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Synchronization in Impulse Radio Ultra-Wideband Communication Systems


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IwouldliketoexpressmygratitudetoProf.TanWong,forhisconstantencourage-ment,supportandpatiencethroughoutthecourseofmygraduatestudy.Hismeticulousguidanceandfrankadvicehavebeeninstrumentalinbringingthisworktoitspresentform.IamthankfultoSaravananVijayakumaranforthenumerousinsightfuldiscussionsandhishelpinreviewingsomeofthiswork.Iwouldalsoliketothankthemembersofmysupervisorycommittee,Prof.JohnShea,Prof.LiuqingYangandProf.YeXia,fortheirguidance,suggestionsandinterestinmywork.Iwouldliketothankmyparentsandmysisterandbrother-in-lawfortheiruncon-ditionallove,supportandpatience,whichhaveconstantlymotivatedmetoconfrontthechallengesfacedduringmygraduatestudies. iii

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page ACKNOWLEDGMENTS ................................ iii LISTOFFIGURES ................................... vii ABSTRACT ....................................... ix CHAPTER 1INTRODUCTION ................................ 1 1.1ObjectivesandMainContributions ..................... 2 1.2DissertationOutline ............................. 4 2BACKGROUNDANDRELATEDRESEARCH ................ 5 2.1AcquisitionMethodsinTraditionalSpreadSpectrumSystems ...... 5 2.2SignalAcquisitioninUWBSystems .................... 9 2.2.1SystemModel ............................ 9 2.2.2CurrentApproachesTowardsUWBSignalAcquisition ...... 12 2.2.2.1Detection-basedapproaches ............... 12 2.2.2.2Efcientsearchstrategies ................ 15 2.2.2.3Searchspacereductiontechniques ........... 16 2.2.2.4Estimation-basedschemes ................ 17 2.2.2.5Miscellaneousapproaches ................ 20 3ISSUESANDCHALLENGESINTHEDESIGNOFUWBACQUISITIONSYSTEMS .................................... 21 3.1HitSet .................................... 21 3.2AsymptoticAcquisitionPerformanceofThreshold-basedSchemes .... 23 3.3TheSearchSpaceinUWBSignalAcquisition ............... 25 3.4GeneralizedLikelihoodRatioTestforUWBSignalAcquisition ..... 26 4ACQUISITIONWITHHYBRIDDS-THUWBSIGNALING ......... 31 4.1SystemModel ................................ 31 4.1.1HybridDS-THSignalFormat .................... 31 4.1.2ReceivedSignal ........................... 32 4.2HitSetFormulation ............................. 33 4.3Stage1:THAcquisition .......................... 36 4.3.1TheDecisionStatistic ........................ 38 iv

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4.3.2MeanandVarianceoftheDecisionStatistic ............ 39 4.3.2.1AveragingovertheDS sequence .......... 41 4.3.2.2AveragingovertheTH sequence ........... 41 4.3.3FalseAlarmandDetectionProbabilities .............. 43 4.4Stage2:DSAcquisition .......................... 44 4.5SettingThresholds 1 and 2 ........................ 46 4.6MeanDetectionTime ............................ 49 4.7NumericalResults .............................. 51 4.8SystemDesignandComplexityConsiderations .............. 55 5ACQUISITIONINTRANSMITTEDREFERENCEUWBSYSTEMS ..... 57 5.1TR-UWBSystems ............................. 57 5.2SystemModel ................................ 58 5.3Two-levelDSSignalingStructure ..................... 59 5.4HitSetDenition .............................. 62 5.5Two-stageAcquisitionSchemeforTR-UWBSignaling .......... 67 5.5.1DecisionStatisticoftheFirstStage ................. 69 5.5.2DecisionStatisticfortheSecondStage ............... 71 5.5.3ProbabilitiesofFalseAlarmandDetection ............. 71 5.6DecisionThresholdSelection ........................ 72 5.7MeanDetectionTime ............................ 77 5.8NumericalResults .............................. 80 6FINETIMINGESTIMATION .......................... 83 6.1SystemModel ................................ 84 6.2Pilot-AssistedTimingEstimation ...................... 85 6.2.1MaximumLikelihoodTimingEstimation .............. 86 6.2.2Cramer-RaoLowerBound ..................... 88 6.3Non-pilot-assistedTimingEstimation ................... 90 6.3.1MaximumLikelihoodTimingEstimation ............. 91 6.3.2CramerRaoLowerBound ..................... 93 6.4Sub-optimalTimingEstimationMethods .................. 97 6.4.1TimingwithDirtyTemplates(TDT) ................ 97 6.4.2TransmittedReference(TR)Signaling ............... 97 6.5NumericalResults .............................. 98 6.5.1ChannelParameterExtraction .................... 98 6.5.2ComputationandSimulationResults ................ 99 7CONCLUSIONS ................................. 110 7.1Conclusions ................................. 110 7.2OpenProblems ............................... 111 REFERENCES ..................................... 112 v

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.............................. 119 vi

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Figure page 2Blockdiagramofaparallelacquisitionsystemfordirect-sequencespreadspec-trumsystems ................................... 7 2Blockdiagramofaserialacquisitionsystemfordirect-sequencespreadspec-trumsystems ................................... 7 2BlockdiagramoftheacquisitionschemeproposedbyBlazquezetal. ..... 13 2BlockdiagramoftheacquisitionschemeproposedbySoderietal. ....... 14 2Autocorrelationfunction(ACF)ofcorrelatoroutputsz[n]oritsFourierseries(FS)coefcientsestimatedviasampleaveragingandusedtoestimatetimingoffset. ....................................... 18 3EffectofreceivedSNRonsizeofhitsetHforNR=5andNR=10. ..... 22 3TheROCswhenthethresholdissetforasingletonhitsetcontainingonlythetruephaseandforahitsetdenedin( 3 )withn=103. .......... 23 3Generalizedlikelihoodratiotestforevaluationofphase^. ........... 30 4ThehybridDS-THsignalformat. ........................ 32 4ConceptualblockdiagramofthehybridDS-THtwo-stageacquisitionscheme. .......................................... 32 4EffectofdeviationfromthetruephaseonBERforNR=5. ........... 37 4SquaringloopforTHpatternacquisition ..................... 38 4AcquisitionsystemforDSstage ......................... 45 4FlowgraphtodeterminemeandetectiontimeforHybridDS-THAcquisitionSystem. ...................................... 51 4EffectofreceivedSNRonsizeofhitsetHforNR=5andNR=10. ..... 53 4MeandetectiontimeforhybridDS-THanddoubledwellsystemsforNR=5. 54 4MeandetectiontimeforhybridDS-THanddoubledwellsystemsforNR=10. 54 5Illustrationofthedelayandmultiplyoperationonthereceivedsignal. ..... 61 vii

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............................... 63 5Blockdiagramoftheautocorrelationreceiver. .................. 64 5Illustrationofsearchstrategyusedbythetwo-stageTRacquisitionsystem. .. 69 5Flowgraphillustratingtheproposedtwo-stageacquisitionscheme. ....... 78 5EffectofreceivedSNRonhitsetsizeH. ..................... 82 5Meandetectiontimefortwo-stageandsingle-stageTR-UWBacquisitionsys-tems. ........................................ 82 6ComparisonbetweennoiselessreceivedsignalinCM1modelandreconstructedsignalafterparameterextraction. ......................... 100 6CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM1channelwithNt=16. .............................. 102 6CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM2channelwithNt=26.Notethegreaternumberofsymbolobservationscom-paredtoCM1. ................................... 103 6CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM3channelwithNt=42. .............................. 104 6CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM4channelwithNt=99. .............................. 105 6CRLBandsimulationresultsofpilot-assistedMLestimatorinatypicalCM1channelwithNt=16,assumingNtmax=100. ................. 106 6CRLBandsimulationresultsofpilot-assistedMLestimatorinatypicalCM2channelwithNt=26,assumingNtmax=100. ................. 107 6CRLBandsimulationresultsofnon-pilot-assistedMLandTDTestimatorsinatypicalCM1channelwithNt=16,assumingNtmax=100. .......... 108 6CRLBandsimulationresultsofnon-pilot-assistedMLandTDTestimatorsinatypicalCM2channelwithNt=26,assumingNtmax=100. .......... 109 viii

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ix

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1 4 ]hasrecentlyreceivedasignicantamountofattentionfromacademicresearchersaswellasfromtheindustry.Ultra-widebandsignalingisbeingconsideredforhighdataratewirelessmultimediaapplicationsforthehomeentertainmentandpersonalcomputerindustryaswellasforlowdataratesensornetworksinvolvinglow-powerdevices.Itisalsoconsideredapotentialcandidateforalternatephysicallayerprotocolsforthehigh-rateIEEE802.15.3andthelow-rateIEEE802.15.4wirelesspersonalareanetwork(WPAN)standards[ 5 6 ].Inanycommunicationsystem,thereceiverneedstoknowthetiminginformationofthereceivedsignaltoaccomplishdemodulation.Thesubsystemofthereceiverwhichperformsthetaskofestimatingthistiminginformationisknownasthesynchronizationstage.Synchronizationisanespeciallydifculttaskinspreadspectrumsystemswhichemployspreadingcodestodistributethetransmittedsignalenergyoverawideband-width.Thereceiverneedstobepreciselysynchronizedtothespreadingcodetobeabletodespreadthereceivedsignalandproceedwithdemodulation.Inspreadspectrumsystems,synchronizationistypicallyperformedintwostages[ 7 8 ].Therststageachievescoarsesynchronizationtowithinareasonableamountofaccuracyinashorttimeandisknownastheacquisitionstage.Thesecondstageisknownasthetrackingstageandisrespon-sibleforachievingnesynchronizationandmaintainingsynchronizationthroughclockdriftsoccurringinthetransmitterandthereceiver.Trackingistypicallyaccomplishedusingadelaylockedloop[ 7 ].TimingacquisitionisaparticularlyacuteproblemfacedbyUWBsystemsasexplainedinthesequel.ThisdissertationaddressesthesignicanceoftheacquisitionprobleminUWBsystemsandthewaystoefcientlytackleit. 1

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Accuratetimingestimationfollowingcoarseacquisitionisrequiredtoperformdemodulationandisusefulinapplicationsrequiringpreciselocalization.Impulseradiosystemsoftendonothaveknowledgeofthereceivedpulseshapeandthechannel.Wepresentpilot-assistedandnon-pilot-assistedmaximumlikelihood(ML)timingestimatorsintheabsenceofsuchinformation.WealsoderivetheCramer-Raolowerbound(CRLB)ontheperformanceofanyunbiasedestimator.WecomparethesimulationperformanceoftheMLestimatorstotheCRLBandtotheperformanceofsub-optimaltimingestimatorsthatdonotrequireknowledgeofthepulseshapeandthechannel. 1 ]employedinUWBsystemsplacestringenttimingrequirementsatthereceiverfordemodulation[ 9 10 ].Thewideband-widthresultsinaneresolutionofthetiminguncertaintyregiontherebyimposingalargesearchspacefortheacquisitionsystem.TypicalUWBsystemsalsoemploylongspreadingsequencesspanningmultiplesymbolintervalsinordertoremovespectrallinesresultingfromthepulserepetitionpresentinthetransmittedsignal.Intheabsenceofanysideinformationregardingthetimingofthereceivedsignal,thereceiverneedstosearchthroughalargenumberofphases

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hasadedicatedportionknownastheacquisitionpreamblewithinwhichthereceiverisexpectedtoachievesynchronization.Howeverforthehighdata-rateapplicationsenvisagedforUWBsignaling,longacquisitionpreambleswouldsignicantlyreducethethroughputofthenetwork.Thetransmittedpulsecanbedistortedthroughtheantennasandthechannelandhencethereceivermaynothaveexactknowledgeofthereceivedpulsesignalwaveform[ 11 ].TheshortpulsesusedinUWBsystemsalsoresultinhighlyresolvablemultipathwithalargedelayspread,atthereceiver[ 12 ].TheUWBreceivercouldthereforesynchronizetomorethanonepossiblearrivingmultipathcomponent(MPC)andstillperformsatisfactorily.Thismeansthattherecouldexistmultiplephasesinthesearchspacewhichcouldbeconsideredacceptableandcouldbeexploitedtospeeduptheacquisitionprocess.ThesechallengesarisingfromthesignalandchannelcharacteristicsuniquetoUWBsystemsindicatethesignicanceoftheacquisitionprobleminUWBcommunicationandtheneedtoaddressitefciently.Addressingsomeoftheseissuesisthefocusofthisdissertation.Weproposeahybriddirectsequence-timehopping(DS-TH)signalingformatandatwo-stageacquisitionschemeforUWBsystemswhichenablesinasignicantreductioninthesizeofthesearchspace.Weevaluatetheperformanceofthetwo-stageschemeintermsofthemeandetectiontime.Wedenethehitsetasthesetofphases,which,followingacquisition,resultinasatisfactoryreceiverBERperformance.Thehitsetthuscharacterizestheeffectofthedenseresolvablemultipathontheacquisitionsystemperformance.Theacquisitionproblemindirectsequencetransmittedreference(TR)UWBsystemsisanalyzed,whereweobservethatthereisasignicantrelaxationinthetimingrequirementwhichcanbeexploitedbytheuseofatwo-stageacquisitionschemepresented.WeproposetostudythetrackingproblemforUWBsystemsasfuturework.

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2 ,webrieysummarizetheacquisitionapproachesadoptedbytraditionalspreadspectrumsystemsandalsoclassifyanddiscussthecurrentresearchonUWBsignalacquisition.ThroughadiscussionofexistingspreadspectrumtechniquesweseektodistinguishtheissuesandchallengesuniquetoUWBacquisitionsystemdesign,whicharepresentedinChapter 3 .Anunderstandingoftheseissues,particularlytheexistenceofmultipleacquisitionphasesandtheasymptoticacquisitionperformance,enablesbetterUWBacquisitionsystemdesign.InChapter 4 ,wepresentthehybridDS-THsignalingformatandatwo-stageacquisitionschemetocombatthelargesearchspaceprobleminUWBacquisitionsystems.InChapter 5 wediscusstheacquisitionprobleminTR-UWBsystemswithDSsignalingandpresentatwo-stageacquisitionschemewhichsignicantlyreducesthemeandetectiontime.Wediscusspilot-assistedandnon-pilot-assistednetimingestimationmethodsforimpulseradio,intheabsenceofreceivedpulseshapeandchannelinformationinChapter 6 .ThedissertationisconcludedinChapter 7

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7 8 13 ].Wewillbringoutthemainissuesbyconsideringthetimingacquisitionofdirect-sequencespreadspectrumsystems.Inadirect-sequencespreadspectrumsystem,thereceiverattemptstodespreadthereceivedsignalusingalocallygeneratedreplicaofthespreadingwaveform.Despreadingisachievedwhenthereceivedspreadingwaveformandthelocallygeneratedreplicaarecorrectlyaligned.Ifthetwospreadingwaveformsareoutofsynchronizationbyevenachipduration,thereceivermaynotcollectsufcientenergyfordemodulationofthesignal.Asmentionedbefore,thesynchronizationprocessistypicallydividedintotwostages:acquisitionandtracking.Intheacquisitionstage,thereceiverattemptstobringthetwospreadingwaveformsintocoarsealignmenttowithinachipduration.Inthetrackingstage,thereceivertypicallyemploysacodetrackingloopwhichachievesne 5

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synchronization.Ifthereceivedandlocallygeneratedspreadingwaveformsgooutofsynchronizationbymorethanachipduration,theacquisitionstageofthesynchronizationprocessisreinvoked.Thereasonforthistwostagestructureisthatitisdifculttobuildatrackingloopwhichcaneliminateasynchronizationerrorofmorethanafractionofachip.Atypicalacquisitionstageattemptstobringthesynchronizationerrordowntobewithinthepull-inrangeofthetrackingloopbysearchingthetiminguncertaintyregioninincrementsofafractionofachip.Asimpliedblockdiagramofanacquisitionstagewhichisoptimalinthesensethatitachievescoarsesynchronizationwithagivenprobabilityintheminimumpossibletimeistheparallelacquisitionsystem[ 7 ]showninFig. 2 .Thisstagechecksallthecandidatephasesintheuncertaintyregionsimultaneouslyandthephasecorrespondingtothemaximumcorrelationvalueisdeclaredtobethephaseofthereceivedspreadingwaveform.InanadditivewhiteGaussiannoise(AWGN)channel,thisacquisitionstrategyproducesthemaximum-likelihoodestimate(fromamongthecandidatephases)ofthephaseofthereceivedspreadingwaveform.However,thehardwarecomplexityofsuchaschememaybeprohibitivesinceitrequiresasmanycorrelatorsasthenumberofcandidatephasesbeingchecked,whichmaybelargedependingonthesizeofthetiminguncertaintyregion.Awidelyusedtechniqueforcoarsesynchronization,whichtradesoffhardwarecomplexityforanincreaseintheacquisitiontime,istheserialsearchacquisitionsystemshowninFig. 2 .Thissystemhasasinglecorrelatorwhichisusedtoevaluatethecandidatephasesseriallyuntilthetruephaseofthereceivedspreadingwaveformisfound.HybridmethodssuchastheMAX/TCcriterion[ 14 ],havealsobeendevelopedwhichemployacombinationoftheparallelandserialsearchacquisitionschemesandreducetheacquisitiontimeatthecostofincreasedhardwarecomplexity.Alltheacquisitionschemesemployavericationstage[ 15 ]whichisusedtoconrmthecoarseestimateofthetruephasebeforethecontrolispassedtothetrackingloop.

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Figure2: Blockdiagramofaparallelacquisitionsystemfordirect-sequencespreadspectrumsystemswhichevaluatesthecandidatephasest1;t2;:::;tn.Intheitharm,thedecisionstatisticcorrespondingtothecandidatephasetiisgeneratedbycorrelatingthereceivedsignalwithadelayedversionofthelocallygeneratedspreadingwaveforms(t). Figure2: Blockdiagramofaserialacquisitionsystemfordirect-sequencespreadspectrumsystemswhichevaluatesthecandidatephasest1;t2;:::;tnseriallyuntilthethresholdisexceeded.Thedecisionstatisticcorrespondingtothecandidatephasetiisgeneratedbycorrelatingthereceivedsignalwithadelayedversionofthelocallygener-atedspreadingwaveforms(t).Ifthethresholdisnotexceeded,thesearchupdatesthevalueofthecandidatephaseandtheprocesscontinues.

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Intraditionalspreadspectrumacquisitionschemes,thesignal-to-noiseratio(SNR)ofthedecisionstatisticimproveswithanincreaseinthedwelltime,whichistheintegrationtimeofthecorrelator.Thustheprobabilityofcorrectlyidentifyingthetruephaseofthereceivedspreadingwaveformcanbeincreasedbyincreasingthetimetakentoevaluateeachcandidatephase.Thistradeoffhasbeenidentiedandexploitedbyseveralresearchersforthedevelopmentofmoreefcientacquisitionschemesandhasledtotheirclassicationintoxeddwelltimeandvariabledwelltimeschemes[ 7 8 ].Thexeddwelltimebasedschemesarefurtherclassiedintosingleandmultipledwellschemes[ 16 ].Thedecisionruleinasingledwellschemeisbasedonasinglexedtimeobservationofthereceivedsignalwhereasamultipledwellschemecomprisesmultiplestageswitheachstageattemptingtoverifythedecisionmadebyapreviousstagebyobservingthereceivedsignaloveracomparativelylongerduration.Variabledwelltimemethodsarebasedontheprinciplesofsequentialdetection[ 17 ]andareaimedatreducingthemeandwelltime.Theintegrationtimeisallowedtobecontinuousandincorrectcandidatephasesaredismissedquicklywhichresultsinasmallermeandwelltime.Severalperformancemetricshavebeenusedtomeasuretheperformanceofacquisitionsystemsforspreadspectrumsystems.Theusualmeasureofperformanceisthemeanacquisitiontimewhichistheaverageamountoftimetakenbythereceivertocorrectlyacquirethereceivedsignal[ 7 8 18 ].Thevarianceoftheacquisitiontimeisalsoausefulperformanceindicator,butisusuallydifculttocompute.Themeanacquisitiontimeistypicallycomputedusingthesignalowgraphtechnique[ 19 ].Forparallelacquisitionsystems,amoreappropriateperformancemeasureistheprobabilityofacquisitionoralternativelytheprobabilityoffalselock[ 20 ].Inthepresenceofmultipath,therecouldexistmorethanonephasewhichcouldbeconsideredtobethetruephaseofthereceivedsignal.However,fewacquisitionschemesforspreadspectrumsystems[ 21 22 ]havetakenthisintoconsideration.

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1.1 ,thedistinguishingfeatureofUWBsystemsisthewidebandwidthandtherelativelylowtransmissionpowerconstraintimposedbyregulatorybodies.ThewidebandwidthenablesnetimingresolutionresultinginalargenumberofresolvablepathsintheUWBchannelresponse.Theremaybemorethanonepathwhereareceiverlockcouldbeconsideredsuccessfulacquisition.Thestringentpowerconstraintnecessitatestheuseoflongspreadingsequenceswhichtogetherwithnetimingresolutionresultsinalargesearchspacefortheacquisitionsystem.SothemaindifferencebetweentheacquisitionproblemsforUWBsystemsandtraditionalspreadspectrumsystemsisthepresenceofmultipleacquisitionstatesandtherelativelylargesearchspaceintheformer.Thelargesearchspaceobviatestheuseofafullyparallelacquisitionsystemduetoitshighhardwarecomplexity.HencemuchoftheexistingworkonUWBsignalacquisitionhasfocusedonserialandhybridacquisitionsystems.Severalresearchershavetackledthelargesearchspaceproblembyproposingschemeswhichinvolvemoreefcientsearchtechniques.However,theexistenceofmultipleacquisitionstateshasreceivedrelativelylessattentionandhasnotbeensufcientlyexploited.Furthermore,asignicantportionoftheexistingworkassumesanAWGNchannelmodelfortheUWBchannelandneglectstheeffectofmultipathinthedevelopmentandevaluationoftheproposedacquisitionschemes.Inthenextsubsection,wedescribegeneralmodelsforthepropagationchannelandtheacquisitionsignalforUWBsystems.ThismodelwillbeusedinthelatersubsectionstodescribethemainfeaturesofsomeoftheproposedschemesforUWBsignalacquisition.

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reducespectrallines.Thepolaritiesofthetransmittedpulsesmayalsoberandomizedus-ingaDSspreadingcodetomitigatemultipleaccessinterference(MAI).ThegeneralizedUWBsignaltransmittedduringtheacquisitionprocessforasingleusercanbeexpressedasaseriesofUWBmonocycles(t)ofwidthTpeachoccurringonceineveryframeofdurationTfas 12 23 ]whichcanbeexpressedinthegeneralformintermsofitsimpulseresponse 24 ].Thefunctionsfk(t)modelthecombinedeffectofthetransmitandreceiveantennasandthepropagationchannelcorrespondingtothekthpathonthetransmittedpulse.Toenabletractableanalysis,weassumethatthepulsesaredistortedidenticallyinalltheMPCs,i.e.,fk(t)=f(t)andthattheexcessdelays

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12 ].Theaverageenergygainforthekthpath[ 23 ]isgivenby 1exp(Tc=).TheNakagami-mrandomvariablehkhastheprobabilitydensityfunctiongivenby 23 ],Etot,randareallmodeledbylognormaldistributions.TheNakagamifadingguresmkforhkaredistributedaccordingtotruncatedGaussiandistributionswhosemeanandvariancevarylinearlywithexcessdelay.Theselong-termstatisticsaretreatedasconstantsoverthedurationoftheacquisitionprocess.Thereceivedsignalfromasingleusercanthenbeexpressedas

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where

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InBlazquezetal.[ 25 ],thetraditionalcoarseacquisitionschemewherethesearchspaceissearchedinincrementsofachipfractionisanalyzedfortheacquisitionofTHUWBsignalsinAWGNnoise.Fig. 2 showsablockdiagramoftheschemewhereaparticularphasetiinthesearchspaceischeckedbycorrelatingthereceivedsignalwithalocallygeneratedtemplatesignalwithdelayti.Iftheintegratoroutputexceedsthethreshold,thephasetiisdeclaredtobeacoarseestimateofthetruephaseofthereceivedsignal.Ifthethresholdisnotexceeded,thesearchcontrolupdatesthephasetobecheckedasti+1=ti+Tpwhere<1andTpisthepulsewidth.Thisprocesscontinuesuntilthethresholdisexceeded. Figure2: BlockdiagramoftheacquisitionschemeproposedbyBlazquezetal. AparallelacquisitionschemeispresentedinYuanjinetal.[ 26 ]forUWBsignalsspreadbyaBarkercodeoflength4,whichisunreasonableconsideringthatlongspreadingsequencesareneededinUWBsystemstoeliminatespectrallines.Theoutputofamatchedltermatchedtothereceivedpulseissampledatthechiprateandthesamplesarethenpassedthroughfourpsuedonoise(PN)matchedlterscorrespondingtothefourpossibledelaysoftheBarkersequence.Thedelaycorrespondingtotheoutputwithlargestenergyischosenasthecoarseestimateofthetruephase.InSoderietal.[ 27 ],theoutputofamatchedlter,whoseimpulseresponseisatime-reversedreplicaofthespreadingcode,isintegratedoversuccessivetimeintervalsofsizemTcwhere1
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evaluatedinstaticmultipathchannelswith2and4pathsandisshowntoimprovemeanacquisitiontimeperformance. Figure2: BlockdiagramoftheacquisitionschemeproposedbySoderietal. InMaetal.[ 28 ],thenon-consecutivesearchproposedinShinetal.[ 21 ]andasimplerversionoftheMAX/TCscheme[ 14 ]calledtheglobalMAX/TCareappliedtotheacquistionofUWBsignalsinthepresenceofmultipathfadingandMAI.Inthenon-consecutivesearch,onlyonephaseineveryDconsecutivesearchspacephasesistestedbycorrelatingthereceivedsignalwithatemplatesignalwiththatparticularphase.ThedecimationfactorDischosentobenotlargerthanthedelayspreadNtap.IntheglobalMAX/TC,aparallelbankofcorrelatorsisusedtoevaluateallthenon-consecutivephasesandthephasecorrespondingtothecorrelatoroutputwithmaximumenergyischosenasthecoarseestimateofthetruephase.InZhangetal.[ 29 ],ahybridacquisitionschemecalledthereducedcomplexityse-quentialprobabilityratiotest(RC-SPRT)ispresentedforUWBsignalsinAWGN,whichisamodicationofthemultihypothesissequentialprobabilityratiotest(MSPRT)forthehybridacquisitionofspreadspectrumsignals[ 30 ].IntheMSPRT,ifthesequentialtestinoneoftheparallelcorrelatorsidentiesthephasebeingtestedasapotentialtruephasethecontrolispassedtothevericationstagewhichveriesitsdecision.IntheRC-SPRT,thesequentialtestineachoftheparallelcorrelatorsisusedonlytorejectthehypothesesbeingtestedassoonastheybecomeunlikelyandreplacesthemwithnewhypotheses.TheRC-SPRTstopswhenallthephasesexceptonehavebeenrejected.ThisschemehasmeritatlowSNRswherethetimerequiredtorejectincorrectphasesmaybemuchsmallerthanthetimerequiredtoidentifythetruephase.

