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Some Challenging Problems in Wireless Communications

Permanent Link: http://ufdc.ufl.edu/UFE0041975/00001

Material Information

Title: Some Challenging Problems in Wireless Communications
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Dong, Xihua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: communications, delay, ofdm, qos, queueing, red, synchronization, wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we study three challenging topics in wireless communications: 1) QoS provisioning for single-user wireless systems, 2) reliability-delay tradeoff in wireless ad hoc networks, 3) frequency synchronization for OFDM systems. We study the performance of a single-user wireless communication system over fading channels. We instigate the maximum transmission data rate over fading channels with both delay and error probability constraints. The system under study consists of 1) a finite-buffer discrete-time queueing system on the link layer, and 2) a rate-adaptive channel coding system on the physical layer. The objective of our work is to analyze the relationship among data rate (R), packet error probability (E), and delay bound (D), under the interaction between the link layer and the physical layer. In our analysis, we consider three types of packet errors, i.e., 1) packet drop due to full buffer, 2) packet error due to delay bound violation, and 3) packet decoding error due to channel noise. We obtain an upper bound on the packet error probability. Furthermore, by minimizing the packet error probability over the transmission rate control policies, we obtain an optimal rate control policy that guarantees the user-specified data rate and delay bound. In the case of constant arrival, the optimal rate control policy results in an RED triplet; then by varying data rate and delay bound, we obtain RED Pareto-optimal surface. Our results provide important insights into optimal rate control policy for joint link layer and physical layer design; the RED Pareto surface represents a major step toward deriving the probabilistic delay-constrained channel capacity of fading channels, which is an unsolved problem in information theory. Though the relationship between capacity and average delay in wireless ad hoc networks has been intensively studied in the literature, statistical delay guarantee provisioning in large scale ad hoc networks has not received enough attention. A realtime application, e.g., interactive game and realtime video, requires stringent delay (delay bound) but may allow a small probability of outage (deadline violation probability). This motivates us to study the relationship between delay bound and deadline violation probability. In this work, we try to answer the following interesting questions: as the network size scales up, 1) given a delay bound $B(n)$, how does the deadline violation probability scale? 2) does the deadline violation probability go to zeros? 3) if the deadline violation probability goes to zeros, how fast is it? For mobile ad hoc networks, based on a simple i.i.d. mobility model, we shown that tradeoff between deadline violation probability and delay bound is given by is given by $P_l(n) =e ^{-\Theta(B(n)/n)}$. This tradeoff explicitly indicates the increase of delay one must tolerate for achieving specified decay rate of deadline violation probability. Since the deadline violation probability can be interpreted as a description of the reliability of delay-sensitive communications, our results provide insights into understanding the reliability-delay tradeoff in large scale wireless ad hoc networks. We switch to PHY layer design in the last part of this dissertation. A recursive maximum likelihood (ML) carrier frequency offset (CFO) estimator is proposed in this work, where redundancy information contained in the cyclic prefix (CP) of multiple consecutive OFDM symbols is exploited in an efficient recursive fashion. Since the estimator is based on multiple OFDM symbols, the time-varying CFO must be considered. We investigate the effect of time-varying CFO on the performance of the estimator and the tradeoff between fast tracking ability and low estimation variance. We show that, without channel noise, the mean squared estimation error (MSE) due to CFO variation increases approximately quadratically with $n$, where $n$ is the number of OFDM symbols used for CFO estimation (estimation window size), while the MSE due to channel noise decreases proportionally to $1/n$ (approximately) if the CFO is constant. A closed-form expression of the optimal estimation window size (approximately) is derived by minimizing the MSE caused by both the time-varying CFO and channel noise. For wireless systems with time-varying rate of change for CFO, the proposed estimator can be implemented adaptively. In addition, typical optimal estimation window sizes for WiMAX, DVB-SH and MediaFLO systems are evaluated as an example.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xihua Dong.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Wu, Dapeng.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041975:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041975/00001

Material Information

Title: Some Challenging Problems in Wireless Communications
Physical Description: 1 online resource (104 p.)
Language: english
Creator: Dong, Xihua
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: communications, delay, ofdm, qos, queueing, red, synchronization, wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we study three challenging topics in wireless communications: 1) QoS provisioning for single-user wireless systems, 2) reliability-delay tradeoff in wireless ad hoc networks, 3) frequency synchronization for OFDM systems. We study the performance of a single-user wireless communication system over fading channels. We instigate the maximum transmission data rate over fading channels with both delay and error probability constraints. The system under study consists of 1) a finite-buffer discrete-time queueing system on the link layer, and 2) a rate-adaptive channel coding system on the physical layer. The objective of our work is to analyze the relationship among data rate (R), packet error probability (E), and delay bound (D), under the interaction between the link layer and the physical layer. In our analysis, we consider three types of packet errors, i.e., 1) packet drop due to full buffer, 2) packet error due to delay bound violation, and 3) packet decoding error due to channel noise. We obtain an upper bound on the packet error probability. Furthermore, by minimizing the packet error probability over the transmission rate control policies, we obtain an optimal rate control policy that guarantees the user-specified data rate and delay bound. In the case of constant arrival, the optimal rate control policy results in an RED triplet; then by varying data rate and delay bound, we obtain RED Pareto-optimal surface. Our results provide important insights into optimal rate control policy for joint link layer and physical layer design; the RED Pareto surface represents a major step toward deriving the probabilistic delay-constrained channel capacity of fading channels, which is an unsolved problem in information theory. Though the relationship between capacity and average delay in wireless ad hoc networks has been intensively studied in the literature, statistical delay guarantee provisioning in large scale ad hoc networks has not received enough attention. A realtime application, e.g., interactive game and realtime video, requires stringent delay (delay bound) but may allow a small probability of outage (deadline violation probability). This motivates us to study the relationship between delay bound and deadline violation probability. In this work, we try to answer the following interesting questions: as the network size scales up, 1) given a delay bound $B(n)$, how does the deadline violation probability scale? 2) does the deadline violation probability go to zeros? 3) if the deadline violation probability goes to zeros, how fast is it? For mobile ad hoc networks, based on a simple i.i.d. mobility model, we shown that tradeoff between deadline violation probability and delay bound is given by is given by $P_l(n) =e ^{-\Theta(B(n)/n)}$. This tradeoff explicitly indicates the increase of delay one must tolerate for achieving specified decay rate of deadline violation probability. Since the deadline violation probability can be interpreted as a description of the reliability of delay-sensitive communications, our results provide insights into understanding the reliability-delay tradeoff in large scale wireless ad hoc networks. We switch to PHY layer design in the last part of this dissertation. A recursive maximum likelihood (ML) carrier frequency offset (CFO) estimator is proposed in this work, where redundancy information contained in the cyclic prefix (CP) of multiple consecutive OFDM symbols is exploited in an efficient recursive fashion. Since the estimator is based on multiple OFDM symbols, the time-varying CFO must be considered. We investigate the effect of time-varying CFO on the performance of the estimator and the tradeoff between fast tracking ability and low estimation variance. We show that, without channel noise, the mean squared estimation error (MSE) due to CFO variation increases approximately quadratically with $n$, where $n$ is the number of OFDM symbols used for CFO estimation (estimation window size), while the MSE due to channel noise decreases proportionally to $1/n$ (approximately) if the CFO is constant. A closed-form expression of the optimal estimation window size (approximately) is derived by minimizing the MSE caused by both the time-varying CFO and channel noise. For wireless systems with time-varying rate of change for CFO, the proposed estimator can be implemented adaptively. In addition, typical optimal estimation window sizes for WiMAX, DVB-SH and MediaFLO systems are evaluated as an example.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Xihua Dong.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Wu, Dapeng.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-02-28

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041975:00001


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Firstofall,myspecialgratitudegoestomyadvisor,ProfessorDapengOliverWu,forhiscontantguidance,supportandencouragementduringmyentirePh.D.study.Hisenthusiasmanddeepthoughtssparkmyinterestinacademicresearch.Hegavemecountlessadviceandinsightsduringthepastfouryears,withoutwhichcompletingthisthesiswouldhavebeenmuchmoredifcult.Healsopatientlytaughtmehowtowritegoodpapersandgiveimpressivepresentations,whichareskillsIwillforeverindebtedtohim.Mydeeplyappreciationgoestomycommitteemembers:Prof.YuguangFang,Prof.JohnSheaandProf.ShigangChen,fortheirinterestinmyworkandthevaluablefeedbacksonmyresearch.IwouldliketothankotherprofessorsintheECEdepartment.Ihavelearntalotfromtheirclassesandelaboratelydesignedcourseprojects.Iwouldliketothankmylab-matesintheMultimediaCommunicationsandNetworkingLaboratory(MCN)hereinUF.Iamfortunatetobeamemberofthisfriendlyandfamily-likegroup.IwouldliketothankDr.XiaochenLiandDr.ChiZhang,fortheirhelpfuldiscussionsontheresearchandcooperationofmanypapers;Dr.BingHan,WenxingYe,Dr.JunXu,ZhifengChen,TaoranLu,YiranLi,YunzhaoLiandDr.JieyanFan,fortheirconstantsupportsandsincerefriendship,andIcherisheveryminutewehavespenttogether;ShanshanRenandZiyiWang,forhostingthepartiesandaddingtheelementoffuntomyPh.D.life.IhavespentwonderfulfouryearsinGainesville.Withoutthem,itisnotevenpossible.IwouldalsoliketothankZongruiDing,LeiYang,QianChen,JiangpingWang,YakunHu,QinChen,QingWang,YounghoJoandChrisPaulson.WishyouallhavesuccessinyourPh.D.studies.Iwouldliketothankmyparentsfortheendlessloveandconstantsupportthey'veprovidedduringmywholelife.SpecialthanksalsogotomygirlfriendLinZhang,forherdeeplove,understandingandsupport. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 12 1.1CapacityofFadingChannels ......................... 12 1.2ThroughputandDelayScalinginWirelessAdhocNetworks ........ 15 1.3FrequencySynchronizationforOFDMSystems ............... 17 1.4OutlineoftheDissertation ........................... 18 2REDTHEORYFORQOSPROVISIONINGINWIRELESSCOMMUNICATIONS 20 2.1Introduction ................................... 20 2.2SystemModel ................................. 22 2.3AnalysisofPacketErrors ........................... 26 2.3.1DecodingErrorProbability ....................... 26 2.3.2PacketDropProbability ........................ 28 2.3.3DelayBoundViolationProbability ................... 30 2.3.4TotalPacketErrorProbability ..................... 34 2.4ThroughputMaximizationProblem ...................... 35 2.5REDTripletandItsProbabilities ........................ 37 2.5.1AdmissionControlSpace ....................... 38 2.5.2ERDfunctionandREDfunction .................... 40 2.5.3RelationshipbetweenREDTheoryandEffectiveCapacity ..... 41 2.6SimulationandNumericalResults ...................... 42 2.6.1Simulationsettings ........................... 43 2.6.2DelayBoundViolationProbability ................... 44 2.6.3TradeoffbetweenDecodingErrorProbabilityandPacketDropProbability ................................ 45 2.6.4AnExampleofParetoSurface ..................... 46 2.6.5OptimalPolicyvs.OptimalFixed-decoding-errorPolicy ....... 46 2.7Summary .................................... 47 3RELIABILITYDELAYTRADEOFFINWIRELESSNETWORKS ......... 49 3.1Introduction ................................... 49 3.2NetworkModelsandDenitions ........................ 52 3.2.1NetworkModelandMobilityModel .................. 52 5

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............. 53 3.3Reliability-DelayTradeoffinMobileNetworks ................ 56 3.4Summary .................................... 60 4FREQUENCYSYNCHRONIZATIONFOROFDMSYSTEMS .......... 61 4.1Introduction ................................... 61 4.2TheOFDMSystemModel ........................... 64 4.3RecursiveMLEstimationforConstantCFO ................. 65 4.3.1DerivationoftheBasicFormRecursiveMLEstimator ........ 65 4.3.2Cramer-RaoLowerBound ....................... 68 4.3.3FrequencySelectiveChannels .................... 70 4.3.4ImplementationfortheUplinkofMultiuserOFDMSystems ..... 70 4.4RecursiveMLEstimationofTime-VaryingCFO ............... 71 4.4.1Performancedegradationduetotime-varyingCFO ......... 71 4.4.2OptimalestimationwindowsizeofRCFOE .............. 73 4.4.3TypicalEstimationWindowSizesforOFDMSystems ........ 76 4.5SimulationResults ............................... 76 4.5.1Time-varyingCFO ........................... 77 4.5.2PerformanceoftheRCFOEestimatorinAWGNchannels ..... 78 4.5.3PerformanceofRCFOEestimatorinfrequency-selectivechannels 82 4.6Summary .................................... 83 5CONCLUSIONS ................................... 86 APPENDIX ASUPPORTINGRESULTSFORCHAPTER2 ................... 88 A.1ProofofLemma 2.2 .............................. 88 A.2ProofofLemma 2.3 .............................. 88 A.3ProofofProposition 2.1 ............................ 89 A.4ProofofProposition 2.2 ............................ 90 A.5ProofofProposition 2.3 ............................ 91 A.6ProofofTheorem 2.4 ............................. 91 BSUPPORTINGRESULTSFORCHAPTER3 ................... 92 B.1ProofofProposition 3.1 ............................ 94 B.2ProofofLemma 3.1 .............................. 95 CSUPPORTINGRESULTSFORCHAPTER4 ................... 96 C.1DerivationoftheLog-likelihoodFunction ................... 96 C.2ProofofLemma 4.1 .............................. 96 REFERENCES ....................................... 98 BIOGRAPHICALSKETCH ................................ 104 6

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Table page 4-1TypicalEstimationWindowSizesforOFDMsystems ............... 77 7

