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tx097 宽带 信号 最佳 接收 问题 分析

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tx097超宽带信号的最佳接收问题分析,tx097,宽带,信号,最佳,接收,问题,分析

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ONDESIGNINGTHEOPTIMALTEMPLATEWAVEFORMFORUWBIMPULSERADIOINTHEPRESENCEOFMULTIPATHAliTaha?KeithM?ChuggCommunicationSciencesInstituteUniversityofSouthernCaliforniaLosAngeles?CA?taha?usc?edu?chugg?usc?eduABSTRACTUsinganappropriatetemplatewaveformmatchedtothereceivedsignalallowsextractingtheenergyofthereceivedsignale?ciently?Thise?ciencybecomesvi?talforUltraWideBandwidth?UWB?ImpulseRa?diointhepresenceofmultipath?whereeachpathundergoesadi?erentchannelcausingdistortioninthereceivedpulseshapeduetoavarietyoffactorssuchasdi?erentamountsofattenuationfordi?er?entfrequencies?Insuchasituation?usingacleanideallineofsightpathsignalasatemplatemaydegradetheperformanceduetothemismatchesbe?tweenthetemplatewaveformandthereceivedsignal?Furthermore?becauseofinherent?lteringintheRFprocessing?i?e?antennas?ampli?ers?etc?itisoftendi?culttodetermineevensuchacleanlineofsightpulse?Inthispaper?algorithmsfordesigningopti?maltemplatewaveformsforUWBImpulseRadioaredevelopedandtheimprovementoveramoretradi?tionaltemplatewaveformusedforthiskindofradioisillustrated?INTRODUCTIONDuetosendingasub?nanosecondpulseineachframeperiod?impulseradioenjoysaveryhighmultipathresolutioncapabilityandaverylowdutycyclesig?nalwithhugespreadspectrumprocessinggain?Ontheotherhandultra?widebandwidthsuggeststhatthehigherfrequenciesattenuatemorethanthelowerfrequencies?causingdistortionintheshapeofthereceivedpulse?Thedelayspreadoftheimpulseradioreceivedsignalismanymanypulsedurationsevenforindoorapplications?Thesephenomenamotivateustodesignanalgorithmwhichderivesanoptimaltemplatewaveformatthereceiverthatcapturesthemostamountofenergywiththeleastnumberofcorrelations?Sincethee?ectsofthechannelaresomehowembeddedinthereceivedsig?nal?wecancomputetheoptimaltemplatewaveformbasedonthereceivedsignalonline?Thismakesouralgorithmadaptive?sincewithchangesinthechan?nel?thereceivedsignalchanges?andsodoesouropti?maltemplatebasedonthereceivedsignal?WeshowtheimprovementachievedbythisoptimaltemplatewaveformcomparedtomoretraditionalsecondorderderivativeofGaussianwaveform?byapplyingouriterativealgorithmtorealdataobtainedfrommea?surementexperimentstakenintheWirelessRadioLaboftheUniversityofSouthernCalifornia?Usingourtemplatewaveformalgorithmhelpsusadaptourtemplatetodi?erentenvironmentsbasedonthere?ceivedsignalwhichembodiesallthechannelcharac?teristics?includingthoseoftheantennasonthewave?formswhicharesometimesnotwell?understood?ThealgorithmdevelopedinthissectionisnotlimitedtoUWBsystemsonly?andcanbeappliedtoanykindofcommunicationsystem?InSectionII?optimallong?tailedtemplatewave?formdesignispresented?whichthenleadsustode?signoptimalsinglepathtemplatewaveforminSec?tionIII?ConclusionremarksaremadeinsectionIV?LONG?TAILEDTEMPLATEUsingthedigitalsamplingoscilloscope?measure?mentshavebeentakenintheWirelessRadioLaboftheUniversityofSouthernCalifornia?Thesemea?surementsareshowninFig?Thesearethere?ceivedsignalsfromapulserthatgeneratesmonocy?cles?Itisworthmentioningthateachmeasurementistheaverageof?receivedpro?lesatthesamelocationtogetamorestablemeasurementandne?glectsometransiente?ects?Wesampleeachmea?surementatarategreaterthanNyquistrateandnormalizethemtohaveunitenergy?Afterthesepro?cedures?werepresenteachmeasurementbyavector?0-7803-7496-7/02/$17.00 2002 IEEE.2002 IEEE Conference on Ultra Wideband Systems and