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5{12 ))ISTHEVECTORINTHESETACORRESPONDINGTOTHEPERMUTATIONR 5{11 ),fork=1;2;:::;H,forallpossiblevaluesofd=GCD(Ns;H).Case1:d=H fori=1;2;:::;Ns.ItiseasytoseethatthepermutationconsistsofHconsecutiveblockseachhavingMelementswherethekthblockcanbewrittenas (k;H+k;2H+k;:::;(M1)H+k);(E{2) fork=1;2;:::;H.SinceanytwopositionsinablockareatleastH1phasesapartinSp,foranypositionpoftherstelementofthehitsetthereisexactlyonepositionintheblockwhereahitsetelementappears.Thus,foraparticularvalueofp2Sp,theithappearanceofahitsetelementisintheithblock,i.e.,(i1)M+1tp;iiMfori2f1;2;:::;Hg.Furthermore,ahitsetelementappearsineverypositionofablockexactlyHwhereeachappearancecorrespondstoadistinctvalueofpinSp.Thenwehave and wherethelastequalityfollowsfromthefactthatNk=MkandNs=NkH=MHk. 108
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Inthiscase,theithpositioninthepermutationisgivenby fori=1;2;:::;Ns. Werstcalculates1=PNsp=1tp;1.Thusweneedtoenumeratethepositionsoftherstappearanceofahitsetelementinthepermutationforallp2Sp.ConsidertherstN1+1positionsofthepermutation,namelyR1;R2;:::;RN1+1.NotethatthelocationsofthesepositionsinthesearchspacearesuchthatanytwoconsecutivepositionsarelocatedH1phasesapartandanyblockofHconsecutivephasesinthesearchspaceSpcontainsoneofthesepositions.Weclaimthatforallp2Sp,therstappearanceofahitsetelementoccursinoneofthesepositions,i.e.,tp;1N1+1.Supposethisisfalse.Thenthereexistsap2Spsuchthatwhentherstelementofthehitsetblockisinp,noneoftheHconsecutivehitsetphasesappearinR1;R2;:::;RN1+1.ThisimpliesthatthereisablockofHconsecutivephasesinSpwhichdoesnotcontainanyofR1;R2;:::;RN1+1;whichisacontradiction.AnytwopositionsintherstN1positionsofthepermutation,namelyR1;R2;:::;RN1,areatleastH1phasesapartinthesearchspace.Thusforaparticularvalueofp,ahitsetelementcanappearonlyinoneofthesepositions.SoeveryappearanceofahitsetelementintheseN1positionsisarstappearanceandcorrespondstoadistinctp2Sp.SinceahitsetelementappearsexactlyHtimesineveryRi,thereareexactlyHrstappearancesineachoneofR1;R2;:::;RN1andtherecannotbeanymoreappearances.ThisaccountsfortherstappearanceofahitsetelementcorrespondingtoN1Hdistinctp'sinSp.Bythefactthattp;1N1+1forallp2Sp,theremainingNsN1HappearanceshavetooccurinRN1+1.Thus 109
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Inordertocalculates2=PNsp=1tp;2,weconsiderthepositionsRN1+2;RN1+3;:::;RN2+1.ThelocationsofthesepositionsinthesearchspacearesuchthatanytwoconsecutivepositionsarelocatedH1phasesapartandanyblockofHconsecutivephasesinthesearchspaceSpcontainsoneofthesepositions.Hencebytheargumentinthepreviousparagraph,ahitsetelementappearsinoneofthesepositionsforallp2Sp.SincetherstappearanceofahitsetelementoccursinR1;R2;:::;RN1+1forallp2Sp,eachappearanceofahitsetelementinRN1+2;RN1+3;:::;RN2+1iseitherthesecondorthird 110
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(E{7) andusing( E{6 )wehave Thevaluesk=PNsp=1tp;kfork>2canbecalculatedusingargumentsverysimilartothoseusedincalculatings2.Finally,weget fork=1;2;:::;H.Case3:1
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d.ThenNK=bNs fork
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fork=1;2;:::;d.Notethatforj=(k1)K+i,wehave d+Nsi H%=M(k1)+Ns Thusforj=(k1)K+i,from( E{15 )and( E{16 )wehave (E{18) Thusforallpossiblevaluesofdwehaveshownthat fork=1;2;:::;H.ThenthevectorinthesetAcorrespondingtothepermutationRisgivenby (sH;sH1;:::;s1)=(rHrH1;rH1rH2;:::;r2r1;r1)=q: 113
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Wewishtoevaluatethepdfofthefollowingrandomvariable wherepkisequallylikelytobe+1or1,hkisaNakagamirandomvariablewithparametersm,andnyisazeromeanGaussianrandomvariablewithvariance2y.Itisalsogiventhatpk,hkandnyareindependent. Forconvenience,letA0=p 2Z101 A0pN(yx)dx+1 2Z101 A0pN(y+x)dx=1 A02m1emx2 22ydx+Z10x A02m1emx2 22ydx#=1 22y A20+1 22yx2+xy 2ydx+Z10x2m1em A20+1 22yx2xy 2ydx#=A1Z10x2m1eA2x2+A3xdx+A1Z10x2m1eA2x2A3xdx=A1(2A2)m(2m)eA23 8A2D2mA3 58 ],respectively.HereDp(z)istheparaboliccylinderfunctionand1F1(;;z)istheconuenthypergeometricfunction. 114
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[1] M.Z.WinandR.A.Scholtz,\Ultrawidebandwidthtimehoppingspreadspectrumimpulseradioforwirelessmultipleaccesscommunications,"IEEETrans.Commun.,vol.48,pp.679{691,Apr.2000. [2] ,\Impulseradio:Howitworks,"IEEECommun.Lett.,vol.2,pp.36{38,Feb.1998. [3] K.Siwiak,\Ultrawidebandradio:Introducinganewtechnology,"inProc.2001SpringVehicularTechnologyConf.,Rhodes,Greece,2001,pp.1088{1093. [4] M.Z.Win,X.Qiu,R.A.Scholtz,andV.O.K.Li,\ATMbasedTHSSMAnetworkformultimediaPCS,"IEEEJ.Select.AreasCommun.,vol.17,pp.824{836,May1999. [5] F.RamirezMireles,\PerformanceofultrawidebandSSMAusingtimehoppingandMaryPPM,"IEEEJ.Select.AreasCommun.,vol.19,pp.1186{1196,Jun.2001. [6] ,\OnperformanceofultrawidebandsignalsinGaussiannoiseanddensemultipath,"IEEETrans.Veh.Technol.,vol.50,pp.244{249,Jan.2001. [7] M.Z.WinandR.A.Scholtz,\Ontherobustnessofultrawidebandwidthsignalsindensemultipathenvironments,"IEEECommun.Lett.,vol.2,pp.51{53,Feb.1998. [8] ,\Characterizationofultrawidebandwidthwirelessindoorcommunicationschannel:Acommunicationtheoreticview,"IEEEJ.Select.Areas.Commun.,vol.20,pp.1613{1627,Dec.2002. [9] J.Foerster,E.Green,S.Somayazulu,andD.Leeper,\Ultrawidebandtechnologyforshortormediumrangewirelesscommunications,"IntelTech.Journal,vol.5,pp.1{11,May2001. [10] R.L.Peterson,R.E.Ziemer,andD.E.Borth,IntroductiontoSpreadSpectrumCommunications.EnglewoodClis,NJ:PrenticeHall,1995. [11] M.K.Simon,J.K.Omura,R.A.Scholtz,andB.K.Levitt,SpreadSpectrumCommunications:VolumeIII.Rockville,MD:ComputerSciencePress,1985. [12] Z.TianandG.Giannakis,\BERsensitivitytomistimingincorrelationbasedUWBreceivers,"inProc.2003IEEEGlobalTelecom.Conf.,Dec.2003,pp.441{445. [13] I.GuvencandH.Arslan,\PerformanceevaluationofUWBsystemsinthepresenceoftimingjitter,"inProc.2003IEEEConf.onUltraWidebandSys.Tech.,Reston,VA,Nov.2003,pp.136{141. [14] S.S.RappaportandD.M.Grieco,\Spreadspectrumsignalacquisition:Methodsandtechnology,"IEEECommun.Magazine,vol.22,pp.6{20,Jun.1984. 115
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[15] G.Corazza,\OntheMAX/TCcriterionforcodeacquisitionanditsapplicationtoinfrequencyselectiveDSSSMAsystems,"IEEETrans.Commun.,vol.12,pp.1173{1182,Sep.1996. [16] D.M.DiCarlo,\Multipledwellserialsearch:Performanceandapplicationtodirectsequencecodeacquisition,"IEEETrans.Commun.,vol.31,pp.650{659,May.1983. [17] A.Wald,SequentialAnalysis.NewYork,NY:JohnWiley,1947. [18] J.HolmesandC.C.Chen,\AcquisitiontimeperformanceofPNspreadspectrumsystems,"IEEETrans.Commun.,vol.25,pp.778{784,Aug.1977. [19] A.PolydorosandC.Weber,\Auniedapproachtoserialsearchspreadspectrumcodeacquisition:PartI.Generaltheory,"IEEETrans.Commun.,vol.32,pp.542{549,May1984. [20] R.R.RickandL.B.Milstein,\ParallelacquisitioninmobileDSCDMAsystems,"IEEETrans.Commun.,vol.45,pp.1466{1476,Nov.1997. [21] O.S.ShinandK.B.Lee,\UtilizationofmultipathsforspreadspectrumcodeacquisitioninfrequencyselectiveRayleighfadingchannels,"IEEETrans.Commun.,vol.49,pp.734{743,Apr.2001. [22] L.L.YangandL.Hanzo,\SerialacquisitionofDSCDMAsignalsinmultipathfadingmobilechannels,"IEEETrans.Veh.Technol.,vol.50,pp.617{628,Mar.2001. [23] R.Blazquez,P.Newaskar,andA.Chandrakasan,\Coarseacquisitionforultrawidebanddigitalreceivers,"inProc.2003IEEEIntl.Conf.onAcoustics,SpeechandSig.Proc.,HongKong,China,Apr.2003,pp.137{140. [24] S.Soderi,J.Iinatti,andM.Hamalainen,\CLPDIalgorithminUWBsynchronization,"inProc.2003Intl.WorkshoponUWBSystems,Oulu,Finland,Jun.2003,pp.759{763. [25] Y.Ma,F.Chin,B.Kannan,andS.Pasupathy,\Acquisitionperformanceofanultrawidebandcommunicationswidebandsystemoveramultipleaccessfadingchannel,"inProc.2002IEEEConf.UltraWidebandSys.Tech.,2002,pp.99{104. [26] H.Zhang,S.Wei,D.L.Goeckel,andM.Z.Win,\Rapidacquisitionofultrawidebandradiosignals,"in36thAsilomarConf.onSignals,SystemsandComputers,Nov.2002,pp.712{716. [27] C.W.BaumandV.V.Veeravalli,\Asequentialprocedureformultihypothesistesting,"IEEETrans.Info.Theory,vol.40,pp.1994{2007,Nov.1994. [28] E.A.HomierandR.A.Scholtz,\Rapidacquisitionofultrawidebandsignalsinthedensemultipathchannel,"inProc.2002IEEEConf.UltraWidebandSys.Tech.,Baltimore,MD,2002,pp.105{109.
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[29] ,\Hybridxeddwelltimesearchtechniquesforrapidacquisitionofultrawidebandsignals,"inProc.Intl.WorkshoponUltraWidebandSystems,Oulu,Finland,June2003,pp.1{5. [30] ,\Ageneralizedsignalowgraphapproachforhybridacquisitionforultrawidebandsignals,"Intl.Journ.ofWirelessInform.Networks,pp.179{191,Oct.2003. [31] H.BahramgiriandJ.Salehi,\MultipleshiftacquisitionalgorithminultrawidebandwidthframetimehoppingwirelessCDMAsystems,"inProc.13thIEEEPersonal,IndoorandMobileRadioCommun.,Lisbon,Portugal,Sept.2002,pp.1824{1828. [32] S.Aedudodla,S.Vijayakumaran,andT.F.Wong,\UltrawidebandsignalacquisitionusinghybridDSTHspreading,"IEEETrans.WirelessCommun.,vol.5,pp.2504{2515,Sep.2006. [33] J.Furukawa,Y.Sanada,andT.Kuroda,\NovelinitialacquisitionschemeforimpulsebasedUWBsystems,"inProc.2004Intl.WorkshoponUltraWidebandSystems,Kyoto,Japan,May2004,pp.278{282. [34] S.Gezici,E.Fishler,H.Kobayashi,H.Poor,andA.Molisch,\Arapidacquisitiontechniqueforimpulseradio,"inProc.2003IEEEPacicRimConf.onCommun.,Comp.andSig.Proc.,Victoria,B.C.,Canada,Aug.2003,pp.627{630. [35] L.ReggianiandG.M.Maggio,\Areducedcomplexityacquisitionalgorithmforimpulseradio,"inProc.2003IEEEConf.onUltraWidebandSys.Tech.,Reston,VA,Nov.2003,pp.131{135. [36] S.Aedudodla,S.Vijayakumaran,andT.F.Wong,\Rapidultrawidebandsignalacquisition,"inProc.2004IEEEWirelessCommun.andNetworkingConf.,Atlanta,GA,Mar.2004,pp.21{25. [37] J.Y.LeeandR.A.Scholtz,\RanginginadensemultipathenvironmentusinganUWBradiolink,"IEEEJournalonSelectedAreasinCommunications,vol.20,no.9,pp.1677{1683,Dec.2002. [38] S.Gezici,Z.Tian,G.B.Biannakis,H.Kobayashi,A.F.Molisch,H.V.Poor,andZ.Sahinoglu,\Localizationviaultrawidebandradios,"IEEESignalProcessingMagazine,vol.22,pp.70{84,July2005. [39] R.Cardinali,L.D.Nardis,M.G.D.Benedetto,andP.Lombardo,\UWBrangingaccuracyinhighandlowdatarateapplications,"IEEETransactionsonMicrowaveTheoryandTechniques,vol.54,no.4,pp.1865{1875,Jun.2006. [40] Y.Qi,H.Kobayashi,andH.Suda,\OnTimeofarrivalPositioninginaMultipathEnvironment,"IEEETrans.Veh.Technol.,vol.55,no.5,pp.1516{1526,Sep.2006.
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[41] S.Gezici,Z.Sahinoglu,A.F.Molisch,H.Kobayashi,andH.V.Poor,\Atwosteptimeofarrivalestimationalgorithmforimpulseradioultrawidebandsystems,"inMERLTechnicalReport,Dec.2005. [42] C.Falsi,D.Dardari,L.Mucchi,andM.Z.Win,\TimeofarrivalestimationforUWBlocalizersinrealisticenvironments,"EURASIPJournalonAppliedSignalProcessing,vol.2006,pp.1{13,2006. [43] I.GuvencandZ.Sahinoglu,\ThresholdbasedTOAestimationforimpulseradioUWBsystems,"inProc.IEEEInternationalConf.UWB,Zurich,Switzerland,Sep.2005,pp.420{425. [44] ,\ThresholdselectionforUWBTOAestimationbasedonkurtosisanalysis,"IEEECommun.Lett.,vol.9,pp.1025{1027,Dec.2005. [45] I.Guvenc,Z.Sahinoglu,andP.V.Orlik,\TOAestimationforIRUWBsystemswithdierenttransceivertypes,"IEEETransactionsonMicrowaveTheoryandTechniques,vol.54,no.4,pp.1876{1886,Jun.2006. [46] I.GuvencandZ.Sahinoglu,\TOAestimationwithdierentIRUWBtransceivertypes,"inProc.IEEEInternationalConf.UWB,Zurich,Switzerland,Sep.2005,pp.426{431. [47] D.Cassioli,M.Z.Win,andA.F.Molisch,\Theultrawidebandwidthindoorchannel:Fromstatisticalmodeltosimulations,"IEEEJ.Select.AreasCommun.,vol.20,pp.1247{1257,Aug.2002. [48] R.J.Cramer,R.A.Scholtz,andM.Z.Win,\Evaluationofanultrawidebandpropagationchannel,"IEEETrans.AntennasPropagat.,vol.50,pp.561{570,May2002. [49] A.F.Molisch,J.F.Foerster,andM.Pendergrass,\Channelmodelsforultrawidebandpersonalareanetworks,"IEEEWirelessCommun.Mag.,vol.10,pp.14{21,Dec2003. [50] D.Cassioli,M.Z.Win,F.Vatalaro,andA.F.Molisch,\PerformanceoflowcomplexityRakereceptioninarealisticUWBchannel,"inProc.IEEEIntl.Conf.onCommunications(ICC2002),NewYork,NY,Apr.2002,pp.763{767. [51] M.K.SimonandM.S.Alouini,DigitalCommunicationoverFadingChannels:AUniedApproachtoPerformanceAnalysis.NewYork,NY:Wiley,2000. [52] H.V.Poor,IntroductiontoSignalDetectionandEstimation,2nded.NewYork,NY:Springer,1994. [53] M.Z.WinandR.A.Scholtz,\Ontheenergycaptureofultrawidebandwidthsignalsindensemultipathenvironments,"IEEECommun.Lett.,vol.2,pp.245{247,Sep.1998.