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Figure page 2-1Systemmodel .................................... 22 2-2Timingdiagram .................................... 24 2-3Bufferstatusinthen-thblock ............................ 31 2-4Delayboundviolationprobabilityanditsupperbound .............. 44 2-5Packeterrorprobability,decodingerrorprobability,andpacketdropprobabilityunderlinearratecontrolpolicies .......................... 45 2-6Paretosurface:maximumdatarate()asafunctionofdelayboundDandpacketerrorprobability" 47 2-7Performancecomparisonbetweentheoptimalpolicyandtheoptimalxeddecodingerrorpolicy(purequeueing) ....................... 48 3-1Mobilenetworkmodel(Fig.1in[ 1 ] ......................... 53 3-2Tradeoffbetweenreliabilityindexanddelayboundformobilenetworks.Thescaleoftheaxesareintermsoftheordersofn. ................. 59 4-1Thediscrete-timebasebandmodeloftheOFDMsystem ............. 64 4-2StructureofthebasicformRCFOEestimator ................... 68 4-3Amobilecommunicationnetwork .......................... 78 4-4Time-varyingCFO .................................. 79 4-5CFOrateofchange ................................. 79 4-6MSEvs.SNRforconstantCFO ........................... 80 4-7MSEvs.nforvariousratesofchangeofCFO ................... 81 4-8TheoptimalestimationwindowsizenoasafunctionofSNR ........... 82 4-9TheoptimalestimationwindowsizenoasafunctionofrateofchangeofCFO 83 4-10PerformanceofRCFOEfortime-varyingCFO ................... 84 4-11PerformanceofRCFOEfortime-varyingCFOinfrequencyselectivechannels 85 8

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Inthisdissertation,westudythreechallengingtopicsinwirelesscommunications:1)QoSprovisioningforsingle-userwirelesssystems,2)reliability-delaytradeoffinwirelessadhocnetworks,3)frequencysynchronizationforOFDMsystems. Westudytheperformanceofasingle-userwirelesscommunicationsystemoverfadingchannels.Weinstigatethemaximumtransmissiondatarateoverfadingchannelswithbothdelayanderrorprobabilityconstraints.Thesystemunderstudyconsistsof1)anite-bufferdiscrete-timequeueingsystemonthelinklayer,and2)arate-adaptivechannelcodingsystemonthephysicallayer.Theobjectiveofourworkistoanalyzetherelationshipamongdatarate(R),packeterrorprobability(E),anddelaybound(D),undertheinteractionbetweenthelinklayerandthephysicallayer.Inouranalysis,weconsiderthreetypesofpacketerrors,i.e.,1)packetdropduetofullbuffer,2)packeterrorduetodelayboundviolation,and3)packetdecodingerrorduetochannelnoise.Weobtainanupperboundonthepacketerrorprobability.Furthermore,byminimizingthepacketerrorprobabilityoverthetransmissionratecontrolpolicies,weobtainanoptimalratecontrolpolicythatguaranteestheuser-specieddatarateanddelaybound.Inthecaseofconstantarrival,theoptimalratecontrolpolicyresultsinanREDtriplet;thenbyvaryingdatarateanddelaybound,weobtainREDPareto-optimalsurface.Ourresultsprovideimportantinsightsintooptimalratecontrolpolicyforjointlinklayerandphysicallayerdesign;theREDParetosurfacerepresentsamajorsteptowardderiving 9

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Thoughtherelationshipbetweencapacityandaveragedelayinwirelessadhocnetworkshasbeenintensivelystudiedintheliterature,statisticaldelayguaranteeprovisioninginlargescaleadhocnetworkshasnotreceivedenoughattention.Arealtimeapplication,e.g.,interactivegameandrealtimevideo,requiresstringentdelay(delaybound)butmayallowasmallprobabilityofoutage(deadlineviolationprobability).Thismotivatesustostudytherelationshipbetweendelayboundanddeadlineviolationprobability.Inthiswork,wetrytoanswerthefollowinginterestingquestions:asthenetworksizescalesup,1)givenadelayboundB(n),howdoesthedeadlineviolationprobabilityscale?2)doesthedeadlineviolationprobabilitygotozeros?3)ifthedeadlineviolationprobabilitygoestozeros,howfastisit?Formobileadhocnetworks,basedonasimplei.i.d.mobilitymodel,weshownthattradeoffbetweendeadlineviolationprobabilityanddelayboundisgivenbyisgivenbyPl(n)=e(B(n)=n).Thistradeoffexplicitlyindicatestheincreaseofdelayonemusttolerateforachievingspecieddecayrateofdeadlineviolationprobability.Sincethedeadlineviolationprobabilitycanbeinterpretedasadescriptionofthereliabilityofdelay-sensitivecommunications,ourresultsprovideinsightsintounderstandingthereliability-delaytradeoffinlargescalewirelessadhocnetworks. WeswitchtoPHYlayerdesigninthelastpartofthisdissertation.Arecursivemaximumlikelihood(ML)carrierfrequencyoffset(CFO)estimatorisproposedinthiswork,whereredundancyinformationcontainedinthecyclicprex(CP)ofmultipleconsecutiveOFDMsymbolsisexploitedinanefcientrecursivefashion.SincetheestimatorisbasedonmultipleOFDMsymbols,thetime-varyingCFOmustbeconsidered.Weinvestigatetheeffectoftime-varyingCFOontheperformanceoftheestimatorandthetradeoffbetweenfasttrackingabilityandlowestimationvariance.Weshowthat,withoutchannelnoise,themeansquaredestimationerror(MSE)dueto 10

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Threerelatedbutdifferenttopics,QoSprovisioningforsingle-userwirelesssystems,statisticaldelayguaranteeforwirelessadhocnetworksandfrequencysynchronizationforOFDMsystems,areinvestigatedinthisdissertation.Inthissection,wewilltrytobuildageneralbackgroundfortheseresearchtopicsandreviewsomerelatedworks.However,weputthespecicbackgroundintroductionandmorecompleteliteratereviewinthecorrespondingchapters. 2 ].Theshannonchannelcapacity,whichisdenedasthechannel'smutualinformationmaximizedoverallpossibleinputdistributions,dictatesthemaximumdataratesthatcanbetransmittedoverchannelswithasymptoticallysmallerrorprobability,assumingnoconstraintsondelayorcomplexityoftheencoderanddecoder.Foradiscrete-timeAWGNchannelwithbandwidthBandreceivedsignal-to-noiseratio(SNR),thechannelcapacityisgivenbyShannon'swell-knowformula:C=Blog2(1+). TheShannoncapacityoffadingchannelsdependsontheavailabilityofchannelstateinformation(CSI)atthetransmitterandreceiver.Nowweconsidertwocases:CSIatreceiveronlyandCSIatboththereceiverandtransmitter.Consideraatfadingchannelwithchannelinput/outputrelationshipyi=p 12

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InthecaseofCSIatreceiver,i.e.,theCSIgiisknowntothereceiverattimei,theShannoncapacityofafadingchannelistheexpectationofinstantaneouschannelcapacity[ 3 ],i.e.,C=Z10Blog2(1+)p()d. AnalternatecapacitydenitionoffadingchannelswithreceiverCISiscapacitywithoutage(outagecapacity).Itisdenedasthemaximumdataratethatcanbetransmittedoverachannelwithanoutageprobabilitycorrespondingtotheprobabilitythatthereceivedsymbolcannotbedecodedwithnegligibleerrorprobability.Capacitywithoutageisgenerallyapplicabletoslowlyvaryingchannels,wheredatacanbetransmittedatrateBlog2(1+)(instantaneouscapacity)withnegligibleprobabilityoferror.Byallowingasmallprobabilityoferror(outage),thefadingchannelcanbeconsideredaequivalentconstantchannel. WhenCSIisavailableatboththetransmitterandreceiver,thechannelcapacityisachievedbyanadaptivetransmissionscheme,wherethetransmissionpowerisadaptedtothechannelstate.Itisshownin[ 4 ]thattheShannoncapacityisgivenbyC=Z10Blog2 0p()d, wherethecutoffvalue0satisesZ101 13

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5 ],HanlyandTseproposedtheconceptofdelay-limitedcapacitywhichisdenedasthemaximumachievableratewithlimiteddelayindependentofhowslowthefadingis.Thedelay-limitedchannelcapacityprovidesanappropriatelimitforreal-timeserviceswithlimiteddecodingdelay.However,theexplicitrelationbetweendelay,decodingerrorprobabilityandthroughputisstillunknown. Inordertoaddressthedelayconstraints,throughputmaximizationwithaveragedelayconstraintisstudiedin[ 6 9 ].Queueingtheorybecomesamajortoolinsolvingthethroughputmaximizationproblemandtheclosed-formresultsdonotexistingeneral.TwofactsrestrainapplicationoftheseworksinQoSprovidingforwirelessnetworks:1)Theaveragedelayguaranteemaynotspecifytherequirementsofsomedelay-sensitiveapplications,i.e.,interactivegamesandrealtimevideo,sincetheseapplicationsrequiredeterministicdelayguaranteebutmayallowasmallprobabilityofoutage.2)Thebuffersizeisoftenassumedtobeinnitewhilepracticalsystemshavelimitedbufferspace. In[ 10 ],WuandNegiproposedtheconceptofeffectivecapacity,whichcanbeinterpretedthemaximumconstantdateratecanbetransmittedsuchthatthedelayboundviolationhassomespecieddecayrate.Theeffectivecapacitybuildsanexplicitrelationshipamongdelaybound,delayboundviolationprobabilityanddatarateandthusyieldssubstantialimprovementinQoSperformance.However,therearethreelimitsinapplyingtheeffectivecapacity:1)Thebuffersizeisassumedtobeinnity.2)Theirresultsareonlyprovedtobevalidforlargedelayregimesincethelargedeviationtheoryisusedforprovingthemainresults.3)Idealcodingisrequiredtoachievetheinstantchannelcapacity. Alongthedirectionofstatisticaldelayguarantee,inChapter 2 ,weinvestigatethemaximumthroughputofanitebuffersystemsubjecttobothdeterministicdelaybound 14

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11 ],GuptaandKumarintroducedarandomnetworkmodelforstudyingthroughputscalinginstaticwirelessadhocnetworks.Theydenedarandomnetworkwhichconsistsofnnodesindependentlyanduniformlydistributedonaunitdisk.Eachnodehasarandomlychosendestinationnode.Theyshowedthatforalmostallrealizationsoftherandomnetwork,throughputscalesas(1=p Notethatthethroughputofstaticnetworkstendstozeroasthenetworksizescalesup.In[ 12 ],GrossglauserandTseshowedthatbyallowingthenodestomove,throughputscalingcanbedramaticallyimproved.Ifnodemotionisindependentacrossnodesandhasauniformstationarydistribution,aconstantthroughputscaling((1))perS-Dpairisfeasible.Thefundamentalreasonforthisimprovementisaformofmultiuserdiversityviapacketrelaying. Theaboveworksdonotconsiderthenetworkdelay.Thatis,toachievethethroughputcapacity,packetsmaytravelarbitrarylongtimefromtransmittertoreceiver.Inmostnetworkingapplications,delayisalsoakeyperformancemetricaswellasthroughput.Thusamoreusefuldescriptionofnetworkwouldbeintermsofdelay-constrainedthroughput.In[ 13 ],ElGamalet.alstudiedthetradeoffbetweenaveragedelayandthroughputinbothstaticandmobilenetworks.ForastaticrandomnetworkwiththroughputT(n)=O(1=p 1 ],NeelyandModianoconsideredthecapacityandaveragedelaytradeoffsformobilenetworks.Basedonasimpliedi.i.d.mobilitymodel,theauthorscalculatedthe 15

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Duringthepasttenyears,manynetworkmodels,mobilitymodelsandcommunicationschemeshavebeenproposed.Sharmaet.alproposedaglobalapproachtostudythethroughputdelaytradeoffinmobileadhocnetworks[ 14 ].Theauthorsdenedanewconcept:criticaldelay,belowwhich,thenodemobilitycannotbeexploitedforimprovingthecapacity.Itisshownthatthecriticaldelaydependsonthenatureofthenodemobilitybutnotsomuchonthenetworksetting. Inallaboveworks,whichaddressnetworkdelay,onlyaveragedelayisconsidered.Thoughthethroughput-delaytradeoffresultsprovidedeepinsightsintounderstandingthefundamentalperformancelimitsofdelayconstrainedcommunicationsoverwirelessnetworks,theyareobviouslyinadequateforrealtimeapplicationssincewithknownaveragedelay,theactualdelaymayhavealargedynamicrange.Forexample,realtimevideotransmissionrequiresstringentdelay(delayboundordeadline)butallowsasmallprobabilityofoutage(packetlossordeadlineviolation)probability.Ifapacketcannotreachitsdestinationwithinagivendelaybound(e.g.,0.1s),itmaybeconsideredasalostpacketatthereceiverandcannotbeusedfordecoding.Thepacketlossrateduetodeadlineviolationcanbeinterpretedasareliabilitymetricofwirelessnetworks.Suchpracticalrequirementofrealtimeapplicationsmotivesustostudytherelationshipbetweendelayboundanddeadlineviolationprobability. InChapter 3 ,weestablishthetradeoffbetweendelayboundanddeadlineviolationprobabilityformobileadhocnetworks.Weconsideramobilewirelessnetworkmodelwithnnodesandagivencommunicationscheme.Let(n)denotesystemthroughput,B(n)denotethegivendelaybound(deadline)andPl(n)denotethecorrespondingdeadlineviolationprobability.WeshowthatifandonlyifthedelayboundhasorderlargerthantheaveragedelayD(n),i.e.,B(n)=!(D(n)),thecorrespondingdeadlineviolationprobabilityPl(n)willgoestozeroasngoestoinnity.Withacarefulchoice 16

