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CrosstalkMitigationinDMTVDSLwithImpulseNoise*

HuaiyuDai,StudentMember,IEEE,andH.VincentPoor,Fellow,IEEE

DepartmentofElectricalEngineering,PrincetonUniversity

Princeton,NJ08540

Tel:(609)258-4634Fax:(609)258-1560

Email:huaiyud@ee.princeton.edu,poor@princeton.edu

Abstract:

Crosstalkandimpulsenoisearetwoprincipalsourcesofdegradationinvery-high-ratedigitalsubscriber

line (VDSL) transmission systems. The traditional single-user data detector for such systems merges

crosstalk into thebackgroundnoise,which is assumed to bewhite andGaussian.Recent researchhas

exploredthenatureofcrosstalksignalsandshownthepotentialbenefitsofmultiuserdetectionforVDSL

signalswith strongcrosstalkers. Impulsenoise isoneof themost difficult transmission impairments to

suppressandispoorlycharacterizedandunderstoodaswell.InDSLtransmissionimpulsenoiseistypically

combated with interleaved forward error correction. However, recent data indicates that a significant

minorityofimpulsenoiseeventsarelongerthanthemaximumerrorcorrectingcapacitiesofthedefault

interleavedforwarderrorcorrection(FEC)providedwithincurrentANSIstandards.Thus,itisofinterest

toconsidersignalprocessingmethodsthatcanjointlymitigatecrosstalkandimpulsivenoise.Inthispaper,

we explore such a technique based on a recently developed robustM-detector structure for multiuser

detectioninnon-Gaussianambientnoise.

IndexTerms

Crosstalk,DMT,DSL,Impulsenoise,M-estimation,Multiuserdetection

*ManuscriptreceivedJune20,2000;revisedMay9,2001.ThisresearchwassupportedbytheNationalScienceFoundationunderGrantCCR-99-80590.

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1. Introduction*

Digital subscriber line (DSL) technology provides transport of high-bit-rate digital information over

telephonesubscriberlines.Phonelines,whichwereoriginallyconstructedtocarryasinglevoicesignal

witha3.4kHzbandwidthchannel,areactuallycapableofcarryingveryhighdataratesifthenarrowband

switchinthephonecompanycentralofficecanbeavoided.VariousDSLtechniques(basicrateintegrated

servicesdigitalnetworks(ISDN),high-bit-rateDSL(HDSL),AsymmetricDSL(ADSL),andvery-high-

rate DSL (VDSL)) which involve sophisticated digital transmission schemes and extensive signal

processing have recently become practical due to advances in microelectronics. The latest in DSL

technologyisVDSL,whichprovidestensofmegabitspersecondtothosecustomerswhodesirebroadband

entertainmentordataservices.AsymmetricVDSLisviewedmoreasaresidentialservice,supportingupto

52 Mb/s downstream and 6.4 Mb/s upstream rates for delivery of digital TV and high definition TV

(HDTV)services. Symmetric application ofVDSLprovides two-waydatarates up to26Mb/sfor data

networkorlocalareanetwork(LAN)extension,mainlyasabusinessservice.Atsuchhighrates,signalson

twistedpairscanbereliablytransmittedatmosttoafewthousandfeet.Thus,VDSLwillprimarilybeused

forloopsfedfromanopticalnetworkunit(ONU)oracentraloffice(CO)toacustomerspremises,theso-

calledlastmileproblem.ThemodulationschemeforVDSLcaneitherbemulticarrier-basedorsingle

carrier-based, typically discrete multitone (DMT) and carrierless amplitude/phase modulation

(CAP)/quadrature amplitude modulation (QAM). The duplexing methods can be either time-division

duplexing(TDD)orfrequency-divisionduplexing(FDD).

TypicalphonelinesthatcarryVDSLsignalsare24-or26-gaugeunshieldedtwistedpairs(UTP).Multiple

telephonepairsmaysharethesamecable.NormallyVDSLsignalsoccupyfrom300kHzto30MHzofthe

twisted-pairbandwidthandareseparatedfromplainoldtelephonesystem(POTS)/ISDNsignalsbysplitter

devices.Noiseonphonelinesnormallyoccursbecauseofimperfectbalanceofthetwistedpair.Thereare

manytypesofnoisesthatcouplethroughimperfectbalanceintophonelines,themostcommonofwhich

*This article isbased ona paper presentedat theIEEE CAS-COMWorkshop'99on high speeddatanetworksheldatPrincetonUniversityonJuly26-28,1999.