Technologies010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)Figure?MeasurementsfromtheUWBradiore?ceivedwaveformtakenattheWirelessRadioLabattheUniversityofSouthernCalifornia?namely?ri?ri?ri?rin?t?fori?Nowwe?ndthevectorw?w?w?wn?tforwhichthefunc?tion?F?NXi?j?ri?w?j?ismaximum?Inthiscase?we?ndthenearestvectortoallthemeasurementvectorsinthesensethatitcapturesthemostenergyoutofthemeasurementsifwejustwanttodoasinglecorrelationatthereceiver?Wesetwtobeofunitenergyfornormalizationpurposes?Thisisaconstrainedoptimizationproblem?e?g?seeappendixCin?withkwk?Solvingthisoptimizationproblem?weget?A?I?w?whereA?M?MtandMisamatrixwhoseithcolumnisri?Therefore?wissimplythenormal?izedeigenvectorofmatrixAcorrespondingtoitslargesteigenvaluesinceF?wtAw?wt?w?kwk?Thisnormalizedeigenvectorissim?plytheoptimaltemplatewaveformwhenwewanttodoonlyonecorrelationagainstthereceivedsignalatthereceiver?Thisproblemcanbegeneralizedinastraightforwardmannertothecasewhenwewanttodesigntwoormoreorthogonaltemplatewaveformsthatcapturetheenergyofthereceivedsignalopti?mally?Thesolutionistheeigenvectorscorrespond?ingtothelargesteigenvaluesofmatrixA?SincethematrixAissymmetric?thesetemplatewaveformscanbeselectedorthogonal?Fig?shows?orthonor?maltemplatewaveformscorrespondingtotheninenonzeroeigenvaluesofmatrixA?The?rsttemplate?ThisisequivalenttoaLScriterionwhenkwkiscon?strainedtobeconstant?010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.20.100.10.2Time (nanoseconds)Amplitude (Volts)010200.20.100.10.2Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.40.200.20.4Time (nanoseconds)Amplitude (Volts)010200.20.100.10.2Time (nanoseconds)Amplitude (Volts)010200.500.51Time (nanoseconds)Amplitude (Volts)Figure?Orthonormaltemplatesastheoutputofthealgorithm?waveformattheupperleftcornerofthe?gurecap?tures?ofthetotalenergycontainedinallthemeasurementsbyjustonecorrelationwitheachmea?surement?Thesecondtemplateintheuppermiddleofthe?gure?captures?ofthetotalenergyoutofallthemeasurementswithjustonecorrelationwitheachmeasurement?Therestofthetemplatescap?ture?respectively?Forninemeasurements?weshouldbeabletocapturethewholeenergywithatmostninesinglelong?tailedorthonormaltemplatewaveforms?BylookingatmatrixA?weseethateachelementsomehowcomputestheaverageofthecorrelationbetweentwospeci?edcomponentsofeachmeasure?ment?overallthemeasurements?SINGLEPATHTEMPLATEAsFig?shows?therearealotofpathsinthere?ceivedsignal?Wewanttoresolvethepathsand?ndtheoptimaltemplatewaveformbasedonthesein?dividualpaths?Inthatcasewedesireashort?tailedtemplatewaveform?i?e?withsupportmuchlessthanthedelayspread?andwemayuseittodoaselectivemultiplecombining?forthemostdominantpaths?De?nethenewobjectivefunctionFasF?jr?LXj?cjw?n?nj?j?whererisatypicalreceivedwaveform?njisthedelayassociatedtoitsjthpathandcjisthecorrespondingamplitude?Astheaboveformulasuggests?weassumeonlythe?rstLdominantpathsinourmodel?Inor?dertominimizeFwithrespecttow?cj?s?andnj?s?we?rstminimizeFconditionedonagivenwaveformwasaninitialestimation?Agoodinitialestima?tioncanbethetruncatedversionofthelong?tailedtemplatewaveformobtainedinthelastsection?Af?ter?ndingtheoptimizedvaluesofcj?sandnj?sforj?Lbasedonthisinitialwaveform?weusethesevaluesofthecoe?cientscj?sanddelaysnj?sto?ndtheoptimizedwaveformwoflengthm?Nowweusethisnewoptimizedwaveformwtocomputethenewvaluesofthecoe?cientsanddelaysandwerepeatthisprocedureagainandagainuntilconver?genceoccursforthewaveformw?Thelengthofw?m?isadesignparameter?Ifweassignaverysmalllengthforthetemplatewaveform?thenitwillnotbee?ective?sinceitrequiresmorecorrelationsagainstthereceivedsignaltocapturethesameamountofenergy?Therefore?wecanchooseaninitialvalueform?andthenobtainthetemplatewaveformandcom?putethenumberofcorrelationstocaptureaspeci?edamountofenergyoutofthereceivedsignal?Thenweincreasem?andrepeatthesameprocedureagain?Ifthereductioninthenumberofcorrelationstocap?tureenergyissigni?cant?weincreasemagainuptothepointwherethereductionisnotworthchoosingalongertemplatewaveformorwegetnegligiblevaluesforthetemplatewaveformaftersomepoint?Forthecasewhenweknowthewidthofonereceivedpulse?wecansimplychoosemsuchthatitmeetsthewidthofthepulse?SoF?j?r?PLj?cjwI?n?nj?j?wheresubscriptImeanstheinitialestimationforw?Assumethereceivedsignalbeforesamplingasr?t?s?t?n?t?where?n?t?istheadditivewhiteGaussiannoisewithpowerspectrallevelofN?Thereceivedsignalr?t?consistsofseveralpathsatspe?ci?cdelaysni?i?n?i?L?andamplitudesci?s?fori?L?Assumingselectivecombiningforthe?rstLdominantpaths?weignoretherestofthepaths?r?t?LXi?ciw?t?ni?n?t?Findingthemaximumlikelihood?ML?estima?torisequivalentto?ndingtheMinimumMeanSquaredEstimates?MMSE?of?ci?sand?ni?s?be?cause?n?t?isAWGN?De?ningc?c?c?cL?tandn?n?n?nL?tandignoringtheirrelevanttermoftheaboveintegralincalculatingtheMMSEofcandnwegetthefollowingestimations?n?argmax?X?n?R?X?n?and?c?R?X?n?whereX?n?ZT?r?t?BBB?w?t?n?w?t?n?w?t?nL?CCCAdt?andthecorrelationmatrixRisR?BBB?R?n?n?R?n?n?R?n?nL?R?n?n?R?n?n?R?n?nL?R?nL?n?R?nL?n?R?nL?nL?CCCA?whereR?ni?nj?RT?w?t?ni?w?t?nj?dt?Usingthevaluesobtainedfordelaysandampli?tudesin?and?tocomputetheoptimalw?werepeatthewholeprocedurewithournewwinsteadofwIuntiltheoptimaltemplatewaveformconvergestoits?nalformat?Inordertocomputetheneww?weneedtominimizeF?jr?PLj?cjw?n?nj?j?Inthisequation?risann?vector?andwisanm?vector?andinordertowritetheaboveequa?tioncorrectly?weneedtoaddzerostoeachw?n?nj?suchthatitbecomesavectorofordern?too?i?e?w?n?nj?w?w?w?wm?twherewehaveadded?njzerosatthebeginningofthevector?andthenwehavetheunknowncoe?cients?wj?s?whicharetobedetermined?and?nallyweaddn?m?njzerostocompletethedimensionasann?vector?InordertominimizeFwithrespecttowj?sforj?m?weneedtotakethederivativesofFwithrespecttoeachwjandequatethemtozero?F?wp?LXk?c?kwp?LXk?Xl?kckcl?wp?lk?wp?lk?LXi?rni?pci?p?m?where?lk?jnk?nlj?In?setwj?forj?m?orj?Theabovelinearsystemofequationscanbesolvedforthegivenvaluesofci?sandnj?susinganystandardalgorithmforsolvingalinearsystemofequationsavailableinanynumericalcomputationbook?Thelaststepistonormalizethetemplatewave?formobtainedfromalltheabovestepsinordertomakeitofunitenergyastheconstraintofourop?timization?sothatwecancompareitsperformancewithanyotherunitenergytemplateintermsoftheamountofthecapturedenergyaftercorrelation?wopt?wkwk?wherewoptistheoptimaltemplatewaveformastheoutputofouralgorithm?Inordertodeterminewhetherthealgorithmhasconvergedtowoptornot?wecanusethefollowingcriterion?Ifkw?k?opt?w?k?optk?forsomepositive?thenstoprunningthealgorithm?otherwisecontinuefromsteptwo?Theoutputofthealgorithmafterthekthiterationhasbeendenotedbyw?k?opt?Here?dependsontheaccuracyneeded?Thesmallerthe?thebettertheapproximation?Fig?demonstratestheoutputofthealgorithm?wopt?alongwiththesecond?orderderivativeofGaussianwaveform?Fig?showsthe?owchartofthealgorithm?00.20.40.60.811.21.41.61.820.30.20.100.10.20.30.40.50.6Time (ns)Amplitude (V)Dash: SecondOrder Derivative of Gaussian WaveformSolid: Optimal Template WaveformFigure?Optimaltemplatealongwiththesecond?orderderivativeofGaussianwaveformAs?suggests?thereisanonlinearcomplexityas?sociatedtotheexhaustivesearchfor?ndingtheop?timalvaluesofthedominantpaths?sarrivaltimes?However?wecansimplyuseasuboptimallinearsearchwhenweassumeanegligibleoverlapbetweenadjacentpaths?Thiscanbeexplainedbylookingat?Inthiscase?wecanseethatmatrixRbecomesstronglydiagonal?sodoesitsinversein?There?fore?suggeststhatweneedtosearchforthosevaluesofnwherekXk?becomesmaximum?Sincewecansearchforeachdominantpathindependentlyinthiscase?thissimplymeansto?ndthosevaluesof