbe possibly implementedinthe GNSSreceiver

[35,36] .However,inthepresenceofMPinterferences,the statisti-calmodels usedtosolve theMPmitigation problemintheGNSS receiverdepend onunknown time-varyingmodel parameter vec-tors (containing the MP signal parameters) that need to be esti-matedjointlywiththestate vector.Thusweproposeinthiswork to investigatean EM algorithm over one observation interval for achievingjointstateandtime-varyingparameterestimationinthe context of MP mitigation in GNSS receivers (as explained in the subsequentsections).

3.1. Expectation-maximizationforMPinterferencemitigationinGNSS receivers

Since thesampled GNSSsignal vectorsy_{1: k} =

{

y₁ ,...,y_k

}

over each observation interval are mutually independent, the log-likelihoodfunctionL_θ

1:k

(

y1: k

)

forasequenceofkobservation

inter-vals withunknown modelparameter vectors

θ

_{1: k} =



θ

1 ,. . .,

θ

k



∈



(



is the feasible set of parameters) can be defined as

L _θ

1:k

(

y 1: k

)

:=logp θ1:k

(

y 1: k

)

=

k

logp _θk

(

y _k

)

(8)

wherek=1,...,∞denotesthekthobservationinterval.Thus the ML estimator 

θ

ML

k for the kth observation interval can be

ob-tainedby maximizing the corresponding log-likelihoodL_θ k

(

yk

)

:= logp_θ k

(

yk

)

,i.e., 

θ

ML k =argmax θk∈  L _θk

(

y _k

)

. (9)

ThekeyideaoftheEMalgorithmistoconsiderasequenceofk ob-servationsy1:k asincompletedataandconstructa completedata

log-likelihoodfunctionfortheunknown modelparametervectors

θ

1: k byintroducinglatentstatevectorsx1: k [33] ,i.e.,

L _θ1:k

(

y 1: k , x 1: k

)

:=logp θ1:k

(

y 1: k , x 1: k

)

(10)

where p_θ

1:k

(

y1: k ,x1: k

)

is the joint pdf of the observations and

states. Since the EM algorithm iteratively estimates

θ

_{1: k} by maximizing the expected log-likelihood of the complete data

L_θ

1:k

(

y1: k ,x1: k

)

,thecorrespondingexpectedlog-likelihoodfora

se-quenceofkobservationintervalscanbedefinedasfollows [30] Q



θ

1: k , 

θ

1: k

(

r

)



:=E _θ 1:k(r)



L _θ1:k

(

y _{1: k} , x _{1: k}

)

|

y _1: k



= 

_

logp _θ1:k

(

y 1: k , x 1: k

)



p  _θ 1:k(r)

(

x 1: k

|

y 1: k

)

dx 1: k (11)

where

θ

_{1: k}

(

r

)

isthe estimator of the unknown model parameter vectorsattherthEMiterationforasequenceofkobservation in-tervals.

Consideringthatthepdfassociatedwiththestatedynamicsin

(5)isindependentoftheunknownmodelparametervector

θ

over each observationinterval, p_θ

1:k

(

y1: k ,x1: k

)

canbe decomposed

us-ingtheMarkovpropertyassociatedwiththestateandobservation equations (5) and (7)

p _θ1:k

(

y _{1: k} , x _{1: k}

)

= k

g _θk

(

y _k

|

x _k

)

f

(

x _k

|

x _{k −1}

)

(12)

wherek=1,...,∞.Afterreplacing (12) into (10) ,weobtain

L _θ1:k

(

y _{1: k} , x _{1: k}

)

= k



logg _θk

(

y _k

|

x _k

)

+log f

(

x _k

|

x _{k −1}

)



∝ k logg _θ k

(

y k

|

x k

)

. (13)

Asaconsequence,using (13) into(11)leadsto

Q



θ

1: k , 

θ

1: k

(

r

)



∝ k  log



g _θk

(

y _k

|

x _k

)



p _θ _k(r)

(

x k

|

y 1: k

)

dx k = k Q



θ

k , 

θ

k

(

r

)



(14) where Q



θ

k , 

θ

k

(

r

)



:=  log



g _θk

(

y _k

|

x _k

)



p  _θk(r)

(

x k

|

y 1: k

)

dx k (15)

andwhere

θ

_k

(

r

)

is the estimatorof theunknown model param-etervector at therth EM iteration forthe kth observation inter-val, p(xk|y1:k) is the posterior pdf of the state vector xk given

all available observations y_{1: k} and is referred to as the filter-ing pdf. According to (14) , the expected log-likelihood function

Q



θ

1: k ,

θ

1: k



forasequenceofobservationintervalsisproportional toasummationoftheexpectedlog-likelihoodfunction Q



θ

k ,

θ

k



for each observation interval when the pdf associated with the statedynamicsisknown.Thus theiterativeEMsolutionfora se-quence ofobservationintervals canbe decomposed intothe cor-responding iterative solutions for each observation sequence. In addition, the algorithm for the kth observation interval is such that _Q



θ

_{k ,}

θ

_k

(

r

)



_>Q





θ

_k

(

r

)

_,

θ

_k

(

r

)



guaranteeing an increase of thelog-likelihood L_θ

k

(

yk

)

>L θk(r)

(

yk

)

at ther+1thiteration (see

Corollary 1). After choosing some initial value of

θ

denoted as 

θ

k

(

0

)

∈



, the EM algorithm generates iteratively a sequence of

estimates

θ

_k

(

r

)

(r=0,1,2...) whosefinalvalue approximatesthe ML estimator of

θ

in (9) . The EM iteration forMP mitigation in GNSSreceiversissummarizedin Algorithm 1 .