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InVijayakumaranetal.[ 31 32 ],theeffectofequalgaincombining(EGC)ontheacquisitionofUWBsignalswithTHspreadingisinvestigatedinamultipathenviron-ment.Theacquisitionproblemisformulatedasabinarycompositehypothesistestingproblemwherethesetofphaseswhereareceiverlockresultsinanominaluncodedbiter-rorprobabilityconstitutethealternatehypothesis.TwoschemesbasedonEGCcalledthesquare-and-integrate(SAI)andtheintegrate-and-square(IAS)areanalyzedandcomparedin[ 32 ].TheIASschemeissimilartotheoneshowninFig. 2 withtheexceptionthatthetemplatesignalisgivenby 2 )withv(t)=PG1k=02r(tkTc).ItisshownthateventhoughEGCimprovestheacquisitionperformanceinSAIatlowSNRs,theperformanceofIASwithnoEGCissuperiortotheSAIatallSNRs. 33 ].AgeneralizedowgraphmethodispresentedinHomieretal.[ 34 35 ]tocomputethemeanacquisitiontimefordifferentserialandhybridsearch

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strategies.ForthecasewhentheacquisitionphasesareKconsecutivephasesintheuncertaintyregion,ithasbeenclaimedthatthelook-and-jump-by-K-binssearchistheoptimalserialsearchpermutationwhenKisknownandthebitreversalistheoptimalsearchpermutationwhenKisunknown.UndertheassumptionthattheprobabilityofdetectioninalltheKconsecutiveacquisitionphasesisthesameandwithmeandetectiontimeastheperformancemetric,theoptimumpermutationsearchstrategyhasbeenfoundin[ 36 ]usingtechniquesinmajorizationtheory.Supposethatthetiminguncertaintyregionisdividedintobinsindexedby0;1;:::;Ns1.Theithpositionintheoptimalpermutationisgivenby (Ns 37 41 ].Thebasicprinciplebehindalltheseschemesisthattherststageperformsacoarsesearchandidentiesthetruephaseofthereceivedsignaltobeinasmallersubsetofthesearchspace.Thesecondstagethenproceedstosearchinthissmallersubsetandidentiesthetruephase.InBahramgirietal.[ 37 ],suchatwo-stageschemeisproposedfortheacquisitionoftime-hoppedUWBsignalsinAWGNnoiseandMAI.ThesearchspaceisdividedintoQmutuallyexclusivegroupsofMconsecutivephaseseach.Intherststage,eachoneoftheQgroupsischeckedbycorrelatingthereceivedsignalwithasumofMdelayedversionsofthelocallygeneratedreplicaofthereceivedsignal.Onceagroupisidentiedascontainingthetruephase,thephasesinthegrouparesearchedbycorrelatingwithjustonereplicaofthereceivedsignal.AschemebasedonthesameprinciplehasbeendevelopedindependentlyinGezicietal.[ 38 ].Bothoftheseschemeshavebeendevelopedunderthe

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assumptionofanAWGNchannelandtheirperformanceislikelytosufferinthepresenceofmultipath.InReggianietal.[ 39 ],anacquisitionschemeforUWBsignalswithTHspreadingcalledn-scaledsearchispresented,wherethesearchspaceisdividedintogroupsofM=Nf=2nwheren1.TheTHsequenceusedtogeneratethereplicaofthereceivedsignalisalsomodiedbyneglectingthenleastsignicantbitsofeachadditionalshiftcl.Althoughtheactualschemeinvolveschip-ratesamplingofamatchedlteroutput,itisequivalenttocorrelatingthereceivedsignalwithMdelayedversionsofthemodiedreplicaofthereceivedsignal.Inthissense,itissimilarinspirittotheschemesdescribedabove. 42 43 ]exploitcyclostationarity,inherentinUWBsignalingduetopulserepetition,toestimatetiminginformationofthereceivedsignal.Theseschemesrequireframe-ratesamplingintheacquisitionstageandpulse-ratesamplingduringthetrackingstage.ThesignalmodelassumesonlyTHspreadingandnopolarityrandomizationofthepulses,i.e.,al=1.Itisalsoassumedthatthereceivedpulsesfromallpathsk=(t),fork=0;1;:::;Ntap1,andtheperiodoftheTHsequenceisequaltoasymbolduration,i.e.,Nb=Nth.Thetimingoffsetisassumedtobeconnedtoasymboldurationandisexpressedas=NTf+,whereN2[0;Nth1]and2[0;Tf)representsthepulse-leveloffset.Theacquisitionsystemestimatestheframe-leveltimingoffsetbyestimatingN.Todothis,aslidingcorrelatorcorrelatesthereceivedsignalwiththetemplate(t)andframe-ratesamplesz(n)=R(n+1)TfnTf(tnTf)r(t)areobtained.Undercertainconditions,itisobservedthattheautocorrelationRz(n;)=Efz(n)z(n+)gofz(n)isperiodicinnwithperiodNthandhencez(n)isacyclostationaryprocess.Estimates^Rz(n;)ofRz(n;)areobtained

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bysampleaveragingandtheframe-leveltimingestimateisobtainedbypickingthepeakoftheperiodicallytime-varyingcorrelationofthesampledcorrelatoroutput[ 43 ]andisgivenby 42 43 ]estimatestheFouriercoefcients^Rz(n;)oftheperiodicsequenceRz(n;)viasampleaveragingwhicharethenusedtoestimatetheframe-leveltimingas 2(^(n;)Nth 2 Figure2: Autocorrelationfunction(ACF)ofcorrelatoroutputsz[n]oritsFourierseries(FS)coefcientsestimatedviasampleaveragingandusedtoestimatetimingoffset. InTianetal.[ 44 ],amaximumlikelihood(ML)timingestimationschemeispresentedfordataaidedandnon-dataaidedmethodsandatradeoffbetweenacquisitionaccuracyandcomplexityisdiscussed.Adata-aidedtimingestimationschemeemployingEGCisanalyzedin[ 44 ],assumingthetimingoffsettobelessthanasymbolduration,whichestimatestheframe-leveltimingoffsetfromtheobservationofMsymboldurationsofthereceivedsignalas

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wherezl(N;bl)=PG1g=0R1r(t)bl(tlNbTfNTfgTc)dtdenotestheoutputofthecorrelatorwiththeEGCwindowoflengthG.Anothersimilardata-aidedtimingestimationschemeisdevelopedin[ 45 ]wherethetimingestimationproblemistranslatedtoanMLamplitudeestimationproblemandageneralizedlikelihoodratiotesttodetectthepresenceorabsenceofaUWBsignalisdevelopedwhichmakesuseoftheMLtimingestimatesinthelikelihoodratiotest.Leastsquaresestimatesofthetimingandthechannelimpulseresponse,usingNyquistratesamplesofthereceivedsignal,areobtainedinCarbonellietal.[ 46 ],undertherestrictiveassumptionthatthe
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techniques.However,theseschemescanestimatetksonlyafterthetimingoffsetisknownandhencecannotbeusedfortimingacquisition.InZhangetal.[ 53 ],theCramer-Raolowerbounds(CRLBs)forthetimedelayestimationproblemarederivedforUWBsignalsinAWGNandmultipathchannels.ItisshownthatalargernumberofmultipathresultsinhigherCRLBsandapotentiallyinferiorperformanceforunbiasedestimators. 54 ]intheabsenceofmultipath.ThisschememaynotbeapplicableinthepresenceofmultipathwhichisusuallythecasewithUWBsystems.AnacquisitionschemeimplementedonUWB-basedpositioningdeviceswhichuseacodedbeaconsequenceinconjunctionwithabankofcorrelatorsispresentedinFlemingetal.[ 55 ]andassumesabsenceofmultipath.AdistributedsynchronizationalgorithmforanetworkofUWBnodes,motivatedbyresultsfromsynchronizationofpulse-coupledoscillatorsinbiologicalsystemssuchassynchronizedashingamongaswarmofreiesandsynchronousspikingofneurons,ispresentedinHongetal.[ 56 ].

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21

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57 ]isemployedfordemodulationhasbeenderivedin[ 32 58 ]andistreatedindetailinChapter 4 .Fig. 3 showsaplotofthenumberofphasesinthehitsetasafunctionoftheSNRwhenn=102andthePRakereceiverhasNR=5andNR=10ngers.ItisobservedthatthecardinalityofthehitsetcouldbesignicantlylargedependingupontheoperatingSNR. Figure3: EffectofreceivedSNRonsizeofhitsetHforNR=5andNR=10. Adesignforanacquisitionsystemwhichdoesnottakethehitsetintoaccountcanresultinasignicantperformancedegradation.Forinstance,inserialacquisitionschemes,suchastheoneshowninFig. 2 ,thedecisionthresholdisusuallysetsuchthattheaverageprobabilityoffalsealarmisconstrainedbyasmallpositiveconstant1,i.e.,

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Fig. 3 showstworeceiveroperatingcharacteristics(ROCs)foranacquisitionschemewherethereceivedsignaliscorrelatedwithatemplatesignalandthecorrelatoroutputissquaredandcomparedtoathreshold.Thedetailedderivationoftheperformanceanalysiscanbefoundin[ 32 ].ForoneoftheROCs,thethresholdwassetassumingthatthehitsetconsistsofonlythetruephaseandfortheotherthehitsetdenitionin( 3 )wasusedassumingaPRakereceiverwithNR=5ngerswiththenominalBERrequirementn=103andtheaverageenergyreceivedperpulsetonoiseratioequalto5dB.Whenthehitsetcontainsonlythetruephase,thethresholdneedstobesetmuchhigherinordertopreventthedecisionstatisticsfortheotherphasesinthemultipathprole,whichhavesignicantenergy,fromexceedingit.Thiscausesthedegradationintheprobabilityofdetectionwhen^=. Figure3: TheROCswhenthethresholdissetforasingletonhitsetcontainingonlythetruephaseandforahitsetdenedin( 3 )withn=103.

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istohavealargedwelltimeforthecorrelator[ 7 ].ThelargedwelltimeincreasestheeffectiveSNRofthedecisionstatisticandintheabsenceofchannelfading,thisresultsinaccurateverication,i.e.,theprobabilitiesofafalsealarmandamisscanbemadearbitrarilysmall.However,forthreshold-basedacquisitionschemesinmultipathfadingchannelsitwasshown[ 59 ]thatnomatterhowlargetheSNRisorhowwechoosethethresholditmaynotbepossibletomaketheprobabilitiesofdetectionandfalsealarmarbitrarilysmall.Inparticular,theasymptoticperformanceoftwotypicalthreshold-basedacquisitionschemesforTHUWBsignalswascalculatedin[ 60 ].Itwasshownthatifthethresholdissuchthattheaverageprobabilityoffalsealarmislessthanagiventolerance,thenthereisanon-triviallowerboundontheasymptoticaverageprobabilityofmiss.Thislowerboundtranslatestoanupperboundontheasymptoticaverageprobabilityofdetection.TheseresultssuggestthatitmaynotbepossibletobuildagoodvericationstageforUWBsignalacquisitionsystemsbyjustincreasingthedwelltime.TheyalsosuggestthattheprinciplesunderlyingthedesignofefcientUWBsignalacquisitionschemesmaybeverydifferentfromthetraditionalspreadspectrumacquisitionschemes.Intraditionalspreadspectrumacquisitionsystems,thedecisionthresholdischosensuchthattheprobabilityoffalsealarmineachofthenon-hitsetphasesissmall.Thevericationstagehelpstheacquisitionsystemrecoverfromfalsealarmeventswhentheyoccur.ConsideringthattheconstructionofavericationstageinsomeUWBsignalacquisitionsystemsmaybedifcult,amoreappropriatechoiceofdecisionthresholdisonewhichrestrictstheprobabilitythattheacquisitionprocessencountersafalsealarmtobesmall.SoifPF()istheaverageprobabilitythattheacquisitionprocessendsinafalsealarm,thenthedecisionthresholddischosensuchthatPF()isconstrainedbyasmallpositiveconstant1, 7 19 ].Inmeanacquisitiontimecalculations,

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afalsealarmpenaltytimeisassumedwhichisthedwelltimeofthevericationstage,i.e.,thetimerequiredbytheacquisitionsystemtorecoverfromafalsealarmevent.Thusmeanacquisitiontimecalculationsimplicitlyassumetheexistenceofavericationstage.ForUWBsignalacquisitionsystems,ifthethresholdissetaccordingto( 3 )themeandetectiontimeisareasonablemetricforsystemperformance.Themeandetectiontimeisdenedastheaverageamountoftimetakenbytheacquisitionsystemtoendinadetection,conditionedonthenon-occurrenceofafalsealarmevent.Thecalculationofthemeandetectiontimethusdoesnotrequireanyassumptiononthevericationstage.Finally,severaldetection-basedschemesforUWBsignalacquisitionhaveproposedusingsomeformofEGCtoimprovetheacquisitionperformancebycombiningtheenergyinthemultipath[ 37 39 ].Theasymptoticperformanceofthreshold-basedUWBsignalacquisitionschemesusingEGChasbeencalculatedin[ 60 ].IthasbeenshownthatEGCmayleadtoasignicantperformancedegradation.

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amountstotradingoffhardwarecomplexityforanincreaseintheacquisitiontimetoachievesimilaracquisitionperformance.However,theperformanceofsuchreducedcomplexityestimation-basedacquisitionschemesintermsofestimationaccuracyandacquisitiontimeisstillanopenresearchdirection.Althoughdetection-basedschemeswhichevaluatethephasesinthesearchspaceoneatatimehaveasimplerhardwareimplementation,theymaysufferfromalargemeandetectiontimewhichmakesthemunsuitableforhighdatarateapplications.Forinstance,themeandetectiontimeoftheserialacquisitionschemein[ 32 ]wasfoundtobeoftheorderofonesecond.Furthermore,itwasshownthatthetimespentbytheacquisitionsysteminevaluatingandrejectingthenon-hitsetphaseswasthedominantpartofthemeandetectiontimecausingittodecreaseonlymarginallywithincreaseinSNR.Thusacquisitiontechniquescapableofreducingthesearchspacearecrucialinthedesignofefcientacquisitionschemes.Forexample,thetwo-stagehybridDS-THscheme[ 58 ]describedinChapter 4 achievesameandetectiontimeoftheorderofamillisecond.Anotherapproachtosolvethesearchspaceproblemisbydesigningthehigherlayersinthenetworkarchitecturecarefully.AmultipleaccessprotocolwhichemployscontinuousphysicallayerlinksinthenetworkinordertoavoidrepeatedacquisitionispresentedinKolencheryetal.[ 61 ].Thetiminguncertaintyregionmaybereducedsignicantlyifabeacon-enablednetworkisemployed,wherethemediumaccessisco-ordinatedbyacentralnodewhichperiodicallytransmitsbeaconstowhichothernodessynchronizeandfollowaslottedmediumaccessapproach.

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likelihoodratiotest(GLRT).ItisinstructivetoexaminethestructureoftheGLRTde-tectorusedbyaserialacquisitionsystemwhichtriestondthetruephasebyevaluatingthephasesinthesearchspaceoneatatime.AlthoughtheGLRTisnotanoptimaltest,ithasbeenknowntoworkquitewellingeneral[ 62 ].TheGLRThasbeenshowntobeasymptoticallyuniformlymostpowerfulamongtheclassofinvarianttests[ 63 ].ThereceivedsignalisobservedoveradurationofMperiodsoftheDSsequence,whichisassumedwithoutlossofgeneralitytobelongerthantheTHsequence,andthisobservationisdenotedbyr.Theacquisitionsystemistodeterminewhetherahypothesizedphase^canbeconsideredthetruephaseofthereceivedsignal.ItisassumedthatthehypothesizedphaseisamultipleofthechipdurationTc.Toenabletractableanalysis,itassumedthatthetruephaseisalsoamultipleofTc.ThenumberofphasesthesearchspaceisthusNdsNf.Fromthedenitionofthehitsetearlier,itisclearthatthereexistmanywaysinwhich^canbeconsideredtobethetruephase.Withoutlossofgenerality,supposethatthehitsetisfbTc;(b1)Tc;:::;+fTcgwherebandfareintegersbetween0andNdsNf=2.Alsosupposethatanall-onesdatatrainingsequenceissentintheacquisitionpreamble.Thisresultsinacompositehypothesistestingproblemwhosehypothesescanbeformulatedasfollows: maxfh;=2S(^)gp(rjh;)H1?H0(3)wherethevectorh,ofchannelgainsfhkg,isassumedtobedeterministicbutunknownandisthedecisionthreshold.Itcanbeshowneasilyusingtechniquessimilartothoseusedin[ 64 65 ]thatwhenn(t)isanAWGNprocesswithpowerspectraldensityN0=2,

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thechoicesofandhwhichmaximizep(rjh;)inthenumeratorin( 3 )aregivenby1=argmax2S(^)CT()C()h1=1 3 )0=argmax=2S(^)CT()C()h0=1 3 )canbewrittenas 3 )and( 3 ),theGLRTin( 3 )reducesto 3 )thattheteststatisticgivenbytheGLRTamountstocorrelatingthereceivedsignalwithNtapdifferenttemplates,eachcorrespondingtoadifferentMPC,summingthesquaredoutputsofeachofthesecorrelators,maximizingthissumfortwodisjointsetsofphases,andcomparingthedifferencetoathresholdasillustratedinFig. 3 .ThisteststatisticthusattemptstocollecttheenergyfromalltheMPCsthroughaformofequalgaincombining.

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Howeveritisimmediatelyclearthatsuchanimplementationisprohibitivelycomplextorealize.Thusothersub-optimalstrategiesneedtobeexploredwhichwouldcollectenergyfromtheMPCsinanalternativeway.Simpleenergydetectionapproachesthusneedtobeconsideredandothertechniquestoreducethesearchspaceandthusthemeandetectiontimeneedtobedesigned.Weproposeandanalyzetwosuchtechniquesinthefollowingtwochapters.

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Figure3: Generalizedlikelihoodratiotestforevaluationofphase^.TheupperandlowerMAXoperationsevaluatethemaximumofPNtap1k=0C2k()over2S(^)and2=S(^),respectively.

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40 58 ]forUWBsystemswhichenablesinasignicantreductioninthesearchspacefacedbytheacquisitionsystem.ThehybridDS-THsignalingformatallowstheacquisitiontobeperformedintwostageswhichresultsinsmallvaluesofthemeandetectiontime. 4.1.1HybridDS-THSignalFormatTheproposedhybridDS-THsignalingformatforUWBusestwolevelsofspreading.ThedatasymbolsarerstspreadusingaTHsequenceofperiodNth.Theresultingtime-hoppedsignalisfurtherspreadusingaDSsequenceofperiodNds.ItisnotedthatNdsischosentoberelativelylongcomparedtoNthandisamultipleofNth,i.e.Nds=DNthwhereDisaninteger.Thetransmittedsignalisatrainofmonocycles(t)ofenergyp Nbcahl Ndsi^(tlTthchl NthiTc)(4)where^(t)=(t)=p

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whereTcisthechipdurationwhichisgreaterthanthewidthofthetransmittedmono-cycle.ThehybridDS-THformatforUWBisillustratedinFig. 4 .Binaryphaseshiftkeying(BPSK)datamodulationisassumed,i.e.,bi2f1;1g.Thetwo-levelspreadingallowsustodividetheacquisitionprocessintotwostages:onefortheTHsequenceandanotherfortheDSsequenceasshowninFig. 4 .Asaresult,thesearchspacecanbesignicantlyreducedaswillbeshowninthefollowingsections.Inaddition,theDSspreadingontopoftheTHspreadingsmoothesanyspectrallinescausedduetotheshorterperiodicTHsequence. Figure4: ThehybridDS-THsignalformat. Figure4: ConceptualblockdiagramofthehybridDS-THtwo-stageacquisitionscheme. 2.2.1 .Thereceivedpulsewaveformisgivenbyr(t)=f(t)^(t).ItisassumedthatthedurationofthereceivedpulsedoesnotexceedthechipdurationTc.Withthechannelresponsein( 2 )thereceivedwaveformcorrespondingtoasinglepulseis

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Thereceivedsignalasasumofthesignalandnoisecomponentsisgivenbyr(t)=rs(t)+n(t).Thesignalcomponentisgivenby Nbcahl Ndsi!r(tlTthchl NthiTc)=p Nbcahl Ndsir(tkTclTthchl NthiTc);(4)denotesthereceivedsignalfromthekthmultipathcomponentandisthesignaldelaythroughthechannel.Thenoisecomponent,n(t),isassumedtobeazero-meanAWGNprocesswithtwo-sidedpowerspectraldensityN0 31 ].ItisassumedthatthechannelestimationblockfollowingacquisitionestimatesthechannelcoefcientsperfectlyandthatapartialRakedemodulator[ 57 ],whichestimatestherstarrivingNR(NRNtap)pathsisused.Perfectchannelestimationisassumedtomaketheanalysisamenableandtofocusonthedenitionofthehitset.Supposethatthereceiverlocksontothehypothesizedphase^,whichisanintegermultipleofTc.Tomaketheanalysistractable,theactualdelayisalsoassumedtobeanintegermultipleofTc.Let=^=Tth+Tc,whereNds1Nds1and

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{z }k16=k2SrXi=Sm| {z }i6=0Nb1Xl=0U(l+i;0)bbl+i Nbcahl Ndsiahl+i Ndsik1k2hk1hk22k1+(l+i)Nh+chl+i Nthi;k2+lNh+chl Nthi; 4 )denotesthecontributiontothesignalpartofthecorrelatorbythedesiredbitb0andthesecondtermdenotesthecontributionarisingfromtheinter-symbolandinter-frameinterference.Conditionedonh,thenoisecomponentofthecorrelatoroutput,

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4 )and( 4 )wouldbezero.Thenthedecisionstatisticybconditionedonthechannelcoefcientsh,isaGaussianrandomvariablewithmeanyb((;h))=8><>:p 66 ,pp.268-270],theaverageprobabilityoferrorcannowbeexpressedasasingleintegralEPe(;h);Eb

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where mkmk:(4)isthemomentgeneratingfunctionof(;h).Foraxednumber,NR,ofRaketaps,theminimumreceivedbit-energy-to-noiseratio,Eb 4 illustratestheeffectofdeviationfromthetruephaseontheBERperformanceofthesystemforNR=5.Fromthisgure,ifnweretobechosentobe102,thecorrespondingEb 4 .TheDSspreadinginthesignalisremovedbythesquaringoperation.Aserialsearchstrategyisemployedbytheclockcontrolwhensearchingforthetruephase.Inordertoevaluatethehypothesizedphase^,thesquaredreceivedsignaliscorrelatedwithalocallygeneratedreferencesignal

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Figure4: EffectofdeviationfromthetruephaseonBERforNR=5. 4 )asr2s(t)=E1Ntap1Xk=0h2k1Xl=2r(tkTclTthchl NthiTc) +E1Ntap1Xk=0Ntap1Xm=0| {z }k6=mkmhkhmsk(t)sm(t):(4)Thereferencesignalfortherststageisgivenbythetrainofsquaredpulses NthiTc^);(4)wherethedwelltimeofthecorrelatorfortherststageisMperiodsoftheTHsequence.Constructingthereferencesignalforcorrelationin( 4 )requiresknowledgeofthe

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receivedpulseshapeandthespreadingcodesfclgandfalgatthereceiver,andthisassumptionhasbeenmadethroughoutthedocument. Figure4: SquaringloopforTHpatternacquisition 4 asz(;h)=Z^+MNthTth^r2(t)s(t^)dt=Z^+MNthTth^r2s(t)s(t^)dt| {z }z1(;h)+Z^+MNthTth^2rs(t)n(t)s(t^)dt| {z }z2(;h)+Z^+MNthTth^n2(t)s(t^)dt| {z }z3(;h): 4 )as {z }k6=mkmhkhmAk;m();(4)

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whererk()=1 Nthi+k+iNh;cl+andAk;m()=MNth1Xl=01Xi=Sm1Xj=Smahl+i+ Ndsiahl+j+ Ndsi3(chl+i+ Nthi+k+iNh;chl+j+ Nthi+m+jNh;chl Nthi+);with3(a;b;c)=1,ifa=b=candzerootherwise.Thetermz1(;h)denotesthecontributiontothedecisionstatisticduetothesignalpartalone.ThenumberoftimesthekthmultipathcomponentiscollectedbythecorrelatoristhusgivenbyMNthrk()andAkm()representsthecoefcientofthecrosstermcorrespondingtothekthandmthmultipathcomponents.Thesecondtermontherighthandsideof( 4 )denotesthecontributionfrominter-frameinterferencecausedduetothemultipathofonepulsespillingoverintotheadjacentTHframe.Similarly,weobtainthesecondtermofthedecisionstatistic,thecontributionfromthesignal-noisecrosstermresultingfromthesquaringoperation,asz2(;h)=2E1Ntap1Xk=0khkBk()ZTc03r(t)n(t)dt;where Ndsi2(chl+i+ Nthi+k+iNh;chl Nthi+):(4)Finally,thethirdtermdenotingthepurenoisecomponentisgivenby

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bandwidthoftheUWBcommunicationsystemisB.TheaboveGaussianapproximationfortheoutputoftheintegratorisquiteaccuratewhenthetime-bandwidthproductoftheintegrator,MNthTthBislarge[ 7 ,pp.240-250],whichisusuallytrueforaUWBsystem.Themeanofthedecisionstatisticconditionedonfalgandfclgcanbeshowntobez(;hjfalg;fclg)=E[z(;h)jfalg;fclg]=MNthE1R2(0)Ntap1Xi=0ri()h2i+E1R2(0)Ntap1Xk=0Ntap1Xm=0| {z }k6=mkmhkhmAk;m()+MNthR(0)N0 E[z23(;h)jfalg;fclg]=M2N2thN20R2(0) 4+N20

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41 4.3.2.1AveragingovertheDS sequence Themeanandvarianceofthedecisionstatistic,averagedovertheDSsequence f a l g ,aretobeevaluated.Itcanbeshownthatfor k 6= m E [ A k;m ( )]=0 where theaveragingisdoneovertheDSsequence.Hencethemeanofthedecisionstatistic conditionedon f c l g foragiven h ,isgivenby z ( ; h jf c l g )= MN th E 1 R 2 (0) N tap 1 X k =0 r k ( ) h 2 k + MN th R (0) N 0 2 : (4) Similarlyitcanbeshownthatthevarianceofthedecisionstatisticconditionedon f c l g for agiven h ,isgivenby 2 z ( ; h jf c l g )= N 2 0 2 MN th R 2 (0) +2 M 2 ( N th + S m +1) N 0 E 1 R 3 (0) N tap 1 X k =0 N tap 1 X m =0 k m h k h m C km ( ) (4) where C km ( )= 1 N th + S m +1 N th X s = S m 1 X i 1 = S m 1 X i 2 = S m 2 ( c h s + N th i + k + i 1 N h ;c h s i 1 N th i + ) (4) 2 ( c h s + N th i + m + i 2 N h ;c h s i 2 N th i + ) : 4.3.2.2AveragingovertheTH sequence Themeanandvariancein( 4 )and( 4 )averagedovertheTHsequenceistobe found.From( 4 )and( 4 ),itisclearthatallweneedistoevaluate E [ r k ( )] and E [ C km ( )] ,wheretheaveragingisdoneoverthesequence f c l g .Theaveragevalues

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31 ]andaregivenby 4 )isasumofBernoullirandomvariables2(c[s+ Nth]+k+i1Nh;c[si1 Nth]+m+i2Nh;c[si2 4 )tobenegligiblecomparedtothersttermatthevaluesofEb

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4 ,ahitfortherststageistheeventwhenthecorrelatoroutputexceedsthedecisionthreshold1forsome^2H.Amissfortherststageoccurswhenthedecisionstatisticdoesnotexceed1forall^2H.Conditioningonaparticularchannelrealizationh,withtheapproximateGaussiandistributionforthedecisionstatisticoftherststageandgiventhedecisionthreshold1,thefalsealarmanddetectionprobabilitiesforaparticularvalueofcanbecomputedasPfa1(1;)=Prfz(;h)>1j^=2Hg;Pd1(1;)=Prfz(;h)>1j^2Hg: 31 ],whichexpressesthefalsealarmprobabilityaveragedoverallchannelrealizationsintheformofasingleintegral 21 2+j1t zR1t zdt;(4)where

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4 inwhichthereceivedsignaliscorrelatedwithareferencesignalandtheoutputofthecorrelatorissquaredandcomparedtoathreshold2.Thereferencesignalforthesecondstageisgivenby Nds]a[l Nds]2c[l+i+ Nth]+k+iNh;c[l Nth]+.Again,whentheDSandTHsequencesaresufcientlylong,r0k()canbereplacedbyitsmeanvalue,E[r0k()],averagedoverthesequencesfalgandfclg,whereitcanbeshownthatE[r0k()]=2(;kNh)if2fSm;:::;1gandzerootherwise.Conditioningonagivenchannelrealizationh,y(;h)isaGaus-sianrandomvariablewithmeany(;h)=R2(;h);whereR2(;h)=M0Ndsp

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Figure4: AcquisitionsystemforDSstage noisevarianceisgivenby2y=M0NdsR(0)N0 2+jp yR2t ydt+1 2jp yR2t ydt;=12 2R2t ysinp ydt 4 )isobtainedfromthefactthat 2kM0Ndsp 4 ),k(!)isthecharac-teristicfunctionoftheNakagami-mrandomvariablehk[ 66 ].Theaveragedetectionprobability,Eh[Pd2(2;)],canbeobtainedsimilarly.Thedecisionthreshold2,forthesecondstageischosenasdescribedinthefollowingsection.

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7 ],asthesystemperformancemetricforUWBacquisitionsystems.ThecalculationofPFAissummarizedinthefollowing.AnupperboundonthisprobabilityPbFA,isalsoderivedsincePbFAiseasiertocalculatethanPFA.SupposethatthehitsetHconsistsofHconsecutivephasesandcanbeassumedwithoutlossofgeneralitytobethesetf0;Tc;2Tc;:::;(H1)Tcg.Giventhethresholds1and2,lettheaverageprobabilitiesofdetectionforthejthphaseinthehitsetfortherst(TH)andsecond(DS)stagesbedenotedbyPD1(j)andPD2(j)respectivelywherej2f1;2;:::Hg.Theaveragefalsealarmprobabilityforthe(nH)thphaseinthesetofnon-hitsetphasesfortheTHstage,fHTc;(H+1)Tc;:::;(N1)Tcg,isdenotedbyPFA1(n)andtheaveragefalsealarmprobabilityforthe(mH)thphaseinthesetofnon-hitsetphasesfortheDSstage,fHTc;(H+1)Tc;:::;(ND1)Tcg,isdenotedbyPFA2(m)wheren2fH+1;H+2;:::Ng,m2fH+1;H+2;:::NDgandN=NthNh.TheaverageprobabilitiesoffalsealarmfortherstandsecondstagescanbeboundedbyPFA1(n)1andPFA2(m)2where1=maxnPFA1(n)and2=maxmPFA2(m).Themaximumprobabilitiesoffalsealarm1and2aretypicallythefalsealarmprobabilitiescorrespondingtophaseswhichliejustoutsidethehitset.Wenotethat1and2aredeterminedbythechoicesof1and2.AnoverallfalsealarmeventforthehybridDS-THacquisitionsystemcanoccurintwoways: 1. AfalsealarmeventintheTHstagewouldmeanafalsealarmeventfortheentiresystem,sincefollowingafalsealarmintheTHstage,theDSstagewouldevaluateonlynon-hitsetphases.

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2. TherecouldoccuradetectionintheTHstagefollowedbyafalsealarmintheDSstage.First,theaverageprobabilitythatafalsealarmeventwouldoccurintheDSstage,followingadetectionintheTHstageatthejthhitsetphase,iscomputed.ThisaverageprobabilityisdenotedbyPF2(j).FollowingthedetectionintheTHstageatthejthhitsetphase,thephaseevaluationintheDSstageisequallylikelytobeginatanyoneofthephasesf(j1)Tc;(j1+N)Tc;:::;(j1+(D1)N)Tcg,andPF2(j)isgivenby:PF2(j)=1 1PM2(j)hQD1k=1(1PFA2(j+kN))i;1(12)DdPD2(j) 1(1PD2(j))(12)D1,Pbf2(j;d):

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ItcanbeshownthatPF1(n)=1"NYk=n(1PFA1(k))#1PF12(1)PM1

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Giventhisupperbound,thethresholds1and2arechosensuchthatPbFA,wheredenotesthetoleranceontheaverageprobabilityoftheoccurrenceofafalsealarmforthehybridDS-THsystem. 7 19 ].Insuchcalculations,afalsealarmpenaltytimeisassumed,whichisthetimerequiredbytheacquisitionsystemtorecoverfromafalsealarmevent.Intypicalacquisitionsystems,thispenaltytimeisthelongdwelltimeofasubsequentvericationstagewhichattemptstoconrmdetectionandfalsealarmeventswithhighprobability.However,ithasbeenshownin[ 59 ]thattheconstructionofsuchavericationstageforthreshold-basedUWBacquisitionsystemsinmultipathfadingchannelsisdifcult.ThisisbecauseoftheexistenceofanupperboundontheprobabilitiesofdetectionevenastheSNRisasymptoticallyincreased.HenceitisnotimmediatelyapparenthowoneshouldassignapenaltytimeforafalsealarmeventinUWBacquisitionsystems.Forthisreasonweusethemeandetectiontimeasthemetricforsystemperformanceinsteadofthemeanacquisitiontime.Themeandetectiontimeisdenedastheaverageamountoftimetakenbytheacquisitionsystemtoendinadetection,conditionedonthenon-occurrenceofafalsealarmevent.Thecalculationofthemeandetectiontimethusdoesnotrequireanyassumptiononthefalsealarmpenaltytime.However,toensurethattheacquisitionsystemrarelyencountersafalsealarm,theprobabilityoftheacquisitionsystemendinginafalsealarmeventPFAisconstrainedbyasexplainedintheprevioussection.ThesearchspacefortherststageconsistsofNphases:S1=f(n1)Tc:1nNg.ThesearchspaceforthesecondstageisthesetofDphases:S2=f^1+(d1)NthTc:d=1;2;:::;Dg,where^1isthehypothesizedphasewhichcausesahittooccurintherststage.ThustheeffectivesizeofthesearchspaceoftheproposedtwostageUWBacquisitionprocessisN+D,whichissignicantlysmallercompared

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tothesearchspacesizeofNDwhenthespreadingisdoneusingonlyeitheraDSorTHsequenceofperiodNds.TheproposedhybridDS-THUWBsignalformatandtheproposedtwo-stageacquisitionsystemwhichexploitsthisformatresultinasignicantreductioninsearchspace.Usingtheowgraphtechnique[ 7 19 ],themeandetectiontimefortheoverallsystemcanbeeasilycalculatedandthenalexpressionhasbeenprovided.Fig. 4 showstheowgraphfordeterminingthemeandetectiontime.Intheowgraph,thejthhitsetphaseisshadedandlabeledj.ThedwelltimesoftherstandsecondstagesaredenotedbyT1=MNthTthandT2=M0NdsTth.Thenthemeandetectiontimefromtheowgraphisgivenby dzG(z)z=1(4)withG(z)=1 1G1M(z)G10(z)PHj=1hQj1m=1G1Mm(z)i[G1Dj(z)]G2j(z);ifi=2f1;:::;Hg;1 1G1M(z)G10(z)PHj=ihQj1m=iG1Mm(z)iG1Dj(z)G2j(z)+G10(z)[QHm=iG1Mm(z)]

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onadetectioneventintherststageoccurringatthephasej2f1;:::;Hgandisgivenby 1G2Mj(z)G20j(z)"1+DXd=2zT2(Dd+1)#;(4)wherethebranchlabelsandfunctionsforthesecondstagearegivenbyG2Dj(z)=PD2(j)zT2;j2f1;:::;Hg Figure4: FlowgraphtodeterminemeandetectiontimeforHybridDS-THAcquisitionSystem.