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[54] J.GilPelaez,\Noteontheinversiontheorem,"Biometrika,vol.38,pp.481{482,1951. [55] W.Rudin,PrinciplesofMathematicalAnalysis,3rded.NewYork,NY:McGrawHill,1976. [56] P.Billingsley,ProbabilityandMeasure,3rded.NewYork,NY:WileyInterscience,1995. [57] A.W.MarshallandI.Olkin,Inequalities:TheoryofMajorizationanditsApplications.NewYork,NY:AcademicPress,1979. [58] I.S.GradshteynandI.M.Ryzhik,TableofIntegrals,Series,andProducts,5thed.London,UK:AcademicPress,2000.
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SaravananVijayakumaranreceivedtheB.Tech.degreeinelectronicsandcommunicationengineeringin2001fromtheIndianInstituteofTechnologyatGuwahatiandtheM.S.degreeinelectricalengineeringin2004fromtheUniversityofFlorida,Gainesville.HeiscurrentlypursuingthePh.D.degreeattheUniversityofFlorida.FromJan.2006toJuly2006,hewasaresearchinternattheLaboratoryforComputerCommunicationsandApplications,EcolePolytechniqueFederaledeLausanne,Lausanne,Switzerland.Hisresearchinterestsincludewirelesscommunicationsandinformationtheory. 120



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DESIGN AND PERFORMANCE OF ULTRAWIDEBAND ACQUISITION SYSTEMS By SARAVANAN VIJAYAKUMARAN 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 2007 Copyright 2007 by Saravanan Vii i,.,umaran To my teachers. ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Tan F. Wong, for his guidance and encouragement throughout my graduate studies at UF. I have learned a great deal from him not only through his instruction but also by imitation. I feel very fortunate for having had the opportunity to work with him for the past few years. I would also like to thank Dr. Michael Fang and Dr. John Shea for their guidance and the many interesting discussions. Finally, I would like to thank the Dr. Paul Robinson and Dr. Alexander Turull of the UF Mathematics Department for encouraging my interest in their courses and for their unlimited patience in answering my questions. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF TABLES . . . . . . . . . 7 LIST OF FIGURES .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. .. 8 A B ST R A C T . . . . . . . . . . 10 CHAPTER 1 INTRODUCTION .................................. 11 1.1 Brief Review of Spread Spectrum Acquisition Systems ........... 13 1.2 Previous Work on UWB Acquisition ..................... 15 1.2.1 Efficient Search Strategies ....................... 17 1.2.2 Search Space Reduction Techniques . . . . 18 1.3 Previous Work on UWB TimeofArrival Estimation .. . .. 20 1.4 Dissertation Outline . . . . . . . 22 2 PROBLEM DEFINITION . . . . . . . 25 2.1 Introduction . . . . . . . . 25 2.2 System M odel . . . . . . . . 25 2.2.1 C !i i,, I M odel . . . . . . . 25 2.2.2 Transmitted and Received Signals . . . . 26 2.3 Hit Set Definition . . . . . . . . 27 2.4 The UWB Acquisition Problem . . . . . . 30 3 ACQUISITION OF TIMEHOPPING UWB SIGNALS . . ...... 32 3.1 Introduction . . . . . . . . 32 3.2 Analysis of SAI . . . . . . . . 33 3.2.1 Derivation of the Decision Statistic . . . . 33 3.2.2 Average Probabilities of Detection and False Alarm . ... 36 3.3 Analysis of IAS . . . . . . . . 37 3.3.1 Derivation of the Decision Statistic . . . . 37 3.3.2 Average Probabilities of Detection and False Alarm . ... 38 3.4 Mean Detection Time Analysis of Serial Search . . . 39 3.5 Numerical Results . . . . . . . . 42 3.6 Conclusions . . . . . . . . 45 4 ASYMPTOTIC PERFORMANCE OF THRESHOLDBASED ACQUISITION SYSTEM \ IS IN MULTIPATH FADING CHANNELS .... . . 54 4.1 Introduction . . . . . . . . 54 4.2 System M odel . . . . . . . . 55 4.3 Asymptotic Performance of Thresholdbased Acquisition Systems . 57 4.4 Asymptotic Performance of Thresholdbased UWB Signal Acquisition 61 4.4.1 Asymptotic Performance of the SAI Approach . . . 62 4.4.2 Asymptotic Performance of the IAS Approach . . . 65 4.4.3 Numerical Results . . . . . . . 66 4.5 Conclusions . . . . . . . . 68 5 A SEARCH STRATEGY FOR UWB SIGNAL ACQUISITION . ... 72 5.1 Introduction . . . . . . . . 72 5.2 System M odel . . . . . . . . 72 5.3 Mean Detection Time Calculation . . . . . 74 5.4 The JumpbyH Permutation Search Strategy . . . . 75 5.5 Numerical Results . . . . . . . . 79 5.6 Conclusions . . . . . . . . 79 6 UWB TIMEOFARRIVAL ESTIMATION STRATEGIES . . . 81 6.1 Introduction . . . . . . . . 81 6.2 UWB TOA Estimation: Known C'!i ii., I Statistics . . ... 82 6.3 UWB TOA Estimation: Unknown C'! i,, I Statistics . . . 84 6.4 Numerical Results . . . . . . . . 87 6.4.1 Dense UWB Channels . . . . . . 87 6.4.2 Sparse UWB C i. . . . . . ... 89 6.5 Conclusions . . . . . . . . 91 APPENDIX A AVERAGE NUMBER OF MPCS COLLECTED .... . . 101 B AVERAGE PROBABILITY THAT THE ACQUISITION PROCESS WILL END IN A FALSE ALARM . . . . . . . 103 C PROOF THAT A,(7; AT) AND B,(7; AT) DEFINED IN (425) SATISFY THE CONDITIONS OF THEOREM 1 . . . . . . 105 D PROOF THAT A,(7; AT) AND B,(7; AT) DEFINED IN (435) SATISFY THE CONDITIONS OF THEOREM 1 . . . . . . 107 E PROOF THAT Q (DEFINED IN (512)) IS THE VECTOR IN THE SET A CORRESPONDING TO THE PERMUTATION R . . 108 F THE PDF THE SUM OF A FLIPPED NAKAGAMI RANDOM VARIABLE AND A GAUSSIAN RANDOM VARIABLE ................... 114 REFERENCES ...................................... 115 BIOGRAPHICAL SKETCH ................................ 120 LIST OF TABLES Table page 51 Serial search for Ns = 8 and H 3. . . . . . .... 80 52 Permutation search (1,4, 7,2, 5, 8, 3, 6) for Ns = 8 and H= 3. . . 80 53 Mean detection time (\!I)T) values for the serial search and hueristic search strategies . . . . . . . . . 80 LIST OF FIGURES Figure page 11 Block diagram of a parallel acquisition system for directsequence spread spectrum system s . . . . . . . . . 23 12 Block diagram of a serial acquisition system for directsequence spread spectrum systems which evaluates the candidate phases t1, t2, ... t,, serially. .. . 23 13 Block diagram of the acquisition scheme proposed by Blazquez et al. ... . 23 14 Block diagram of the acquisition scheme proposed by Soderi et al. . . 23 15 Template signals used in the twostage acquisition scheme proposed by Bahramgiri et al . . . . . . .. .. . . . 24 16 Block diagram of the twostage acquisition scheme proposed by Aedudodla et al. 24 17 Transmitted signal along with its component signals used by Furukawa et al. 24 21 The hit set size as a function of the average energy received per pulse to noise ratio for Np = 5 and 10 . . . . . . . . 31 31 Block diagram of the SAI acquisition system. . . . . 46 32 Block diagram of the IAS acquisition system. . . . . 46 33 Effect of EGC window length on the probability of a miss for SAI when Np = 5 46 34 Effect of EGC window length on the probability of a miss for SAI when Np 10 47 35 Effect of EGC window length on the probability of a miss for IAS when Np =5 48 36 Effect of EGC window length on the probability of a miss for IAS when Np 10 49 37 Effect of EGC window length on the mean detection time for SAI when Np = 5 50 38 Effect of EGC window length on the mean detection time for SAI when Np 10 51 39 Effect of EGC window length on the mean detection time for IAS when Np =5 52 310 Effect of EGC window length on the mean detection time for IAS when Np =10 53 41 Best AROC of the SAI approach to UWB signal acquisition. . . 70 42 Best AROC of the IAS approach to UWB signal acquisition. . . 70 43 IAS AROC corresponding to hit set phases other than the LOS path when G 1. 71 61 Illustration of the packet exchange scheme used to estimate the TOA. .. . 92 62 Mobile positioning based on TOA measurements. . . . . 92 63 The location of the observations used for TOA estimation. . . ... 93 64 Probability of incorrect estimation in dense channels for the rule which minimizes the error probability when the channel statistics are known. . . 93 65 Mean estimation error in dense channels for the rule which minimizes the error probability when the channel statistics are known. . . . . 94 66 Probability of incorrect estimation in dense channels for the rule which minimizes the average estimation error when the channel statistics are known. .. . 95 67 Mean estimation error in dense channels for the rule which minimizes the average estimation error when the channel statistics are known. . . . 96 68 Probability of incorrect estimation in dense channels for the heuristic rule when the channel statistics are unknown. . . . . . . 97 69 Mean estimation error in dense channels for the heuristic rule when the channel statistics are unknown . . . . . . . 98 610 Probability of incorrect estimation in sparse channels for the heuristic rule when the channel statistics are unknown. . . . . . . 99 611 Mean estimation error in sparse channels for the heuristic rule when the channel statistics are unknown . . . . . . . . 100 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 DESIGN AND PERFORMANCE OF ULTRAWIDEBAND ACQUISITION SYSTEMS\ [ By Saravanan Viji .v umaran May 2007 Ch! ,i': Tan F. Wong Major: Electrical and Computer Engineering The acquisition of ultrawideband (UWB) signals is a potential bottleneck for system throughput in a packetbased network employing UWB signaling format in the physical 1iv.r. The problem is mainly due to the fine time resolution and the low received signal power which forces the acquisition system to process the signal over long periods of time before getting a reliable estimate of the timing of the signal. In this dissertation, we focus on the development of more efficient acquisition schemes by taking into account the signal and channel characteristics. The presence of dense multipath in the UWB channel ~.;; i the presence of multiple acquisition states which could be exploited to speed up the acquisition process. In this dissertation, we give a precise characterization of the set of phases in the uncertainty region where a receiver lock can be considered successful acquisition. We call this set of phases the hit set. We design and compare the performance of two schemes for the acquisition of timehopping UWB signals which attempt to exploit the energy in the multipath to improve the acquisition performance. We prove a general result characterizing the .~,, ii 1.1. ic performance of thresholdbased acquisition schemes in multipath fading channels. We use this result to characterize the performance limits of the aforementioned UWB acquisition schemes. We then consider the problem of finding efficient search strategies when there are multiple elements in the hit set. We use the insights gained in the design of UWB acquisition schemes in the development of efficient schemes for the closely related problem of timeofarrival estimation. CHAPTER 1 INTRODUCTION The Federal Communications Commision (FCC) defines ultrawideband (UWB) technology as any wireless transmission scheme that operates with a fractional bandwidth of at least 211'. or occupies more than 500 MHz of absolute bandwidth. Ultrawideband signaling [14] is under evaluation as a possible modulation scheme for wireless personal area network (PAN) protocols. The features of UWB radio which make it an attractive choice are its multiple access capabilities [1, 5], lack of significant multipath fading [68], ability to support high data rates [9] and low transmitter power resulting in longer battery life for portable devices. 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 bandwidth. 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 [10, 11]. 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 responsible 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 [10]. Timing acquisition is a particularly acute problem faced by UWB systems due to the following reasons. Short pulses and low duty cycle signaling [1] employ, ,1 in UWB systems place stringent timing requirements at the receiver for demodulation [12, 13]. The wide bandwidth 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 packetbased network, each packet has a dedicated portion known as the acquisition preamble within which the receiver is expected to achieve synchronization. However for the high datarate applications envisaged for UWB signaling, long acquisition preambles would significantly reduce the throughput of the network Hence there is a need to develop more efficient acquisition schemes by taking into account the UWB signal and channel characteristics. Closely related to the problem of timing acquisition in UWB systems is the problem of localization using UWB signals. The absence of a carrier in UWB signals obviates the use of energybased localization methods. Localization based on roundtrip timeofflight measurements is an ideal candidate for UWB localization systems due to its simplicity and the high time resolution of the UWB signals. However, the dense multipath in the UWB channel is a hindrance to the accuracy of such systems. This chapter is organized as follows. In the next section, we briefly review the main features of acquisition methods used in traditional spread spectrum systems to put the problem of UWB signal acquisition in perspective. In Section 1.2, we briefly describe 1 Traditionally, in directsequence spread spectrum systems the chiplevel timing of the PN sequence is referred to as the phase of the spreading signal. In this document, we use phase and timing interchangeably. the existing literature on UWB acquisition. In Section 1.3, we review the existing work on timeofarrival estimation of UWB signals. An outline of this dissertation is given in Section 1.4. 1.1 Brief Review of Spread Spectrum Acquisition Systems Ultrawideband communication falls in the category of spread spectrum communication systems. There has been extensive research on spreading code acquisition and tracking for spread spectrum systems with directsequence, freC ii .,hopping and hybrid modulation formats [10, 11, 14]. We will bring out the main issues by considering the timing acquisition of directsequence spread spectrum systems. In a directsequence 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 down the synchronization error to within the pullin 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 [10] shown in Fig. 11. This acquisition system checks all the candidate phases in the uncertainty region simultaneously. In the ith arm, the decision statistic corresponding to the candidate phase ti is generated by correlating the received signal with a d. 1 i, .1 version of the locally generated spreading waveform s(t) 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 maximumlikelihood 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. 12. 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. The decision statistic corresponding to the candidate phase ti is generated by correlating the received signal with a d.1 i,, i1 version of the locally generated spreading waveform s(t). If the threshold is not exceeded, the search updates the value of the candidate phase and the process continues. Hybrid methods such as the MAX/TC criterion [15] 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 which is used to confirm the coarse estimate of the true phase before the control is passed to the tracking loop. In traditional spread spectrum acquisition schemes, the signaltonoise 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 [10, 11]. 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 [10, 11, 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. 1.2 Previous Work on UWB Acquisition In this section, we describe the existing approaches to UWB acquisition which take a detectiontheoretic approach to the problem. 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 prevents 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 either an AWGN or a flat fading channel model for the UWB channel and neglects the effect of multipath in the development and evaluation of the proposed acquisition schemes. In Blazquez et al. [23], the traditional coarse acquisition scheme where the search space is searched in increments of a chip fraction is analyzed for the acquisition of timehopped UWB signals in AWGN noise. Fig. 13 shows a block diagram of the scheme where a particular phase t, 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+j = ti + cTp where e < 1 and Tp is the pulse width. This process continues until the threshold is exceeded. In Soderi et al. [24], the output of a matched filter, whose impulse response is a timereversed replica of the spreading code, is integrated over successive time intervals of size mTc, where m is an integer greater than one but not exceeding the number of taps in the channel response 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 mT, and compared to a threshold as illustrated in Fig. 14. 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. In Ma et al. [25], the nonconsecutive search proposed by Shin et al. [21] and a simpler version of the MAX/TC scheme [15] called the global MAX/TC are applied to the acquisition of UWB signals in the presence of multipath fading and multiple access interference (\ AI). In the nonconsecutive 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 of the channel. In the global MAX/TC, a parallel bank of correlators is used to evaluate all the nonconsecutive 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. [26], a hybrid acquisition scheme called the reduced complexity sequential probability ratio test (RCSPRT) is presented for UWB signals in AWGN, which is a modification of the multihypothesis sequential probability ratio test (\ ISPRT) for the hybrid acquisition of spread spectrum signals [27]. 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 RCSPRT, 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 RCSPRT 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. 