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arecrosstalknoise,radionoiseandimpulsenoise.Crosstalkiscausedbyelectromagneticradiationofother

phonelinesincloseproximity,inpracticewithinthesamecable.Suchcouplingincreaseswithfrequency

andcanbecausedbysignalstravelingintheoppositedirection,callednear-endcrosstalk(NEXT),andby

signals traveling in thesame direction, called far-end crosstalk (FEXT).Radionoise is the remnantof

wireless transmission signalscoupling into phone lines, particularly AM radio broadcasts andamateur

(HAM)operatortransmissions.Impulsenoiseisanonstationarycrosstalkfromtemporaryelectromagnetic

events(suchastheringingofphonesonlinessharingthesamebinder,andatmosphericelectricalsurges)

thatcanbenarrowbandorwidebandandthatoccursrandomly.Impulsenoisescanbetensofmillivoltsin

amplitudeandcanlastaslongashundredsofmicroseconds[6],[16].

WhiletheradionoiseproblemcanbesolvedoratleastalleviatedbyrestrictingtheVDSLtransmissionin

radiobands,crosstalkandimpulsenoiseare twoprincipalsourcesofdegradationinVDSLtransmission

systems.Thetraditionalsingle-userdetector(SUD)forsuchsystemsmergescrosstalkintothebackground

noise,whichisassumedtobewhiteandGaussian.Actually,crosstalkistheresultofthesumofseveral

filtereddiscretedatasignals.ItsdistributiondeviatesfromGaussian,anditspowerspectraldensity(PSD)

isalsosignificantlygreaterthanthatofbackgroundaddedwhiteGaussiannoise(AWGN).Recentresearch

hasexploredthenatureofcrosstalksignalsandhasshownthepotentialbenefitsofmultiuserdetectionfor

VDSL signals with strong crosstalkers [2], [3], [4]. In DSL transmission impulse noise is typically

combatedwithinterleavedforwarderrorcorrection (FEC).However,recentdataindicatesthatasignificant

minorityofimpulsenoiseeventsarelongerthanthemaximumerrorcorrectingcapacitiesofthedefault

interleavedFECprovidedwithincurrentANSIstandards[9],[10].Thus,itisofinteresttoconsidersignal

processingmethods thatcan jointlymitigatecrosstalkand impulsivenoise. Recentworkhas examined

multiuser detection (MUD) innon-Gaussian ambientenvironments for wireless code-divisionmultiple-

access(CDMA)systems[22].Inparticular,thisworkhasshownthatstandardlinearmultiuserdetectorsare

notrobustto certain typesofnon-Gaussianambientnoise(particularly impulsivenoise),whereas low-

complexitynonlinearmodificationsprovideexcellentperformanceinsuchenvironments.Itisthepurpose

of thispaper toexaminesimilar techniques for crosstalk and impulsenoisemitigation inDMT VDSL

systems.NotethatthecrosstalksignalsinDSLtransmissionareofvarioustypesandcannotberepresented

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undera uniformframework,to thebest of theauthorsknowledge.In ourapplicationofMUDtosignal

detectioninDSLsystems,wedealmainlywithNEXTofothertypes(incontrasttotheselfNEXTcoming

fromthephonelinescarryingthesameVDSLservice).Thereasonisgivenasfollows.FEXTexperiences

the same line attenuation as the desired signal while NEXT does not, which makes NEXT the most

detrimentaltypeof interference,especially athigh frequencies.SelfNEXT canbe largely alleviatedby

duplexingmethodsthatseparate theupstreamanddownstreamdata intimeorfrequency.Therefore,the

other-typeNEXT provides the best opportunity for performance gain. However, the idea of multiuser

detection is valid and applicable tomitigation of crosstalkof all types, althoughmodifications of the

techniquesproposedheremaybenecessaryforeachspecificsituation.Notethatwedonotconsidercoding

inthispaper,butthisissueistreatedinasequel[8].

Thispaperisorganizedasfollows.InSection2asignalmodelfortheDMTVDSLcommunicationsystem,

aswellas theimpulsechannelnoisemodel,is described.InSection3weproposea robustMUD-based

scheme for DMTVDSLsignaldetection. Inorderto reducethe receiver complexity whilemaintaining

goodperformance, a suboptimum receiver is introduced in Section4, together with its robustversion.

SimulationresultsaregiveninSection5,andSection6concludesthepaper.

2. SystemModel

Figure1depictsabasiccrosstalkingchannelwithonedesiredVDSLsignalandK-1crosstalkers.Theloop

transferfunctionH ofthedesiredVDSLsignaland thecrosstalkcouplingfunctionsareassumedto be

known.Atthechanneloutputbackgroundnoiseisadded,amodelforwhichwillbeintroducedshortly.