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periodNth=128,theperiodoftheDSsequenceNds=1024,i.e.,D=8,thenumberofchanneltapsNtap=100,Nh=16,Nb=8,M=1,M0=1=2andTc=2ns.TherequirednominaluncodedBERperformancehasbeensetasn=102whichcorrespondstoEb 23 ].TheNakagamifadingguresmk=3:5kTc 23 ].Inordertoverifytheapproximationsemployedinthecalculation,aMonteCarlosimulationwasrunontheacquisitionsystemandtheaveragedetectiontimewasnoted.Thesimulationsmakeuseoftheanalyticallycomputedthresholds1and2.Theperformanceoftheproposedtwo-stage,hybridDS-THacquisitionsystemwascomparedtoaconventionaldoubledwellacquisitionsystem[ 7 ],withmeandetectiontimebeingtheperformancemetric.Inordertomakeafaircomparisonbetweenthetwosystems,thesamehybridDS-THsignalingformatwasusedinthedoubledwellsystem.Thus,boththesystemsundercomparisonhavesimilarhitsets.Eachofthetwostagesinthedoubledwellsystemissimilartothesecond(DS)stageofourproposedhybridtwo-stagesystemwithaserialsearchemployed.However,unliketheproposedtwo-stagehybridDS-THsystem,theordinarydoubledwellsystemdoesnotexploitthestructureinherentinthehybridDS-THsignalingformattoaidacquisition.Moreover,theperformanceofthedouble-dwellsystemisoptimizedbyatwo-dimensionalsearchoverthevariabledwelltimesofthetwostages.Boththesystemsundercomparisonconstraintheaverageprobabilityoftheacquisitionsystemendinginafalsealarm,PFA<=0:05.Fig. 4 showstheeffectofthereceivedEb

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thetruephaseandthisresultsinincreasingsizesofthehitset.ThemeandetectiontimeresultsforthehybridDS-THandthedoubledwellsystemsareshowninFig. 4 andFig. 4 forNR=5andNR=10respectivelywherethehybridDS-THsystemisseentooutperformtheoptimumdoubledwellsystembyagainwhichisoftheorderofD.ThemeandetectiontimeofthehybridDS-THsystemobtainedanalyticallyaswellasfromsimulationarepresented.Thecloseaccordancebetweenthesimulationandtheresultsfromthecalculationindicatethattheapproximationsmadeintheanalysisarejustied.EvenasuboptimumchoiceofdwelltimesforthetwostagesinthehybridDS-THacquisitionsystemresultsinamuchsmallermeandetectiontimewhencomparedtotheoptimumdoubledwellacquisitionsystemasseenfromFig. 4 andFig. 4 .ThisadvantageisprimarilyduetothereductioninthesizeofthesearchspaceachievedbyusingthehybridDS-THsignalingformatandtheproposedhybridtwo-stageacquisitionscheme. Figure4: EffectofreceivedSNRonsizeofhitsetHforNR=5andNR=10.

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Figure4: MeandetectiontimeforhybridDS-THanddoubledwellsystemsforNR=5. Figure4: MeandetectiontimeforhybridDS-THanddoubledwellsystemsforNR=10.

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Thecomputationalcomplexityoftheproposedacquisitionsystemisminimal.ThecomputationofthethresholdsfortheTHandDSstagescanbedoneusingsingle-variableintegrationasin( 4 )and( 4 )respectively.Onanalnote,thehardwarerequiredtoimplementboththestagesoftheacquisitionsystemasshowninFig. 4 andFig. 4 wouldneedbothanaloganddigitalcomponents.ThesquaringoperationinFig. 4 wouldneedtobeimplementedinanalogwhereasthesquaringoperationinFig. 4 couldbedonedigitallyaftersamplingtheintegratoroutput.Thesamplingraterequiredfortherststagewouldbeoftheorderof1

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64 67 75 ].InTR-UWBsystems,areferencepulseistransmit-tedforeverydata-modulatedpulsewhichisthenusedbythereceiverfordemodulation.Demodulationistypicallyperformedusinganautocorrelationreceiver[ 69 ]whichcor-relatesthereceivedsignalwithadelayedversionofitself.Thedelayischosensuchthatchannelresponsecorrespondingtothereferencepulseiscorrelatedwiththechannelresponsecorrespondingtothedata-modulatedpulseresultinginacaptureofalltheenergyinthemultipath.AlthoughtheTRschemesuffersaperformancedegradationatlowSNRsduetousageofthenoisyreceivedsignalasthecorrelatortemplate[ 70 75 ],thefactthatitenablestheuseofalow-complexityreceivercapableofexploitingthedensemultipathintheUWBchannelmakesitanattractivealternativetosystemswhichuseconventionalRakereceivers.MuchoftheexistingliteratureonTR-UWBsystemshasfocusedontheperformanceevaluationofoptimalandsuboptimalreceivers[ 64 69 71 72 75 ].TheproblemoftimingacquisitioninTR-UWBsystemshasreceivedlittleattention.Ithasbeenclaimedbysomeauthors[ 67 71 ]thattheuseofTRschemesmitigatesthetimingacquisitionprobleminUWBsystems.However,therehasnotbeenanyanalysistosupporttheseclaims.Inthischapter,weinvestigatetheproblemofacquisitionforTR-UWBsignals 57

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withDSsignaling.Weshowatwo-levelDSsignalingstructureisessentialtoachievegoodacquisitionperformance.Wealsoproposeatwo-stageacquisitionschemewhichexploitstheTRsignalstructuretoreducetheacquisitionsearchspace. 2.2.1 .InTR-UWBsystems,thetransmittedsignalconsistsofasequenceofpulsepairs,eachpairconsistingofareferencepulsefollowedbyadata-modulatedpulse.Themodulationformatcanbeeitherpulseamplitudemodulation[ 64 ]orpulsepositionmodulation[ 69 ].Thesignalmayalsobespreadusingtime-hoppingordirect-sequencesignalingtoeliminatespectrallinesandhelpcombatMAI.Inthischapter,weconsiderTR-UWBsystemswithDSsignalingandantipodalmodulation.Thetransmittedsignalisgivenby

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Thereceivedsignalisgivenbyr(t)=h(t)x(t)+n(t)=rs(t)+n(t)=p 5 illustratesthepolaritiesduetoDSsignalingandmodulationofthereceivedsignal,thedelayedversionofthereceivedsignaland

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theproductofthethereceivedsignalanditsdelayedversion 69 ].Thismethodofachievingdemodulationwithoutknowledgeoftheoutercodefdlgisadvantageousbecauseofthefollowingreason.ThedistinguishingfeatureofUWBsystemsisthewidebandwidthandthestringentspectralmaskconstraintimposedbyregulatorybodies.Thewidebandwidthresultsinaneresolutionofthetiminguncertaintyregionwhilethespectralmaskconstraintnecessitatestheuseoflongspreadingsequencesspanningmultiplesymbolintervalsinordertoremovespectrallinesresultingfromthepulserepetitionpresentinthetransmittedsignal.Thesetwofeaturestogetherresultinalargespacefortheacquisitionsystemwhichresultsinalargeacquisitiontimeiftheacquisitionsystemevaluatesphasesinaserialmannerandinaprohibitivelycomplexacquisitionsystemifthephasesareevaluatedinaparallelmanner.However,byemployingthetwo-levelDSsignalingtheburdenofeliminatingthespectrallinescanbeplacedontheoutercodefdlgbychoosingitslengthMdstobelarge.Thisdoesnotincreasethesizeofthesearchspacewhichisproportionaltothe

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Figure5: Illustrationofthedelayandmultiplyoperationonthereceivedsignal.

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lengthoftheinnercodeNds.Theinnercodefalgresultsinapeakyambiguityfunctionasintraditionalspreadspectrumsystemsandhenceisessentialforgoodacquisitionperformance.Inaddition,theinnercodehelpscombatMAI.Thusitcanbechosentobeofrelativelysmallerlengthcomparedtotheoutercode.Fromtheabovediscussion,itisclearthatthepurposeoftheacquisitionsystemistoalign(atleastapproximately)theRZgatingwaveformwiththeusefulpartoftheproductsignal.Althoughthesizeofthesearchspaceisnowproportionaltothelengthoftherelativelyshortinnercode,itcanstillbelargeowingtothenetimingresolutionoftheUWBsystem.Soweproposeatwo-stageacquisitionsystemwhichsolvestheproblemofRZgatingwaveformalignmentintwosteps.Intherststage,theacquisitionsystemattemptstondthephaseoftheinnercodemodulatingtheproductsignalbycorrelatingtheproductsignalwithalocallygeneratedreplicaofthenon-return-to-zero(NRZ)innercodeDSwaveformwithchipdurationTf.AsillustratedinFig. 5 ,thephaseofthelocallygeneratedNRZinnercodeDSwaveformcansufferalargemarginoferrorandstillbesuccessfulindespreadingtheinnercodecorrespondingtotheusefulpartoftheproductsignal,whentheDScodeshaveidealautocorrelationproperties.Thisallowstherststagetoevaluatephasesinincrementsproportionaltothemarginoferror.Oncetheapproximatephaseoftheinnercodeisfoundintherststage,thesecondstageoftheacquisitionsystemproceedstoaligntheRZgatingwaveformwiththeusefulpartoftheproductsignalbysearchingseriallythroughthephasesaroundtheestimatedphase.Amoreprecisedescriptionofthetwo-stageacquisitionsystemcanbefoundinSection 5.5

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Figure5: Illustrationofthemarginoferrortolerableinthedespreadingoftheinnercodeintherststage.

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ofncanbeconsideredagoodestimateofthetruesignalphase.Wedenethehitsettobethesetofsuchhypothesizedphases.Thiswayofdeningthehitsetwasrstproposedin[ 32 58 ].WeassumethattheautocorrelationreceiverofFig. 5 isusedfordemodulation.Foragiventruephase,letPe()denotetheBERperformanceoftheautocorrelationreceiverwhenitlockstothehypothesizedphase^,where=^.LetEb Figure5: Blockdiagramoftheautocorrelationreceiver. Tocompletelycharacterizethehitset,weneedtocalculatetheerrorperformanceofanautocorrelationreceiverwhichislockedtoaparticularhypothesizedphase^.Sincethegoaloftheacquisitionstageiscoarsesynchronization,thehypothesizedphasecanbeassumedtobeanintegermultipleofTc.Tomaketheanalysistractable,weassumethatthetruephaseisalsoanintegermultipleofTc.Thus=(Nf+)Tcwhere;areintegersand0Nf1.Thiscorrespondstotheassumptionthatthetrackinglooplockstothetruephaseifthehypothesizedphase^iswithinTcofit,i.e.,
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Inordertodemodulatethebitbk,thereceivercorrelatesthereceivedsignalwithaproductofthedelayedreceivedsignalandtheRZgatingwaveformck(t^)=P(k+1)Nb1l=kNba[l=Nds]pm(tlTfTd^),where^isthereceiver'sestimateofthetruephaseandpm(t)=1fort2[0;Tm)andzerootherwise.AsexplainedinSection 5.3 ,theRZgatingwaveformdoesnotdependontheouterDScodefdlg.Thisisbecausefdlgmodulatesboththereferenceanddata-modulatedpulse(see( 5 ))andisthusremovedbythedelayandmultiplyoperation.Todeterminethehitset,itisnecessarytoobtaintheprobabilityofbiterrorasafunctionofthetimingoffset.Withoutlossofgenerality,considerthedemodulationofbitb0.Thedecisionstatisticatthecorrelatoroutput,conditionedonaparticularchannelrealizationh,isgivenby {z }Y1(;h)+1 {z }Y2(;h)+1 {z }Y3(;h)+1 {z }Y4(;h) 69 73 ]withmeanb(;h)andvariance2b(;h),whicharetobeevaluated.

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Themeanofthedecisionstatistic,conditionedonthechannelrealizationh,canbeshowntobeapproximatelyb(;h)=8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:E1 sinceEN[Y2(;h)]=EN[Y3(;h)]=EN[Y4(;h)]=0.NotethatEN[]intheaboveequationdenotesexpectationwithrespecttothenoisedistribution.Itcanbeeasily

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shownthat 5.3 .Wealsoderivethedecisionstatisticsfortherstandsecondstages.Sincethedemodulationstagedoesnotdependonthephaseoftheoutercodefdlg,thetiminguncertaintyregionisequaltothedurationoftheinnercodeandisgivenby

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5.4 ,boththetruephaseandthehypothesizedphase^areassumedtobeintegermultiplesofTctomaketheanalysistractable.Then;^2Sp=fnTc:n2Zand0nNs1gand=^=Tf+TcwhereandareintegerssuchthatNds+1Nds1and0Nf1.ThehitsetHtypicallyconsistsofhypothesizedphasesintheneighborhoodofthetruephase,i.e.,H=fbTc;(b1)Tc;:::;+fTcg,wherebandfarenon-negativeintegers.ThehitsetthusconsistsofH=b+f+1phases.ThephasesinthesearchspacecanbelabeledwithoutlossofgeneralityasSp=f0;1;:::;Ns1g,wherethelabelidenotesthephase+iTc.Whenlabeledinthismanner,thehitsetisgivenbyH=fNsb;:::;Ns1;0;1;:::;fg.TherststageoftheacquisitionsystemevaluatesthephasesinthesearchspaceSpinincrementsofJ=2NtapNd+1.ThereasonforchoosingthisvalueforJwillbecomeapparentinthenextsubsection,whereitwillbeshownthatthisisexactlythesizeofthemarginoferrortolerableindespreadingtheinnercode.ThusbyevaluatingthesearchspaceinincrementsofsizeJtherststagewillevaluateatleastonephasewheretheinnercodeisdespread.IftherststagebeginsitssearchinphasejofthesearchspaceSp,thephasesevaluatedbytherststagearegivenbyS1(j)=fj;(j+J)Ns;(j+2J)Ns;:::;(j+DJ)Nsg,whereD=b(Ns1)=Jcandthenotation(i)Ns=imodNs.ThephasejwherethesearchbeginscanbeanyelementofthesearchspaceSp.AparticularphaseiinthesearchspaceisevaluatedbycorrelatingtheproductsignalwithareplicaoftheinnercodeDSwaveformwhichisdelayedbyiTcandthecorrelatoroutputiscomparedtoathreshold.Thustheeffectivesearchspace(ESS)S1(j)oftherststageconsistsofD+1phasesinthesearchspacewhichareevaluatedsequentiallyuntilthethresholdisexceeded.Oncethethresholdisexceededatparticularphasei2Spintherststage,thecontrolispassedtothesecondstagewhichthenevaluatesthephasesinthesetS2(i)=fiNd;iNd+1;:::;i1;i;i+1;:::;i+Nd1g.ThesetS2(i)

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consistsof2NdphasescenteredatthephaseiandistheESSforthesecondstage.Inthesecondstage,aparticularphaseiisevaluatedbycorrelatingtheproductsignalwithagatingwaveformwithdelayiTcandcomparingthecorrelatoroutputtoathreshold.Therationalebehindthenesearchinthesecondstageisthatthethresholdcrossingintherststagemayoccurataphasewheretheinnercodeisdespreadbutthegatingwaveformisnotsufcientlyaligned.Thenesearchtriestoalignthegatingwaveformsuchthatasignicantportionoftheusefulpartoftheproductsignaliscollected.TheoverallsearchstrategyisillustratedinFig. 5 Figure5: Illustrationofsearchstrategyusedbythetwo-stageTRacquisitionsystem.

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wherethedwelltimeofthecorrelatorisequaltoM1periodsoftheinnercode.Again,thedecisionstatisticZ(;h)canbeapproximatedbyaGaussiandistributionwithmeanz(;h)andvariance2z(;h),whicharetobeevaluated.Onceagain,bytherandomsequenceassumptionontheDSsignalingsequencesfalgandfdlgwhenNdsissufcientlylarge,themeanofthedecisionstatisticiszerofor=2f1;0gandcanbeshowntobeapproximately 5.4 ,thenoisevarianceofthedecisionstatisticfortherststage,conditionedonh,canbeshowntobegivenby

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5.4 ,thedecisionstatisticforthesecondstageY(;h)conditionedonhhasaGaussiandistributionwithmean

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Conditionedonthechannelcoefcientshandgiventhethreshold1,theprobabili-tiesoffalsealarmanddetectionfortherststagearegivenbyPf1(1;jh)=PrfZ(;h)>1j^=2H0g;Pd1(1;jh)=PrfZ(;h)>1j^2H0g: 7 ],asthesystemperformancemetricforUWBacquisitionsystems.Theuseoftheaverageprobabilityoftheacquisitionsystemendinginafalsealarmasthecriterionfor

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decisionthresholdselectionwasrstdonein[ 32 58 ].ThecalculationofPFA(1;2)issummarizedinthefollowing.Inordertoalloweaseofnotationinthesequel,werelabel,withoutlossofgenerality,thephasesinthesearchspaceasfollows.ThephasesinthehitsetoftherststagearenowdenotedbyH0=f0;1;:::;2Nd+H2g.Theotherconsecutivephasesinthesearchspacearedenotedbyf2Nd+H1;:::;Ns1g.ThephasesbelongingtothehitsetofthesecondstagecanthenbedenotedbyH=fNd1;Nd;:::;Nd+H2g.Theaverageprobabilityoffalsealarmforanon-hitsetphasei=2H0fortherststageisdenotedbyPf1(i)andthatforanon-hitsetphasej=2HforthesecondstageisdenotedbyPf2(j).Theaverageprobabilityofdetectionforthehitsetphasem2f0;:::;2Nd+H2gfortherststageisdenotedbyPd1(m)andtheaverageprobabilityofdetectionforthehitsetphasen2fNd1;:::;Nd+H2gforthesecondstageisdenotedbyPd2(n).Anoverallfalsealarmeventforthetwo-stageTRacquisitionsystemcanoccurintwoways: 1. Afalsealarmeventintherststagewouldmeanafalsealarmeventfortheentiresystem,sincefollowingafalsealarmintherststage,thesecondstagewouldevaluateonlynon-hitsetphases. 2. Therecouldoccuradetectioneventintherststagefollowedbyafalsealarminthesecondstage.TheseeventsneedtobeenumeratedwhilecomputingPFA(1;2).First,theaverageprobabilitythatafalsealarmeventwouldoccurinthesecondstage,followingadetectioneventintherststageatthejthhitsetphase(0j2Nd+H2),iscomputed.ThisaverageprobabilityisdenotedbyPF2(j)andcanbeshowntobegivenby 1PM2(j)Phf2(j)(5)wherePnf2(j)denotestheprobabilityofnon-occurrenceofafalsealarminthesecondstagepriortotheevaluationofahitsetphase,followingadetectioneventintherststage

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atitsjthhitsetphase,andisgivenby

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rststage,istobedetermined.Thefalsealarmprobabilityconditionedoneachstartingphasewillbethesumoftheprobabilitiesofmutuallyexclusiveeventsleadingtoafalsealarm.Therearisetwocases:Case1:Searchstartsoutsidehitset,i.e.,k2f2Nd+H1;:::;Ns1g.Itcanbeshownthat 1PM1(k)Phf1(k)(5)wherePnf1(k)denotestheprobabilityofnon-occurrenceofafalsealarmintherststagepriortotheevaluationofahitsetphase,conditionedontherstphaseevaluationoftherststagebeingatthekthphase,andisgivenby JcandPF12(j;k)istheaverageprobabilityoftheacquisitionsystemendinginafalsealarminthesecondstage,followingadetectionintherststage,conditionedonthephaseevaluationintherststagestartinginthejthhitsetphase(k+(n(k)+j)J)NsandisgivenbyPF12(j;k)=LXi=j"i1Yl=j(1Pd1((k+(n(k)+l)J)Ns))#Pd1((k+(n(k)+i)J)Ns)PF2((k+(n(k)+i)J)Ns):

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Theprobabilityofmissingallthehitsetphasesfallinginthesearchspacefortherststage,conditionedontherstphaseevaluationbeginningatphasek,isgivenby 1PM1o(k)Pb(k)

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andtheprobabilityofmissingallthehitsetphasesinthesearchspacegivenbyPM1(k)=DYl=L0(k)+DL+1[1Pd1((k+(n(k)+l)J)Ns)]: 5 showstheowgraphforthetwo-stageacquisitionscheme.AsdiscussedinSection 5.5.3 ,thehitsetfortherststagecomprisesthosephaseswhereathresholdcrossingintherststageresultsinatleastonephasefromHbeingevaluatedinthesecondstage.ThedwelltimesoftherstandsecondstagesaredenotedbyT1=M1NdsTfandT2=M2NdsTf.Thenthemeandetectiontimeisgivenby dzG(z)z=1(5)

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Figure5: Flowgraphillustratingtheproposedtwo-stageacquisitionscheme. withG(z)=1 1z(D+1L)T1G1Mk(z);(5)Thefunctionscorrespondingtotherststageusedintheabovearegivenby 1zn3(j)T2G2Mj(z);(5)

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wheren1(j)denotesthenumberofnon-hitsetphasesevaluatedbythesecondstagepriortotheevaluationofahitsetphase,conditionedonahitoccurringintherststageatphasej,andisgivenby 1z(D+1L)T1G1M0k(z);(5)

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wherethetransferfunctioncorrespondingtothepathsleadingtotheACQstateuntiltherstnon-hitsetphaseisencounteredisgivenby

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23 ].TheNakagamifadingguresmk=3:5kTc 23 ].TherequirednominaluncodedBERperformancehasbeensetasn=103.ThenumberofphasesinthehitsetisshownasafunctionofEb 5 .AsEb 5 .Inordertoverifytheapproximationsmadeintheanalysis,wealsodeterminethemeandetectiontimeforthetwosystemsthroughMonteCarlosimulation,inwhichwemakeuseoftheanalyticallycomputedthresholds1and2.Weobservethattheapproximationsmadeintheanalysisarereasonable.Theimprovementinperformanceachievedbythetwo-stageschemeisduetoasignicantreductioninthesizeoftheeffectivesearchspace.Thesingle-stagesystemfacesalargesearchspaceofsizeNsanditsmeandetectiontimeisdominatedbythetimespentbythesearchinevaluatingandrejectingthelargenumberofnon-hitsetphases.Thuseventhoughtheprobabilitiesofdetectionforthesingle-stageschemeimprovewithSNR,theimprovementinthemeandetectiontimeisimperceptible.Ontheotherhand,thesizeoftheeffectivesearchspaceofthetwostageschemeisjust2Nd+D+1,animprovementoftheorderofNds.Thusthetimespentbytheacquisitionsysteminevaluatingandrejectingthenon-hitsetphasesismuchsmallerandtheimprovementinthemeandetectiontimewithincreaseinSNRisapparent.

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Figure5: EffectofreceivedSNRonhitsetsizeH. Figure5: Meandetectiontimefortwo-stageandsingle-stageTR-UWBacquisitionsystems.

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76 ],accuratetiminginformationisalsoessentialinUWBsystemsincorporatingpreciserangingcapabilities.AnimportantchallengefacedbyUWBsystemsisthatthetransmittedpulsecanbedistortedthroughtheantennasandthechannel.DuetothefrequencyselectivityoftheUWBchannel,thepulseshapesreceivedatdifferentexcessdelaysarepath-dependent[ 24 ].Moreover,theshortpulsesusedinUWBsystemsresultinhighlyresolvablemultipathwithalargedelayspread[ 12 ].Inthischapter,weaddressthetimingestimationprobleminUWBsystemswhenthereceiverdoesnothaveknowledgeofthereceivedpulseshapesandthechannel.Wederivemaximumlikelihood(ML)timingestimatorsandtheCramer-Raolowerbound(CRLB)forbothpilot-assistedandnon-pilot-assistedscenarios.Wefocusonnetimingestimation,wherethetiminguncertaintyregioniswithinapulsewidth,andassumethatcoarsesynchronizationhasalreadybeenachieved.EfcienttimingacquisitionschemesforUWBsystemshavebeendeveloped[ 77 ],whichachievecoarsesynchronizationtowithinapulseduration.WecomparetheCRLBtotheperformanceoftheMLtimingestimatorandsub-optimaltimingestimationmethodssuchasthedirty-templatemethod(TDT)[ 78 ]andtransmittedreference(TR)[ 67 69 71 ]signaling,bothofwhichdonotrequireknowledgeofthepulseshapesorthechannelatthereceiver.WeevaluatetheCRLBandsimulatetheperformanceoftheMLtimingestimatorandthesub-optimalschemesundertheIEEE802.15.3aUWBchannelmodelsdescribedin[ 12 ]. 83

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12 23 ]whichcanbeexpressedinthegeneralformintermsofitsimpulseresponse

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Thereceivedsignalfromasingleusercanthenbeexpressedasr(t)=rs(t)+n(t)with

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wheretheorthonormalbasisfunctionsaregivenbyi(t)aregivenby 6 )withcl=0foralll,isobservedoveradurationofNssymbolsandthisobservationisdenotedbyr. 2NsNbhTh(6)

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wherethevectorB(p;)=[B0(p0;);B1(p0;);:::;BNt1(pNt1;)]TandBk(pk;)=pTkCk().Intheabove,Ck()=[Ck;0();Ck;1();:::;Ck;L01()]Twith 6 )byrstmaximizingtheargumentoverp0;p1;:::;pNt1andthenover.Foranyvalueof>0,argmaxfp0;p1;:::;pNt1kpkk=1gNt1Xk=0pTkCk()2=argmaxfp0;p1;:::;pNt1:>0;kpkk=1gNt1Xk=0pTkCk()Ctk()| {z }Ak()pkwithpTkpk=18k: 6 ),wenotethatforanyvalueof,foreachk2f0;1;:::;Nt1g,thematrixAk()isreal,symmetricandpositivedenite.Thereforethekthterminthesummationis

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maximizedbysettingpk=u1k(),whereu1k()istheeigenvectorcorrespondingtothemaximumeigenvalue1k()ofAk().Theoptimumvalue^isthusgivenby 6 )areasfollows:1.J0;0=Eh@2L(r;)

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WecanobtainasimpliedexpressionforJ0;0byobservingthesecondderivative,withrespecttotime,ofthebasisfunctionsof( 6 )tobe 6 )and( 6 ),wecanshowthatJ0;0=42 @hkCTk()=JTk+1;0isa1L01vectorwhoseithelementisgivenby[J0;k+1]i=8>>>>>>><>>>>>>>:1

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Fromtheabove,theFisherinformationmatrixin( 6 )canthenbeexpressedinthesimpliedpartitionedform CRLB1()=J10;0=J0;0EG1ET:(6)TheinverseofthematrixGcanbecomputedbynotingfrom( 6 )thatfor0kNt,J1k+1;k+1=2 6 )canbeexpressedas CRLB()=1

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6 )overthesymbols,yielding 2KNsNbhTh(6)wherethematrixofpulsecoefcientsp=[p0;p1;:::;pNt1],K=2 6 )as 2KNsNb2(6)TheMLestimatesforthetiming,pulseshapesandthechannelcoefcientscanthenbeexpressedas[E;pE;^hE;E]=argmaxf;p;^h;gNs1Xn=0lncoshhK^hTLn(p;)i1 2KNsNb2 6 ),wenotethattheoptimizationproblemmaynotbewell-dened,asforcertainchannelrealizations,theargumentin( 6 )maybestrictlyincreasingin.However,foraxed>0,theoptimizationproblemin( 6 )over;pand^hiswell-dened.Thusforeachxedvalueof>0,wewouldneedtomaximizethersttermoftheargumentin( 6 )over;pand^h.AthighSNRs,adecisionfeedbackapproachmaybeadopted,whoseanalysiswouldbesimilartothatofthepilot-assistedcase.Hence,fornon-pilot-assistedtimingestimation,wefocusonrelativelylowSNRs.Moreover,wenotethatforjxj1,

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2x2forjxj1,wecanwrite( 6 )atlowSNRsasL(r;;p;h)K2 2KNsNbhTh=K2 {z }A(p;)h1 2KNsNbhTh=K2 2KNsNbhTh=K2 6 ),wenotethatforanygivenvalueof,weneedtodeterminef;p;^hgtomaximize^hTA(p;)^h.Wecanwrite^hTA(p;)^h=Ns1Xn=0"Ntap1Xk=0^hkpTkFk;n()!Ntap1Xj=0^hjpTjFj;n()!#=Ntap1Xk=0Ntap1Xj=0^hk^hjpTk"Ns1Xn=0Fk;n()FTj;n()#| {z }Dk;j()pj=uT266666664D0;0():::D0;Nt1()D1;0():::D1;Nt1().........DNt1;0():::DNt1;Nt1()377777775| {z }E()u

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wherethevectoru=h^h0pT0;^h0pT0;:::;^hNt1pTNt1iT.ThematrixS()issuchthatE()=S()ST()andisgivenbyS()=266666664F0;0():::F0;Ns1()F1;0():::F1;Ns1().........FNt1;0():::FNt1;Ns1()377777775 6 ),wenotethat^hTA(p;)^h=uTE()u,ismaximizedif^handparesuchthatuistheeigenvectorcorrespondingtothemaximumeigenvalue1E()ofthematrixE().Theestimateofthetimingisthengivenby^=argmax2[0;Tw]21S(): 6.2.2 ,wedenotetheparametervectorby=[;qT0;qT1;:::;qTNt1;hT]T.Usingtheaveragedlikelihoodfunctionin( 6 ),wecomputetheFisherinformationmatrixof( 6 ).

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@@ @Ns1Xn=0lncoshKhTLn(p;)!#=E"Ns1Xn=0@ @sinhKhTLn(p;) @KhT@Ln(p;) 6 ),wecandetermine@Ln(p;) 6 )that @[Fk;n()]i=ZT0r(t)(n+1)Nb1Xl=nNbal_i(tlTfkTw)dt:(6)and 6 )and( 6 ).IfwedenotetheGaussianrandomvariablesXn=KhTLn(p;),Yn=KhT@Ln(p;) 6 )canbeexpressedasJ0;0=Ns1Xn=0EY2nsech2Xn+ZntanhXn: 6 ).Similarly,wederivetheotherelementsoftheFisherinformationmatrixinthefollowing.