1.2.1 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 is more than one acquisition phase 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 "lookandjumpbyK oi, search and bit reversal search are analyzed in the noiseless scenario with mean stopping time as the performance metric in Homier et al. [28]. Suppose that the timing uncertainty region is divided in to bins indexed by 0,1,..., Ns 1. In lookandjumpbyKbins search, starting in bin 0, the search continues on to bin K, then to 2K and so on. So for Ns = 9 and K = 3, the lookandjumpbyKbins search searches the bins in the following order {0, 3, 6,t1, 4, 7, 2, 5, 8}. In bit reversal search, the order in which the bins are searched is obtained by reversing the bits in the binary representation of the linear search variable. For instance, when Ns = 9 the linear search has the binary representation {000, 001, 010, 011,... 111} and the bit reversal search is obtained by 'bit reversal' by {000, 100, 010, 110,... 111}. It then corresponds to the search order {0, 4, 2, 6, 1, 5, 3, 7}. A generalized flow graph method is then used to compute the mean acquisition time for different serial and hybrid search strategies [29, 30]. For the case when the acquisition phases are K consecutive phases in the uncertainty region, it has been claimed that the lookandjumpbyKbins search is the optimal serial search permutation when K is known and the bit reversal is the optimal search permutation when K is unknown. 1.2.2 Search Space Reduction Techniques Some acquisition schemes attempt to solve the large search space problem by employing a twostage acquisition strategy [3135]. 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. [31], such a twostage scheme is proposed for the acquisition of timehopped UWB signals in AWGN noise and multipleaccess interference (\ AI). The search space is divided into 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 d. 1 , .1 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. This is illustrated in Fig. 15 in the absence of noise and MAI. A scheme based on the same principle has been developed independently by Gezici et al. [34]. 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 Ri .i et al. [35], an acquisition scheme for UWB signals with timehopping (TH) spreading called nscaled search is presented, where the search space is divided into groups of M = Nf/2" where Nf is the frame size and 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 chiprate sampling of a matched filter output, it is equivalent to correlating the received signal with M d 1 ', ,1 versions of the modified replica of the received signal. In this sense, it is similar in spirit to the schemes described above. A twostage scheme which achieves search space reduction by employing a hybrid DSTH spreading signal format is described by Aedudodla et al. [32, 36]. In the first stage, the DS spreading is removed by squaring the received signal and the timing of the TH spreading code, which has a relatively small length, is acquired. Once this is done, the acquisition of the DS spreading code is performed by searching the search space in increments equal to the length of the TH code. Fig. 16 shows a conceptual block diagram of this system. Another twostage acquisition scheme for UWB signals with DS spreading which employs a special signal format is presented by Furukawa et al. [33]. The signal transmitted during the acquisition process is a sum of two signals, a periodic pulse train and a pulse train with DS pi 'In,i as shown in Fig. 17. In the first stage, the timing of the periodic pulse train is acquired by correlating the received signal with a replica of the periodic pulse train. This is an easy task considering that the uncertainty region is just twice the pulse repetition time Tf. Once this is done, the chip boundaries of the DS spreading sequence are known and the second stage needs to only search in increments of 2Tf to acquire the timing of the DS spreading sequence. 1.3 Previous Work on UWB TimeofArrival Estimation There has been a recent explosion in the existing literature on UWB timeofarrival (TOA) estimation. However, a significant number of these papers describe experimental results obtained from hardware testbeds employing UWB signals to perform TOA estimation. The actual algorithms used to estimate the TOA of the UWB signals have also been developed independently in papers which employ mathematical models and computer simulations to make their case. We will now briefly present the key contributions in the latter portion of the existing literature. One of the earliest contributions to TOA estimation was made by Lee and Scholtz [37], who used a generalized maximum likelihood approach to estimate the multitude of nuisance parameters in addition to the TOA to get a better estimate. In Gezici et al. [38], a survey of the UWB localization methods based on signal strength (SS) measurements, angleofarrival (AOA) measurements and TOA measurements is given. The problems arising out of the dense multipath in UWB channels are discussed and the Cramer Rao lower bounds (CRLBs) for the TOA estimation problem are derived. TOA estimation schemes based on correlation of the received signal with a noisy template (which itself is a part of the received signal) are presented. In Cardinali et al. [39], the CRLBs for the two high data rate signal formats proposed by the IEEE 802.15.3a Task Group, i.e., the direct sequence UWB (DSUWB) and the multiband orthogonal frequencydivision multiplexing (M\ IOFDM), are calculated. By optimizing over the set of synchronization sequences, it is shown that the MBOFDM format can provide potentially better performance. Also, the CRLB for the low data rate signal format proposed by the IEEE 802.15.4a Task Group is analyzed as a function of the pulse shape. In Qi et al. [40], the CRLBs in the presence and absence of prior knowledge of the nonlineofsight (NLOS) delay statistics are calculated. The maximum likelihood and maximum a posteriori detectors are presented and modified to account for the fact that strong multipath components can help achieve better accuracy for TOA estimation. A twostep TOA estimation scheme is presented by Gezici et al. [41], where the first step uses an energy detector to coarsely estimate the position of the multipath profile and the second step uses a hypothesis testing approach to locate the LOS path by casting as a change detection problem. The unknown channel parameters are estimated using maximum likelihood and method of moments estimators and these estimates are used in the calculation of the likelihood ratios. In Falsi et al. [42], several suboptimal algorithms based on detecting the peaks in the matched filter output are ain &i. .1 The first algorithm calculates the position of the N matched filter outputs of largest magnitude and picks the earliest arriving position as the TOA estimate. In the second algorithm, the largest matched filter output is estimated and its contribution is subtracted from the received signal. The remaining signal is passed through the matched filter and the largest output is calculated and its contribution subtracted. This process is repeated N times and the earliest arriving position of the N largest matched filter outputs is taken as the TOA estimate. The third algorithm is similar to the second in the iterative process of estimation and subtraction, with the exception that the ith step involves the estimation of the i largest matched filter outputs. Energy detectionbased approaches to TOA estimation are considered in Guvenc et al. [4345]. In the first paper by Guvenc et al. [43], the received signal is passed through an energy detector and the samples of the energy detector output are compared to a threshold. The threshold is selected to be between the maximum and minimum values of the outputs and the first threshold crossing gives the location of the LOS path. In the second paper by Guvenc et al. [44], for the same system model the threshold is chosen using the kurtosis of the energy detector output samples. In the third paper [45], the decision statistics and performance of storedreference, transmittedreference and energydetection based schemes are analyzed under the assumption of an AWGN channel. For realistic multipath channels, a maximum likelihood approach is taken. In another paper by the same authors [46], the received signal is either passed through an energy detector or processed by correlating it with a stored reference signal or a transmitted reference signal. In each case, the outputs are then used to perform TOA estimation via a hypothesis testing approach. 1.4 Dissertation Outline This dissertation is organized as follows. In C'!i Ipter 2, we describe the UWB system model which will be used in the design and evaluation of the acquisition schemes proposed in this document. We evaluate and compare two schemes for the acquisition of TH UWB signals in C'!i Ipter 3. We prove a general result characterizing the .,i~I, ,i1.. ic performance of thresholdbased acquisition schemes in multipath fading channels in C'! Ipter 4. This result is used to evaluate the .,i~I, ,ii.. ic performance of the two schemes proposed in C'!I Ipter 3. The problem of finding efficient search strategies in the set of all search strategies which are permutations of the search space is addressed in C'!I Ipter 5. We develop and evaluate schemes for timeofarrival estimation of UWB signals in C'! Ipter 6. filter detector corresponding Verification To code Received to largest stage tracking loop signal energy s(tt) SBandpass Energy  filter detector  s(ttd Figure 11: Block diagram of a parallel acquisition system for directsequence spread spectrum systems. Received Bandpass Energy Is threshold Yes Verification SuccessTo cod signal filter 1 detector exceeded? stage tracking loop signa" tracking loop s(ttT) No Failure Spreading waveform Search generator control Figure 12: Block diagram of a serial acquisition system for directsequence spread spectrum systems which evaluates the candidate phases t1, t2... t serially. Received Is threshold Yes Declare ti to be signal X Integrator exceeded? coarse estimate of true phase s(tti) No Template signal Search generator control Figure 13: Block diagram of the acquisition scheme proposed by Blazquez et al. mTc Received T Th odod Rcesignalved PN matched Integrator with Threshold To code filter dwell time mTc comparison tracking loop Figure 14: Block diagram of the acquisition scheme proposed by Soderi et al. Received signal A Fi rst stage template signal Second stage template signal Figure 15: Template signals used in the twostage acquisition scheme proposed by Bahramgiri et al. Received Squaring TH spreading Hit DS spreading Hit signal operation code acquisition code acquisition Figure 16: Block diagram of the twostage acquisition scheme proposed by Aedudodla et al. Periodic pulse train I I I IK I n1,. Tf Pulse train with DS spreading Transmitted signal I___^ _______ t I \ ______ r _______ \1 _______ ______ r .____ Uy  ^\r  ^0  ^r1  v\I  ^~f Figure 17: Transmitted signal along with its component signals used by Furukawa et al. Figure 17: Transmitted signal along with its component signals used by Furukawa et al. CHAPTER 2 PROBLEM DEFINITION 2.1 Introduction The timing information of the received signal is essential for the performance of a receiver in a wireless communication system. In a multipath channel, the energy corresponding to the true signal phase is spread over several MPCs. The main difference between the acquisition problems in a multipath channel and a channel without multipath is that there are more than one hypothesized phases which can be considered a good estimate of the true signal phase. In a dense multipath environment, which is the typical scenario under which UWB systems operate, the receiver may lock onto a nonlineofsight (NLOS) path and still be able to perform adequately as long as it is able to collect enough energy. From the viewpoint of postacquisition 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. In this chapter, we propose a definition of the set of hypothesized phases which correspond to a good estimate of the true signal phase by considering the demodulation performance subsequent to acquisition. We call this set of hypothesized phases the hit set. The hit set concept enables us to give a precise definition of the acquisition problem for UWB systems. We note that such a definition is applicable for any multipath channel. In the next section, we describe the UWB system model. In Section 2.3, we calculate the hit set for this system, followed by the definition of the UWB acquisition problem in Section 2.4. 2.2 System Model 2.2.1 Channel Model We assume that the propagation channel is modeled by the UWB indoor channel model described in Cassioli et al. [47]. This model gives a statistical distribution for the path gains based on a UWB propagation experiment but does not address the issue of characterization of the received waveform shape. Due to the frequency sensitivity of the UWB channel, the pulse shapes received at different excess dv1 are pathdependent [48]. To enable tractable analysis, we assume that the pulse shapes associated with all the propagation paths are identical. The channel is then a stochastic tapped d. 1 ,v line model expressed as the impulse response Ntap1 h(t) pkhkf (t kT), (21) k0 where Ntap is the number of taps in the channel response, To = 2 ns is the tap I' iil hk is the path gain at excess delay kTU, Pk is equally likely to be 1 to account for signal inversion due to reflections [49] and f(t) models the combined effect of the transmitting antenna and the propagation channel on the transmitted pulse. The path gains are independent but not identically distributed with N ,1 :inim distributions. The average energy gains 2k = E[h ] of the path gains normalized to the total energy received at one meter distance are given by Etot for k 0 1k l+rF(c) (22) 1Ett re((k1)Tc/), for k 1 2, ...Ntap 1, Il+rF (c) "' t where Etot is the total average energy in all the paths normalized to the total energy received at one meter distance, r is the ratio of the average energy of the second MPC and the average energy of the direct path, c is the decay constant of the power delay profile and F(e) 1exp[ (Ntap 1kTc]1. According to Cassioli et al. [47], Etot, r and c are all modeled by lognormal distributions. The N ,1: iii fading figures {mk} are distributed according to truncated Gaussian distributions whose mean and variance vary linearly with excess d. 1 iv. These longterm statistics are treated as constants over the duration of the acquisition process. 2.2.2 Transmitted and Received Signals The transmitted signal is given by OO lOO where Q(t) is the UWB monocycle waveform, P is the transmitted power, Tf NfTc is the pulse repetition time, {al} is the pseudorandom directsequence (DS) code with period Nds taking values 1, {cf} is the pseudorandom timehopping (TH) sequence with period Nth taking integer values between 0 and Nh 1, and Tc is the step size of the additional time shift provided by the TH sequence. The pulse repetition time Tf is chosen to be not less than (Nh + Ntap)Tc to avoid overlap between the multipath responses corresponding to distinct transmitted pulses. Note that the transmitted signal is periodic with period NperTf where Nper is the lowest common multiple of Nth and Nds. If u(t) = h(t) x(t), the received signal is given by r(t) u(t) + n(t) E1 E "(t( ITf clTc T) + n(t), (24) S00 where Ntap1 w(t)= ,',(t kTc). (25) k0 Here El is the total received energy at a distance of one meter from the transmitter, br(t) = f(t) b(t) is the received UWB pulse of duration Tw < Tc normalized to have unit energy, r is the propagation delay, and n(t) is an additive white gaussian noise (AWGN) process with zero mean and power spectral density . 2.3 Hit Set Definition As mentioned earlier, we will use demodulation performance subsequent to acquisition to define the hit set. We need to describe the receiver structure in order to quantify demodulation performance. The presence of a high degree of path diversity in the UWB channel motivates the use of a Rake receiver to improve demodulation performance. The three main Rake receiver structures considered for UWB signal demodulation are the all Rake (ARake), the selective Rake (SRake) and the partial Rake (PRake) receivers [8, 50]. The large number of resolvable multipaths in the UWB channel obviates the use of the ARake receiver due to the complexity involved in its implementation. We assume that the receiver uses a partial Rake (PRake) receiver to perform demodulation. Our choice is guided by the fact that the PRake receiver has lower complexity and still achieves comparable bit error performance relative to the SRake receiver [50]. 