Algorithm1 TheiterativesolutionoftheEMalgorithmforan ob-servationinterval.

1:E-Step.ComputeQ



θ

k,

θ

k

(

r

)



2:M-Step.Compute

θ

_k

(

r₊1

)

₌argmax_θ

k∈ Q



θ

k ,

θ

k

(

r

)



3.2.E-step:computingQ



θ

k ,

θ

k



basedonaparticlefilter

According to (15) ,computing Q



θ

k ,

θ

k



in theE-step requires to evaluate the expectation under the posterior pdf p_θ

k

(

xk

|

y1: k

)

,

where 

θ

k has been estimated within the previous M-step and

thereforeis known. According to the Bayesian estimation princi-ple,theposteriorpdfp_θ

k

(

xk

|

y1: k

)

ofthestatevectorxk (containing

thedirectsignal parameters)givenall availableobservations y_{1: k}

canberecursivelyupdatedasfollows

p  _θk

(

x k

|

y 1: k

)

∝ p _θk

(

y k

|

x k

)

p

(

x k

|

y 1: k −1

)

(16)

with

p

(

x _k

|

y _{1: k −1}

)

= 

p

(

x _k

|

x _{k −1}

)

p  _θk−1

(

x k −1

|

y 1: k −1

)

dx k (17)

where (17) representsapredictionstepresultinginthepriorpdfof thestate vectorforthekthobservationinterval.Accordingto (7) , the observationrelatedto thedirect LOSsignal parameter vector is definedby a highlynon-linear equation. Thus, it isdifficult to obtainananalyticsolutionoftheposteriorpdfin (16) .Inthis sit-uation,itisclassicaltoconsideraparticlefilterapproximatingthe posteriordistributionofinterestbyusingasetofweighted parti-clesleadingto [37] p  _θk

(

x k

|

y 1: k

)

≈ N s  i =1

ω

i k

δ



x _k − xi k



(18)

whereNs isthe numberofparticles,

δ

(· )istheDiracdelta func-tion,xi _k istheithstateparticle,

ω

_k i isanappropriateweight associ-atedwiththeithparticleandN s

i =1

ω

k i =1forthekthobservation

interval.

As aconsequence,Q



θ

k ,

θ

k



in theE-stepisapproximated by replacing (18) into (15) ,i.e.,

Q



θ

k , 

θ

k



≈ Q



θ

k , 

θ

k



= N s  i =1

ω

i k logg θk



y k

|

x ik



(19)

where the approximation quality can be enhanced by increasing thenumberofparticlesNs .According totheliterature,many par-ticlefiltermethods havebeen investigatedforapproximating the posteriorpdfp(xk|y1:k) [38] .Allthesemethodscanbeusedin

or-dertoobtain (19) .

3.3. M-step:maximizingQ



θ

k ,

θ

k



usingNewton’smethod

IntheM-step,theapproximationQ



θ

k ,

θ

k



needstobe maxi-mizedwithrespectiveto

θ

_k inordertoobtainanewiterativeML estimateofthemodelparametervector(containingtheMPsignal parameters)for thekth observationinterval. Assumingthat a set of weightedstate particle



ω

i _k ,xi _k



N s

i =1 havebeen obtained in the

E-step, (19) canbemaximizedbysettingthepartialderivativesof 

Q



θ

k ,

θ

k



with respectiveto

θ

_k tozero,i.e.,

∂

_Q



_θ

_{k ,} 

_θ

k



∂θk

= N s  i =1

ω

i k

∂

logg _θk



y k

|

xik



∂θk

=0 (20) where

θ

_k =



θ

1 ,k ,. . .,

θ

M,k



T, xi k =



ai _{0 ,k} ,

τ

i 0 ,k ,

ϕ

0 i ,k ,f0 d ,k,i



T is the ith particleassociatedwiththedirectLOSsignalparameters. Consider-ingthat thecomplexamplitude

α

in (3) containsthe correspond-ing amplitudeandcarrierphaseofthe receivedsignal, (20) leads to N s  i =1

ω

i k

∂

logg _θk



y _k

|

xi k



∂αm,k

=0 (21) N s  i =1

ω

i k

∂

logg _θk



y _k

|

xi k



∂τm

,k =0 (22) wherem=1,...,M.