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@[Ln(p;)]ktanhXn

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Also,thediagonalelementsofJk+1;k+1canbeshowntobegivenby[Jk+1;k+1]i;i=Ns1Xn=0E"K2h2k[Fk;n()]ipk;i

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FromtheFisherinformationmatrix,theCramerRaoboundcanthenbeobtainedasCRLB1()=[J1]0;0. 78 ]makesuseofcross-correlationsbetweenadjacentsymbols,intheabsenceofinter-symbolinterference,toestimatetiminginformationofthereceivedsignal.Thetimingcanbeachievedusingeitherpilot-assistedornon-pilot-assistedmethods.Inthisscheme,asymbol-lengthsegmentofthereceivedwaveformisusedasatemplateandcorrelatedwiththesubsequentsymbollengthsegment,andthesymbol-ratecorrelatoroutputsamplesaresummedoverKpairsofsymbolstoestimatethetiminginformationas 78 ]. 79 ].Basedontheacquisitionsystemproposedin[ 79 ],weconsiderapilot-assistedtimingestimatorforTRsignaling,basedonthedelay-correlateoperation,givenby

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wherethereturn-to-zerogatingwaveformc(t)=P1l=bbl=Nbca[l=Nds]pm(tlTfrTd)andpm(t)=1fort2[0;Tm)andzerootherwise.TheintervalbetweenthereferenceanddatapulsesisdenotedbyTdandTm=NtTwisthemultipathdelayspread.TdischosentobegreaterthanTmtoavoidinter-frameinterference.Eachframecomprisesareferenceanddatapulse,withtheframedurationdenotedbyTfr=2TdandtheobservationdurationT0=NsNbTfr.ThecompletedetailsoftheTRsystemmodelcanbefoundin[ 79 80 ]. 12 ]inthefollowingnumericalcomputations.WecalculatetheCRLBandobtaintheperformanceoftheMLtimingestimator,theTDTandTRestimatorsthroughsimulation.Weemploythemethodofleastsquarestoextracttheparametersofthespecularmultipathmodelin( 6 )fromchanneldatageneratedbasedontheIEEEchannelmodels.Theparameterextractionenableseaseofanalysiswhilestillmakinguseofavailablemodeldata.WecomparetheperformanceoftheMLtimingestimatorwiththeCRLB,whenthereceivedpulseshapesareunknown.Wealsocomparetheperformanceofpilot-assistedTDTmethodandthedelay-correlateestimatorforTRsignaling. 81 ]as 6 )canthenbeexpressedusing( 6 )as

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wherewehaveabsorbedhk'sintotheotherparametersforconvenience.WemakethesimplifyingassumptionthatJk=Jforallk.ThereceivedpulsegeneratedusingtheIEEEchannelmodelisdenotedbyw(t).Wewillassumethatthesig-nalissampledataratefsanddenotetheNsignalsamplesofw(t)inthevectorformw=[w[0];w[1];:::;w[N1]]T.Wedenotetheparametervector=[0;0;0;0;:::;0;J1;0;J1;:::;Nt1;J;Nt1;J].Similarly,weusetheN1vectorsjkand~jktodenotethesampledversionsof(j)(tkTc)and~(j)(tkTc),respectively.Inmatrixnotation,( 6 )canbeexpressedaswr=X,wheretheN2NtJmatrixX=[00;~00;:::;J10;~J10;:::;JNt1;~J1Nt1].Usingthemethodofleastsquares[ 82 ],weobtaintheestimateofas^=(XTX)1XTw.WeestimatethenumberoftapsNtas 6 )andthechannelcoefcientshcanbeeasilydeterminedusing( 6 )andtheFourierseriescoefcientsof(j)(t)and~(j)(t). 12 ].Weassumethetransmittedpulsetobethesecond-orderderivativeGaussianpulse[ 65 ]withrmspulsewidthof0.78125ns.Thesamplingrateforthechannelparameterextractionischosentobefs=5GHz.Forsimplicity,weassumeitisknownthatJ=2.ThisvalueofJhasbeenobservedtogivegoodestimatesduringchannelparameterextractionfrommodeldata.Fig. 6 comparesthenoiselessreceivedsignalinatypicalCM1channelandthereconstructedsignalafterthechannelparameterextractionstepsdescribedintheprevioussubsection.Asnotedearlier,weassumethatcoarsesynchronizationhasalreadybeenaccomplishedandthetiminguncertaintyregion

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Figure6: ComparisonbetweennoiselessreceivedsignalinCM1modelandrecon-structedsignalafterparameterextraction. iswithinachipwidth.Wechoosethefollowingparametersduringthecomputationsandsimulations:Tw=Tc=2:2ns,thenumberofbitspersymbolNb=32,theperiodoftheDSsequenceNds=32,L=10andNf=Nt+10.Forafaircomparisonamongthethreetimingestimationschemes(ML,TDTandTR)andtheCRLB,wemakeuseofthesametrainingsequencerequiredbythepilot-assistedTDTmethod,bk=(1)bk=2c[ 78 ],inallthesimulations.Also,thesamerandomlygeneratedDSsequencewasusedintheschemes.WefurthernotethattheTRschemeinherentlysuffersa3dBlossduetotheenergyspentonthereferencepulses.Also,forafaircomparisonamongalltheschemes,wexthenumberofobservedsymbolsNstobethesame.Thismeansthatforthepilot-assistedTDTscheme,thenumberofsymbolpairsKin( 6 )wouldbehalfthenumberofobservedsymbols.TheCRLBandtheMSEfromsimulationarenormalizedtoT2w.

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Fig. 6 showstheperformanceofthethreetimingestimationschemesandtheCRLBforatypicalCM1channel,whichisaline-of-sight(LOS)channelwithina4mrange.WeobservethattheMLestimationschemesignicantlyoutperformstheTDTandTRschemesatmedium-to-highSNRs.Fig. 6 showstheperformanceoftheschemesinatypicalCM2channel,whichisanon-line-of-sight(NLOS)channelwithin4mwithalargerdelayspread.WeobservethattheperformanceoftheTDTandtheTRschemesdegradessignicantlyintheCM2channelandtheMLschemerequiresagreaternumberofobservedsymbolsforaperformancecomparabletothatinCM1.TheMLestimatorneedstoestimateagreaternumberofchannelandpulseshapeparametersintheCM2channelduetothelargerdelayspread.Fig. 6 andFig. 6 showtheresultsfortypicalCM3andCM4channelsrespectively.TheCM3channelisaNLOSchannelintherangeof4-10mandtheCM4channelisahighlydispersivechannelwithalargedelayspread[ 12 ].ThereasonforthedegradationofperformanceoftheTDTandtheTRschemesintheCM2-CM4channelsisduetotheabsenceofastrongpath,whichresultsinarelativelyatambiguityfunctioncomparedtotheCM1channel.AsnotedinSection 6.2.1 ,wecancomputeCRLBandtheperformanceoftheMLtimingestimatorsevenwhenthenumberofchanneltapsNtisunknown,byassumingamaximumnumberoftapsNtmax.Fig. 6 andFig. 6 showtheCRLBandtheMLestimatorperformanceforthepilot-assistedcasefortheCM1andCM2channelsrespectivelywithNtmax=100.Notethedegradationintheperformanceduetothelargernumberofparameterstobeestimated.However,theMLestimatorstilloutperformsthesub-optimalestimationmethods,especiallyintheNLOSCM2channel.AlthoughtheTDTandTRschemesenablelower-complexityimplementation,theymayonlyremainoperationalinLOSchannels.Fig. 6 andFig. 6 showtheCRLBandthesimulationperformanceoftheMLestimatorforthenon-pilot-assistedcaseandthenon-pilot-assistedTDTestimator.Itisinterestingtonotethatthetimingestimationperformanceofthenon-pilot-assistedML

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Figure6: CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM1channelwithNt=16. timingestimatorisbetterthanthatofthepilot-assistedMLtimingestimator,especiallyatlowSNRs.FromthedevelopmentinSection 6.3 ,wenotehoweverthatthenon-pilot-assistedMLestimatordoesnotprovideanestimateofthechannelgain,unlikethepilot-assistedMLestimator.

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Figure6: CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM2channelwithNt=26.NotethegreaternumberofsymbolobservationscomparedtoCM1.

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Figure6: CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM3channelwithNt=42.

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Figure6: CRLBandsimulationresultsofML,TDTandTRestimatorsinatypicalCM4channelwithNt=99.

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Figure6: CRLBandsimulationresultsofpilot-assistedMLestimatorinatypicalCM1channelwithNt=16,assumingNtmax=100.

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Figure6: CRLBandsimulationresultsofpilot-assistedMLestimatorinatypicalCM2channelwithNt=26,assumingNtmax=100.

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Figure6: CRLBandsimulationresultsofnon-pilot-assistedMLandTDTestimatorsinatypicalCM1channelwithNt=16,assumingNtmax=100.

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Figure6: CRLBandsimulationresultsofnon-pilot-assistedMLandTDTestimatorsinatypicalCM2channelwithNt=26,assumingNtmax=100.

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110

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simplereceiverstructureandfastacquisition,canpresentacompellingalternativetoconventionalimpulseradiosystemsemployingRakereceivers.TR-UWBsignalingmaybeanattractivechoicetoasystemdesignerwillingtotrade-offlossinBERperformanceforasignicantreductioninreceivercomplexity.Wehavederivedpilot-assistedandnon-pilot-assistedMLtimingestimatorsandtheCRLBforimpulseradioUWBsystemswhenthereceiverdoesnothaveknowledgeofthereceivedpulseshapeandthechannel.WecomparedtheperformanceoftheMLestimatortotheCRLBandtosub-optimaltimingestimationmethods,whichoperateintheabsenceofchannelandpulseshapeinformation.WeobservethattheMLestimatorssignicantlyoutperformsthesub-optimalmethods,especiallyinNLOSchannels.AlthoughtheimplementationcomplexityoftheMLestimatorsisaconcern,thesuperiorperformanceoftheseestimatorsespeciallyinNLOSchannels,enablesthemtoserveasabenchmarkforthecomparisonofothersub-optimalmethods.

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SandeepAedudodlareceivedhisB.Tech.degreeinelectronicsandcommunicationengineeringattheIndianInstituteofTechnology,Guwahati,in2002andtheMasterofSciencedegreeinelectricalandcomputerengineeringfromtheUniversityofFloridain2004.Hisresearchinterestsincludespreadspectrumandultra-wideband(UWB)communications.AsaPh.D.studentattheUniversityofFlorida,hehasbeeninvolvedwithresearchindevelopingtimingsynchronizationschemesforimpulseradioUWBcommunicationsystems.InSummer2004,hewasaninternwiththeMitsubishiElectricResearchLabsinCambridge,MA,wherehewaspartoftheteamdevelopingaUWB-basedphysicallayerfortheIEEE802.15.4astandard. 119


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SYNCHRONIZATION IN IMPULSE RADIO ULTRA-WIDEBAND
COMMUNICATION SYSTEMS

















By

SANDEEP AEDUDODLA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Sandeep Aedudodla















ACKNOWLEDGMENTS

I would like to express my gratitude to Prof Tan Wong, for his constant encourage-

ment, support and patience throughout the course of my graduate study. His meticulous

guidance and frank advice have been instrumental in bringing this work to its present

form. I am thankful to Saravanan Vijayakumaran for the numerous insightful discussions

and his help in reviewing some of this work. I would also like to thank the members of

my supervisory committee, Prof. John Shea, Prof. Liuqing Yang and Prof Ye Xia, for

their guidance, suggestions and interest in my work.

I would like to thank my parents and my sister and brother-in-law for their uncon-

ditional love, support and patience, which have constantly motivated me to confront the

challenges faced during my graduate studies.















TABLE OF CONTENTS
page

ACKNOW LEDGM ENTS ...................... . ........ iii

LIST OF FIGURE S . . . . . . . . vii

ABSTRACT .... ............................... .. ix

CHAPTER

1 INTRODUCTION .............................. 1

1.1 Objectives and Main Contributions ........ . ...... 2
1.2 D issertation Outline . . . . . . . 4

2 BACKGROUND AND RELATED RESEARCH ........ . .. 5

2.1 Acquisition Methods in Traditional Spread Spectrum Systems . 5
2.2 Signal Acquisition in UWB Systems ....... . ...... 9
2.2.1 System M odel ...... .. ..... ... .... ......... 9
2.2.2 Current Approaches Towards UWB Signal Acquisition . 12
2.2.2.1 Detection-based approaches .... . ..12
2.2.2.2 Efficient search strategies . . . . 15
2.2.2.3 Search space reduction techniques . ..... 16
2.2.2.4 Estimation-based schemes . . . . 17
2.2.2.5 Miscellaneous approaches . . . . 20

3 ISSUES AND CHALLENGES IN THE DESIGN OF UWB ACQUISITION
SY STEM S . . . . . . . . . 21

3.1 H it Set . . . . .. . .. . 2 1
3.2 Asymptotic Acquisition Performance of Threshold-based Schemes . 23
3.3 The Search Space in UWB Signal Acquisition . . 25
3.4 Generalized Likelihood Ratio Test for UWB Signal Acquisition . 26

4 ACQUISITION WITH HYBRID DS-TH UWB SIGNALING . . 31

4.1 System M odel . . . . . . . . 31
4.1.1 Hybrid DS-TH Signal Format . . . . . 31
4.1.2 Received Signal . . . . . . 32
4.2 Hit Set Formulation . . . . . . . 33
4.3 Stage 1: TH Acquisition . . . . . . 36
4.3.1 The Decision Statistic . . . . . . 38










4.3.2 Mean and Variance of the Decision Statistic . . . 39
4.3.2.1 Averaging over the DS sequence . . 41
4.3.2.2 Averaging over the TH sequence . . 41
4.3.3 False Alarm and Detection Probabilities . . . 43
4.4 Stage 2: DS Acquisition . . . . . . 44
4.5 Setting Thresholds 71 and 72 . . . . . 46
4.6 Mean Detection Time . . . . . . . 49
4.7 Numerical Results . . . . . . . 51
4.8 System Design and Complexity Considerations . . . 55

5 ACQUISITION IN TRANSMITTED REFERENCE UWB SYSTEMS . 57

5.1 TR-UWB Systems . . . . . . . 57
5.2 System M odel . . . . . . . . 58
5.3 Two-level DS Signaling Structure . . . . . 59
5.4 Hit Set Definition . . . . . . . 62
5.5 Two-stage Acquisition Scheme for TR-UWB Signaling . . 67
5.5.1 Decision Statistic of the First Stage . . . . 69
5.5.2 Decision Statistic for the Second Stage . . . 71
5.5.3 Probabilities of False Alarm and Detection ... . ...... 71
5.6 Decision Threshold Selection . . . . . . 72
5.7 Mean Detection Time . . . . . . . 77
5.8 Numerical Results . . . . . . . 80

6 FINE TIMING ESTIMATION . . . . . . 83

6.1 System M odel . . . . . . . . 84
6.2 Pilot-Assisted Timing Estimation . . . . . 85
6.2.1 Maximum Likelihood Timing Estimation . . . 86
6.2.2 Cramer-Rao Lower Bound . . . . . 88
6.3 Non-pilot-assisted Timing Estimation . . . . 90
6.3.1 Maximum Likelihood Timing Estimation . . . 91
6.3.2 Cramer Rao Lower Bound . . . . . 93
6.4 Sub-optimal Timing Estimation Methods . . . . 97
6.4.1 Timing with Dirty Templates (TDT) . . . . 97
6.4.2 Transmitted Reference (TR) Signaling ..... . . 97
6.5 Numerical Results . . . . . . . 98
6.5.1 Channel Parameter Extraction . . . . . 98
6.5.2 Computation and Simulation Results . . . . 99

7 CONCLUSIONS . . . . . . . . 110


7.1 C conclusions . . . . . .
7.2 Open Problem s . . . . .

REFERENCES . .....................


. . . 1 10
.. . 111

.. . 112









BIOGRAPHICAL SKETCH ............................. 119















LIST OF FIGURES
Figure page

2-1 Block diagram of a parallel acquisition system for direct-sequence spread spec-
trum system s . . . . . . . . 7

2-2 Block diagram of a serial acquisition system for direct-sequence spread spec-
trum system s . . . . . . . . 7

2-3 Block diagram of the acquisition scheme proposed by Blazquez et al. .. 13

2-4 Block diagram of the acquisition scheme proposed by Soderi et al. ...... 14

2-5 Autocorrelation function (ACF) of correlator outputs z[n] or its Fourier series
(FS) coefficients estimated via sample averaging and used to estimate timing
o ffset . . . . . . . . . 1 8

3-1 Effect of received SNR on size of hit set H for NR 5 and NR 10. .. 22

3-2 The ROCs when the threshold is set for a singleton hit set containing only the
true phase and for a hit set defined in (3-1) with An 10. .. . 23

3-3 Generalized likelihood ratio test for evaluation of phase .. .. . .30

4-1 The hybrid DS-TH signal format . . . . . . 32

4-2 Conceptual block diagram of the hybrid DS-TH two-stage acquisition scheme.
. . . . . . . . . . 3 2

4-3 Effect of deviation from the true phase on BER for NR = 5 . .... 37

4-4 Squaring loop for TH pattern acquisition .................. .. 38

4-5 Acquisition system for DS stage .................. ...... .. 45

4-6 Flowgraph to determine mean detection time for Hybrid DS-TH Acquisition
S y stem . . . . . . . . . 5 1

4-7 Effect of received SNR on size of hit set H for NR = 5 and NR= 10. .. 53

4-8 Mean detection time for hybrid DS-TH and double dwell systems for NR = 5. 54

4-9 Mean detection time for hybrid DS-TH and double dwell systems for NR = 10. 54

5-1 Illustration of the delay and multiply operation on the received signal. .. 61









5-2 Illustration of the margin of error tolerable in the despreading of the inner
code in the first stage . . . . . . . 63

5-3 Block diagram of the autocorrelation receiver. ........... ........ 64

5-4 Illustration of search strategy used by the two-stage TR acquisition system. 69

5-5 Flowgraph illustrating the proposed two-stage acquisition scheme. ...... 78

5-6 Effect of received SNR on hit set size H... ............ . 82

5-7 Mean detection time for two-stage and single-stage TR-UWB acquisition sys-
tem s . . . . . . . . . . 82

6-1 Comparison between noiseless received signal in CM1 model and reconstructed
signal after parameter extraction . . . . . . 100

6-2 CRLB and simulation results of ML, TDT and TR estimators in a typical CM1
channel w ith = 16 . . . . . . . 102

6-3 CRLB and simulation results of ML, TDT and TR estimators in a typical CM2
channel with N = 26. Note the greater number of symbol observations com-
pared to CM 1 . . . . . . . . 103

6-4 CRLB and simulation results of ML, TDT and TR estimators in a typical CM3
channel with N = 42 . . . . . . . 104

6-5 CRLB and simulation results of ML, TDT and TR estimators in a typical CM4
channel with N = 99 . . . . . . . 105

6-6 CRLB and simulation results of pilot-assisted ML estimator in a typical CM1
channel with N = 16, assuming Ntmax = 100....... . . 106

6-7 CRLB and simulation results of pilot-assisted ML estimator in a typical CM2
channel with N = 26, assuming Ntmax = 100....... . . 107

6-8 CRLB and simulation results of non-pilot-assisted ML and TDT estimators in
a typical CM1 channel with N = 16, assuming Ntmax = 100. . . 108

6-9 CRLB and simulation results of non-pilot-assisted ML and TDT estimators in
a typical CM2 channel with N = 26, assuming Ntmax = 100. . . 109















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SYNCHRONIZATION IN IMPULSE RADIO ULTRA-WIDEBAND
COMMUNICATION SYSTEMS

By

Sandeep Aedudodla

December 2006

Chair: Tan F. Wong
Major Department: Electrical and Computer Engineering

Ultra-wideband (UWB) communication has recently generated a significant

amount of interest and is a primary physical layer candidate for wireless personal area

network (WPAN) standards aimed at both high data rate and low data rate applications.

Ultra-wideband spread spectrum systems employing impulse radio signaling are being

considered for these applications. The UWB channel is a dense multipath channel with

a large delay spread and highly resolvable multipath components, owing to the narrow

pulses employed. In any communication system, the receiver requires knowledge of the

timing information of the received signal to accomplish demodulation. The process of

obtaining this timing information is known as synchronization. Spread spectrum systems

typically achieve synchronization in two stages: acquisition and tracking. The acquisition

stage attempts to achieve coarse synchronization to within a chip duration by evaluating

the phases in the search space and the tracking stage employs a code tracking loop to

maintain fine synchronization. Ultra-wideband systems employ long spreading sequences

to eliminate spectral lines resulting from pulse repetition. Moreover, the short pulses

employed result in a fine resolution of the search space. Due to these reasons, UWB

acquisition systems face a large search space which results in a large amount of time for









the system to successfully acquire the signal. This results in long preambles in a packet-

based WPAN which adversely affects the network throughput. The dense resolvable

multipath results in the existence of multiple phases which can be considered to be a

good estimate of the true phase. In this dissertation, we define a hit set to be the set of

phases which guarantee a nominal demodulation performance subsequent to acquisition.

We focus on signal design and the development of efficient acquisition schemes which

significantly reduce the mean detection time through search space reduction. One such

design is the proposed hybrid direct sequence-time hopping (DS-TH) signaling format

and a two-stage acquisition scheme which combats the large search space problem in

impulse radio UWB systems. We study the acquisition problem in transmitted reference

(TR) UWB systems with DS signaling and design a two-stage acquisition scheme which

greatly reduces the mean detection time. We also consider the fine timing estimation

problem in impulse radio systems in which the receiver does not have knowledge of

the received pulse shapes or the channel. We derive the Cramer-Rao lower bound and

maximum likelihood estimators for pilot-assisted and non-pilot-assisted cases, and

compare their performance to sub-optimal methods with lower complexity.















CHAPTER 1
INTRODUCTION

A class of spread spectrum techniques known as ultra-wideband (UWB) commu-

nication [1-4] has recently received a significant amount of attention from academic

researchers as well as from the industry. Ultra-wideband signaling is being considered for

high data rate wireless multimedia applications for the home entertainment and personal

computer industry as well as for low data rate sensor networks involving low-power

devices. It is also considered a potential candidate for alternate physical layer protocols

for the high-rate IEEE 802.15.3 and the low-rate IEEE 802.15.4 wireless personal area

network (WPAN) standards [5, 6].

In any communication system, the receiver needs to know the timing information

of the received signal to accomplish demodulation. The subsystem of the receiver which

performs the task of estimating this timing information is known as the synchronization

stage. Synchronization is an especially difficult task in spread spectrum systems which

employ spreading codes to distribute the transmitted signal energy over a wide band-

width. The receiver needs to be precisely synchronized to the spreading code to be able to

despread the received signal and proceed with demodulation. In spread spectrum systems,

synchronization is typically performed in two stages [7, 8]. The first stage achieves coarse

synchronization to within a reasonable amount of accuracy in a short time and is known

as the acquisition stage. The second stage is known as the tracking stage and is respon-

sible for achieving fine synchronization and maintaining synchronization through clock

drifts occurring in the transmitter and the receiver. Tracking is typically accomplished

using a delay locked loop [7]. Timing acquisition is a particularly acute problem faced by

UWB systems as explained in the sequel. This dissertation addresses the significance of

the acquisition problem in UWB systems and the ways to efficiently tackle it.









Accurate timing estimation following coarse acquisition is required to perform

demodulation and is useful in applications requiring precise localization. Impulse radio

systems often do not have knowledge of the received pulse shape and the channel.

We present pilot-assisted and non-pilot-assisted maximum likelihood (ML) timing

estimators in the absence of such information. We also derive the Cramer-Rao lower

bound (CRLB) on the performance of any unbiased estimator. We compare the simulation

performance of the ML estimators to the CRLB and to the performance of sub-optimal

timing estimators that do not require knowledge of the pulse shape and the channel.

1.1 Objectives and Main Contributions

Short pulses and low duty cycle signaling [1] employed in UWB systems place

stringent timing requirements at the receiver for demodulation [9, 10]. The wide band-

width results in a fine resolution of the timing uncertainty region thereby imposing a

large search space for the acquisition system. Typical UWB systems also employ long

spreading sequences spanning multiple symbol intervals in order to remove spectral lines

resulting from the pulse repetition present in the transmitted signal. In the absence of

any side information regarding the timing of the received signal, the receiver needs to

search through a large number of phases1 at the acquisition stage. This results in a large

acquisition time if the acquisition system evaluates phases in a serial manner and results

in a prohibitively complex acquisition system if the phases are evaluated in a parallel

manner.

Moreover the relatively low transmission power of UWB systems requires the

receiver to process the received signal for long periods of time in order to obtain a

reliable estimate of the timing information. In a packet-based network, each packet



1 Traditionally, in direct-sequence spread spectrum systems the chip-level timing of the
PN sequence is referred to as the phase of the spreading signal. In this document, we use
phase and timing interchangeably.









has a dedicated portion known as the acquisition preamble within which the receiver

is expected to achieve synchronization. However for the high data-rate applications

envisaged for UWB signaling, long acquisition preambles would significantly reduce the

throughput of the network.

The transmitted pulse can be distorted through the antennas and the channel and

hence the receiver may not have exact knowledge of the received pulse signal waveform

[11]. The short pulses used in UWB systems also result in highly resolvable multipath

with a large delay spread, at the receiver [12]. The UWB receiver could therefore

synchronize to more than one possible arriving multipath component (MPC) and still

perform satisfactorily. This means that there could exist multiple phases in the search

space which could be considered acceptable and could be exploited to speed up the

acquisition process.

These challenges arising from the signal and channel characteristics unique to UWB

systems indicate the significance of the acquisition problem in UWB communication

and the need to address it efficiently. Addressing some of these issues is the focus

of this dissertation. We propose a hybrid direct sequence-time hopping (DS-TH)

signaling format and a two-stage acquisition scheme for UWB systems which enables

in a significant reduction in the size of the search space. We evaluate the performance

of the two-stage scheme in terms of the mean detection time. We define the hit set as

the set of phases, which, following acquisition, result in a satisfactory receiver BER

performance. The hit set thus characterizes the effect of the dense resolvable multipath

on the acquisition system performance. The acquisition problem in direct sequence

transmitted reference (TR) UWB systems is analyzed, where we observe that there is a

significant relaxation in the timing requirement which can be exploited by the use of a

two-stage acquisition scheme presented. We propose to study the tracking problem for

UWB systems as future work.









1.2 Dissertation Outline

This dissertation is organized as follows. In Chapter 2, we briefly summarize the

acquisition approaches adopted by traditional spread spectrum systems and also classify

and discuss the current research on UWB signal acquisition. Through a discussion of

existing spread spectrum techniques we seek to distinguish the issues and challenges

unique to UWB acquisition system design, which are presented in Chapter 3. An

understanding of these issues, particularly the existence of multiple acquisition phases

and the asymptotic acquisition performance, enables better UWB acquisition system

design. In Chapter 4, we present the hybrid DS-TH signaling format and a two-stage

acquisition scheme to combat the large search space problem in UWB acquisition

systems. In Chapter 5 we discuss the acquisition problem in TR-UWB systems with

DS signaling and present a two-stage acquisition scheme which significantly reduces

the mean detection time. We discuss pilot-assisted and non-pilot-assisted fine timing

estimation methods for impulse radio, in the absence of received pulse shape and channel

information in Chapter 6. The dissertation is concluded in Chapter 7.















CHAPTER 2
BACKGROUND AND RELATED RESEARCH

In this chapter, acquisition methods in conventional spread spectrum systems

are summarized and acquisition techniques for UWB communication systems are

discussed. Through a discussion of spread spectrum acquisition techniques, we wish

to highlight that UWB signaling is a class of spread spectrum communication and thus

acquisition techniques for UWB systems could employ similar concepts with appropriate

modifications. We also discuss the existing work on acquisition in UWB systems.

2.1 Acquisition Methods in Traditional Spread Spectrum Systems

Ultra-wideband communication falls in the category of spread spectrum communica-

tion systems. In this section, we briefly review the main features of acquisition methods

used in traditional spread spectrum systems to put the current approaches to UWB

signal acquisition in perspective. There has been extensive research on spreading code

acquisition and tracking for spread spectrum systems with direct-sequence, frequency-

hopping and hybrid modulation formats [7, 8, 13]. We will bring out the main issues by

considering the timing acquisition of direct-sequence spread spectrum systems.

In a direct-sequence spread spectrum system, the receiver attempts to despread the

received signal using a locally generated replica of the spreading waveform. Despreading

is achieved when the received spreading waveform and the locally generated replica are

correctly aligned. If the two spreading waveforms are out of synchronization by even

a chip duration, the receiver may not collect sufficient energy for demodulation of the

signal. As mentioned before, the synchronization process is typically divided into two

stages: acquisition and tracking. In the acquisition stage, the receiver attempts to bring

the two spreading waveforms into coarse alignment to within a chip duration. In the

tracking stage, the receiver typically employs a code tracking loop which achieves fine









synchronization. If the received and locally generated spreading waveforms go out of

synchronization by more than a chip duration, the acquisition stage of the synchronization

process is reinvoked. The reason for this two stage structure is that it is difficult to build

a tracking loop which can eliminate a synchronization error of more than a fraction of a

chip.

A typical acquisition stage attempts to bring the synchronization error down to

be within the pull-in range of the tracking loop by searching the timing uncertainty

region in increments of a fraction of a chip. A simplified block diagram of an acquisition

stage which is optimal in the sense that it achieves coarse synchronization with a

given probability in the minimum possible time is the parallel acquisition system [7]

shown in Fig. 2-1. This stage checks all the candidate phases in the uncertainty region

simultaneously and the phase corresponding to the maximum correlation value is declared

to be the phase of the received spreading waveform. In an additive white Gaussian noise

(AWGN) channel, this acquisition strategy produces the maximum-likelihood estimate

(from among the candidate phases) of the phase of the received spreading waveform.

However, the hardware complexity of such a scheme may be prohibitive since it requires

as many correlators as the number of candidate phases being checked, which may be

large depending on the size of the timing uncertainty region. A widely used technique

for coarse synchronization, which trades off hardware complexity for an increase in the

acquisition time, is the serial search acquisition system shown in Fig. 2-2. This system

has a single correlator which is used to evaluate the candidate phases serially until the

true phase of the received spreading waveform is found. Hybrid methods such as the

MAX/TC criterion [14], have also been developed which employ a combination of the

parallel and serial search acquisition schemes and reduce the acquisition time at the cost

of increased hardware complexity. All the acquisition schemes employ a verification

stage [15] which is used to confirm the coarse estimate of the true phase before the

control is passed to the tracking loop.





















X--- filter deter corresponding Verification o code
Received Ito largest stage tracking loop
signal IN energy




Bandpass Energy
X filter detector

s(t-t,)

Figure 2-1: Block diagram of a parallel acquisition system for direct-sequence spread
spectrum systems which evaluates the candidate phases 1 12, ... I. In the ith arm, the
decision statistic corresponding to the candidate phase t, is generated by correlating the
received signal with a delayed version of the locally generated spreading waveform s(t).