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. To simplify the analysis, we assume that the true phase 7 is an integer multiple of Tc. By the periodicity of the transmitted signal, we have 0 < r < (NperNf 1)Tc. The hypothesized phase r is also an integer multiple of T, with the same range as 7. Then AT = T 7 = aTf + /3T where a and 3 are integers such that Nper + 1 < a < Nper 1 and 0 < 3 < Nf 1. For a given true phase T, let PE(AT) denote the BER performance of the PRake receiver when it locks to the hypothesized phase r. Let Tm be the minimum SNR at which the PRake 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(O) > An for all SNRs less than Tn. Then for an SNR T > Tn and true phase 7, the hit set is given by Sh {r: PE(AT) < An}. (26) To completely characterize the hit set, we need to calculate the error performance of a partial Rake (PRake) receiver which is locked to a particular hypothesized phase r. We assume that the modulation format is BPSK with Nb consecutive UWB monocycles modulated by one bit. The signal received during the demodulation stage is given by 00 rb(t) L= b b 1"I(t ITf cjTc T) + n(t), (27) 1 oc where bi c {1, 1} for each i, [x] is the largest integer not greater than x, Ntap1 w(t)= ',(t kT), (28) k O and n(t) is a zeromean AWGN process with power spectral density io. The PRake receiver is assumed to have Np fingers where Np < Ntap. When the receiver estimates 7 to be the true phase in the acquisition stage, the PRake receiver estimates and combines the paths arriving at d. 1' ,r + kTU (k = 0,1,..., Np 1) to obtain the decision statistic. Since = + (aNf + 3)Te, the PRake receiver is estimating the values of p]Nf+3+ihNf+3+i for i 0,1,...,Np 1, where we define pk hk =0 for k {0,1,...,Ntap 1}. To make the analysis tractable, we assume that PRake is able to estimate these path gains and inversions perfectly. The decision statistic for the mth bit, Zm, can be obtained by correlating the received signal with the following template signal, (m+l)Nb 1 Sb (t) alv(t lTf CTc ), (29) l= mNb where Np1 V(t) = PoNf++ihc ', (t iT). (210) i=0 Then we have Np I i=0 where nb is a zeromean Gaussian random variable with variance o 2 No ZNp 1 h2 b _b2Nb i 0 aNf+3+i Then from Simon et al. [51, pp.268269], the average probability of error is given by PE(AT) Eh Q[ No2ENb aNf+3+i 1j2 2f+3+p1N , Nf+ p ( 2ENb dO, (212) Tr 0 .No sin 2o iaNf +3 where Mi(.), the moment generating function of hf, is given by forie {0,1,...,Ntap 1} 1 otherwise. Fig. 21 shows the hit set size as a function of the average energy received per pulse to noise ratio E t for the nominal uncoded BER requirement An = 103, the number of monocycles modulated by one bit Nb 8, the length of the channel response Ntap 100, Nf 116 and Np = 5, 10. We assume that Etot = 20.4 dB which is its mean value when the transmitterreceiver separation is 10 m [47]. We choose the power ratio r = 4 dB, decay constant e 16.1 dB and fading figures mk = 3.5 , 0 < k < Ntap 1, which are their mean values given in Cassioli et al. [47]. This plot confirms our claim in the beginning of this chapter about the existence of multiple phases where a receiver lock can guarantee adequate demodulation performance. 2.4 The UWB Acquisition Problem For a particular value of T, the hit set Sh is obtained from (26) using (212). The acquisition process can then be formulated as a composite binary hypothesis testing problem [52] with the following hypotheses: Ho : 5ASh H, : f Sh. (214) Our goal is to design efficient acquisition schemes which take into account the UWB signal and channel characteristics, and characterize their performance. S60  50)  E 40 30  20  10  0 0 2 4 6 8 10 12 14 16 18 20 Average energy received per pulse to noise ratio (dB) Figure 21: The hit set size as a function of the average energy received per pulse to noise ratio fr N 30 5 and 10 ratio for N/p 5 and 10 CHAPTER 3 ACQUISITION OF TIMEHOPPING UWB SIGNALS 3.1 Introduction The UWB channel is a dense multipath channel without significant fading [7, 53]. In a dense multipath environment, there will be a considerable amount of energy available in the multipath components (\!PCs). It seems reasonable to expect that an acquisition scheme which utilizes the energy in the MPCs would perform better than one which does not. In this chapter, we consider the acquisition of UWB signals having only TH spreading. The system model is same as that described in C'!i pter 2, except that DS code is absent in the transmitted signal1 . Considering that we have no information regarding the channel state, there are essentially two v in which we can attempt to utilize this energy in order to develop a more efficient acquisition scheme. In the first approach, the received signal is first squared to eliminate the channel inversion and then equal gain combining (EGC) is performed to exploit the rich path diversity present in UWB channels. In the second approach, EGC is performed first and the integrator output is then squared to generate the decision statistic. In the sequel, we will refer to the former as squareandintegrate (SAI) and to the latter as integrateandsquare (IAS). It is not exactly clear which approach is more efficient. Also, the choice of the length of the EGC window is not apparent. For instance, in SAI, a small window will not collect enough energy and thus will result in a low probability of detecting the correct signal phase. A large window may collect a considerable amount of energy even when the true phase does not match the hypothesized phase, resulting in a high probability of false 1 The expressions for the transmitted and received signals can be obtained by setting a = 1. alarm. In this chapter, we derive and compare the performance of both SAI and IAS as a function of the EGC window length. This chapter is organized as follows. We derive expressions for average probabilities of detection and false alarm for SAI and IAS in Sections 3.2 and 3.3, respectively. In Section 3.4, we give a design criterion for choosing the decision threshold and derive the mean detection time for a serial search strategy as a function of the average probabilities of detection and false alarm. In Section 3.5, the mean detection time and the probability of a miss are used as performance metrics to compare the two approaches. Section 3.6 has some concluding remarks. 3.2 Analysis of SAI 3.2.1 Derivation of the Decision Statistic The acquisition system correlates the squared received waveform with a locally generated replica and compares the correlator output to a threshold to determine whether the hypothesized phase of the replica is correct (as shown in Fig. 31). If the threshold is exceeded, the hypothesized phase becomes the estimate of the true phase. We assume that the normalized received monocycle waveform Qr(t) and the TH sequence {cf} are known to the receiver. The received signal is the same as in (24) with the exception of the DS code and is given by OO r(t) = E wt ITr cTe r) + n(t). (31) We propose to use an equal gain combiner of window size G. The receiver template signal Wr(t) is given by G1 Wr(t) ri(t kTU). (3 2) k 0 The reference TH signal can be obtained by combining the receiver template signal Wr(t) and the known time hopping sequence as MNth1 st)= wrl(t I0Tf T, ), (33) l=0 where M specifies the number of TH waveform periods2 in the dwell time and 7 is the hypothesized phase. The correlator output is given by M ft+MNthTf r2(t) (t)t M lNth f t +MNth u2(t) s(t) dt +  +MNhTfu(t)s(t)n(t)dt MN(th J MNth + +NthTfn2(t)s(t)dt. (34) MNth f The first term in (34) can be simplified to Ntap 1 s(AT; h) E R2(0) E rk (AT) h, (35) k0 where h is an Ntap x 1 vector containing the channel gains, R',y(V) f= _,, (t>,,, (t + v)dt and rk(AT), the average number of times the energy in the kth MPC is collected by one period of the reference TH signal, is given by t Nth1 G1 rk (AT) xc + + Cl+i+ + k + iNf), (36) th 0 i=0 j=0 where x(a, b) = 1 if a = b, and 0 otherwise. The value of rk(AT) depends on the particular pseudorandom TH sequence chosen. To simplify the analysis we assume that the TH sequence is random and that Nth is large. Under these assumptions, the mean value of rk(AT) is a reasonable approximation to the actual value. The mean value of rk(AT) is calculated in Appendix A by averaging over the set of random TH sequences. 2 In the absence of the DS spreading code, the period of the transmitted signal is Nper = Nth. Conditioned on the random vector h, the second term in (34) is a zeromean Gaussian random variable with variance S(AT; h) 2 Nthf 2(t)(t)dt (37) th 2E1R 3(0)N Ntap 1 MNth rk (AT)hk (38) k0 where the second equality is obtained by exploiting the similarity between the integral in (37) and the first term of (34). We have also used the fact that MNth1G1 2(t) JY kT IT, cT), (39) 1l0 k=0 which differs from s(t) only in the exponent of the received pulse waveform /br(t). We approximate the third term in (34) by a Gaussian random variable with mean Py and variance v2 which are given by 1 f+MNthTf No +MNthTf S MNthE L nt (t)dt 2MNth js +NthT GRr(0)No GNo (3 2 2 and 1 [ Tf+MNthTf 2( ) 21 2' 7N2 J(+MNthTf 2 ( t+MNthTf )2 0 S 2 (N s(t) 2 P 2 2MNth respectively. Note that the expectation in the derivation of py and v2 is only with respect to the noise process n(t). This approximation is accurate provided that the product of the integration time MNthTf and the bandwidth of the system B is large [10, pp. 240250], which is the case for the scenarios we consider. Then the correlator output can be written as y = s(AT;h) + ny, (312) where, conditioned on h, ny is a Gaussian random variable with mean py and variance a (AT; h) + v2. 3.2.2 Average Probabilities of Detection and False Alarm For a particular channel realization h and fixed AT, the decision statistic y in (312) has a Gaussian distribution with probability density function S_____1______ (y s(AT;h) y)2 Pv y = iexp 2. (313 /27r(o (Ar; h) + vy) L 2(7 (AT; h) + vy2) The probabilities of false alarm and detection conditioned on the particular channel realization and given the decision threshold 7 are given as PFA (7,ATh) Pr[y > 7 Sh]= Q (A ,h T Sh (314) 2(AT; h) + Vy PD(7,Arh) Pr[y> 7eSh] Q 7 (ATh) Y) ,T Sh. (315) a (AT; h) + Vy From (35) and (38), one sees that the conditional probabilities of false alarm and detection depend on h only through si(Ar; h) = Y:N k (AT)h which is a scaled version of s(Ar; h). Using (35) and (38) we define (7 El R72 (0)s I(AT; h) /y I(sl(Ar; h)) = Q I (316) S2E1R 3(0)No I S MNt (AT; h) + v2 Since the path gains hk (k = 0, 1,..., Ntap 1) are independent, the characteristic function of si(Ar; h) is given by Ntap 1 I4(w; AT) 7J Mk(jrk(AT)w), (317) kO where AMk(') is defined in (213). The probability density function (pdf) of si(Ar; h) is given by fs(x; Ar) fl s(w; A7)e> wd. Then for r i Sh, the probability of false alarm averaged over the channel realizations is given by EH[PFA(7, ArTh)] EH [I(si(AT; h))] j I(t)fs(t; Ar)dt. (318) Similarly, for r e Sh, the average probability of detection is given by EH[PD(7, Arh)] I(t) fs(t; Ar)dt. (319) The structure of AMk(') prevents from evaluating fs(.) in closed form. So we resort to numerical integration to calculate fs(.) and the average probabilities of false alarm and detection. 3.3 Analysis of IAS In this section, we analyze an acquisition system which takes the IAS approach. The derivation of the decision statistic in this case is very similar to the decision statistic derivation in the previous section. All the relevant assumptions made in the previous section, to enable tractable analysis, still hold unless stated otherwise. To avoid repetition, we only define those quantities which have not already been defined in the previous section. 3.3.1 Derivation of the Decision Statistic In this approach, the acquisition system correlates the received waveform with a locally generated template signal and squares the integrator output to generate the decision statistic (as shown in Fig. 32). The receiver template signal vr(t) is given by G1 vr(t) Y= r(t kTc). (320) k0 The reference TH signal is given by MNth1 q(t) = vr(t T lci ). (321) l0 The decision statistic is given by 2 1 f+MNthTf E2 Ntap1 z = T r(t)q(t)dt E r )Pkhk+n, (322) LAM]th Jk 0 V(Ar;h) where nz is a zeromean Gaussian random variable with variance Z = N and th(A) is given in (36). 3.3.2 Average Probabilities of Detection and False Alarm For a particular channel realization h and fixed AT, the decision statistic z in (322) has a noncentral chisquare distribution with probability density function pz(z) 1 1 e (+V2(;h)) /2 cOsh VzV(AT; h)). (3 23) The probabilities of false alarm and detection conditioned on the particular channel realization and given the decision threshold 7 > 0 are given by PFA(7,ATh) Pr[z > 7 Sh] Q(7 V(AT; h) +Q /7 + V(AT; h) Sl, (324) = +e ,f^^ ^Sh, (324) PD(7, Arh) Pr[z > 7 Tc Sh Q 7 V(AT;h) + Q 7+ V(AT;h) ,e Sh. (325) Before we derive the average probabilities of detection and false alarm, it is instructive to look at the characteristic function 4v(w; AT) of V(AT; h). Since the polarities pk and path gains hk are independent, we have Nv I (V I i rk(A T) ) + ',, (r/ (A T) )) (3 2 6) +v(L; AT) (326) kO0 where i (.) is the characteristic function of the N .1: .; i,:inm distributed hk [51]. Since the hk's are realvalued, the ., (.)'s are conjugate symmetric functions and hence 4v(') is a realvalued function. The GilPelaez lemma [54] gives an alternative form of the Q function as 1 1 f" 1 Q(x) = e /2 sin(tx)dt. (327) 2 7T J0 t Substituting this form of the Q function in (324), the probability of false alarm averaged over the channel realizations, for 7 ( Sh, is given by EH[PFA(7, Arh)] (328) 1 / t (2 t V(AT; h)) t(/ + V(AT; h))] = 1 e EH sin +sin ] dt 7 0 t z Oz 2 01 2/2 /t r V(AT;h)t] = 1 e /2 sin m t EH sin V(; h)dt 7T o t Cz / z S 1 2 r e2/2 sin I) E (L;A2 dt, (3 29) 2 Jot where the last equality follows from our observation that v(.) is realvalued. Similarly, for 7 e Sh, the average probability of detection is given by EH[PD(Q, Ah)] =1 2 l t2/2 sin (; AT dt. (330) S t z / (3z30) 3.4 Mean Detection Time Analysis of Serial Search We define a hit or detection event as the event when the decision threshold is exceeded for some 7 c Sh. We define a miss as the event when the decision threshold is not exceeded for all ce Sh. Although the average probability of a miss is a potential indicator of acquisition system performance, the mean acquisition time is usually the metric used to evaluate the performance of acquisition systems [10]. The mean acquisition time of an acquisition system depends on the particular search strategy used in evaluating the phases in the search space. We consider a serial search strategy for the evaluation of the acquisition schemes developed in this chapter. The design of better search strategies is considered in C'! Ipter 5. The calculation of the mean acquisition time enumerates all false alarms which occur before a detection event and associates a false alarm penalty time Tfa to each one of them. The false alarm penalty time is equal to the dwell time of a verification stage in the acquisition system which aids in the confirmation of detection events and rejection of false alarm events with high probability. In other words, a good verification stage simultaneously achieves low probabilities of miss and false alarm. The choice of the mean acquisition time as the performance metric implicitly assumes that one can construct such a verification stage. However, we will show in C'!i Ipter 4 that for thresholdbased UWB acquisition systems the average probabilities of false alarm and miss cannot be made arbitrarily small even in the .,,:iii:i ic scenario of the SNR approaching infinity. Thus it is not apparent how one would build a good verification stage for such systems. We propose to deal with this problem by choosing the decision threshold 7 such that the average probability that the acquisition process will end in a false alarm, PF(7), is small. The justification for this design is that a false alarm is a more serious problem in the absence of a verification stage. Then if PF(7) is small enough, we can use the mean detection time as the performance metric. The mean detection time is defined as the average time it takes for the acquisition process to end in a detection event in the absence of false alarms. In Appendix B, we calculate PF(7) as a function of the average probabilities of detection in the hit set and the average probabilities of false alarm at the phases not in the hit set. We choose the decision threshold 7d to be the minimum threshold such that Pp(7) is not greater than a given positive constant 6 < 1, 7d inf{71 PF(7) < 6} (3 31) If the correlator outputs for different phase evaluations are assumed to be independent, then the average probability of a hit for a particular r is Eh[PD(7d, ArIh)] and the average probability of a miss is given by PM = (1 Eh[PD(7d, Arh)]). (332) TESh Owing to our definition, the hit set Sh consists of a contiguous set of H hypothesized phases within the search space. The search space is the set Sp = {nT n E Z and 0 < n < Ns 1} where Ns = NthNf. Let the first phase of the hit set be at position A in the search space Sp. Then the hit set consists of the phases {(A 1)Te, ATE,..., (A + H 2)TI}. The initial value of the hypothesized phase which corresponds to the starting point of the search is chosen at random from the set Sp. Thus there is no loss of generality in assuming that A 1. We need to consider all possible sequences of events leading to a hit or detection event. The mean detection time can then be calculated as the average time taken for each of the detection events. A detection event is defined by a particular position n of the initial value of the hypothesized phase in Sp, the position i of the hypothesized phase in Sh where we have a hit and a particular number of misses j of Sh. Let Tdet (f) be the mean detection time conditioned on the event that the serial search starts at the nth position in Sp i.e. the initial value of the hypothesized phase is (n 1)Te. Then the mean detection time is SNs Tdet Ns Tdet ( (333) n l First, suppose that the initial value of the hypothesized phase lies to the right of the hit set, i.e., n C {H + 1, H + 2,..., Ns}. The total detection time for a particular detection event defined by (n,j, i) is then T(n, j,i) = (N Nn+1)T+jNT+ iT S(s n+ + jNs+ i)T (334) where T is the dwell time for the evaluation of one hypothesized phase. Let Pd(i) denote the average probability of detection of the ith phase of the hit set. The average probability of the serial search missing the hit set is PM = H1 [1 Pd(i)]. Then the probability of j misses of Sh followed by a hit at the phase in Sh which is at the ith position of the hit set is P/Ph(i) where Ph(i) Pd(i) iH[1 Pd(r)]. The mean detection time conditioned on the starting point of the serial search is given by H oc Tdet ( T nj )P ) Ph i=1 j=0 H NsPM + ( PMTH h) (Ns n + 1)T + P + 1 PM (335) 1 PM 1PM where we have used the identities i1 Ph(i) PM in obtaining the third equality. Now suppose that the initial value of the hypothesized phase falls in the hit set, i.e., n C {1, 2,..., H}. Let m be the total number of phases evaluated for a particular detection event. We can partition the set of detection events into two sets, one containing those events for which m < H n + 1 and the other containing those events for which m > H n + 1. The mean detection time for events in the first set is just mT and for events in the second set it is Tdet(H + 1) + (H n + 1)T where Tdet(H + 1) is obtained from (335). Averaging over the total number of phases evaluated we get H i1 Tdet(n) ( + 1)T Pd() (1 Pd(j)) in jin H +(Tdet(H + 1) + (H n + 1)T) (1 Pd(j)). (336) jin From (335) and (336), we obtain the conditional mean detection times Tdet(n) for all values of n E {1, 2,..., Ns}. The mean detection time is obtained by substituting these values in (333). 