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IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.18,NO.7,JULY20001153AdaptiveMultidimensionalCodedModulationOverFlatFadingChannelsKjellJ.Hole,Member,IEEE,HenrikHolm,StudentMember,IEEE,andGeirE.ØienWeintroduceageneraladaptivecodingschemeforNakagamimultipathfadingchannels.Aninstanceofthecodingschemeutilizesasetof -dimensional(2 -D)trelliscodesorig-inallydesignedforadditivewhiteGaussiannoise(AWGN)chan-nels.Anysetof -DtrelliscodesforAWGNchannelscanbeused.Setsforwhichallcodescanbegeneratedbythesameencoderand -Dtrelliscodes.Asanillustrativeexample,wecal-culatetheaveragespectralefficiencyofanadaptivecodecutilizingeight4-Dtrelliscodes.TheexamplecodecisbasedontheInterna-tionalTelecommunicationsUnion’sITU-TV.34modemstandard.IndexTerms—Adaptivetrelliscoding,linkspectralefficiency,Nakagamimultipathfading.I.INTRODUCTIONHEINFORMATIONrateofamobileradiolinkmaybedefinedasthenumberoftransmittedinformation [bits/s],isallowedtovarywiththereceivedCNR,theaverageinformationrate, [bits/s],isobtainedbyaveraging overalldifferentCNR’s.Ifwedenotethebandwidthofthechannelby [Hz],thentheaverageinformationrateperunitbandwidth,calledtheaveragespectralefficiency,isgivenby [bits/s/Hz].Let denotetheobtainableinformationrateonthefadingchannelforagivenreceivedCNRandarbitrarilysmallManuscriptreceivedFebruary26,1999;revisedSeptember14,1999andJan-uary5,2000.K.J.HoleiswiththeDepartmentofInformatics,UniversityofBergen,HiB,N-5020Bergen,Norway(e-mail:Kjell.Hole@ii.uib.no).H.HolmandG.E.ØienarewiththeDepartmentofTelecommunica-tions,NorwegianUniversityofScienceandTechnology,O.S.Bragstadsplass2B,N-7491Trondheim,Norway(e-mail:Henrik.Holm@tele.ntnu.no; overallreceivedCNR’s.IfwedenotetheShannoncapacityby [bits/s],thentheimumavespectralefficiency(MASE)ofthefadingchannel [bits/s/Hz].Theradiospectrumavailableformulti-mediawirelesscommunicationsisscarce,andavariable-ratetransmissiontechniqueforfuturewirelessnetworksmustthere-forehaveaveragespectralefficiency closetotheMASE GoldsmithandVaraiya[1]havedeterminedtheMASEofageneral,single-user,flat-fadingchannelwithperfectchannelstateinformation(CSI)atthetransmitterandreceiverforthreedifferentadaptivetransmissionpolicies:optimalsimultaneoustransmitpowerandinformationrateadaptation,constantpowerwithoptimalrateadaptation,andchannelinversionwithfixedrate.AlouiniandGoldsmith[2]havesinceusedthegeneraltheoryin[1]toobtainclosed-formexpressionsfortheMASEofthethreeadaptivetransmissionpoliciesoverNakagamimulti-pathfading(NMF)channels.Numericalresultsshowedthatthetwoinformationrateadaptationpoliciesyieldnearlythesame -dimensional( -D)trelliscodeswhere issomeposi-tiveinteger.Theindividualcodesshouldbedesignedforaddi-tivewhiteGaussiannoise(AWGN)channels.Anysetof trelliscodesforAWGNchannelscanbeused,butinpracticeonlycodeswithdifferentspectralefficienciesareofinterest.For0733–8716/00$10.00©2000IEEE 1154IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.18,NO.7,JULY2000 ,thecodingschemereducestothe2-Dcodingschemein[4]withconstanttransmitpower.Weconcentrateonthemulti- versionsofthecodingscheme.Thehardwarecomplexityofthecodingschemeissignifi-cantlyreducedifallcodescanbegeneratedbythesameencoderanddecodedbythesameViterbidecoder.UnlikeotherfadingresistantcodingschemesbasedontrelliscodesforAWGN[5],[6],theschemeconsideredheredoesnotutilizeinterleaving.GoodperformanceoftheschemerequiresaccurateestimatesoftheCSIatthedecoderandareliablefeedbackchannelbetweentheencoderandthedecoder.Bufferingoftheinputdataisalsoneededsincetheinformationratevarieswiththechannelcon-InSectionIIIwedevelopageneraltechniquetocalculatetheaveragespectralefficiency