Received
signal


Figure 2-2: Block diagram of a serial acquisition system for direct-sequence spread
-"--p~im ilS systems which evaluates the candidate 1 .l. --- tl, 12, ... ,n serially until the
threshold is exceeded. The decision statistic corresponding to the candidate phase 1i is
generated by correlating the received signal with a delayed version of the locally gener-
ated -, .; ,v-i. waveform s(t). If the threshold is not exceeded, the search updates the
value of the candidate phase and the process continues.


STo code
tracking loop









In traditional spread spectrum acquisition schemes, the signal-to-noise ratio

(SNR) of the decision statistic improves with an increase in the dwell time, which is

the integration time of the correlator. Thus the probability of correctly identifying the

true phase of the received spreading waveform can be increased by increasing the time

taken to evaluate each candidate phase. This tradeoff has been identified and exploited

by several researchers for the development of more efficient acquisition schemes and

has led to their classification into fixed dwell time and variable dwell time schemes

[7, 8]. The fixed dwell time based schemes are further classified into single and multiple

dwell schemes [16]. The decision rule in a single dwell scheme is based on a single

fixed time observation of the received signal whereas a multiple dwell scheme comprises

multiple stages with each stage attempting to verify the decision made by a previous

stage by observing the received signal over a comparatively longer duration. Variable

dwell time methods are based on the principles of sequential detection [17] and are aimed

at reducing the mean dwell time. The integration time is allowed to be continuous and

incorrect candidate phases are dismissed quickly which results in a smaller mean dwell

time.

Several performance metrics have been used to measure the performance of

acquisition systems for spread spectrum systems. The usual measure of performance

is the mean acquisition time which is the average amount of time taken by the receiver

to correctly acquire the received signal [7, 8, 18]. The variance of the acquisition time

is also a useful performance indicator, but is usually difficult to compute. The mean

acquisition time is typically computed using the signal flow graph technique [19]. For

parallel acquisition systems, a more appropriate performance measure is the probability

of acquisition or alternatively the probability of false lock [20].

In the presence of multipath, there could exist more than one phase which could be

considered to be the true phase of the received signal. However, few acquisition schemes

for spread spectrum systems [21, 22] have taken this into consideration.









2.2 Signal Acquisition in UWB Systems

As discussed in Section 1.1, the distinguishing feature of UWB systems is the wide

bandwidth and the relatively low transmission power constraint imposed by regulatory

bodies. The wide bandwidth enables fine timing resolution resulting in a large number

of resolvable paths in the UWB channel response. There may be more than one path

where a receiver lock could be considered successful acquisition. The stringent power

constraint necessitates the use of long spreading sequences which together with fine

timing resolution results in a large search space for the acquisition system. So the main

difference between the acquisition problems for UWB systems and traditional spread

spectrum systems is the presence of multiple acquisition states and the relatively large

search space in the former.

The large search space obviates the use of a fully parallel acquisition system due

to its high hardware complexity. Hence much of the existing work on UWB signal

acquisition has focused on serial and hybrid acquisition systems. Several researchers

have tackled the large search space problem by proposing schemes which involve more

efficient search techniques. However, the existence of multiple acquisition states has

received relatively less attention and has not been sufficiently exploited. Furthermore, a

significant portion of the existing work assumes an AWGN channel model for the UWB

channel and neglects the effect of multipath in the development and evaluation of the

proposed acquisition schemes.

In the next subsection, we describe general models for the propagation channel

and the acquisition signal for UWB systems. This model will be used in the later

subsections to describe the main features of some of the proposed schemes for UWB

signal acquisition.

2.2.1 System Model

The transmitted UWB signal consists of a train of short pulses monocycless) which

may be dithered by a time-hopping (TH) sequence to facilitate multiple access and to









reduce spectral lines. The polarities of the transmitted pulses may also be randomized us-

ing a DS spreading code to mitigate multiple access interference (MAI). The generalized

UWB signal transmitted during the acquisition process for a single user can be expressed

as a series of UWB monocycles b(t) of width Tp each occurring once in every frame of

duration Tf as
00
x(t) = bi1/Nbja[l/Nds] t(t Tf C[INh]T), (2-1)
1 --00
where Nb is the number of consecutive monocycles modulated by each data symbol

bi, Tf is the pulse repetition time, T, is the chip duration which is the unit of additional

time shift provided by the TH sequence and [.], [-L denote the integer division remainder

operation and the floor operation respectively. The pseudorandom TH sequence {ct(l);h 1

has length Nth where each cl takes integer values between 0 and Nh 1 where Nh is less

than the number of chips per frame Nf = Tf/Tc. The DS sequence {ai)}N has length

Nds with each al taking the value +1 or -1. Some UWB systems may employ only TH

or DS spreading and may not send any data during the acquisition stage. In those cases,

the transmitted signal is obtained by setting c = 0, al = +1 and bi = +1 accordingly.

The UWB indoor propagation channel can be modeled by a stochastic tapped delay

line [12, 23] which can be expressed in the general form in terms of its impulse response

Ntap
h(t) S pkhfk(t- tk), (2-2)
k-0

where Ntap is the number of taps in the channel response, hk is the path gain at excess

delay tk corresponding to the kth path and pk denotes the polarity of the path gain. Due

to the frequency sensitivity of the UWB channel, the pulse shapes received at different

excess delays are path-dependent [24]. The functions fk(t) model the combined effect

of the transmit and receive antennas and the propagation channel corresponding to the

kth path on the transmitted pulse. To enable tractable analysis, we assume that the pulses

are distorted identically in all the MPCs, i.e., fk(t) = f(t) and that the excess delays









tk = kTe. Under these assumptions, the impulse response can be expressed as
Ntap-1
h(t) S pkhkf(t- kTe), (2-3)
k-0

where f(t) represents the distortion to the transmitted pulse due to the antennas and

the channel, Ntap is the number of MPCs and hk is the path gain at the excess delay

(k + 1)T, normalized by the received power at one meter. The path gains are modeled

as independent, but not identically distributed random variables with Nakagami-m

distributions. The pk's are modeled as independent random variables equally likely to

take the values 1 [12]. The average energy gain for the kth path [23] is given by

Etot for k = 0
-k l+rF(c) (2-4)
Et re-(k-1)Tc/ fork 1...Ntap -
l+rF(c) "

where Etot is the total average energy in all the paths normalized to the total average

energy El received at a distance of Im, r is the ratio of the average energy of the second

MPC to the average energy of the direct path, c is the decay constant of the power delay

profile and F(e) = 1-exp[ (Nti1)T The Nakagami-m random variable hk has the

probability density function given by

2mmk h2mc-1 mITk h2
Pk (h) exp (2-5)
k F(mfk) Q k

According to Cassioli et al. [23], Etot, r and c are all modeled by lognormal distributions.

The Nakagami fading figures mk for hk are distributed according to truncated Gaussian

distributions whose mean and variance vary linearly with excess delay. These long-term

statistics are treated as constants over the duration of the acquisition process.

The received signal from a single user can then be expressed as
oO
r(t) = b /Nbja[l/Nd]Wr(t- ITf C[/Nth]Tc T) + n(t), (2-6)
1 --oc









where
Ntap -
Wr(,t)= ,, ,'.-(t- kTr) (2-7)
k-0
is the received waveform corresponding to a single pulse. Here r (t) f(t) ((t) is

the received UWB pulse. The duration of the received pulse T, is assumed to be less than

the chip duration T,. The propagation delay is denoted by 7 and n(t) is a zero mean noise

process. Given the received signal, the acquisition system attempts to retrieve the timing

offset 7.

2.2.2 Current Approaches Towards UWB Signal Acquisition

Acquisition schemes for UWB systems in the literature can be broadly classified

into those which follow detection-based approaches and those which rely on estimation-

theoretic strategies. The acquisition methods which employ a detection based approach

typically evaluate a candidate phase by first obtaining a measure of correlation between

the received signal and a locally generated template signal offset by the candidate phase.

This measure of correlation is then compared to a threshold in order to make a decision.

These candidate phases could be evaluated in a serial, parallel or hybrid manner. Among

the detection-based schemes for UWB acquisition some schemes focus exclusively on

the development of efficient search strategies to quickly evaluate the candidate phases

in the search space and certain other schemes propose two-stage acquisition methods

that achieve a reduction in the search space itself. In the estimation-based methods,

an estimate of the timing is typically obtained by maximizing a statistic over a set of

candidate phases. This statistic is usually obtained from correlation of the received signal

with a template signal. These schemes thus do not involve a threshold comparison. Most

of the estimation-based schemes attempt to exploit the cyclostationarity inherent in UWB

signaling owing to pulse repetition.

2.2.2.1 Detection-based approaches

Some of the acquisition schemes proposed for UWB signal acquisition involve the

straightforward application of traditional spread spectrum acquisition techniques.









In Blazquez et al. [25], the traditional coarse acquisition scheme where the search

space is searched in increments of a chip fraction is analyzed for the acquisition of TH

UWB signals in AWGN noise. Fig. 2-3 shows a block diagram of the scheme where

a particular phase ti in the search space is checked by correlating the received signal

with a locally generated template signal with delay ti. If the integrator output exceeds

the threshold, the phase ti is declared to be a coarse estimate of the true phase of the

received signal. If the threshold is not exceeded, the search control updates the phase

to be checked as ti+ = ti + cTp where c < 1 and Tp is the pulse width. This process

continues until the threshold is exceeded.

Received Is threshold Yes Declare ti to be
Signal Integrator exceeded? coarse estimate
signal y exceededr
of true phase
s(t-t No


Template signal Search
generator control

Figure 2-3: Block diagram of the acquisition scheme proposed by Blazquez et al.


A parallel acquisition scheme is presented in Yuanjin et al. [26] for UWB signals

spread by a Barker code of length 4, which is unreasonable considering that long

spreading sequences are needed in UWB systems to eliminate spectral lines. The output

of a matched filter matched to the received pulse is sampled at the chip rate and the

samples are then passed through four psuedonoise (PN) matched filters corresponding to

the four possible delays of the Barker sequence. The delay corresponding to the output

with largest energy is chosen as the coarse estimate of the true phase.

In Soderi et al. [27], the output of a matched filter, whose impulse response is a

time-reversed replica of the spreading code, is integrated over successive time intervals

of size mTo where 1 < m < Ntap and To is the chip duration in an attempt to combine

the energy in the multipath. The integrator output is then sampled at multiples of mTo

and compared to a threshold as illustrated in Fig. 2-4. The performance of this scheme is









evaluated in static multipath channels with 2 and 4 paths and is shown to improve mean

acquisition time performance.

mTc
Received PN matched ith Threshold To code
signal PN matched Integrator with Threshold To code
filter dwell time mT c comparison tracking loop

Figure 2-4: Block diagram of the acquisition scheme proposed by Soderi et al.


In Ma et al. [28], the non-consecutive search proposed in Shin et al. [21] and a

simpler version of the MAX/TC scheme [14] called the global MAX/TC are applied

to the acquisition of UWB signals in the presence of multipath fading and MAI. In the

non-consecutive search, only one phase in every D consecutive search space phases is

tested by correlating the received signal with a template signal with that particular phase.

The decimation factor D is chosen to be not larger than the delay spread Ntap. In the

global MAX/TC, a parallel bank of correlators is used to evaluate all the non-consecutive

phases and the phase corresponding to the correlator output with maximum energy is

chosen as the coarse estimate of the true phase.

In Zhang et al. [29], a hybrid acquisition scheme called the reduced complexity se-

quential probability ratio test (RC-SPRT) is presented for UWB signals in AWGN, which

is a modification of the multihypothesis sequential probability ratio test (MSPRT) for the

hybrid acquisition of spread spectrum signals [30]. In the MSPRT, if the sequential test

in one of the parallel correlators identifies the phase being tested as a potential true phase

the control is passed to the verification stage which verifies its decision. In the RC-SPRT,

the sequential test in each of the parallel correlators is used only to reject the hypotheses

being tested as soon as they become unlikely and replaces them with new hypotheses.

The RC-SPRT stops when all the phases except one have been rejected. This scheme

has merit at low SNRs where the time required to reject incorrect phases may be much

smaller than the time required to identify the true phase.









In Vijayakumaran et al. [31, 32], the effect of equal gain combining (EGC) on the

acquisition of UWB signals with TH spreading is investigated in a multipath environ-

ment. The acquisition problem is formulated as a binary composite hypothesis testing

problem where the set of phases where a receiver lock results in a nominal uncoded bit er-

ror probability constitute the alternate hypothesis. Two schemes based on EGC called the

square-and-integrate (SAI) and the integrate-and-square (IAS) are analyzed and compared

in [32]. The IAS scheme is similar to the one shown in Fig. 2-3 with the exception that

the template signal is given by

Nth-1
(t)= v(t- ITf ciT), (2-8)
l=0

where v(t) = O t r(t kTc), G is the length of the EGC window and rb(t) is the

receiver's estimate of the received pulse shape. Thus in IAS, EGC is done first and then

the correlator output is squared to generate the decision statistic. In SAI, the received

signal is first squared to eliminate the pulse inversion and then EGC is performed to

utilize the energy in the multipath. In this case, the template signal is once again given

by (2-8) with v(t) = J o 22(t kT). It is shown that even though EGC improves

the acquisition performance in SAI at low SNRs, the performance of IAS with no EGC is

superior to the SAI at all SNRs.

2.2.2.2 Efficient search strategies

A search strategy specifies the order in which the candidate phases in the timing

uncertainty region are evaluated by the acquisition system. When there are more than one

acquisition phases in the uncertainty region the serial search which linearly searches the

uncertainty region is no longer the optimal search strategy. More efficient nonconsecutive

search strategies called the "look-and-jump-by-K-bins" search and bit reversal search are

analyzed in the noiseless scenario with mean stopping time as the performance metric

in Homier et al. [33]. A generalized flow graph method is presented in Homier et al.

[34, 35] to compute the mean acquisition time for different serial and hybrid search









strategies. For the case when the acquisition phases are K consecutive phases in the

uncertainty region, it has been claimed that the look-and-jump-by-K-bins search is the

optimal serial search permutation when K is known and the bit reversal is the optimal

search permutation when K is unknown. Under the assumption that the probability of

detection in all the K consecutive acquisition phases is the same and with mean detection

time as the performance metric, the optimum permutation search strategy has been found

in [36] using techniques in majorization theory. Suppose that the timing uncertainty

region is divided in to bins indexed by 0,1, ,... 1. The ith position in the optimal

permutation is given by


R (i 1)K (mod N) + +1, (2-9)


where i E {1, 2,... N} and d is the greatest common divisor (GCD) of Ns and K.

2.2.2.3 Search space reduction techniques

Some acquisition schemes attempt to solve the large search space problem by

employing a two-stage acquisition strategy [37-41]. The basic principle behind all these

schemes is that the first stage performs a coarse search and identifies the true phase of

the received signal to be in a smaller subset of the search space. The second stage then

proceeds to search in this smaller subset and identifies the true phase. In Bahramgiri et

al. [37], such a two-stage scheme is proposed for the acquisition of time-hopped UWB

signals in AWGN noise and MAI. The search space is divided in to Q mutually exclusive

groups of M consecutive phases each. In the first stage, each one of the Q groups is

checked by correlating the received signal with a sum of M delayed versions of the

locally generated replica of the received signal. Once a group is identified as containing

the true phase, the phases in the group are searched by correlating with just one replica

of the received signal. A scheme based on the same principle has been developed

independently in Gezici et al. [38]. Both of these schemes have been developed under the









assumption of an AWGN channel and their performance is likely to suffer in the presence

of multipath.

In Reggiani et al. [39], an acquisition scheme for UWB signals with TH spreading

called n-scaled search is presented, where the search space is divided into groups of

M = Nf/2 where n > 1. The TH sequence used to generate the replica of the received

signal is also modified by neglecting the n least significant bits of each additional shift

cl. Although the actual scheme involves chip-rate sampling of a matched filter output, it

is equivalent to correlating the received signal with M delayed versions of the modified

replica of the received signal. In this sense, it is similar in spirit to the schemes described

above.

2.2.2.4 Estimation-based schemes

Certain approaches towards acquisition in UWB systems have employed estimation-

theoretic methods to obtain timing information of the received signal. The non-data

aided timing estimation approaches [42, 43] exploit cyclostationarity, inherent in UWB

signaling due to pulse repetition, to estimate timing information of the received signal.

These schemes require frame-rate sampling in the acquisition stage and pulse-rate

sampling during the tracking stage. The signal model assumes only TH spreading

and no polarity randomization of the pulses, i.e., al 1. It is also assumed that the

received pulses from all paths bk = (t), for k = 0, 1,..., Ntap 1, and the period

of the TH sequence is equal to a symbol duration, i.e., Nb = Nth. The timing offset is

assumed to be confined to a symbol duration and is expressed as 7 = N6Tf + c, where

N, E [0, Nth 1] and c E [0, Tf) represents the pulse-level offset. The acquisition

system estimates the frame-level timing offset by estimating N6. To do this, a sliding

correlator correlates the received signal with the template b(t) and frame-rate samples

z(n) = J( Tf !(t nTf)r(t) are obtained. Under certain conditions, it is observed that

the autocorrelation Rz(n; v) = E{z(n)z(n + v)} of z(n) is periodic in n with period Nth

and hence z(n) is a cyclostationary process. Estimates R (n; v) of R (n; v) are obtained









by sample averaging and the frame-level timing estimate is obtained by picking the peak

of the periodically time-varying correlation of the sampled correlator output [43] and is

given by

N = round{[argmax (n, v) + f]N]th (2-10)

where round{-} denotes the rounding operation. A slightly more robust approach [42, 43]

estimates the Fourier coefficients 7Ri(n, v) of the periodic sequence R (n; v) via sample

averaging which are then used to estimate the frame-level timing as

N= round [(-v (n; v) Nth) (2-11)
L2 7I Nth

where 0(n; v) = Z, (n, v). The estimation of the pulse-level timing offset C, is done

using a similar method but however requires pulse-rate sampling of the correlator output.

These schemes are conceptually illustrated in Fig. 2-5.

Sliding z(n) Estimate
r(t)-- Correlator ACF/FS Coefts. Estimate
via sample averaging N and E


V (t)

Figure 2-5: Autocorrelation function (ACF) of correlator outputs z[n] or its Fourier series
(FS) coefficients estimated via sample averaging and used to estimate timing offset.


In Tian et al. [44], a maximum likelihood (ML) timing estimation scheme is

presented for data aided and non-data aided methods and a tradeoff between acquisition

accuracy and complexity is discussed. A data-aided timing estimation scheme employing

EGC is analyzed in [44], assuming the timing offset to be less than a symbol duration,

which estimates the frame-level timing offset from the observation of M symbol

durations of the received signal as
M-1
N, argmax Y z,(N,, bl) (2-12)
l=0









where zi(N, bl) = E-1 f oJ- r(t\. (t INbTf NTf gT)dt denotes the output

of the correlator with the EGC window of length G. Another similar data-aided timing

estimation scheme is developed in [45] where the timing estimation problem is translated

to an ML amplitude estimation problem and a generalized likelihood ratio test to detect

the presence or absence of a UWB signal is developed which makes use of the ML timing

estimates in the likelihood ratio test.

Least squares estimates of the timing and the channel impulse response, using

Nyquist rate samples of the received signal, are obtained in Carbonelli et al. [46], under

the restrictive assumption that the r < Tf and is thus far from being practical. A non-data

aided timing estimation method called timing with dirty templates (TDT) is presented

in Yang et al. [47] which in the absence of inter-symbol interference (ISI), makes use

of cross-correlations between adjacent symbols to estimate timing information of the

received signal. In this scheme, a symbol-length segment of the received waveform is

used as a template and correlated with the subsequent symbol-length segment, and the

symbol-rate correlator output samples are summed over K pairs of symbols to estimate

the timing information 7, which is assumed to be within a symbol duration, as

K /2kNbTf \ 2
7 arg max S r(t)r(t NbTf)d (2-13)
E[O,NbTf) ~=1 (2k-1)NbTf

A training sequence design method for a similar data-aided scheme is presented in Yang

et al. [48]. In Wu et al. [49], a method is presented for optimizing allocation of pulses in

training and information symbols used for acquisition, channel estimation and symbol

detection.

Transform-domain methods, which obtain estimates of channel parameters employ-

ing sub-Nyquist sampling rates, are presented in MAravic et al. [50-52] where the joint

channel and timing estimation problem is translated into a harmonic retrieval problem.

These methods obtain samples Fr [n] of the Fourier transform, Fr () of the received sig-

nal and use them to estimate the excess delays tk employing standard spectral estimation









techniques. However, these schemes can estimate tkS only after the timing offset 7 is

known and hence cannot be used for timing acquisition.

In Zhang et al. [53], the Cramer-Rao lower bounds (CRLBs) for the time delay

estimation problem are derived for UWB signals in AWGN and multipath channels. It

is shown that a larger number of multipath results in higher CRLBs and a potentially

inferior performance for unbiased estimators.

2.2.2.5 Miscellaneous approaches

An acquisition strategy for impulse radio which makes use of relative timing

between pulses in specially chosen TH sequences is presented in Zhang et al. [54] in the

absence of multipath. This scheme may not be applicable in the presence of multipath

which is usually the case with UWB systems. An acquisition scheme implemented on

UWB-based positioning devices which use a coded beacon sequence in conjunction

with a bank of correlators is presented in Fleming et al. [55] and assumes absence

of multipath. A distributed synchronization algorithm for a network of UWB nodes,

motivated by results from synchronization of pulse-coupled oscillators in biological

systems such as synchronized flashing among a swarm of fireflies and synchronous

spiking of neurons, is presented in Hong et al. [56].















CHAPTER 3
ISSUES AND CHALLENGES IN THE DESIGN OF UWB ACQUISITION SYSTEMS

We present, in this chapter, the significance of the acquisition problem in UWB

communication systems and discuss the issues that distinguish the acquisition problem

in UWB systems from traditional spread spectrum systems. We discuss some of the

issues and challenges in UWB signal acquisition which may not have received sufficient

attention in the existing literature.

3.1 Hit Set

In a multipath channel, the energy corresponding to the true signal phase is spread

over several MPCs. The primary difference between the acquisition problems in a multi-

path channel and a channel without multipath is that there is more than one hypothesized

phase which can be considered a good estimate of the true signal phase. In a multipath

environment, the receiver may lock onto a non-line-of-sight (non-LOS) path and still

be able to perform adequately as long as it is able to collect enough energy. From the

viewpoint of post-acquisition receiver performance, a receiver lock to any one of such

paths can be considered successful acquisition. Thus we require a precise definition of

what can be considered a good estimate of the true signal phase.

A typical paradigm for transceiver design is the achievement of a certain nominal

uncoded bit error rate (BER) An. Then all those hypothesized phases such that a receiver

locked to them achieves an uncoded BER of An can be considered a good estimate of the

true signal phase. We define the hit set to be the set of such hypothesized phases. For a

given true phase 7, let PE(AT) denote the BER performance of the receiver when it locks

to the hypothesized phase 7 where AT = 7 7. Let Tm be the minimum SNR at which

the receiver achieves a BER of An when it locks to the LOS path, that is, PE(O) < An

when the SNR is Tn and PE(0) > An for all SNRs less than Tn. Then for an SNR










T > To and true phase r, the hit set is given by


{H = { : PE(AT) < A}. (3-1)


The hit set when a partial Rake (PRake) receiver [57] is employed for demodulation has

been derived in [32, 58] and is treated in detail in Chapter 4. Fig. 3-1 shows a plot of the

number of phases in the hit set as a function of the SNR when An = 10-2 and the PRake

receiver has N = 5 and N = 10 fingers. It is observed that the cardinality of the hit set

could be significantly large depending upon the operating SNR.

50 C----

45 NR=

40 -

35 -

30
T 25
U)


Eb/No (dB)


Figure 3-1: Effect of received SNR on size of hit set H for NR


5 and NR = 10.


A design for an acquisition system which does not take the hit set into account

can result in a significant performance degradation. For instance, in serial acquisition

schemes, such as the one shown in Fig. 2-3, the decision threshold is usually set such that

the average probability of false alarm is constrained by a small positive constant 6 < 1,

i.e.,

d = argminmax Eh[PFA(7, AT)] < (3-2)
7 T~r ri











Fig. 3-2 shows two receiver operating characteristics (ROCs) for an acquisition scheme

where the received signal is correlated with a template signal and the correlator output is

squared and compared to a threshold. The detailed derivation of the performance analysis

can be found in [32]. For one of the ROCs, the threshold was set assuming that the hit

set consists of only the true phase 7 and for the other the hit set definition in (3-1) was

used assuming a PRake receiver with NR = 5 fingers with the nominal BER requirement

An = 10-3 and the average energy received per pulse to noise ratio equal to 5 dB. When

the hit set contains only the true phase 7, the threshold needs to be set much higher in

order to prevent the decision statistics for the other phases in the multipath profile, which

have significant energy, from exceeding it. This causes the degradation in the probability

of detection when = r.



09

08-



S06

05

|04

2
|03-



01

0
ROC for singleton hit set
ROC for defined hit set
0 01 02 03 04 05 06 07 08 09 1


Figure 3-2: The ROCs when the threshold is set for a singleton hit set containing only the
true phase and for a hit set defined in (3-1) with An 10-3.



3.2 Asymptotic Acquisition Performance of Threshold-based Schemes

A typical threshold-based timing acquisition system consists of a verification stage

in which a threshold crossing at a candidate phase is checked to see if it was a false alarm

or a true detection event. The usual procedure for implementing the verification stage









is to have a large dwell time for the correlator [7]. The large dwell time increases the

effective SNR of the decision statistic and in the absence of channel fading, this results

in accurate verification, i.e., the probabilities of a false alarm and a miss can be made

arbitrarily small. However, for threshold-based acquisition schemes in multipath fading

channels it was shown [59] that no matter how large the SNR is or how we choose the

threshold it may not be possible to make the probabilities of detection and false alarm

arbitrarily small. In particular, the asymptotic performance of two typical threshold-based

acquisition schemes for TH UWB signals was calculated in [60]. It was shown that if the

threshold is such that the average probability of false alarm is less than a given tolerance,

then there is a non-trivial lower bound on the asymptotic average probability of miss.

This lower bound translates to an upper bound on the asymptotic average probability of

detection. These results suggest that it may not be possible to build a good verification

stage for UWB signal acquisition systems by just increasing the dwell time. They also

suggest that the principles underlying the design of efficient UWB signal acquisition

schemes may be very different from the traditional spread spectrum acquisition schemes.

In traditional spread spectrum acquisition systems, the decision threshold is chosen

such that the probability of false alarm in each of the non-hit set phases is small. The

verification stage helps the acquisition system recover from false alarm events when

they occur. Considering that the construction of a verification stage in some UWB signal

acquisition systems may be difficult, a more appropriate choice of decision threshold is

one which restricts the probability that the acquisition process encounters a false alarm to

be small. So if PF(7) is the average probability that the acquisition process ends in a false

alarm, then the decision threshold 7d is chosen such that Pp (7) is constrained by a small

positive constant 6 < 1,

7d = argmin PF(7) < 6. (3-3)

The performance of spread-spectrum acquisition systems has typically been characterized

by the calculation of mean acquisition time [7, 19]. In mean acquisition time calculations,









a false alarm penalty time is assumed which is the dwell time of the verification stage,

i.e., the time required by the acquisition system to recover from a false alarm event.

Thus mean acquisition time calculations implicitly assume the existence of a verification

stage. For UWB signal acquisition systems, if the threshold is set according to (3-3) the

mean detection time is a reasonable metric for system performance. The mean detection

time is defined as the average amount of time taken by the acquisition system to end in a

detection, conditioned on the non-occurrence of a false alarm event. The calculation of

the mean detection time thus does not require any assumption on the verification stage.

Finally, several detection-based schemes for UWB signal acquisition have proposed

using some form of EGC to improve the acquisition performance by combining the

energy in the multipath [37-39]. The asymptotic performance of threshold-based UWB

signal acquisition schemes using EGC has been calculated in [60]. It has been shown that

EGC may lead to a significant performance degradation.

3.3 The Search Space in UWB Signal Acquisition

The large search space in UWB signal acquisition poses significant challenges in

the design and implementation of practical systems. Most estimation-based schemes

are based on the ML principle and hence involve the simultaneous calculation of

the likelihood function corresponding to each one of the phases in the search space

followed by a maximum operation. When the search space is large, a fully parallel

implementation of this scheme is not feasible and one may have to resort to a serial or

hybrid implementation where the system calculates the likelihood functions for small

groups of phases in the search space sequentially. The likelihood functions calculated at

each intermediate step need to be stored until all the phases are evaluated. The likelihood

functions calculated at each step correspond to different noise realizations and so a simple

maximum operation may not be a good method to find the true phase especially at low

SNRs. A more robust approach might be repeated calculation of the likelihood function at

each phase followed by averaging to reduce the variations due to noise. This effectively









amounts to trading off hardware complexity for an increase in the acquisition time to

achieve similar acquisition performance. However, the performance of such reduced

complexity estimation-based acquisition schemes in terms of estimation accuracy and

acquisition time is still an open research direction.

Although detection-based schemes which evaluate the phases in the search space one

at a time have a simpler hardware implementation, they may suffer from a large mean

detection time which makes them unsuitable for high data rate applications. For instance,

the mean detection time of the serial acquisition scheme in [32] was found to be of the

order of one second. Furthermore, it was shown that the time spent by the acquisition

system in evaluating and rejecting the non-hit set phases was the dominant part of the

mean detection time causing it to decrease only marginally with increase in SNR. Thus

acquisition techniques capable of reducing the search space are crucial in the design of

efficient acquisition schemes. For example, the two-stage hybrid DS-TH scheme [58]

described in Chapter 4 achieves a mean detection time of the order of a millisecond.

Another approach to solve the search space problem is by designing the higher

layers in the network architecture carefully. A multiple access protocol which employs

continuous physical layer links in the network in order to avoid repeated acquisition

is presented in Kolenchery et al. [61]. The timing uncertainty region may be reduced

significantly if a beacon-enabled network is employed, where the medium access is

co-ordinated by a central node which periodically transmits beacons to which other nodes

synchronize and follow a slotted medium access approach.

3.4 Generalized Likelihood Ratio Test for UWB Signal Acquisition

There has not been much effort in the direction of finding the optimal detectors

for the acquisition problem in UWB systems. Most detection-based schemes for UWB

signal acquisition have been ad hoc schemes based on the principles of traditional spread

spectrum acquisition systems. In the context of the hit set and the dense multipath in

UWB systems, a reasonably systematic approach to detector design is the generalized









likelihood ratio test (GLRT). It is instructive to examine the structure of the GLRT de-

tector used by a serial acquisition system which tries to find the true phase by evaluating

the phases in the search space one at a time. Although the GLRT is not an optimal test,

it has been known to work quite well in general [62]. The GLRT has been shown to be

asymptotically uniformly most powerful among the class of invariant tests [63].