3.5 Numerical Results In this section, we compare the performance of SAI and IAS in terms of the average probability of a miss PM and the mean detection time Tdet. We also investigate the effect of increasing the EGC window length on these performance metrics for both schemes. We choose the following values for the system parameters: the TH sequence period Nth = 256, Nh = 16, the length of the channel response Ntap = 100, Nf 116 and the number of monocycles modulated by one bit Nb = 8. The nominal uncoded BER requirement is set to be An = 103. The decision threshold 7d is chosen to be the minimum threshold such that the bound on PF(7d) is 6 = 0.05. We assume that Etot 20.4 dB which is its mean value when the transmitterreceiver separation is 10 m [47]. We choose the power ratio r = 4 dB, decay constant e 16.1 dB and fading figures mk = 3.5 0 < k < Ntap which are their mean values given in Cassioli et al. [47]. Figs. 33 and 34 show the effect of increasing G on the average probability of a miss PM for SAI when the number of PRake fingers are Np = 5 and 10, respectively. For each value of Np, we plot PM for the average energy received per pulse to noise ratio =EEtot 7, 10, 15 and 20 dB. When EtNt is low, PM decreases at first as G increases and then begins to increase. When E,0tNt is low, increasing G helps combat the effect of the noise by collecting more energy when the hypothesized phase belongs to the hit set. This is the reason for the initial decrease in PM. At the same time, the energy collected in the nonhit set phases begins to increase resulting in a much higher threshold being chosen to ensure that PF(7d) does not exceed 6. Consequently, the probabilities of detection decrease causing PM to increase. When E,0tt is high, the detrimental effect of the noise is not very significant and hence increasing G does not improve the probabilities of detection significantly. But the probabilities of detection suffer from the stringent choice of threshold required to keep PF(7d) small. This is the reason for the increase in PM with G for high values of EEtot The values of PM show a slight decrease for large G in the case when Np = 10 compared to the case when Np = 5. This is because the hit set is larger when Np = 10 but the probabilities of detection in the additional phases becomes significant only for large G. Figs. 35 and 36 show the effect of increasing G on the average probability of a miss PM for IAS when the number of PRake fingers are Np = 5 and 10, respectively. For all values of E,1E,, we considered, PM increases with G from a value close to zero to a value close to one. Thus performing EGC in the IAS approach actually results in a degradation in performance. As G increases, the EGC window collects multiple paths which may have opposing polarities resulting in cancellations and hence a decrease in the probabilities of detection in the hit set phases. This is the reason for the increase in PM with G. This cancellation effect is also present in the nonhit set phases resulting in a less stringent choice for the threshold needed to keep PF(7d) small. This effect is absent in the case of SAI because the squaring operation eliminates the path polarities. The small values of PM (for small G) .:: I that IAS does a better job of averaging out the effect of the noise than SAI. Squaring the received signal before integrating seems to be preventing this averaging effect in SAI, resulting in higher values of PM when E1Ett is low. When ElE tt is high, the less stringent threshold results in smaller values of PM for IAS in comparison to SAI. Once again, the presence of a larger hit set for Np 10 results in smaller values of PM for IAS in comparison to the case of Np = 5. To compare the performance of the two schemes in terms of the mean detection time Tdet, we assume that the dwell time is equal to one period of the TH sequence i.e. M 1 and T = NthNfTc. Figs. 37 and 38 show the mean detection time in seconds (at different values of lot) for SAI as a function of G for Np = 5 and 10, respectively. Figs. 39 and 310 show the corresponding plots for IAS. For both schemes, the effect of increasing G on the mean detection time mirrors its effect on PM for the same reasons mentioned in the previous paragraph. Once again, performing EGC is beneficial in the SAI approach and causes performance degradation in the IAS approach. For SAI, we observe that the minimum mean detection time is achieved for some value of G larger than one. This value of G changes with EEltot but is the same even when the number of PRake fingers Np is increased from 5 to 10. Increasing Np keeping the EEtot fixed increases the size of the No hit set. But the probabilities of detection in the additional phases becomes significant only for large values of G where the probabilities of detection have already been lowered by the stringent choice of threshold. On the other hand, the IAS approach achieves mean detection times which are significantly lower than the corresponding values for SAI when EE0tot is low and hence is a more efficient scheme. Even though the probabilities of detection get better as increases, the mean detection time does not change significantly since it is dominated by the time the acquisition system spends in the nonhit set phases. The minimum mean detection time is seen to be of the order of a second which is too high from a practical system viewpoint. This is due to the large search space and the fact that the serial search has to evaluate a considerable number of phases on the average before it encounters the hit set. This issue can be alleviated by a parallel search strategy. 3.6 Conclusions We have analyzed two approaches, namely SAI and IAS, for the acquisition of UWB signals which perform EGC to utilize the energy in the multipaths. By considering system performance subsequent to acquisition, the set of phases which can be considered a hit was obtained. In the SAI approach, performing EGC improves acquisition performance at low SNRs while it causes performance degradation in the IAS approach. With mean detection time as the metric for system performance, we observe that the IAS approach outperforms the SAI significantly. Thus EGC may not be a good method to utilize the energy available in the multipaths to improve acquisition performance. Finally, the far from practical values of the mean detection time obtained motivate the need for a parallel search strategy and the development of acquisition schemes capable of reducing the search space. R(Ax;h) s(tT ) Reference Signal Generator Figure 31: Block diagram of the SAI acquisition system. r(t) orrelator Squaring R(A ;h) r(t) s Correlator Operation 1Operation s(t ) Reference T Signal Generator Figure 32: Block diagram of the IAS acquisition system. 0.6 0.5 0.4 0.3 S . E 7 + ./ / ,;/ 1 ilotN 0 0 d i / ."/ / / / //  \ / "/ .E/ E1EtotN 0= 15 dB "" / E 10dB S....... E Eto 15dB ...... .. E ._ E1E totN =20dB 2 4 6 8 10 12 14 EGC Window Length Figure 33: Effect of EGC window length on the probability of a miss for SAI when Np 5 0.7 Figure 34: Effect 10 2 4 6 8 10 12 14 EGC Window Length of EGC window length on the probability of a miss for SAI when Np 2 4 6 8 10 12 14 EGC Window Length Figure 35: Effect of EGC window length on the probability of a miss for IAS when Np 5 2 4 6 8 10 12 14 EGC Window Length Figure 36: Effect of 10 EGC window length on the probability of a miss for IAS when Np 9 8 7. 6 S5/ 4 CI 0/ . 2 / 1./ Figure 37: Effect of EGC window length on the mean detection time for SAI when N/ ) / /I'" :. EEN =7dB 2 46 8 10 12 14 Figure 37: Effect of EGC window length o7 the mean detection time for SAI when Np 2 2 Figure 38: Effect of 10 4 6 8 10 12 14 EGC Window Length EGC window length on the mean detection time for SAI when Np 9 8 7ii 7 I .' 6 5 I' 4 3 i.;; 31 2 / E 1 SEEtotN = 7 dB E E1EtotN = 10 dB SE1EtotN o 15 dB EEtotN =20dB 2 4 6 8 10 12 14 EGC Window Length Figure 39: Effect of EGC window length on the mean detection time for IAS when Np 5 2  1 1  0 2 Figure 310: Effect of 10 4 6 8 10 12 14 EGC Window Length EGC window length on the mean detection time for IAS when Np CHAPTER 4 ASYMPTOTIC PERFORMANCE OF THRESHOLDBASED ACQUISITION SYSTEMS IN MULTIPATH FADING CHANNELS 4.1 Introduction In this chapter, we investigate the .,vmptotic error performance of thresholdbased timing acquisition systems having fixed dwell time in multipath fading channels. We restrict our attention to acquisition systems with fixed dwell time because it represents the case of packetized mobile communication systems. This is a scenario where good acquisition performance is crucial, since the timing needs to be repeatedly estimated for every packet as it may change due to node mobility. And since throughput considerations limit the length of the preamble which can be prepended to a particular packet, there might be a limit to the accuracy with which the timing can be estimated. Thus it is of interest to get an estimate of the best possible acquisition performance which can be achieved by using a finitelength preamble. In the absence of channel f lii_ it is a wellknown result that the probabilities of occurrence of false alarms and misses, which are due to the noise alone, can be made arbitrarily small by operating at a higher SNR, which is typically done by increasing the dwell time of the correlator [10]. As the SNR increases, even a suboptimally chosen threshold, located between the means of the distributions of the decision statistic when the hypothesized symbol timing is correct and incorrect, forces the probabilities of false alarm and miss to become arbitrarily small. It is, however, reasonable to expect that the presence of channel fading can cause errors to occur, irrespective of how high the average SNR is. This is due to the fact that a high average SNR only guarantees that the detrimental effect of the noise is negligible and the channel fading can still induce errors in the acquisition process. In this chapter, we isolate the detrimental effect of the multipath channel fading on the acquisition performance of a finite dwell time thresholdbased acquisition system, by considering the .i i, ii ic performance as the average SNR increases without bound. We show that no matter how large the average SNR is or how we choose the threshold, there exist fading scenarios with a nonzero and sometimes restrictive average probability of occurrence of false alarms and misses. We describe the system model in Section 4.2 which is general enough to encompass most thresholdbased timing acquisition systems. In Section 4.3, we state and prove the main result of the chapter which basically zvi that if there is a threshold which restricts the average probability of false alarm to be smaller than a fixed tolerance, then no matter how large the average SNR is, there is a possibly nontrivial lower bound on the .~,vill I .1 ic average probability of miss. In Section 4.4, we apply the result to evaluate and compare the .', ii i I ic acquisition performance of the two acquisition schemes developed in Ch'! pter 3. Section 4.5 has some discussion and conclusions. 4.2 System Model Let s(t) be the transmitted signal and h(t) be the channel response which is assumed to be random but fixed during the acquisition process. Then the received signal is given by r(t) = x(t 7) + n(t) where x(t) = s(t) h(t) ( denotes convolution), 7 is the true symbol timing and n(t) is a zeromean widesense stationary (WSS) additive noise process. Let f be the hypothesized symbol timing. Then the decision statistic generated by the acquisition system is given by R(Ar; h) = g(r(t), f) where g is some bivariate functional, AT = r 7 and assuming that the channel fading effects can be characterized by a finitedimensional vector h. Let FN(; ArTh) be the conditional cumulative distribution function (CDF) of R(Ar; h) conditioned on h. In a multipath channel, the receiver need not lock to the lineofsight (LOS) path to perform successful demodulation. Depending on the performance criteria chosen, there will be a set of hypothesized symbol timings r called the hit set (which we will denote by Sh) where a receiver lock can be considered successful acquisition. Since the goal of the acquisition process is to achieve coarse synchronization, the true symbol timing 7 can be assumed to belong to a finite set Sp of timings which is an adequately quantized version of the timing ambiguity region. The hypothesized symbol timings are chosen from this finite set and hence the hit set is also finite. Note that it is the distance Ar of a hypothesized symbol timing r from the true symbol timing 7 which determines if r belongs to Sh or not. In this sense, the actual value of the true symbol timing is irrelevant. For a particular value of 7, the acquisition process can be formulated as a composite binary hypothesis testing problem with the following hypotheses: Ho : rSh H1 : r Sh. (41) The probabilities realization h and of false alarm and detection conditioned on the particular channel given the decision threshold 7 are given, respectively, by PFA (; A7h) PD(Q; Ar h) Pr[R(Ar; h) > 7h, Sh] Pr[R(Ar; h) > 7h, c Sh] FN(7; ATh),r e Sh, FN(7; Ar h) FN(7; Ar h), Sh, FN(7; Arh) (42) where F(.; Arh) is the complementary conditional CDF of R(Ar; h) conditioned on h. Then the probabilities of false alarm and detection averaged over the channel realizations are given by PFA (7; Ar) PD(; Ar) EH [PFA (7; Arh)] EH[PD(7; Arh)] EH[FN(7; Ar h)], r S EH[FN(7; Ar h)], r Sch. The average probability of miss is then given by PM(7;Ar) 1 PD (7;Ar), rE Sh. Henceforth, whenever we write PFA(7; A7) or PFA(7; Arh) it is implicit that r ( Sh. Similarly, PD(7; A7), PD(7; Arh), PM(7; Ar) and PM(7; Arh) all imply that r e Sh. (43) (44) (45) 4.3 Asymptotic Performance of Thresholdbased Acquisition Systems Let a.2 be the power (variance) of the noise process n(t). Let H be the set of all possible channel parameter vectors h. Note that PFA(7; AT) and PM(7; AT) defined in the previous section are functions of a. To avoid cumbersome notation, we write lim,,o+ PFA (7; AT) to mean lim,,o+ PFA (, ,o; AT). Furthermore, for a positive sequence {or,} with limit 0, we write limsupo,,o+ PM(7; AT) to mean limsup PM,(7, oa; AT). The following theorem is the main result of this chapter. Theorem 1. Consider a thresholdbased acquisition system with decision statistic R(AT; h) with the 1j.*/p '/;/ that for every threshold 7 and c > 0, there is an T/(7, c) > 0 such that when 0.2 < T/(7, ) there exist subsets Ae(7; AT),Be(7; AT) of 7, for every fT Sp, such that (i) Pr(A,(7; AT) U B,(7; AT)) > 1 e. (ii) For all h c A,(7; AT), FN(7; AT h) > 1 e. (iii) For all h c B,(7; AT), FN(7; AT h) < e. For some 6 > 0, if there exists an 11(6J) > 0 such that PFA(7; AT) < 6 for all a2 < 1 (6) and for all T Sh, then lim>o+ PM(7; AT) > limo+ Pr(A,(7m(6); AT)) where 7m(6) inf{7 : limo+ Pr(B('(7; AT)) < 6, for all T Sh}. Furthermore, given > 0 there exists a K(6,0) > 0 such that PM(7; AT) > lim,,o+ Pr(A,(7m(5); AT)) for all a.2 < K(OI. Proof. By the hypothesis, for every 7 and c > 0 there is an T/(7, c) > 0 such that when a 2 < /Q(7, c) there exists a subset A,(7; AT) of 7 such that for all h e A,(7; AT), FN(7; AT h) > 1 c. Then for all JO2 < ( 7(, ), we have PM(7; AT) > EH[FN(7; AT h)IAh() A )(h)] > (1 ) Pr(A,(7; AT)) > Pr(A(7; AT)) (46) where IA(7;Ar)() is the indicator function of the set A,(7; AT). Furthermore, when a2 < Ty(7, e) and De(7; Ar) = A(7; Ar) U B,(7; Ar) we have PM(7; AT) < EH[FN(7; AT h)(ID(,A) + I,(Y;AO))(h))] < EH[FN(7; ATh)(IA(;A, ) + IBQ(;A) + ID(,;A ))(h))] < Pr(A,(7; AT)) + c Pr(B,(7; AT)) + c < Pr(A,(7; AT)) + 2c. (47) Consider any convergent positive sequence {or} with limit zero. For any c > 0, from (46) and (47), we have Pr(A,(7; AT)) e < liminf PM(; AT) < limsup PM(7; AT) a0+ 7n^0+ < Pr(A,(7; AT)) + 2c. (48) Now consider a convergent positive sequence {e,} with limit zero. Since (48) holds for every e > 0, we have limsupPr(A, (7; AT)) limsup[Pr(A, (7; AT)) ] < liminf PM (7; AT) c 0+ cn0+ ffnO+ < limsup PM(7; AT) < liminf[Pr(A, (7; AT)) + 2e,] a7n0+ cn 0+ liminf Pr(A, (7; AT)). (49) c 0+ But for any sequence {eI}, liminfPr(A, (7; AT)) < limsupPr(AI (7; AT). (410) n0+ n 0+ So we have liminf Pr(A n(7; AT)) limsup Pr(AI (7; AT)). (411) n0+ e0+ Thus lim 0'o+ Pr(A6n (7; AT)) exists for every positive sequence {en} converging to zero. Furthermore, all the inequalities in (49) are actually qualities and lim 0o+ PM('7; AT) exists for all sequences {on}. By fixing the sequence {In} and considering all possible positive sequences {of,} converging to zero, we see that lim,,0o+ PM(7; AT) = lime0o+ Pr(A,, (7; AT)) for all sequences {o,}. Thus by the definition of the limit of a function [55] we have lim PM(7; AT) lim Pr(A,7 (; AT)). n^0 Since the left hand side in (412) is fixed for all sequences {en}, by the definition of the limit of a function we have lim PM (7; AT) r0+ lim Pr(A( 7; AT)). (O+ Similarly, for a2 < T7(7, c) we have Pr(B,(7; AT)) c < PFA(7; AT) < Pr(B,(7; AT)) + 2c and hence we can show that lim PFA(7; AT) lim Pr(B (7; AT)). 