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Reliable Data Communication and Storage in Underwater Acoustic Networks

Permanent Link: http://ufdc.ufl.edu/UFE0043581/00001

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Title: Reliable Data Communication and Storage in Underwater Acoustic Networks
Physical Description: 1 online resource (117 p.)
Language: english
Creator: Cao, Rui
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: acoustic -- codes -- communications -- cooperative -- data -- data-centric -- fountain -- networks -- relay -- sensor -- storage -- underwater
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: Underwater acoustic communications (UAC) and networking (UAN) are promising paradigms for various oceanic applications. However, acoustic signal transmissions are characterized by long propagation delay, frequency-dependent attenuation, and doubly-selective fading. These pose great challenges to the design of reliable underwater communication and networking protocols. In this dissertation, two major techniques are investigated to enhance data communication and storage reliability in underwater acoustic networks. The first one is cooperative relay communications. To improve communication reliability and to extend range, relay communications have been extensively studied in terrestrial environments by exploiting spatial diversity in a distributed manner. Therefore the technique is feasible in UAC. However, due to the unique features of the UAC channel, underwater cooperative relay communications exhibit peculiar characteristics. Thus we will carefully explore the underwater application from both information-theoretic and protocol design perspectives. Based on empirical underwater acoustic signal attenuation and noise models, the capacity of underwater relay communications is analyzed. The capacity gain over the direct-link system is quantified to illustrate the benefits of relaying, and the factors affecting system capacity are also revealed. Then, an asynchronous cooperative relaying protocol is specially designed for underwater relay communications. The proposed scheme addresses the synchronization difficulty and frequency-selectivity issue inherent in UAC, and takes advantage of the sparse nature of the UAC channel to facilitate reliable and efficient data transmissions. The second technique is rateless fountain codes. Forward error correction (FEC) is commonly adopted to improve higher-layer packet transmission reliability with a few number of retransmissions. Thus FEC is beneficial for large-latency UAC and networking. Among all error-correcting erasure codes, fountain codes are favorable for underwater acoustic networks due to their low complexity and rate adaptability to channel-fading dynamics. In addition, the protocol design of underwater cooperative communications and distributed data storage requires multi-layer reliability. Multi-layer reliable schemes based on traditional fountain codes induce a large computation cost. In order to reduce energy consumption while retaining multi-layer reliability, we will explore decomposed fountain codes (DFCs), which feature multi-layer encoding but a single layer of decoding. Analyses are carried out to develop general decomposed Luby Transform (DLT) codes and decomposed Raptor codes (DRC). Practical algorithms are also designed for DFC construction. Then, a reliable cooperative relay communication scheme is proposed with the DLT codes, and a reliable underwater data-centric storage (DCS) protocol is designed with the assistance of DRC. The performance of the reliable data communication and storage schemes is then analyzed and simulated.
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 Rui Cao.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Yang, Liuqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043581:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043581/00001

Material Information

Title: Reliable Data Communication and Storage in Underwater Acoustic Networks
Physical Description: 1 online resource (117 p.)
Language: english
Creator: Cao, Rui
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: acoustic -- codes -- communications -- cooperative -- data -- data-centric -- fountain -- networks -- relay -- sensor -- storage -- underwater
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: Underwater acoustic communications (UAC) and networking (UAN) are promising paradigms for various oceanic applications. However, acoustic signal transmissions are characterized by long propagation delay, frequency-dependent attenuation, and doubly-selective fading. These pose great challenges to the design of reliable underwater communication and networking protocols. In this dissertation, two major techniques are investigated to enhance data communication and storage reliability in underwater acoustic networks. The first one is cooperative relay communications. To improve communication reliability and to extend range, relay communications have been extensively studied in terrestrial environments by exploiting spatial diversity in a distributed manner. Therefore the technique is feasible in UAC. However, due to the unique features of the UAC channel, underwater cooperative relay communications exhibit peculiar characteristics. Thus we will carefully explore the underwater application from both information-theoretic and protocol design perspectives. Based on empirical underwater acoustic signal attenuation and noise models, the capacity of underwater relay communications is analyzed. The capacity gain over the direct-link system is quantified to illustrate the benefits of relaying, and the factors affecting system capacity are also revealed. Then, an asynchronous cooperative relaying protocol is specially designed for underwater relay communications. The proposed scheme addresses the synchronization difficulty and frequency-selectivity issue inherent in UAC, and takes advantage of the sparse nature of the UAC channel to facilitate reliable and efficient data transmissions. The second technique is rateless fountain codes. Forward error correction (FEC) is commonly adopted to improve higher-layer packet transmission reliability with a few number of retransmissions. Thus FEC is beneficial for large-latency UAC and networking. Among all error-correcting erasure codes, fountain codes are favorable for underwater acoustic networks due to their low complexity and rate adaptability to channel-fading dynamics. In addition, the protocol design of underwater cooperative communications and distributed data storage requires multi-layer reliability. Multi-layer reliable schemes based on traditional fountain codes induce a large computation cost. In order to reduce energy consumption while retaining multi-layer reliability, we will explore decomposed fountain codes (DFCs), which feature multi-layer encoding but a single layer of decoding. Analyses are carried out to develop general decomposed Luby Transform (DLT) codes and decomposed Raptor codes (DRC). Practical algorithms are also designed for DFC construction. Then, a reliable cooperative relay communication scheme is proposed with the DLT codes, and a reliable underwater data-centric storage (DCS) protocol is designed with the assistance of DRC. The performance of the reliable data communication and storage schemes is then analyzed and simulated.
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 Rui Cao.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Yang, Liuqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043581:00001


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RELIABLEDATACOMMUNICATIONANDSTORAGEINUNDERWATERACOUSTICNETWORKSByRUICAOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011RuiCao 2

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Idedicatethistomywifeandfamily. 3

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ACKNOWLEDGMENTS MyPhDstudyattheUniversityofFloridahasbeenamagnicentandchallengingexperience.Agreatmanypeoplehavecontributedtotheaccomplishmentofthisdissertation.Iowemygratitudetoallthosepeoplewhohavemadethispossible.Firstandforemost,Iwouldliketoexpressmydeepandsinceregratitudetomysupervisor,Dr.LiuqingYang.WhenIenteredmyrst-yearPhDstudywithlittleengineeringbackground,shebroadenedmyvisionofwirelesscommunicationsandnetworks.Herin-depthknowledgeandtirelessguidancehavehelpedmerealizetheturningpointofmyresearch.Sheinspiredmycuriosity,andgavemethefreedomtoexplorenewareasofresearch.Herinsightfulcommentsandconstructivecriticismshavehelpedmedevelopnewthoughtsandrenerawideas.Herlogicalwayofthinkingandexpressingideaswillbeagreatfortuneinmyfuture.IamindebtedtoDr.ShigangChen,Dr.YuguangFang,andDr.JaniseMcNairfortheirconstantsupportinmyPhDstudy,andfortheirvaluabletimeandeffortinservingonmyPhDsupervisorycommittee.MythanksalsogotomycolleaguesintheSignalProcessing,CommunicationandNetworking(SCaN)group.IamgratefultohaveconductedmyrstresearchtopicwithDr.WoongCho.Hiskindnessandpatienceeasedmystartofresearchinwirelesscommunications.MycollaborationwithDr.FengzhongQuintheunderwatercommunicationsprojecthasbeenagreatpleasureandfruitful.Iamparticularlythankfultohimforhisvaluableandinsightfuldiscussionsduringmyresearch.IalsowanttothankXilinChengandRobertGrifnfortheirprecioushelpinresearchandpaper-writing.Inaddition,Iwanttoextendmythankstoallothergroupmembersandalumni:Dr.HuilinXu,DongliangDuan,WenshuZhang,PanDeng,BoYu,andNingWang.Icherishthegreatmomentswiththem.Finally,Iowemylovingthankstomywife,WeiZang,andmyfamily.TheirloveandencouragementhavesupportedmethroughoutmyPhDstudy. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1UnderwaterAcousticNetworks ........................ 13 1.2ProtocolDesignChallenges .......................... 14 1.3Motivations ................................... 15 1.3.1CooperativeRelayCommunications ................. 15 1.3.2DecomposedFountainCodes ..................... 18 2CAPACITYOFUNDERWATERRELAYCOMMUNICATIONS .......... 21 2.1AcousticSignalAttenuationandNoiseModels ............... 21 2.2CapacityAnalysis ............................... 22 2.2.1SystemCapacity ............................ 22 2.2.2End-to-endSNR ............................ 22 2.2.3SignalBandwidth ............................ 23 2.3NumericalResults ............................... 24 2.3.1OptimalFrequencyand3dBBandwidth ............... 24 2.3.2SystemCapacity ............................ 25 2.3.3AffectingSystemFactors ........................ 25 2.4Summary .................................... 26 3ASYNCHRONOUSUNDERWATERRELAYCOMMUNICATIONS ........ 30 3.1SystemandChannelModel .......................... 30 3.2AsAPRelayProtocol .............................. 31 3.2.1SourceTransmission .......................... 31 3.2.2RelayForwarding ............................ 32 3.2.3DestinationDecoding .......................... 35 3.3PerformanceEvaluation ............................ 36 3.3.1PEPofAsAP .............................. 36 3.3.2DiversityofAsAP ............................ 38 3.3.2.1MCDachievability ...................... 40 3.3.2.2Underwateracousticchannel ................ 40 3.3.2.3Effectoftheamplicationfactor .............. 40 3.4Simulations ................................... 41 5

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3.4.1Simulationsetup ............................ 41 3.4.2Resultsandcomparisons ....................... 41 3.4.2.1Benetsofrelayingandprecoding ............. 41 3.4.2.2Effectofnumberofchanneltaps .............. 42 3.4.2.3EffectofnumberofrelaysR ................ 42 3.4.2.4Effectofamplicationfactor ................. 43 3.5Summary .................................... 43 4DECOMPOSEDFOUNTAINCODES ........................ 48 4.1Background ................................... 48 4.1.1LTCodes ................................ 49 4.1.2RaptorCodes .............................. 50 4.2DecomposedLTCodes ............................ 51 4.2.1ProblemFormulation .......................... 52 4.2.2ValidChoiceof(x) .......................... 55 4.2.3DecompositionAlgorithm ....................... 58 4.2.4RSDDecomposition .......................... 58 4.2.5H-DLTCodes .............................. 61 4.2.5.1Encodingscheme ...................... 61 4.2.5.2Hybriddistributiondecomposition ............. 62 4.2.5.3HybridRSDdecomposition ................. 63 4.2.6DLTCodePerformance ........................ 64 4.2.6.1Degreedistributions ..................... 64 4.2.6.2Decodingperformance ................... 65 4.2.6.3Computationcost ...................... 66 4.3DecomposedRaptorCodes .......................... 66 4.3.1ProblemFormulation .......................... 66 4.3.2EfcientDecomposition ........................ 68 4.3.2.1Validchoiceof(x) ..................... 68 4.3.2.2Decompositionsolutionof!(x) ............... 69 4.3.3FiniteLengthDRC ........................... 70 4.3.4PerformanceofDRC .......................... 71 4.3.4.1DRCdistributions ...................... 71 4.3.4.2Decodingperformance ................... 72 4.3.4.3Effectofgroupsizemontheperformance ........ 73 4.3.4.4EffectofwandDontheDRC-ANA ............ 73 4.4Summary .................................... 74 5UNDERWATERAPPLICATIONSOFDECOMPOSEDFOUNTAINCODES ... 81 5.1ReliableUnderwaterCooperativeRelayCommunications ......... 81 5.1.1RelatedWorks ............................. 81 5.1.2H-DLT-assistedCooperativeRelayCommunications ........ 83 5.1.2.1Sourceencodingandbroadcast .............. 83 5.1.2.2Relayencodingandcooperativeforwarding ........ 83 6

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5.1.2.3Destinationdecoding .................... 84 5.1.3PerformanceofDLT-CC ........................ 84 5.1.4End-to-endLatency ........................... 84 5.1.4.1Communicationcost ..................... 86 5.1.4.2Computationcomplexity ................... 87 5.2ReliableUnderwaterData-CentricStorage .................. 88 5.2.1RelatedWorks ............................. 88 5.2.1.1Data-centricstorage ..................... 88 5.2.1.2Fountaincodesbaseddistributedstorage ......... 89 5.2.2DRC-basedUnderwaterDCS ..................... 90 5.2.2.1Protocoldescription ..................... 90 5.2.2.2Homezoneencodingandooding ............. 90 5.2.2.3Inter-zonedatatransport .................. 91 5.2.2.4Storagezoneencoding ................... 92 5.2.2.5Dataretrieval ......................... 92 5.2.3PerformanceAnalysis ......................... 93 5.2.3.1Inter-zonedatatransport .................. 93 5.2.3.2Networkcommunicationcost ................ 93 5.2.4Simulationresults ............................ 94 5.2.4.1Simulationsetup ....................... 94 5.2.4.2Networksize ......................... 95 5.2.4.3Zonesize ........................... 95 5.3Summary .................................... 96 6CONCLUSIONSANDFUTUREWORKS ..................... 101 6.1Conclusions ................................... 101 6.2FutureWorks .................................. 103 APPENDIX APROOFOFNONNEGATIVESOLUTIONREQUIREMENTSOFDLTCODES 104 BPROOFOFTHEVALIDRANGEOFFIRST-LAYERDISTRIBUTION ...... 106 CPROOFOFNONNEGATIVESOLUTIONOFDRC ................ 108 REFERENCES ....................................... 110 BIOGRAPHICALSKETCH ................................ 117 7

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LISTOFTABLES Table page 4-1Theaverageencodingdegree ........................... 75 5-1Theaveragecomputationcost ........................... 97 8