The VDSL signal studied here uses a DMT transmission system, whose transmitter and receiver are

depicted inFig. 2andFig. 3,respectively.The typical twistedpair isan intersymbol-interference(ISI)

channel.However,ifthenumberofsubchannelsislargeenough,thecontinuoustransferfunctionofthe

channelresponsecanbeapproximatedbydiscretesubchannelgains,asillustratedinFig.4.Thenwecan

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effectivelydecomposetheoriginalchannelintoasetofN parallelindependentchannelswithnoISI.For

eachsubchannelinthefrequencydomain,theoutputisgivenby

=

++=

K

k

nknnnnNCXHY

2

, , Nn ,...,1= , (1)

where nH isthechannelgain, nX isthetransmitted(complex)DMTsymbol, knC , istheinterferencefrom

thekthcrosstalker, Kk ,,2 = ,and nN isthebackgroundnoiseatthenthsubchannel[5].

Impulsenoise is a severe impairment toDSLtransmission, especiallyafter long loop attenuation (ata

residentiallocation)andinhighfrequencies(wheretheDSLsignalismoreseverelyattenuated).However,

thearea of impulsenoisemodeling remainsunsettled.Cook presented an analyticalmodel in [7]. The

ADSLstandard,however,usesstoredrepresentativeimpulsewaveforms,whicharemeasuredempirically.

ValentietalcollectedimpulsenoiseandbackgroundnoisedataonADSLloopsatNewJerseyresidences

anddidanalysisonthedatainthreeways:aspowerandenergyspectraldensities,asprobabilitydensity

functionsofthetimewaveformvoltageamplitudes,andasimpulsearrivalandinterarrivaltimestatistics

[12], [18], [19]. So far there areno suchmodels for impulse noise in VDSL, but similar results are

anticipated[20].Ourkey observationfromtheseanalyses is:thereare significantimpulsespikesin the

PSDofthemeasuredwidebandnoise,whichisotherwiseessentially flat.Tomodel thisbehaviorofthe

impulse noise we use a two-term Gaussian mixture model in the frequency domain. The first-order

probabilitydensityfunctionofthisnoisemodelhastheform

),0(),0()1(22

+ (2)

with 0> , 10 , and 1 . Here, the ),0(2

term represents thenominal background noise

(Gaussianwithzeromeanandvariance 2 ),andthe ),0(2

termrepresentsanimpulsecomponent

(Gaussianwithzeromeanandvariance 2 ),with representingtheprobabilitythatimpulsesoccur[22].

Itisassumedthatnoisesamplesindisjointfrequencybinsareindependent.

3. RobustMaximumLikelihoodMultiuserDetectionReceiver

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AswementionedinSection1,itispossibletoapplymultiuserdetectiontojointlydetecttheVDSLsignal

andthecrosstalksignalsandthusgreatlyimprovethesystemperformance.Forsimplicityletusassumethe

backgroundnoiseto beGaussian(i.e.(2)with 0= )forthemoment.Wewillreintroducetheimpulse

noise model below. According to the systemmodel given in Fig. 1, the optimalmaximum likelihood

multiuser detector (ML-MUD) forGaussiannoise isone thatestimatestheVDSLinputandcrosstalker

inputs inunison soas tominimize thedistancebetweenthe channeloutput received signal and allthe

possiblediscretewaveformoutcomes.Itispossiblethatthecrosstalksignalsarewronglyestimated,butthe

probabilityoferroneousselectionofthedesiredVDSLsignalwillbelessforsuchadetectorthanwhenthe

crosstalk signals are simply assumed to be Gaussiannoise.We will expect a greater improvement in

performanceifthedifferencebetweenthePSDlevelofcrosstalksignalsandbackgroundnoiseislarger.

Generally speaking, crosstalk strength increaseswith frequency: NEXT with 5.1f andFEXT with 2f .

Fortunately,FEXTexperiencesthesamelineattenuationasthedesiredsignal;butunfortunately,NEXT

doesnot.ForVDSLsystems,high-frequencyNEXTisthemostdetrimentaltypeofcrosstalk,butwillalso

bemostpromisingforreductionviaMUD.ThetypicalbackgroundnoiselevelofVDSLtransmissionis

140dbm,while thetypicalNEXT is90~110dbm;thuswecanexpect substantialgain frommultiuser

detectionrelativetotraditionalsingleuserdetectioninthissituation.Besides,inDMTVDSLsubchannels

wheretherearesubstantiallystrongercrosstalksignals(typicallyinhighfrequencybandsonlongloops),

theso-called"near-far"probleminwirelessCDMAsystems,SUDwillfailtoworkproperlywhileoptimal

MUDshouldessentiallyachievethesingleuserlowerbound.