Initial Estimation for w(t) Non-Linear Exhaustive Search for nLinear Algorithm for Finding c LinearAlgorithm for w(t)Figure?FlowchartofthealgorithmnforwhichthemagnitudeofeachcomponentofXismaximum?ThisisalinearcomplexsearchintermsofthenumberofcomponentsofX?Itisworthmention?ingthatthissuboptimalalgorithmbecomesoptimalforthecasewhenthereisnooverlapbetweentheadjacentpathsatall?Becauseoftheexcellentmulti?pathresolutioncapabilityofimpulseradioduetoitsultrawidebandwidth?wecanemploythesuboptimalalgorithmwithsomecon?dence?Sincetheresultsob?tainedbythesuboptimalalgorithmmatchthoseofoptimalalgorithmwithahighprecision?theresultspresentedherere?ectthoseobtainedbyrunningthefastlinearsuboptimalalgorithm?Runningthesub?optimalalgorithmonavariousgenerateddatausingcomputersimulationhasshownthatthealgorithmresolvesthepathssuccessfullyunderdi?erentmul?tipathscenarioswheretwodi?erentpathscanevenoverlapwitheachother?Foranygivenw?t?ateachiterationofthealgo?rithm?ci?sandni?sarethemaximumlikelihoodesti?matesoftheamplitudesanddelaysofdi?erentpaths?Speci?cally?ni?sareobtainedthroughanexhaustivesearchtominimizethemagnitudeofthedi?erencebetweenPLi?ciw?t?ni?andthereceivedwaveformr?t?Alsoatthesametime?duetotheAWGNnatureoftheproblem?ci?saremeansquaredestimationswhichminimizethemeansquarederror?Bytheseexplanations?weseethatafterestimatingtheampli?tudesanddelaysofdi?erentpaths?themean?squarederrorbecomessmallerduringanyiteration?Forthesecondpartofeachiteration?giventheestimatesofamplitudesanddelays?wecomputetheshapeofthetemplatewaveformbytakingderivativestominimizethemean?squarederror?therefore?wegetasmallermean?squarederrorafterthesecondhalfofeachiter?ation?Sincethesequenceofmean?squarederrorsisadecreasingsequenceboundedfrombelowbyzero?weconcludethatthissequenceisconvergent?RunningthealgorithmwhenL?forthemea?surementshownintheupperleftofFig?andwiththeinitialestimationastheinputtothealgorithmtobethesecond?orderderivativeofGaussianwaveform?demonstratesabout?dBimprovementintermsofthecapturedenergyoutofthethreemostdominantpathswithrespecttothatofsecondorderderivativeofGaussian?Similarresultsareobtainedbyrunningthealgorithmontherestofthemeasurements?Also?thealgorithmconvergesveryfast?andinfactaftertheseconditeration?thereisnomoreimprovement?Thisveri?estheoptimalityofourtemplatewaveformshowninFig?overthesecondorderderivativeofGaussian?Todemonstratetherobustnessofthealgorithm?weconsideraverybadinitialestimationofthetem?platewaveform?whichisjustaunitenergyrectangu?larpulse?FlatTemplateWaveform?overtheinterval?t?nanoseconds?Thecapturedenergyus?ingthistemplatewaveformisonly?percentofthetotalenergy?Fig?showsthetemplatewaveformafteronlytwoiterationsofthealgorithm?Itcap?tures?percentoftheenergyanditisidenticaltothetemplatewaveformobtainedusingthesecond?orderderivativeofGaussianastheinitialestimation?The?dBimprovementinextractingtheenergyoutofthereceivedsignalusingtheoptimaltemplatewaveformcomparedtothesecondorderderivativeofGaussian?canevenfurthermitigatethealreadylowfadingmargininUWBimpulseradio?CONCLUSIONTwoalgorithmstodesignoptimaltemplatewave?formswereintroduced?Oneforoptimalone?timecorrelationlong?tailedtemplatesandtheotherformulti?correlationshort?tailedtemplates?Weshowedhowwecandesignoptimaltemplatesinthesenseofcapturingthemostamountofenergywiththeleastnumberofcorrelationsagainstthereceivedsignalinthepresenceofmultipath?Simulationresultsveri?estherobustnessandaccuracyofouriterativeal?gorithm?Applyingouralgorithmtorealdataob?tainedfrommeasurements?wenoticethecapabilityofthealgori