VDSL Paper
Transcript of VDSL Paper
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CrosstalkMitigationinDMTVDSLwithImpulseNoise*
HuaiyuDai,StudentMember,IEEE,andH.VincentPoor,Fellow,IEEE
DepartmentofElectricalEngineering,PrincetonUniversity
Princeton,NJ08540
Tel:(609)258-4634Fax:(609)258-1560
Email:[email protected],[email protected]
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).