ofthecodingschemeforanysetof -Dtrelliscodes.Asanillustrativeexample,wedeterminetheaveragespectralefficiencyofacertainvari-able-rateencoderanddecoder(codec)inSectionIV.Theex-amplecodecisbasedontheInternationalTelecommunicationsUnion’sITU-TV.34modemstandard.ThecodecutilizeseightnestedM-QAMsignalconstellationscontaining4,8,16,32,64,128,256,and512signalpointstoencodeanddecodeeight4-Dtrelliscodes.Theaveragespectralefficiency iscom-paredtotheMASE .AsummaryandadiscussionarepresentedinSectionV.II.SODELANDTheend-to-endsystemforvariable-rateconstant-powerTCMoverNMFchannelswasfirststudiedin[1]–[4]and[7].Itisas-sumedthatthetime-varyingphaseshiftandchannelgain(fadingenvelope)ofthetransmittedsignalareperfectlyknownatthere-ceiver.Consequently,thereceiverisabletofullycompensateforthephasevariation,i.e.,weassumeperfectcoherentdetection.Adecisiondeviceutilizesthechannelgaintoselectanappro-priateinformationratebychoosingacodeamongasetof trelliscodes.Thedecisiondevicetheninformsthetransmitteraboutitsdecisionviaafeedbackchannel.AsmentionedintheIntroduction,the -DtrelliscodesshouldinpracticebegeneratedbythesameencoderanddecodedbythesameViterbidecoder.Hence,weassumethatthetrelliscodesareimplementedasavariable-rateTCMcodecwhereitispossibletochangerateatanytime.Forsatisfactoryoperationofthecommunicationsystem,boththevariable-rateencoderandthevariable-rateViterbidecodermustusethesamecodeatanyinstant.Anefficienterrorcon-trolschemeisthereforeneededtoensureanerror-freefeedbackchannel.Wesimplyassumethatthefeedbackchannelisnoise-Letthetransmitted2-Dsignalhavecomplexbasebandrep- attime .Thestationaryander-godicchannelgain,representedby ,isindependentofthechannelinputandhasthesamevalueinbothsignaldi-mensions.Thereceivedbasebandsignalisgivenby where denotescomplexAWGN.Therealandimaginarypartsofthenoisearestatisticallyindependent,Weremarkthattheprotocolsneededtochangebetweenthedifferentcodesarenotconsideredexplicitly,butareassumedtoworkperfectly.bothwithvariance where [W/Hz]isthetwo-sidednoisepowerspectraldensity. [W]denotetheconstanttransmitsignalpower.TheinstantaneousreceivedCNRofthesystemisequalto ,anditsexpectedvalueis where istheaveragereceivedpowergain[1],[3].Intheremainderofthepaperwewillomitthetimereference andreferto and Thereal-valuedchannelgain ismodeledasacontinuousstochasticvariablewithaNakagamiprobabilitydensityfunc-tion(pdf)controlledbytheNakagamifadingparameter,areal .Theparameter ishereassumedtobeapos-itiveinteger.TheNakagamidistributionrepresentsawiderangeofmultipathfadingchannelsviadifferentintegervaluesof .When thepdfistheRayleighpdf.For theNakagamidistributioncloselyapproximatestheRicedistri-bution[8,p.48].TheinstantaneousreceivedCNR onanNMFchannelisacontinuousstochasticvariablewithgammadistribution.ThePDFfor isgivenby[2],[7] (1)where isthegammafunctionwhichequals !when apositiveinteger.Thefadingbecomeslesssevere increasesbecausethevarianceof ,isThedelayinthefeedbackchannelofthesystemmodelmayreducetheachievableaveragespectralefficiencyforagiventargetBERwhenthechannelgain or,equivalently,there-ceivedCNR changesrapidly[9].Weassumethat ataratemuchslowerthanthesymbolrate,sothatthechannelremainsroughlyconstantoverseveralhundredsymbols,andwethereforeignorethepropagationdelay.TheMASEofthesystemmodelwithconstanttransmitpowerandadelay-freefeedbackchannelis [bits/s/Hz](2)for apositiveinteger[2,eq.