Letusconsiderthedetectionproblemforthedatamodelgivenin(1).Thetraditionalsingleuserdetector

performsQAMdemodulationand detection.Ontheotherhand, jointmaximum-likelihooddetectionof

both VDSL and crosstalk signals selects a set of N inputs { }nX and the crosstalk sequence

{ } ,,,2,,...,, )(,

)(,2

)(,1

)(KkCCC

i

kN

ik

ik

ik

==C tosatisfy

},minarg{2

1 2

)(

,}{},{

)(,

== =

N

n

K

k

i

knnnnCX

n CXHYX iknn

Nn ,...,1= , (3)

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where theminimization is searched over the DMTsignal alphabet and all possible crosstalk sequences

{ } ,,,2,,...,1,)( Kki kikk === CCC thatcanoccurwithintheVDSLsymbolperiodofinterest.Thesize

,,,2, Kkk =C oftheallpossiblecrosstalksequencessetcanbelargebutisalwaysfinitewhenallthe

crosstalkersaredigitalsignalsorarederivedfromdigitalsignals.

ForwhiteGaussiannoise,maximumlikelihooddetectionisthesameasleast-squares(LS)curvefitting,as

canbeseenfrom(3).ItiswellknownfromtheclassicworkofTukey[17]thatleast-squaresestimatesare

verysensitivetothetailbehavioroftheprobabilitydensityofmeasurementerrors(representedherebythe

additive noise). Its performance depends significantly on the Gaussian assumption, and even a slight

deviation of the noise density from the Gaussian distribution can, in principle, cause a substantial

degradationoftheLSestimate.TheLSestimatecorrespondingto(3)canberobustifiedbyusingtheclass

ofM-estimatorsproposed byHuber [11]. Insteadofminimizing overa sumofsquared residuals, Huber

proposedtousealessrapidlyincreasingpenaltyfunction soastoalleviatetheeffectoftheimpulses.

},)(minarg{1 2

)(

,}{},{

)(,

== =

N

n

K

k

i

knnnnCX

n CXHYX iknn

Nn ,...,1= . (4)

Theusualrequirementsforthepenaltyfunctionanditsderivative = are:

(1) issub-quadraticfunctionforlargevaluesofresiduals,inordertode-emphasizetheerrorcausedby

noise"outliers"(inthiscasecausedbyimpulsenoise);

(2) isboundedandcontinuous;

(3) kxx )( forsmallx ,soastoachievehighefficiencyintheGaussiancase;

(4) 0)}({ =jNE to get a consistent estimate; and for symmetric noise densities is usually odd

symmetric.

A good choice for Gaussianmixture noise is the Huber penalty shown in Fig. 5 together with its

derivative .Thesefunctionsaregivenexplicitlyby

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>

=

222

2

2

2

||2

||

||2)(

kxk

xk

kxx

x (5)

and

>

=

kxxk

kxx

x

||)sgn(

||)(

2

2 , (6)

where k , ,and (see(2))arerelatedby

)1(2

)()(

= kQ

k

k, (7)

with 22

21)( xex

and t

x dxetQ 22

21)(

(see,[11],[22]).

Inthispaper,wewillconsiderthisparticularchoiceof ,andtheresultingDMTVDSLdetectorwillbe

calledtherobustmaximumlikelihoodmultiuserdetectionreceiver(ML-MUD-R).Theperformanceofthis

detectorwillbecomparedwithML-MUDandSUDinSection5.

4. AnInterferenceCancellationMultiuserDetectorandItsRobustVersion

Justas itscounterpartinwirelessCDMA,themaximum likelihoodmultiuserdetectorachievesoptimum

performance but suffers from very high complexity. A full search in the input domain requires

approximately N |C||M| squarederror computations,where N is thenumberofsubchannels,|C|isthe

numberofpossiblecrosstalksequences,and|M|istheaveragesizeofthetransmittedalphabet.Inpractice

N and especially |C| can be very large, introducing prohibitive computational complexity. The large

number ofpossible crosstalksequences alsomeansan exponentially greater number of states, making

dynamicprogramminginappropriate.Soweneedtoconsiderasimplifiedreceiverstructurethatmaintains

satisfactoryperformancewhilerequiringfarlesscomputationcomplexity.

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Onelower-complexityapproachistoemployalinearmultiuserdetectiontechnique,suchasdecorrelating

(zeroforcing)or minimum-mean-square-error(MMSE)multiuser detection.However, unlikeCDMAor

space-divisionmultiple-access(SDMA)wherelineardetectionhasbeeneffective,thereisnoidentifying

signaturesuchasthespreadingcodeforCDMAorthesteeringvectorforSDMA,toaidlineardetectionin

VDSL.Instead, desiredsignalsandcrosstalksignalsareoftenofdifferent data format.Another popular

approachistoemployinterferencecancellation,i.e.,toattemptremovalofthecrosstalkfromthereceived

signalbeforemakingthetraditionalDMTVDSLsignaldetection[21].Thisistheapproachweadopthere.