The received signal is observed over a duration of M periods of the DS sequence,

which is assumed without loss of generality to be longer than the TH sequence, and

this observation is denoted by r. The acquisition system is to determine whether a

hypothesized phase r can be considered the true phase of the received signal. It is

assumed that the hypothesized phase is a multiple of the chip duration T,. To enable

tractable analysis, it assumed that the true phase is also a multiple of T. The number of

phases the search space is thus NdsNf. From the definition of the hit set earlier, it is clear

that there exist many ways in which r can be considered to be the true phase. Without

loss of generality, suppose that the hit set is {r AbTc, r (Ab 1)T,..., r + AfTc}

where Ab and Af are integers between 0 and NdsNf/2. Also suppose that an all-ones

data training sequence is sent in the acquisition preamble. This results in a composite

hypothesis testing problem whose hypotheses can be formulated as follows:

Ho : r is not an acceptable phase, i.e., 7r SO()

HI : r is an acceptable phase, i.e, r E S(r),

whereS(r) {r AfTc, r (Af 1)T,..., + AbTc}. The GLRT is given by

max{h,vS(f)}p(r h, v) H1
A (r) = (3-4)
) max{h,v~s(f)}p(r h, v) Ho (

where the vector h, of channel gains {hk}, is assumed to be deterministic but unknown

and 7 is the decision threshold. It can be shown easily using techniques similar to those

used in [64, 65] that when n(t) is an AWGN process with power spectral density No/2,









the choices of v and h which maximize p(r I h, v) in the numerator in (3-4) are given by


T = arg max CT(v)C(v)
VES(t)
1
hi =C(Tr) (3-5)
MNds Rs,

and similarly for the denominator in (3-4)


To argmaxCT(v)C(v)
vS(t)
1
ho C(To), (3-6)
MNd&s R,

where R,, foTw (t) and C(u) = [Co(), Ci(),..., C~a (u) with

/MNdTf
Ck ) r(t)Sk(t- ) (3-7)

where
MNds-l
Sk(t V) a[1l/Nds]r(t kTc ITf cTc v)dt. (3-8)
l=0
Also, it can be easily shown that the test in (3-4) can be written as

MNdsR T hT H' No0
A(r) [h ) h o Nds [hih- hhoh] >I -<7. (3-9)
2 Ho 2

Using (3-5) and (3-6), the GLRT in (3-9) reduces to

Ntap Ntap1 H
A(r) = max C (v) max > C(v) d -N'7 (3-10)
VES(T) v S() Ho 2

The threshold 7' can be set such that the probability of false alarm PFA Pr{A(r) > 7' I

Ho} < 6, where 6 is a specified false alarm tolerance. It can be observed from (3-10) that

the test statistic given by the GLRT amounts to correlating the received signal with Ntap

different templates, each corresponding to a different MPC, summing the squared outputs

of each of these correlators, maximizing this sum for two disjoint sets of phases, and

comparing the difference to a threshold as illustrated in Fig. 3-3. This test statistic thus

attempts to collect the energy from all the MPCs through a form of equal gain combining.






29


However it is immediately clear that such an implementation is prohibitively complex to

realize. Thus other sub-optimal strategies need to be explored which would collect energy

from the MPCs in an alternative way. Simple energy detection approaches thus need to be

considered and other techniques to reduce the search space and thus the mean detection

time need to be designed. We propose and analyze two such techniques in the following

two chapters.















S+ ATc)


2






s (t-t + AT)
Np-1 : MAX

s t-t A o)



S(trtT)






s t t--- --1 ) -












Ntap-




MAX

s t- 4+ (A+)+ )








s (t- + +T )



Figure 3-3: Generalized likelihood ratio test for evaluation of phase The upper and
lower MAX operations evaluate the maximum of "I Z C k (v) over v S(T) and
v S(I), respectively.















CHAPTER 4
ACQUISITION WITH HYBRID DS-TH UWB SIGNALING

In this chapter we present a hybrid DS-TH signaling format [40, 58] for UWB

systems which enables in a significant reduction in the search space faced by the

acquisition system. The hybrid DS-TH signaling format allows the acquisition to be

performed in two stages which results in small values of the mean detection time.

4.1 System Model

4.1.1 Hybrid DS-TH Signal Format

The proposed hybrid DS-TH signaling format for UWB uses two levels of spreading.

The data symbols are first spread using a TH sequence of period Nth. The resulting time-

hopped signal is further spread using a DS sequence of period Nds. It is noted that Nds is

chosen to be relatively long compared to Nth and is a multiple of Nth, i.e. Nds = DNth

where D is an integer. The transmitted signal is a train of monocycles b(t) of energy

E/P and can be expressed as
00

l=-oO

where b(t) = (t)/ /p is a unit energy monocycle, [-] and [-J denote the integer divi-

sion remainder operation and the floor operation, respectively. The sequence {ai}1'

is a random periodic DS spreading sequence with period Nds, where al is equally likely

to be -1 or +1 and a,, and at, are independent if /1 / 12. The sequence {c} th-1 is a

random periodic TH sequence with period Nth where each sequence element ci is equally

likely to take any value in the set {0, 1,..., Nh 1} and c, and Q, are independent if

11 / 12. The data bits are denoted by the sequence {bi}. The number ofmonocycles

modulated by one bit of data is Nb. The duration of a time-hopping frame is Tth = NhTo










where To is the chip duration which is greater than the width of the transmitted mono-

cycle. The hybrid DS-TH format for UWB is illustrated in Fig. 4-1. Binary phase shift

keying (BPSK) data modulation is assumed, i.e., bi E {-1, 1}. The two-level spreading

allows us to divide the acquisition process into two stages: one for the TH sequence

and another for the DS sequence as shown in Fig. 4-2. As a result, the search space

can be significantly reduced as will be shown in the following sections. In addition, the

DS spreading on top of the TH spreading smoothes any spectral lines caused due to the

shorter periodic TH sequence.







Na h
TNd



Th

Figure 4-1: The hybrid DS-TH signal format.




Received Squaring TH spreading Ht DS spreading it
signal operation code acquisition code acquisition


Figure 4-2: Conceptual block diagram of the hybrid DS-TH two-stage acquisition
scheme.


4.1.2 Received Signal

We assume that the channel is modeled as detailed in Section 2.2.1. The received

pulse waveform is given by br (t) = f(t) (t). It is assumed that the duration of the

received pulse does not exceed the chip duration T,. With the channel response in (2-3)

the received waveform corresponding to a single pulse is

Ntap-1
(t)= ,k,,' (t -kT). (42)
k-0









The received signal as a sum of the signal and noise components is given by r(t) =

rs(t) + n(t). The signal component is given by

00 Ntap-1
rs(t)= E > bL[ja[i]Lr(t- lTth-c[1TC -T') = >E pkhksk(t), (4-3)
1=-oo k-0

where
So00
Sk(t) = bL ja [](t kTe ITth c[ Tc T), (4-4)
7b [Nd [Nth}

denotes the received signal from the kth multipath component and r is the signal delay

through the channel. The noise component, n(t), is assumed to be a zero-mean AWGN

process with two-sided power spectral density m-. It is assumed that the received pulse

waveform r (t) and the TH and DS spreading sequences are known at the receiver.
LE A NbEjEtotR b(0)
The received bit-energy-to-noise ratio is then given by a NbE Et l(0) where

R1n (u) = j~0 ,, (t ,, (t + v)dt denotes the autocorrelation of a power of the received

pulse waveform.

4.2 Hit Set Formulation

When the transmitted signal is passed through a multipath channel, more than one

hypothesized phase at the receiver may be considered a good estimate of the true phase.

A hypothesized phase can be considered as belonging to the hit set if a receiver locked

to that phase performs successful demodulation in the sense of achieving a relatively

low probability of bit error following acquisition. This way of defining the hit set by

considering post-acquisition performance of the receiver of the hit set was previously

done in [31]. It is assumed that the channel estimation block following acquisition

estimates the channel coefficients perfectly and that a partial Rake demodulator [57],

which estimates the first arriving NR (NR < Ntap) paths is used. Perfect channel

estimation is assumed to make the analysis amenable and to focus on the definition of the

hit set. Suppose that the receiver locks onto the hypothesized phase 7, which is an integer

multiple of T. To make the analysis tractable, the actual delay 7 is also assumed to be an

integer multiple of T,. Let AT = aTth +/3T, where -Nds 1 < a < Nd, 1 and









0 < ,3 < Nh 1. The decision statistic of the partial Rake demodulator for a particular

data bit (without loss of generality, consider the demodulation of bit bo) following perfect

channel estimation is given by


yb(AT; h) Rb(AT; h) + nb, (4-5)

where Rb(AT; h) is the signal component and nb is the noise component. The signal

component of the correlator output is given by
min{Ntap-1,aNh+3+NR-1}
Rb(Ar;h) = boNbVE h R(O)
k=max{0,aNh+3}
Ntap-1 minNtap-l,aNh+/+NR-1} S, Nb-1
U(1 + ,i,0)
+ Y ZY I ,o)
k1=0 k2 max{0,aNh+/3} i=-Sm 1=0
kick2 i 0

b a ar ~ ar+i]PklPk2hk" hkiX2 k( + ( +i)Nh+ C[ ,]k2 Nh+ C[ )

(4-6)

where X2(a, b) = 1 if a = b and zero otherwise, U(a, b) = 1 if a > b and zero otherwise,

Sm = [N~j 1 + 1 denotes the number of time-hop frames the multipath spread occupies

and S, = [Nl 1 + 1 denotes the number of frames occupied by the NR taps. The vector

h is a vector containing the channel parameters {pkhk}. The first term on the right-hand

side of (4-6) denotes the contribution to the signal part of the correlator by the desired

bit bo and the second term denotes the contribution arising from the inter-symbol and

inter-frame interference. Conditioned on h, the noise component of the correlator output,









nb, is a Gaussian random variable of zero mean and variance given by
min{Ntap- ,aNh ++NR-1}
I 2 S hkaJ (O)
k=max{0,aNh+3}
min{Ntap-1,aNh+3+NR-1} Nb-1
1 7 L Ids I dsI
+ Y a[ il la[ 12 1Pk Pk2hk hk2
kj,k2 max{O,cNh+P} 11,12=0
k l k2 11712

SX2(ki + llNh + c[i ], k2 + 2Nh + C[]). (4-7)
Nth Nth

Due to the random nature of the DS spreading sequence {al} for sufficiently large Nds,

the mean values of Rb(AT; h) and or averaged over all possible random sequences {a}

reasonably approximate their true values. When this averaging is done, the second term

in both (4-6) and (4-7) would be zero. Then the decision statistic yb conditioned on the

channel coefficients h, is a Gaussian random variable with mean


((AT, h)) / A NbR (0)0 (AT, h) if bo 1
E -/NbR9 (0)e(AT, h) if bo -1

and variance ab (0(AT, h)) NbR9 (0)0(AT, h), where e(AT, h) is the sum

representing the channel energy gains collected by the partial Rake demodulator,
min{Ntap- l,aNh ++NR-1}
O(AT, h) hk.
k=max{0,aNh +3}

The probability of bit error (for BPSK modulation) can then be expressed as

Pe(O(AT, h), Eb/A ) Q (p((AT h)) Q 2Ebo(AT, h) (4-8)
(yb (e(AT, h)) NoEt

where Q(x) 1/ /2 J:0 et/2dt. Using the result from [66, pp. 268-270], the average

probability of error can now be expressed as a single integral

Ee P (AT, h), 1 j e O E dii, (4-9)
E P N( Nh),o o NoEtot sin2









where
min{Ntap- 1,aNh +3+NR-1} -mlk
e(s)= n (1 -- (4-10)
k=max{0,aNh+3} 'k
is the moment generating function of e(AT; h). For a fixed number, NR, of Rake
taps, the minimum received bit-energy-to-noise ratio, (k- required to achieve a
certain nominal uncoded BER performance when the receiver locks to the strongest
path is determined. In the channel model considered, the power delay profile exhibits a
decaying path loss and the strongest path corresponds to the first arriving path. Hence

Eo [P (0((0, h), )i =) An, where An is the nominal desired uncoded BER.
Then, for Eb > (b ) the hit set is defined, for a given value of true phase 7, as the set

of phases
H {= :Ee [Pe' (AT,h), ) < An (4-11)

This definition of the hit set implies that at = () the hit set consists of

one element, i.e., the true phase of the received signal. However, for higher 1, there
could exist multiple phases in 'H which would still guarantee at least the nominal BER
performance. Hence, the cardinality of 'H increases in general with increasing -.
Fig. 4-3 illustrates the effect of deviation from the true phase on the BER performance

of the system for NR = 5. From this figure, if A, were to be chosen to be 10-2, the
corresponding (o) is approximately 11 dB.

4.3 Stage 1: TH Acquisition

The first step of the acquisition process is to acquire timing information of the time

hopping pattern. Without loss of generality, an all-ones training sequence (i.e. bi = 1) is

assumed to be transmitted in the acquisition preamble. The acquisition of the TH pattern
can be accomplished by the squaring loop illustrated in Fig. 4-4. The DS spreading in
the signal is removed by the squaring operation. A serial search strategy is employed by
the clock control when searching for the true phase. In order to evaluate the hypothesized
phase 7, the squared received signal is correlated with a locally generated reference signal














10,



10-2











4 6 8 10 12 14 16 18 20 22
(Eb/No) dB

Figure 4-3: Effect of deviation from the true phase on BER for NR = 5.


s(t T). The squared received signal can be written as


r2(t) r(t) + n(t) + 2r,(t)n(t). (4-12)


The squared signal component can be obtained from (4-3) as

Ntap-1 o
2(E) ,1 ) h 7 2 krc -1rth U[,]rc I)
th N h I
k0 -ooth

Nt,,ap-1 Ntp-1
+E1 ) )1 PkPmhkhmSk(t)Sm(t). (4-13)
kO0 mO0
kmrn
The reference signal for the first stage is given by the train of squared pulses

MNth-1
s(t- ) 2 '(t-lTth -c C T ), (4-14)
l=0th

where the dwell time of the correlator for the first stage is M periods of the TH sequence.

Constructing the reference signal for correlation in (4-14) requires knowledge of the










received pulse shape and the spreading codes {cl} and {al} at the receiver, and this

assumption has been made throughout the document.


r(t) z z' hit


s(t- T) miss

Reference TH Clock
Signal Generator Control


Figure 4-4: Squaring loop for TH pattern acquisition


4.3.1 The Decision Statistic

The decision statistic z(Ar; h) for the first stage, conditioned on the channel

coefficients h and the sequences {a } and {cl}, can be expressed as a sum of three

random variables resulting from the output of the correlator of Fig. 4-4 as

T-+MNthTth
z(A; h) r2(t)s(t )dt
T +MNth Tth fT+MNthTth


zI (AT;h) Z2 (Ar;h)
fT+MNthTth
+ n2(t)s(t- T)dt. (4-15)

Z3(A-T;h)

We can compute z (Ar; h) using (4-13) as

Ntap-1 Ntap-1 Ntap-1
zl(ATr;h) MNthE lR2(0) > rk (AT)hk+EIR 2(0) pkPm khmAk,m(AT),
k=0 k=0 m=0
kmrn
(4-16)










where rk(Ar) Nth-1 -S X2 ( + k + + and


Ak,m(AT) -
MNth-1 1 1
Sa a +iadi,+j x3(C[+i+] + k + k+i, hc[,+j] +m+ jN, c[I +,3),
l=0 i=-Sm j=-Sm Ldh

with X3(a, b, c) = if a b c and zero otherwise. The term zl(Ar; h) denotes the

contribution to the decision statistic due to the signal part alone. The number of times

the kth multipath component is collected by the correlator is thus given by MNthrk(AT)

and Ak, (AT) represents the coefficient of the cross term corresponding to the kth and

mth multipath components. The second term on the right hand side of (4-16) denotes

the contribution from inter-frame interference caused due to the multipath of one pulse

spilling over into the adjacent TH frame. Similarly, we obtain the second term of the

decision statistic, the contribution from the signal-noise cross term resulting from the

squaring operation, as

Ntap- 1 Tc
z2(AT; h) 2El, pkhkBk(AT) j '(t))n t)dt
k=

where
MNth-1 1
Bk(AT) ar[+i+.]X2(crl+i+] + k + iNh, c[L I +/3). (4-17)
[ Nd,\ Nth [Nth\
l= 0 i= -m

Finally, the third term denoting the pure noise component is given by

MNth-1
z3(Ar; h)= n2 (t) ''(t-lTth- 1 )dt. (4-18)
1=0

4.3.2 Mean and Variance of the Decision Statistic

For a given AT, conditioned on a particular channel realization h and the sequences

{al} and {cl} we approximate the decision statistic z(AT; h) by a Gaussian distribution

with mean p (Ar; h) and variance oa,(Ar; h) which are to be evaluated. Suppose that the









bandwidth of the UWB communication system is B. The above Gaussian approximation

for the output of the integrator is quite accurate when the time-bandwidth product of the

integrator, MNthTthB is large [7, pp. 240-250], which is usually true for a UWB system.

The mean of the decision statistic conditioned on {ai} and {cl} can be shown to be

pI(Ar;h {al}, {c}) =E[z(Ar; h) {a}, {cl}]
NtMp-1 NtMp-1 NtMp-1
MNthEIR,2(O) ri(AT)h + EIR,2(O) S PkOhkhmAk,(A)
i=O k=0 m=0
kem
SMNthR, (0) N
S 2(4-19)
2

The following (conditional) moments will be useful in computing the conditional

variance of the decision statistic:


E[z(AT; h) {ai}, {ci}] (E[zi(AT; h) | {al}, {c}])2,
/Ntap- 1 \ 2
E[z2(AT; h) {az}, {ci}] 2NoR93 (0) p PkhkBk(AT)) ,
\ k-

m2 2 2 T2 2
(A1; h) 1 {a}, {cfl] 4 + 0^02 MNthlM 2(O). (4-20)

It can also be shown that E[zz2] = 0 = E[z2Z3] and E[ziZ3] = E[zi]E[z3]. Using the

above results, we obtain the conditional variance of the decision statistic as

Nt/p-l )2
2(AT; h {a}, {c}) =2NoEIR 3 (0) pkhkBk(AT) + MNthR2(0).
\ k=o0
(4-21)

It is noted that the averaging done in the preceding analysis was only with respect to the

noise process n(t). To simplify the analysis, we make use of the assumptions that the

TH and DS sequences are random sequences and that Nth and Nds are sufficiently large.

Under these assumptions, the values of rk(AT), p (AT; h) and oa,(AT; h), averaged over

{al} and {cI}, reasonably approximate their actual values.









4.3.2.1 Averaging over the DS sequence

The mean and variance of the decision statistic, averaged over the DS sequence

{al}, are to be evaluated. It can be shown that for k / m, E[Ak,m(AT)] = 0 where
the averaging is done over the DS sequence. Hence the mean of the decision statistic

conditioned on {cl} for a given h, is given by

Ntp-1
p,(AT; h I {cf}) =-MNthEIR2(0) rk(AT)h + MNthR9(O)-. (4-22)
k-0

Similarly it can be shown that the variance of the decision statistic conditioned on {cl} for

a given h, is given by


2(AT;h {cI}) 2 MNaRN2(O)
Ntap-1 Ntp- 1
+ M2(Nth + Sm + 1)NoEIR0 3(0) Y Y pkpmnhkhCkmn(AT)
k=0 m=0
(4-23)

where

Nth Nth 1 1
CkmA7) Nth +Sm+ 1 X2(c[]+ k+ilN, c[i] +3)
s -Sm iI=-sm i2=-s-m
(4-24)

X2(C[Rl +fl+i2Nh2 C[r-is+1 3).
LNth] [Nth J

4.3.2.2 Averaging over the TH sequence

The mean and variance in (4-22) and (4-23) averaged over the TH sequence is to be

found. From (4-22) and (4-23), it is clear that all we need is to evaluate E[rk(AT)] and

E[Ckm(A-T)], where the averaging is done over the sequence {cl}. The average values









E[rk(A-)] have been calculated in [31] and are given by
1- min{#,k+iNh}+N/h-I1 if -a G I-SI
X2(3, k- aNh) + Ei min{ /3m=max,k+iNh} 1N if { S ,
E[rk(A-T) i i =
1i Ymin{3,k+iNh}+Nh-i1 2 otherwise.
li=-S,,m m=max{4,k+iNh} N--
(4-25)

The values E[Ckm,(A-)] can be calculated similarly, from the observation that Ckm(A-T)

in (4-24) is a sum of Bernoulli random variables X2(C[lI+ + k + ilNh, C [il +
Nth Nth
3)X2 (c[ ] + m + i2Nh, c -_ + 3), and are given by
Nth

Y1 min{/3, k+ilNh, m+i2Nh}+Nh-l 1
2 ,_--hm .Z-r=max{3, k+ilNh, m+i2Nh} Nh
I1 YminP{, k+iNh, m+iNh}+Nh- +iN1 +h-
i -,, Zrmax{3, kiNh, Tm m+iN,} if ..

[Ckm (AT) 1
2(3, k aNh) Z!2- S^ 2Zlmax{?3, m+4;iN} 7
E[CkM) X2(0 k -1 a Ymin{f3 m +iNh }+Nh -1 I
+X2(P, m alIh) Yi_ 2ina3, m+iNhl+Nh-I 1
i= -Sr Eq=maxf{3, m+iNhl }

+X2(P, k aNh)X2(/3, m aNh) otherwise.

Then the average variance of the decision statistic for a given channel realization h can be

expressed as


o2(AT; h)-
S2 Ntap-1 Ntap-1
o M NR2(0) + 2M2(Nth + S + NOEIR3() PkP khE[Ckm(AT)
k=O m=0
(4-26)

The typical values of E[Ckm(AT)] are observed to be small causing the second term of

the average variance on the right hand side of (4-26) to be negligible compared to the first

term at the values of 1 considered and the second term can be safely ignored, greatly

simplifying further analysis.









4.3.3 False Alarm and Detection Probabilities

As shown in Fig. 4-4, a hit for the first stage is the event when the correlator output

exceeds the decision threshold 71 for some E c H. A miss for the first stage occurs when

the decision statistic does not exceed 71 for all E c H. Conditioning on a particular

channel realization h, with the approximate Gaussian distribution for the decision statistic

of the first stage and given the decision threshold y7, the false alarm and detection

probabilities for a particular value of AT can be computed as

Pfal(7, AT) = Pr{z(AT; h) > 71 | i ,

Pdl(7, AT) = Pr{z(Ar; h) > 71 | e H}. (4-27)

Then the false alarm probability averaged over all channel realizations is given by

Eh[Pfal.l(71, AT)] [Q (7 /-1(AT;h)) V H. (4-28)

The average false alarm probability is computed using a similar technique employed in

[31], which expresses the false alarm probability averaged over all channel realizations in
the form of a single integral

1 1 foo IM 1 t2 t \ ji
Eh[Pf1(71, AT)- Im -e J dt, (4-29)
h /T 2 7T.I F

where
Ntp-1 0- m
OR,(W ) IH I jw.MNthElR2(O)E[rk(ATr)] Qk (4-30)
k-

is the characteristic function of Ri(AT; h) = MNthER1 2(0) Z~o 1 E[rk ( h ,

which is a weighted sum of the squares of channel taps and thus a weighted sum of

independent Gamma random variables. A similar expression can be obtained for the

average probability of detection, Eh [Pdl (1, Ar)]. The decision threshold 71 for the first

stage is chosen as discussed in Section VI.









4.4 Stage 2: DS Acquisition

When the first stage encounters a hit for a particular hypothesized phase Ti, the

acquisition process enters the second stage which deals with the acquisition of the DS

sequence. The search space for the second stage of the acquisition system consists only of

the D phases: Ti + (d 1)NthT, d 1,, 2,..., D, because one period of the DS sequence

encompasses D periods of the TH sequence and one of the above D phases would be

the true phase of the received signal unless a false alarm has occurred in the TH stage.

A false alarm occurring in the first stage would mean a false alarm event for the hybrid

DS-TH system, since the search space for the DS stage would then comprise only non-hit

set phases. The acquisition system for the second stage is shown in Fig. 4-5 in which

the received signal is correlated with a reference signal and the output of the correlator

is squared and compared to a threshold 72. The reference signal for the second stage is

given by
M'Nds-1
s'(t f E a[l/Nds]r(t lTth C[/Nh]To ), (4-31)
l=0
where the dwell time of the correlator is M' periods of the DS sequence. The output of

the integrator can be expressed as

r-+M'NdsTth Ntap-1
y(AT; h) ] r(t)st(t- f)dt M'Nd VEIR9(O) r'(AT)hk + ny,
Sk=0
(4-32)

where r (AT) E Z1 N L- a ]a[ ]X2 (c[ k i+ N, i + ).

Again, when the DS and TH sequences are sufficiently long, r'(AT) can be replaced

by its mean value, E[r'(AT)], averaged over the sequences {al} and {cl}, where

it can be shown that E[Ir(AT)] = -2(3, k aNh) if -a {-Sm,...,1} and

zero otherwise. Conditioning on a given channel realization h, y(AT; h) is a Gaus-

sian random variable with mean p,(Ar; h) R2(AT; h), where R2(AT; h) =

M'Nds ER (0) Zj"N-1 E[rI(A)]pkhk. It is straightforward to show that the average

















Figure 4-5: Acquisition system for DS stage


noise variance is given by a2 M'NdsR) (0) -. Conditioned on the channel coefficients
h and given the threshold 72, the false alarm and detection probabilities for the second
stage are given by


Pf2 72, AT)

Pd2 (72, AT)


Pr{y2(AT; h) > 72 H},

Pr{y2(AT; h) > 72 T }.


The false alarm probability averaged over the channel realizations for the second stage is
given by,


Eh [Pfa2


where ti


(2, AT)] Eh -(AT;h)Y (\//-2 ++(AT;h) -
71 Im -e tv -- dt + Im -{e 2 dr,
J \1 2 t (__ t _
te t2 12t) -35)
:1 -e O2 in ) dt (4-35)
7J t ty )

ie third equality in (4-35) is obtained from the fact that


Ntap-1
t ( I- (M'NdsVE7R(O)E[r'(AT)]u) + ( (M'NdVER()E[K(AT)w))
k=0
(4-36)
is the real-valued characteristic function of R2(Ar; h). In (4-36), ., (w) is the charac-
teristic function of the Nakagami-m random variable hk [66]. The average detection
probability, Eh [Pd2 (72, AT)], can be obtained similarly. The decision threshold 72, for the
second stage is chosen as described in the following section.


(4-33)

(4-34)









4.5 Setting Thresholds 71 and 72

The thresholds 71 and 72 are chosen such that the average probability of the two-

stage acquisition system ending in a false alarm, PFA < 6, where 6 is chosen to be

some small number. The reason for imposing this constraint on the system is made

clear in the following section which describes the use of mean detection time, instead

of mean acquisition time [7], as the system performance metric for UWB acquisition

systems. The calculation of PFA is summarized in the following. An upper bound on

this probability PjA, is also derived since PjA is easier to calculate than PFA. Suppose

that the hit set H consists of H consecutive phases and can be assumed without loss

of generality to be the set {0, Te, 2T, ..., (H 1)Tc}. Given the thresholds 71 and

72, let the average probabilities of detection for the jth phase in the hit set for the first
(TH) and second (DS) stages be denoted by PD1(j) and PD2 (j) respectively where

j E {1, 2,... H}. The average false alarm probability for the (n H)th phase in the set

of non-hit set phases for the TH stage, {HTe, (H + 1)T, ..., (N 1)To}, is denoted

by PFA1 (n) and the average false alarm probability for the (m H)th phase in the set

of non-hit set phases for the DS stage, {HTe, (H + 1)T,..., (ND 1)T}, is denoted

by PFA2(m) where n E {H + 1, H + 2,... N}, m E {H +1, H + 2,... ND} and

N = NthNh. The average probabilities of false alarm for the first and second stages

can be bounded by PFA (n) < 61 and PFA2(m) < 62 where 61 = max, PFA1(n) and

62 = maxm PFA2(m). The maximum probabilities of false alarm 61 and 62 are typically
the false alarm probabilities corresponding to phases which lie just outside the hit set.

We note that 61 and 62 are determined by the choices of 71 and 72. An overall false alarm

event for the hybrid DS-TH acquisition system can occur in two ways:

1. A false alarm event in the TH stage would mean a false alarm event for the entire

system, since following a false alarm in the TH stage, the DS stage would evaluate

only non-hit set phases.









2. There could occur a detection in the TH stage followed by a false alarm in the DS

stage.

First, the average probability that a false alarm event would occur in the DS stage,

following a detection in the TH stage at the jth hit set phase, is computed. This average

probability is denoted by PF2(j). Following the detection in the TH stage at the jth hit

set phase, the phase evaluation in the DS stage is equally likely to begin at any one of the

phases {(j 1)Tc, (j 1 + N)T,,..., (j- 1 + (D- 1)N)T}, and PF2(j) is given by:
D-1
1 1 D-I
PF2(j) PM2(J)Pf2(, 1) + Pf2(j, d) (4-37)
d=l
D-1
< -PM2(j)P2(j ) + E 2( d) 2(j). (4-38)
d=2

where PM2(j) 1 PD2(j) is the probability of miss in the DS stage at the jth hit

set phase and Pf2 (j, d) is the probability of occurrence of a false alarm in the DS stage

conditioned on a detection in the TH stage at the jth hit set phase and the first evaluation

in the DS stage beginning at the phase (j 1 + dN)To (d = 1,..., D 1), and can be

shown to be
1D-
Pf2, d) 1- PFA2(j + kN)) 1 PM2(j)
k=d i PM2(J) [ i (- PFA2(j + kN))

< 1 -(1 D-d PD2 j1 P2( d). (4-39)


Now, the average probability PF (n), of the acquisition system ending in a false

alarm conditioned only on the starting phase (n 1)To of the serial search of the TH

stage, is to be determined. The false alarm probability conditioned on each starting phase

will be the sum of the probabilities of mutually exclusive events leading to a false alarm.

There arise two cases:

Case 1: Search starts outside hit set, i.e., n E {H + 1, H + 2,..., N}









It can be shown that

S) 1 [( 1 F12(1) PM1
PFI(M) 1 (1 PFA I(k)) (
k=n 1 PM1 [kH+1( PFAI(k))

< 1 -(1 6)N-n+l 1 2() M1 P (), (4-40)
1 -M1( 61)N-H

where PF12(j) is the average probability of the DS-TH system ending in a false alarm in

the DS stage, following a detection in the TH stage, conditioned on the phase evaluation

in the TH stage starting in the jth hit set phase and is given and bounded as
H i-1
PFI2(j)= l D(k)) PDPF2(
i=j k=j
H i-1
S (1- PDk)) PD- D1 F((i) A (j), (4-41)
i=j k=j

and PM1 I= I(1 PDl(j)) is the average probability of miss for the TH stage.

Case 2: Search starts inside hit set, i.e., n E {1, 2,..., H}

In this case, it can be shown that


PiF(n) PF12(n)+ F PD

INIH 1- PF1()) PMN
[- (1- PDI() (1- PFAl(k )) 1 -- 2 -- M1
k=n k=H+1 1 PM1 (I fkH+(1 PFAI(k4))


< P N
k=n I


\ -/
-H _1 Pl2 APM1 pb1().
l -pMI(t- 61)N-H] F


(4-42)


From the bounds Pb1(n), an upper bound can be obtained on the average probability of

the hybrid DS-TH acquisition system ending in a false alarm, PFA, as


N N
PFA PF (n) < p(n) A b
n=l n=l


(4-43)









Given this upper bound, the thresholds 71 and 72 are chosen such that PjA < 6, where 6

denotes the tolerance on the average probability of the occurrence of a false alarm for the

hybrid DS-TH system.