(23)].Here isthecomplementaryincompletegammafunctiondefinedby[10,eq.(8.350.2),p.949] Thefunctioniscommonlyavailableinnumericalsoftware.Assumethat quantizationlevels(orfadingregions)areusedtorepresenttheinstantaneousreceivedCNR .Thethe-oreticaltransmissionschemethatachievesMASEhas infinitelysmallfadingregions[1].Ourcodingstrategyforafi-nitenumberoffadingregionsistoassignone -Dtrelliscodetoeachfadingregion.Thetrelliscodesareintroducedtocom-pensateforthefinitequantizationof andthelimitationsonthecodingalphabetassociatedwithrealizablesignalconstellations.Letthe regionsbedefinedbythethresholds .Code , ,isusedwhen etal.:ADAPTIVEMULTIDIMENSIONALCODEDMODULATIONOVERFLATFADINGCHANNELS1155theinstantaneousreceivedCNR fallsinregion ,i.e.,when .Weassumethatfadingregion1representsthesmallestvaluesof forwhichinformationistransmitted.When ,noinformationissent.WeusetrelliscodesforAWGNchannels.Tounderstandwhy,observethatthewidth ofeachfadingregiongoestozerowhenthenumberoffadingregions goestoinfinity.Con-sequently,forlarge thereceivedCNRmaybeapproximatedbyaconstantforeachfadingregionandwemayviewtheNMFchannelatagiventimeinstantasanelementinasetofAWGNchannelswithdifferentbutconstantCNR’s. -Dsignalconstellationforcode isgivenbythe -foldCartesianproductofa2-Dsignalsetwith signalpoints(orsymbols).Eachtimetheencoderforcode ceives informationbits,itgenerates codedbits.Thecodedbitsthendetermine transmittable2-Dmodulationsymbols.Ifthetime betweenthetransmissionoftwoconsecutive2-Dsymbolsissuchthatthereisnointersymbolinterferenceinthesamplesattheoutputofthechannel,thentheinformationrateforcode isequalto [bits/s].SincetheNyquistbandwidthis ,themaximumspectralefficiencyforcode is [bits/s/Hz].Thecodesshouldhavespectralefficiencies whichincreasewith ,i.e., for Thismakesitpossibletotransmitathighinformationrateswhenthereislittleornofading,andtoreducetheinformationrateasthefadingincreases.III.SFFICIENCYOFTHEInthissectionwedetermineageneralexpressionfortheav-eragespectralefficiencyofthecodingscheme.Theexpressionisvalidforanysetof -Dtrelliscodes.ItisfirstarguedthattheBERofa -DtrelliscodeonanAWGNchannelcanbeapproximatedbyanexpressionofthe (3)where istheCNRandtheconstants and dependontheweightdistributionofthecode.Theencoderforthecodegenerates 2-Dmodulationsym-bolseachtimeitreceives informationbits.The2-Dsignalconstellationconsistsof signalpointsdrawnfromaregularlatticewheretheminimumEuclideandistancebetweentwopointsis .When iseven,weassumethattheconstella-tionisquadratic,otherwisetheconstellationhasa“crossshape”asdescribedin[11].TheVoronoiregionofasignalpointisasquarewitharea .Theaveragetransmitenergycanbeex-pressedas where issomepositiverealnumber.Toobtainanexpressionofthedesiredform(3)forequiprob-ablesignaling,weadopttheintegralapproximationtechniqueof[11]toexpress intermsof .When iseven,thesquarecon-stellationhaslength resultinginatotalareaof Weremarkthatthefollowinganalysisdoesnotrequirethatalltrelliscodesaregeneratedbythesameencoderanddecodedbythesamedecoder. .Theaverageenergyofthe signalpointsisapproxi-matelyequaltotheaverageenergy ofallpoints