Todoso,weneeda schemeto detectthe crosstalk signals first.Aswementionedbefore, thecrosstalk

signalsinDSLtransmissionareofvarious typesand cannotberepresentedundera uniformframework.

Thetypeweexaminehereisthedominantnear-endQAM-likecrosstalk(e.g.[3]).

At first glance, it seems quite difficult to detect the crosstalk correctly with reduced computational

complexity.Afterall,itisthehugesetofpossiblecrosstalksequencesthatcomplicatesthecomputationin

(3). Let us consider the power spectrum of the DMTVDSL signal and the crosstalk signals. As we

mentionedbefore,eachsubchannelhasindependenttransmitteddataandaddedbackgroundnoise,sothe

energyisfairlyspreadacrossthefrequencydomainofinterest,althoughitisnotequaleverywheresince

differentbitsmaybeassignedtodifferentsubchannelstoachievetheoptimumperformance.Incontrast,

thePSDof theQAM-like crosstalk signalsareoftenclusteredaroundseveral relativelynarrow spectral

components(called "tones" inDMTmodulation).Anatural ideais tozero these tones inDMT-VDSL

transmission,i.e.,donottransmitDMTVDSLsignalsonthesetones.ThisisaformofCDMAwherethe

DMTVDSLsignalisorthogonalto thecrosstalksignalsonthesetones.Because thedataratesofthe

crosstalksignalsareusuallylowcomparedtotheVDSLsignalandtheSNRsareexcellent(thecrosstalk

signals are treated asthe signalsof interest here), only a fewzeroed tones are necessary to detect the

crosstalksignalfairlywell.Thusthecomputationcomplexityisgreatlyreducedto Nz|C|,whereNzisthe

numberofzerotones(e.g.,Nz=5),inadditiontothenearlytrivialconventionaldemodulation.Thechoice

oftones tobe zeroeddepends onthe knowledge ofwhere theenergy ofcrosstalk signalsconcentrates,

whichgenerally is known.Advancesin digitalsignal processingmake tone zeroingeasyto implement.

Furthermore, spectral compatibility with otherDSL transmission and radio broadcast oftennecessitates

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someparticulartonesbeingzeroed.Finally, zeroingofthese tonesalsoleadstoa reduction inFEXTon

thesetones.

Figure6givesthestructureofthisinterferencecancellationmultiuserdetector(IC-MUD).Thedetailofthe

IC-MUDalgorithmisgivenasfollows.

1. Choosethetonestobezeroedbasedontheknowledgeofaspecificcrosstalksignal;

2. Thecrosstalksignalisdetectedandreconstructedin theseDMT-symbol-freechannels;e.g.,

forahomephonenetworkofAmerica(HPNA)signal(see[3]),itcanbedetectedvia

}minarg{2

,,}{

,,

=

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5. SimulationResults

Inthissectionweexaminethebehaviorandtheperformanceoftheproposedmultiuserdetectionreceivers

forDMT-VDSLsignalswithcrosstalkandimpulsenoiseviacomputersimulations.Bit-error-rate(BER)is

adopted as the performance measure withr espect to the average signal-to-noise ratio (SNR), which is

definedas

.

1

2

1

2

=

=

=N

n

n

N

n

nn

N

XH

SNR (11)

Inthesimulation,theDMTVDSLsignalisassumedtooccupy0-25.6MHzwith256subchannelsinan

frequency-divisionmultiplexed(FDM)design.ThesymbolrateforeachVDSLsubchannelis100kHz.In

eachsubchannel,2bitsareassignedsothesignalsare4-QAM.Nobitallocationalgorithmsareusedhere,

although extensionto this case is straightforward. The transfer function of the DMT VDSL signal is

simulatedby

3

2

2

10539.050.1965.0

242)(

+

=

jj

jjj

ee

eeeH . (12)

WeassumeoneNEXTcrosstalksignalwithaknowncouplingfunctiongivenas

4/3)( = KeF j , (13)

where Kis a constantusedtoadjust thePSDofthecrosstalk signal.Thesesettingsaremadeto roughly

approachthePSDshapesindicatedin[1].Weassumethatthesetransferfunctionsstayfixedforthewhole

simulationinterval,whichisreasonableforwirelinecommunicationsenvironments.Thecrosstalksignalis

BPSKmodulatedon8MHzcarrierfrequencywitha1Msymbol-per-secondrate.Suchasituationwould

arise,forexample,duetothecoexistenceofhome-phoneLANsandasymmetricDMTVDSLsignalsinthe

samecableinthecustomerpremises.Thus,thereare210

possiblecrosstalksequencesinoneVDSLsymbol.