After replacing (3) into (21) , the ML solution of the complex amplitudeforthemthMPsignalcanbeexpressedas

R _SC



τm

_,k



− N s  i =1

ω

i k





τm

,k −

τ

0 i ,k



α

i 0 ,k− M  n =1 ,n  = m





τn

,k −

τm

,k



α

n,k −Ta

α

m,k =0. (23)

Accordingly,a bankofpartialderivativeswithrespectto

α

m (m=

1,...,M)canbegatheredintothefollowingcompactexpression

A _k

α

_k =b _k (24) with A _k =

⎛

⎜

⎜

⎜

⎝

Ta





τ2

,k −

τ1

,k



. . .





τM

,k −

τ1

,k







τ1

,k −

τ2

,k



T a . . .





τM,k

−

τ2

,k



.. . . . . ... ...





τ1

,k −

τM

,k







τ2

,k −

τM

,k



. . . Ta

⎞

⎟

⎟

⎟

⎠

(25) and b k =



b 1 ,k , . . . , b M,k



T (26) where b_m,k =R_SC



τ

_m,k



−N s i =1

ω

i k





τ

m,k −

τ

0 i ,k



α

i 0 ,k and m=1,

...,M. As a consequence,the complex amplitudes ofMP signals canbesolvedinclosedformasfollows



α

k =



A T k A k



−1 A T k b k (27) where 

α

k=





α

1,k ,. . .,

α

M,k



T

.Therefore, the corresponding ampli-tude and carrier phase of MP signals can be extracted fromthe estimatedcomplexamplitude

α

_{k ,}i.e.,



a _m,k =



_α

_m,k



ϕ

m,k =∠

α

m,k (28)

where m=1,...,M. Since it is difficult to separate the carrier Doppler frequencyfromthe correlation measurements in (3) ,the carrierDopplerfrequencyofMPsignalsisextractedfromtwo suc-cessivecarrierphaseestimationasfollows

_f d m,k= 

ϕ

m,k − 

ϕ

m,k −1 Ta (29) wherem₌1_,...,M.

RegardingtheMLsolutionofthecodedelays,replacing (3) into

(22)yields R



α

∗ m,k

∂

R _d



τm

_,k



∂τm



=0 (30) with R _d



τm,k



= R SC



τm,k



− N s  i =1

ω

i k





τm,k

−

τ

i 0 ,k



α

i 0 ,k − M  n =1 ,n  = m





τn

,k −

τm

,k



α

n =  kT 0



s

(

t

)

− N s  i =1

ω

i k

α

i 0 c



t −

τ

i 0



− M  n =1 ,n  = m

α

n c

(

t −

τ

n

)



c

(

t −

τ

m

)

dt (31)

where m=1,...,M. Considering that the correlation function in

(31) dependson the code delayparameter though the PRNcode

c

(

t−

τ

)

,thereisnoclosedformexpressionfortheparameter

τ

_m,k . WeproposetouseNewton’smethodin [20] toiterativelycompute theML estimatorofthecodedelay.Thusthe iterativesolutionof thecodedelayofthemthMPsignalis



τm

,k

(

r +1

)

=

τm

,k

(

r

)

− R



R ∗_d





τm,k

(

r

)



∂Rd

(

 τm,k(r )

)

∂τm



R



R ∗ d



_

τm

,k

(

r

)



∂ 2_R d

(

 τm,k(r )

)

∂τ2 m



(32)

wherem=1,...,M andr=0,1,2... isthe numberofiterations. More details about the derivation of the derivatives ∂R_∂τ(τ ) and

∂2_{R(τ )}

∂2_τ are given in[20] .

Notethat the estimatedcomplex amplitudein (27) is a func-tionofthecodedelayofMPsignalsforthekthobservation inter-val,whereasthecodedelayestimatorin (32) requirestoknowthe complex amplitude ofMP signals.Therefore, the amplitude esti-matesin (27) atthe

(

r+1

)

thiterationarecalculatedbyusingthe codedelayestimatesattherthiteration.Thenthecodedelay esti-matesatthe

(

r₊1

)

thiterationcanbeimplementedbasedonthe lastestimatesofcomplexamplitudes.

ConsideringthatMPsignalsdependontheenvironmentwhere thereceiver is located, itis difficult to accurately obtain the ini-tialvalues ofthe MP signal parameters atthe beginning ofeach EM iteration. Generally, MP signals do not affect positioning re-sults inside the receiver when the code delay offsets of the MP signals withrespect to the direct LOS signal are equal or larger than2Tc (whereTc isthechipdurationofthePRNcode).Thus,the estimation ofthe MP signal parameters is only performed when

τ

m∈

(

τ

0 ,

τ

0 +2Tc

)

wherem=1,...,M.Inthispaper,thecode

de-lays of the MP signals are supposed to be random values uni-formlydistributedintheinterval

τ

m ∈

(

τ

0 ,

τ

0 +2Tc

)

withthe con-dition

τ

_{0 <}

τ

_{1 <...<}

τ

_{M <}

τ

_{0 +}2T_c atthebeginningoftheEM it-erationsforeachobservationinterval.Finally,theEM-basedMP in-terferencemitigationapproachinGNSSreceiversissummarizedin

Algorithm 2 .