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LISTOFFIGURES Figure page 2-1Therelaysystemsetupforthecapacityanalysisofunderwaterrelaysystems. 27 2-2Thecomparisonofoptimalcarrierfrequencyand3dBbandwidthbetweenunderwaterrelaysystemsanddirect-linksystemswithdifferentS-Ddistances. ....... 27 2-3Thecomparisonofsystemcapacitybetweenunderwaterrelaysystemsanddirect-linksystemswithdifferentS-Ddistances.ThetotaltransmitpowerisP=100dBrePa. .................................. 28 2-4Thecapacitygainofunderwaterrelaysystemsovercorrespondingdirect-linksystemswithdifferentS-Ddistances.ThetotaltransmitpowerisP=100dBrePa. 28 2-5Thecapacitycontourofanunderwaterrelaysystemwithdifferentrelaylocationratiosandpowerallocationratios.ThetotaltransmitpowerisP=100dBrePa,andS-DdistanceisD=2km. ............................ 29 3-1ThesystemtopologyfortheAsAPprotocol. .................... 44 3-2AnexampleofunderwateracousticchannelestimatedfromRACE08seatrialatatransmissiondistanceofD=1000m. ..................... 44 3-3TheoptimalGLCPgrouping.ThemthelementfromeachoftheKblocksisselectedtoformgroupm. .............................. 45 3-4TheBERperformanceforthedirect-linksystemsandtheAsAPsystemswithandwithoutprecoding.Thewaterdepthis30m. .................. 45 3-5TheBERperformancefortheAsAPsystemswithdifferentnumberofchanneltaps.Considerasinglerelayandnodirect-link. .................. 46 3-6TheBERperformancefortheAsAPsystemswithdifferentnumberofrelaysandwithnodirectlink.Thewaterdepthis30m. .................. 46 3-7TheBERperformancefortheAsAPsystemswithdifferentamplicationschemes.Thewaterdepthis30m,R=1. ........................... 47 4-1TheencodingdiagramforaDLTcodewithtwoencodingDDPsof(x)and!(x). .......................................... 75 4-2TheencodingdiagramoftherstencoderoftheSD-DLTcodes. ........ 75 4-3TheencodingdiagramofthesecondencoderoftheSD-DLT/h-DLTcodes. .. 75 4-4Theencodingdiagramoftherstencoderoftheh-DLTcodes. ......... 76 4-5ThecomparisonoftheresultantdegreedistributionoftheSD-DLTcodeandtheRSD.Thesystemparametersare:k=1000,c=0.08and=0.05. .... 76 9

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4-6TheaveragerecoveryratiowithrespecttodifferentoverheadfortheSD-DLT,distributedLT,h-DLTandprimitiveLTcodes.Thesystemparametersare:k=1000,c=0.08and=0.05. ............................ 77 4-7Thecomplementarycumulativedistributionfunctionofrequiredoverhead()forsuccessfuldecodingfortheSD-DLT,distributedLT,h-DLTandprimitiveLTcodes.Thesystemparametersare:k=1000,c=0.08and=0.05. ..... 77 4-8ThecomparisonoftheresultantdistributionsofDRCsandtheRaptordistribution.Theparametersare=0.05,g=100,Dk=5andw=2.ForDRC-LP,=0.02andc=0.05. ................................ 78 4-9ThecomparisonofaveragerecoveryratioforbothDRCsandtheRaptorcode.Theparametersare=0.05,m=21,g=100andw=2.ForDRC-LPandDRC-MIN,=0.02andc=0.05. ......................... 78 4-10Theeffectsofgroupsizemontheoverheadrequiredforfullrecovery.Theparametersare=0.05,mg=10000,w=2andDk=5.ForDRC-LPandDRC-MIN,=0.02andc=0.05. ....................... 79 4-11TheeffectsofwandDontheDRCperformance.Theparametersare=0.05,m=100,g=100andD=84. ........................ 79 4-12TheeffectsofwandDontheaverageencodingratio!0(1).Theparametersare=0.05,m=100,g=100andD=84. .................... 80 5-1AcooperativerelaysystemwithLrelays. ..................... 97 5-2TheaveragetransmissionlatencyperpacketfortheARQ,LTandDLTbasedcooperativetransmissionschemes. ......................... 97 5-3TheaveragenumberoftransmissionsperpacketfortheARQ,LTandDLTbasedcooperativetransmissionschemes. ..................... 98 5-4Thesensornetworktopology.ThenetworkisvirtuallydividedintoNzgeographiczones.EachzonehasNnsensornodes.Alldatasensedinthehomezonewillbedeliveredtothestoragezone. ........................ 99 5-5ThecommunicationcostforanetworkwithxedzonesizeNz=4anddifferentnetworksizes.Eachsensornodehasm=10P0packetstotransmit. ...... 100 5-6ThecommunicationcostforanetworkwithxednetworksizeN=100anddifferentzonesizes.Eachsensornodehasm=10P0packetstotransmit. .. 100 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyRELIABLEDATACOMMUNICATIONANDSTORAGEINUNDERWATERACOUSTICNETWORKSByRuiCaoDecember2011Chair:LiuqingYangMajor:ElectricalandComputerEngineering Underwateracousticcommunications(UAC)andnetworking(UAN)arepromisingparadigmsforvariousoceanicapplications.However,acousticsignaltransmissionsarecharacterizedbylongpropagationdelay,frequency-dependentattenuation,anddoubly-selectivefading.Theseposegreatchallengestothedesignofreliableunderwatercommunicationandnetworkingprotocols. Inthisdissertation,twomajortechniquesareinvestigatedtoenhancedatacommunicationandstoragereliabilityinunderwateracousticnetworks.Therstoneiscooperativerelaycommunications.Toimprovecommunicationreliabilityandtoextendrange,relaycommunicationshavebeenextensivelystudiedinterrestrialenvironmentsbyexploitingspatialdiversityinadistributedmanner.ThereforethetechniqueisfeasibleinUAC.However,duetotheuniquefeaturesoftheUACchannel,underwatercooperativerelaycommunicationsexhibitpeculiarcharacteristics.Thuswewillcarefullyexploretheunderwaterapplicationfrombothinformation-theoreticandprotocoldesignperspectives.Basedonempiricalunderwateracousticsignalattenuationandnoisemodels,thecapacityofunderwaterrelaycommunicationsisanalyzed.Thecapacitygainoverthedirect-linksystemisquantiedtoillustratethebenetsofrelaying,andthefactorsaffectingsystemcapacityarealsorevealed.Then,anasynchronouscooperativerelayingprotocolisspeciallydesignedforunderwaterrelaycommunications.Theproposedschemeaddressesthesynchronizationdifcultyandfrequency-selectivity 11

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issueinherentinUAC,andtakesadvantageofthesparsenatureoftheUACchanneltofacilitatereliableandefcientdatatransmissions. Thesecondtechniqueisratelessfountaincodes.Forwarderrorcorrection(FEC)iscommonlyadoptedtoimprovehigher-layerpackettransmissionreliabilitywithafewnumberofretransmissions.ThusFECisbenecialforlarge-latencyUACandnetworking.Amongallerror-correctingerasurecodes,fountaincodesarefavorableforunderwateracousticnetworksduetotheirlowcomplexityandrateadaptabilitytochannel-fadingdynamics.Inaddition,theprotocoldesignofunderwatercooperativecommunicationsanddistributeddatastoragerequiresmulti-layerreliability.Multi-layerreliableschemesbasedontraditionalfountaincodesinducealargecomputationcost.Inordertoreduceenergyconsumptionwhileretainingmulti-layerreliability,wewillexploredecomposedfountaincodes(DFCs),whichfeaturemulti-layerencodingbutasinglelayerofdecoding.AnalysesarecarriedouttodevelopgeneraldecomposedLubyTransform(DLT)codesanddecomposedRaptorcodes(DRC).PracticalalgorithmsarealsodesignedforDFCconstruction.Then,areliablecooperativerelaycommunicationschemeisproposedwiththeDLTcodes,andareliableunderwaterdata-centricstorage(DCS)protocolisdesignedwiththeassistanceofDRC.Theperformanceofthereliabledatacommunicationandstorageschemesisthenanalyzedandsimulated. 12

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CHAPTER1INTRODUCTION 1.1UnderwaterAcousticNetworks Advancesinunderwatertechnologiespropelvariousoceanicapplications,suchasenvironmentalmonitoring,off-shoreexploration,underwaternavigationandetc.[ 25 34 ].Inmission-basedapplications,aockofautonomousunderwatervehicles(AUVs)orrobotsaresentdowntotheoceantoperformunderwatertasks.Theseunderwaterdevicesneedtocoordinatetheiroperationsbydistributingcontrolcommands,exchanginglocationandmovementinformation,etc.Inlong-termapplications,asensornetworkisdeployedontheseaoororinthewatertoconstantlymonitortheenvironment.Thesensorsrequiredatacommunicationabilitytoexchangedataandprovidedataservicetopotentialoutsideusers.Therefore,underwatercommunicationandnetworkingaretheenablingtechniquesformostoceanicapplications. Duetothedifcultyofunderwatercabledeployment,wirelesscommunicationispreferableforunderwaternetworks.Inaddition,sparenetworktopologyismorepracticalforthedeploymentchallengesandhighcostinmostoceanicapplications.Thuslongrangeunderwaterwirelesscommunicationisfavorable.Amongallavailablewirelesssignalingtechniques,electromagnetic(EM)wavesundergoverystrongabsorptioninwater;theradio-frequency(RF)wavesusedinterrestrialwirelesscommunicationscanonlypropagateaboutonemeter,andtheopticalwavesareabletosupportcommunicationsofaroundten-meterrange[ 25 34 ].Fortunately,theslowattenuationofacousticwavesunderwaterenablescommunicationsoverseveralkilometers,whichmakesacousticwavescomparativelymoreattractiveinformationcarriers[ 43 ].Thus,underwateracousticcommunications(UAC)isthepromisingunderlyingcommunicationtechniqueforunderwaternetworks. 13

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1.2ProtocolDesignChallenges Intheprotocoldesignofunderwateracousticnetworks,besidescommonconsiderationsforwirelesssystems(constrainednodeenergy,fadingchannels,etc.),manyuniquechallengesneedtobeaddressed.Thesechallengesareinducedbythespecialcharacteristicsofunderwateracousticwavepropagationandtheharshseaenvironment. First,thecommunicationdelayislongandvariant.Thepropagationofacousticwavesunderwaterisveryslow,about1500m/s,thusthepropagationdelayisontheorderofonesecond,whichismuchlongerthanthatofEMwaves.Inaddition,thedelayisaffectedbytheenvironmentconditionssuchasoceanwaves,watertemperature,salinity,etc.,whichinducelargedelayvariance.Thelongandvariantdelaywillhamperaccuratetimesynchronization,round-triptime(RRT)estimationandclosed-loopfeedbackcontrol.Consequently,thedesignofreliablemediumaccesscontrol,routingandtransportlayerprotocolsneedstobereconsidered[ 34 47 49 ]; Secondly,underwateracousticlinksaredynamicandunreliable.Duetotheoccurrenceofinternalwaves[ 46 ],theenvironment-dependentacousticsignalpropagationandrandomhumanormarineanimalactivities,theUACchannelisfeaturedwithintermittentconnectionanddouble-selectivity,whichinduceerror-pronetransmissions[ 41 ].Inaddition,thelongRTTandcomparablyfastchannelvariationinduceunavailabilityoftransmitter-sidechannelinformation.Thusopen-loopreliablecommunicationsarerequiredforUAC; Thirdly,thecommunicationcapacityandnetworkthroughputareextremelylimited.Theunderwateracousticsignalischaracterizedbylowfrequency(kHz)anddistance-dependentselectiveattenuation.Thustheavailablebandwidthisverysmall,andstronglydistance-dependent[ 25 45 ],thisresultsinlimitedcapacityofpoint-to-pointcommunications.Inaddition,theend-to-endnetworkthroughputisfurtherreducedbythelonground-tripcontroltime; 14

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Finally,theharshseaenvironmentcausesvulnerableunderwaternodes.Themarineanimalinterferenceandfouling,saltwatercorrosion,etc.renderhighernodefailureprobability[ 25 ].Thusnodereliability,andmoreimportantlydatareliability,areessentialtounderwaternetworkdesign. Insummary,underwateracousticnetworksdifferfromterrestrialRFwirelesssystemsinmanyaspects.ThusthetransmissionandnetworkingprotocolsdesignedforRFsystemscannotbedirectlyadoptedbyunderwateracousticsystems.Redesignofexistingprotocolsanddevelopmentofnewtechniquesarenecessaryforunderwateracousticnetworks.Inaddition,RELIABILITYisthekeyissuetobeaddressedintheunderwatersystemdesign.InUAC,communicationreliabilityneedstobeenhancedwithefcientbandwidthutilizationandrelaxedsynchronizationrequirements.Inunderwateracousticsensornetworks(UASN),higher-layerdatatransportandstoragereliabilitywithmoreefcientresourceusageisrequired. 1.3Motivations Toenhancethereliabilityofunderwateracousticcommunicationandnetworking,twotechniquesarepromisingduetotheirgreatsuccessinterrestrialenvironmentsandfeasibilityinunderwaternetworks:cooperativerelaycommunicationsandfountaincodes.However,theuniquefeaturesofunderwateracoustictransmissionrequireadaptationsofthesetechniquesandnovelprotocoldesigns. 1.3.1CooperativeRelayCommunications CooperativerelaycommunicationshavebeenintensivelystudiedinRFwirelesscommunicationstoimprovecommunicationreliabilityandincreasenetworkcoverage[ 7 14 21 ].Incooperativerelaycommunications,anumberofgeographicallyseparatedrelaynodesbetweenthesourceandthedestinationareemployedtoformavirtualantennaarray.Withtheassistanceofrelayforwarding,spatialdiversitygaincanbeachievedfortheend-to-endtransmission.Inaddition,byintelligentlyallocatingthesystemenergyandcontrollingtherelaytopology,resourceoptimizationcan 15

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furtheradvancecommunicationreliabilityandrange[ 1 2 30 ].Thuscooperativerelaycommunicationsfurnishapromisingsolutionforreliablelong-rangeUAC. However,duetothepeculiaracousticsignalattenuationandnoisepowerspectrumdensity,thebenetsofrelaycommunicationsinunderwaterenvironmentsneedtobere-evaluated.Inaddition,theunderwatercooperativerelayingprotocolrequiresmoredelicatedesignstoaddressthechallengesoftimesynchronizationandvariantdelay.Intheliterature,[ 45 ]showsthatthesystemcapacityofdirect-linkUACmanifestsdifferentcharacteristicsthanRFcommunicationsbecauseoftheuniquefrequency-dependentattenuationofacousticsignalsandcolorednoise.In[ 44 ],thecapacityofamultihopunderwaterrelaycommunicationsystemisstudied.Intheanalysis,amultihoprelaysetupisconsidered,whereallrelaysarelocatedonalineandareequallyspacedwiththesametransmitpower.Inmobileunderwateracousticnetworks,however,therelaypositionscanbedynamicandthenodetransmitpowercanbeheterogeneous.Inaddition,throughresourceoptimization,systemperformancecanbeoptimized.Inmywork,thecapacityofadual-hoprelaysystemwitharbitraryresource(locationandpower)allocationwillbeinvestigated.Throughanalyticalandnumericalresults,thecapacitybenetsarerevealedbycomparisonwithtraditionaldirect-linksystems,andtheaffectingfactorsofthecapacityofunderwaterrelaysystems,suchaspowerallocation,relaylocationaswellasend-to-enddistance,arealsounveiled. Intheliteratureofunderwaterrelaycommunications,severalrelayingprotocolshavebeenproposedtoaddressthetimesynchronizationdifculty.In[ 49 ],acooperativeschemewithtime-reversaldistributedspace-timeblockcode(TR-STBC)isproposedtoprovidereliablecommunicationswithresistancetotimingerrors.ThisAlamouti-typecooperativetransmissionretainsspatialdiversityatthecostofreducedcapacity.Inaddition,thisschemeisalsosensitivetothepropagationdelayvariancewithineachAlamoutipair.Anotherschemetermedaswavecooperation(WC)ispresentedin[ 33 ]. 16