4.6 Mean Detection Time

Typically the performance of spread-spectrum acquisition systems has been

characterized by the calculation of mean acquisition time [7, 19]. In such calculations, a

false alarm penalty time is assumed, which is the time required by the acquisition system

to recover from a false alarm event. In typical acquisition systems, this penalty time is the

long dwell time of a subsequent verification stage which attempts to confirm detection

and false alarm events with high probability. However, it has been shown in [59] that the

construction of such a verification stage for threshold-based UWB acquisition systems in

multipath fading channels is difficult. This is because of the existence of an upper bound

on the probabilities of detection even as the SNR is asymptotically increased. Hence

it is not immediately apparent how one should assign a penalty time for a false alarm

event in UWB acquisition systems. For this reason we use the mean detection time as the

metric for system performance instead of the mean acquisition time. The mean detection

time is defined as the average amount of time taken by the acquisition system to end in a

detection, conditioned on the non-occurrence of a false alarm event. The calculation of

the mean detection time thus does not require any assumption on the false alarm penalty

time. However, to ensure that the acquisition system rarely encounters a false alarm, the

probability of the acquisition system ending in a false alarm event PFA is constrained by

6 as explained in the previous section.

The search space for the first stage consists of N phases: S1 {(n 1)Tc : 1 <

n < N}. The search space for the second stage is the set of D phases: S2 = 1i +

(d 1)NthTc : d = 1, 2,..., D}, where Ti is the hypothesized phase which causes a

hit to occur in the first stage. Thus the effective size of the search space of the proposed

two stage UWB acquisition process is N + D, which is significantly smaller compared









to the search space size of ND when the spreading is done using only either a DS or

TH sequence of period Nds. The proposed hybrid DS-TH UWB signal format and the

proposed two-stage acquisition system which exploits this format result in a significant

reduction in search space. Using the flowgraph technique [7, 19], the mean detection time

for the overall system can be easily calculated and the final expression has been provided.

Fig. 4-6 shows the flowgraph for determining the mean detection time. In the

flowgraph, the jth hit set phase is shaded and labeled j. The dwell times of the first and

second stages are denoted by Ti = MNthTth and T2 = M'NadsTh. Then the mean

detection time from the flowgraph is given by


Tdet [= G(z)] (4-44)
dz

with G(z) = L 1 Gi(z), where the functions Gi(z) represent the sum of the branch

labels of all paths leading from the ith cell in the first stage to the acquisition (ACQ) state,

and are given by

NI Glom(z) jH [H G1Mm() [GDj(z)] G2(z) ifi {1,.
1- G IM (Z)GIO (Z) j "n "'l 2"
Gi(z) I ( z)o() i [ i GlMm(Z) GlDj(z)G2j(z)

+ 1M()GIO() C 1 [H 1 G1Mm(Z)] G1Dj(z)G2(z),if {1,...


(4-45)
The functions corresponding to the first stage used in the above are given by GiM(z)

=l G1Mmn(z) and Gio(z) _= HNH+1 Giom(z), where the branch labels for the first
stage are given by


G1Mmr ()

G1Dm (Z)

Giom(z)
GloDrnW


(4-46)

(4-47)

(4-48)


(1 PDI(m))z1, me {,..., H}

PDI(m)T1, me {1,..., H}

ZT1 mC{ H + 1t,... N}.


The functions corresponding to the second stage are as follows. The function G2j(z)

represents the sum of the branch labels of all paths leading to the ACQ state conditioned


,H}.


H}.


,










on a detection event in the first stage occurring at the phase j E {1,..., H} and is given

by
1 G2Dj(z) D d
GD 1 G2 ()Go() T2(D-d+l) (4-49)
D 1 G22Mj(z)G20j[) t+

where the branch labels and functions for the second stage are given by


G2Dj(Z) D2(j)z, j {1,...,H} (4-50)

G2Mj() (1- PD2(j))T2, j {1,...,H} (4-51)

G20j(z) T2(D-1), j {H+ 1,...,ND}. (4-52)



C. e




ION(



/1" z) G z
1+(d-) N GD \i
H+(d-1)N
Sj+(d-1)N
Stg IZ z / U J i/,T i
G /D



1+D-1)N G ) H+(D- )N

G2Dj)Z)



( ACQ

Figure 4-6: Flowgraph to determine mean detection time for Hybrid DS-TH A i ii -.iti, *i'
System.


4.7 Numerical Results

The performance of the proposed UWB acquisition system has been evaluated

through the calculation of the mean detection time of the overall system. The following

values for the system parameters were chosen for the calculations. The TH sequence









period Nth = 128, the period of the DS sequence Nds = 1024, i.e., D = 8, the number

of channel taps Ntap 100, Nh = 16, Nb = 8, M = 1, M' = 1/2 and To = 2 ns.

The required nominal uncoded BER performance has been set as AX = 10-2 which

corresponds to (-) = 11 dB when the number of taps in the Rake demodulator

NR = 5 and ( ) 8.5 dB when NR 10. The tolerance on the average probability
No mmi
of the acquisition system ending in a false alarm has been set has 6 = 0.05. The power

ratio has been set as r = -4 dB, the decay constant e = 16.1 dB and Etot = -20.4 dB

which corresponds do the distance of 10m between the transmitter and receiver [23]. The

Nakagami fading figures mk = 3.5 -, 0 < k < Ntap 1 are their mean values given

in [23]. In order to verify the approximations employed in the calculation, a Monte Carlo

simulation was run on the acquisition system and the average detection time was noted.

The simulations make use of the analytically computed thresholds 71 and 72.

The performance of the proposed two-stage, hybrid DS-TH acquisition system was

compared to a conventional double dwell acquisition system [7], with mean detection

time being the performance metric. In order to make a fair comparison between the

two systems, the same hybrid DS-TH signaling format was used in the double dwell

system. Thus, both the systems under comparison have similar hit sets. Each of the two

stages in the double dwell system is similar to the second (DS) stage of our proposed

hybrid two-stage system with a serial search employed. However, unlike the proposed

two-stage hybrid DS-TH system, the ordinary double dwell system does not exploit the

structure inherent in the hybrid DS-TH signaling format to aid acquisition. Moreover,

the performance of the double-dwell system is optimized by a two-dimensional search

over the variable dwell times of the two stages. Both the systems under comparison

constrain the average probability of the acquisition system ending in a false alarm,

PFA < 6 = 0.05.

Fig. 4-7 shows the effect of the received m on the size of the hit set. As the received

b is increased, the system achieves the nominal desired BER at more phases other than











the true phase and this results in increasing sizes of the hit set. The mean detection time

results for the hybrid DS-TH and the double dwell systems are shown in Fig. 4-8 and

Fig. 4-9 for NR = 5 and NR = 10 respectively where the hybrid DS-TH system is

seen to outperform the optimum double dwell system by a gain which is of the order

of D. The mean detection time of the hybrid DS-TH system obtained analytically as

well as from simulation are presented. The close accordance between the simulation and

the results from the calculation indicate that the approximations made in the analysis

are justified. Even a suboptimum choice of dwell times for the two stages in the hybrid

DS-TH acquisition system results in a much smaller mean detection time when compared

to the optimum double dwell acquisition system as seen from Fig. 4-8 and Fig. 4-9. This

advantage is primarily due to the reduction in the size of the search space achieved by

using the hybrid DS-TH signaling format and the proposed hybrid two-stage acquisition

scheme.

50
| N R5
45 -- N =10

40

35 -

30

S25


12 13
Eb/No (dB)


Figure 4-7: Effect of received SNR on size of hit set H for NR = 5 and NR =10.





















Optimum Double Dwell
Hybrid Simulation
Hybrid Analytic


14 15 16 17 18 19 20 21 22 23
E /N (dB)


Figure 4-8: Mean detection time for hybrid DS-TH and double dwell systems for Nj

5.












100
SHybrid Simulation
Hybrid Analytic
S Optimum Double Dwell





j3 101
10





| ~-- _
0i










10
14 15 16 17 18 19 20 21 22 23
E .' (dB)



Figure 4-9: Mean detection time for hybrid DS-TH and double dwell systems for Ni\

10.









4.8 System Design and Complexity Considerations

The number of monocycles modulated by one bit of data, Nb would determine

the transmission rate of the system. However the DS and TH spreading codes can

span multiple symbols, which is typically the case with UWB systems which use long

spreading codes to keep the transmit power spectral density under the FCC limit. The

ratio D = Nda/Nth would determine the reduction of the search space achieved by using

hybrid DS-TH spreading compared to a system that only uses a single long spreading

code of length Nas. The value of Nb can be independent of the ratio D.

The hit set has been defined assuming that the receiver comprises a partial Rake

demodulator which estimates NR consecutive channel taps. It is crucial to note that

although channel estimation cannot be done during the acquisition phase, the hit set

can still be defined assuming perfect channel estimation would be done subsequent

to acquisition. The effect of estimation errors on the performance of the acquisition

system is subject to further research. The threshold calculation in both the stages of

the acquisition system has been constructed such that the channel need not be known

exactly at the acquisition stage; only knowledge of the distribution of the channel taps, the

received pulse shape and the received signal-to-noise ratio 1 would suffice.

The design parameter M is the number of periods of the TH sequence NthTth

comprising the dwell time of the correlator of the first stage and M' is the number of

periods of the DS sequence NdaTth comprising the dwell time corresponding to the

second stage. In any acquisition system there is always a trade-off in increasing the dwell

time of the correlator. Although increasing the dwell time would improve the SNR of the

decision statistic thereby resulting in lower probabilities of miss and false alarm, it would

also contribute towards an increase in the detection time. In the TH stage, the designer

would typically need to choose values of M > 1 to obtain a reliable decision statistic, due

to the short period of the TH sequence. However M' can be a fraction less than 1 due to

the relatively longer period of the DS sequence.









The computational complexity of the proposed acquisition system is minimal. The

computation of the thresholds for the TH and DS stages can be done using single-variable

integration as in (4-29) and (4-35) respectively. On a final note, the hardware required

to implement both the stages of the acquisition system as shown in Fig. 4-4 and Fig. 4-5

would need both analog and digital components. The squaring operation in Fig. 4-4

would need to be implemented in analog whereas the squaring operation in Fig. 4-5 could

be done digitally after sampling the integrator output. The sampling rate required for the

first stage would be of the order of 1 Hz and that of the second stage would be of
MNthNhTc
the order of 1 Hz which are significantly lower than the symbol sampling rate of
Hz due to the TH and DS spreading sequences spanning multiple symbols.
1 Hz due to the TH and DS spreading sequences spanning multiple symbols.
Nb Nh Tc















CHAPTER 5
ACQUISITION IN TRANSMITTED REFERENCE UWB SYSTEMS

In this chapter, we discuss the acquisition problem in TR-UWB systems employing

DS spreading. When DS spreading is employed, we show that there exists a significant

relaxation in the timing requirement of the autocorrelation receiver. We exploit this

relaxation by designing a two-stage acquisition scheme for TR-UWB systems.

5.1 TR-UWB Systems

Recently, there has been a significant amount of interest in TR schemes for UWB

communication systems [64, 67-75]. In TR-UWB systems, a reference pulse is transmit-

ted for every data-modulated pulse which is then used by the receiver for demodulation.

Demodulation is typically performed using an autocorrelation receiver [69] which cor-

relates the received signal with a delayed version of itself. The delay is chosen such that

channel response corresponding to the reference pulse is correlated with the channel

response corresponding to the data-modulated pulse resulting in a capture of all the

energy in the multipath. Although the TR scheme suffers a performance degradation at

low SNRs due to usage of the noisy received signal as the correlator template [70, 75], the

fact that it enables the use of a low-complexity receiver capable of exploiting the dense

multipath in the UWB channel makes it an attractive alternative to systems which use

conventional Rake receivers.

Much of the existing literature on TR-UWB systems has focused on the performance

evaluation of optimal and suboptimal receivers [64, 69, 71, 72, 75]. The problem of

timing acquisition in TR-UWB systems has received little attention. It has been claimed

by some authors [67, 71] that the use of TR schemes mitigates the timing acquisition

problem in UWB systems. However, there has not been any analysis to support these

claims. In this chapter, we investigate the problem of acquisition for TR-UWB signals









with DS signaling. We show a two-level DS signaling structure is essential to achieve

good acquisition performance. We also propose a two-stage acquisition scheme which

exploits the TR signal structure to reduce the acquisition search space.

5.2 System Model

We assume that the propagation channel is modeled as detailed in Section 2.2.1.

In TR-UWB systems, the transmitted signal consists of a sequence of pulse pairs, each

pair consisting of a reference pulse followed by a data-modulated pulse. The modulation

format can be either pulse amplitude modulation [64] or pulse position modulation

[69]. The signal may also be spread using time-hopping or direct-sequence signaling to

eliminate spectral lines and help combat MAI. In this chapter, we consider TR-UWB

systems with DS signaling and antipodal modulation. The transmitted signal is given by
00
x(t) = d[/t i, 1 [(t ITf) + b/Nbja[l/Nds](- ITf Td)] (5-1)
1 --00

where b(t) is the UWB monocycle waveform, P is the transmitted power, Tf NfTo

is the frame duration containing one pulse pair and [.], [-L denote the integer division

remainder operation and the floor operation, respectively. The two DS sequences {di} and

{al} take values in {-1, 1} and have lengths Mds and Nds, respectively. The number of
frames modulated by a bit bi E {-1, 1} is given by Nb. The delay Td = NdTc between

the reference and modulated pulses is chosen to be larger than the multipath spread

Tm = NtapTc to avoid interference between the multipath responses of the reference

and data-modulated pulses. Here we assume that the duration of the received pulse

,', (t) = fk(t) (t) corresponding to the kth path is less than T,. The frame duration Tf

is chosen to be equal to 2Td to avoid interframe interference.










The received signal is given by


r(t) = h(t) x(t) + n(t) = rs(t) + n(t)


l=-00
(5-2)

where
Ntap-1
Wr(t) = ,.,,,,.', (t kTe). (5-3)
k-0
Here El is the total received energy at a distance of one meter from the transmitter, ,', (t)

is the received UWB pulse corresponding to the kth path which is normalized to have

unit energy, 7 is the propagation delay, and n(t) is an AWGN process with zero mean

and power spectral density _. For analytical simplicity, the MAI is modeled as a white

Gaussian random process whose effect is included in n(t). The received bit energy to

noise ratio is thus given by Eb A 2NbElEtot
No No
5.3 Two-level DS Signaling Structure

In this section, we explain why a two-level structure on the DS signaling is essential

for achieving good acquisition performance. We also briefly describe a two-stage

acquisition system which exploits the TR signal format to reduce the search space.

The main advantage of using a TR signal format for UWB systems is that the energy

in the multipath can be collected using a simple autocorrelation receiver, which correlates

the received signal with a delayed version of itself. The DS signaling consists of an outer

code {di} which modulates both the reference pulse and the data-modulated pulse and an

inner code {al} which modulates only the data-modulated pulse. To see the advantage

of this two-level DS signaling, we need to consider the effect of the delay and multiply

operation on the received signal. Fig. 5-1 illustrates the polarities due to DS signaling

and modulation of the received signal, the delayed version of the received signal and









the product of the the received signal and its delayed version1 in the absence of noise.

Note that the polarity of every alternate received pulse in the product signal depends

only on the modulating bit and the inner code {al} and is independent of the outer code

{di}. Thus if the phase of the inner code is known, the bit can be demodulated without

knowledge of the phase of the outer code {d,} by using the return-to-zero (RZ) gating

waveform shown in the figure. The product signal is multiplied with the RZ gating

waveform and integrated over a bit duration to get the decision statistic for a particular

bit. The purpose of the RZ gating waveform is two-fold: to despread the inner code {al}

and to restrict the period of integration to those times where the product signal does not

depend on the outer code. Thus the RZ gating waveform accomplishes a function similar

to the multiple integration windows employed in [69].

This method of achieving demodulation without knowledge of the outer code

{di} is advantageous because of the following reason. The distinguishing feature of

UWB systems is the wide bandwidth and the stringent spectral mask constraint imposed

by regulatory bodies. The wide bandwidth results in a fine resolution of the timing

uncertainty region while the spectral mask constraint necessitates the use of long

spreading sequences spanning multiple symbol intervals in order to remove spectral

lines resulting from the pulse repetition present in the transmitted signal. These two

features together result in a large space for the acquisition system which results in a

large acquisition time if the acquisition system evaluates phases in a serial manner and

in a prohibitively complex acquisition system if the phases are evaluated in a parallel

manner. However, by employing the two-level DS signaling the burden of eliminating

the spectral lines can be placed on the outer code {d,} by choosing its length Mds to be

large. This does not increase the size of the search space which is proportional to the



1 We shall henceforth refer to the product of the received signal and its delayed version
as the "product signal".













Polaritie




i v A \t I \A A


J
s


Transmitted


K


: do b d d1 a
I I I I


Received




Polarities
d d2 b d d a

Delayed Received
SDelayed Received


Polaritier
aobo d doaobo ab dda b a 1 a2b0 d3d2a2bo a3b


Product Si





Polaritie;
ao a1 a2 a3
SI I I I I I I I


Gating Waveform


Figure 5-1: Illustration of the delay and Itmuliptl operation on the received signal


s


s


d, d, a, t1


d da, l


d3 d3 a,


I


s



gnal









length of the inner code Nds. The inner code {al} results in a peaky ambiguity function

as in traditional spread spectrum systems and hence is essential for good acquisition

performance. In addition, the inner code helps combat MAI. Thus it can be chosen to be

of relatively smaller length compared to the outer code.

From the above discussion, it is clear that the purpose of the acquisition system is to

align (at least approximately) the RZ gating waveform with the useful part of the product

signal. Although the size of the search space is now proportional to the length of the

relatively short inner code, it can still be large owing to the fine timing resolution of the

UWB system. So we propose a two-stage acquisition system which solves the problem

ofRZ gating waveform alignment in two steps. In the first stage, the acquisition system

attempts to find the phase of the inner code modulating the product signal by correlating

the product signal with a locally generated replica of the non-return-to-zero (NRZ) inner

code DS waveform with chip duration Tf. As illustrated in Fig. 5-2, the phase of the

locally generated NRZ inner code DS waveform can suffer a large margin of error and

still be successful in despreading the inner code corresponding to the useful part of the

product signal, when the DS codes have ideal autocorrelation properties. This allows the

first stage to evaluate phases in increments proportional to the margin of error. Once the

approximate phase of the inner code is found in the first stage, the second stage of the

acquisition system proceeds to align the RZ gating waveform with the useful part of the

product signal by searching serially through the phases around the estimated phase. A

more precise description of the two-stage acquisition system can be found in Section 5.5.

5.4 Hit Set Definition

In a multipath channel, there may be multiple phases where a receiver lock can be

considered successful acquisition. Thus we require a precise definition of what can be

considered a good estimate of the true signal phase. A typical paradigm for transceiver

design is the achievement of a certain nominal uncoded bit error rate (BER) An. Then all

those hypothesized phases such that a receiver locked to them achieves an uncoded BER
























aobo dldoaobo albe
I I I


d2dialbo a2bo d3d2a2bo a3b
I I I I


Polarities


I I


Product Signal


l I/ / ,Polarities

a, a2 a3
I I I I


Inner code DS waveform


Figure 5-2: Illustration of the margin of error tolerable in the despreading of the inner
code in the first stage.


Margin of error


\--,









of An can be considered a good estimate of the true signal phase. We define the hit set to

be the set of such hypothesized phases. This way of defining the hit set was first proposed

in [32, 58].

We assume that the autocorrelation receiver of Fig. 5-3 is used for demodulation.

For a given true phase 7, let Pe(AT) denote the BER performance of the autocorrelation

receiver when it locks to the hypothesized phase T, where AT = 7. Let (b mi

be the minimum received bit energy to noise ratio at which the autocorrelation receiver

achieves a BER of An when it locks to the LOS path, i.e., Pe(O) < An when the SNR

is E )b and Pe(0) > An for all SNRs less than b) Then for an SNR >
is ) min N min No
(b )i and true phase 7, the hit set is given by

H { : Pe(AT) < An}. (5-4)


Received Integrate over \ Decision
Signal bit duration


Delay Td x)


Gating waveform

Figure 5-3: Block diagram of the autocorrelation receiver.


To completely characterize the hit set, we need to calculate the error performance of

an autocorrelation receiver which is locked to a particular hypothesized phase 7. Since

the goal of the acquisition stage is coarse synchronization, the hypothesized phase can be

assumed to be an integer multiple of T. To make the analysis tractable, we assume that

the true phase is also an integer multiple of T. Thus AT = (Nf + 3)Tc where a, f3 are

integers and 0 < f < Nf 1. This corresponds to the assumption that the tracking loop

locks to the true phase 7 if the hypothesized phase 7 is within T, of it, i.e., AT < T,. If

AT > T, it is assumed that the tracking loop locks to the nearest multipath component.









In order to demodulate the bit bk, the receiver correlates the received signal with

a product of the delayed received signal and the RZ gating waveform Ck(t 7)

lktNb a[L/Nds]Pm(t Tf Td T), where 7 is the receiver's estimate of the true
phase and pm(t) = 1 for t E [0, Tm) and zero otherwise.

As explained in Section 5.3, the RZ gating waveform does not depend on the outer

DS code {di}. This is because {di} modulates both the reference and data-modulated

pulse (see (5-2)) and is thus removed by the delay and multiply operation.

To determine the hit set, it is necessary to obtain the probability of bit error as a

function of the timing offset AT. Without loss of generality, consider the demodulation of

bit bo. The decision statistic at the correlator output, conditioned on a particular channel

realization h, is given by

1 +NbTf
Y(A; h) r(t)r(t Td)co(t T)dt (5-5)
Yb Jf

which can be expressed as a sum of signal and noise components as


Y(AT;h) r(t)(t Td)co(t T)dt + +NbTf (t)n(t Td)CO(t )dt

Yi(Ar;h) Y2(AT;h)
1 f-+NbTf +NbTf


Y3(Ar;h) Y4(ALr;h)
(5-6)

with Yi(Ar; h) denoting the signal component and {Y (Ar; h)}i-2 comprising the

noise components. Suppose that the bandwidth of the UWB system is B. When the time

bandwidth product of the integrator NbTfB is large, the decision statistic Y(AT; h) can

be approximated by a Gaussian distribution [69, 73] with mean pb(AT; h) and variance

o-b(AT; h), which are to be evaluated.









The mean of the decision statistic, conditioned on the channel realization h, can be

shown to be approximately


/fb (AT; h)

El -Nb -l b ,N tap-1 2
Nb 1= 0 b(l+)/NbJa[l/Nds]a[(l+ )/Nds] zk=7t3


E NNb- tap-
b 2 l=0o 1 b[(l+ /N /Nbbja[l \ [ I "/Nd ]d[( / i I k=)/3-



E -b-1 b bj-Nf+Ntap-1 2
E E 1o b(l(++l)/Nbja[l/Nd,]a[(+a+l )/Nds] Zk=N+Ntap


0


-P3-Nd+Ntap-1

if 0 < 3 < Nd,
-1
Nd h2
-Nd nk,


if Nd <3Nf t,

otherwise.


To simplify the analysis, we assume that the codes {ai} and {di} have been ran-

domly generated with each element equally likely to be 1. Then when Nb is sufficiently

large, the mean of the decision statistic, conditioned on the channel realization h, can be

approximately expressed as

boEl YNta h2 if a = 0 and 0 < 3 < Nd 1

Pb(AT; h) boE, ENf+Nti-1 h2 if a -1 and Nf Ntap < 3 < Nf 1

0 otherwise.

The variance of the decision statistic, conditioned on the channel realization h can be

expressed as


o1 (AT; h)


EN [Y22(AT; h) + Y2 (A; h) + 2(AT; h)]

+2EN [Y2(AT; h)Y3(AT; h) + Y2(AT; h)Y4(AT; h) + Y3(AT; h)Y4(ATr()7)


since EN[Y, (AT; h)] EN[Y3 (AT; h)] EN [Y4(AT; h)] 0. Note that EN[-] in the

above equation denotes expectation with respect to the noise distribution. It can be easily








shown that
NT ++NbTf 2t tN E Ntap-1
E[Y22(AT; h)] E[Y32(AT; h)] 2= r (tct )dt = 2 h
2Nb 2 Nb k-0
(5-8)
and E[Y42 (AT; h)] NTm. Also it is easy to show that

E[Y2(AT; h)Y4(AT; h)] = E[Y3(AT; h)Y4(AT; h)] E[Y2(AT; h)Y3(AT; h)] = 0.

The noise variance of the decision statistic conditioned on a particular channel real-
ization is thus given by CT (A-; h) NT + E : h The conditional
probability of bit error is then given by Pe(Ar; h) Q b b(T ;h) )where

Q(x) (1/ ) f e-t2 /2dt. The average probability of bit error Pe(ArT)
P[ b (AT;h) I|fpo i
LH [ b (z.T;h) )J -
Here, we briefly describe the general method used to evaluate this average probabil-
ity of error. The average probabilities of false alarm and detection in the following section
are also calculated similarly. Let the function I(A, B) = Q (A + be a function of
p( v'q( -)I
the two positive random variables B z= y Nt" h and A(AT) = AT) h2, where
0 < p(AT), q(A-) < Ntap 1. The average value of the function I(.) is thus given by
E[I(.)] fo0 fo0 I(a, b)fAB(a, b)dadb, where fAB(a, b) is the joint probability density
function (p.d.f) of the random variables A and B. The random variables A and B are
each a sum of independent gamma random variables. The joint p.d.f of A and B can be
evaluated easily as fAB(a, b) = fBA(b a)fA(a) by noting that B can be expressed as
B P(A)- h2 + A + Nt%"p-1 h .
1 k-0 kk kq(A-)+l
5.5 Two-stage Acquisition Scheme for TR-UWB Signaling
In this section, we give a detailed description of the two-stage acquisition scheme
mentioned in Section 5.3. We also derive the decision statistics for the first and second
stages.
Since the demodulation stage does not depend on the phase of the outer code {di},
the timing uncertainty region is equal to the duration of the inner code and is given by









NsT, where Ns = NdsNf. As in Section 5.4, both the true phase r and the hypothesized

phase r are assumed to be integer multiples of To to make the analysis tractable. Then

r, S E Sp {nT : n E Z and 0 < n < N, 1} and Ar = r = aTf + 3To where

a and 3 are integers such that -Nds + 1 < a < Nds 1 and 0 < 3 < Nf 1. The hit

set '-H typically consists of hypothesized phases in the neighborhood of the true phase, i.e.,

- = {17 AbTc, T (Ab 1)Tc,..., r + AfTc}, where Ab and Af are non-negative

integers. The hit set thus consists of H = Ab + Af + 1 phases. The phases in the search

space can be labeled without loss of generality as Sp {0, 1,... N 1}, where the

label i denotes the phase 7 + iT. When labeled in this manner, the hit set is given by

-H= {Ns Ab,..., Ns 1,0, 1,...,Af}.
The first stage of the acquisition system evaluates the phases in the search space

Sp in increments of J = 2Ntap Nd + 1. The reason for choosing this value for

J will become apparent in the next subsection, where it will be shown that this is

exactly the size of the margin of error tolerable in despreading the inner code. Thus

by evaluating the search space in increments of size J the first stage will evaluate at

least one phase where the inner code is despread. If the first stage begins its search

in phase j of the search space Sp, the phases evaluated by the first stage are given by

S(j) = j, (j + J), (j + 2J) s,..., (j + DJ)N}, where D = [(Ns 1)/JJ and the

notation (i), = i mod Ns. The phase j where the search begins can be any element of

the search space Sp. A particular phase i in the search space is evaluated by correlating

the product signal with a replica of the inner code DS waveform which is delayed by

iTc and the correlator output is compared to a threshold. Thus the effective search space

(ESS) Si(j) of the first stage consists of D + 1 phases in the search space which are

evaluated sequentially until the threshold is exceeded.

Once the threshold is exceeded at particular phase i E Sp in the first stage,

the control is passed to the second stage which then evaluates the phases in the set

S2(i) Nd, Nd + 1,... + ,..., + d 1}. ThesetS2(i)









consists of 2Nd phases centered at the phase i and is the ESS for the second stage. In the

second stage, a particular phase i is evaluated by correlating the product signal with a

gating waveform with delay iT, and comparing the correlator output to a threshold. The

rationale behind the fine search in the second stage is that the threshold crossing in the

first stage may occur at a phase where the inner code is despread but the gating waveform

is not sufficiently aligned. The fine search tries to align the gating waveform such that a

significant portion of the useful part of the product signal is collected. The overall search

strategy is illustrated in Fig. 5-4.

First stage search strategy

| 2J Search


space
H- s(j) 2-
Second stage ESSs S2 (+J) --

Figure 5-4: Illustration of search strategy used by the two-stage TR acquisition system.


5.5.1 Decision Statistic of the First Stage

The first stage of the acquisition system attempts to despread the inner code {al}

modulating the product signal by correlating it with a replica of the inner code DS

waveform with chip duration Tf given by
00
sa(t) = a[l/Nds]pT(t -ITf) (5-9)


where pT(t) = 1 for t E [0, Tf) and zero otherwise. The decision statistic of the first stage

for a particular hypothesized phase r, conditioned on a particular channel realization h, is

given by

Z(Ar; h) + dTf r(t)r(t Td)Sa(t r)dt (5-10)
M V Nds ?









where the dwell time of the correlator is equal to M1 periods of the inner code. Again, the

decision statistic Z(AT; h) can be approximated by a Gaussian distribution with mean

pI(Ar; h) and variance Cr2(AT; h), which are to be evaluated.

Once again, by the random sequence assumption on the DS signaling sequences

{al} and {di} when Ad, is sufficiently large, the mean of the decision statistic is zero for
a I {-1, 0} and can be shown to be approximately

El Y -t- h2 if a 0 and 0 < f3 < Nd

E Np d h2 ifa 0 and Nd < < Nd + ap 1
z(AT; h) < E Nd-'1 h if a = -1 and Nd < 3 < Nd + Nap 1

El ZNtp-1 hI ifa = -1 and Nd + Ntap < 3 < f 1

0 otherwise.

From the above expression, it is clear that the signal part of the decision statistic is the

maximum when a = 0, 0 < 3 < Nd and a = -1, Nd + Ntap < 3 < Nf 1. Thus

the signal part of the decision statistic is maximum when -Tf + Tm + Td < AT < Td,

i.e. for all 7 E [7 (Td Tm), r + Td]. This set of hypothesized phase around the true

phase also corresponds to the region in the search space where the inner code modulating

the useful part of the product signal is despread. So the estimate of the true phase of the

inner code can be anywhere in this contiguous set of phases and still enable successful

despreading of the inner code. This property of first stage decision statistic allows us to

perform a coarse search of the search space in increments of size J = 2Nd Ntap + 1,

which is the number of phases in this maximum energy region. This coarse search is then

guaranteed to hit the maximum energy region atleast once no matter where in the search

space it begins.