Algorithm 2 ProposedEM-based MPinterferencemitigation ap-proachinGNSSreceivers. Step1:Initialization(r₌0). 1: Calculate xi _k ∼ f



xk

|

xi_{k −1}



according to (5) where xi _k =



ai _{0 ,k} ,

τ

i 0 ,k ,

ϕ

0 i ,k ,f0 d ,k,i



T andi₌1_,...,N_s .

2: Generate the initial code delay of the mth MP signal 

τ

m,k

(

0

)

∼ U

(

τ

0 ,k ,

τ

0 ,k +2Tc

)

where

τ

_{0 ,k} = max 1 <i<N s

{

τ

i 0 ,k

}

and 

τ

1 ,k

(

0

)

<...<

τ

M,k

(

0

)

and U

(

a,b

)

denotesthe

uniformdistributionontheinterval

(

a,b

)

.

3: Calculate the corresponding complex amplitude

α

m,k

(

0

)

for

m₌1_,...,M

using(27).

Step2:Iteration.Forr=1,2,...

%E-step

4:Usingthemodelparametervector

θ

_k

(

r− 1

)

,updateeach parti-cleweight

ω

i

k

based on a particle filter approach (e.g., the bootstrap particle filter[38])and

computeQ



θ

k ,

θ

k

(

r− 1

)



accordingto(19). %M-step

5: Using the ML estimate of the code delay

τ

_m,k

(

r− 1

)

where

m=1,...,M,

computethecomplexamplitude

α

_k

(

r

)

ofMPsignalsaccordingto (27).

6 Extract the amplitude, carrier phase and Doppler frequency associatedwith

MPsignalaccordingto(28)and(29).

7: Using the ML estimate of the complex amplitude

α

k

(

r

)

,

com-putethecodedelay

oftheMPsignal

τ

_m,k

(

r

)

accordingto(32)form₌1_,...,M.

%Stoppingrule 8:If  _Q   _θ r+1,  θr  − Q  θr,  θr  _Q   _θ r+1,  θr <

δ

orr_{≥ r}max where0<

δ

 1,stop the

EMiteration elsesetr=r+1.

4. Convergenceanalysis

Since it is difficult to establish convergence of the sequence of estimates 

θ

(

r

)

(r₌0_,1_,2_...), it is classical to prove that the iterativesolution ofthe EM algoritm verifies Q





θ

(

r+1

)

,

θ

(

r

)



> Q





θ

(

r

)

,

θ

(

r

)



guaranteeing a strict increase of the log-likelihood

L_{ θ(r+1 )}

(

y_{1: k}

)

_>L_{θ( r )}

(

y_{1: k}

)

for a sequence of observation intervals atthe

(

r+1

)

th iteration, asdemonstrated inTheorem 2 of [39] . In the context of MP mitigation in GNSS receivers, this non-decreasingpropertyoveranobservationcanbeobtainedbasedon Theorem2of [39] .

Corollary1. Let

θ

_k

(

r+1

)

begeneratedfrom

θ

_k

(

r

)

byaniterationof Algorithm1.Then

L  _θ

k(r+1 )≥ L θk(r )

∀

r =0, 1, 2, . . . (33) withequalityifandonlyif

Q





θ

k

(

r +1

)

, 

θ

k

(

r

)



=Q





θ

k

(

r

)

,

θ

k

(

r

)



(34) and p  _θk(r+1 )

(

x k

|

y 1: k

)

= p  _{θk(r )}

(

x k

|

y 1: k

)

(35)

foralmostall(withrespecttoLebesguemeasure)x_k . Proof.See Appendix A . _

Accordingto (3) ,thelikelihoodoftheMPsignalparameters in theM-stepdependsonthecodedelayofreceivedsignalsvia the crosscorrelation function.Forthe C/A codeinthe GPS L1signal, thecorrelation functionisaconvextriangularfunction.Therefore, the strict monotonicity of any EM iteration is guaranteed inthe contextofMPmitigationinGNSSreceivers.

In the proposed EM-basedMP mitigation approach, _Q



θ

_{k ,}

θ

_k



is approximated by using the particle-based approximation  QN s



θ

k ,

θ

k



. Accordingto Lemma 9.2in [32] , QN s



θ,



_θ



provides an accurate approximation of_Q



θ,



θ



when the number of par-ticleNs is sufficiently large. Therefore,using a particle-based

ap-proximationQN s



θ,



_θ



instead ofQ



θ,



_θ



is a reasonablewayof implementingthe ML estimation of MPsignal parameters inthe proposedEM-basedMPmitigationapproach.

5. Algorithmassessment

InordertoevaluatetheperformanceoftheproposedEM-based MPmitigationapproach,theGPSL1C/Asignalwastakeninto ac-countandtherelatedparametersusedinalltestscenariosare pro-videdin Table 1 .TheMPmitigationapproachwasimplementedfor two kindsof test scenarios:(a) a static scenariowhere the joint estimateaccuracyofthe statevector (containingthedirectsignal parameters)andmodelparametervector(containingtheMPsignal parameters)was evaluated underthe condition ofa setof spec-ifiedmodel parameters (i.e., the MP signal parameter offsets are constantwithrespecttothedirectLOSsignal);(b)adynamic sce-narioinwhichthejointestimateaccuracywasevaluatedunderthe

Table 1

Related parameters in test scenarios.