insidetheconstellationarea.Thisintegralapproximationgivesaverageenergyequalto When isodd,theapproximationisequalto Inbothcases,theaverageenergyisapproximatedbyanexpres-sionoftheform for apositivenumber.TheminimumsquaredEuclideandistanceofthe -Dcodecanlikewisebeexpressedas forsome .WhiletheinstantaneousCNRisarandomvariableinthecaseofafadingchannel,theCNRforanAWGNchannelisaconstant .ForahighCNR,theBERcanbeapproximated (4)where istheaveragetotalnumberofinformationbiterrorsassociatedwiththecodewordsindistance fromthecorrectcodeword[12,eq.(14–21),p.376].Wenowsettheaveragetransmitenergy where isanarbitrarypositiveconstant.Itfollowsthat .Ifwethenusetheupperbound andrewrite ,wehavefrom(4)that whichhasthedesiredform(3)for and Wehaveonlyshownthattheapproximationin(3)isgoodforahighCNR.However,plotsofBERfoundintheliteraturestronglyindicatethatwecanusecurvefittingtechniquestode-terminevaluesfor and suchthattheexpressioncanbeusedwithgoodaccuracyalsoformediumandlowCNR’s.WewillillustratethisinSectionIV.WearenowreadytodetermineageneralexpressionfortheaveragespectralefficiencyoftheadaptivecodingschemeonanNMFchannel.Thecodingschemeutilizes trelliscodeswhose2-Dsignalconstellationshave signalpointsandwhoseBERcanbeapproximatedby(3)forsomevalues and for .Code istobeusedwhentheinstantaneousreceivedCNR fallsinfadingregion ,i.e., .Thelowerthreshold isequaltotheleastCNRrequiredtoachieveagiventargetBERdenotedbyBER Weobtainthefollowingthresholdsbyassumingequalityin(3), BER ,andinvertingtheexpres-sionwithrespecttotheCNR: (5)where BER Itmaynotbecompletelyobviousthat for since and varywith ,butthismonotonicityTheintegralapproximationsarecalculatedin[11]for 1156IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.18,NO.7,JULY2000followsfromthefactthat,foragivenBER,theminimumre-quiredCNRforcode islargerthantherequiredCNRfor when .Inpractice, BER suchthat andallthresholds Theaveragespectralefficiency isequaltothesumoverall ofthespectralefficiencies oftheindividualcodes,eachweightedbytheprobability thatthe fallsinregion [bits/s/Hz] Theprobabilities where isdefinedby(1),aregivenby[7,eq.(10)] Forthespecialcaseof apositiveinteger,nonumericalsoftwareprogramisneededtocalculate in(7)sincethereexistsaclosed-formexpression[13,eq.(11.6),p.278] Itfollowsfrom(8)that(7)isgivenby for (Rayleighfading).IV.SFFICIENCYOFInthefollowing,wecalculatetheaveragespectralefficiencyofaparticularinstanceoftheadaptivecodingscheme.Usingtechniquesdescribedin[4]and[14],wedesignedanadaptivecodecwitheight4-Dtrelliscodesutilizingtheeightnested2-DsignalconstellationsinFig.1.Ateacheven ,thevariable-rateencoderreceives tionbits, ,andgenerates codedbits.Theoverallencodercontainstherate2/3trellisen-coderandthebitconverter,butnotthe4-Dblockencoder,de-pictedin[14,Fig.6].Therate2/3trellisencoderandbitcon-verterarealsousedintheInternationalTelecommunicationsUnion’sITU-TV.34modemstandard[15]. codedbitsdefinetwosymbolnumbers,eachintherange ,whichistherangeofthesymbolnumbersforthe2-Dconstellationwith symbols.Thetwochosensymbolsaresentthroughthechannelattimes and Thenumberofinformationbits shiftedintotheencoderdeterminewhich2-Dconstellationandhencewhichcodeisused.Theinstantaneousspectralefficiency maythereforebevariedsimplybychangingthenumberofin-formationbitsshiftedintotheencoder.Sincetheoverallencodergeneratestwosymbolsatthetime,thespectralefficiencymayonlybechangedateventimeinstants.Allcodeshave