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TheWCprotocolonlyworksundertheidealassumptionthatthesignalsasynchronouslyreceivedfromthesourceandtherelaysareperfectlyseparable. Toavoidtheseproblems,weproposeapracticalandefcientrelayingprotocoltailoredforUAC:Asynchronousamplify-and-forward(AF)relayingwithPrecodedOFDM(AsAP).TheAsAPsystemhasthefollowingfeatures:(1)Allrelaysworkinanasynchronousmode,andthereceivedsignalisforwardedtothedestinationassoonastheprocessingiscompleted;(2)WithanAFrelayingscheme,eachrelaysimplyampliesthereceivedsignalwithoutdecoding;(3)Thetransmittedsignalsaremodulatedwithgroupedlinearconstellationprecoding(GLCP)OFDM[ 38 ];(4)Full-duplextransmissioncanbeimplementedattherelays.AllthesecomponentsareuniquelydesignedforunderwaterrelaycommunicationstoaddressthedifcultiesofUAC,whiletakingadvantageofUACfeatures.First,theasynchronousAFrelaydesignavoidsthetimesynchronizationdifcultyandrealizessimplerelayfunctionality.Secondly,theprecodedOFDMresolveschannelfrequencyselectivitywhilecollectingspaceandmultipathdiversity.Duetothesparsityofunderwaterchannels[ 28 ],theasynchronismnaturallyconvertsspacediversityintoresolvablemultipathdiversity,whichcanbecollectedbyprecodedOFDM.Finally,full-duplexcommunicationsattherelayscanboostthecapacityofunderwaterrelaycommunications.ThankstotheseparationofhydrophoneandtransducerinUACdevicesandthedirectionaltransmissionofthetransducer[ 33 ],therelayscanoperatewithsimultaneousreceptionandtransmission.TodemonstratethemeritsofourAsAPprotocol,wederivetheaveragepair-wiseerrorprobability(PEP)ofthesystemandthemaximumcollectablediversitygain.Inaddition,simulationsandcomparisonsareprovidedtoverifythecommunicationreliabilityofourAsAPprotocolandtheaffectingfactors,suchasthenumberofrelaysandchanneltaps,andamplicationfactors. 17

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1.3.2DecomposedFountainCodes Fromanetworkingperspective,end-to-enddatatransmissionreliabilityneedstobeensuredwithcontrolschemesinthelink-layerandthetransportlayer.Traditionally,anautomaticretransmissionrequest(ARQ)mechanismisdevelopedtoguaranteethereliabilitybyrequestingretransmissionsoferroneouslyreceivedpacketsfromthesourceuntilallpacketsarecorrectlyreceived.Toreducethelargeend-to-endlatencyinducedbyfrequentretransmissions,hybridARQschemesarefurtherdesigned.Bytransmittingsomeredundantpacketsatthesourcewithforwarderrorcorrection(FEC)codes[ 59 60 ],thedestinationrequiressignicantlyfewerretransmissions.Intheliterature,variousFECcodeshavebeendeveloped.ExamplesincludesReed)]TJ /F1 11.955 Tf 9.3 0 Td[(Solomon(RS)codesandTornadocodes[ 61 ].Morerecently,ratelessfountaincodes[ 62 ]areproposedtoreducecomputationcostandenablecoderateadaptability.Transmissionschemesbasedonfountaincodesrequiretheleastamountofretransmissionsandnochannel/routeerasureinformationisrequiredatthetransmitter.Thusfountaincodesareespeciallybenecialforreliableunderwateracousticnetworks.Intheunderwaterliterature,fountaincodesandothererasurecodesareshowntobebenecialinbroadcastsystemsandgeneralmultihoptransmissionwithenhanceddatatransportreliability,[ 29 50 ].Inmywork,Iwillfurtherexplorethepotentialoffountaincodesinmulti-levelunderwateracousticcommunicationsandnetworks. Inunderwatercooperativerelaycommunications,eachpacketdeliveryconsistsoftwosteps:thesource-relayandrelay-destinationtransmission.Thisinherentdual-hopnaturerequirestwoconsecutivereliabilitycontrolsontheprotocoldesign[ 53 ].Intheliterature,twomajorapproachesareadoptedtoaddressthisissue:independentfountainencoding[ 15 53 63 ]andconcatenatedfountainencoding[ 9 54 64 65 ].Intheformerone,therelayswilldecodeandre-encodeeachreceivedpacketwithanotherfountaincode,andsendacknowledgementsbacktoconrmeachcorrectreception.Clearly,highcomputationcostisincurredattherelaysandlargetransmissionlatency 18

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isinduced.Whileinthelatterapproach,therelaynodessimplyapplyasecond-layerofcodingtothefountain-codedsourcedatawithoutdecoding.Thusthisapproachrequiressignicantdecodingcomplexityatthedestination.Inordertoreducecomputationalcomplexityandlatencywhileretainingcommunicationreliabilityonbothlinksusingfountaincodes,decomposedfountaincodes(DFCs)consistingoftwolayersofdataencodingareofgreatinterest. Duetotheharshseaenvironmentandnodeenergyconstraints,in-networkdatastorageispreferableinUASN.Intheliterature,data-centricstorage(DCS)isproposedasthemostefcientin-networkstorageprotocolfordatawithfrequentquery[ 42 ].InDCS,alldataofthesametypearestoredatonespeciclocation.Thustheusercansenddataqueriesdirectlytothestorageplacetoretrievethedata.Inthisscheme,dataqueryisbothtimeandenergyefcient.However,inharshwirelessenvironments,thedistributeddatastorageandmulti-hopdata-centricdeliveryrenderDCSavulnerableprotocolduetoitslackofprotectionofdatadeliveryandstorage.Intheliterature,severalprotocolshavebeendesignedtoaddresseachoftheseproblems.In[ 42 75 ],GreedyPerimeterStatelessRouting(GPSR)basedroutingprotocolsaredevelopedforreliabledatadelivery.In[ 76 77 ],zone-basedstoragewithdatareplicationisimplementedtoenhancedatastoragereliabilityaswellastoimprovedatacommunicationandstorageuniformity.Clearly,thestorageoverheadislargeanddatacommunicationreliabilityisnotguaranteed.Fountaincodesarealsoimplementedindistributeddatastoragetoenhancestoragereliabilitywithreducedstorageoverheadandbetteruniformity,[ 35 37 ].However,thecommunicationcostishugeintheseprotocolsinbothdatastorageandretrievalstages,andthetransmissionreliabilityissueisnotaddressed.Byapplyingfountaincodesintothezone-basedDCSdesign,bothdatacommunicationandstoragereliabilitycanbeassured.Inthistypeofsystem,two-leveldatatransmissionandone-leveldatastoragewillbeentailed.Thus 19

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decomposedfountaincodes(DFCs)withmulti-layerencodingarealsofavorableforunderwaterzone-basedDCS. Therefore,inmydissertation,generalDFCswillbeinvestigated.Morespecically,decomposedLubyTransform(DLT)codesforcooperativerelaycommunications,anddecomposedRaptorcodes(DRC)forunderwaterDCSsystems.FortheDLTcodes,ageneralDLTcodeconstructionapproachisrstinvestigatedbasedonpolynomialdecomposition.DuetothespikyfeatureoftheclassicalrobustSolitondistribution(RSD)[ 68 ],ahybriddecompositionalgorithmisthendeveloped.TheresultanthybridDLT(h-DLT)codesenableexiblecomputationcostallocationbetweentwoencoderstocopewiththenodeenergyheterogeneityissueincooperativenetworks.Basedontheh-DLTcodes,areliablecooperativecommunicationscheme(DLT-CC)isdesigned,withimplementsthetwo-layerh-DLTencodinginthedual-hopcommunications.Thebenetsofthedesignedprotocolareillustratedbyevaluatingthesystemperformanceintermsofcommunicationlatency,energyconsumptionandcomputationallocation.ForDRC,amoreefcientdecompositionmethodisdeveloped,andanumericaldecompositiontechniqueisalsodesignedfornitelengthDRC.Then,aDRC-assistedreliablezone-basedDCSprotocolisproposed(DCS-DRC).Byseamlesslyincorporatingthemulti-layerencodingintoeachstageofdatatransmissionandstorageintheDCSprotocol,wecanachievebothtransmissionandstoragereliability.AnalysesandNS2simulationsareconductedtorevealthecommunicationcostoftheproposedprotocol. Therestofthedissertationisorganizedasfollows.ThesystemcapacityofunderwaterrelaycommunicationsisinvestigatedinChapter 2 .TheAsAPprotocolisdescribedandevaluatedinChapter 3 .Afterthat,thedecomposedfountaincodesaredevelopedinChapter 4 .TheapplicationsoftheseDFCsinbothcooperativerelaycommunicationsandunderwaterDCSschemesareinvestigatedinChapter 5 .ConclusionsandfutureworksarelistedinChapter 6 20

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CHAPTER2CAPACITYOFUNDERWATERRELAYCOMMUNICATIONS Inthischapter,wewillinvestigatethebenetsofunderwaterrelaycommunicationsfrominformation-theoreticperspective.Consideradual-hoprelaysystemwithasinglerelayinFigure 2-1 ,andamplify-and-forward(AF)relayingprotocolisadoptedattherelay.Assumethechannelisnon-fadingwiththesignalattenuationandnoisepowerspectrumdensity(PSD)obeyingtheempiricalformulas[ 45 ].Thesystemcapacityiscomputedforageneralrelaysetupwitharbitrarytransmitpowerandrelaylocation.Numericalresultsareillustratedtorevealthecapacityfeaturesofunderwaterrelaycommunicationsandthecapacitygainoverdirect-linksystems. 2.1AcousticSignalAttenuationandNoiseModels ComparedtotheterrestrialRFsignalpropagation,theunderwateracousticsignalpropagationisfeaturedwithfrequency-dependentattenuation.Thenoiseatthereceiverisalsouniquewithfrequency-dependentPSD.Intheliterature,empiricalsignalpropagationandnoisemodelsareproposed[ 27 ],andlaterintroducedintoUAC[ 45 ]. UnderwateracousticsignalattenuationAisdependentonbothpropagationdistanceDandsignalfrequencyf,whichiscomputedas: A(D,f)=Dka(f)D. (2) Intheaboveequation,kisthepathlossexponent,whichreectsthegeometryofacousticsignalpropagation.Forexample,k=2isusedforsphericalspreading,k=1isforcylindricalspreading,andk=1.5isreferredtoaspracticalspreading.Inthischapter,k=1.5isusedifnototherwisespecied.Thefrequencydependencyisreectedina(f),whichisgivenbyThorp'sformula[ 27 ]: 10loga(f)=0.11f2 1+f2+44f2 4100+f+2.7510)]TJ /F10 7.97 Tf 6.58 0 Td[(4f2+0.003. (2) 21

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InEquations( 2 )and( 2 ),theunitsare:Disinkm,fisinkHz. ThenoisePSDN(f)ismodeledas[ 45 ]: 10logN(f)=N1)]TJ /F6 11.955 Tf 11.96 0 Td[(log(f), (2) whereN1andareconstantswithempiricalvaluesN1=50dBrePaand=18dB/decade. Theuniquefeaturesofunderwateracousticchannelhavegreatimpactonthesystemcapacityofunderwaterrelaycommunications.Thefrequency-dependentnatureofthesignalattenuationandnoisePSDnecessitatescarefulevaluationofthecapacitybenetsofunderwaterrelaycommunicationswithrespecttodirect-linktransmissions. 2.2CapacityAnalysis WiththegivensignalattenuationandnoisePSDformulas,werstcomputetheend-to-endsignal-to-noiseratio(SNR).Basedontheresult,optimumcarrierfrequencyandavailablesignalbandwidtharedetermined.Finally,thesystemcapacitycanbecalculatedcorrespondingly. 2.2.1SystemCapacity ThesystemcapacityCforastationaryfrequency-selectivesystemiscomputedbyintegratingtheShannoncapacityoverthesignalbandwidthB[ 32 ].Accordingly,foradual-hopAFrelayingsystem,theend-to-endsystemcapacitycanbecalculatedas: C=ZBlog2(1+eq(f))df, (2) whereeqistheequivalentend-to-endsignal-to-noiseratio(SNR). 2.2.2End-to-endSNR Inpoint-to-pointUAC,theSNRatreceiverjwithtransmitteriiscomputedas: j,i(f)=Pi(f) A(Di,j,f)N(f), (2) 22

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wherePi(f)isthetransmitpowerdensityatnodei,andDi,jisthedistancebetweennodeiandj. Foradual-hopAFrelaysystem,theend-to-endequivalentSNReqisgivenin[ 13 ]: eq=r,sd,r r,s+d,r+1, (2) wherer,sandd,rrepresenttheSNRsattherelaynoderandthedestinationnoded,withtransmissionsfromthesourcesandtherelayr,respectively.Thus,theequivalentSNRatthedestinationforunderwaterrelaycommunicationscanbeapproximatedathighSNR(r,s+d,r1)as: eq(f)Ps(f)Pr(f)=N(f) Ps(f)A(Drd,f)+Pr(f)A(Dsr,f). (2) Furthermore,tofacilitatethecapacityanalysis,wedeneapowerallocationratioPastheportionofsourcetransmitpowerPsoutofthetotallytransmitpowerP,i.e.Ps=PP,thusthetransmitpowerattherelaycanbecomputedas:Pr=(1)]TJ /F6 11.955 Tf 10.61 0 Td[(P)P.Likewise,wealsodenearelaylocationratioDasthesource-relay(S-R)distanceoversource-destination(S-D)distanceD,i.e.Ds,r=DDandDr,d=(1)]TJ /F6 11.955 Tf 10.38 0 Td[(D)D.Inourrelaysystems,nochannelstateinformationisassumedatthetransmitterside,thusthetransmitpowerateachnodeisuniformlydistributedovertheentiresignalbandwidth;thatisPi(f)=Pi=B,i2fs,rg.ThentheequivalentSNReq(f)canbereexpressedas: eq(fjD,P,D)P=B=N(f) A(DD,f)=P+A((1)]TJ /F6 11.955 Tf 9.96 0 Td[(D)D,f)=(1)]TJ /F6 11.955 Tf 9.96 0 Td[(P). (2) Clearly,theequivalentSNRisdeterminedbytheS-DdistanceDandtheresourceallocation:PandD. 2.2.3SignalBandwidth NoticefromEquation( 2 ),theend-to-endSNRisfrequency-dependent.Weadopttheempirical3dBbandwidthasthesignalbandwidthinouranalysis.SupposethebestSNRthatarelaysystemcanachieveiseq(fc)attheoptimalfrequencyfc,then 23