Amplitude noise σa = 0 . 01

Code delay noise στ= 0 . 01 chips/s

Doppler frequency noise σf = 5 Hz/s

Integration time T a = 25 ms

Correlation spacing δ= 0 . 15 chips

conditionoftime-varyingmodelparameters(i.e.,theMPsignal pa-rameteroffsetsaretime-varyingwithrespecttothedirectLOS sig-nal).The bootstrapparticlefilterwasimplementedforrecursively approximatingthefilteringpdfintheproposedEM-basedMP mit-igationapproachandthenumberofparticleswassettoNs =100. Finally,therootmeansquare error(RMSE)oftheestimator asso-ciatedwiththedirectLOSsignalwasusedasaperformance mea-sure.Thismeasureisdefinedas

RMSE=



1 N m N m  i =1



 x (i )_{− x}



2 ₍₃₆₎

where x(i ) is the ith run estimate. 100 Monte Carlo (MC) simu-lations (N_{m =}100) were run for anytest scenario. All algorithms havebeencodedusingMATLABandrunonalaptopwithInteli-5 and8GBRAM.

5.1. MPmitigationperformancecomparisoninstaticscenarios

Thereceivedcomplexbasebandsignalswasgeneratedbyusing thesignalmodelin (1) andweassume thatonereflectedMP sig-nalaffects thedirectLOS signal(i.e., M=1). Inorder toevaluate the estimation performance of the proposed EM-based MP miti-gationapproach andtocompare itwithother MP mitigation ap-proaches in the static scenario, we considered the proposed MP mitigation approach, the fast iterative maximum-likelihood algo-rithm(FIMLA)studiedin [20] andthecoherentnarrowcorrelator DLLforthisscenario.Themaximumnumberofiterationsinboth theproposed MPmitigation approachandthe FIMLAwere setto 15 (i.e., rmax =15). The carrier-to-noise ratio (CNR) of the direct LOS signal was setto 50dB-Hz and the multipath-to-directratio of the MP signal amplitude was maintained to a constant value 0.5.Inaddition,theRMSEenvelops(i.e.,theRMSEversusthecode delayoffsetofMPsignal)forthe codedelayandcarrierphaseof thedirectLOSsignalwasusedtoevaluatetheperformance ofthe proposedMPmitigationmethod.Asetofcodedelayoffsetsofthe MPsignal can be obtainedby settingthe difference betweenthe code delays of the direct LOS signal and those of the MP signal intheinterval(0,1.5Tc ),i.e.,

τ

₁ =

τ

1 −

τ

0 where

τ

1 ∈(0,1.5Tc ). ThecarrierphaseoffsetoftheMPsignal(i.e.,

ϕ

1=

ϕ

1−

ϕ

0)was

set to0◦ (in-phase withthe direct LOSsignal) andto 180◦ (out-of-phase with the direct LOS signal) when computing the RMSE envelopsassociatedwiththedirectLOScodedelay,andto90◦and −90◦ _{(orthogonalwiththedirectLOSsignal)whencomputingthe}

RMSEenvelopsforthedirectLOScarrierphase,respectively.1

Asshownin Fig. 1 ,theRMSEenvelopesofthedirectLOScode delayandcarrierphaseobtainedwiththeproposedEM-basedMP mitigation approach and the FIMLA are smaller than those ob-tainedwiththenarrowcorrelatorDLL,whichindicatesthattheMP mitigation performance of the ML-based methods is better than that ofthenarrowcorrelator-based method.Inaddition,the esti-mationperformance forthedirectLOScarrierphaseissimilarfor theproposed approachandFIMLA, whereasthe RMSEenvelop of thedirectLOScodedelayobtainedwiththeproposedapproachis obviouslysmaller than theother ones. Accordingly, estimated er-rors of theMP code delay (i.e.,

δτ

₁ =

τ

1 −

τ

1 ) obtainedwith the

proposed approach and the FIMLA over 100 MC simulations are depicted as box plots in Fig. 2 (the figure only displays the es-timated error for the in-phase situation since a similar result is

1 In a conventional tracking loop of the GPS receiver, the tracking errors of the direct LOS code delay due to MP interferences were maximized when the carrier phase offset of MP signal was 0 ◦ _{and 180} ◦ _{and the tracking errors of the direct} LOS carrier phase due to MP interferences were maximized when the carrier phase offset of MP signal was −90 ◦_{and 90} ◦_.