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15 ]foritshighdataratetransmissioncapabilityandintersymbolinterferencemitigationability.Moreover,incontrastwithsinglecarriersystems,channelequalizationcanbeeasilyaccomplishedinfrequency-domain.RecentlydevelopedmobileOFDMtechnologiesincludeWiMAX[ 16 ],DVB-SH[ 17 ]andMediaFLO[ 18 ]. However,successfulimplementationofOFDMsystemsrequirestheorthogonalityofsubcarriers,whichisassuredbythesubcarrierseparationf=1=TN,whereTNisthesymboltime.Inpractice,thefrequencyseparationofthesubcarriersisimperfectandsofisnotexactlyequalto1=TN.Thisisgenerallycausedbymismatchedoscillators,Dopplereffects,ortimingerrors.Ifthereisfrequencyoffset,itisshownin[ 19 ]thatthetotalintercarrierinterference(ICI)poweronasubcarriericanbeapproximatedbyICIiC0(TN)2 whereC0issomeconstant,isthefrequencyoffset.Itisalsoshownin[ 19 ],acarrierfrequencysynchronizationerrorof1partpermillioncandestroytheorthogonalityofsubcarriers.Thus,frequencysynchronizationalgorithmsareexpectedtohavestrongtrackingability. Generally,thecarrierfrequencyoffset(CFO),whichiscausedbyasymmetricDopplerspectrum,isslowlytime-varying[ 20 ].Thus,frequencysynchronizationalgorithmsareexpectedtohavestrongtrackingability. 17

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21 22 ];2)multiple-symbolbasedapproaches,e.g.,[ 23 25 ].Thesingle-symbolbasedapproachesusesonereceivedOFDMsymbolforCFOestimation.Thustheyhavestrongtrackingability.However,duetosmallobservationdataset,theirestimationmaybenotaccurate.Themultiple-symbolbasedapproachesutilizemultipleOFDMsymbolsforCFOestimationthushavelowestimationvariance.However,tothebestofourknowledge,thereisnoexistingworkprovidessimplemethodsforchoosingtheoptimalestimationwindowsize. InChapter 4 ,weproposeamultiple-symbolbasedCFOestimationalgorithm.Incontrast,byinvestigatingtheeffectoftime-varyingCFO,weprovideaclosed-formexpressionfortheestimationwindowsizeasafunctionofcyclicprex(CP)length,SNRandrateofchangeofCFO. 2 investigatesthestatisticalQoSprovisioningforend-to-endwirelesscommunicationsystems.Basedontheassumptiononanitebufferqueueingsystem,weanalyzethepacketerrorprobabilityofaqueueing/codingsystem.Basedonouranalyticalresultforpacketerrorprobability,weobtainanoptimalratecontrolpolicythatguaranteestheuser-speciedQoS,byminimizingthepacketerrorprobabilityoverthetransmissionrate.TheoptimalratecontrolpolicyresultsinaREDtriplet;thenbyvaryingdatarateanddelaybound,weobtainREDPareto-optimalsurface,whichservesastheperformancelimitofthesystemunderstudy. Chapter 3 studiesthestatisticaldelayQoSprovisioningformorecomplexwirelessnetworks,i.e.,wirelessadhocnetworks.Weinvestigatethetradeoffbetweendelayboundanddeadviolationprobability. WeswitchtothePHYlayerdesignofwirelesscommunicationsystemsinChapter 4 .WeinvestigatethefrequencysynchronizationproblemforOFDMsystemsandpropose 18

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5 summarizesthedissertation. 19

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Datacommunicationoverfadingchannelswithoutdelayconstrainthasbeenextensivelystudiedintheliterature.Whenthedelayconstraintisabsent,themaximumthroughputisgenerallytheShannonergodiccapacity,whichisoftenderivedbyaninformationtheoreticalapproach.Dependingontheassumptionsofchannelstateinformation(CSI)availabilityatthetransmitter(CSIT)andatthereceiver(CSIR),existingworksfallintoseveralcategories.AnincompletelistincludeschannelswithperfectCSITandCSIR[ 26 ],andthenite-stateMarkovchannels(FSMC)withoutCSI[ 27 ],andchannelswhereCSITisadeterministicfunctionofCSIR[ 28 ],andchannelswithcausalCSI[ 29 ].SomeworkshavealsoaddressedthemorerealisticcaseofnonperfectCSI[ 30 31 ].Practicaladaptivemodulationandcodingschemesfordatacommunicationoverfadingchannelsarestudiedin[ 32 34 ]. Theaboveworksdonottaketransmissiondelayintoconsideration,thusmaybenotapplicablewhenthereisdelayconstraint.Recently,delay-constrainedcommunicationhasreceivedmoreattention.In[ 5 ],HanlyandTseproposedtheconceptofdelay-limitedcapacitywhichisdenedasthemaximumachievableratewithlimiteddelayindependent 20

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6 9 ].TwofactsrestrainapplicationoftheseworksinQoSprovidingforwirelessnetworks:1)Theaveragedelayguaranteemaynotspecifytherequirementsofsomedelay-sensitiveapplications,i.e.,interactivegamesandrealtimevideo,sincetheseapplicationsrequiredeterministicdelayguaranteebutmayallowasmallprobabilityofoutage.2)Thebuffersizeisassumedtobeinnitewhilepracticalsystemshavelimitedbufferspace.Thismotivesustoinvestigatestatisticaldelayguaranteeforwirelesscommunicationsystemswithnitebuffer. Inthiswork,wedealwithasingle-usercommunicationsystemwithnitebuffer.Packettransmissionissubjecttoadeterministicdelayboundconstraint.Besides,differentfrommostexistingworkswhichaddresstransmissiondelay,e.g.,[ 6 7 ],wedonotassumeperfectchannelcoding.Thusapacketmaysufferfromthreetypesoferrors:1)delayboundviolation,2)packetdropduetobufferoverow,and3)decodingerrorduetochannelnoise. Themaximumsystemthroughputisachievedbychoosingaratecontrolpolicywhichminimizesthetotalerrorprobability.Then,inthecaseofconstantarrival,byvaryingdatarateanddelaybound,weobtainanRED(rate-error-delay)Pareto-optimalsurface.Wefurtherstudythestructuralproperties,i.e.,monotonicityandconvexity,oftheParetosurface.Ourresultsprovideimportantinsightsintooptimalratecontrolpolicyforjointlinklayerandphysicallayerdesign;theREDParetosurfacerepresentsamajorsteptowardderivingtheprobabilisticdelay-constrainedchannelcapacityoffadingchannels,whichisanopenproblemininformationtheory. OurworkisclosedrelatedtothatofHoangandMotaniin[ 35 ],sincebothconsidernite-bufferqueueandimperfectchannelcodingandusesimilartechniqueinderivingtheoptimaltransmissionschemes.Packetdropanddecodingerrorarealsoaddressedin[ 36 ]and[ 37 ].Oneofthemajordifferencesofourworkfrom[ 35 37 ]isthatweaddressthedeterministictransmissiondelayboundanddelayboundviolation 21

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10 ],aneffectivecapacityapproachwasproposedtoanalyzetherelationshipamongdatarate,delayboundanddelayboundviolationprobability.However,thequeueingmodelin[ 10 ]assumesinnitebufferspace.Besides,largedeviationtheorymethodisappliedtoderivedthedelayboundviolationprobabilitythustheresultsin[ 10 ]areonlyprovedtoholdinlargedelayregimewhileourresultsholdforarbitrarydelay. Theremainderofthechapterisorganizedasbelow.Section 2.2 describesthesystemmodel.InSection 2.3 ,wepresentouranalysisfordelayandpacketerrorprobability.Section 2.4 presentsthethroughputmaximizationproblem.Section 2.5 describesourREDtheory.SimulationandnumericalresultsaregiveninSection 2.6 .Section 2.7 concludesthechapter. 2-1 .Datapackets,whosesizeisassumedtobeLbits,arrivefromsomeupperlayerandarebufferedatthelinklayer.TimeisdividedintoblocksofequallengthTb.Aratecontrolunitremovessomehead-of-line(HOL)packetsfromthebufferandconveythemtotherateadaptivechannelencoderinthephysicallayer.Thentheencodeddataismodulatedandtransmittedthroughafadingchannelchannel. Systemmodel 22

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38 ]andisalsoadoptedin[ 6 ]and[ 35 ].LetWdenotethebandwidth,thenN=WTbmodulatedsymbolscanbetransmittedduringoneblock.Letx=(xn1,xn2,...,xnN)denotetheoutputoftherateadaptivechannelencoder(codedsymbols),whichistheinputofthemodulator.Lety=(yn1,yn2,...,ynN)denotetheoutputofthedemodulatorinthen-thblock.Thenwehaveynk=gnxnk+znk,k=1,2,...,N whereznk,k=1,2,...,Nareindependentandidenticallydistributed(i.i.d.)circularlysymmetriccomplexGaussianrandomvariableswithzeromeanandvarianceN0. Notethatwedonotspecifytheencoding/decodingschemes,sinceourobjectiveistobuildageneralframeworkforstudyingdelay-constrainedcommunicationproblems,thoughdifferentschemesmayaffectthedelayanderrorperformances.WeassumethecodewordlengthcannotexceedN,thenumberofchannelusesperblock,andacodewordneedtobedecodedwithinoneblock.Thustheencoding/decodingdelayshouldbelessthan2blocks. Thequeueingsubsystemismodeledasadiscrete-timenite-bufferqueuewithbuffersizeMpackets.Whennewlyarrivedpacketsndthatthebufferisfull,somepacketsneedtobedroppedfromthebuffer.Therearethreestrategiesforpacketdropping:Strategy1(tail-dropping)dropsthenewlyarrivedpackets;Strategy2(tailpushout)dropsthetail(end-of-line)packetsinthequeueandappendstheincomingpacketstothetailofthequeue;andStrategy3(HOLpushout)dropsthehead-of-line 23

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Letfangbeani.i.d.randomprocesswithstatespaceAR+whichrepresentsthenumberofpacketsarrivingatthebufferduringthen-thblock.WeassumeEan=and,forthespecialconstantarrivalcase,an=.AsshowninFig. 2-2 ,attheverybeginningofthen-thblock,abatchofanpacketsarrive,followedbythedepartureofrn(whichwereferasservicerateortransmissionrate)packets;thequeuelengthqnisobservedimmediatelyafterthedeparture.LetQdenotethebufferstatespace,i.e.,Q=f0,1,...,Mg. Timingdiagram Inordertocopewithchannelvariation,weassumethetransmissionrateisadaptivewhilethetransmissionpowerisconstant.Weassumethechannelstateinformation,bufferstateinformationandarrivalstateinformationareavailableatboththetransmitterandthereceiver.Thusthetransmissionrate(orservicerate)rnisspeciedbyaratecontrolpolicyR:QGA!Q,i.e.,rn=R(qn1,gn,an). Notethatrndependsonqn1insteadofqnbecauseqnisnotavailablewhenthen-thdeparturetakesplace.Sotheevolutionofthequeueingsystemisgivenbyqn=min(qn1+an,M)R(qn1,gn,an) 24

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Recallthatweareinterestedindelay-constrainedcommunicationoverfadingchannels.LetDmaxdenotethemaximumtolerabledelay,i.e.,ifonepacketcannotreachitsdestinationwithinDmaxblocks,itwillbeconsideredasanerroneouspacket.LetDdenotethetotaldelayexperiencedbyapacketinthesysteminFig. 2-1 .ThenDisthesumofthedelayinthebufferplustheencoding/decodingdelay.Aswementionedabove,theencoding/decodingdelayisconnedtobeatmost2blocks.Thuswewillomittheencoding/decodingdelaybutfocusonthequeueingdelayinthereminderofthiswork. Ourobjectiveistondthemaximumsystemthroughputwhilesatisfyingthedelayandpacketerrorprobabilityconstraints,whichisequivalenttominimizingthepacketerrorprobabilitywithdatarate(averagearrivalrate)anddelayconstraints.NowconsiderapacketentersthesysteminFig. 2-1 .Itmayexperiencethreetypesoferrors:1)packetdropduetofullbuffer,2)delayboundviolation(failtoreachdestinationwithinDmaxblocks),and3)packetdecodingerrorduetochannelnoise.Itiseasytoseethatthereisatradeoffbetweenthedecodingerrorandtheothertwotypesoferrors.Ifweincrease(resp.,decrease)theservicerate,thebufferwillbeclearedmorequickly(resp.,slowly),resultinginasmaller(resp.,larger)dropprobabilityanddelayboundviolationprobability;however,thedecodingerrorprobabilitywillincrease(resp.,decrease)sincemore(resp.,less)bitsaretransmittedthroughthechannel.Sotheoptimalratecontrolpolicyshouldbalancepacketdroperrorprobability,decodingerrorprobabilityanddelayboundviolationprobabilitysoastominimizethetotalerrorprobability.Inthenextsection,webeginwithanalyzingthethreetypesoferrors. 25

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2-1 mayexperiencethreetypesofpacketerrors:1)packetdropduetofullbuffer,2)delayboundviolation,and3)packetdecodingerror.Inthissection,werstanalyzetheseerrorsseparately,thenwegivetheexpressionofthetotalpacketerrorprobability. 34 39 ].Nowweconsiderbothblockingcodeandconvolutionalcode. Forblockcode,weonlyconsiderrandomcodinganddecoding.Aswementionedbefore,acodewordshouldbedecodedattheendofablock.Ifrpacketsaretobetransmittedinoneblock,thentheserverrateisrL=Nbitsperchannelsuse.NotethatLandNarethepacketsizeandthenumberofchannelusesperblockrespectively.AssumethetransmissionenergyisPtpersymbolwhichisconstantforallsymbols.Weencodealldatatobetransmittedinoneblocktoonesinglecodeword.Thenthefollowingrandomcodingbound[ 40 ]ontheprobabilityoferror(symbolerror)holds,forany2(0,1]:Psexp(N(rL=Nlog2E0(,g))) whereE0(,g)=ln1+Ptjgj2 Sincealldatatobetransmittedinoneblockisencodedintoonesinglecodeword,ifacodewordisdecodedcorrectly,thenthereisnopacketdecodingerrorhappens; 26