0+ Let 7* be a threshold such that PFA(7*; AT) < 6 for all a2 < ]1i(6) and T Sh. Since the complementary conditional CDF FN(7; Arh) is a nonincreasing function of the threshold 7, the average probability of false alarm PA (7; AT) is a nonincreasing function of 7. For all a 2 < T11(6) we have 7* > inf{7 : PFA(7; AT) < 6, for all T i Sh}. (416) Since (416) holds for all a2 < 7,i(6), we have 7* > inf{7 : lim PFA(7; AT) < 6, for all T i Sh} ,70+ inf{7 : lim Pr(B,(7; AT)) < 6, for all T i Sh}, (417) where the equality follows from (415). Note that the expression on the right hand side of the equality in (417) is equal to 7m(6) defined in the statement of the theorem. (412) (413) (414) (415) Since 7* > 7m(6), for r c Sh we have PM(7*; AT) EH[FN(7*; ATh)] > EH[FN(7m(J); ATrh)] PM(7m(); Ar), (418) where the inequality follows from the fact that FN(7; Arh) being a conditional CDF is an increasing function of 7. From (413) and (418), we have lim PM(7*; AT) > lim Pr(Ac(7m(7); AT)), (419) c0+ 0+ for all r E Sh, which proves the first statement of the theorem. From (413), given ( > 0 and a threshold 7, there exists a u72(Q,) > 0 such that PM(7; AT) > lim Pr(A,(7; Ar)) (420) for all a 2 < 72(7, ). Then from (418) and (420), we have PM(7*; Ar) > Pm(7m(6); Ar) > lim Pr(A,(7m(6); Ar)) (421) for all a2 < T72(7m(J), ) (6, ). This proves the second statement of the theorem. E We present some discussion regarding the conditions and statement of the above theorem. From (42) and (42), it is clear that the set Ae(7; Ar) corresponds to a subset of H where PFA(7; Arh) or PD(7; Arh) (depending on whether r Sh or SC Sh) do not exceed e. Similarly, the set Be(7; Ar) corresponds to a subset of H where PFA (7; Arh) or PD(7; Arh) exceed 1 e. So the conditions of Theorem 1 require the decision statistic to be such that when the noise variance is small enough (or equivalently at high enough SNRs), the conditional probabilities of false alarm and detection are (with probability close to one) either close to zero or close to one. Furthermore using conditions (i)(iii) of the theorem and (43)(44), it is easy to see that at high SNRs PFA(7; A) Pr(B6(7; Ar)) and PM(7; Ar) Pr(A(7Q; AT)). Any threshold 7 which restricts PFA(7; Ar) to be less than some 6 will be larger than the smallest threshold 7m(6) which restricts Pr(Be(7; Ar)) to not exceed 6. So the theorem states that this lower bound on the threshold translates to a lower bound on PM (7; Ar) which may be nontrivial even in the .I'I :, 1 1 .ic scenario. If the threshold 7 is chosen carefully, then we have PM(7; Ar) > limo0+ PM (7; AT), but this is not true in general for all 7. So the lower bound on the .,vmptotic average probability of miss may not alv, i be a lower bound on the average probability of miss at finite SNRs. Nevertheless, the last statement of the theorem states that the lower bound in the .,vmptotic case is a good approximation for the lower bound on the average probability of miss at large (finite) SNRs. Thus the tradeoff between the PFA(7; Ar) and PM(7; Ar) at large SNRs can be characterized by the tradeoff between 6 and lim_0o+ Pr(Ac(7m(6); AT)). The main advantage of the theorem is that this tradeoff can be calculated using sets defined according to the conditional probabilities of detection and false alarm, which are usually easier to obtain. 4.4 Asymptotic Performance of Thresholdbased UWB Signal Acquisition In this section, we evaluate and compare the .i,iiiil 1 ic acquisition performance of the SAI and IAS approaches to the acquisition of UWB signals with timehopping spreading. We assume that the PRake receiver has Np fingers where Np < Ntap. Then for true phase 7, we choose the hit set as Sh { (N 1))T, (Np 2)Tc,..., T+ (Ntap )Tc}. The phases in the hit set correspond to those phases from which the PRake receiver can collect at least one resolvable path of the channel response corresponding to the true phase. This is not a reasonable definition for the hit set at finite SNRs since some of the resolvable paths may be too weak to enable good demodulation performance. Hence a receiver lock to such a path may not be considered successful acquisition. However, Sh defined as above contains any path where good demodulation performance can be achieved at a finite SNR. Thus it represents the largest possible hit set and consequently SC is the smallest possible nonhit set. This corresponds to the least restrictive choice of 77m(6) in Theorem 1. For this choice of Sh, the lower bound on the .,vmptotic average probability of miss is the smallest and hence it results in the best possible ., i,'l,". ic acquisition performance over all choices of Sh. 4.4.1 Asymptotic Performance of the SAI Approach In this subsection, we derive the .iv ii l ic performance of an acquisition system which takes the SAI approach. From (312), the decision statistic of the SAI approach is given by R(Ar; h) s(AT; h) + ny, (422) where, conditioned on h, ny is a Gaussian random variable with mean py and variance a 2(AT; h) + vf. The expressions for the mean and variance can be found in Section 3.2.1. The probabilities of false alarm and detection conditioned on the particular channel realization and given the decision threshold 7 are given as PA(7, ArTh) F(7; Arh), Sh = Q (A; h) Sh, S2 (Ar; h)+ Y PD(7,Arh) F(; Arh),r Sh = Q , Sh a 2 (AT; h)+ y In order to be able to apply Theorem 1 to this case, we need to first verify that the required conditions hold. Since the path gains are distributed according to N i.1 .; I, im distributions, h has an absolutely continuous distribution [56] and hence s(Ar; h) has an absolutely continuous distribution. Since Sp is finite, for any threshold 7 > 0 and every e > 0, there exists a K(7, c) > 0 such that for all fr Sp we have Pr({h : 7 K(7, ) < s(Ar; h) < 7 + (7Q, e)}) < (423) 2 Note that s(Ar; h) is a nonnegative random variable for all fr Sp. Then by choosing a positive integer n such that n1 < e/2 and a positive real number Ks > max{mean(s(Ar; h)) : p E p}, for all E Sp we get Pr({h : s(AT; h) > nKs}) < mean(( h)) < (424) nK, n 2 In Appendix C, we show that Ae(7; AT) and Be(7; AT) defined below satisfy the conditions of Theorem 1. A,(7; AT) {h: s(AT; h) < 7K (7, e)}. B,(7; AT) {h : 7 + (7, ) < s(AT; h) < nKs}. (425) Then for all e > 0, Pr(A,(7; AT)) Pr({h : s(Ar; h) < 7}) Pr({h : 7 K(7, e) < s(Ar; h) < 7}) > Pr({h:s(Ar; h) < 7}) Pr({h : 7 K(7, e) < s(Ar; h) < 7 + K(7, e)}) > Pr({h : s(Ar; h) < 7}) e, (426) where the last inequality follows from (423). Since Pr(Ae(7; AT)) < Pr({h : s(Ar; h) < 7}) for all e > 0, we have lim Pr(A,(7; AT)) Pr({h : s(AT; h) < 7}). (427) Similarly, we can show that lim Pr(B,(7; AT)) Pr({h : s(Ar; h) > 7}). (428) Then by Theorem 1, for any 6 > 0 if there exists a threshold 7 and an 1 i(5) > 0 such that PFA(7; AT) < 6 for all J.2 < T11(5) and for all T Sh, then lim PM(7; AT) > Pr({h : s(AT; h) < 7nm()}) (429) where m() inf{: Pr({h ( h) > }) < for all Sh0+ where 7m(6) inf{7 : Pr({h : s(A; h) > 7}) < 6, for all r i Sh}. Note that the lower bound on the ..imptotic average probability of miss in (429) results in the following upper bound on the ..imptotic average probability of detection, lim PD(7; AT) < Pr({h : s(Ar; h) > 7m(J)}), (430) ,70+ where f c Sh. By evaluating this upper bound as a function of 6, we obtain an ..i',ii! .1 ic receiver operating characteristic (AROC) which characterizes the best achievable tradeoff between the average probabilities of false alarm and detection. From the definition of 7m(6) and the expression for the upper bound in (430), we observe that the AROC for a particular f c Sh depends on the separation between the corresponding distribution of s(AT; h) and the distributions of s(AT; h) for all T Sh. For instance, if the distribution of s(AT; h) for some T c Sh is close to the distribution of s(AT; h) for any T Sh, then the upper bound on the .. i ,lI. ic average probability of detection for that f C Sh will be close to 6. The CDF of s(Ar; h) is needed to calculate the AROC. From (35), s(Ar; h) is a linear combination of independent random variables and hence its characteristic function is given by Ntap 1 Os(w;AT) H J (E1/R (0)rk(AT)w), (4 31) k0 where / (.)'s are the characteristic functions of the Gamma distributed hi's [51]. By the GilPelaez lemma [54], the CDF of s(Ar; h) is given by 1 1f e~itx s(t; AT)} Fs(x) + Imdt 2 Jo t 1 2f + IM jxtanO s( tan O; AT) dO, 2 7 o sin 20 (432) where the second equality is obtained by the change of variable t tan 0. The second integral has finite limits of integration and hence is more suitable for numerical evaluation. 4.4.2 Asymptotic Performance of the IAS Approach In this subsection, we derive the ..,",iii!i ll ic performance of an acquisition system which takes the IAS approach. From (322), the decision statistic is given by R(AT; h) [V(AT; h) + nz] (4 33) where nz is a zeromean Gaussian random variable with variance = and (A) given in (36). The probabilities of false alarm and detection conditioned on the particular channel realization and given the decision threshold 7 > 0 are given by PFA(7, AT h) N F(7; Arh), T Sh (/ V(Ar;h)\ / + V(Ar; h)\ =  + Q , ^ h,^ PD(7,ArT h) F F(7;AT h),T c Sh /y V(Ar;h)\ / + V(Ar;h) Sh As before, V(AT; h) has an absolutely continuous distribution and hence for any threshold 7 > 0 and every c > 0, there exists a K(7, c) > 0 such that for all T E Sp we have Pr({h : V(AT; h) V/ < Kt(, c) or V(AT; h) + / < t(7, c)}) < e. (434) In Appendix D, we show that Ae(7; AT) and Be(7; AT) defined below satisfy the conditions of Theorem 1. A(7;Q AT) {h : + K(7, c) < V(AT; h) < V (7, c)}. B,(7; AT) {h : V(AT; h) > / + K(7, e) or V(AT; h) < t(7, c)}. (435) We can also show that lim Pr(A,(7; AT)) Pr({h : 7 < V(AT; h) < }), lim Pr(B,(7; AT)) Pr({h: V(AT; h) > 7 or V(AT; h) < }). c*O+ Then by Theorem 1, for any 6 > 0 if there exists a threshold 7 and an rqi(6) > 0 such that PFA(7; Ar) < 6 for all a'2 < r1(6) and for all r Sh, then lim PM(7; AT) > Pr({h : 7(J) < V(AT; h) < ym)}) (436) 70+ where 7m(6) inf{7 : Pr({h : V(Ar; h) > /7 or V(Ar; h) < /7}) < 6, for all r i Sh}. Finally, we have the following upper bound on the ,imptotic average probability of detection, lim PD(7; AT) < Pr({h : V(Ar; h) > /7m(6) or V(Ar; h) < }), ,70+ where ce Sh. Once again, the AROC calculation requires the CDF of V(Ar; h). Since the polarities pk and path gains hk are independent, the characteristic function of V(Ar; h), in this case, is given by Nta p (1 Erk(AT)a;) + , (/Elrk(AT)w;)] Qv(w; Ar) ii rElrk(Arw)2 (4 37) where / (.) is the characteristic function of the N .1;, i,,nim distributed hk [51]. Substitution of the above equation in (432) yields the CDF of V(Ar; h). 4.4.3 Numerical Results To calculate the AROC for the SAI and IAS acquisition schemes, we choose the following values for the system parameters: the TH sequence period Nth = 1024, Nh = 16, M = 1, the length of the channel response Ntap = 100, the number of PRake fingers Np = 5 and Nf 116. We assume that Etot = 20.4 dB which is its mean value when the transmitterreceiver separation is 10 m [47]. We choose the power ratio r = 4 dB, decay constant e 16.1 dB and fading figures mk = 3.5 , 0 < k < Ntap 1, which are their mean values given in Cassioli et al. [47]. The best AROC, which is again a plot of limo0+ PD(7m(6); 0) versus 6, does not depend on the received power and hence we set El =1. For the UWB channel model we have chosen, the .,,:iiiII I ic average probability of detection is largest when AT = 0. Thus the best AROC is a plot of limo0+ PD(7m(6); 0) versus 6. Fig. 41 shows the best AROC of the SAI approach for EGC window sizes G 1, 2, 5, 10 and 15. The AROC becomes worse as the EGC window length increases and is best for G 1, which is equivalent to the case when there is no EGC. This is consistent with the finite SNR results where we found that performing EGC for acquisition is not advantageous at high SNRs. As G increases the signal energy collected by the EGC window s(Ar; h) increases both when r = 7 and r Sh. For r = T, the additional energy collected is from the NLOS paths which are weaker in comparison to the LOS path and thus the increase in signal energy is relatively small. The increase is more significant when r Sh since the additional energy is comparable to the energy collected when G 1. Thus the separation between the distributions of s(Ar; h) when r = and r 4 Sh decreases, causing the AROC to get worse. Fig. 42 shows the best AROC of the IAS approach for EGC window sizes G 1, 2,5, 10 and 15. The upper bound on the .,vmptotic average probability of detection is almost trivial for G 1 and becomes significantly restrictive as G increases. As G increases, for r = r the EGC window collects multiple paths which may have opposing polarities resulting in cancellations and hence a decrease in the probability of detection. For G 1, this cancellation is absent when r = 7 but still occurs when r Sh since the random timehopping sequence facilitates collection of multiple paths. Thus the signal energy collected when r Sh is much smaller than the signal energy collected when r = r, resulting in a significant separation between the corresponding distributions of V(Ar; h). Hence the AROC is not restrictive for G 1. Since the best AROC is just an upper bound on the AROCs of all the hit set phases, we plot for G = 1 the AROCs of the phases e c Sh corresponding to AT 5Te, 10Te, 15Tc, 20T, and 30Tc in Fig. 43. We see that even for IAS with G 1 the bound on the ., in', ii ic average probability of detection becomes increasingly restrictive as the distance of the hit set phase from the LOS path increases. This is because the energy in the paths decays with increase in distance from the LOS path. 4.5 Conclusions A typical 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 [10]. The large dwell time increases the effective SNR of the decision statistic and in the absence of channel f ,dii. this results in accurate verification. In this chapter, we evaluated the .,vmptotic performance of thresholdbased timing acquisition systems in the presence of multipath fading and found that, no matter how large the SNR is or how we choose the threshold, there are fading scenarios in which false alarms and misses occur with nonzero and sometimes significant average probability. Thus it may not be possible to build a good verification stage for thresholdbased acquisition systems operating in such channels by just increasing the dwell time. We found that if we choose a threshold such that the average probability of false alarm is less than a given tolerance, then there is a possibly nontrivial lower bound on the .,i~iHii.l, I ic average probability of miss. This lower bound translates to an upper bound on the .i,iiiil ,lI ic average probability of detection. We evaluated this upper bound for two thresholdbased approaches, namely SAI and IAS, for the acquisition of UWB signals with timehopping spreading. For SAI, we found that the upper bound on the .,vmptotic average probability of detection was significantly restrictive for all values of EGC window size. But for IAS, the upper bound was almost trivial atleast for some hit set phases when there was no EGC being done. Nevertheless, there were still some hit set phases where the upper bound was restrictive. These results seem to ,. 1 that EGC may not be a good strategy to improve acquisition performance. More importantly, they ,.; 1 that acquisition might be a potential bottleneck on throughput in any UWBbased packet network employing thresholdbased acquisition systems. Z50.6 . 0.5 E > 0.4 'C Figure 41: Best AROC of the SAI approach to UWB signal acquisition. Z0.6 / E 0.5 / / >0.4. / 0.3  0.2  0.1 0 I 0 0.1 0.2 0.3 0.4 0.5 5 0.6 0.7 0.8 0.9 Figure 42: Best AROC of the IAS approach to UWB signal acquisition. 50.6 C O. 0.5 0.5 S0.4: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 43: IAS AROC corresponding to hit set phases other than the LOS path when G 1. CHAPTER 5 A SEARCH STRATEGY FOR UWB SIGNAL ACQUISITION 5.1 Introduction When there are multiple elements in the hit set, the serial search may no longer be the optimal sequential search strategy. In this chapter, we consider the problem of finding efficient search strategies in the set of all search strategies which are permutations of the search space. Finding the optimal permutation search strategy which minimizes the mean detection time when the search space is large and the probabilities of detection of the hit set elements are arbitrary turns out to be prohibitively complex. However, if we assume the probabilities of detection of all the hit set phases to be equal then there exists a permutation search strategy which minimizes the mean detection time. Since the actual probabilities of detection are not equal, this search strategy although not optimal serves as a useful heuristic solution to an otherwise intractable problem. Furthermore, we see that this search strategy has a simple JumpbyH structure and improves the mean detection time by a significant amount compared to the serial search. The features of the UWB system model relevant to the problem considered are briefly described in Section 5.2. The mean detection time of an arbitrary permutation search strategy is calculated in Section 5.3 and the best permutation search strategy under the assumption of equal probabilities of detection is found in Section 5.4. We present some numerical results in Section 5.5 quantifying the improvement in mean detection time performance followed by some concluding remarks in Section 5.6. 