Fig. 1. RMSE envelopes of the LOS code delay and carrier phase with different approaches. Proposed approach: black solid line; FIMLA: red dashed line; Narrow correlator DLL: blue dash-dotted line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Estimate error of MP code delay versus the code delay offset of MP signal.

obtainedfortheout-of-phasecomponent).Itisclearthatthe esti-matederroroftheMPcodedelayobtainedwiththeproposed ap-proachissignificantly smallerthanthat obtainedfromtheFIMLA when the code delayoffset of the MP signal is smaller than 0.2 chips.Thisdifferencebetweenthetwoapproachesgradually disap-pearsasthecodedelayoffsetofMPsignalincreases.Althoughthe direct LOSand MPsignal parameters are jointly estimatedbased ontheiterativefashion bothinthesetwoapproach,theproposed

approachcaneffectivelyimprovetheestimationperformance due tothe factthat theprior informationaboutthe directLOSsignal parametersistakenintoaccountbyusingtheEMiteration. Fig. 3

showstheestimatedMPcodedelayoffset(i.e.,

τ

₁ =

τ

1 −

τ

0 )

ver-sus the EM iteration times over 100 MC simulations. Thanks to theconvexpropertyofthecorrelationfunctionassociatedwiththe C/AcodeoftheGPSL1signal,theestimatedMPcodedelayoffsets convergerapidlyto valuescloseto thetrueparameterswhen the

Fig. 3. Estimated MP code delay offset versus the number of EM iterations for 100 MC simulations. Estimated value: blue line; True value: black line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. RMSE of the LOS code delay for different CNRs.

correspondinginitialvalueatthebeginningoftheEMiterationsis uniformlychoseninthefixedinterval.

In order to evaluate the effect of different CNR levels on the estimationperformance of the proposed approach, the three MP mitigation approaches were implemented using code delay and carrierphase offsets of the MP signal fixed to 0.5 chips and 0◦, respectively. Fig. 4 displaystheRMSEofthedirectLOScodedelay versusCNR. It is knownthat the narrowcorrelator DLLis insen-sitive to CNR changes, which is confirmed in Fig. 4 . Considering that a smaller value of CNR leadsto a larger noise variance im-pairingtheMLestimationaccuracy,theFIMLAisverysensitiveto theCNR andrequiresalongintegrationtime(resultinginasmall noisevariance)toobtainreliableestimationresults.Theproposed approachusingprior informationaboutthe directLOSsignal can

efficientlyreduce theinfluenceofnoiseandMPontheestimation performance.

Consideringthat more thanone MP signal entersthe receiver atthe same time insome MP scenarios, we assume in a second scenariothattwo reflectedMPsignalsaffectthedirectLOSsignal (i.e., M₌2). The multipath-to-directratios of the MP signal am-plitudeswere setto0.7and0.4,respectively.Theestimatedmean andstandard deviation ofthe code delayoffsets forthe two MP signalsobtainedwiththeproposedapproachandFIMLAover100 MC simulationsare reportedin Table 2 . Table 2 showsthat simi-larresultsareobtainedwhenthetwoMPsignalsarein-phaseand orthogonal.It isclearthat theestimationperformance ofthe ap-proaches isdegraded whenthecode delayoffsetsofthetwo MP signalsare relativelyclose, whereas thecorresponding estimation accuracyis clearlyimprovedasthecode delayoffsetsofthe two MPsignalsaregradually separated.Inaddition,theestimated ac-curacy obtained with the proposed approach is better than that obtainedwithFIMLA,especiallyforthefirstMPsignal(i.e.,ashort MP signal).Thus a better MPmitigation performance canbe ob-tainedbyusingtheproposedEM-basedMPmitigationapproachin thepresenceofmultipleMPsignals.

5.2. MPmitigationperformanceindynamicscenarios

Inordertodemonstratethejointestimationperformanceofthe proposedEM-basedMPmitigationapproachindynamicscenarios, weconsideredonereflectedMPsignalwiththefollowing dynam-ics [23,25] :

(i) The code delayof the MP signal is assumedto follow the process

τ1

,k =

τ1

,k −1 +

τ1

˙ ,k −1 T a +

ω

τ

˙

Table 2

Estimated means and standard deviations for the code delay offsets of the two MP signals.