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the3dBbandwidthisdenedasacontinuousfrequencyrangearoundfc,whereanyfrequencyinthisrangeachievesatleasthalfofthebestSNR,i.e.B3=ff:fminffmaxandeq(f)eq(fc)=2g,where eq(fmin)=eq(fmax)=eq(fc)=2. (2) InanunderwaterrelaysystemwithparametersD,P,D,thebestsystemSNRandthecorrespondingoptimalfrequencyfccanbecomputedfromEquation( 2 ).The3dBbandwidthB3canbedeterminedbyndingfminandfmaxinEquation( 2 ).DuetothecomplexformoftheThorp'sformulainEquation( 2 ),concavityproofofeq(f)andclosed-formresultsoffcandB3arenotmathematicallytractable.However,extensivenumericalsimulationsindicatethateq(f)isconcaveinf,whichwillbeshowninthenextsubsection.Thus,bydeterminingfcandB3numerically,wecancalculatetheend-to-endcapacityofanunderwaterrelaysystemas: C(D,P,D)=Zfmaxfminlog21+P=B=N(f) A(DD,f)=P+A((1)]TJ /F6 11.955 Tf 9.96 0 Td[(D)D,f)=(1)]TJ /F6 11.955 Tf 9.96 0 Td[(P)df, (2) Inaddition,althoughclosed-formresultsarenotanalyticallytractable,weobservefromtheanalysesthattheoptimalfrequencyfc,3dBbandwidthB3andsystemcapacityCareallaffectedbytheend-to-endtransmissiondistanceD,powerallocationratioP,andrelaylocationratioD.Therefore,wewillinvestigatetheeffectsofthesefactorsnumerically.Besides,toillustratethebenetsofrelaying,underwaterrelaysystemswillbecomparedwithcorrespondingtraditionaldirect-linksystems. 2.3NumericalResults 2.3.1OptimalFrequencyand3dBBandwidth Todemonstratethedistancedependencyoftheoptimalfrequencyand3dBsignalbandwidth,wesimulateanunderwaterrelaysystemwithuniformpowerallocation(P=1=2)andmid-pointrelay(D=1=2).Theresultsarecomparedwiththoseofthedirect-linksysteminFigure 2-2 .Thisgurerevealsthattheunderwaterrelaysystem 24

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canoperateathighercarrierfrequencywithlarger3dBsignalbandwidthincomparisonwiththedirect-linksystem.AstheS-Ddistanceincreases,theoptimalfrequencyand3dBbandwidthdecreaseforbothunderwaterrelaysystemanddirect-linksystems.Inaddition,thesetwovaluesshrinkalmostlinearlyinlogarithmscaleatlargeS-Ddistances. 2.3.2SystemCapacity WerstinvestigatetheeffectofS-Ddistanceonthesystemcapacity.Forauniformsystemsetup(P=D=1=2)withtotaltransmitpowerP=100dBrePa,wesimulatethecapacityfortherelaysystemandthedirect-linksystematvarioustransmissiondistancesinFigure 2-3 .Theresultsaredisplayedinlogarithmscale.Thisgureshowsthat,similartofcandB3,thecapacityofbothunderwaterrelaysystemsanddirect-linksystemdecaysrapidlywiththeS-Ddistanceandalmostlinearlyinlogarithmscaleatlongdistances.Thisobservationindicatesthatthedatarateisverylimitedforlong-distancetransmissions. Todemonstratethebenetsofunderwaterrelaysystemsquantitatively,wedeneacapacitygainasthedifferenceratiobetweenthecapacityofanunderwaterrelaysystemandthatofthecorrespondingdirect-linksystem:CapacityGain(%)=((Crelay)]TJ -448.93 -23.9 Td[(Cdirect)=Cdirect)100%,whereCrelayandCdirectrepresentthecapacityoftheunderwaterrelaysystemandthedirect-linksystemrespectively.TheresultisplottedinFigure 2-4 .Prominentcapacitygainisobservedfromthegure:atallS-Ddistances,theassistanceoftherelayimprovesthesystemcapacitybymorethan40%.TheminimumcapacitygainhappensatD=2km. 2.3.3AffectingSystemFactors NowweevaluatetheeffectsofpowerallocationratioPandrelaylocationratioDonsystemcapacity.ForanunderwaterrelaysystemwithxedS-DdistanceD=2kmandtotaltransmitpowerP=100dBrePa,weplotthesystemcapacitycontourwithrespecttoPandDinFigure 2-5 .Fromthisgure,weobservethat: 25

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1. Themaximumsystemcapacityisachievedwithuniformpowerallocation(P=1=2)andmid-distancerelay(D=1=2).Thusinahomogenousnetworkwherethetransmitpowerforallnodesisthesame,itisbenecialtochoosetherelaylocatedinthemiddleofthesourceandthedestinationortomovetherelaytothatposition; 2. Foreachxedrelaylocation(D),thesystemcapacityvariesveryslightlywithP;whereasforxedpowerallocation(P),thesystemcapacitychangesdramaticallywithD.Thisobservationrevealsthatrelaylocationisamuchmorecriticalfactoraffectingthesystemcapacitythanpowerallocation.Therefore,relaylocationoptimizationisveryimportantinunderwaterrelaycommunications. 2.4Summary Inthischapter,weanalyzedthecapacityofunderwaterrelaycommunicationsfrominformation-theoreticperspective.BasedontheempiricalacousticwavepropagationandnoisePSDmodels,thecapacityofunderwaterrelaycommunicationsiscalculated.Numericalresultsrevealthatunderwaterrelaysystemsachievemorethan40%capacitygaincomparedwithdirect-linksystemsatalltransmissiondistances.Theaffectingfactoranalysisshowsthatrelaylocationisamuchmorecriticalfactortocapacitythanpowerallocation.Thusacarefulchoiceofrelaylocationisnecessary. 26

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Figure2-1. Therelaysystemsetupforthecapacityanalysisofunderwaterrelaysystems. Figure2-2. Thecomparisonofoptimalcarrierfrequencyand3dBbandwidthbetweenunderwaterrelaysystemsanddirect-linksystemswithdifferentS-Ddistances. 27

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Figure2-3. Thecomparisonofsystemcapacitybetweenunderwaterrelaysystemsanddirect-linksystemswithdifferentS-Ddistances.ThetotaltransmitpowerisP=100dBrePa. Figure2-4. Thecapacitygainofunderwaterrelaysystemsovercorrespondingdirect-linksystemswithdifferentS-Ddistances.ThetotaltransmitpowerisP=100dBrePa. 28

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Figure2-5. Thecapacitycontourofanunderwaterrelaysystemwithdifferentrelaylocationratiosandpowerallocationratios.ThetotaltransmitpowerisP=100dBrePa,andS-DdistanceisD=2km. 29

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CHAPTER3ASYNCHRONOUSUNDERWATERRELAYCOMMUNICATIONS Inthischapter,wewilldesignapracticalunderwaterrelayingprotocol.Inunderwaterrelaycommunications,accuratetimesynchronizationamongallrelaysishinderedbytheslowandvariantsignalpropagation,thusasynchronousrelaydesignisadoptedinourprotocol.InRFenvironmentswithdensemultipath,suchasynchronismwilleasilycausediversitylossduetomultipathcollision.Inourunderwaterrelaycommunicationssetup,however,wedeliberatelychoosetoallowsuchasynchronismresultedfromdifferentpropagationdelaystotakeadvantageofthesparsenatureofunderwateracousticchannel.Infact,bythissimpleasynchronousapproach,spacediversityistransformedintomultipathdiversity.Tocollectrichmultipathdiversityinunderwaterrelaycommunications,wewillresorttothegroupedlinearconstellationprecoding(GLCP)OFDMtechnique.Hence,thereisnoneedforanyspecialmeansofdealingwithspacediversity.TheresultantschemeistermedasAsynchronousAmplify-and-forwardrelayingwithPrecodedOFDM(AsAP). 3.1SystemandChannelModel InAsAP,weconsideradual-hoprelaysystemwithonesourcenodes,Rrelaynodesr2f1,,Rgandonedestinationnoded,showninFigure 3-1 .TheUACchannelisfeaturedwithlongyetsparechanneltaps.Duetotheslowpropagationofacousticwavesunderwater(vacoustic),themulti-reectionsfromtheseasurfaceand/orbottomgeneratethesparsemultipathUACchannel[ 26 ].EachreectionpathobeysthepropagationruleinEquation( 2 ).ThusaUACchanneloflengthLcanberepresentedash:=[h0,,hL)]TJ /F10 7.97 Tf 6.59 0 Td[(1],withinwhichLnztapsarenon-zero.Eachnon-zerotaphl,l2fl0,,lLnz)]TJ /F10 7.97 Tf 6.59 0 Td[(1gcorrespondstoonearrival,andisindependentlycomplexGaussiandistributedwithzeromeanandvariance2l,i.e.hlCN(0,2l)[ 49 ].Thevarianceiscomputedas: l=)]TJ /F8 7.97 Tf 6.18 -1.77 Td[(l p A(dl,f), (3) 30

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whereA(dl,f)isthesignalattenuationinEquation( 2 )anddlisthepropagationdistanceoftheltharrival,whichisdeterminedbythenumberofreectionsfromseasurfaceandbottominthechannelgeometrydescribedin[ 51 ].)]TJ /F8 7.97 Tf 6.78 -1.79 Td[(listhereectionlossofacousticwaves.Inaddition,eachnon-zerotapischaracterizedbyitsdelaytimel,whichiscalculatedasl=dl=vacoustic.InFigure 3-2 ,weplotasnapshotofthesparseacousticchannelinseatrialRACE08ata1000mtransmissiondistance[ 41 ]. 3.2AsAPRelayProtocol TheAsAPend-to-endtransmissionconsistsofthreephases.First,thesourcegeneratesinformationsymbolsmodulatedwithGLCPOFDM,andbroadcaststhem.Inthesecondphase,eachrelayampliesitsreceivedsignalandasynchronouslyforwardsittothedestinationwithoutanytimecoordinationwithotherrelays.Finally,thedestinationcollectsallasynchronouslyarrivingsignalsanddecodesthem.Thesignalprocessingowchartforoneequivalentsource-relay-destination(S-R-D)pathisshowninFigure 3-1 .Thedetailedprocessingateachstepwillbeelaboratedinthefollowingsubsections. IntheAsAPsystem,weassumethatthechannelforeachlinkisblock-stationary,andtheadditivenoiseiscomplexGaussiandistributedCN(0,N0)[ 49 ]withzeromeanandvarianceN0.Inaddition,tofacilitateoursystemdescription,weadoptthefollowingnotations.FdenotestheN-pointfastfouriertransformation(FFT)matrixwithF(p,q)=(1=p N)exp()]TJ /F5 11.955 Tf 9.3 0 Td[(j2pq=N),p,q2f0,,N)]TJ /F3 11.955 Tf 12.59 0 Td[(1g,andFHistheinverseFFT(IFFT)matrix.Inaddition,hi,j,i,j2fs,r,dgrepresentsthechannelvectorbetweennodeiandj,and~hi,j=Fhi,jisthefrequencydomainchannelresponse. 3.2.1SourceTransmission Atthesource,eachOFDMsymbol~xisprecodedwithGLCPandthenbroadcasttotherelaysandthedestination.TheGLCPconsistsoftwosteps:groupingandprecoding.Weadopttheoptimalgroupingrulein[ 38 ],whichachievesthemaximumcodinggainanddiversitygain.Thegroupingisperformedasfollows(Figure 3-3 ). 31

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ForaprecodingsizeK,thesymbol~xoflengthNisrstdividedintoconsecutiveKblocks,eachofsizeM,thenthemth(m2f1,,Mg)elementfromeachblockisselectedtoformagroup,denotedbyavector~xm.AllpossiblegroupvectorsformaniteconstellationsetAK.Thisprocedurecanbemathematicallyrepresentedas:~xm=m~x,wherem:=IN(Im,:)isaKNselectionmatrixwiththeKrowschosenfromanNNidentitymatrixIN,andtheindexoftheKrowsaredenedinthesetIm.Foroptimalgrouping,eachrowsetischosenasIm:=fm,m+K,,m+(K)]TJ /F3 11.955 Tf 10.19 0 Td[(1)Mg.Inthesecondstep,eachgroupvector~xmisencodedwithprecodermatrixKofsizeKK.Toachievethemaximumdiversitygain,Kisdesignedasin[ 38 ].Then,thecodedgroupsarereassembledtoformtheprecodedsymbol~xs.TheentireGLCPprocesscanberepresentedas: ~xs=MXm=1Tmm~x. (3) Toovercomeintersymbolinterference(ISI),cyclicprex(CP)isinsertedafterIFFT.Thenthegeneratedtime-domainsymbolxsisbroadcasttotherelaysanddestinationwithtransmitpowerPs.TheCPlengthisproperlychosentoaddressthepossibleasynchronousdelay,whichisspeciedinthefollowingpart.Toachievebetterspectrumefciency,coarsetimesynchronizationcanbeimplementedforsmallerasynchronousdelay. 3.2.2RelayForwarding Afterpropagatingthroughthesource-to-relay(S-R)channel,thereceivedsignalyratrelayrcanberepresentedas: yr=Hsrp Psxs+nr, (3) whereHsristhetoeplitzmatrixoftheS-Rchannelhsrandnristhenoiseatrelayr. InAsAPrelayingprotocol,tocompensatechannelfading,eachrelayrstampliesthereceivedsignalandthenforwardsittothedestination.TheamplicationfactorAris 32