andtrelliseswith16statesand parallelbranches.TheViterbidecoderforthecodewith isusedtodecodethecodeswith sincethese Fig.1.Nestingofthe4-QAM,8-STAR,16-QAM,32-CROSS,64-QAM,128-CROSS,256-QAM,and512-CROSSsignalconstellations.Thefilledblackcirclesconstitutethe8-STARconstellation.codesareallcontainedinthe code.Aroughestimateofthescheme’shardwarecomplexityisthereforegivenbythehardwarecomplexityofthelargest Theindividualcodes’BERperformancesonanAWGNchannelweresimulatedforvariousCNR’s.Thesimulationpoints(representedbyboxes)areshowninFig.2.ForeachCNR,thesimulationprogramgeneratedatleast200decodedbitsinerrorbeforecalculatingtheBER.Allsimulationpointswerecalculatedusingarelativelyshortpathmemoryoflength9.ThetruesquaredEuclideandistancewasusedasmetricintheViterbidecoder.Tocalculatetheaveragespectralefficiencyoftheexamplecodec,letthe th4-Dtrelliscode, ,bethecodebasedonthe2-DsignalconstellationinFig.1with signalpoints.TheBERforcode canbeapproximatedbytheexpression Curvefittingwiththeleastsquaresmethod[16,Ch.10]isusedtoobtaintheparameters and listedinTableI.TheresultingBERapproximations(9)areplottedinFig.2.Notethattheap-proximatedBERcurvesareveryclosetothesimulatedBERpointsforallcodes.Thethresholds ,calculatedfrom(5),arealsotabulatedinTableI.Theaveragespectralefficiency,ob-tainedfrom(6)and(7),isplottedinFig.3forNakagamifading andaveragereceivedCNR ThedifferencebetweentheMASE(2)andtheaveragespec-tralefficiencyisshowninFig.4.Observethatthedifferencedecreasesas increases.Onereasonforthismaybethatthe8-STARsignalconstellationinFig.1hasaratherlargeaverage etal.:ADAPTIVEMULTIDIMENSIONALCODEDMODULATIONOVERFLATFADINGCHANNELS1157 Fig.2.TheboxesareBERestimatesobtainedfromsoftwaresimulationandthecurvesareestimatesobtainedfrom(9).TABLEI ANDb ODECANDALCULATED (dB)FORTARGETBER =10 energyequalto .Hence,thecodebasedonthe8-STARconstellationrequiresaratherlargereceivedCNR toachievethetargetBER.As increases,theprobabilitythatthissub-optimalcodeisuseddecreases,anditsnegativeeffectontheaveragespectralefficiencydiminishes.V.DISCUSSIONANDUMMARYWehaveintroducedageneraladaptivecodingschemeforsingle-userchannelswithfrequency-flat,slowlyvaryingNMF.Aninstanceoftheschemeutilizes -Dtrelliscodesorig-inallydesignedforAWGNchannels.Ageneralexpressionforthecodingscheme’saveragespectralefficiencywasobtained.Asanexample,weapproximatedtheaveragespectraleffi-ciencyofacodeccontainingeight4-Dtrelliscodes(seeFig.3).Theaveragespectralefficiencymaybeincreasedifweusemorecircularversionsofthelargesignalconstellations.Amoderateshapinggainmayalsobeobtainedbyutilizingextendedcon-stellationsandasimpleblockcode[14].Byturningtheshapingonandoff,itispossibletoincreasethespectralefficiencyinstepsof0.5[bits/s/Hz]insteadof1[bits/s/Hz]ashasbeendoneinthispaper.Theexamplecodecutilizeseightnested2-Dsignalconstella-tionswithfourormoresignalpoints(seeFig.1).TherelativelyThereexistotherconstellationswith8pointsthathavesmalleraverageen-ergythan8-STAR.Unfortunately,theseconstellationsarenotsubconstellationsofthelargerconstellationsinFig.1. Fig.3.Averagespectralefficiency(bits/s/Hz)ofexamplecodecfortargetBER =10 Fig.4.DifferencebetweenMASE(bits/s/Hz)andaveragespectralefficiency(bits/s/Hz)ofexamplecodecfortargetBER =10 