True MP signal parameter offsets Estimated MP code delay offsets

code delay (chips) carrier phase ( ◦_{) (mean ± standard deviation)}

τ1 τ2 ϕ1 ϕ2 Proposed approach FIMLA

 τ1  τ2  τ1  τ2 0.18 0.025 ± 0.122 0.173 ± 0.043 0.058 ± 0.003 0.076 ± 0.004 0.25 0.088 ± 0.057 0.208 ± 0.039 0.176 ± 0.059 0.207 ± 0.030 0.30 0.080 ± 0.043 0.283 ± 0.056 0.027 ± 0.027 0.236 ± 0.032 0.45 0.086 ± 0.026 0.434 ± 0.036 0.034 ± 0.008 0.387 ± 0.038 0.10 0.60 0 ◦ ₀ ◦ _{0.097 ± 0.019 0.595 ± 0.031 0.039 ± 0.006 0.554 ± 0.036} 0.82 0.096 ± 0.017 0.818 ± 0.026 0.041 ± 0.006 0.777 ± 0.036 0.98 0.094 ± 0.026 0.970 ± 0.042 0.049 ± 0.013 0.949 ± 0.052 1.19 0.097 ± 0.032 1.183 ± 0.069 0.046 ± 0.010 1.151 ± 0.043 1.48 0.096 ± 0.012 1.479 ± 0.020 0.040 ± 0.006 1.478 ± 0.026 0.18 0.124 ± 0.095 0.109 ± 0.059 0.001 ± 0.029 0.148 ± 0.064 0.25 0.068 ± 0.049 0.239 ± 0.051 0.006 ± 0.031 0.172 ± 0.078 0.30 0.076 ± 0.033 0.291 ± 0.045 0.023 ± 0.019 0.257 ± 0.033 0.45 0.097 ± 0.017 0.451 ± 0.025 0.034 ± 0.008 0.439 ± 0.029 0.10 0.60 0 ◦ ₉₀ ◦ _{0.097 ± 0.015 0.604 ± 0.024 0.039 ± 0.007 0.593 ± 0.025} 0.82 0.095 ± 0.015 0.822 ± 0.027 0.042 ± 0.007 0.820 ± 0.026 0.98 0.096 ± 0.014 0.979 ± 0.025 0.043 ± 0.007 0.979 ± 0.022 1.19 0.099 ± 0.011 1.185 ± 0.023 0.049 ± 0.012 1.190 ± 0.018 1.48 0.098 ± 0.012 1.482 ± 0.019 0.046 ± 0.006 1.478 ± 0.018

where

τ

˙_{1 ,k} refers tothe change rateoftheMP codedelay,

ω

τ and

ω

_τ˙ arezeromeanGaussianwhitenoiseswith vari-ances

σ

_τ2 and

σ

_τ2 _˙ . Inaddition,theMP codedelayis initial-izedwith

τ

1 , 0 =

τ

0 , 0 +

|

τ

1 , 0+

ω

τ

|

(38)

where

τ

_1,0 istheinitialcodedelayoffsetoftheMPsignal withrespecttothedirectLOSsignal.

(ii) ThecomplexamplitudeoftheMPsignal

α

_{1, k} dependsonthe previousamplitude

α

_{1 ,k −1} throughtheparametricmodel

α

1 ,k =e j2 πf caT aτ˙ 1,k

α

_{1 ,k −1} +

ω

_{α (39)}

where

ω

_α is azeromeanadditive complexGaussianwhite noisewithvariance

σ

_α2 .

This modelwasmotivatedby its efficiencyin MPprone envi-ronmentssuchastheurbancanyons [27] .Weproposetocompare the proposed approach with FIMLA, the coherent narrow corre-lator DLL and particle filter-based MP mitigation approach

stud-ied in [22] for this scenario. Accordingly, the propagation

mod-els forthedirect LOSandMPsignal parameters usedinthe par-ticle filter-basedMP mitigation approach were defined using (5), (37) and(39) . The simulation time Twas set to 5 s. The CNR of the directLOS signaland themultipath-to-directratio oftheMP signal amplitudes were set to 46dB-Hz and 0.5, respectively. A fast-fadingMPconditionwasconsideredinthisdynamicscenario. Accordingly, the initial code delay offset of the MP signal and the corresponding rate of change were set as

τ

_{1 , 0} =0.2 chips and

τ

˙_{1 , 0} =0.01chips/s (whichisequivalenttoa relativespeed of about 4m/s between the receiver and the reflector). The carrier Doppler frequencyoffsetoftheMPsignal withrespecttothe di-rectLOSsignal(i.e.,thefadingfrequency)wasapproximatelyequal to 5 Hz. The standard deviations were fixed to

σ

_τ=10−3 chips,

σ

τ˙ =10−4 chip/s and

σ

_α=0.01, which isa reasonablechoice to resembleatypicalurbansatellite navigationchannelenvironment

[40] .

Fig. 5 showsthe RMSEsofthedirectLOS codedelayobtained with different approaches in the fast-fading MP condition. Since the prior information about the LOS signal parameters is con-sidered both in the proposed EM-based and particle filter-based MP mitigation approaches, the RMSEs obtained with these two

Fig. 5. RMSE of the LOS code delay obtained with the different approaches.

approaches are obviously smaller than those obtained with the coherentnarrow correlator DLLandFIMLA.Moreover, the results obtainedbyusingtheproposedEM-basedandparticlefilter-based MPmitigation approachesareverysimilar. Althoughthe propaga-tionmodelassociatedwithMPsignalparametersisnotconsidered intheproposedapproach,theEMiterationcanreducetheimpact ofthe modeluncertainty resultingfromthe unknown model pa-rametervector (containingtheMPsignal parameters)onthe esti-mateaccuracyforthestatevector(containingthedirectsignal pa-rameters).The averaged valuesofMPcode delayoffsetestimates over100MCsimulationsaredepictedin Fig. 6 .SincetheEM iter-ationisimplementedovereachobservationinterval,thechangeof MPcode delaywithtime can be accurately trackedby using the proposed approach andthe corresponding estimation accuracy is slightlyinferiortothatobtainedwiththeparticlefilter.Thisisdue tothefactthatthepriorpropagationmodelforMPsignal parame-tersisnottakenintoaccountintheproposedapproach.Whenthe numberofEMiterationsismaximum,thecomputational complex-itiesofthe proposedEM-based andparticle filter-basedMP miti-gationapproachesareO(rmax Ns T) andO(Ns T),respectively. Table 3

showstheexecutiontimesfor100MonteCarlorunsby using dif-ferentnumbersofparticlesintheproposedEM-basedandparticle filter-basedMP mitigation approaches.The computational cost of