By a calculation similar to Section 5.4, the noise variance of the decision statistic for

the first stage, conditioned on h, can be shown to be given by

NTf 3NoE1 Ntap-1
S4(AT; h) 1N 2 13 h 2 (5-11)
kO0









5.5.2 Decision Statistic for the Second Stage

In the second stage, the acquisition system tries to align the gating waveform with

the product signal so that a significant portion of the energy in the useful part of the

product signal is collected. The decision statistic for the second stage is given by


Y(AT; h) 1 J N r(t)r(t Td)c(t- -)dt (5-12)
1 Nds Jr

where c(t) is the gating waveform given by

M2Nds-l
c(t) = a[i/Nd]Pm(t- ITf Td) (5-13)
1=0

and the dwell time of the correlator of the second stage is 3 _. periods of the DS sequence

{al}. Following an analysis similar to the one in Section 5.4, the decision statistic for the

second stage Y(AT; h) conditioned on h has a Gaussian distribution with mean

E Ntp-1 h ifa 0 and <3 < Nd -

py(Ar; h) El -f+Nt-10 h2 ifa -1 and Nf Ntap < < Nf- 1

0 otherwise.

and variance
N2T N01 Ntap 1
2c(AT; h) h (5-14)
411.'Nds 1-. Nd
k=0

5.5.3 Probabilities of False Alarm and Detection

The hit set for the acquisition system and hence the second stage is H. The hit

set for the first stage is defined as the set of all those hypothesized phases which,

upon a threshold crossing event in the first stage, result in at least one phase from '

being evaluated in the second stage. The hit set for the first stage is thus a larger set of

2Nd + H 1 elements given by 'H' = {N, Nd Ab+ i,..., N 1,0,1,..., Nd+Af}.








Conditioned on the channel coefficients h and given the threshold 71, the probabili-
ties of false alarm and detection for the first stage are given by

Pfl(71, A|h) = Pr{Z(Ar; h) > <71 I ,H'},

Pdl(71, Ar|h) = Pr{Z(Ar; h) > <7 1 e -H'}. (5-15)

For the first stage, the probabilities of false alarm and detection averaged over the channel
realizations are given by,

Pfl(71, AT)- EH[Pfl(71, ATrh)] -EH Q (71 z(AT; h) H/

Pddl((11, AT) EH Pd1 (71, AT ] h)] EH 7- (A; h) IT -. (5-16)

Similarly, for the second stage the conditional probabilities of false alarm and detection
for a threshold 72 are given by

Pf2(72, Arlh) = Pr{Y(AT; h) > 72 i T },

Pd2(72, Arlh) = Pr{Y(AT; h) > 72 | e -H}. (5-17)

Then the average probabilities of false alarm and detection are given by

Pf2(72, AT) -= EH [Pf2 (72, Ar|h)] EH 2 (Ar; h)

Pd2 (72, AT) -= EH[Pd2(72, Ar|h)] = EH Q 72 py(Ar; h) (5-18)


5.6 Decision Threshold Selection
The thresholds of the two stages, 71 and 72, are chosen such that the average
probability of the two-stage acquisition system ending in a false alarm, PFA(71, 72), is
small. The reason for setting the thresholds in this manner is made clear in the following
section which describes the use of mean detection time, instead of mean acquisition
time [7], as the system performance metric for UWB acquisition systems. The use of the
average probability of the acquisition system ending in a false alarm as the criterion for









decision threshold selection was first done in [32, 58]. The calculation of PFA(71,72) is

summarized in the following. In order to allow ease of notation in the sequel, we relabel,

without loss of generality, the phases in the search space as follows. The phases in the

hit set of the first stage are now denoted by 7-' = {0, 1,..., 2Nd + H 2}. The other

consecutive phases in the search space are denoted by {2Nd + H 1,... Ns 1}.

The phases belonging to the hit set of the second stage can then be denoted by H =

{Nd 1, Nd,... Nd + H 2}. The average probability of false alarm for a non-hit set

phase i 7-' for the first stage is denoted by Pfl (i) and that for a non-hit set phase j H-

for the second stage is denoted by Pf2 (j). The average probability of detection for the hit

set phase m E {0,..., 2Nd + H 2} for the first stage is denoted by Pdl(m) and the

average probability of detection for the hit set phase n E {Nd 1,..., Nd + H 2} for

the second stage is denoted by Pd2 (n).

An overall false alarm event for the two-stage TR acquisition system can occur in

two ways:

1. A false alarm event in the first stage would mean a false alarm event for the entire

system, since following a false alarm in the first stage, the second stage would

evaluate only non-hit set phases.

2. There could occur a detection event in the first stage followed by a false alarm in

the second stage.

These events need to be enumerated while computing PFA(71, 72). First, the average

probability that a false alarm event would occur in the second stage, following a detection

event in the first stage at the jth hit set phase (0 < j < 2Nd + H 2), is computed. This

average probability is denoted by PF2(j) and can be shown to be given by


1 PM2(j)Phf
PF2(j) 1- Pnf2(j)1 P(i)P(i) (5-19)

where Pnf2(j) denotes the probability of non-occurrence of a false alarm in the second

stage prior to the evaluation of a hit set phase, following a detection event in the first stage









at its jth hit set phase, and is given by

S2Nd-j1 Pf2 ((j Nd + k)N)], if0 < j < 2Nd 2
Pnf2(j)
1, if2Nd- 1 < < 2Nd+H-2

and Phf2(j) denotes the probability of non-occurrence of a false alarm in the second stage
during one complete evaluation of all non-hit set phases in its search space and is given
by

Nd--1 [1 f2 (( Nd + k)N)] if0 < j < H 1

2 (H2Nd-j- 1 [ Pf2 Nd+ k) )] (n [1 P( ( 1 + k))
if H
S2j-3Nd [1 ((Nd + k)j)], if2Nd- 1
Following a detection event at the jth hit set phase in the first stage, the probability of
miss for the second stage PM2(j) is the probability of missing all the hit set phases falling
in the search space of the second stage and is given by

n 0 o[1- Pd2(Nd+k-1)], if0 PM2 (j) H +f [ P-d2(Nd + k -1), if H j < 2Nd- 1

k j-2Nd+I [1 Pd2(Nd + k -1)] if2Nd < j < 2Nd + H -2.

Using the values PF2(j), the probability of the two-stage acquisition system encountering
a false alarm can now be calculated. The first phase k evaluated by the first stage is
assumed to lie randomly in the set of phases {0,, .., N, 1} which corresponds to
the timing uncertainty region in the absence of any timing side-information. The first
stage then evaluates the D + 1 phases in the set Si(k). Since the number of phases in
the hit set of the first stage is 2Nd + H 1 and since the first stage evaluates only one
phase among every J phases, the number of phases evaluated that lie in the hit set of the
first stage is L + 1 where L = 2Nd+H-l. Now, the average probability PFI(k), of the
acquisition system ending in a false alarm conditioned only on the starting phase k of the









first stage, is to be determined. The false alarm probability conditioned on each starting

phase will be the sum of the probabilities of mutually exclusive events leading to a false

alarm. There arise two cases:

Case 1: Search starts outside hit set, i.e., k E {2Nd + H 1,..., Ns 1}.

It can be shown that

1 P 1PF2(1, k)- Mi(k)
PFI(k) = 1 fl(k), (5-20)
1- PMl( ) Phfl(k)

where PfI (k) denotes the probability of non-occurrence of a false alarm in the first stage

prior to the evaluation of a hit set phase, conditioned on the first phase evaluation of the

first stage being at the kth phase, and is given by


Pnfl(k) (1- Pfl ((k+iJ)N)) (5-21)
i=0

and PhfI (k) denotes the probability of non-occurrence of a false alarm in the first stage

during one complete evaluation of all non-hit set phases in its search space, conditioned

on the first phase evaluation of the first stage being at the kth phase, and is given by


Phfl(k) (I P fl((k + iJ))) ] (1- P((k iJ)N)) (5-22)
i=n(k)+L+l i=0

The number of non-hit set phases evaluated by the first stage prior to the evaluation of

the first phase in the hit set is n(k) + 1 where n(k) = [LN- and PF12(j, k) is the

average probability of the acquisition system ending in a false alarm in the second stage,

following a detection in the first stage, conditioned on the phase evaluation in the first

stage starting in the jth hit set phase (k + (n(k) + j)J)Ns and is given by


PF12(j, k) = Pd ((k (n(k) + 1)J)N))

S ((k + ((k) + N P((k + ((k) + ). (523)
Pdl ((k + (n(k ) + i)J)N ) PF2((k + (n(k ) + i)J)N ). (5-23)









The probability of missing all the hit set phases falling in the search space for the first

stage, conditioned on the first phase evaluation beginning at phase k, is given by

L
PMi(k) = (1 Pdl ((k + (n(k) + 1)J)N)). (5-24)
l 1

Case 2: Search starts inside hit set, i.e., k {0, 1,..., 2Nd + H 2}.

In this case, the number of hit set phases evaluated before the first stage evaluates a

non-hit set phase is L'(k) + 1 where L'(k) = L2Nd+-k- ] It can then be shown that the

probability of the system ending in a false alarm is given by

L'(k)
PF(k) =Pa(k) + [1 dl (( +J))] b(k)) 1(k)
o 0 t PMio(k)Pb(k)
(5-25)

where the probability that the system ends in a false alarm prior to a non-hit set phase

evaluation is given by

L'(k) i-1
Pa(k) (1 dl ((k 1J)Ns)) Pdl ((k + iJ)Ns) PF2 ((k + iJ)N) (5-26)
i=0 L.=0

and the probability of non-occurrence of a false alarm during one complete evaluation of

all the non-hit set phases in the first stage is given by

L'(k)+D-L
Pb (k) n (1- Pf (( + J)N)). (5-27)
i=L'(k)+1

The probability of the system ending in a false alarm during one complete evaluation of

all the hit set phases for the first stage is given by

L'(k)+D+1 i-l
e(k) ( i- d ((k + 1J)N)) Pd ((k + J)N) PF2 ((k + J)N)
i=L'(k)+D-L+l l=0


(5-28)









and the probability of missing all the hit set phases in the search space given by

D
PMI(k) [1 Pdl ((k + (n(k) + ) J)N)] (5-29)
l=L'(k)+D-L+1

The average probability of the two-stage acquisition system ending in a false alarm,

PFA (1, 72) can now be expressed as

SNis-1
PFA(71,l72) Y PFl(k). (5-30)
k-0

The thresholds 71 and 72 are chosen such that PFA(71, 72) < 6, where 6 denotes the

tolerance on the average probability of the occurrence of a false alarm for the two-stage

acquisition system.

5.7 Mean Detection Time

The mean detection time is the average amount of time taken by the acquisition

system to end in a detection, conditioned on the non-occurrence of a false alarm event.

The calculation of the mean detection time thus does not require any assumption on the

false alarm penalty time. However, to ensure that the acquisition system rarely encounters

a false alarm, the probability of the acquisition system ending in a false alarm event

PFA(71, 72) is constrained by 6 as explained in the previous section.

The first phase evaluated by the first stage is assumed to be picked equally likely

from any of the elements in the set Sp. Conditioned on the starting phase k E Sp,

the set of phases evaluated by the first stage is thus given by Si (k), which consists of

D + 1 phases. Fig. 5-5 shows the flowgraph for the two-stage acquisition scheme. As

discussed in Section 5.5.3, the hit set for the first stage comprises those phases where a

threshold crossing in the first stage results in at least one phase from 'H being evaluated

in the second stage. The dwell times of the first and second stages are denoted by

T1 = M1NdsTf and T2 3= .VNdsTf. Then the mean detection time is given by


Tdet= -IG(z) (5-31)
dz z=1









Stage 1


Ns-1 0 1 Nd-l Nd+H-2 2Nd+I-2



J Hit Set / \ \

N +H-2
d Nd+i_2

Stage 2\ Stae2 Stage2 Stage 2 ;

Nd-1



ACQ

Figure 5-5 Flowgraph ill itr..-i;, the proposed twc J.t..-r, e -iq Ir.;it;,-,,, scheme.


with G(z) = ,'o Gk(z), where the functions Gk(z) represent the sum of the branch

labels of all paths leading from the kth cell in the first stage to the acquisition (ACQ)

state, and are given as follows

Case 1: k e {2Nd + H 1, 2Nd + H,..., Ns 1}.

G(n(k)+l)TI G1Dk (Z)
Gk(z) (5-32)
1 z(D+1-L)TIG1Mk(Z)'

The functions corresponding to the first stage used in the above are given by

L i-1
G1Dkc() ()Pdk(k)+i)) (- d(k((k)+)+)), (5-33)
i= 1 l 1

and G1MkM(Z) ZTPM1(k), where for convenience of notation, we denote k = (k +

iJ)Ns. Conditioned upon a hit in the first stage in the phase j E {0, 1,..., 2Nd + H 2},

the sum of the branch labels leading to the ACQ state are given by


G" znl(J)T2G2Dj ( (34
G2() ()T2G2 (5-34)
1 zR3(j)T G2Mj(X









where ni(j) denotes the number of non-hit set phases evaluated by the second stage prior
to the evaluation of a hit set phase, conditioned on a hit occurring in the first stage at
phase j, and is given by


i() 2Nd-j if0 nl(j) -
0 if2Nd- 1
The total number of non-hit set phases evaluated by the second stage, conditioned on a hit
occurring in the first stage at phase j, is given by

ni () if0 < < H-
n3(j) n(j) + j H + 2 if H < j < 2Nd -1
3Nd j H + 2 if 2Nd
The functions corresponding to the second stage used in the above are given by

G2Dj(z)

o z(k+)TPdNd + 2 k -1)H Io 1 [1-Pd2(Nd +k -1)], if0 -1 z(k+)TPd2(Nd + k 1) [1 -Pd2(Nd + -1)], if H Sj--2Nd+1 Z(k-2Nd+j)T2d2( k 1) -2Nd1 d2
:-2+l ^PdNd + k I- i j-2Nd+l [1 Pd2(NVd + i- 1)],
if2N2d < < 2d + H- 2.

and
Sz(J+)T2PM2(j) if0 G2Mj(z) ZHTPM2) ifH < j < 2Nd 1

z2Nd-j+H- M2(j) if 2Nd < j < 2Nd + H 2.
Case 2: k e {0,1,...,2Nd + H- 2}.

G1M k(z)G1D'k(Z) (5-
Gk(z) G1Dpk(z) Z (D+1-L)TIG1M'k()' (









where the transfer function corresponding to the paths leading to the ACQ state until the

first non-hit set phase is encountered is given by

L'(k) i-1
G1Dpk(Z) zG1, i(z)Pdl(kj) H (1 Pdl(j1)) (5-36)
i=0 1=0

and the transfer function corresponding to the paths leading to the ACQ state during one

sweep of all the hit set phases encountered for the first stage is given by

D i-1
G1D'k(Z) = zG (z)Pdl(kJ) H T1P(1- Pdl(kj)). (5-37)
i=L'(k)+D-L+1 l=L'(k)+D-L+1

The transfer function corresponding to missing the hit set phases, until the first non-hit set

phase is encountered, is given by

L'(k)
G1Mpk(Z) ZT1( Pdl(kj)), (5-38)
1=0

and the function corresponding to missing all the phases in the hit set during a sweep is

given by
L'(k)+D+l
GIM'k(Z) H ZT(l- Pdl(kf)). (5-39)
l=L'(k)+D-L+2

5.8 Numerical Results

The mean detection time Tdet is used as a performance metric to evaluate the

performance of the proposed two-stage acquisition scheme. The performance of the

proposed system is compared to that of a single-stage acquisition system which performs

a serial search over the whole search space Sp utilizing a detector which has a structure

similar to the second stage of the proposed system. It is noted that the hit set for this

single-stage acquisition system is the same as the proposed two-stage system. The

following values of the system parameters were chosen during the calculations. The

period of the inner DS sequence Nds = 32, the period of the outer DS sequence

Mds = 256, Nd = 110, Ntap = 100, M1 = 1, = 1 and To = 2 ns. The tolerance on

the average probability of the acquisition system ending in a false alarm has been set as









6 = 0.05. The power ratio has been set as r = -4 dB, the decay constant c = 16.1 dB and

Etot = -20.4 dB which corresponds do the distance of 10m between the transmitter and

receiver [23]. The Nakagami fading figures mk = 3.5 0-[-, 0 < k < Ntap 1 are their

mean values given in [23]. The required nominal uncoded BER performance has been set

as An 10-3.

The number of phases in the hit set is shown as a function of m in Fig. 5-6. As

b increases, the system achieves the nominal desired BER at more phases other than

the true phase and this results in increasing sizes of the hit set. The mean detection time

performance of the proposed two-stage system and the single-stage acquisition system is

shown in Fig. 5-7. In order to verify the approximations made in the analysis, we also

determine the mean detection time for the two systems through Monte Carlo simulation,

in which we make use of the analytically computed thresholds 71 and 72. We observe that

the approximations made in the analysis are reasonable. The improvement in performance

achieved by the two-stage scheme is due to a significant reduction in the size of the

effective search space. The single-stage system faces a large search space of size Ns and

its mean detection time is dominated by the time spent by the search in evaluating and

rejecting the large number of non-hit set phases. Thus even though the probabilities of

detection for the single-stage scheme improve with SNR, the improvement in the mean

detection time is imperceptible. On the other hand, the size of the effective search space

of the two stage scheme is just 2Nd + D + 1, an improvement of the order of Nds. Thus

the time spent by the acquisition system in evaluating and rejecting the non-hit set phases

is much smaller and the improvement in the mean detection time with increase in SNR is

apparent.













































0'
10 12 14 16 18 20 22
EbN0 (dB)


Figure 5-6: Effect of received SNR on hit set size H.






100


24 26 28 30


Single-stage Analytic
Two-stage Analytic
Single-stage Simulation
.... Two-stage Simulation


10'1


E

0
h




S 10-2


10-3
10 12 14 16 18 20 22 24 26 28 30
Eb 0 (dB)


Figure 5-7: Mean detection time for two-stage and single-stage TR-UWB acquisition

systems.















CHAPTER 6
FINE TIMING ESTIMATION

Accurate timing estimation is crucial to the performance of impulse radio ultra-

wideband (UWB) systems. The narrow pulses and low duty cycle signaling in UWB

systems place stringent timing requirements at the receiver for demodulation. In addition

to affecting the receiver's bit error rate performance [76], accurate timing information is

also essential in UWB systems incorporating precise ranging capabilities. An important

challenge faced by UWB systems is that the transmitted pulse can be distorted through

the antennas and the channel. Due to the frequency selectivity of the UWB channel, the

pulse shapes received at different excess delays are path-dependent [24]. Moreover, the

short pulses used in UWB systems result in highly resolvable multipath with a large delay

spread [12].

In this chapter, we address the timing estimation problem in UWB systems when

the receiver does not have knowledge of the received pulse shapes and the channel. We

derive maximum likelihood (ML) timing estimators and the Cramer-Rao lower bound

(CRLB) for both pilot-assisted and non-pilot-assisted scenarios. We focus on fine timing

estimation, where the timing uncertainty region is within a pulse width, and assume that

coarse synchronization has already been achieved. Efficient timing acquisition schemes

for UWB systems have been developed [77], which achieve coarse synchronization to

within a pulse duration. We compare the CRLB to the performance of the ML timing

estimator and sub-optimal timing estimation methods such as the dirty-template method

(TDT) [78] and transmitted reference (TR) [67, 69, 71] signaling, both of which do not

require knowledge of the pulse shapes or the channel at the receiver. We evaluate the

CRLB and simulate the performance of the ML timing estimator and the sub-optimal

schemes under the IEEE 802.15.3a UWB channel models described in [12].









6.1 System Model

The transmitted UWB signal consists of a train of short pulses monocycless) which

may be dithered by a time-hopping (TH) sequence to facilitate multiple access and to

reduce spectral lines. The polarities of the transmitted pulses may also be randomized

using a direct-sequence (DS) spreading code to mitigate multiple access interference

(MAI). Such a signal can be expressed as a series of UWB monocycles 0(t) of width Tp,

each occurring once in every frame of duration Tf as
Co
x(t) = bL /Nbja[l/Nd]O(t -( d Tf C[/Nth]TC), (6-1)


where Nb is the number of consecutive monocycles modulated by each data symbol bl,

Tf is the pulse repetition time, To is the chip duration which is the unit of additional time

shift provided by the TH sequence and [.], [-L denote the integer division remainder oper-

ation and the floor operation, respectively. The pseudorandom TH sequence {ci}l 0-1 has

length Nth where each Qc takes integer values between 0 and Nh 1 with Nh less than the

number of chips per frame Nf Tf/T,. The DS sequence {a}lIN-1 has length Nds with

each al taking the value +1 or -1. Some UWB systems may employ only TH (al = +1) or

only DS (c = 0) spreading and may not send any data (bl = +1) during the acquisition

stage.

The UWB indoor propagation channel can be modeled by a stochastic tapped delay

line [12, 23] which can be expressed in the general form in terms of its impulse response

Nt-1
h(t) = hkfk(t kT), (6-2)
k-0
where Nt is the number of taps in the channel response, hk is the path gain at excess

delay kT, corresponding to the kth path. The functions fk(t) model the combined effect

of the transmit and receive antennas and the propagation channel corresponding to the kth

path on the transmitted pulse. We assume that Nf > Nt, so that there is no inter-frame

interference.










The received signal from a single user can then be expressed as r(t) = rs(t) + n(t)

with
00
rT(t) = blb/Nbja[l/Nds]Wr(t -Tf C[/Nh]Tc 7), (6-3)
1 00
where
Nt-1
wr(t)= ,, ', (t kTc) (6-4)
k-0
is the received waveform corresponding to a single pulse. Here ,', (t) = fk(t) (t) is

the received UWB pulse from the kth path normalized to unit energy. The duration of

the received pulse Tw is assumed to be equal to the chip duration Tc. The propagation

delay is denoted by 7 and n(t) is a zero-mean additive white Gaussian noise process with

variance .2. Given the received signal, the synchronization system attempts to retrieve

the timing offset 7. We consider a single-user system during the following development.

The analysis can be easily extended to the case of multiple users, if the multiple-access

interference can be modeled as an additional Gaussian noise component.

6.2 Pilot-Assisted Timing Estimation

In this section, we derive a maximum likelihood timing estimation algorithm for a

UWB radio receiver in the absence of information regarding the channel and received

pulse shape, but having knowledge of the training symbols {bl}. We ignore time-hopping

for notational simplicity and the analysis can easily be extended to incorporate time-

hopping.

We approximate the received pulse in the kth path by a truncated Fourier series

expansion using L' = 2L + 1 coefficients as

L'-1
St) p, ,.(t) (6-5)
i=0









where the orthonormal basis functions are given by Oi(t) are given by

wTpw(t) ifi 0
S(t) cos(2rit/Tw)pw(t) if 1 < < L (6-6)

S sin(2r(i L)t/Tw)pw(t) if L + 1 < i < 2L.

where Eci = fT cos2(2rit/Tw)dt, Esj = fT sin2(27rit/Tw)dt for i = 1,..., L

and pw(t) = 1 for t E [0, Tw] and zero otherwise. The unit energy condition on

,', (t) therefore translates to ppkP = 1 where the vector of L' pulse coefficients

Pk [= k,0,Pk,cl, ,Pk,cL,Pk,sl, ,Pk,sL].
The received waveform corresponding to a single transmitted pulse can then be

modeled as
Nt-1 Nt-1 L'-1
wr(t) ,,(t kTw) hk .. (t kT,), (6-7)
k=O kO0 i=O
where the pulse received from the kth path is normalized to unit energy. The received

signal, given by (6-3) with c = 0 for all 1, is observed over a duration of N, symbols and

this observation is denoted by r.

6.2.1 Maximum Likelihood Timing Estimation

We denote the channel coefficient vector by h =[ho, hi,. ht- _]T. Recall

that the pulse coefficient vector of the kth path is pk [Pk,o,Pk,1, .. Pk,L'-1]T for

k E {0, 1,..., Nt 1}. For ease of notation we denote the matrix of pulse coefficients

by p = [po, P, ... PNt-1]. We note that during the development of the ML timing

estimators and the CRLB for the pilot-assisted and non-pilot-assisted cases, we assume

that the number of channel taps is known at the receiver. When Nt is not known at the

receiver, the analysis can be easily extended by assuming a maximum number Ntmax such

that Ntmax is always larger than Nt. The conditional log-likelihood function of r, given

the parameters 7, p and h is

1 1
AL(r; r, p, h) = const + hTB(p,7) -NNh'h (6-8)
O- 2









where the vector B(p, 7) = [Bo(po, T), B(po, r),..., B, N-1(pN, 1, T)]T and

Bk(pk 7) pk Ck(T). In the above, Ck(T) [Ck,O(r), Ck,(T),..., CLk,'-IT

with
Ns -
Ck,i(r) = b,(k,i,n(r)


(6-9)


where
(n+l)Nb-1
(k,i,n(T) rt) a[/\, jt f -kTw )dt. (6-10)
"To l=nNb
The ML estimates for the timing and the pulse and channel coefficients are then given by


[T0, po0.,I Nt-1, h] arg max [2hTB(p, r) NNbhTh]
{T,Po,Pl,...,PNt ,h : T>0, IIP 1 ll1}

argmax U(, p,h)- V(r,p), (6-11)
{T,Po,Pl,...,P Nt ,h : r>0, HIIP7 1}

where U(r, p, h) NN h N B(p, T) h N B(p, )] and V(T, p)

N BT(p, r)B(p, r) are quadratic forms and U(r, p, h) is minimized for h

l1Nb B(p, 7). Hence, the ML estimates for the pulse coefficients and the timing are
given by


[T, o, p1, PNt-1]


arg max
{TP,PO,P1...,P -1: r>0, HIIP llk 1}


Nt-1
arg max p Ck()]2 (6-12)
{T,po,P ...,PNt : T>0, Hpklll 1} k 0

We solve the optimization problem in (6-12) by first maximizing the argument over

po, Pi,., PNt-1 and then over r. For any value of > 0,
Nt-1
arg max C T] 2
{pO,Pl,...,PNt-l|lPk| 1} k 0
Nt-1
argmax p Ck(T)C(T) Pk with p k 1 Vk. (6-13)
{PO,P1,...,PNt-1: T>0, p kll 01} k=o
Ak (T)

In (6-13), we note that for any value of for each k E {0, 1,..., Nt 1}, the matrix

Ak T() is real, symmetric and positive definite. Therefore the kth term in the summation is


V(r, p)









maximized by setting pk = Ulk(T), where Ulk(T) is the eigenvector corresponding to the

maximum eigenvalue Alk (T) of Ak (-). The optimum value 7 is thus given by

Nt-1
7= argmax Ai~lk(). (6-14)
e[o0,Tw] k 0

The ML estimates of the pulse coefficients are then given by pk = Ulk(T) for 0 < k <

Nt-1.

6.2.2 Cramer-Rao Lower Bound

We now compute the CRLB for estimating the timing information. The parameters

to be estimated are the timing 7, the pulse coefficients po, pl,... p-1 and the channel

gain coefficients h. Owing to the unit-energy condition on the pulses, we have pk,L'-

/1 t_ 2 p ,j, and hence we instead consider the reduced-dimension vector of pulse

coefficients qk = Pk,o, ... Pk,L'-2]T for 0 < k < Nt 1. The vector of parameters

is denoted by E = [7, q qT,..., qT_ hT]T. The parameters are assumed to be

deterministic but unknown. The Fisher information matrix for the estimation of E is

given by

Jo,o Jo,i .. Jo,Nt Jo,Nt+l

Jl,0 J1,l J1,Nt J1,Nt+l
J = (6-15)

JNt,O JNt,1 .. JNt,Nt JNt,Nt+l

JNt+1,0 JNt+l,1 JNt+l,Nt JNt+l,Nt+l

The elements of the Fisher information matrix in (6-15) are as follows: 1. Jo, =

-E a 2- ] : Since n(t) is a zero-mean noise process, Jo,o is given by


Joo = r,(t; E)r,(t; O)dt. (6-16)
a, 0 To








We can obtain a simplified expression for Jo,o by observing the second derivative, with
respect to time, of the basis functions of (6-6) to be

0 ifi 0

i( = T i (t) if l 4w2(i-L)2 i L(t) ifL+l
Using (6-16) and (6-17), we can show that
4 2 Nt-1 L 2L
Jo,o = Ns22 b [ L) p (6-18)
k=O j=1 j=L+1

2. JOkl -E l(r ) for 0 < k < Nt 1 Making use of the relation
2 J o___l --LL [q'L 'c l J- --
Pk,L'-l 1 2/- 0 j, we can show that Jo,kc+l 1 l C T J+1,0 is a
1 x L' 1 vector whose ith element is given by

NsN.hb P, 2LPL if i =
,7 /_1 Tw
Ns,,. 1, 2 +ip+ p 2LpL) if 1 < i < L
[Jo,k+l1i T2 Tw Pk Tw (6-19)
-1 NNbh h 2k7(i-L)pkiL + Pki 27LPk,L
T2 TPk,L -1Tw
if L +1
where we have made use of the fact that Opk,L'- 1/pk,i -Pk,i/Pk,L'- -
3. Jo,Nt+l = -E 2AL ( J It can be shown that Jo,Nt+l JT+, ,0 0.

4. Ji+1+i -E A forO < kl,k2 < t -t 1 It can be shown that
4. Jkl+l,k2+l L q q J -- -
Jk1+1,k2+1 = 0 for ki / k2. For k = k2 k we can show that

1 F 1
Jk+1,k+ 1 s 1 I + qkqk (6-20)
S Pk,L'-1

5. Jk+1,N+[ -E A(r;l] for0 < k < N 1 : We can show that

Jk+1,Nt+l Nt+l,k+l 0.
6. JNt+l,Nt+l -E a2l(r : It is straightforward to show that the Nt x Nt
matrix, JNt+i,Nt+l= 2I.









From the above, the Fisher information matrix in (6-15) can then be expressed in the

simplified partitioned form
Jo,o E 0

J- ET G 0 (6-21)

S0OH

where E = [Jo,, Jo,2,... Jo,Nt], H = JNt+,Nt+l and

Jl,1 0 0 ... 0

0 J2,2 0 ... 0
G (6-22)


0 0 ... 0 JNt,N,

By using a standard result on the inversion of partitioned matrices, the CRLB on the

variance of any unbiased timing estimator can be expressed as


CRLB-1() [J-1], Jo,o EG-ET. (6-23)

The inverse of the matrix G can be computed by noting from (6-20) that for 0 < k < Nt,

2
-^1,f+2 [I + q(--PkL'-i q q Y) qk]


NNb -2 [I qkq] (6-24)

Hence (6-23) can be expressed as

1
CRLB(T) Nt (6-25)
k JO,kJk,k O,k

6.3 Non-pilot-assisted Timing Estimation

In this section we consider the timing estimation problem, when the training symbols

are unknown and derive the ML timing estimator and compute the CRLB.