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Forconvolutioncodes,thedecodingerrorprobabilitydependsonthespeciedmodulation/codingscheme.AssumeBPSKisused(ouranalysiscanbeeasilyextendedtoanyotherlinearmodulationscheme).LetPbcdenotethebiterrorprobability.Itisboundedby[ 39 ]Pbc1Xd=dfreeBdPd whereBdisthetotalnumberofnonzeroinformationbitsonallweight-dpathsdividedbythenumberofinformationbitsperunittime.dfreeistheminimalHammingdistancebetweendifferentencodedsequences.Pdisthepairwiseerrorprobabilitywhichisdenedastheprobabilitythatthedecoderselectsanerroneouspathatthedistancedfromthetransmittedpath.Pddependsonthechanneltype,modulationschemeanddecodingtype(hardorsoftdecisions).ForadditivewhiteGaussianchannelandharddecisiondecoding,thepairwiseerrorprobabilityis[ 39 ]Pd=8>>>>>>>><>>>>>>>>:dXe=(d+1)=20B@de1CA(pb)e(1pb)de,dodd1 20B@dd=21CA(pb)d=2(1pb)d=2+dXe=d=2+10B@de1CA(pb)e(1pb)de,deven, wherepbisthechannelbiterrorrate;forBPSK,pbisgivenbypb=Q(q Inordertoutilizetheaboveupperbound,weneedtochooseappropriateconvolutioncodewhichhasrequiredcoderate.ForBPSKmodulation,thenumberoftransmitted 27

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Wedeneaveragedecodingerrorprobabilityastheratioofthelong-termaveragenumberofincorrectlydecodedpacketstotheaveragenumberofarrivingpackets.ThentheaveragedecodingerrorprobabilityisgivenbyPce=limsupT!11 where(x)+,max(x,0).Notethattheaveragepacketdropprobabilitydependsonratecontrolpolicywhichgovernsthequeuestate. If,bychoosingsomeratecontrolpolicy,thesteadystateoffsngexists,thepacketdropprobabilityisgivenbyPqe=1 whichcanalsobedirectlycalculatedbythestandardFSMCapproach,i.e.,rstcalculatingthesteadystatedistributionoffsng,thenobtainingthepacketdropprobabilitybasedon( 2 ).Nowweconsiderthespecialcaseofconstantarrivaland 28

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whereikl,Pr(qn=ljgn=Gi,qn1=k)=Pr(min(k+,M)R(k,Gi)=l). LetBkl=[iklaij]Ki,j=1,(k,l2M),thentheprobabilitytransitionmatrixofthebivariateMarkovchainfqn1,gngbecomesT=0BBBBBBB@B00B01B0MB10B11B1M............BM0BM1BMM1CCCCCCCA. Sincewehaveassumedthatbychoosingappropriateratecontrolpolicies,thebivariateMarkovchainfqn1,gngisergodic;thusitsstationarydistributionexists(oursimulationshowsthatformostcases,thisassumptionisvalid).Let=(0,1,...,M)denotethestationarydistributionwherek=(k1,k2,...,kK)wherekj=limn!1Pr(qn1=k,gn=Gj),k2M,j=1,...K. 29

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Sothesteadystatedistributionofqnisgivenbylimn!1Pr(qn=k)=KXj=1limn!1Pr(qn1=k,gn=Gj)=KXj=1kj,k2M. Then,from( 2 ),wegetthepacketdropprobabilityPqe=1 41 ],isdenedastheprobabilityofthatapacketfailstoreachitsdestinationwithinagivendelaybound.Generally,itisdifculttoderivethequeueingdelayexceptforsomebasicformqueueingsystems[ 42 ].Byanalyzingthearrivalanddepartureprocesses,numericalmethodsareproposedtocalculatethequeueingdelayin[ 41 ]and[ 43 ].Sinceweareconsideringanite-bufferdiscrete-timequeueingsystem,thedeparture,queuelengthandarrivalofwhicharecorrelated,thecalculationofqueueingdelayis,ifnotimpossible,aformidabletask.Thusweproposeanalternativeupperboundapproach.Sincethebuffersizeifnite,apacketcanbeeitherdroppedduetobufferoverowortransmitted(removedbytheratecontrolunit).Sincethepacketdrop 30

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2.3.2 ,nowweonlyconsiderthetransmittedpackets.Wehavethefollowingusefullemma. 2-2 .LetDdenotethequeueingdelayofapacketwhichdepartsinthen-thblock,thenthefollowinginequalitieshold.PrnXi=nDmax+1aiDmax)PrnXi=nDmax+1aiDmax)PrnXi=nDmax+1aiDmax.Thustherstinequalityholds. Bufferstatusinthen-thblock 31

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Nowweconsiderthreetypesofarrivalprocesses:constantarrival,Poissonarrivalandgenerali.i.d.arrivalandderivethecorrespondingupperbounds. 2 )becomesPr(D>Dmax)Pr(Dmax><>>:0,ifDmaxM=,1,ifDmax
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41 ]indetail. 44 ].AnupperboundonPrPDmaxi=1aiMcanbederivedbydirectlyapplyingtheGartner-Ellistheorem. 2 ),wereplacedqn+rnin( 2 )withM.Whentheloadofthequeueingsystemislow,thequeueisgenerallyfarfromfull.However,weseethateventheproposedupperboundsdecreaserapidly,especiallywhenthebuffersizeischosentobelargerthanM=.Thus,inpractice,wecanchooseappropriatedelayboundandbuffersize,whichresultinneglectabledelayboundviolationprobability,andsimplifytheanalysisintheremainderofthischapter. 33

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where^PdedenotestheupperboundonPr(D>Dmax)wederivedaboveandthelastequalityholdsbecauselimn!1Ern=(1Pqe),thatis,thetotalarrivalequalstothedroprateplusthetransmissionrate. 2.3.3 holdsforanyblock,whichguaranteesthecalculationin( 2 ). 34

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wherePce,PqeandPdearegivenin( 2 ),( 2 )and( 2 )respectively.Specially,ifthearrivalisconstantandthedelayboundischosentobelargerthanM=,wehavePe(R,Dmax,)=Pqe+Pce, wheretheequalityholdsbecausepacketdropanddecodingerroraremutualexclusive,i.e.,ifonepacketisdropped,itcannotbetransmitted. Wehaveconductedpacketerroranalysis.Thetotalpacketerrorprobabilityisexpressedasafunctionofratecontrolpolicy,delayboundandaveragearrivalrate.Inthenextsection,wewillinvestigatetheoptimalratecontrolpolicywhichminimizethetotalpacketerrorprobability. 35 ],thethroughputofthesysteminFig. 2-1 isdenedasthelong-termaveragedatarateatwhichpacketsaresuccessfullytransmitted.GivenarandomarrivalprocessfangwithmeananddelayconstraintDmaxthethroughputcanbecalculatedby(1Pe(R,Dmax,)). Thus,maximizationofthroughputisequaltominimizationofpacketerrorprobability,whichisminR2RPe(R,Dmax,) whereRistheadmissionratecontrolspace.Sincetheexactpacketerrorprobabilityisunavailable,weseektondaratecontrolpolicewhichminimizesthefollowingupper 35

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whichisderivedbasedon( 2 ).Fromtheabovediscussion,theproblemofndingtheratecontrolpolicywhichmaximizethethroughputisanaveragecostMarkovdecisionproblemwithstatespaceQGAandper-stagecostPce(gn,rn)rn+(1^Pde)(qn1+anM)++^Pde.Suchoptimizationproblemcanbesolvedviathepolicyiterationalgorithm[ 45 ]However,thecomputationalcomplexityofpolicyiterationishigh.ThecomputationalcomplexityisO(jQj3jGj3jAj3).Fortunately,inoursimulations,wendthepolicyiterationalgorithmconvergesinasmallnumberofiterations,e.g.,inabout10iterations.Thusthecomputationalburdenisacceptableingeneral. AsinSection 2.3.2 ,weconsiderthespecialcaseofconstantarrivalandshowhowtondtheoptimalratecontrolpolicyviapolicyinteractionalgorithm.WemakethesameassumptionsaboutthechannelgainprocessasinSection 2.3.2 .Theminimizationproblemin( 2 )canbeformulatedasaninnite-horizonMarkovdecisionproblemwithaveragecostandstatespaceMG.Thestatetransitionprobabilityispljki(R),Pr(qn=l,gn+1=Gjjqn1=k,gn=Gi,R) whereikl,Pr(qn=ljgn=Gi,qn1=k)=Pr(min(k+,M)R(k,Gi)=l). Thestagecostisgivenbyf(qn1,gn,R),Pce(gn,rn)rn+(1^Pde)(qn1+anM)++^Pde. 36

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45 ],whichwedesignasfollows.Letht(k,i)andtdenotethedifferentialandaveragecostsatthet-thstep. 1. InitialStep:RandomlychooseafeasibleinitialpolicyR0. 2. Policyevaluationstep:t+ht(k,i)=f(k,i,Rt(k,i))+MXl=0KXj=1pljki(Rt(k,i))ht(l,j),k2M,i=1,2,...,K, 3. Policyimprovementstep:Rt+1(k,i)=argminrf(k,i,r)+MXl=0KXj=1pljki(r)ht(l,j)! 4. Terminationconditionst+1=t,ht+1(k,i)=ht(k,i),k2M,i=1,...,K. 46 ]andeffectivecapacity[ 10 ]approaches.Thisproblemcanalsobeinterpretedaslookingforanequivalentconstantratechannelforatime-varyingchannel.Weproposetouseatriplet(,Dmax,")tocharacterizethedelay-constrainedthroughputofafadingchannel,whereisthemaximumdatarateofaowwithdelayboundDmaxandpacketerrorprobability".Specically,givendelayboundDmaxanderrorprobability",wewanttondthemaximumdatarate.Byvaryingthedelayandpacketerrorprobabilityconstraints,weobtainaParetooptimalsurface.Suchrate-error-delay 37

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Aswementionedabove,ifthearrivalisconstant,thereisanexplicitrelationbetweendelayboundviolationprobabilityandbuffersize:ifDmaxM=,thenthecorrespondingdelayboundviolationprobabilityiszero.Sincethebuffersizeisoftenadesignparameterforpracticalsystems,wemayalwayschooseM=Dmax.Inthereminderofthiswork,wealwaysmakethefollowingtwoassumptions:1)constantarrival,2)buffersizeM=Dmax.Thus,theratecontrolpolicyonlydependsonthechannelgainandqueuelength,i.e.,thedeparturerateinthenthblockisrn=R(qn1,gn). WerstinvestigatetheadmissioncontrolspaceinSection 2.5.1 ,thenwestudythestructuralpropertiesoftheParetooptimalsurfaceinSection 2.5.2 .InSection 2.5.3 ,weshowconsistencybetweenREDfunctionandeffectivecapacityproposedinRef.[ 10 ]. LetU(qn1,gn)Qdenotethesetoffeasiblevaluesofun,i.e.,un(qn1,gn)2U(qn1,gn).NowwedetermineU(qn1,gn).Sincethequeuelengthisavailableatthetransmitter,thedeparturerate(inunitofpackets/block)cannotexceedthetotalnumberofpacketsinthebuffer,sowehaveun(qn1,gn)min(qn1+,M).Moreover,sincetheCSIisavailableatthetransmitter,itisnotreasonabletotransmitatarate 38

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Llog21+jgnj2Pt ThuswehaveU(qn1,gn)=frjr2M,rmin(qn1+,M),rC(gn)g=[0,min(M,qn1+,C(gn))] Wedenotebythesetofadmissiblepolicies,whichisthesetofallsequencesoffunctionsu=fu1,u2,...gwhereun:QG!Qandun(qn1,gn)2U(qn1,gn),n=1,2,...Thentheminimumpacketerrorprobabilityisminu2Pe(u,Dmax,). AswediscussedinSection 2.4 ,theabovemaximizationproblemisaninnite-horizonMarkovdecisionproblemwithaveragecost.From[ 45 ],itisknownthatforanaveragecostproblemwithnitestateandcontrolspace,therealwaysexistsastationarypolicywhichisoptimal.Forinnitestateandcontrolspace,withsomemildconditions,therealsoexistsanoptimalstationarypolicy.Withoutlossofgenerality,throughoutthischapter,weassumetherealwaysexistsanoptimalstationaryratecontrolpolicy. Nowwepresenttwolemmasaboutthestructureoftheadmissioncontrolspace.ConsiderasystemasshowninFig. 2-1 ,givenarealizationofchannelgainsequenceH=fHngandbuffersizeM,wesayasequenceofcontrolactionsfrngisfeasibleifrn2U(qn1,Hn),wherefqngisthecorrespondingqueuelengthsequence.Thefeasibleconditionguaranteesqn0,n=0,1,...Let)]TJ/F8 7.97 Tf 6.77 4.33 TD[(HMdenotethesetoffeasiblecontrolactionsequences.Wehavethefollowingresults.