largeaverageenergyofthe8-STARconstellationresultedinalargerdifferencebetweentheMASEandtheobtainedaveragespectralefficiencyforsmallCNR’s(seeFig.4).Theextralossinaveragespectralefficiencyincurredbythe8-STARcodeisnotdramatic.However,wedonotrecommendthatitisusedinarealcodec.AninterestingalternativetoourexamplecodecisacodecbasedonUngerboeck’s2-Dtrelliscodeswith16states[17].Thesecodeshave whileourcodeshave Unfortunately,Ungerboeck’scodeshaveonly180 invariance.Ourcodes,ontheotherhand,canbemadetrans-parentto90 ,180 ,and270 phaserotationsofthesignalcon-stellationsbyintroducingadifferentialencoderandadifferen-tialdecoder.Hence,theexamplecodeceliminatestheneedforacoherentphasereference[18].AlouiniandGoldsmith[2]showedthataslongastheinfor-mationratecanbevaried,onlyaverymodestincreaseinMASEisobtainedbyalsovaryingthetransmitpower.TheyconcludedthattheinformationrateadaptionratherthanthetransmitpoweradaptionisthekeytoincreasingtheaveragespectralefficiencyonNMFchannels.However,eventhoughverylittleisgainedVariable-ratecodecscontainingUngerboeckcodeswith4,8,32,and128stateswereinvestigatedin[4]. 1158IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.18,NO.7,JULY2000intermsofMASEbyvaryingthetransmitpower,thisdoesnottellusanythingabouthowtobestachieveanaveragespectralefficiencyclosetotheMASE.Aninformalcomparisonofplotsinthispaperandplotsin[4]for (Rayleighfading)in-dicatesthatmorecodesareneededtoachieveagivenaveragespectralefficiencywhenthetransmitpowerisfixedthanwhenthepowerisallowedtovary.Theaboveresultfor4-Dcodes,aswellastheresultsobtainedbyGoldsmithandChua[4]for2-Dcodes,indicatethat,forroughlythesamehardwarecomplexity,variable-rateTCMmayachievehigheraveragespectralefficiencythanmanyfixed-ratetrelliscodes.However,beforethesespectralefficienciescanbeobtainedinpractice,moreworkisneededtodeterminenewchannelestimationtechniquesthatwillallowtheadaptivecodingschemetomeetBERspecificationswithimperfectchannelestimationandnonzerofeedbackdelay.Someinterestingpreliminaryresultsinthisdirectionmaybefoundin[19].Weendthispaperwithacommentonthetimedelayinthefeedbackchannel.Goeckel[20]hasrecentlyshownthattheoutputBERofanadaptivecodingschememayincreaserapidlywithgrowingdelayinthefeedbackchannel.Wehaveassumedthatthefeedbackdelayiszero.MoreworkisthereforeneededtodeterminetheBERperformanceofouradaptivecodingschemefornonzerofeedbackdelay.CKNOWLEDGMENTTheauthorswouldliketothanktheanonymousreviewersformanyhelpfulsuggestionsthatconsiderablyimprovedthispaper.[1]A.J.GoldsmithandP.P.Varaiya,“Capacityoffadingchannelswithchannelsideinformation,”IEEETrans.Inform.Theory,vol.43,pp.1986–1992,Nov.1997.[2]M.-S.AlouiniandA.J.Goldsmith,“CapacityofNakagamimultipathfadingchannels,”inProc.47thIEEEVeh.Technol.Conf.(VTC’97)Phoenix,AZ,May1997,pp.358–362.[3]A.J.GoldsmithandS.-G.Chua,“Variable-ratevariable-powerMQAMforfadingchannels,”IEEETrans.Commun.,vol.45,pp.1218–1230,October1997. ,“Adaptivecodedmodulationforfadingchannels,”IEEETrans.,vol.46,pp.595–602,May1998.[5]J.Ventura-Traveset,G.Caire,E.Biglieri,andG.Taricco,“Impactofdiversityreceptiononfadingchannelswithcodedmodulation—PartI:Coherentdetection,”IEEETrans.Commun.,vol.45,pp.563–572,May[6]V.K.N.LauandM.D.Macleod,“VariablerateadaptivetrelliscodedQAMforhighbandwidthefficiencyapplicationsinRayleighfadingchannels,”inProc.48thIEEEVeh.Technol.Conf.(VTC’98),Ottawa,Canada,May1998,pp.348–351.[7]M.-S.AlouiniandA.J.Goldsmith,“AdaptiveM-QAMmodulationoverNakagamifadingchannels,”inProc.6thCommun.TheoryMiniconf.