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adiagonalmatrixwiththeamplicationmagnitudeforeachsubcarrier/bitasitsdiagonalentry,whosevalueischosentoassurethattheaveragerelayingsignalpowerperbitcomplieswiththerelaytransmitpowerPr.Dependingondifferentperformanceandcomputationrequirements,theamplicationcanbeimplementedeitherintimedomain(TD)orinfrequencydomain(FD). InTDamplication,theforwardingsignalxrisgeneratedbymultiplyingtheamplicationfactordirectlytothetime-domainreceivedsignal,i.e.xr=Aryr.TheamplicationfactorArisdeterminedtosatisfythepowerconstraintoftheentiretransmitsymbol.ThustheamplicationmagnitudeAristhesameforeachbitintimedomain,namelyAr=ArI,whereIisanidentitymatrix.WeconsidertwoTDamplicationschemes:thexedamplication(TD-FA)andtheinstantaneousamplication(TD-IA).InTD-FA,theamplicationfactoriscomputedbycomplyingwiththeaveragepowerconstraint:E)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jArj2jjyrjj2=(LCP+N)Pr.Thustheamplicationmagnitudecanbepre-computedwiththeaveragechannelpowerE(jjhsrjj2)andnoisevarianceN0: Ar,TD)]TJ /F8 7.97 Tf 6.59 0 Td[(FA=s Pr PsE(jjhsrjj2)+N0. (3) ForTD-IA,theamplicationfactorwillsatisfytheinstantaneouspowerconstraintforeachOFDMsymbol,i.e.,jArj2jjyrjj2=(LCP+N)Pr.Thustheamplicationmagnitudeiscomputedas, Ar,TD)]TJ /F8 7.97 Tf 6.59 0 Td[(IA=p (LCP+N)Pr kyrk. (3) WithTDamplication,themaximumtolerableasynchronousdelay(MTAD)matthedestinationwillbethedifferencebetweentheCPlengthandthegreatestS-R-Dchannellength:m=LCP)]TJ /F3 11.955 Tf 10.93 0 Td[(maxr2f1,,RgfLs,r+Lr,d)]TJ /F3 11.955 Tf 10.94 0 Td[(1g.Toincreasethetolerancetotimingasynchronism,eachrelaycanupdatetheforwardingsignal'sCPwiththelastLCPbitsoftheampliedsignal.Inthisscheme,theMTADwillbethedifferencebetweentheCP 33

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lengthandthegreatestrelay-destination(R-D)channellength,i.e. m=LCP)]TJ /F3 11.955 Tf 22.51 0 Td[(maxr2f1,,RgLr,d. (3) IntheFDamplication,thereceivedsignalisrsttransformedtothefrequencydomainrepresentation,andtheneachsubcarrierisampliedaccordingtoitssignalattenuation.ThroughnormalCPremovalandFFT,thefrequency-domainsignal~yrisrepresentedas: ~yr=Dsrp Ps~xs+~nr, (3) whereDsr=diag(~hsr)=diag(Fhsr)isthediagonalS-Rchannelmatrix,and~nr=FnristheFDnoisevector.TheFDampliedsignaliscomputedas:~xr=Ar~yr,andtheTDforwardingsignalisgeneratedbyIFFTof~xrandCPinsertion.InFDamplication,theMTADatthedestinationisthesameastheTDamplicationschemewithCPupdateasinEquation( 3 ). WealsoconsidertwoFDamplicationschemes:thesubcarrieramplication(FD-SA)andthegroupamplication(FD-GA).InFD-SAscheme,thepowerconstraintisimposedoneachsubcarrieri,i.e.jAr(i,i)~yr(i)j2=Pr.Thustheamplicationfactorisobtainedas Ar,FD)]TJ /F8 7.97 Tf 6.58 0 Td[(SA=diagp Pr j~yrj. (3) InFD-GA,becauseoftheGLCPschemeadoptedinourprotocol,itisreasonabletoconstrainthetransmitpowerofeachprecodinggrouptoKPr,i.e.Amjjm~yrjj2=KPr,whereAmistheamplicationmagnitudeforgroupm.Thenwecanobtaintheamplicationfactoras: Ar,FD)]TJ /F8 7.97 Tf 6.58 0 Td[(GA=MXm=1Tmp KPr jjm~yrjjeK, (3) whereeK=[1,,1]Tisanall-onecolumnvectorofsizeK. 34

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Thereareprosandconsfortheseamplicationschemes.TheTD-FAschemepossessthefeatureofsimpleimplementationwithlesssignalprocessing;theotherthreeschemesarebasedoninstantaneoussignalinformation,thusmoresignalprocessingisneeded,whichinducesextraenergyconsumptionandprocessingdelay.However,theinstantaneousschemesareexpectedtoachievebetterperformance. Finally,aftertheamplication,allrelayswillforwardthesignalstothedestinationinthesametimeslotinanasynchronousmode. 3.2.3DestinationDecoding Atthedestination,therelayedsignalsarrivewithrandomdelays,whichoriginatefromrelaylocationdifference,signalpropagationvarianceandrandomsignalprocessingdelayateachrelay.Denotethedelayofthesignalfromrelayrwithrespecttothearrivaltimeofthedirectlinksignalasr.AssumethatthemaximumrandomdelayforallrelaysislessthanMTADm.ThenafternormalOFDMprocessing,wecanexpresstheFDreceivedsymbol~ydas: ~yd=Deqp Ps~xs+~neq, (3) whereDeqistheequivalentend-to-enddiagonalchannelmatrix,whichiscomputedasDeq=Dsd+PRr=1rDrdArDsr.Inthisequation,Dsd=diag(~hsd)=diag(Fhsd)andDrd=diag(~hrd)=diag(Fhrd)arethediagonalsource-to-destination(S-D)andR-Dchannelmatricescorrespondingly.Besides,risanNNdiagonalasynchronousdelaymatrixforrelayr,whichmultiplieseachsubcarrieriwithacorrespondingphasefactorr(i,i)=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(j2ir=N).Inaddition,~neqistheequivalentFDnoisevectorcomputedas: ~neq=~nd+RXr=1rDrdAr~nr, (3) where~nd=FndistheFDnoiseatthedestination.Noticethattheequivalentnoise~neqiscoloredwithcovariancematrixN=I+PRr=1jDrdArj2. 35

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Assumeperfectchannelstateinformationisknownatthedestination,thedecisionstatisticiscomputedbywhiteningthenoiseas: ~yd=)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2N~yd=)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NDeqp Ps~xs+)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2NNeq. (3) Thesymboldetectoratthedestinationestimateseachtransmittedsymbolaccordingtothegroupingschemedenedbym.EveryKbitsineachgrouparedecodedtogetherwithmaximumlikelihood(ML)criterion: ^~xm=argmin~xi2AKjjm~yd)]TJ /F18 10.909 Tf 9.09 0 Td[((m)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NDeq)p Ps~xijj,m2f1,,Mg. (3) Finally,thedecodedgroups^~xm,m2f0,,M)]TJ /F3 11.955 Tf 10.79 0 Td[(1garereassembledtoformthedecodedsymbolas:^~x=PMm=1Tm^~xm. 3.3PerformanceEvaluation TheAsAPsystemisdesignedtoattainreliablecommunicationsbycollectingbothspaceandmultipathdiversityfromthesparseUACchannel.Inthissection,wewillanalyzetheperformanceoftheproposedAsAPsystem.Theaveragepair-wiseerrorprobability(PEP)isadoptedastheperformancemetric,andthemaximumcollectablediversityisderivedaccordingly. 3.3.1PEPofAsAP Assumeasymbol~x2AKistransmitted,anditiserroneouslydecodedas^~x,where^~x6=~xand^~x2AK.WithMLdetection,thecorrespondingPEPPeiscomputedas: Pe=P(~x!^~x)=Q0@s d2(~yd,^~yd) 2N01A, (3) whered(~yd,^~yd)=jj~yd)]TJ /F3 11.955 Tf 13.46 2.66 Td[(^~ydjjistheEuclideandistancebetween~ydand^~yd,and~yd=)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NDeqp Ps~x,^~yd=)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NDeqp Ps^~x.Bydeningtheerrorvector~xe=~x)]TJ /F3 11.955 Tf 12.16 2.66 Td[(^~xandtheerrormatrixDe=diag(~xe),theEuclideandistancesquarecanbecomputedas: d2(~yd,^~yd)=jj)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NDeqp Ps~xejj2=jjp Ps)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NDe~heqjj2=~hHeqPs)]TJ /F10 7.97 Tf 6.58 0 Td[(1NjDej2~heq, (3) 36

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where~heqistheequivalentchannelvectorinfrequencydomain: ~heq=~hsd+RXr=1ArDrdr~hsr. (3) ByusingthealternativerepresentationofQfunction[ 31 ]: Q(x)=(1=)Z=20exp()]TJ /F5 11.955 Tf 9.3 0 Td[(x2=(2sin2))d, (3) thePEPcanbereexpressedas: P(~x!^~x)=1 Z 20"exp )]TJ /F18 10.909 Tf 9.88 10.71 Td[(~hHeqPs)]TJ /F10 7.97 Tf 6.59 0 Td[(1NjDej2~heq 4N0sin2!#d. (3) AndtheaveragePEPPefortheAsAPsystemcanbecomputedbyaveragingover~heq: Pe=E~heq[P(x!^x)]=1 Z 20E~heq"exp )]TJ /F18 10.909 Tf 9.88 10.7 Td[(~hHeqPs)]TJ /F10 7.97 Tf 6.59 0 Td[(1NjDej2~heq 4N0sin2!#d. (3) TondtheaveragePEP,thestatisticalpropertyof~heqneedstobeexaminedrst.FromEquation( 3 ),weobservethatconditionedontheR-DchannelDrd,~heqisacomplexGaussianvectorwithmeaneq=0andcovariancematrix~Veq.Duetotheindependencyamongallchannelshi,j,i,j2fs,r,dg,~Veqcanbecalculatedas: ~Veq=E~hsdh~hsd~hHsdi+RXr=1DrdE~hsrArr~hsrArr~hsrHDHrd. (3) Noticethatthecovariancematrixisamplicationdependent.FortheTD-FAscheme,theamplicationfactorAr=ArIhasthesamediagonalvaluesandisirrelevanttotheinstantaneouschannel,thusEquation( 3 )canbefurthersimpliedas: ~Veq=FVsdFH+RXr=1jArj2(Drd)FVsr(r)FH(Drd)H, (3) whereVsdandVsr(r)arethediagonalcovariancematricesoftheS-DchannelhsdandS-Rchannelshsrdelayedbyr,respectively.TheranksofVsdandVsr(r)areLsd,nzandLsr,nz. 37

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Furthermore,thecharacteristicfunctionofthequadraticformofcomplexGaussianvectorsin[ 48 ]givesrisetotheresult:Evexp)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F16 11.955 Tf 9.3 0 Td[(vHBv=det(I+VvB),wherevisaGaussianvectorwithcovariancematrixVvandBisanarbitrarymatrixindependentofv.Therefore,theaverageterminEquation( 3 )canbecomputedas: E~heq"exp )]TJ /F18 10.909 Tf 9.88 10.7 Td[(~hHeqPs)]TJ /F10 7.97 Tf 6.59 0 Td[(1NjDej2~heq 4N0sin2!#=E~hrd"detI+Ps 4N0sin2~VeqjDej2)]TJ /F10 7.97 Tf 6.59 0 Td[(1N)]TJ /F10 7.97 Tf 6.58 0 Td[(1#. (3) ThentheaveragePEPcanbereadilyexpressedas: Pe=1 Z 20E~hrd"detI+Ps 4N0sin2~VeqjDej2)]TJ /F10 7.97 Tf 6.58 0 Td[(1N)]TJ /F10 7.97 Tf 6.59 0 Td[(1#d. (3) Intheaboveequation,DeandNarediagonalmatrices.However,duetothecorrelationamongdelaytapsoftheequivalentchannelatthedestination,thecovariancematrix~Veqisnotdiagonalingeneral.ThisrenderstheexplicitexpressionfortheaveragePEPinEquation( 3 )mathematicallyintractable.Therefore,weresorttodiversityanalysistoillustratethesystemperformance. 3.3.2DiversityofAsAP Ingeneral,theaverageerrorperformancePeofacommunicationsystemcanbeapproximatedintermsofP=N0asfollows: Pe/GcP N0)]TJ /F8 7.97 Tf 6.59 0 Td[(Gd, (3) wheretheexponentGdisreferredtoasdiversitygainandthefactorGcistermedascodinggain.TheperformanceoftheAsAPsystemshowninEquation( 3 )indicatesthatthediversitygainofthesystemisrelatedtothedeterminantterm.Thoughtheexactexpressionofthedeterminantisnotavailable,wecandeterminethemaximumcollectablediversity,whichisstatedinthefollowingproposition. Proposition1(MaximumCollectableDiversity(MCD)). ThemaximumdiversitygainoftheAsAPsystemwithxedamplicationfactoristheminimumoftheprecodingsizeand 38

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thesummationofthenumberofnon-zerotapsoftheS-DandS-Rchannels,i.e. GdminfK,Lsd,nz+RXr=1Lsr,nzg (3) Proof. LetT=~VeqjDej2)]TJ /F10 7.97 Tf 6.59 0 Td[(1N,thenthedeterminantinEquation( 3 )canbewrittenas: detI+Ps 4N0sin2T)]TJ /F10 7.97 Tf 6.59 0 Td[(1=K)]TJ /F10 7.97 Tf 6.59 0 Td[(1Yi=01 1+Ps N0(4sin2)i, (3) wherei,i2f0,,K)]TJ /F3 11.955 Tf 12.91 0 Td[(1garethenon-increasingeigenvaluesofthematrixT.Supposetherankofthematrixisr,thentherearernonzeroi,thusEquation( 3 )canberewrittenas: detI+Ps 4N0sin2T)]TJ /F10 7.97 Tf 6.59 0 Td[(1=r)]TJ /F10 7.97 Tf 6.59 0 Td[(1Yi=01+Ps N0(4sin2)i)]TJ /F10 7.97 Tf 6.58 0 Td[(1 r)]TJ /F10 7.97 Tf 6.59 0 Td[(1Yi=0i 4sin2!Ps N0)]TJ /F8 7.97 Tf 6.59 0 Td[(r. (3) RecallthedenitioninEquation( 3 ),weknowthatthediversitygainoftheAsAPsystemisr,whichisdeterminedbytherankofT. IntheTmatrix,)]TJ /F10 7.97 Tf 6.59 0 Td[(1NisdiagonalwithfullrankofK.Inaddition,accordingtotheprecodingmatrixdesign[ 38 ],De=diag(xe)isalsooffullrankforanyerrorpairs.ThustherankofTisdeterminedbytherankof~Veq,i.e.rank(T)=rank(~VeqjDej2)]TJ /F10 7.97 Tf 6.58 0 Td[(1N)=rank(~Veq).ForTDamplication,~VeqisshowninEquation( 3 ).NoticethatbothFandDrdbothareoffullrankandtherankofVsdandVsrareLsd,nzandLsr,nz,respectively.Accordingtotherankpropertyofmatrixsummation,wecanobtainthatrank(~Veq)Lsd,nz+PRr=1Lsr,nz.Inaddition,thesizeof~VeqisKK,thusthemaximumcollectablediversityislimitedbytheprecodingsizeK.Therefore,themaximumcollectablediversityistheminimumofKandrank(~Veq). TheresultinProposition1isobtainedwithoutanyrequirementonthedistributionoftheR-Dchannels,thusitissuitableforgeneralR-Dchannelstatistics.Severalremarksforthispropositionarelistedinthefollows. 39