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2.2 tellsthatwithalargerbuffersize,wewillhavemorefreedomtochoosecontrolpolicies.Nowwestudytheextremecase,thatis,innitebuffersize.Inthiscase,thebuffersizeconstraintonthefeasiblecontrolspaceislifted,i.e.,U(qn1,gn)=[0,min(qn1+,C(gn)].Wehavethefollowingresultforinnitebuffersystems. 2-1 withinnitebuffersize.Thesetoffeasiblecontrolactionsequences)]TJ/F8 7.97 Tf 6.77 4.34 TD[(H1isconvex. 2.3 maynothold. andtheREDfunctionisanexpressionofthemaximumachievabledataratewithdelayandpacketerrorprobabilityconstraints:RED(",Dmax),maxfjERD(,Dmax)"g where"isthegivenconstraintofpacketerrorprobability.NowweinvestigatethepropertiesofERDandREDfunctions.WehavethefollowingpropositionaboutmonotonicityoftheERDfunction. Nowassumethephysicalbuffersizeisinnite,i.e.,thequeuecangrowtoinnite.ThevirtualbuffersizeMisstillnite.ItiseasytoseethatinthiscaseERD(,Dmax)is 40

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InProposition 2.2 ,therequirementthatPce(g,r)isaconvexfunctionofrcanbesatisedwithrandomcoding,whichcanbeseenfrom( 2 ).Proposition 2.2 tellsthatalthoughtheminimumpacketerrorprobabilityisnotaconvexfunctionofdelayboundingeneral,anupperboundontheminimumpacketerrorprobabilityisconvex. ThefollowingtheorempresentsthemonotonicpropertyoftheREDfunctionandtheREDParetosurface. 10 ]providesaformulathatcharacterizestherelationshipamongthroughput,delaybound,anddelayboundviolationprobability.However,unliketheREDtheoryofthischapter,theeffectivecapacityapproachassumesthedecodingerrorprobabilityiszeroduetotheuseofanidealchannelcode.Inaddition,theeffectivecapacityapproachuseslargedeviationtheorywhiletheREDtheoryusesthetheoryofMarkovdecisionprocess.Moreover,theeffectivecapacityapproachassumesinnitephysicalbuffersizeinsteadofnitephysicalbuffersize;hence,forthequeue,theeffectivecapacityapproachstudiesthedelaybound 41

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). Foraproof,seetheAppendix.TheimplicationofTheorem 2.4 istwo-folded.First,itallowsustousethedelayboundviolationprobabilityestimatedbytheeffectivecapacityapproach[ 10 ]topredict(orapproximate)thepacketdropprobabilityofthesystemstudiedbytheREDtheory.Notethatinpractice,itismucheasiertoestimatethedelayboundviolationprobabilityusingtheeffectivecapacityapproachthanestimatingthepacketdropprobabilityofSystemA.Second,ittellsthattheREDtheoryisconsistentwiththeeffectivecapacitytheorywhendecodingerrorisnotconsidered. 2.6.1 ,weusesimulationresultstoverifytheupperboundondelayboundviolationprobabilityproposedinSection 2.3.3 .Thenweuseanexampletoshowthetradeoffbetweenpacketdropprobability(linklayer)anddecodingerrorprobability(physicallayer),whichjustiesamajormeansinourREDtheory,i.e.,anoptimalratecontrolpolicythatoptimallybalancesthelinklayerandthephysicallayerperformance.Then,wegiveanexampleoftheParetosurface,resultingfromtheoptimalratecontrolpolicy.Finally,wecomparetheoptimalpolicywiththeoptimalxed-decoding-errorpolicy. 42

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2-1 .Weconsidertwotypesofarrival:constantarrivalandPoissonarrival,bothofwhichhavemeanpacketsperblock.ThedepartureisdeterminedbyaratecontrolpolicyR.ThechannelgainsequencefgngismodeledasaMarkovchainwithstatetransitionmatrixQ.Inallourexperiments,thetimeblocklengthissettoTb=0.005s.ThechannelbandwidthisW=20kHz,thusthenumberofcomplexchannelusesperblockN=WTb=100.TheSNRwithoutfading(channelgainjgnj=1)is10dB.Thelinklayerpacketsizeissetto10bits.Thesetofsimulationsettingcanbearbitrary,however,wejustchooseappropriateparameterstoclearlyshowourresults. Weconsiderthreetypesofratecontrolpolicies:linearpolicy,optimalpolicy,andoptimalxed-decoding-errorpolicy.ThelinearratecontrolpolicyisoneofthesimplestpoliciesandisdenedasbelowRLinear(m,g)=min(m+,M,bC(g)c) whereC(g)istheinstantaneouschannelcapacity,and2[0,1].Wecallitlinearpolicysincetherateisupperboundedbyalinearfunctionoftheinstantaneouschannelcapacity,i.e.,C(g). Theoptimalpolicycanbeobtainedbythepolicyiterationalgorithm[ 45 ].Theoptimalxed-decoding-errorpolicyisanoptimalpolicyundertheconstraintofxeddecodingerrorprobability.Foratargetdecodingerrorprobabilitydec2(0,1),theoptimalxed-decoding-errorpolicyisdenedbyRDEC(m,g)=minm+,M,bmaxPce(g,r)decrc. Theoptimalxed-decoding-errorpolicytakesapurequeueingapproachsinceitonlyoptimizesthequeueingperformancewhilekeepingxeddecodingerrorprobability;inotherwords,itdoesnotjointlyoptimizethephysicallayerandthelinklayer. 43

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Theratecontrolpolicyisanlinearpolicywithparameter=0.4asdenedin( 2 ).Fig. 2-4 showsthesimulationresultofdelayboundviolationprobabilitycomparedwiththeupperboundproposedin( 2 )inSection 2.3.3 .Fromthegure,wecanseethat Delayboundviolationprobabilityanditsupperbound boththesimulationresultandtheupperbounddecreaserapidlyasthedelayboundbecomeslarger,especiallywhenthedelayboundislargerthanM=.Thuswecan 44

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2 ).Fig. 2-5 showspacketdrop Figure2-5. Packeterrorprobability,decodingerrorprobability,andpacketdropprobabilityunderlinearratecontrolpolicies probability,averagedecodingerrorprobability,andaveragepacketerrorprobabilityvs.(theparameterofthelinearcontrolpolicy).Fromthegure,itcanbeobservedthatasincreases,packetdropprobabilitydecreaseswhiledecodingerrorprobabilityincreases.Hence,thereisatradeoffbetweenpacketdropprobabilityanddecodingerrorprobability.Inaddition,thegureshowsthattheminimumpacketerrorprobability 45

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Thisexampleindicatesthattheoptimalperformancecanonlybeachievedthroughcrosslayerdesignduetotheconictingnaturebetweenthelinklayerandthephysicallayerperformance(i.e.,packetdropprobabilityvs.decodingerrorprobability). 2.3.3 .ItiseasytocalculatethattheShannonergodiccapacity(withoutpowercontrol)is44kb/s;forratecontrol,weusetheoptimalpolicy.Fig. 2-6 showstheParetosurface,i.e.,maximumdatarateasafunctionofdelayboundDmaxandpacketerrorprobability.Inthisandthenextsimulation,whenplottingParetosurfaces,weincreasethedelayboundby2blockstoincludetheencoding/decodingdelay.Fromthisgure,itcanbeobservedthatthemaximumdatarateisamonotonicallyincreasingfunctionofdelayboundandpacketerrorprobability.Moreover,themaximumdatarateisalwayssmallerthantheergodiccapacity. 2 ),andwechoosedec=105.Fig. 2-7 showsthemaximumdatarateasafunctionofdelayboundDmaxandpacketerrorprobability,undertheoptimalpolicyandtheoptimalxed-decoding-errorpolicy,respectively.Fromthisgure,itcanbeseenthattheoptimalpolicyachieveshigherdataratethantheoptimalxed-decoding-errorpolicy.Thisisbecausetheoptimalpolicybalancespacketdropprobability(linklayer)anddecodingerrorprobability(physicallayer)whiletheoptimalxed-decoding-errorpolicyisapurequeueing(linklayer) 46

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Paretosurface:maximumdatarate()asafunctionofdelayboundDandpacketerrorprobability" 47

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Performancecomparisonbetweentheoptimalpolicyandtheoptimalxeddecodingerrorpolicy(purequeueing) resultsprovideimportantinsightsintostatisticalQoSprovisioninginwirelesssystems;theREDParetosurfacerepresentsamajorsteptowardderivingtheprobabilisticdelay-constrainedchannelcapacityoffadingchannels.Inourfuturework,bothrateadaptationandpoweradaptationwillbeusedtoachieveahigherdatarate. 48

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SincethepublicationoftheoriginalworkofGuptaandKumar[ 11 ]in2000,therehasbeenalotofworksininvestigatingthethroughput,withorwithoutdelayconcern,ofwirelessadhocnetworks.Inthisworks,networkmodelswithnnodesareconsidered.In[ 12 ],byallowingnodestomoveandbasedona2-hoprelayingscheme,GrossglauserandTseshowedthat,ifthenodemotionisindependentacrossnodesandhasauniformstationarydistribution,thethroughputpersource-destination(S-D)paircanbedramaticallyincreasedto(1)from(1=p 11 ].Thedelaycausedbytherelayingschemeisnotaddressedin[ 12 ].Thatis,toachievethethroughputcapacity,packetsmaytravelarbitrarylongtimefromtransmittertoreceiver. Sinceboththroughputanddelayareimportantnetworkperformancemetricsfromanapplicationviewpoint,signicantefforthasbeendevotedtoinvestigatingtherelationshipbetweenthroughputanddelayinadhocnetworks.BansalandLiu[ 47 ]wereamongthersttostudytherelationshipbetweendelayandthroughputinwirelessnetworks.Ageographicroutingschemewhichachievesnearoptimalcapacityisproposedin[ 47 ]andthecorrespondingdelayperformanceisstudiedaswell.Delaylimitedcapacityofmobileadhocnetworks(MANET)isstudiedin[ 48 ]. 49

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1 49 50 ];therandomway-pointmobilitymodel[ 51 ];theBrownianmobilitymodel[ 51 54 ];andtherandomwalkmobilitymodel[ 13 52 53 ].Withdifferentmodelsandsettings,thethroughput-delaytradeoffsreportedintheaboveworksdifferconsiderably.In[ 14 ],Sharmaetal.proposedaglobalapproachtounifypartofpreviousworksonthethroughput-delaytradeoffinmobileadhocnetworks.Theauthorsdenedanewconcept:criticaldelay,belowwhichthenodemobilitycannotbeexploitedforimprovingthecapacity.Itisshownthatthecriticaldelaydependsonthenatureofthenodemobilitybutnotsomuchonthenetworksetting.Recently,networkcoding,whichisrstintroducedbyAhlswedeetal.[ 55 ],isusedtofurtherimposethethroughputanddelayperformancesinwirelessnetworksin[ 56 61 ]. Inallaboveworks,whichaddressnetworkdelay,onlyaveragedelayisconsidered.Thoughthethroughput-delaytradeoffresultsprovidedeepinsightsintounderstandingthefundamentalperformancelimitsofdelayconstrainedcommunicationsoverwirelessnetworks,theyareobviouslyinadequateforrealtimeapplicationssincewithknownaveragedelay,theactualdelaymayhavealargedynamicrange.Forexample,realtimevideotransmissionrequiresstringentdelay(delayboundordeadline)butallowsasmallprobabilityofoutage(packetlossordeadlineviolation)probability.Ifapacketcannotreachitsdestinationwithinagivendelaybound(e.g.,0.1s),itmaybeconsideredasalostpacketatthereceiverandcannotbeusedfordecoding.Thepacketlossrateduetodeadlineviolationcanbeinterpretedasareliabilitymetricofwirelessnetworks.Suchpracticalrequirementofrealtimeapplicationsmotivesustostudytherelationshipbetweendelayboundanddeadlineviolationprobability.ThistopicbelongstotheareaofstatisticalQoSprovisioningandhasbeenextensivestudiedinsingle-userormulti-userwirelesssystems[ 10 41 62 65 ].However,tothebestofourknowledge,weareamong 50

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Weconsideramobilewirelessnetworkmodelwithnnodesandagivencommunicationscheme.Let(n)denotesystemthroughput,B(n)denotethegivendelaybound(deadline)andPl(n)denotethecorrespondingdeadlineviolationprobability.ThereexistsaexplicittradeoffbetweenB(n)andPl(n),i.e.,ifwepicklarger(smaller)delaybound,wemayexpectasmaller(larger)deadlineviolationprobability.Thusseveralinterestingquestionsarise:asthenetworksizescalesup,1)givenadelayboundB(n),howdoesthedeadlineviolationprobabilityscale?2)doesthedeadlineviolationprobabilitygotozeros?3)ifthedeadlineviolationprobabilitygoestozeros,howfastisit? Inordertoanswertheabovequestions,westudytheend-to-endpacketdelayinadhocnetworksbyaqueueingtheoreticalapproach.Ingeneral,itisverydifculttoanalyzethestatisticaldelayperformanceofwirelessnetworks.Forexample,evenacoarseupperboundondeadlineviolationprobabilityprovidedbytheChebyshev'sinequalityrequiresthevarianceofpacketdelay,whichisunknownformostqueueingsystems.Thusinthisrstwork,weconsiderasimplei.i.d.mobilitymodel.Considerationsofmorerealisticmobilitymodelsareproposedforfuturework.WeshowthatifandonlyifthedelayboundhasorderlargerthantheaveragedelayD(n),i.e.,B(n)=!(D(n)),thecorrespondingdeadlineviolationprobabilityPl(n)willgoestozeroasngoestoinnity.Withanappropriatechoiceofdelaybound,thedeadlineviolationprobabilitycanhavearequireddecayrateandthetradeoffbetweendeadlineviolationprobabilityanddelayboundisgivenbyPl(n)=e(B(n)=n).Sincethedeadlineviolationprobabilitycanbeinterpretedasadescriptionofthereliabilityofdelay-sensitivecommunications,ourresultsprovideinsightsintounderstandingthereliability-delaytradeoffinlargescalewirelessadhocnetworks. 51

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3.3 ,weestablishthereliability-delaytradeoffinmobileadhocnetworks.Section 3.4 concludesthischapter. 1 ]asshowninFig. 3-1 .Theshapeandlayoutofcellregionisarbitrary,althoughthecellsareassumedtohaveidenticalarea,donotoverlapandcompletelycoversthenetworkarea.Therearenmobilenodesonthenetworkarea,eachofwhichcanbebothtransmitterandreceiver.Timeisslottedforpacketsizedtransmission.Forsimplicity,thetimeslotsareassumedtohaveunitlength.PackettransmissionisonlyallowedbetweentwonodesinthesamecellasshowninFig. 3-1 .Iftwonodesarewithinthesamecellduringatimeslot,onecantransferasinglepackettotheother.Duringoneslot,onlyonenodecantransmitineachcell.Thiscellpartitionednetworkmodelreducesschedulingcomplexityandfacilitatesanalysis.Two-dimensionali.i.d.MobilityModel 1 14 53 61 ].Inthiswork,weadoptsuchaoversimpliedmobilitymodelto 52

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Mobilenetworkmodel(Fig.1in[ 1 ] facilitateouranalysis.Considerationsofmorerealisticmobilitymodels,e.g.,randomwalk,Browniemotion,areproposedforfuturework.Wereferto[ 1 ]formorediscussionsaboutthei.i.d.mobilitymodel. 13 ]and[ 53 ]respectively. 53

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WedenotethepacketdelaybyD(n),whichisafunctionofnetworksize. 54

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Whentransmittingdataoverwirelessnetworks,onemayexpectareliableservice,i.e.,mostofthepacketscanreachtheirdestinationwithinagivendelaybound.Sinceweareinterestedinthescalingpropertyofwirelessnetworks,itisnaturaltorequirethedeadlineviolationprobabilitygoestozeroorhassomedecayrateasthenetworksizengoestoinnity.Thusweproposeanewdenitionofdelay:reliabledelay. Areliabledelayguaranteesdecreasingdeadlineviolationprobabilityasthenetworksizescaleup.Alongthisdirection,wewillfurtherstudytherequireddelayboundsuchthatthedeadlineviolationprobabilityhasspecieddecayrate,whichwenameReliabilityIndexandisdenedasfollows. Adelayboundisreliableifandonlyifthecorrespondingreliabilityindexgoestoinnityasthenetworksizescalesup.Thereliabilityisadescriptionofthedecayrateofthedeadlineviolationprobability.