5.2 System Model In this section, we briefly describe those aspects of an UWB acquisition system which are relevant to the problem of finding efficient search strategies. A more detailed description can be found in earlier chapters. It was found that the IAS approach without EGC (i.e. with EGC window size equal to one) was the better strategy i.. lii; that EGC may not be a good method to utilize the energy in the multipath to improve acquisition performance. In this chapter, we consider the IAS acquisition system without EGC which has the structure shown in Fig. 32. The transmitter transmits a periodic signal with period NsT, during the acquisition process, where To is the UWB pulse duration and Ns is a positive integer. We assume that the pullin range of the tracking loop is To and hence the acquisition search only needs to search the timing ambiguity region in increments of Tc. The timing ambiguity region is equal to the period of the transmitted signal and hence the search space, which is the set of all hypothesized phases, is given by {0, To, 2Te,..., (Ns 1)To}. The received signal is correlated with a locally generated reference signal and the correlator output is squared to generate the decision statistic R(Ar; h) where AT = r 7, the difference between the hypothesized phase r and the true phase 7 of the received signal, and h is a random vector containing the channel taps. The decision statistic R(Ar; h) is compared to a threshold 7 and the hypothesized phase r used to generate the reference signal is accepted as an estimate of the true phase of the received signal if the threshold is exceeded. If the threshold is not exceeded, the process is repeated with a new value for the hypothesized phase. A search strategy is then the sequence of hypothesized phases which are checked until the threshold is exceeded. We will find it convenient to represent the search space by S = {1, 2, 3,... Ns}, where the integer n indexes the hypothesized phase (n 1)Tc. As mentioned earlier, there may be multiple phases in a dense multipath environment which can be considered a good estimate of the true phase. Once again, we assume that a partial Rake (PRake) receiver [50] is employ, ,1 for demodulation and hit set in this case has been derived in C'! Ipter 3. The hit set Sh is typically a block of H consecutive phases in the search space Sp where two elements i, j are considered to be consecutive if ji j (mod Ns) = 1 or Ns 1. For a particular value for the true phase of the received signal, the position of the first element of the hit set block is p, which is assumed to be equally likely to be any element of Sp. Given p, the positions of all the hit set elements are completely specified. When p > Ns H + 1, the last p Ns + H 1 hit set phases wrap around and are represented by the first p Ns + H 1 phases of the search space. This is due to the periodicity of the transmitted signal. 5.3 Mean Detection Time Calculation The problem of finding the optimal permutation search strategy when the probabilities of detection are arbitrary is complex. But if we assume that the probabilities of detection in the hit set elements are equal, we are able to find a suboptimal search strategy which reduces the mean detection time significantly. This permutation search strategy serves as a useful heuristic solution to the otherwise intractable problem. So we proceed to find the permutation search strategy which minimizes the mean detection time under the assumption of equal detection probabilites in all hit set elements. We first calculate the mean detection time when the search strategy is an arbitrary permutation R of the search space. Let Pd be the average probability of detection in any hit set element. For a particular initial position p of the hit set in the search space, let the positions of appearance of elements of hit set elements in the sequential search be {tp,i i 1, 2,... H}. So the first appearance of a hit set element is at tp,i, the second appearance is at tp,2 and so on. Table 5.6 illustrates this for the serial search starting in position 1 of Sp when A = 8 and H = 3, where the positions in boldface indicate the presence of a hit set element. The last three columns of the table contain the positions of the first, second and third appearances of a hit set element for a particular value of p. Table 5.6 shows the positions of appearance of the hit set elements for the permutation search strategy (1, 4, 7, 2, 5, 8, 3, 6) when A = 8 and H = 3. Note that the columns indicating the presence of hit set elements in Table 5.6 are obtained by permuting the corresponding columns of Table 5.6. Also note that a hit set element appears in every position of the permutation exactly H times where each appearance corresponds to a distinct value of p in Sp. It is easy to see that this is true for any permutation search strategy and for all values of NA and H. A detection event is defined by the position tp,j where we have a hit and a particular number of misses j of Sh. Let T be the dwell time of the correlator. The time taken for a miss event is Ns. The time for a particular detection event defined by (p, i, j) is then T(p, i, j) = tpiT + jNsT. (51) The probability that there is a hit in position tp,j is given by Phi) = Pd(l Pd)'1. The probability of j misses of Sh is equal to Pj1 where PM (1 Pd)H. The mean detection time conditioned on the fact that the first element of the hit set is in position p of the search space is given by H oc Tdet(p) T 9 h(p, 1,)P Ph(1) i=1 j=0 H oo [t,iT jNsTPPh( T3H71 tp,i) NsTPM ZH1 jPh 1 PM (1 PM)2 T I tp, Ph(i) NsTPM .(52) 1 PM 1 PM The mean detection time is then given by 1 N T EH1 ( tp,) Ph (i) NsTPM Tdet = Tdet (p) + (53) Ns ,Ns (t PM) 1 PM Note that the second term in the right hand side of (53) does not depend on the permutation R. Then any optimization with respect to R can only hope to minimize the first term. 5.4 The JumpbyH Permutation Search Strategy We want to minimize g(s) E= 1 siPh(i), where s = (SH, HI ..., 1) and si = Y lt N p,j, over all permutations of S. Note that SH > SH1 > ... > si. By the fact that H1 i 2N N +1)H and that Ph(i) is a decreasing function of i for all Pd, we have the following result. Lemma 1. g(s) is Schurconcave [57] on A ={s = (SH,..., S) Si pNltp,i i = 1, 2,..., H, for some permutation R of S}. Proof. Let ) = {(x1,. .., x,) : x1 > ... > x,}. Note that A is a subset of D. For all s C D and k= H,H 1,...,2, H 9(SH,..., Sk+, Sk+ C,Sk1 C, Sk2,...,SI) siPh(i) e[Ph(k 1) Ph(k)], i= 1 which is decreasing in c since Ph(k 1) > Ph(k). The Schurconcavity of g(s) on D follows from Lemma 3.A.2 in Marshall and Olkin [57]. Since A is a subset of D, g(s) is Schurconcave on A. D Thus g(s) is minimized if s is the maximal vector of A. If x < y, i.e. if x is in i i .ed by y for some x, y c A, then k k xi > y, k = ,...,H (54) i= 1 i= 1 and H H Xi yi (55) i= 1 i 1 where xi and yi are the (H i + 1)th components in x and y respectively. Lemma 2. Let rk be not greater than the minimum value of >i s over all permutations of S for k= 1,...,H1 and rH N(N+1)H. If r+2 ri+ > r+ ri for i 0,...,H2, then the vector q (rH rH1,rH1 rH2,... ,r2 r,rl), (56) ,,,;../.,. all the vectors in A. Proof. By hypothesis, we have qi < q2 < ... < qH and H i N (N+1)H where qi is the (H i + 1)th component of q. Let s c A and let si be its (H i + 1)th component. Since s1 < S2 < ... < SH, the sum of the k smallest components of s is k k se >1 rk (rk rk1) + + (r2 rl) + rl (57) i= 1 i= 1 for k = 1, 2,... H 1. Furthermore, 1 Si (+1). Thus q i .es s and since the choice of s was arbitrary, q in, i ii.... all the vectors in A. D We now proceed by finding one particular set of rk's which satisfy the conditions of Lemma 2 and then exhibit a permutation R of S whose corresponding vector x C A is equal to the vector q defined by these rk's. This vector x then i i i.i., all the vectors in A. Hence the permutation search strategy R minimizes the mean detection time. Theorem 2. The minimum value of Z1 si over all permutations of S is Nk(Nk+1)H (Nsk NkH)(Nk + 1) for k = 1,..., H, where Nk = [N These minima are all simulton ...'l,' achieved by a permutation R of S given by R (i 1)H (mod Ns) + [ +1, (58) where Ri is the element in its ith position and d is the greatest common divisor (GCD) of Ns and H. Thus the search str l. ,i/ R is optimal in the set of permutation search strategies. Proof. First, we note that it is not entirely obvious but easy to show that (58) does indeed define a permutation. Suppose Ri and Rj are equal for some integers i, j such that i = () + n andj = m() + nj where 1 < n, nj < W and 0 < 1, m < d 1. If I m, then R Rj = 0 if and only if (i j)H (mod Ns) = 0. Since In, nj\ < () 1, this implies i = j. Now suppose (without loss of generality) that I > m. Then RiR, = [(1mn)(N)+ninj]H (mod Ns) + I m = (n n)H (mod Ns) + m, (59) which is not equal to zero since the first term is a multiple of H and the second term I m is not greater than d 1 < H. Thus the Ri are distinct for i = 1,...,Ns and it then follows by the pigeonhole principle that R is a permutation of S. There are NA number of tp,j's in Z 1I si, where each tp,j is in the set {1, 2,..., Nsj with the restriction that each distinct value of tp,j appears at most H times. We can obtain a lower bound of Z i s by assigning the smallest values in {1,... Ns} to the tp,'s such that each value is assigned H times. Then the elements in the set {1,..., Nk} are each assigned H times and Nk + 1 is assigned Ns NkH times where Nk = [J. Thus we have k si > H+..+NkH+(Nk+1)(NsNkH) i= 1 Nk(Nk2 + )H + (Nsk NkH)(Nk + 1), (510) 2 for k 1,...,H. Let rk be equal to the lower bound obtained in (510), i.e., S N(Nk + )H + (N NkH)(Nk + ) (5 11) 2 fork= 1,...,H. Then H = N(N and rk+1 k (Nk + ).(H Nsk+NkH) + (Nk + 2).H+...+Nk+.H +(Ns(k + 1) Nk+H). (Nk + + 1), for k = 0,1,..., H 1. For each k E {0, 1,..., H 1}, rk+1 rk is a sum of Ns terms belonging to the set {Nk + 1,... Nk+ + 1} with each distinct value appearing at most H times. Since Nk+1 > Nk + M, rk+2 rk+1 > rk+1 rk for i = 0, 1,... H 2. Thus the rk's satisfy the conditions in Lemma 2 and consequently q (rH TH1, IH1 rH2,..., 72 r1, 1) (512) i: ii i.. P all the vectors in A. In Appendix E, we show that the vector x E A corresponding to the permutation R defined in (58) equals q. E It is easy to see that R has a JumpbyH structure. R consists of d consecutive blocks each containing m elements, where the ith block consists of the elements (i, H + ,..., ( 1)H+i) for 1 < i< d. 5.5 Numerical Results In order to compare the mean detection time performance of the heuristic permutation search strategy with the serial search strategy, we chose the following values for the system parameters: Size of the search space Ns = 29696, To = 2 ns, dwell time T = Nsc, and SNR = 7,10 dB. The hit set was obtained under the assumption that the PRake receiver has 5 fingers and the nominal uncoded BER requirement is An = 103. The threshold was set according to (331) with 6 = 0.05. Table 5.6 shows the mean detection times for the serial search and heuristic search strategies. The mean detection time of the serial search strategy does not change much with increase in SNR even though the size of the hit set increases significantly. This is because the mean detection time is dominated by the time spent by the acquisition system in evaluating and rejecting the nonhit set phases before it reaches the hit set. The heuristic permutation strategy provides an improvement of more than 71 in the mean detection time compared to the serial search. 5.6 Conclusions We began with the observation that the serial search may no longer be the optimal search strategy when the hit set consists of multiple phases which is the case for the dense UWB channel. We provided a heuristic suboptimal solution to the generally intractable problem of finding the permutation search strategy which minimizes the mean detection time by assuming that the detection probabilities of all hit set elements are equal. We also found that the heuristic search strategy has a simple JumpbyH structure and hence it can be generated easily obviating the need to store the whole permutation. Table 51: Serial search for N, P t_____p, tp,2 tp,3 8 and H = 3. m 1 r I 1 i i Table 52: Permutation search (1, 4, 7, 2, 5, 8, 3, 6 = 8 and n = 3. SNR Hit set size Serial Search MDT Heuristic Search MDT 7 dB 25 0.8808 s 0.2498 s 10 dB 40 0.8803 s 0.1993 s Table 53: Mean detection time (MDi)T) values for the serial search and hueristic search strategies. p tp,1 tp,2 tp,3 1 1 4 7 2 5 8 3 6 1 4 7 2 1 4 7 2 5 8 3 6 2 4 7 3 1 4 7 2 5 8 3 6 2 5 7 4 1 4 7 2 5 8 3 6 2 5 8 5 1 4 7 2 5 8 3 6 3 5 8 6 1 4 7 2 5 8 3 6 3 6 8 7 1 4 7 2 5 8 3 6 1 3 6 8 1 4 7 2 5 8 3 6 1 4 6 1 ,'i A fl r^ no r ti / \ C ~AT if 1 IF CHAPTER 6 UWB TIMEOFARRIVAL ESTIMATION STRATEGIES 6.1 Introduction The high time resolution of UWB signals i . I the possibility of building precise location estimation systems based on timeofarrival (TOA) measurements. Impulse radio is one of the two optional physical 1v.r specifications identified by the IEEE 802.15.4a standardization group for low data rate communications combined with high precision ranging/location capability (1 meter accuracy and better). The lowpower and lowcost implementation of UWB ranging systems will enable a wide range of applications, including logistics (package tracking), security applications (localizing authorized persons in highsecurity areas), medical applications (monitoring of patients), search and rescue (locating survivors in avalanche/earthquake rubble), and military applications. The simplest method of calculating the distance between two .I, chronous transceivers consists of using a packet exchange to get a measure of the signal roundtrip timeofflight (TOF) and using this time to calculate the distance. A terminal (the requester) which wants to estimate the roundtrip TOF sends packets to the other terminal (the responder) which responds after a predetermined delay. The d. 1iv enables the requester terminal to switch from the transmitting mode to the receiving mode. Once the responder terminal's packets are received by the requester terminal, it can estimate the roundtrip TOF and hence the TOA. This scheme is illustrated in Fig. 61. If the TOAs between a mobile terminal and three distinct anchors (nodes whose positions are known a priori) are available at a fusion center, the mobile position can be easily computed in the twodimensional plane by calculating the intersection of the circles with radii corresponding to the individual distance estimates of the mobile terminal from the anchors (as shown in Fig. 62). The goal of the TOA estimation algorithm is to find the TOA of the earliest path, which we will henceforth refer to as the LOS pathI In a packetbased TOA estimation protocol, the acquisition of the packet is the first operation which is performed. As we have seen in earlier chapters, the receiver may not lock to the LOS path. In this chapter, we propose strategies to locate the LOS path after successful acquisition under different assumptions about the knowledge of channel statistics. The chapter is organized as follows. In Section 6.2, we develop a TOA estimation algorithm under the assumption that the channel statistics are completely known. For the case of unknown channel statistics, we develop a heuristic TOA estimation algorithm in Section 6.3. In Section 6.4, we evaluate the performance of these estimation algorithms using probability of incorrect estimation and mean estimation error as performance metrics. Section 6.5 has some concluding remarks. 