Table 3

Execution times using the different number of particles. Number of particles ( N s ) Execution times (s)

Proposed approach with the maximum number of iterations Proposed approach with the stopping rule Particle filter

50 191.34 105.16 36.58

100 351.85 192.20 69.29

200 662.38 360.47 138.91

400 1295.66 714.87 271.31

Fig. 6. Averaged values of MP code delay offset estimates over 100 MC simulations.

theproposed approachwasevaluatedin thefollowingtwo situa-tions:(a)thealgorithmisstoppedwhenthestoppingruleinLine

8of Algorithm 2 issatisfied(b)thenumberofEMiterationsisset

toitsmaximumvalue r_{max =}15.Since theimplementationofthe EMiteration ismadefor each observationinterval, the computa-tionalcostoftheproposed approachismuchhigherthantheone associatedwiththeparticlefilter.Thankstothegoodconvergence propertyof the proposed EM-based MP mitigation approach,the actual numberof EMiterations ofthe proposed approach is less than the maximum number of iterations. As a consequence,the correspondingcomputationalcostcanbeefficientlyreducedby in-troducing thestopping rule ofLine 8 of Algorithm 2 inthe pro-posedapproach.

Accordingtothepreviousexperiments,theparticlefilter-based approachhasabetter MPmitigationperformance duetothefact thatthe propagationmodelforMP signalparameters followsthe actualdynamics ofMPsignals.However, an accurate propagation modelcannot always be definedto capture the dynamics of MP signalsin real applications. In the next experiments, we assume that the differences between consecutive code delays of the MP signalareindependentandfollowanexponentialdistribution [41] , i.e.,

μ1

,k =

τ1

,k −

τ1

,k −1 (40) with p



μ1

_,k



=

_μ0

1 e− IR(μ1,k) μ0 (41)

whereIR + istheindicatorfunctiononR+ and

μ

0 isa predefined

decayingparameter,whichwassetto2× 10−3 _chipsinthis

exper-iment.Inaddition,thecomplexamplitudeoftheMPsignalvaries accordingto (39) . In this experiment, the code delayof the MP signalwasgeneratedusing (37) duringthetimeinterval[0s,2s) andusing (40) during the time interval [2 s, 5s]. Fig. 7 displays the RMSEs of the direct LOS code delay obtained with the pro-posedEMandparticlefilter-basedMPmitigationapproaches.Itis clearthat theperformance oftheparticle filter-basedapproach is significantlyreducedduringthesecondtimeinterval[2s,5s]due toaninaccuratemodelforthedynamicsofMPsignals.Conversely,

Fig. 7. RMSE of the estimated LOS code delay in the presence of changes affecting the dynamics of MP signals.

Table 4

MP signal parameters in dynamic scenarios.

Signal parameters MP signals

MP #1 MP #2

Appearance time (s) 0–5 0–2.5 4–5

Initial code delay offset (chips) 0.1 0.4 0.3

Initial change rate of the MP code delay (chips/s) 0.01 −0.03 0

Multipath-to-direct 0.7 0.3 0.5

Fading frequency (Hz) 2 5 0

theproposedEM-basedapproachislessaffectedbyan inaccurate propagationmodelfortheMPsignalparameters,sincethismodel is not used inthe EM algorithm. Theseresults indicate that the proposed EMalgorithmprovidesabetter robustnessinconstraint environments,suchasurbancanyons.

Inthe last experiments,we consider two reflected MPsignals simultanouslyaffectingthedirectLOSsignaldynamically.The pa-rameters ofthe two MP signals are provided in Table 4 . The es-timatedcode delayoffsetsandmultipath-to-directratiosofthese MP signals are displayed in Fig. 8 . As shown in Fig. 8 , the MP signal parameters can be accurately tracked by the proposed ap-proacheveninthepresenceofmultipleMPsignals.Notethatthe estimatedvalueofthe multipath-to-directratiofortheMP signal #2islessthan0.05chipswhenthecorresponding MPsignal dis-appears. Thus a thresholdon the estimated MPsignal amplitude canbeusedforeliminatingthefalseMPsignalpathestimationin theproposedMPmitigationapproach.Inotherwords,theMP sig-nalpath isnottakenintoaccount whentheestimatedMP signal amplitudeissmallerthanthethreshold.Consideringthat two re-flected signals very close intime can be considered as only one perturbation inthe LOS signal parameter estimation [19] , a two-pathmodel(i.e.,M=2)intheproposed EM-basedMPmitigation approachisoftensufficientforpracticalapplications.