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10 ].Thedenitionofreliabilityindexsimpliesnotationsandcapturesthekeycharacteristicofthedeadlineviolationprobability. 13 52 54 ],whichonlyconsidertheaveragedelayguarantee,sincewithknownaveragedelay,theactualpacketdelaymayhavealargedynamicrange. 1 ]asshowninFig. 3-1 .Forthesakeofcompleteness,webeginwithabriefoverviewofthenetworkcapacityanddelayresultsinSectionIIof[ 1 ]beforepresentingourreliability-delaytradeoffresult. ThenetworkispartitionedintoCcellsofequalarea.Therearennodesindependentlyroamingovercellsinthenetwork.Letd=n=Cdenotethedensityofnodes.Asthenetworksizescalesup,weassumethenodedensityofxed,i.e.disaconstant.Underthei.i.d.mobilitymodel,inthesteadystate,eachnodeisuniformlydistributedoverallcells.Letirepresenttheexogenousarrivalrateofpacketstonodei(inunitofpacketsperslot).ThepacketarrivalisassumedtobeaBernoulliprocess,sothatwithprobabilityi,asinglepacketarrivesduringthecurrentslot,andotherwise,nopacketarrives.Assumeallnodeshavethesamearrivalrate,i.e.,i=foralli.Aschedulingschemeisstableifthequeuesinthenetworkdonotgrowtoinnityandtheaveragequeueingdelayisbounded.Thecapacityofthenetworkisdenedthemaximumratesuchthatthenetworkcanstablysupport. 56

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1 ],Neelyetal.provedthatthecapacityofamobilenetworkasshowninFig. 3-1 isc=p+q wherep=1(11 C(11 A2-hoprelayschemeisproposedin[ 1 ],andisshowntoachievethenetworkcapacitywithaveragedelayn1 =(n). Takengoestoinnitywhilekeepdxed,thenetworkcapacitytendstothexedvalue(1edded)=(2d).Thusthenetworkhascapacity(1),whichcanbeachievedwith(n)averagedelay. Nowweproceedtostudytherelationbetweendelayboundanddeadlineviolationprobability.Weslightlyrevisethecellpartitionedrelayalgorithmproposedin[ 1 ]tofacilitateourdelayanalysis.Therevisedschemeisdescribedasbelow. 1 ],revised):Ineverytimeslotandforeachcellcontainingatleasttwonodes: 1. IfthereexistsanS-Dpairwithinthecell,randomlychoosesuchapair.Ifthesourcecontainsanewpacketintendedforthatdestination,transmit.Elseremainidle. 2. IfthereisnoS-Dpairinthecell,designatearandomnodewiththecellassender.Independentlychooseanothernodeasreceiveramongtheremainingnodeswithinthesamecell.DividethetimeslottotwosubslotsAandB.Thefollowingisdone. 57

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(b) 1 ]isthateachtimeslotisdividedtotwosubslotsthussendingarelaypackettoitsdestinationandsendinganewrelaypacketcanbeconductedindependently.Bymakingsuchrevision,thearrivalandserviceoftherelayqueuesareindependent(intheoriginalscheme,arrivalandservicearemutuallyexclusive),whichmakestherelayqueuesBernoulliqueuesaswellasthesourcequeues. d=(1), p2dp=(n). TheproofofProposition 3.1 utilizessimilartechniquesin[ 1 ]andisputintheAppendix. 3 ).ThisisbecausewhendirectS-Dtransmissionisavailable,onlyonepacketistransmittedintwosubslots.Asthenetworksizescalesup,theprobabilityofdirecttransmissionismuchlessthanrelayedtransmission,thusthethroughputlossofScheme1isneglectable. 58

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3-1 isPl(n)=e(B(n)=n) TheproofofTheorem 3.1 isputintheAppendix.Fromtheproof,wecanseethatwehaveactuallyobtainedtheexactvalueofthedeadlineviolationprobability.However,sincewearemoreinterestedinthescalingpropertyofthedeadlineviolationprobability,weuseasymptoticnotationsinTheorem 3.1 Tradeoffbetweenreliabilityindexanddelayboundformobilenetworks.Thescaleoftheaxesareintermsoftheordersofn. Fig. 3-2 illustratestherelationshipbetweenreliabilityindexanddelayboundformobilenetworks.Fromthegure,wecanseethatinordertoachieveasymptoticallysmalldeadlineviolationprobabilityasthenetworksizescalesup,onemustchooseadelayboundB(n)=!(n).Moreover,appropriatedelayboundmaybechosen 59

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3.1 andFig. 3-2 ,weseethatbychoosingappropriatedelaybound,wecanobtainrequireddeadlineviolationprobability(inavalidrange).Thus,ourresultprovideaguidelineforprovidingstatisticalQoSguaranteeforlargescalewirelessnetworks. Asabeginning,thischapterbuildsageneralframeworkforstudyingthestatisticaldelayperformanceofwirelessnetworks.Futureworkmayextendtoinvestigatingthereliability-delaytradeoffinmorecomplexscenarios,suchas,morerealisticmobilitymodelsandschedulingschemeswithnetworkcodingetc. 60

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15 ]foritshighdataratetransmissioncapabilityandintersymbolinterferencemitigationability.Moreover,incontrastwithsinglecarriersystems,channelequalizationcanbeeasilyaccomplishedinfrequency-domain.RecentlydevelopedmobileOFDMtechnologiesincludeWiMAX[ 16 ],DVB-SH[ 17 ]andMediaFLO[ 18 ]. SuccessfulimplementationofOFDMrequiresorthogonalityofsubcarriers.ThusOFDMsystemsarequitesensitivetocarrierfrequencyoffset(CFO)whichmaycausethelossoforthogonalityofsubcarriers[ 19 ].AcomprehensivestudyoftheeffectoftimingandfrequencysynchronizationerrorsontheperformanceofOFDMsystemsispresentedin[ 66 ].VariousapproacheshavebeenproposedforCFOestimationduringthepastabouttenyears.CurrentCFOestimationtechniquescanbedividedintotwoclasses:pilot-assistedmethods[ 67 70 ]andblindones[ 21 25 71 75 ].Recently,blindestimationhasreceivedmuchattentionforitshighbandwidthefciency. In[ 21 ],anMLestimatorisdesignedtojointlyestimatethetimingandfrequencyoffsetbyexploitingtheinformationcontainedinthecyclicprex(CP)andisextendedtothemultiuserscenarioin[ 73 ].Thismethodisfurtherdiscussedandextendedin[ 71 ]and[ 74 ].Advancedsignalprocessingtechniques,e.g.,expectationmaximization(EM),expectation-conditionalmaximization(ECM)andspace-alternatinggeneralizedexpected-maximization(SAGE),areusedtoaccomplishdatadetectionandCFOestimationsimultaneouslyin[ 22 ]and[ 75 ].Byanalyzingthepowerspectrumofreceivedsignal,aclosed-formblindCFOestimatorisderivedin[ 72 ]. MostoftheaforementionedtechniquesestimatetheCFObasedononeOFDMsymbol.Asweknow,theperformanceofanestimatorstronglydependsonthesizeofobserveddataset.EstimatorsbasedononeOFDMsymbolmaysufferfromlow 61

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21 ],theCFOestimationerrorprobabilitycouldbeashighas5%(normalizedbysubcarrierspacing)forfrequencyselectivechannelswithSNRof10dB.Thisestimationerrorcouldbelargeenoughtoruintheorthogonalityofsubcarriers[ 19 ]. Inthischapter,weproposeanMLestimator,whichwereferastherecursiveCFOestimator(RCFOE),tousemultiplesuccessiveOFDMsymbolstomakeCFOestimation.Inordertoavoidlargeprocessingdelayandreducebufferrequirement,theproposedestimatorisimplementedinarecursivefashion.TheCFOestimationisupdatedatthetimeofreceivinganewOFDMsymbol,whilethepreviouslyCFOestimationsaretakenintoaccountaswell.Weshowthat,whentheCFOisconstant,themeansquarederror(MSE)ofCFOestimationdecreasesproportionallyto1=n(approximately),wherenisthenumberofsymbolsusedforCFOestimation(estimationwindowsize). Lately,time-varyingCFO,whichiscausedbyasymmetricDopplerspectrum,hasbeenseriouslyconsidered[ 20 ].Generally,whenderivingCFOestimators,CFOisassumedtobeconstantorslowtime-varying.However,sincetheRCFOEestimatorusesmultipleOFDMsymbols,attentionmustbepaidtotime-varyingCFO.Inthissection,weanalyzetheeffectoftime-varyingCFOandinvestigatetheerrorvariance.WeshowthatthemeansquaredestimationerrorduetoCFOvariationincreasesquadraticallywithn(approximately)ifthechannelnoiseisabsent;whilethemeansquaredestimationerrorduetochannelnoisedecreasesproportionallyto1=n(approximately)iftheCFOisconstant.Therefore,thereisanexplicittradeoffbetweentwotypesoferrors,whichcanalsobeconsideredasthetradeoffbetweenfasttrackingabilityandlow-varianceestimates[ 76 ].IftheCFOchangesslower,wecanusemoreOFDMsymbolsforCFOestimationtoachievehighprecision,whileiftheCFOchanges 62

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CFOestimation,whichisbaseduponmultipleOFDMsymbols,andthetradeoffbetweenfasttrackingabilityandlow-varianceestimateshavealsobeenstudiedinpreviousliteratures[ 23 25 73 ].In[ 73 ],CFOestimationsbasedonsingleOFDMsymbolaremoving-averagedtogetanewestimate.However,averageofmultipleestimatesisnotoptimalingeneral.AlowcomplexitynonlinearleastsquareCFOestimatorbasedonmultipleOFDMsymbolsisproposedin[ 23 ].In[ 24 ],ablindCFOestimatorisproposedbyminimizingthepowerofnondiagonalelementsofthecovariancematrix,whichisestimatedbytime-averageoftheCFOcompensatedreceivedvector.Byinvestigatingthein-bandrippleoftheOFDMpowerspectraldensity(PSD),anadaptiveblindcarrierfrequencytrackingalgorithmisproposedin[ 25 ].Thechoiceofthestep-sizeofthetrackingalgorithmservesasimilarroletothatofadjustingtheaveragewindowsize.Onecommonlimitationoftheaboveworksisthatthereisnoclosed-formexpressionfortheaveragewindowsizeorstep-size.Incontrast,byinvestigatingtheeffectoftime-varyingCFO,weprovideaclosed-formexpressionfortheestimationwindowsizeasafunctionofCPlength,SNRandrateofchangeofCFO. Therestofthischapterisorganizedasfollows.WeintroducetheOFDMmodelinSection 4.2 .TherecursiveMLestimatorforOFDMsystemswithconstantCFOisderivedandanalyzedinSection 4.3 .Weinvestigatetheerrorvarianceduetotime-varyingCFOandobtaintheoptimalestimationwindowsizeninthesenseof 63

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4.4 .SimulationresultsareshowninSection 4.5 whileconcludingremarksaregiveninSection 4.6 4-1 .Thetransmitteddatainthelthblock,whichisdenedasxl=[xl(0),xl(1),...,xl(N1)]T,ismodulatedbyinversediscreteFouriertransform(IDFT).Thenacyclicprex(CP)oflengthLisinsertedtoavoidtheintersymbolinterference(ISI).Theresultingvectorsl=[sl(0),sl(1),...,sl(N+L1)]Tisthentransmittedoverthechannel.Thechannelimpulseresponse(CIR)isassumedtobestaticoveranOFDMsymbolwhileitmayvaryfromblocktoblock.TheCIRisdenotedbyhl=[hl(0),hl(1),...,hl(1)]T,wherelL.Weassumesometypeofcoarsetimingestimationhasbeenconducted.SotheremainingtimingoffsetisonlyafractionalpartofthesampleperiodTsthuscouldbeincorporatedintotheCIR,asexplainedin[ 67 ]. Figure4-1. Thediscrete-timebasebandmodeloftheOFDMsystem WeassumetheCFOisconstantduringatleastoneblockbutmaybetime-varyingoverconsecutiveblocks.Let"l2(1=2,1=2]representtheCFOnormalizedbytheintercarrierspacinginthelthblock.Let^"lbetheCFOestimationwhichcanbeusedforthenexttransmission.Forconvenience,welet^"0=0.ThentheCFOestimationerrorinthelthblockisl="l^"l1. 64