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3.3.2.1MCDachievability Tocollectthemaximummultipathdiversity,therankofthecovariancematrix~VeqinEquation( 3 )needstobethesumLsd,nz+PRr=1Lsr,nz,andtheprecodingsizeKshouldbelargerorequaltothesum.InaspecialcasewheretheR-Dchannel~hrdisdeterministicandnon-frequency-selective,namelyDrd=DrdI,wehave ~Veq=F"Vsd+RXr=1jArDrdj2Vsr(r)#FH. (3) IftheS-Dchannel~hsdandtheS-Rchannels~hsrwithdelayrareperfectlyseparatedwithnooverlap,thentherankof~Veqisthesumofallthenon-zerotapsofS-DandS-Rlinks.Thusthemaximumdiversitycanbecollected.ForRayleigh-fadingR-Dchannels,theachievabilityofMCDisnotanalyticallyexplicit.Itwillberevealedbysimulations.However,thelargerprecodingsizeKrequiresmorecomputationenergy.Henceforth,inpracticalsystems,thereisatrade-offbetweendiversitycollectionandenergyconsumption. 3.3.2.2Underwateracousticchannel Inunderwateracousticchannels,thechanneltapsaresparseinnature[ 26 ].ThusintheAsAPdesign,allrelaychannelsassociatedwithrandomdelaysareverylikelytobeseparatedintimeduetotheasynchronism.Thisallowsthesystemtocollectthespacediversityintheformatofmultipathdiversity.Therefore,theAsAPprotocolisinherentlysuitableforUAC.Furthermore,forasingle-relaysystemwithoutdirectlink,asthenumberofnon-zeroS-Rchanneltapsincreases,i.e.thechannelbecomesdenser,theMCDincreaseslinearly.However,foramulti-relaysystem,asthenumberofrelaysincreases,thechanceofchanneltapcollisionincreases,thusthecollectablediversitywillnotincreaselinearlyasthenumberofrelays. 3.3.2.3Effectoftheamplicationfactor TheresultinProposition1isonlyvalidfortheAsAPsystemwithxedamplicationfactor.Forgeneralamplicationschemes,TheMCDisnotexplicitfromtheaverage 40

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PEPformula,sincetherankofthecovariancematrix~VeqinEquation( 3 )ismathematicallyintractable.However,noticethattheamplicationfactorwillnotchangethenumberoffreedomofthechanneltaps,andtheMCDforinstantaneousamplicationschemeswillnotexceedthevaluederivedforxedamplicationinEquation( 3 ).Inthesimulationpart,wewillverifytheeffectsofdifferentamplicationschemes. 3.4Simulations Inthissection,wewillsimulatetheperformanceoftheAsAPsystemwithunderwateracousticchannels.Throughcomparisonsamongdifferentscenarios,thebenetsofrelaycommunicationsandprecodingareveried.Inaddition,theeffectsofnumberofrelays,channeltapsandamplicationfactorsarealsorevealed. 3.4.1Simulationsetup ConsideranAsAPsystemwithS-DdistanceofD=1000mandmid-pointrelay,assuggestedbytheoptimumrelaylocationinSection 2.3.3 .Inallsimulations,binaryphaseshiftkeying(BPSK)signalingisusedandthebiterrorrate(BER)isrecordedtodemonstratetheperformance.TheOFDMprecodingsizeischosentobeK=8.TheOFDMsymbolsizeisN=1024.Tofacilitatethesimulation,weusetheUACchannelmodeldescribedinSection 3.1 .Thereectionlossissettobeaconstantforallpaths)]TJ /F8 7.97 Tf 6.77 -1.79 Td[(l=1=p 2asin[ 49 ]. 3.4.2Resultsandcomparisons 3.4.2.1Benetsofrelayingandprecoding Comparedtothetraditionaldirect-linkcommunication,ourAsAPprotocoladoptsrelaystoassistthecommunicationsandchoosesOFDMprecodingtocollectmultipathdiversity.Thuswerstverifythebenetsofrelaying,andthediversitygainoftheOFDMprecoding.Forawaterdepthof30m,whichgeneratesallchannelswith2non-zerotaps,wesimulatetheBERperformanceofthedirect-linksystemandourAsAPsystemwithandwithoutprecoding.TheresultsareshowninFigure 3-4 .Thecurvesmarkedwithcirclesrepresenttheperformanceofthedirect-linksystems,whilethecurves 41

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markedwithdiamondsshowtheAsAPsystemswithonerelay.Bycomparingthedirect-linksystemandtherelaysystemwithoutprecoding,wenoticethattherelaysystemachievesmuchbetterperformancewithabout8dBcodinggain,thoughbothofthemhavethesamediversitygainof1.Byaddingtheprecoding,thedirect-linksystemachievesadiversitygainof2,whichequalstothenumberofchanneltaps.ForAsAPsystem,theprecodingcollectsadiversitygainof3,whichislessthantheMCDof4.ThecomparisonshowsthattheAsAPsystemexcelsthedirect-linksystemwithbothhigherdiversitygainandlargercodinggain. 3.4.2.2Effectofnumberofchanneltaps ThediscussionsofProposition1indicatethat,forasingle-relaysystemwithoutdirectlink,theMCDincreaseslinearlywiththenumberoftapsoftheS-Rchannel.Sincethenumberofchanneltapschangeswiththewaterdepth,wesimulatetheAsAPsystemswithonerelayatdifferentwaterdepthstoverifythediversitycollection.TheperformancesareshowninFigure 3-5 .Asthewaterdepthdecreasesfrom50mto5m,thenumberofnon-zerochanneltapsincreasefrom1to5.ObservefromthegurethattheAsAPsystemsindeedcollectthediversityorderthesameasthecorrespondingnumberofnon-zerotapsoftheS-Rchannel.ThisobservationalsoprovesthattheMCDisachievableunderrayleighfadingR-Dchannel. 3.4.2.3EffectofnumberofrelaysR RecallthatthesystemMCDwillnotincreaselinearlywiththenumberofrelaysifchanneltapcollisionexists.Inmulti-relayAsAPsystems,therelayedsignalsfromdifferentrelayswillarriveatthedestinationwithrandomdelay,thusthechanneltapwilloverlapswithsomeprobability,whichdecreasestheachieveddiversitygain.InFigure 3-6 ,weplottheBERcurvesforAsAPsystemswithdifferentrelaysatawaterdepthof30m.TheresultsshowthatthediversitygainsfortheAsAPsystemswithR=1,2and3are2,3and4,respectively.Thisindicatesthatthediversitygainincreaseswiththe 42

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numberofrelay,yetnotlinearly.Theattainablediversitygainisaffectedbychanneltapcollisions. 3.4.2.4Effectofamplicationfactor Inallsimulationsabove,wehaveonlyconsideredxedamplicationscheme:TD-FA.InFigure 3-7 ,weplottheperformanceofAsAPsystemswithallfouramplicationschemes:TD-FA,TD-IA,FD-SAandFD-GAatawaterdepthof30m.Thegurerevealsthatallschemescollectsimilardiversityorder,whichconrmsourremarkofProposition1thatthediversityorderofotheramplicationschemewillnotexceedtheMCDresultforTD-FA.Theonlydifferenceoftheseschemesisthecodinggain.Interestingly,exceptTD-FA,allotherthreeschemeshaveexactlythesameperformanceandtheyexhibitabout2dBcodinggainoverTD-FAscheme.Thereasonisthatallthesethreeschemesamplifythesignalsusingtheinstantaneoussignalinformation,andbetterchannelfadingcompensationisachievedcomparedwiththeTD-FAscheme.However,sincetheinstantaneousschemesrequirehighercomputationcomplexityandmoresignalprocessingdelay,thereisatrade-offbetweensystemperformanceandcomplexity. 3.5Summary Inthischapter,wedesignedapracticalasynchronousunderwaterrelaycommunications(AsAP)systemtailoredforunderwaterenvironments.Thenewprotocolresolvestimesynchronizationdifcultyandfrequency-selectivityoftheUACchannel.Inthemeantime,itexploitsthechannelsparsityfeatureandfull-duplextransmissions.ThesystemPEPperformancewasevaluatedandthemaximumcollectablediversitywasproved.Finally,simulationswereconductedtoverifythemeritsoftheAsAPprotocolandeffectsofseveralfactors.ResultsshowthatbothdiversitygainandcodinggainareachievedfromOFDMprecodingandrelaying.Inaddition,thecollectablediversityincreaseslinearlywiththenumberofchanneltapsbutnotlinearlywiththenumberofrelays.Theamplicationfactoraffectsonlycodinggainnotdiversitygain. 43

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Figure3-1. ThesystemtopologyfortheAsAPprotocol. Figure3-2. AnexampleofunderwateracousticchannelestimatedfromRACE08seatrialatatransmissiondistanceofD=1000m. 44

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Figure3-3. TheoptimalGLCPgrouping.ThemthelementfromeachoftheKblocksisselectedtoformgroupm. Figure3-4. TheBERperformanceforthedirect-linksystemsandtheAsAPsystemswithandwithoutprecoding.Thewaterdepthis30m. 45

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Figure3-5. TheBERperformancefortheAsAPsystemswithdifferentnumberofchanneltaps.Considerasinglerelayandnodirect-link. Figure3-6. TheBERperformancefortheAsAPsystemswithdifferentnumberofrelaysandwithnodirectlink.Thewaterdepthis30m. 46

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Figure3-7. TheBERperformancefortheAsAPsystemswithdifferentamplicationschemes.Thewaterdepthis30m,R=1. 47

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CHAPTER4DECOMPOSEDFOUNTAINCODES Ratelessfountaincodes[ 62 ]havebeenproposedasenergyefcienterasurecodeswithreducedcomputationcostandimprovedcoderateadaptability.Thesecodesarewellsuitedforsingle-levelcommunicationandstoragesystemswithunknownerasurerates[ 40 ].Intheunderwaterliterature,fountaincodesandothererasurecodesarealsoshowntobebenecialinbroadcastsystemsandgeneralmultihoptransmissionwithenhanceddatatransportreliability,[ 29 50 ].However,formulti-levelsystems,thenaiveimplementationoffountaincodeswillinducehighcomputationcomplexityorlargelatency[ 9 53 63 64 ].Thusdecomposedfountaincodes(DFCs)havebeenproposedintheliterature.Typically,theDFCsconsistoftwolayersofrandomdataencodingbutasinglelayerofdecoding.Inparticular,analysisin[ 66 ]showsthattheasymptoticperformanceofDFCwithtwo-layerrandomencodingisthesameasthatofthecorrespondingnon-decomposedfountaincodes.TherstDFCisthesotermeddistributedLTcode[ 67 ],whichisessentiallyaspecialDLTcodewiththesecond-layerencodingdegreexedto2or4.However,thexed-degreeencodingatthesecondlayerencodinglimitstheapplicationofthedistributedLTcodesingeneralmulti-layersystems.Therefore,generalDFCswithtwolayersofrandomencodingneedtobedeveloped. Inthischapter,wewillinvestigatebothdecomposedLTcodes(DLT)anddecomposedRaptorcodes(DRC).First,thegeneralLTcodedecompositionproblemisformulated.Then,practicalapproximatedecompositionmethodsaredevelopedfortheclassicalrobustSolitondistribution(RSD)[ 68 ].Finally,anefcientdecompositionisdesignedforDRC. Notation:denotesanemptyset;f0(x)istherst-orderderivativeofthefunctionf(x). 4.1Background Fountaincodesareratelesserasurecodes[ 62 ].Comparedtotraditionalerasurecodes,fountaincodeshavemanyadvantages.First,theencodingisperformedonthe 48

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yandcanpotentiallygenerateunlimitednumberofcodedpackets.Thusthecoderatecanbeadaptedwithoutchannelerasureinformation.Secondly,therawpacketscanberecoveredwithhighprobabilityforanysetofcodedpacketswithsmallredundancy.Thisindicatesthatthecodesarerobusttoanyerasurepatterns.Finally,thecodeshaveasymptoticallyefcientencodinganddecodingalgorithms.Therefore,fountaincodeshavepotentialapplicationsinmanyareas,suchasreliablebroadcast,datastorage,imagecompression,etc.[ 40 ]. 4.1.1LTCodes LTcodes[ 68 ]aretherstpracticalrealizationoffountaincodes.ThebasicfeaturesofLTcodesaresummarizedasfollows. TheencodingofanLTpacketconsistsoftwosteps.Consideratotalofkinputpackets, 1. theencoderrstrandomlychoosesanintegerd2[1,k]asthedegree(numberofinputpackets)ofthecodedpacketaccordingtoadegreedistributionprobability;and 2. dinputpacketsareindependentlyandrandomlyselectedfromthebatchofkpackets.ThesedpacketsareXORedtogethertogenerateoneLTcodedpacket. Torecovertheoriginalpackets,theLTdecodingadoptsthebeliefpropagation(BP)technique.Withtheencodingdegreeandpacketindexinformationofeachcodedpacket,abipartitegraphisformed.Thedecoderstartsbyreleasingpacketswithdegreeone.Thenalledgesconnectedtothedegree-onepacket(s)areremoved.Thisisdonerecursivelyuntilnodegree-onepacketisleft.Ifallkinputpacketsarerecovered,thenthedecodingissuccessful,otherwise,afailureisreported. ForanLTcodetoachievehighdecodingsuccessprobability,thekeyistodesignagoodencodingdegreedistribution.Asprovenin[ 68 ],theidealSolitondistributionguaranteesastatisticallyconstantdecodingrateof1packetpereachBPiteration.TheidealSolitondistributionis(x)=Pki=1ixiwithirepresentingtheprobabilityof 49