Thisnumberischosenforsimulationsimplicity.Inreality,thisnumbercouldbemuchlarger.ForIC-MUD

and IC-MUD-R, the five zeroed tones are }2.8,1.8,0.8,9.7,8.7{=zT MHz, around whichmost ofthe

crosstalkenergyisconcentrated.Theimpulsenoiseisassumedtohaveparameters 1.0= and 100= ,

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whichmeanstheimpulsespikeis20dBhigherthanthebackgroundnoisefloor.TheaveragePSDlevelsof

the crosstalk signal and background noise floor are fixed while that of the desired signal is varied,

corresponding to different line length (the signal attenuation is increasing with the line length). In our

simulation,theaveragePSDofthecrosstalkis27dBhigherthanthatofthebackgroundnoisefloorandthe

peak PSD of the crosstalk is 40 dB higher. These settings seem to agree roughly with empirical

measurements.

IntheDSLenvironment,BERvaluesaslowas10-7

areoftenrequired.ForMonteCarlo(MC)simulation,

approximately eP/10 simulation trials are required to have a 95 percent confidence interval of

]5/8,5/2[ ee PP [13]. To alieviate this computational burden, we use importance sampling (IS) [13],

[14], [15]. Thebasic ideaofimportancesamplingis tobiastheprobability densityfunction (pdf)from

whichthedataaregeneratedsothaterrorsofdetectionaremorelikelytohappen,thenweighteacherror

such thatan unbiased BER estimateis obtained. Assumean erroroccurs when thereceiveddataR fallswithinsomeregionZ.ThentheBERisgivenby

= drrfrP RZe )()(1 , (14)

where )(1 Z istheindicatorfunctionoverZand )(Rf isthepdfofR.

TheMCestimatorof eP isgivenby

==

M

iiZMC R

MP

1

)(11 , (15)

whereMisthenumberoftrialsofthesimulationandthe iR sdenotedatasamples.Whenthedatasamples

areindependentandidenticaldistributed(i.i.d.), MCP isanunbiasedestimatorwithvariance

M

PPP eeMC

)1()var(

= . (16)

TheISestimatorof eP isgivenby

==

M

iiiZIS RWR

MP

1

** )()(11 (17)

with

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)(

)()(

* rf

rfrW

R

R= , (18)

where*iR isthe ith data sample from biased density )(* Rf and )(W is the weight function. Ifthe new

generateddataarei.i.d., ISP isanunbiasedestimatorwithvariance

M

PWP eIS

2

)var(

= , (19)

where W isdefinedas

drrfrWW RZ

)()(

= . (20)

When )(* Rf isappropriatelyselected,thevarianceoftheISestimatorwillbefarlessthanthatoftheMC

estimator.Thus thenumberof trialsneededfora givenestimator variance isgreatly reducedfor theIS

estimatorcomparedtotheMCestimator.Theoptimalbiasdistributionisgivenby

e

RZ

R P

rfrrf

opt

)()(1)(* = , (21)

which achieves zero estimation variance but is degenerative since it requires theknowledge of eP .A

widelyusedmethodofdesigningsuboptimal )(* Rf ismeantranslation(MT).Thisclassofbiaseddensity

functionsisoftheform

)()( *** Trfrf RR += , (22)

whereT ischosentobe themode (atwhichmaximumvalueofa pdf isachieved)of )(* optR

f .Forthe

multiuser communication system of (1), let ),,,( 2 KCCX = , impose the restr iction

)()(* = ff andconditionallyshiftthemeanofthenoise

)())(()(2

******

|**

++=+==

K

kkNNN

CXHnfmnfnf

. (23)

TheISestimatorofBERisthengivenby

= ===

M

iiN

iNii

M

iiiiiIS Nf

NfXX

MNWXX

MP

1*

|

***

1

****

)(

)(|)(|

1),(|)(|

1

**

, (24)

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whereweassumetheindependenceof and N ,*iX isthedetecteddataof

*iX withtheoriginaldecision

rule,and

>=

00

01)(

x

xx . (25)

Whenthenear-farproblemoccurs,i.e.,

)sgn())(sgn(**

XHm = , (26)

weneedtoadjusttheISerrorestimatorasfollows:

+==

M

iiN

iNiiiiIS

Nf

NfXXXX

MP

1*

|

*****

))(

)(1|))((|1(|)(|

1

**

. (27)

Notethatinthissituation,theIStechniqueisusedtocountcorrectdetections(whichhappenwithsmall

probability),whichthengives(see(24))

==

M

iiN

iNiicorrectIS

Nf

NfXX

MP

1*

|

***

)(

)(|)(|

1

**

. (28)

Thequantityof(27)isthenobtainedthrough correctISIS PP =1 .In oursimulations,theIStechniqueis

uniformly better than the MC technique. It achieves great variance reduction for optimal detection

(maximumlikelihood)methodsandalsogetssubstantialgainsforothers.