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wherel=ej2(N+P) istheaccumulatedphaseshiftandwl(k)CN(0,2w)istheindependentadditivewhitenoise. Thetransmittedsignalsl(k)istheIDFTofthedatasymbolsxl(k),whichareassumedtobeindependent,identicallydistributed(i.i.d.).Hencesl(k)islinearcombinationofi.i.d.randomvariables.Asdiscussedin[ 21 ],ifthenumberofsubcarriersNissufcientlylarge,sl(k)isapproximatelyacomplexGaussianprocess,whoserealandimaginarypartsareindependent,withpower2s.SincethesymbolsintheCParecopiedfromthedatasymbol,sl(k)isnotawhiteprocess.TheredundancyinformationcontainedintheCPandtheircopiescouldbeutilizedtoestimatetheCFO. 4.3.1DerivationoftheBasicFormRecursiveMLEstimator 21 ],isderivedbasedontheAWGNchannelandconstantCFOassumptions.ImplementationinfrequencyselectivechannelsisaddressedinSection 4.3.3 ,multiuserOFDMisdiscussedinSection 4.3.4 ,time-varyingCFOisconsideredinSection 4.4 Undertheaboveassumptions,AWGNchannelandconstantCFO,thechannelmodelin( 4 )canbesimpliedasrl(k)=lej2lk=Nsl(k)+wl(k) 65

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21 ]butwithouttimingoffset. ConsidertherstnreceivedOFDMsymbolsrl,l=1,2,...,n.Weassumethattherstn1CFOestimationshavebeenmade,whichare^"l,l=1,2,...,n1.Forconvenience,welet^"0=0.WenowderivetheMLestimationof"basedonthereceivednOFDMsymbols. Asin[ 21 ],wedenetwoindexsetsI=f0,1,...,L1gandI0=fN,N+1,...,N+L1g. ThesetI0containstheindicesofthedatasamplesthatarecopiedtotheCPandthesetIcontainstheindicesoftheCP.Sincethesamplesinthecyclicprexandtheircopiesarepairwisecorrelated,itiseasytoverifyErl(k)rl(k+N)=2sej2l,k2I,l=1,2,...,n wherel="^"l1,l=1,2,...,n, whiletheothersamplesaremutuallyuncorrelatedandareindependentofthesamplesinI[I0. Thelog-likelihoodfunctionfor"canbewrittenas(n)(")=logf(r1,r2,...,rnj")=lognYl=1Yk2If(rl(k),rl(k+N))Yk=2I[I0f(rl(k))!=nXl=1Xk2Ilog(f(rl(k),rl(k+N)))+ wheref()denotetheprobabilitydensityfunctionandissometermindependentof".Notethattheconditioningon"issuppressedfornotationclarity.Bysomemanipulations 66

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wheren=nXl=1ej2^"l1Xk2Irl(k)rl(k+N) and=2s SNR+1. Obviously,theMLestimationof"attimenisgivenby^"n=1 2\n. Fromtheaboveanalysis,wecanseeanewCFOestimationcouldbemadebasedonthenewobservationandpreviousestimations.ThereisanexplicitrecursivestructurefortheCFOestimationscheme.HencewegetaRecursiveCFOEstimatorwhichwereferastheRCFOEestimatorandsummarizeitasfollows. Initialization:"0=0,0=0. Update:Sn=L1Xk=0rn(k)rn(k+N) 2\n. ThestructureofthebasicformrecursiveestimatorisillustratedinFig. 4-2 67

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StructureofthebasicformRCFOEestimator ThecomputationalcomplexityofRCFOEisonlyO(L),whereListhelengthofCP.Besides,thebufferrequirementisverylow,attheendof(n1)-thblock,onlynandlastestimation^"n1needtobestoredfornextstepestimation(However,aswillbeshowninSection 4.4 ,thisisnottrueforestimationoftime-varyingCFO).Thus,itissuitableforreal-timeapplications. 77 ].Theasymptoticallyunbiasedpropertycanalsobeeasilyshownbythefollowingargument.AsLgoestoinnity,E\n=E\nXl=1ej2^"l11 74 ],thisMLestimatorisnotunbiasedsincetheexpectationoperationcannotcommutewiththenonlinearangleoperation. Oneimportantperformancemetricforanestimatorisitsmeansquarederror(MSE).Fromclassicestimationtheory[ 78 ],thevarianceofunbiasedestimation^"nis 68

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wheretheequalityholdsifandonlyiftheunbiasedestimatorisefcient.Letl(")=logf(rlj"),l=1,2,...,n,thenitiseasytoseeE@2(n)(") 79 ],theCRLBforMLestimationofCFOusingsingleOFDMsymbolhasbeenevaluated:E@2l(") ThuswegettheCRLBfortheRCFOEestimatorVar(^"n)12 Thatis,theCRLBisinverselyproportionaltotheestimationwindowsize.NotethattheaboveCRLBholdsonlyforunbiasedestimations,whiletheRCFOEestimatorisonlyasymptoticallyunbiasedandasymptoticallyefcient.However,fromoursimulations(seeSection 4.5 ),foralmostallsettings(e.g.,theCPlengthL8),thevarianceoftheMLestimationisveryclosedtotheCRLBandthemeanisveryclosedtotruevalue.Lete(1)ndenotetheestimationerror,i.e.,e(1)n="^"n thenthefollowingapproximationholdsE(e(1)n)2c112 wherec1isafactordenotesthedifferencebetweentheactualestimationerrorandtheCRLB.c1shouldbelargerthanorequalto1.Fromoursimulations,wefoundc1=1is 69

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4.4 ConsideronereceivedOFDMsymbolwithCPrl.From( 4 ),wehavethefollowingcorrelationsimilarto( 4 )Erl(k)r(k+N)=l1Xm=0jhl(m)j22sej2l,k2I. Inthiscase,rl(k),k=0,1,...,N1arenolongerindependent,whichisnotreectinthethelog-likelihoodfunction( 4 ).Thisisthemainsourceofperformancedegradation.However,fromoursimulationworks,weseethattheRCFOEestimatorcanstillprovideaccurateCFOestimationinfrequencyselectivechannels. 73 ],VandeBeeketal.extendtheMLestimator[ 21 ]totheuplinkofmultiuserOFDMsystem.SincetheRCFOEestimatorisbasedontheMLestimatorproposedin[ 21 ],itcanalsobeimplementedtothemultiuserscenario.Herewesketchthemainideaandreferto[ 73 ]fordetails. InmultiuserOFDMsystems,eachuserisallocatedagroupofadjacentsubcarriers.Thereceivedsignalisthesuperimpositionofthesignalsfromallusers.Alterbankisusedtoseparatethesubcarriergroups.ThentheMLestimatorcanbeappliedforeachuser.Themaindisadvantageofthisschemeisthatgroupingthesubcarrierstogetherpreventsthepossibilityofexploitingthechanneldiversity. 70

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4.4.1Performancedegradationduetotime-varyingCFO wherelisarandomvariablewhichdenotesthechangeofCFObetweentwoblocks.vlisunknownatthereceiver.Weassumeelisupperbounded:jlj, whereisaconstantpositivenumberandcanbeinterpretedastherateofchangeofCFO.Notethatwedonotassumeanyprobabilisticstructureof"l,thatistheCFOcouldbeanystochasticprocessaslongastheCFOvariationinoneblockisupperboundedbyaconstant. Tostudytheeffectoftime-varyingCFO,werstassumeltobeavailableatthereceiver.Thenweestimatetheperformancelossduetounknownl. Considerl,l=1,2,...,nasparameters,itiseasytoseethatinthiscasethefollowingcorrelationpropertyholdsErl(k)rl(k+N)=2sej20l,k2I,l=1,2,...,n where0l="l^"l1="nnXi=l+1ei^"l1. TheMLestimationof"kisgivenby^"0n=1 2\0n 71

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Itiseasytoseethattheestimation^"0nalsosatisestheCRLB( 4 ),i.e.,E(e(1)n)2c112 wheree(1)n="n^"0n. Sincel,l=1,2,...,nisactuallyunknownatreceiver,theoutputoftheRCFOEestimatorisgivenby( 4 ).Lete(2)n=^"0n^"n denotethedifferencebetweentheMLestimationwithCFOknownatreceiverandtheactualRCFOEoutput.e(2)ncanbeinterpretedastheperformancedegradationcausedbytheunknowntimevaryingCFO.FromLemma 4.1 below,weget AnapproximationofE(e(2)n)2canbegivenbyE(e(2)n)2c22n2 wherec2isapositivenumberandc21.Thatis,theestimationerrorcausedbytimevaryingCFOincreasesapproximatelyproportionallytotheestimationwindowsize.Inthenextsection,wewilldiscussthechoiceofc2indetail. 72

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< 4.1 canbesatised. Asdiscussedabove,ifweincreasetheestimationwindowsizen,e(1)nwilldecreasewhilee(2)nwillincrease.Thusthereisanexplicittradeoffbetweene(1)nande(2)n.Thiscanalsobeinterpretedasthetradeoffbetweenfasttrackingabilityandlowestimationvariance[ 76 ].Thatis,withsmallerestimationwindowsize,theestimatorcantrackfasterCFOvariation,whilewithlargerestimationwindowsize,theestimatorcanachievelowerestimationerror(ifCFOisconstant).Inordertoachievethebestperformance,weneedtominimizetheMSEEe2n=E(e(1)n+e(2)n)=E(e(1)n)2+(e(2)n)2 wherethelastequalityholdsbecausee(1)nande(2)narecausedbyindependentphysicalsources(channelnoiseandDopplereffect)andhenceareregardedasuncorrelated.Sincetheexactexpressionofenisunknown,weminimizeitsapproximationinstead, 73

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162SNR2L21=3 wherec=(c1=c2)1=3. Whentheestimationwindowsizetakestheoptimalvalueno,theMSEofestimationoftime-varyingCFOcanbeapproximatedbyEe2nc32 3L2 32SNR+1 3 wherec3=3 821 34 3c1 31c2 32. 4 )givesanexplicitexpressionoftheoptimaloperationvalue(approximately)oftheestimationwindowsizeasafunctionofCPlengthandtherateofchangeofCFO. 4 )showstheapproximatedMSEofCFOestimationasafunctionofSNR,therateofchangeofCFO,andcyclicprexlength,thusweprovideaguidelineforsystemdesign. 4 )is

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4 )isbasedonworst-caseanalysis,c2canbechosentobealittlebitlessthan1.However,weprefertousec2=1sincetheresultingsystemcanaccommodatehighdegreeofuctuationfortime-varyingCFO.So,wecanchoosec=c1=c2=1. 20 ],therateofchangeofCFOisincreasingwhenthemobilesubscriber(MS)isapproachingthebasestation(BS).Inthiscase,theRCFOEalgorithmcanbeimplementedinanadaptivewaysincetherateofchangeofCFOcanbeestimatedbasedthepreviousCFOestimations.Attimen,therateofchangeofCFOcanbeestimatedby^n=1 NotethatMistheaveragewindowsizetoreducethevarianceofestimation. Themainresultinthissectioncanbesummarizedinthefollowingalgorithm. Initialization:"0=0,0=0. 162SNR2L21=3. Update:Sn=L1Xk=0rn(k)rn(k+N), 2\n. 75

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20 ]).Inmostcases,amoderatevalueoftherateofchangeofCFOwillleadtosatisfyingestimationperformance;howtochoosetherateofchangeofCFOdependsonthecommunicationscenarioandSection 4.4.3 willdiscussthis. Inthecommoncommunicationscenariopresentedin[ 20 ],therateofchangeofCFOisshowntobebetween0and2.58KHzpersecond.Normalizedbythesubcarrierspacing(2.232KHz)andOFDMsymboltime(4096+128)7=64sofDVB-SH[ 17 ]system,weget5.3104.AssumingtheSNRis10dB,by( 4 ),wegettheoptimalestimationwindowsizeisabout3. NowweassumethattherateofchangeofCFOisbetween0.2and2KHzpersecondandconsiderthreetypesofOFDMsystems:Wimax[ 16 ],DVB-SHandMediaFLO[ 18 ].Wegetthefollowingtableoftypicalestimationwindowsizes,whereBscdenotesthesubcarrierspacing,FsdenotesthecarrierfrequencyandTsdenotesthesampleinterval. 4.5.1 .TheperformanceoftheRCFOEestimatorisillustratedinSection 4.5.2 .Wealsoverifytheproposedschemeforthechoiceofoptimalestimationwindowsize 76

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TypicalEstimationWindowSizesforOFDMsystems OFDMParameters 26-123 N=4096,L=128 3-18 N=4096,L=96 2-10 4.5.2 .TheperformanceoftheRCFOEestimatorinfrequencyselectivechannelsisshowninSection 4.5.3 20 ].ConsideracellularasshowninFig. 4-3 .Theradiusofunitcellissettobe5000m.InthePHYlayer,weemploytheWiMAXsystemshowninTable 4-1 .Consideramobilesubscriber(MS)movesfromtheinitialposition(10000,6300)withspeed120km/handdirection(0.7902,0.6129).HandoffisexecutedwhenanotherBSisclosertotheMSthanthecurrentBS.Forsimplicity,onlythelineofsight(LOS)signalisconsidered.TheDopplershiftiscalculatedbyvFccos((t))=cl,where(t)istheanglebetweenthedirectionofarrival(DOA)andthemovingdirectionattimetandclisthespeedoflight. Fig. 4-4 showsthetime-varyingLOSDopplershift.Fig. 4-5 showstheCFOvariationperOFDMsymboltimenormalizedbysubcarrierspacing.Fromthesegures,wecanseethattheCFOchangesfasterwhentheMSismovingclosertotheBS.TherateofchangeofCFOrangesfrom0to105. 77

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Amobilecommunicationnetwork 4-6 showstheMSEoftheRCFOEestimatorcomparedwiththeCramer-Raolowerbound(CRLB)fordifferentSNRvalues.ThetrueCFOis"=0.1,therateofchangeofCFOis=0,thatistheCFOisassumedtobeconstant.n=10OFDMsymbolsareusedforestimationandcalculatingtheCRLB.OneOFDMsymbolestimation(n=1)isalsoshownforcomparison.Notethatifn=1,theRCFOEestimatorreducestotheMLestimatorproposedin[ 21 ].WecanseethattheMSEofRCFOEisveryclosedtothatoftheCRLB,whichveriestheapproximation 78