6.2 UWB TOA Estimation: Known Channel Statistics Like most receiver operations, the TOA estimation algorithm will be executed after the acquisition operation. We proceed with the design of the TOA estimation algorithms under the assumption that the acquisition system has successfully locked to a multipath component in the hit set. This simplifies the design and enables isolation of the TOA estimation algorithm performance from the performance of the acquisition step preceding it. In the case of known channel statistics, the hit set is known and the successful acquisition assumption leads us to a Mary hypothesis testing problem where M is the number of multipath components in the hit set. We will be collecting a number of observations around the path the acquisition system has locked to. The distribution of these observations depends on which hit set element was captured by the acquisition 1 The earliest path may be one which passes through several obstacles and hence is not a LOS path in the conventional sense. system. A correct resolution of the true hypothesis generating the observations results in the identification of the hit set element the acquisition system has locked to. We can then obtain the location of the beginning of the multipath profile since the position of the earliest path relative to each hit set element is known. We obtain a vector of W1 + Wr + 1 observations with a chip spacing of T, seconds around the path the acquisition system has locked to. As shown in Fig. 63, Wi of these observations are taken to the left of the acquisition lock position and Wr of them are taken to the right of the acquisition lock position. Including the observation taken at the lock position, we have a total of Wi + Wr + 1 observations. Each observation is obtained by correlating the received signal at the observation position with a reference signal over a duration of Np, frames. As in the derivation of the IAS decision statistic, the reference TH signal is given by Npl1 q(t) = Y r(t IT, cT T), (61) l=0 where r is chosen such that the pulse rr(t) is aligned with the observation location. The observation is given by 1 fT+N Tf Ntap 1 y= N I r(t)q(t)dt E qk (AT)pkhk + ny (62) Np J k0 where ny is a zeromean Gaussian random variable with variance o2y = 7N and qk (AT) is equal to 1 if the observation location corresponds to the kth path of the multipath profile and it is 0 otherwise. Since the channel taps are placed Tc apart, qk(Ar) can be 1 for at most one value of k. Thus each observation is either Gaussian distributed or has a distribution of a random variable which is the sum of a flipped N ,1: .;, ,iii random variable and a Gaussian random variable. The pdf of the observation for the latter case is derived in Appendix F. Let Nobs W1 + Wr + 1. Let the observation vector be Y [Y Y2 ... YNobs]T. Let M be number of multipath components in the hit set the acquisition system can lock to. Let the likelihood of observing Y when the acquisition system has locked to the ith hit set element be pi(Y), i = 1, 2,..., M. Thus we have a Mary hypothesis testing problem with hypotheses 7i : Y ~ pi. (63) for 1 < i < M. Then under the assumption that the acquisition system is equally likely to lock to any one of the M hit set elements, the decision rule d(Y) which minimizes the probability of incorrect decision is d(Y) j if pj(Y) > p(Y) V 1 < < M. (64) If Li is the location of the ith hit set element, the error induced by deciding on 7i when %j is the true hypothesis is given by Ci = Li Lj Under the assumption of equally likely hypotheses, the average error induced by deciding on Hi when Y is observed is given by M QC(Y) CGp(Y) (65) j1 for 1 < i < M. Then the decision rule which minimizes the average error is given by d(Y) if Cj(Y) < C(Y) V 1 < i < M. (66) 6.3 UWB TOA Estimation: Unknown Channel Statistics In this case, we once again assume that the acquisition system has locked to a multipath component in the hit set. Using the method described in the previous section, we collect Nobs observations around this position. When the channel statistics are not known, the distribution of the observation vector Y can be modeled as a Gaussian vector with mean p and covariance matrix oa where I is the Nobs x Nobs identity matrix. The ith component of the mean vector tt, pi, is nonzero if the ith observation location contains a multipath component and is zero otherwise. Since the hit set is not known, we cannot use the Mary hypothesis testing formulation of the previous section. However, if the number of observations to the left of the locked path, Wi, is large enough, we can hope that the observation window starts from a position which is to the left of the multipath profile. In this case, the observations to the left of the multipath profile will have zero mean. We could try to cast this problem as a (Nobs 1)ary hypothesis testing problem with the following hypotheses.2 i: p~j 0 for 1 < j < i, t+1 / 0 (67) for 1 < i < Nobs 1. However, the values of the nonzero mean components are still unknown. One way to solve this problem is an extension of the generalized likelihood ratio test (GLRT) to multiple hypothesis testing. In the GLRT, one substitutes the values of the unknown parameters with their maximum likelihood estimates. Unfortunately, this approach is not viable for the situation here. To see this, consider the following hypotheses for the situation Nobs 3. RH : g = [ 0 o ]T, 2 / 0 R2 : [ 0 0 ]T 3 / 0. The density of the observation vector Y =[ y1 2 Y3 ]T under the hypotheses is v + 1 i + (22 2 + 2 pi(Y; P2, P3) 3 exp 2 (68) P2(Y; P3) exp 2 (69) (2(2 7 72 2L 7 For the case when both Y2 and y3 are nonzero, the maximum likelihood estimates of the unknown parameters are P2 Y2 and p3 Y3. Then pi(Y; P2, p3) > P2(Y; i3), for all such 2 Note that the formulation implicitly assumes that at least one observation falls to the left of the multipath profile. Y. Since a nonzero observation vector occurs with probability one under both hypotheses, for equally likely hypotheses the test which minimizes the probability of error will aliv choose i1. Thus the GLRT approach to dealing with the unknown means is not feasible for this situation. We propose to deal with the problem of the unknown parameters by performing local decisions on each of the observation vector components and using these local decisions to locate the LOS path. Even though this is a heuristic solution, it performs reasonably well as evidenced by the numerical results in the next section. For each component of the observation vector, we consider the following binary hypothesis testing problem 'Ho : pi 0 HI : pi / 0 where yi ~ AV(pj, ,a). Under the usual conventions, a false alarm is the event of choosing H1I when 'Ho is true and a detection event is the event of choosing 71 when 71 is true. We constrain the probability of false alarm to a small value 6 < 1 and seek the uniformly most powerful (UMP) test, i.e., a test which maximizes the probability of detection for all nonzero values of the unknown parameter pi. Unfortunately, a UMP test does not exist for this situation. This is because the most powerful test for positive values of p~ does not coincide with the most powerful test for negative values of Pi. However, if we restrict our attention to unbiased tests, i.e. tests for which the probability of detection is at least 6 for all values of the unknown parameter p~, there exists a UMP unbiased test which is given by the following C'! ... i if yi > 7 (610) where 7 yQ'(6/2). Applying the above binary test on each component of the observation vector Y results in a vector of binary decisions where the positions corresponding to the hypothesis 1Hi give an approximate indication of the location of the multipath components. One way to estimate the location of the LOS path is to choose it to be the position of the earliest HiH decision in the binary decision vector. In the next section, we evaluate this and other heuristic methods to locate the beginning of the multipath profile using this binary vector. 6.4 Numerical Results In this section, we evaluate the performance of the TOA estimation schemes developed in the previous sections using the probability of incorrect estimation and the mean estimation error as the performance metrics. We investigate the effect of the channel model and the location and size of the observation window on the estimation performance. For each channel model and observation window specification, the number of pulses used in generating the correlation statistic, Npl, is increased which in turn results in a linear increase in the signaltonoise ratio. 6.4.1 Dense UWB Channels For dense UWB channels, the multipath profile can be modeled as a tapped delay line with regular tap spacings. The channel model described in Section 2.2 is an example of a dense UWB channel and will be used for evaluating the performance of the TOA estimation schemes. We choose the following values for the system parameters: the length of the channel response Ntap = 100, Nf 116, Nh = 16. The hit set is obtained under the assumption that the PRake receiver has 5 fingers and the nominal uncoded BER requirement is An = 103. The channel statistics are set to the values used in Section 3.5. For the case of known channel statistics, Figs. 64 and 65 show the probability of incorrect estimation of the LOS path location and the mean estimation error as a function of the number of pulses in the correlation, Npl, for the decision rule described in (64), respectively. These results are for the case when the average energy received per pulse to noise ratio E1Etot 5 dB. The number of multipath components in the hit set for this case is 13. The performance metrics are plotted for different values of W\ and Wr, the number of observations taken to the left and to the right of the acquisition lock position, respectively. The performance does not vary with changes in Wr as long as the value of W1 is larger than the hit set size. But once the value of W1 falls below the hit set size the performance degradation is significant, as seen in the cases when W = 10 and WI = 5. This is because the left edge of the multipath profile is the location of a sudden change in channel statistics in dense multipath channels. For the values of WI which are larger than the hit set size, this left edge ahv falls within the observation window. Thus a decision rule based on the channel statistics is able to perform better for these values of W1. A similar trend is observed in Figs. 66 and 67 which show the performance metrics for the decision rule described in (66). For the case of unknown channel statistics, we perform the binary hypothesis test of (610) on each observation vector component with 6 = 0.01. The location of the LOS path is chosen to be the leftmost position in the observation vector where the binary test chooses 'Hi\ three times consecutively.. This is a valid heuristic in a dense UWB channel where the LOS path is immediately followed by other multipaths. We evaluate this decision rule using the same channel model as the previous decision rules to enable a fair comparison. Figs. 68 and 69 show the performance metrics for this rule which requires a larger number of pulses in the correlation to achieve performance comparable to the previous decision rules. Once again, the performance is severely degraded if the value of WI is smaller than the hit set size. This is because the test will ahv i fail if the beginning of the multipath profile does not fall in the observation window and the chance of this event occurring increases when WI is smaller than the hit set size. The problem, however, is that the size of the hit set is unknown when the channel statistics are not known. So larger than necessary observation window sizes might be required to guarantee good performance of this heuristic estimator. 6.4.2 Sparse UWB Channels A sparse UWB channel consists of clusters of arriving paths [49]. The impulse response of a sparse UWB channel can be expressed as L Ki ssp(t) Y Pk,l k,l (t TI k,l), (6 11) where L is the number of clusters, K, is the number of multipath components in the lth cluster, Pk,l, hk,l are the sign and amplitude of the kth component of the Ith cluster, T, is the arrival time of the Ith cluster and Tk,l is the delay of the kth component of the Ith cluster relative to the lth cluster arrival time. The number of clusters L is Poisson distributed with probability mass function L(1) (L)i exp(L) where L is the mean of L. The distribution of the cluster arrival times is given by the Poisson process p(Ti IT_) Aexp[A(Ti TI1)], I > 0, (6 13) where A is the cluster arrival rate. The distributions of the ray arrival times are given by p(Tk,l T(k1),i) = Aexp[A(Trk T(k1),1)], k > 0, (614) where A is the ray arrival rate within each cluster. As in the dense UWB channel case, we model the sign of the a ray component, Pk,i, to be equally likely to be 1 or 1 and its amplitude hk,l to be N ,1: ini distributed. For this channel model, the decision rule which assumes knowledge of the channel statistics becomes prohibitively complex. To see this, let T denote the set of all possible cluster delay and ray delay realizations. The fading figures and energies of the rays are assumed to be delay dependent with this dependence known. Given the realization r E T and observation vector Y, the likelihood of the ith hypothesis is given by pi(Y; r). Then the actual likelihood of the ith hypothesis is given by ETcrPi(Y; r). However, the number of realizations in T increases exponentially with the observation window size. Also, the sparseness of the channel warrants a large observation window as the acquisition lock might occur quite far from the LOS path. So for the case of sparse UWB channels, we restrict our attention to decision rules which do not assume knowledge of channel statistics. Once again, we perform the binary hypothesis test of (610) on each observation vector component with 6 = 0.01. We locate the leftmost position in the observation vector where see a pattern of three consecutive 'Ho decisions followed by a HiH decision and at least one more Hi1 decision in the next two binary decisions. The location of the LOS path is decided to be the first 1HI decision in this pattern. Figs. 610 and 611 show the probability of incorrect estimation of the LOS path location and the mean estimation error as a function of the number of pulses in the correlation for this decision rule. We choose the mean number of clusters L = 3, the cluster arrival rate A = 0.047 ns1 and the ray arrival rate A = 1.54 ns'. The decay constant of the energy of a path is F 12.53 ns, i.e., a path at delay 7 is weaker than the LOS path by a factor of exp(r/F). We neglect paths which are 30 dB weaker than the LOS path. The probability of incorrect estimation is higher than that for the case of dense channels and increases significantly for values of Wi less than 50. The mean estimation error has the same trend but there is a slight increase for large values of Npl. An increase in the number of pulses used in the correlation reduces the variance of the noise and hence smaller thresholds are sufficient to constrain the probability of false alarm under Ho by 6. But a smaller threshold results in weaker paths being detected. When the value of W1 is small, the observation window might start in a position between two clusters and the weaker paths of the first cluster prevent the consecutive 'Ho decisions from occurring until the beginning of the second cluster. This results in the increase in estimation error since the second cluster is farther from the LOS path. 6.5 Conclusions We have developed TOA estimation schemes under the assumptions of known and unknown channel statistics. These schemes have been evaluated in dense and sparse UWB channels. For the case of dense UWB channels, the schemes developed under the assumption of known channel statistics are capable of achieving probabilities of incorrect estimation less than 0.01 and mean estimation error less than 0.1 ns, while the schemes developed under the assumption of unknown channel statistics achieve a probability of incorrect estimation of less than 0.02 and mean estimation error less than 1 ns. However, sparse UWB channels turn out be challenging with the schemes using the channel statistic information becoming prohibitively complex and the schemes which do not use this information resulting in probabilities of incorrect estimation around 0.1 and mean estimation error around 7 ns. Roundtrip TOF SResponse Delay TOA Response packet Acquisition header Transmitted packets Communication payload D Received packets Figure 61: Illustration of the packet exchange scheme used to estimate the TOA. Di = Speed of light X TOAi Figure 62: Mobile positioning based on TOA measurements. Request packet IUA / A 1 W 1 bins Wr bins lIlll I lll I Jlll Multipath profile 4 Acquisition lock Figure 63: The location of the observations used for TOA estimation. 1 1 W = 2( + W= 2( 0.9 x W =2( W = 1! 0.8  W= 1( WI = 5, g 0.7 ( 0.6 0.5 . 0.4 o n 50 100 150 200 250 300 Number of pulses in correlation Figure 64: Probability of incorrect estimation in dense channels for the rule which minimizes the error probability when the channel statistics are known. 10  W = 20, Wr = 15 + W= 20, Wr = 10 9  W, = 20, W = 5 SW = 15, Wr = 15 8 W= 10, Wr= 15 W,= 5, W= 15 6 3 2 1 3 0 50 100 150 200 250 300 Number of pulses in correlation Figure 65: Mean estimation error in dense channels for the rule which minimizes the error probability when the channel statistics are known. S0.7 0.6 o 0.5  0.4  n 0.3  0.2  0.1  0 50 100 150 200 250 300 Number of pulses in correlation Figure 66: Probability of incorrect estimation in dense channels for the rule which minimizes the average estimation error when the channel statistics are known. 6 = 1= 5,W= 15 WI= 10, W = 15 .... W = 5, W = 15 5 4 2 3 C 2 01 0 50 100 150 200 250 300 Number of pulses in correlation Figure 67: Mean estimation error in dense channels for the rule which minimizes the average estimation error when the channel statistics are known. 0.7 0.6 o 0.5 0.4 S0.3  0.2  0.1  50 100 150 200 250 300 Number of pulses in correlation Figure 68: Probability of incorrect estimation in dense channels for the heuristic rule when the channel statistics are unknown. 100 150 200 Number of pulses in correlation Figure 69: Mean estimation error in dense channels for the heuristic rule when the channel statistics are unknown. 60 50 40 C 0 E CU 20 10 0 0.7  E S0.6 8 0.5  0.  O 0.3 0.1  50 100 150 200 250 Number of pulses in correlation Figure 610: Probability of incorrect estimation in sparse channels for the when the channel statistics are unknown. 300 heuristic rule 5 50 0 S40  20  10 0 0 50 Figure 611: Mean estimation channel statistics are unknown 100 150 200 250 300 Number of pulses in correlation error in sparse channels for the heuristic rule when the 100 