Figure7shows theperformance ofvarious detectors forDMTVDSLsystemswithone crosstalkerand

impulsenoise.Aswecansee,thereisasignificantgapbetweentheperformanceofthetraditionalsingle

userdetectorandthesingleuserlowerbound(corresponding toa crosstalk-freechannel), indicatingthe

ineffectivenessof the single-user detector.Whilethe maximumlikelihoodmultiuser detector essentially

achievesthesingleuserlowerbound,itsuffersfromprohibitivecomplexity.Theinterferencecancellation

multiuserdetectoroffersafavorableperformanceandcomplexitytradeoffcomparedwiththesingle-user

andMLmultiuserdetectors.

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Figure8shows theperformanceof theM-estimator-based robustdetectors in thecrosstalkand impulse

noiseenvironment.WhilethereisnotmuchdifferencebetweentheMLmultiuserdetectoranditsrobust

version,bothofwhichapproximatethesingleuserlowerbound,thereissignificantimprovementforthe

robust interferencecancellationmultiuser detection comparedwith itsGaussian-based counterpart. The

crosstalkdetectionerrorsare 41042.3 forIC-MUD, 51010.6 forIC-MUD-Randalmost0 forML-

MUDandML-MUD-R.ItshouldbenotedthattheexpectedimprovementfromusingM-estimatorsisdue

tobetterestimationof thecrosstalk signals intheDMTVDSLcase.ThedesiredDMTVDSLsignalsin

differentsubchannelsareindependentwhilethecrosstalksignalsarecorrelatedinthefrequencydomain,

which means that M-estimators are especially applicable to impulse-noise-contaminated DMT VDSL

systemswithcrosstalksignalsstronglycorrelatedinthefrequencydomain.However,morecrosstalkerrors

donotnecessarilymeanworseperformance,especiallyfortheMLjointdetectionscheme.Thisisbecause

the whole set of possible crosstalk sequences is usually divided into many small subsets. While the

correspondingsequencesofasubsetcanbelargelydifferent,theirspectralcomponentsaresimilar.Infact,

wefoundfromoursimulationsthattheICschemeismuchmoresensitivetocrosstalkdetectionerrorsthan

istheMLscheme.Forexample,ifwelowerthepowerofthecrosstalkby15dB(whichcanbethoughtof

asaFEXT)whilekeepingtheothersettingsunchanged,thecrosstalkdetectionerrorsare 11063.1 for

IC-MUD,2

1065.3

forIC-MUD-R,2

1091.3

forML-MUDand4

1095.1

forML-MUD-R.ButML-

MUD still almost approaches the single user lower bound, which can be seen from Fig. 9. Since the

crosstalksignalsareestimatedfirstinonlyafewtone-freesubchannelsforIC-MUD,moregainofIC-

MUD-RoverIC-MUDisachievedascomparedwiththegainofML-MUD-RoverML-MUD.

Finally,Fig.10showsthat,fortheinterferencecancellationmultiuserdetector,strongcrosstalkactually

improvesthesituation.Welowerthestrengthofthecrosstalk12dBandcomparetheperformancesofthe

traditionalsingleuserdetectorandrobustinterferencecancellationmultiuserdetectorappliedtothetwo

differentcrosstalkenvironments.Theimpulsenoisesettingsremainunchanged.ItisseenthatIC-MUD-R

performsbetterwith thestronger crosstalk.This isno surprise, sinceforsuccessive cancellation, strong

interference isalmost as good as no interference.These results also indicate that for crosstalkwithout

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significantlygreaterPSDlevelthanthatofthebackgroundnoise,athighSNR,theICschemedoesnotget

muchgainoverSUD.

6. Conclusions

InthispaperwehaveshownthepotentialbenefitsofmultiuserdetectionforcrosstalkmitigationinDMT

VDSLsystemssubjecttoimpulsenoise.WeseethatML-MUDcanessentiallyeliminatecrosstalksignals

inDMTsystemsatacostofhighcomplexity.Asatradeoff,IC-MUDcansignificantlyoutperformSUD,

with lower complexity than ML-MUD.We have also shown the effectiveness of the M-estimator in

combatingtheimpulsenoise.