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Time-varyingCFO Figure4-5. CFOrateofchange ( 4 ).Comparedwithonesymbolestimation,theMSEperformanceenhancementisnotable.With10symbolsusedforestimation,wegetaprocessgainofabout10dB. Fig. 4-7 showstheMSEoftheRCFOEestimatorfordifferentnandrateofchangeofCFO.TheactualCFOattimenis"n=0.1,SNR=10dB.TheCPlengthisL=16.TherateofchangeofCFOissettobe=0,=0.001or=0.002.Fromthisgure, 79

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MSEvs.SNRforconstantCFO wecansee,iftheCFOisconstant,theMSEisamonotonicdecreasingfunctionofn.IftheCFOisnotconstant,thereisanexplicitminimumpoint.Moreover,iftheCFOchangesfaster,theminimumpointofMSEshiftstotheleft.ThisphenomenoncanbeexplainedbyouranalysisinSection 4.4 .Ifislarge,theeffectoftime-varyingCFOisemphasized,andtheoptimalsystemtendstouseasmallnumberofOFDMsymbolstocatchtheCFOvariation. Fig. 4-8 illustratestheoptimalestimationwindowsizenoasafunctionofSNR.ThetrueCFOis"=0.1(atcurrenttime).TherateofchangeofCFOis=0.001or=0.005.LengthofCPisL=16.Thetheoreticalvalueofoptimalestimationwindowsizenoiscalculatedby( 4 ).Thescaleparametercissetto1.8.Fromthisgure,wecanseetheexperimentalresultsmatchwellwiththeoreticalresults.ThisveriestheexpressionofnoasafunctionofSNRin( 4 ).AsSNRincreases,nodecreases.ThisisbecauseifSNRislarge,withasmallnumberofOFDMsymbols,theestimationerrorisalreadysufcientlysmall,whiletheeffectoftime-varyingCFOisemphasized. 80

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MSEvs.nforvariousratesofchangeofCFO Fig. 4-9 illustratestheoptimaloperationpointnooftheRCFOEestimatorasafunctionoftherateofchangeofCFO.ThetrueCFOis"=0.1(atcurrenttime).TherateofchangeofCFOis=0.001or=0.005.LengthofCPisL=16.SNRis20dB.Thetheoreticalvalueofnoiscalculatedby( 4 ).Thescaleparametercissetto1.8.Fromthisgure,wecanseetheexperimentalresultsmatchwellwiththeoreticalresults.ThisveriestheexpressionofnoasafunctionoftherateofchangeofCFOin( 4 ). Fig. 4-10 illustratetheMSEperformanceoftheRCFOEestimatorinthetime-varyingCFOscenario.Inthissimulation,thenumberofOFDMsymbolsusedforestimationiscalculatedby( 4 ),wherecissettobe1.ThetrueCFOis0.1(atcurrenttime).CPlengthL=8.TherateofchangeofCFOis=0.001or=0.005.Performanceofonesymbolestimationisalsoshownforcomparison.Fromthisgure,wecansee,iftheCFOchangesfasterortheSNRislarger,theprocessgainofRCFOEissmaller.ThisisbecausewhenCFOchangesfasterorSNRislarger,thenumberofOFDMsymbolsusedforestimationisreduced,aswehaveshowninthelastsimulations.IftheCFOchangessufcientlyfastortheSNRissufcientlylarge,theperformanceofRCFOEis 81

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TheoptimalestimationwindowsizenoasafunctionofSNR thesameasthatofonesymbolestimationsinceinthiscase,thebestchoiceistouseonlyonesymbolforestimation. 21 ],weconsideranoutdoordispersivefadingenvironment.Thewirelesssystemoperatesat2GHzwithabandwidthof5MHz.Thechannelhasanexponentiallydecayingpowerdelayproleandtheratioofthersttaptothelasttapissetto20dB.Themaximumdelayspreadis3s,whichcorrespondsto15samples.Itismodeledby15independentRayleighfadingtapsplusadditivecomplexwhiteGaussiannoise.ThetotalnumberofsubcarriersissettoN=256,CPlengthL=16.TherateofchangeofCFOis=0.0005.PerformanceofCFOestimationinAWGNchannelsis 82

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TheoptimalestimationwindowsizenoasafunctionofrateofchangeofCFO alsoshownforcomparison.Fig. 4-11 showsthesimulationresults.FSmeansfrequencyselectivechannel. FromFig. 4-11 ,wecanseethatalthoughtheperformanceofCFOestimationisdegraded,theRCFOEestimatorstillprovidesreasonablyaccurateCFOestimation.Infrequencyselectivechannels,theestimationofnoin( 4 )maybenotaccurate.Thus,theMSEofRCFOEestimationinfrequencyselectivechannelsmaynotbeamonotonicfunctionofSNRsinceasub-optimalnumberofOFDMsymbolsmaybechosenforestimation. 83

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PerformanceofRCFOEfortime-varyingCFO andlowestimationvariance.Byminimizingthemeansquarederror,weprovidedaclosed-fromexpressionoftheoptimalestimationwindowsizeasafunctionoftherateofchangeofCFO,SNRandCPlength.Simulationresultsagreewellwithourtheoreticalresults,indicatingtheaccuracyofouranalysis. TheproposedMLestimatorisderivedbasedontheAWGNchannelassumption,thusitmaysufferfromperformancedegradationinfrequencyselectivechannels.However,becauseofthepervasivenessofthetradeoffbetweenfasttrackingabilityandlowestimationfortrackingproblems,theresearchmethodologyinthisworkmaybeextendtootherCFOestimationschemes,suchasEMbasedmethod[ 22 ]andspectralmethod[ 25 72 ].Thiswillbeourfuturework. 84

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PerformanceofRCFOEfortime-varyingCFOinfrequencyselectivechannels 85

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Inthisdissertation,weinvestigatedthestatisticalQoSguaranteeforsingle-userwirelesscommunicationsystemsandadhocnetworks.Wealsoexploitedoneofthemostimportantissuesinbroadbandwirelesscommunicationsystems:frequencysynchronizationforOFDMsystems. InChapter 2 ,westudiedtheproblemofdatacommunicationwithbothdelayandpacketerrorprobabilityconstraints.Thetransmissiondatarateisadaptedtochannelstate,bufferstateandarrivalstatetominimizethetotalpacketerrorprobabilitythusmaximizethesystemthroughput.Differentfrommostofpreviousworks,weconsideredasystemwithnitebufferspacethusweaddressedthreetypesoferrors:1)packetdropduetofullbuffer,2)delayboundviolation,and3)packetdecodingerrorduetochannelnoise.Wederivedanupperboundonthetotalpacketerrorprobability.Byminimizingthepacketerrorprobabilityoverthetransmissionrate,weobtainedanoptimalratecontrolpolicythatguaranteestheuser-specieddatarateanddelaybound.Thenbyvaryingdatarateanddelaybound,weobtainedREDPareto-optimalsurface.OurresultsprovideimportantinsightsintostatisticalQoSprovisioninginwirelesssystems;theREDParetosurfacerepresentsamajorsteptowardderivingtheprobabilisticdelay-constrainedchannelcapacityoffadingchannels.Inourfuturework,bothrateadaptationandpoweradaptationwillbeusedtoachieveahigherdatarate. InChapter 3 ,motivatedbytheQoSprovisioningrequirementofrealtimeapplications,wehavestudiedthestatisticaldelayguaranteeforlargescalemobileadhocnetworks,whichhasnotreceiveddeservedattentionintheliterature.Forthersttime,weestablishedtherelationshipbetweendelayboundanddeadlineviolationforlargescalenetworks.Specically,foracellpartitionednetworkunderthei.i.d.mobilitymodel,thedeadlineviolationprobabilityisgivenbyPl(n)=e(B(n),foranydelayboundB(n)=!(n).Sincethedeadlineviolationprobabilitycanbeinterpretedasadescription 86

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InChapter 4 ,weinvestigatedthePHYlayerdesignproblem.WeinvestigatedthepossibilityofutilizingmultiplereceivedOFDMsymbolstoestimatethetime-varyingcarrierfrequencyoffset.Byinvestigatingtheeffectofbothtime-varyingCFOandestimationwindowsizeontheperformanceofCFOestimation,wehaveshownthatthereisanexplicittradeoffinchoosingtheestimationwindowsize,whichcanalsobeconsideredasthetradeoffbetweenfasttrackingabilityandlowestimationvariance.Byminimizingthemeansquarederror,weprovidedaclosed-fromexpressionoftheoptimalestimationwindowsizeasafunctionoftherateofchangeofCFO,SNRandCPlength.Simulationresultsagreewellwithourtheoreticalresults,indicatingtheaccuracyofouranalysis.TheproposedMLestimatorisderivedbasedontheAWGNchannelassumption,thusitmaysufferfromperformancedegradationinfrequencyselectivechannels.However,becauseofthepervasivenessofthetradeoffbetweenfasttrackingabilityandlowestimationfortrackingproblems,theresearchmethodologyinthisworkmaybeextendtootherCFOestimationschemes,suchasEMbasedmethodandspectralmethod.Thiswillbeourfuturework. 87

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2.2 2-1 butdifferentbuffersizes.SystemAhasabufferofsizeM1andSystemBhasabufferofsizeM2,whereM1
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Letqi,i=1,2,denotethequeuelengthsequencesofaninnite-bufferqueueingsystemwithsamearrivalratebutdifferentdeparturesequencesfring,i=1,2,.Thenwehaveqin=qin1+rin,i=1,2,. 2.1 2.2 ,weseethatuisalsoanadmissiblecontrolactionsequenceforSystemB.Sincetheaveragenumberofpacketdropinoneblockequalstothearrivalrateminustheaveragedeparturerate.Moreover,withsameratecontrolpolicy,thenumberofpacketsincorrectlydecodedisalsothesame,sowehavePe(u,D1,)=Pe(u,D2,)

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2.2 2.1 Now,weprovethesecondpartwithsamplepathargument.SincethedelayboundandbuffersizeisrelatedbyM=Dmax,itissufcetoshowthatERD(M=,)isconvexinMforxed.LetH=fHngbeagivenrealizationofchannelgainsequence.Consideraninnite-buffersystemandtwovirtualbuffersizesMi,i=1,2.Let01andM=M1+(1)M2.WeneedtoshowthatERD(M=,)ERD(M1=,)+(1)ERD(M2=,). AssumethecontrolactionsequencesthatattainERD(Mi=,),i=1,2areui=fring,i=1,2andthecorrespondingqueuelengthsequencesarefqing,i=1,2.NowconsidervirtualbuffersizeM.Letthecontrolactionsequencebefrng,wherern=r1+(1)r2.FromLemma 2.3 ,weseethatfrngisalsofeasibleandthecorrespondingqueuelengthsequenceisqn=q1n+(1)q2n.Since()+isaconvexfunction,wehave(qn+M)+(q1n+M1)++(1)(q2n+M2)+. 2 ),weobtainERD(M=,)ERD(M1=,)+(1)ERD(M2=,).

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2.3 2.1 ,weseeERD(D2,1)ERD(D1,1)" 2.4 ) 10 ]).2

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BeforeprovidingproofsforProposition 3.1 andTheorem 3.1 ,westartwithseveralusefullemmas.ThefollowinglemmaisadirectconsequenceofTheorem2.12in[ 80 ]. B.1 ,weknowthegenerationfunctionofthedistributionofSisM(S)=() 92

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Nowweanalyzethequeuesatthesourceandrelaynodes.Weusethesimilarargumentasthatin[ 1 ].Duetothei.i.d.mobilitymodel,bothandsourcequeueandtherelayqueueareBernoulli/Bernoulliqueues.Thearrivalratetothesourcequeueis.Letdenotetheservicerate(transmissionprobability)ofthesourcequeue.Ineachtimeslot,thetotaltransmissionopportunitiesscheduledfortransmittingpacketfromsourcenodeoverthenetworkisCp,wherepistheprobabilityofndingatleasttwousersinaparticularcell.Sinceallnodesareidentical,thetransmissionprobabilityforeachnodeisCp=n=p=d,thus=p=d,whichisthesupremumofthroughput. BecauseofthereversibilityofBernoulliqueue[ 80 ],theoutputprocessofthesourcequeueisalsoaBernoullistreamofrate.Nowconsidertherelayqueue,whichisalsoaBernoulliqueue.SincetheprobabilityofdirecttransmissionisCq=n=q=dandthereare 93

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(n2)=(pq) (n2)p. d(n2). whereIAistheindictorfunction. 3.1 +pq p1r p2dp=(n).

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3.1 p Itiseasytosee,forlargenandconstantx,Pr(IADr>x)>Pr(Ds>x).FromLemma B.3 ,wehavePl(n)2Pr(IA)Pr(Dr>K=2),andPl(n)Pr(IA)Pr(Dr>K). B.2 ,wehavePr(Dr>K)=1r d(n2) (n2)p!K n, wherec0=(pq)(1 Baseonabovediscussions,weseethatthedeadlineviolationprobabilityPl(n)=e(B(n)=n)andthereliabilityindexI(n)=(B(n)=n).2

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21 ],weknowforanyk2Iandl=1,2,...,nf(rl(k),rl(k+N))=1 4 ). 4.1

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XihuaDongreceivedhisB.S.degreeinMathematicsfromShandongUniversity,Jinan,Shandong,China,in2000.HereceivedhisM.S.degreeinmathematicsfromChineseAcademyofSciencein2003andM.S.degreeinelectricalengineeringfromWashingtonUniversityin2006.HewillreceivehisPh.D.degreeinelectricalandcomputerengineeringfromtheUniversityofFloridainAugust,2010.From2003to2004,heworkedasanengineerintheInstituteofSoftware,ChineseAcademyofSciences.Hisresearchinterestsincludecrosslayerdesign,wirelessadhocnetworksandfrequencysynchronizationforOFDMsystems. 104