(CTMCVI),Phoenix,AZ,Nov.1997,pp.218–223.inconjunctionwithIEEEGlobalCommun.Conf.(GLOBECOM’97)[8]G.L.Stüber,PrinciplesofMobileCommunication.Norwell,MA:KluwerAcademic,1996.[9]H.Viswanathan,“CapacityofMarkovchannelswithreceiverCSIanddelayedfeedback,”IEEETrans.Inform.Theory,vol.45,pp.761–771,Mar.1999.[10]I.S.GradshteynandI.M.Ryzhik,TableofIntegrals,Series,andProd-,5thed.SanDiego,CA:Academic,1994.[11]G.D.Forney,Jr.,R.G.Gallager,G.R.Lang,F.M.Longstaff,andS.U.Qureshi,“Efficientmodulationforband-limitedchannels,”IEEEJ.Select.AreasCommun.,vol.2,pp.632–647,Sept.1984.[12]S.B.Wicker,ErrorControlSystemsforDigitalCommunicationandStorage.EnglewoodCliffs,NJ:Prentice-Hall,1995.[13]N.M.Temme,SpecialFunctions—AnIntroductiontotheClassicalFunctionsofMathematicalPhysics.NewYork:Wiley,1996.[14]L.-F.Wei,“Trellis-codedmodulationwithmultidimensionalconstella-IEEETrans.Inform.Theory,vol.33,pp.483–501,July1987.[15]D.J.Costello,Jr.,J.Hagenauer,H.Imai,andS.B.Wicker,“Applica-tionsoferror-controlcoding,”IEEETrans.Inform.Theory,vol.44,pp.2531–2560,Oct.1998.[16]W.CheneyandD.Kincaid,NumericalMathematicsandComputing2nded.Monterey,CA:Brooks/ColePublishing,1985.[17]G.Ungerboeck,“Trellis-codedmodulationwithredundantsignalsets—PartII:Stateoftheart,”IEEECommun.Mag.,vol.25,pp.12–21,Feb.1987.[18]M.D.Trott,S.Benedetto,R.Garello,andM.Mondin,“Rotationalin-varianceoftrelliscodes—PartI:Encodersandprecoders,”IEEETrans.Inform.Theory,vol.42,pp.751–765,May1996.[19]T.Eyceoz,A.Duel-Hallen,andH.Hallen,“Deterministicchannelmod-elingandlongrangepredictionoffastfadingmobileradiochannels,”IEEECommun.Lett.,vol.2,pp.254–256,Sept.1998.[20]D.L.Goeckel,“Adaptivecodingfortime-varyingchannelsusingout-datedfadingestimates,”IEEETrans.Commun.,vol.47,pp.844–855,June1999. KjellJ.Hole(S’89–M’90)wasborninMolde,Norway,onJune1,1961.HereceivedtheB.Sc.,M.Sc.,andPh.D.degreesincomputersciencefromtheUniversityofBergen,Norway,in1984,1987,and1991,respectively.FromAugust1988toMay1990,hewasaVis-itingScholarinProf.J.K.Wolf’sgroupattheCenterforMagneticRecordingResearchattheUniversityofCalifornia,SanDiego.During1993,heworkedinDr.P.Siegel’sgroupatIBMAlmadenResearchCenter,SanJosé,CA.Since1995,hehasbeenaResearchScientistattheUniversityofBergenwithfundingfromtheNorwegianResearchCouncil.Hiscurrentresearchinterestsareintheareasofcodingtheoryandwire-lesscommunications. HenrikHolm(S’96)wasborninSarpsborg,Norway,onJanuary25,1972.HereceivedtheM.Sc.degreeinelectricalengineeringfromtheNorwegianUniversityofScienceandTechnology(NTNU)inIn1998,hejoinedtheDepartmentofTelecom-municationsatNTNUasaResearchFellow,andheiscurrentlyworkingtowardthePh.D.degreeinelectricalengineering.Hismainresearchinterestiscodingandmodulationtechniquesforwireless GeirE.ØienwasborninTrondheim,Norway,onDecember29,1965.HereceivedtheM.Sc.andPh.D.degreesinelectricalengineeringfromtheDepartmentofTelecommunicationsattheNorwegianInstituteofTechnology,Norway,in1989and1993,respectively.FromAugust1994toAugust1996,hewasanAssociateProfessorinsignalprocessingattheStavangerCollegeinStavanger,Norway.InAugust1996,Dr.ØienbecameanAssociateProfessorininformationtheoryattheDepartmentofTelecom-municationsattheNorwegianUniversityofScienceandTechnology(NTNU),Trondheim,Norway.HeservedasPresidentfortheNorwegianSignalPro-cessingAssociation(NORSIG)intheperiod1997–1999.Hiscurrentresearchinterestsareintheareasofinformationtheoryandwirelesscommunications.