Therearesomeissuesoverlookedinthispaper,whichmightbeofinterestforfurtherstudy.Forexample,

wehaveassumedknowledgeofthelinetransferfunctionandthecrosstalkcouplingfunctions.Inreality,

however,channelidentificationisneeded.Also,wehavenotconsideredtheissueofoptimalbitallocation

tosubchannelswithdifferentSNRs.Finally,inoursimulation,onlyonecrosstalksignalisassumed.The

treatmentofmultiplecrosstalksignalsfollowsstraightforward,althoughhighercomplexityisinevitable.

Infuturework,weplantostudyothercrosstalkapplicationswheremultiuserdetectiontechniquescanbe

appliedmoredirectly(e.g.combatingself-FEXT).Weadmitthatinrealitycrosstalksignalsvarywidelyin

modulationformatsanddataratesandsofarthereisnouniformframeworkformitigationofcrosstalkin

DSL.What we address here is the combating of a special class of crosstalk signal (QAM-modulated

signals),butwebelievethatthegeneralideaofmultiuserdetectionisapromisingtechniqueforcrosstalk

mitigation inDSL.Also, iterative (Turbostyle) joint decodingandmultiuser detection (see [23]) isan

attractivetechniquewhoseapplicationonDSLisofinterest.

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Fig.1.SystemmodelforDMTVDSL.

Fig.2.DMTtransmitter.

CrosstalkSignals

DMTSignal

crosstalkcoupling

crosstalkcoupling

)(txK

)(2

tx

)(1

tx

n(t)

)(2

tC

)(tCK

y(t)

{

VDSLlinechannelH

.

.

.

.

.

.

.

.

.

Encoder IFFTInputbitStream .

.

.

1X

2X

NX

Parallel

To

Serial

Converter

.

.

.

1x

2x

Nx

NN 2=

DMTsignal

)(tx

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Fig.3.DMTreceiver.

Fig.4.Multichanneldecompositionofachannelresponse.

Serial

To

Parallel

Converter

FFTChannelOutput .

.

.

1y

2y

Ny

Decoder.

.

.

1Y

2Y

NY

NN 2=

Detecteddata

)(ty

......

|H(f)|Transferfunctionofchannelresponse

f1f 2f 3f 4f 2Nf 1Nf Nf

||0H

|| 1H|| 2H

|| 3H

|| 4H

||2N

H||

1NH

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Fig. 5. Huber penalty function and its derivative for the Gaussian mixturemodel used in this paper.

1.0= , 100= , 12 = , 14.1=k .

Fig.6.InterferencecancellationmultiuserdetectorforDMTVDSLsystemwithcrosstalks.

FFT

CrosstalkDetector

+ ..... +DMT

Decoder

)(ty Y

2C KC

X

Encoder CrosstalkDetector

Encoder

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12 13 14 15 16 17 18 19 20 21 22 2310

-7

10-6

10-5

10-4

10-3

10-2

10-1

SNR

BER

Fig.7.Biterrorrate(BER)versussignal-to-noiseratio(SNR)fordifferentdetectors(x-mark:SUD,circle:

IC-MUD,diamond:ML-MUD,dashed:singleuserlowerbound).

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12 13 14 15 16 17 18 19 20 21 22 2310

-7

10-6

10-5

10-4

10-3

10-2

10-1

SNR

BER

Fig.8.Biterrorrate(BER)versussignal-to-noiseratio(SNR)fordifferentdetectors(x-mark:SUD,circle:IC-MUD, plus: IC-MUD-R,diamond:ML-MUD, star: ML-MUD-R, dashed: singleuser lower bound).

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12 13 14 15 16 17 18 19 20 21 2210

-7

10

-6

10-5

10-4

10-3

10-2

10-1

SNR

BER

Fig.9.Biterrorrate(BER)versussignal-to-noiseratio(SNR)fordifferentdetectorswith15dBweaker

crosstalk(circle:IC-MUD,plus:IC-MUD-R,diamond:ML-MUD,star:ML-MUD-R,dashed:singleuserlowerbound).

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6 8 10 12 14 16 18 20 2210

-6

10-5

10-4

10-3

10-2

10-1

100

SNR

BER

6 8 10 12 14 16 18 20 2210

-6

10-5

10-4

10-3

10-2

10-1

100

SNR

BER

Fig.10.Effectofcrosstalkstrengthfortraditionalandrobustinterferencecancellationmultiuserdetection.

left:SUD;right:IC-MUD-R(solid:strongercrosstalk;dashdot:(12dB)weakercrosstalk).