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choosingdegreed=i,and(x)=x k+kXi=2xi i(i)]TJ /F3 11.955 Tf 11.95 0 Td[(1).Intuitively,thisdistributionisvulnerabletopracticaldecodingvariations,sinceaniterationofzeropacketreleasewillceasetheBPprocess.ThustherobustSolitondistribution(RSD)isdesignedin[ 68 ]byincreasingtheaveragedecodingratetoR=cp kln(k=),whereistheallowablefailureprobabilityandcisadesignparameter.Furthermore,aprobabilityspikeisaddedtothedegreed=k=RtoensurefullcoveragewhenRinputpacketsremainunrecovered.Withthisrationale,theRSDisobtainedbyaddinganextradistribution(x)=k=R)]TJ /F10 7.97 Tf 6.58 0 Td[(1Xi=1R ikxi+Rln(R=) kxk=R,totheidealSolitondistribution. Denition1(RobustSolitonDistribution). Withtwoparameters2[0,1]andc0,theRSDisgiveninpolynomialform(x): (x)=(x)+(x) (4) where=(1)+(1)isanormalizingconstant. OneexampleoftheRSDisshowninFigure 4-5 withc=0.08and=0.05.Noticethatthedegreedistributionisafastdecayingfunctionofdwithmostdistributionconcentratingonthelowdegreeorders,exceptforaspikeatdegreed=k=R.Asprovedin[ 68 ],withasetofk+O(p kln2(k=))outputpacketsencodedwiththeRSD,theBPdecodercansuccessfullyrecoverallkinputpacketswithprobabilityofatleast1)]TJ /F6 11.955 Tf 11.96 0 Td[(. 4.1.2RaptorCodes Raptorcodes[ 69 ]arethelatestfountaincodesthatachievelinearencodinganddecodingcomplexity.ItisaconcatenationofonetraditionalerasurecodewiththeLTcodes. 50

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Raptorencodingconsistsoftwophases:traditionalerasureprecodingandLTencoding.Foratotalofkpackets,theRaptorencoderwillrstencodethemwitharate-rerasurecodetogeneraten=k=rprecodedpackets.Then,considerthesenpacketsasinputs,therandomencoderperformstheLTencodingwithaRaptorDDP(x),where(x)=PDi=1ixiwithDbeingthemaximumdegreeorder. TheouterprecodingwillrelaxtherequirementontheBPdecodingoftheinnerrandomLTcodes.Extendedfromtheperformanceforrandomerasurecodesin[ 61 ],therequirementontheDDP(x)forasymptoticallygoodperformancecanbesummarizedasfollows.Ifthedesigned(x)satisesthefollowinginequality, 0(x)>)]TJ /F3 11.955 Tf 11.29 0 Td[(ln(1)]TJ /F5 11.955 Tf 11.96 0 Td[(x)=(1+),x2[0,1)]TJ /F6 11.955 Tf 11.95 0 Td[(], (4) thentheprobabilitythattheBPdecodercannotrecovernormoreoftheinputnpacketswithany(1+)n+1outputpacketsisupperboundedbye)]TJ /F8 7.97 Tf 6.59 0 Td[(cnforsomepositiverealnumberc.Thustheparametersandwillindicatethequalityof(x). In[ 69 ],aDDPwithgoodasymptoticperformanceisdesigned: (x)=1 1+ x+DXi=2xi (i)]TJ /F3 11.955 Tf 11.96 0 Td[(1)i+xD+1 D!, (4) where=(=2)+(=2)2,D=d4(1+)=eandisasystemparameter.Forpresentationclarity,wetermthe(x)inEquation( 4 )asRaptordistribution,whichisdifferentfromtheRSDdistribution[ 68 ].Asprovenin[ 69 ],theinnerLTcodeswiththisdistributioncanachieve=(=4)=(1+). Inthefollowing,wewillstarttoinvestigatethedecompositionoffountaincodes. 4.2DecomposedLTCodes DifferentfromtheprimitiveLTcodes,theDLTcodesgenerateapacketwithtwolayersofrandomencoding.Inaddition,theoutputcodedpacketsstillpreservethefeaturesofLTcodestofacilitatesimpleBPdecoding.ThekeyoftheDLTcodesistodesignappropriateencodingdegreedistributions.Inthissection,wewillrstformulate 51

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theproblem,andthendevelopageneraldecompositiontechniqueforobtainingencodingdegreedistributions. 4.2.1ProblemFormulation Foratotalofkinputpackets,theencodingprocessoftheDLTcodes,asshowninFigure 4-1 ,canbedescribedasfollows. 1. Intherstlayer,thekinputpacketsarerstrandomlyencodedinthesamemannerastheLTcodesinSection 4.1.1 ,butwithadifferentdegreedistributionpolynomial(DDP)(x),theoutputpacketsaretermedasDLT-1packets; 2. Then,theDLT-1packetsareconsideredastheinputtothesecondlayerrandomencoderwithanotherDDP!(x).ThenaloutputpacketsarecalledDLT-2packets. TheDLTdecoderutilizesthesameBPalgorithmastheLTdecoder.ThusthekeyissueofDLTistondtwoappropriateencodingDDPs(x)and!(x)withgoodBPdecodingperformance.Mathematically,theresultantDDPofconcatenatedrandomcodeswith(x)and!(x)canbecomputedas^(x)=!((x)).SincetheLTcodeDDP(x)hasbeenuniquelydesignedforgoodBPdecoding,anintuitiveapproachforobtaining(x)and!(x)istodecompose(x)intotwovalidpolynomials.Thedistributiondecompositionproblemcanbeformulatedasfollows. ProblemStatement1. ForanLTcodedegreedistribution(x)withsizek,determinetwoDDPs(x)and!(x)withmaximumdegreesDandD!,whereD!D=k,suchthat, !((x))=(x), (4) underconstraints:(1)=1,0,=[1,2,,D],and!(1)=1,!0,!=[!1,!2,,!D!]. Generalpolynomialdecompositionisverychallenging.Intheliterature,existingresearcheshaverevealedthat,forauni-variablepolynomialf(x),analyticaldecompositionsolutionsdonotnecessarilyexistforarbitrarydegreeorders[ 70 ],whilenumericaldecompositionalgorithmscannotguaranteeperfectmatchforhighordercoefcients 52

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[ 71 72 ].Inaddition,tothebestofourknowledge,noneofexistingmethodscanguaranteenonnegativedecompositionsolutions.Furtherexplorationrevealsthefollowingobservation. Lemma1. ForatargetRSD,theexactsolutionsforProblem 1 onlyexistfortrivialcases:D=1orD!=1. Proof. ByexpandingEquation( 4 )foreachcoefcienti,wenoticethatifD>1andD!>1,thelastD!equalityonlyconsistsofD)]TJ /F10 7.97 Tf 6.59 0 Td[(1,Dand!D!,i.e., k)]TJ /F8 7.97 Tf 6.59 0 Td[(D!+1=!D!DD!D)]TJ /F10 7.97 Tf 6.58 0 Td[(1 (4) ...k)]TJ /F10 7.97 Tf 6.59 0 Td[(2=!D!D!2D!)]TJ /F10 7.97 Tf 6.59 0 Td[(2D2D)]TJ /F10 7.97 Tf 6.59 0 Td[(1k)]TJ /F10 7.97 Tf 6.59 0 Td[(1=!D!D!1D!)]TJ /F10 7.97 Tf 6.59 0 Td[(1DD)]TJ /F10 7.97 Tf 6.59 0 Td[(1k=!D!D!D. Thelastthreeequationsproducetwoequalities:D)]TJ /F26 5.978 Tf 5.75 0 Td[(1 D=k)]TJ /F26 5.978 Tf 5.76 0 Td[(1 k1 D!andD)]TJ /F26 5.978 Tf 5.76 0 Td[(1 D=k)]TJ /F26 5.978 Tf 5.76 0 Td[(2 k)]TJ /F26 5.978 Tf 5.76 0 Td[(12 D!)]TJ /F10 7.97 Tf 6.59 0 Td[(1.Tohavebothofthemhold,thefollowingequalitymustbesatised:D!)]TJ /F10 7.97 Tf 6.59 0 Td[(1 2D!=k2)]TJ /F10 7.97 Tf 6.59 0 Td[(3k+2 k2)]TJ /F10 7.97 Tf 6.59 0 Td[(3k.However,thefactindicatesthatD!)]TJ /F10 7.97 Tf 6.59 0 Td[(1 2D!<11andD!>1. ForD=1orD!=1,wecanobtainacorrespondingexacttrivialsolution.WhenD!=1,thesolutionis(x)=(x)and!(x)=x,whichcorrespondstoonlyrst-layerLTencoding;whileforD=1,theexactsolutionis!(x)=(x)and(x)=x,whichonlyrequiressecond-layerLTencoding. Sincetrivialsolutionsdonotprovidethebenetsofdual-layerreliabilityingeneralDLTcodes,weneedtoexploreapproximateapproachestotackletheproblem.Byexpandingthepolynomialcoefcients,Equation( 4 )canbewritteninamatrixformas: !=, (4) 53

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where=[1,2,,k],andisakD!matrix, =26666666666666666641000221003212310...............D!1...............0000D!D3777777777777777775. (4) Thisequationsethasnonlinearterms,whichrenderthedirectsolutionmathematicallyintractable.Noticethatthenonlinearityonlycomesfrom.Ifonecanrstdetermineanappropriatetentativesolutionof,then!canbesolvedfromalinearequationset.AsobservedfromEquation( 4 )andFigure 4-5 ,thecoefcientsof(x)aredecayingintermsoftheencodingdegree(i2).Thusitisreasonabletomatchthedominantlowerordertermsinsteadofthehigherorderones,whichgivesrisetothefollowingproblemstatementforapproximatenonnegativedistributiondecomposition. ProblemStatement2. [ApproximateDecomposition]ForagivenLTDDP(x)withmaximumdegreek,determineapositive-coefcientpolynomial(x)withmaximumdegreeDand(1)=1,suchthatthefollowinglinearequationsethasanonnegativesolution!0, ~!=~, (4) 54

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where~=[1,2,,D!]and~isaD!D!lower-triangulartruncatedmatrixoffull-rank, ~=2666666666641000221003212310...............D!1377777777775. (4) Therefore,toobtaintheapproximatedecomposition,weneedtorstdetermineanappropriate(x),whichguaranteesafeasiblesolutionof!(x). 4.2.2ValidChoiceof(x) Intheliterature,theanalysisonnonnegativesolutionstolinearsystemsisconductedin[ 74 ],andtherequirementsonthecoefcientmatrixareprovided.ByapplyingthisapproachtoourapproximatedecompositionprobleminEquation( 4 ),onecanobtaintheconditionsonthevalidchoiceof(x).Theanalyticalresultin[ 74 ]issummarizedinthefollowingtheorem. Theorem4.1. ForahomogeneouslinearsysteminEquation( 4 )tohaveallnonneg-ativesolutions,thecoefcientmatrixmustsatisfythefollowingsufcientandnecessaryconditions: 266666664a11a12a1na21a22a2n............an1an2ann377777775266666664x1x2...xn377777775=26666666400...0377777775 (4) 1. Eachrowofthecoefcientmatrixmusthavebothpositiveandnegativeterms,i.e.Pi6=andNi6=,wherePiandNiarethesetsofcolumnnumberforpositiveandnegativetermsinrowi,thatis,aik>0,k2Piandai,l<0,l2Ni; 55

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2. Ifn2,reducethenequationsinton)]TJ /F3 11.955 Tf 12.15 0 Td[(1asfollows.Rearrangetherstequationtohavepositivetermsandnegativetermsonbothsideoftheequation,andchangetheformationofotherequationsinthesamemanner: Xk2P1aikxk=Xl2N1ailxl,i2f1,2,,ng, andmultiplyingthepositiveandnegativesidesofthe1stequationtotheoppositesidesoftheithequation:0@Xl2N1a1lxl1AXk2P1aikxk=Xl2N1ailxl0@Xk2P1a1kxk1A. Theresultantn)]TJ /F3 11.955 Tf 10.95 0 Td[(1equationsalsoneedtosatisfytheconditionin1).Step2)continuesuntiln=1. Thetheoremstatesthateachreducedequationshouldcontainbothpositiveandnegativecoefcients.ThistheoremcanbereadilyextendedtoanonhomogeneouslinearequationsetasinEquation( 4 ),andtheresultisshowninthefollowinglemma. Lemma2. Toguaranteenonnegativesolutionsof!inEquation( 4 ),thefollowingconditionmustbesatised: f(r,j)<0,j1,r>j, (4) wheref(r,j)=f(r,j)]TJ /F3 11.955 Tf 11.95 0 Td[(1)j1)]TJ /F5 11.955 Tf 11.96 0 Td[(f(j,j)]TJ /F3 11.955 Tf 11.95 0 Td[(1)~r,jandf(r,0)=)]TJ /F6 11.955 Tf 9.29 0 Td[(r. Proof. SeeAppendix A Duetothelower-triangularfeatureofthe~matrix,wecanobtaintheexpressionof!intermsoff(r,j)asfollows. Lemma3. Thesolutionsof!toEquation( 4 )canberepresentedinthefollowingform: !j=)]TJ /F5 11.955 Tf 9.3 0 Td[(f(j,j)]TJ /F3 11.955 Tf 11.96 0 Td[(1))]TJ /F8 7.97 Tf 6.58 0 Td[(j(j+1)=21. (4) 56

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Proof. IntheproofofLemma 2 ,thejtheliminationstepresultsinasetofequationsinEquation( A ).Therstequationr=j+1canbeexpressedas: )]TJ /F5 11.955 Tf 9.96 0 Td[(f(j,j)]TJ /F3 11.955 Tf 11.95 0 Td[(1)~j+1,j+1!j+1=)]TJ /F5 11.955 Tf 7.3 0 Td[(f(j+1,j)!j, (4) whichimpliesthat!j+1=!j=f(j+1,j)=f(j,j)]TJ /F3 11.955 Tf 12.29 0 Td[(1))]TJ /F10 7.97 Tf 6.59 0 Td[((j+1)1.Consequently,wecanobtaintheexpressionfor!jinthelemma. Thesolutionof!needstobewithintherangeof[0,1).ThusbycombiningtherequirementsinLemmas 2 and 3 ,wecannallyobtaintherulesforvalidchoicesof(x)inthefollowing. Proposition2. ToguaranteethatEquation( 4 )hasvalidsolutions,thecoefcientsof(x)mustobeythefollowingrules: Forj=1,1needstosatisfy 31)]TJ /F6 11.955 Tf 9.97 0 Td[(21+1>0; (4) Forj=2,2shouldbechosenintherange[min,2,max,2],where min,2=max(12)]TJ /F6 11.955 Tf 9.96 0 Td[(21 1,12)]TJ /F13 11.955 Tf 9.96 10.39 Td[(p 22)]TJ /F3 11.955 Tf 9.96 0 Td[(21(3)]TJ /F6 11.955 Tf 9.96 0 Td[(1=1)]TJ /F6 11.955 Tf 9.96 0 Td[(31) 21), (4) max,2=min(21 1,12)]TJ /F13 11.955 Tf 9.96 10.4 Td[(p 22)]TJ /F3 11.955 Tf 9.96 0 Td[(213 21); (4) Forj>2,jmustliewithintherange[min,j,max,j],where min,j=max8<:g(j,j)]TJ /F3 11.955 Tf 9.97 0 Td[(1))]TJ /F6 11.955 Tf 7.97 0 Td[(j(j+1)=21 1(j)]TJ /F10 7.97 Tf 5.18 0 Td[(2)(j+1)=21,(j)]TJ /F10 7.97 Tf 5.18 0 Td[(1)(j+2)=21[1+j+21])]TJ /F5 11.955 Tf 9.96 0 Td[(h(j,j)]TJ /F3 11.955 Tf 9.96 0 Td[(1) h(2f(2,1)+j12)(j2+j)]TJ /F10 7.97 Tf 5.17 0 Td[(4)=21i9=;, (4) max,j=min8<:g(j,j)]TJ /F3 11.955 Tf 9.96 0 Td[(1) 1(j)]TJ /F10 7.97 Tf 5.18 0 Td[(2)(j+1)=21,)]TJ /F5 11.955 Tf 7.31 0 Td[(h(j,j)]TJ /F3 11.955 Tf 9.96 0 Td[(1) h(2f(2,1)+j12)(j2+j)]TJ /F10 7.97 Tf 5.18 0 Td[(4)=21i9=;. (4) 57

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Inthesetwolimits, g(j,j)]TJ /F3 11.955 Tf 9.97 0 Td[(1)=jj(j)]TJ /F10 7.97 Tf 5.17 0 Td[(1)=21+j)]TJ /F10 7.97 Tf 5.17 0 Td[(2Xi=1f(j)]TJ /F5 11.955 Tf 9.96 0 Td[(i,j)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F5 11.955 Tf 9.96 0 Td[(i)~j,j)]TJ /F8 7.97 Tf 5.17 0 Td[(i(i)]TJ /F10 7.97 Tf 6.59 0 Td[(1)(2j)]TJ /F8 7.97 Tf 6.58 0 Td[(i)=21, (4) h(j,j)]TJ /F3 11.955 Tf 7.97 0 Td[(1)=(j)]TJ /F10 7.97 Tf 5.17 0 Td[(1)(j+2)=21[j+11)]TJ /F5 11.955 Tf 9.96 0 Td[(jj2]+j)]TJ /F10 7.97 Tf 5.18 0 Td[(2Xi=1h~j+1,j)]TJ /F8 7.97 Tf 5.17 0 Td[(i1)]TJ /F5 11.955 Tf 9.96 0 Td[(j~j,j)]TJ /F8 7.97 Tf 5.17 0 Td[(i2if(j)]TJ /F5 11.955 Tf 7.97 0 Td[(i,j)]TJ /F5 11.955 Tf 7.97 0 Td[(i)]TJ /F3 11.955 Tf 7.98 0 Td[(1)i(j)]TJ /F10 7.97 Tf 5.18 0 Td[((i)]TJ /F10 7.97 Tf 5.18 0 Td[(1)=2))]TJ /F10 7.97 Tf 5.18 0 Td[(11. (4) Proof. SeeAppendix B 4.2.3DecompositionAlgorithm AccordingtoProposition 2 ,wecanchooseasetofvalidvalues,suchthatthe!computedfromEquation( 4 )satises!(1)=1.ThedecompositionalgorithmforProblemStatement 2 canbesummarizedasfollows. Algorithm1:Degreedistributiondecomposition Input: Thetargetdegreedistribution(x) Result: ThedecomposedDDPs(x)and!(x) Initialization:Setsomeinitialvaluefor2f0,1g; while<1do forj=1toDdo ifj=1thenChooseavaluefor1thatsatisesEquation( 4 );elseComputethevalidrange[min,j,max,j]forjaccordingtoProposition 2 ;Calculatej=min,j+(1)]TJ /F6 11.955 Tf 11.96 0 Td[()max,j;endCompute!fromEquation( 4 )anddeterminethetotalprobability!(1);if!(1)=1thenoutputthecoefcientsof(x)and!(x),andbreak;elseIncrease=+;end 4.2.4RSDDecomposition WithAlgorithm 1 ,onecanpotentiallyndthedecompositionforanyLTdistributiontoconstructaDLTcode.AsintroducedinSection 4.1.1 ,RSDisapracticalandrobustLTdistribution.RecallfromFigure 4-5 thattheRSDhasaspikeatdegreed=k=R.AccordingtoEquations( 4 )and( 4 ),theabruptincreaseofdwillinducelargeg(d,d)]TJ /F3 11.955 Tf 12.79 0 Td[(1)andnegativeh(d,d)]TJ /F3 11.955 Tf 12.79 0 Td[(1),whichmayresultinmax,d
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emptyrangeford.Therefore,directapplicationofAlgorithm 1 withoutspecialtreatmentofthespikemayfailtoestablishavaliddecompositionfor(x).Anintuitiveideaistoremovethisspikefrom(x)anddecomposethesmoothportionofthedistributionwithAlgorithm 1 .ThespikedistributionisthenassignedonlytotherstDLTencoderwithoutsecond-layerencoding.Basedonthisidea,wehavetheseparatedecomposition(SD)schemetailoredfortheRSD. Algorithm2:SeparateDecomposition Input: ThetargetRSD(x) Result: ThedecomposedDDPs(x)and!(x) Initialization:Constructasmoothdistribution~(x)=((x)+~(x))=with~(x)=Pk=Ri=1(R=ik)xi; (1)Decomposition~(x)into!(x)and~(x)withAlgorithm 1 ; (2)Compute(x)=0xk=R+(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)^(x),where^(x)=~(x)=(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0),and0=1)]TJ /F3 11.955 Tf 10.69 2.66 Td[(~(1). TheDLTcodesobtainedfromtheSDalgorithmaretermedasSD-DLTcodes.DuetothemixeddistributionsoftheSD-DLTcodes,theencodingschemeoftheSD-DLTcodesneedstobeslightlymodied,asshowninFigures 4-2 and 4-3 .Attherstencoder,withprobability0,theencodergeneratesacodedpacketofdegreed=k=R,andwithprobability1)]TJ /F6 11.955 Tf 12.18 0 Td[(0,theencoderwillencodeaDLT-1packetaccordingto~(x).Aone-bitIDisattachedtothecodedpackettoindicatewhichprobabilityisusedforthispacket.Atthesecondencoder,theID=0packetsaredirectlyoutputasDLT-2packets,whileallID=1packetsarefurtherencodedtogenerateDLT-2packets. WiththisSD-DLTencodingscheme,theresultantdegreedistributionoftheoutputpacketsandtheaverageencodingdegreeofthebothencoderscanbeobtainedasfollows. Proposition3. ForanSD-DLTcodewitharst-layerencodingDDP(x)=0xk=R+(1)]TJ /F6 11.955 Tf -453.91 -23.91 Td[(0)^(x)andasecond-layerDDP!(x),theresultantoutputdegreedistribution^(x)is 59

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computedas: ^(x)=!((1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)^(x))+(1)]TJ /F6 11.955 Tf 9.96 0 Td[(!(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0))xk=R. (4) Theaverageencodingdegreesoftherstencoder(C1)andsecondencoder(C2)are: C1=0k=R+(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)^0(1) (4) C2=1)]TJ /F6 11.955 Tf 9.96 0 Td[(!(1)]TJ /F6 11.955 Tf 11.96 0 Td[(0)+(1)]TJ /F6 11.955 Tf 11.95 0 Td[(0)!0(1)]TJ /F6 11.955 Tf 11.95 0 Td[(0). Proof. Withtwolayersofrandomencoding,theoutputdegreedistributionis^(x)=!((x)),whichisexpandedas: ^(x)=D!Xi=0!i0xk=R+(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)^(x)i. (4) Atthesecondencoder,anyselectionofID=1packetswilldirectlyresultinaDLT-2packetofdegreed=k=R.Thus, ^(x)=D!Xi=0!i"(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)i(^(x))i+iXm=1imm0(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)i)]TJ /F8 7.97 Tf 5.17 0 Td[(mxk=R#. (4) Aftersomemanipulations,^(x)canbereexpressedinthefollowingequation. ^(x)=D!Xi=0!i(1)]TJ /F6 11.955 Tf 9.97 0 Td[(0)i(^(x))i+(1)]TJ /F8 7.97 Tf 12.75 14.94 Td[(D!Xi=0!i(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)i)xk=R, (4) whichresultsinEquation( 4 ). Byevaluating0(1),wecanreadilyobtainC1.FromEquation( 4 ),wecancomputetheaverageencodingcostforthesecondencoderas: C2=D!Xi=0i!i(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)i+(1)]TJ /F8 7.97 Tf 12.76 14.95 Td[(D!Xi=0!i(1)]TJ /F6 11.955 Tf 9.96 0 Td[(0)i), (4) whichreadilyresultsinEquation( 4 ). Insummary,theSD-DLTcodeshavesimilarperformancewiththeLTcodes,andfacilitatereducedcomputationcostatbothlayers.ThustheSD-DLTcodescanbe 60

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appliedtocooperativerelaycommunicationstoachievebetterenergyefciencythantheconcatenatedfountaincodesbasedschemes.However,nodeenergyistypicallynonuniformincooperativenetworks.Inordertoprolongthenetworklifetime,theenergyconsumptionneedstobeadaptivetotheresidualenergyofeachnode.Duetothexedcomputationallocationbetweentwoencoders,theSD-DLTcodesarenotsuitableinthisperspective.Therefore,wewillnextdevelophybridDLT(h-DLT)codeswithexiblecomputationallocationtailoredforcooperativerelaycommunications. 4.2.5H-DLTCodes Recallthat,withtheSD-DLTcodesobtainedinSection 4.2.4 ,theoutputpacketsaregeneratedwithtwodifferentmethods:single-layerencodingforthespikedistributionofdegreed=k=Randtwo-layerencodingfortherestofthedistribution.Suchanencodingschemewillshiftsomeencodingburdenfromthesecondencodertotherstone,becausethepacketswithsingle-layerencodingdonotinduceanyadditionalcomputationatthesecondencoder.Motivatedbythisobservation,wedesigntheh-DLTcodeswithmoreexiblecomputationcostallocation. Intheh-DLTcodes,thedataencodingisconductedinhybridmodes:two-layercooperativeDLTmodeandone-layerdirectLTmode.InthecooperativeDLTmode,eachpacketwillbeencodedbybothencodersastheDLTcodes;whileinthedirectLTmode,thepacketsaregeneratedonlybytherstencoder.Byadjustingthemoderatio,theh-DLTcodescancontroltheencodingcostallocationbetweentherstandsecondencoders. 4.2.5.1Encodingscheme Withhybridencoding,twodegreedistributions1(x),2(x)andanencodingratioareassociatedwiththerstencodertogeneratebothDLT-1andLTpackets.Thesecondencoderwilldeterminetheencodingmodebasedonthetypesofselectedpackets. 1. Attherstencoder,abinaryrandomnumbergeneratorisadoptedtoselectanencodingmode,asshowninFigure 4-4 .Withprobability,theencoderwill 61

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choosethecooperativeDLTmodeandgenerateaDLT-1packetwiththeencodingDDP1(x);withprobability1)]TJ /F6 11.955 Tf 12.4 0 Td[(,theencoderwilloperateinthedirectLTmode,andanLTpacketisencodedwiththeDDP2(x).Allcodedpacketsaretheinputstothesecondencoder. 2. Atthesecondencoder,anencodingdegreedisrstrandomlychosenwithdistribution!(x).Thendpacketsarechosenfromtheinputs.IfallselectedpacketsarelabeledasDLT-1,theyareXORedtogethertogenerateanh-DLTpacket;otherwise,oneLTpacketisoutputasanh-DLTpacket,asshowninFigure 4-3 SimilartotheSD-DLTcodes,wecancomputetheaverageencodingdegreesatbothencodersoftheh-DLTcodesas: C1=0(1)=01(1)+(1)]TJ /F6 11.955 Tf 9.96 0 Td[()02(1) (4) C2=1)]TJ /F6 11.955 Tf 9.96 0 Td[(!()+!0(). Andtheresultantdegreedistributionoftheoutputh-DLTpacketscanbeobtained, ^(x)=!(1(x))+(1)]TJ /F6 11.955 Tf 9.96 0 Td[(!())2(x). (4) DenotetheresultantDDPsofthecooperativeDLTmodeandthedirectLTmodeas1(x)and2(x).FromEquation( 4 ),weknowthat1(x)=!(1(x))and2(x)=(1)]TJ /F6 11.955 Tf 10.23 0 Td[(!())2(x).DenemoderatioastheportionoftotaldistributionassignedtothecooperativeDLTencoding,thus=1(1)=^(1)=!(). 4.2.5.2Hybriddistributiondecomposition InordertofacilitateasinglelayerBPdecodingontheh-DLTpackets,wecandecomposeanLTdistribution(x)intothreedistributions:1(x),2(x)and!(x).NoticethatonlytheDLTmodeDDP1(x)needstobefurtherdecomposed.Hencewecandetermineaproperdecomposable1(x)forthecooperativeDLTencodingand 62

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decomposeit,thenassigntheremainingdistributionto2(x)forthedirectLTmode.Thisgivesrisetothehybriddegreedecompositionalgorithm. Algorithm3:HybridDecomposition Input: Thetargetdegreedistribution(x)anddesiredmoderatiod Result: ThedecomposedDDPs(x)and!(x) Initialization:Determineadecomposablepolynomial1(x)<(x)with1(1)=(1)=d; (1)Decompose1(x)into~1(x)and!(x)usingAlgorithm 1 ; (2)Compute=~1(1),and1(x)=~1(x)=; (3)Calculate2(x)=((x))]TJ /F6 11.955 Tf 11.96 0 Td[(!(~1(x)))=(1)]TJ /F6 11.955 Tf 11.96 0 Td[(!()); (4)Determine(x)=1(x)+(1)]TJ /F6 11.955 Tf 11.95 0 Td[()2(x). Inthehybriddistributiondecompositionmethod,themostimportantstepistodetermineafeasible1(x).Variousmethodscanbedesignedtoobtain1(x)accordingtothetarget(x).Inthefollowing,wewillproposeauniqueschemefortheRSD. 4.2.5.3HybridRSDdecomposition SimilartotheSDalgorithm,thehybridRSDdecompositionrstconstructsasmoothdistribution~(x)byremovingthespikeoftheRSD.Thenafractionof~(x)isallocatedtotheDLTmode1(x)fordecomposition.Byadjustingthefraction,wecanobtainahybridRSDdecompositionthatsatisesthetargetmoderatio.ThedecompositiontechniqueisdescribedinAlgorithm 4 NoticefromthehybriddistributiondecompositioninAlgorithm 3 ,allnon-decomposeddistributionsareallocatedtothedirectLTmodeasshowninstep(3).Therefore,accordingtoEquation( 4 ),theresultantdistributionoftheh-DLTcodesobtainedwithAlgorithm 3 isidenticaltothetargetLTdistribution,i.e.^(x)=(x).Consequently,theresultantdistributionobtainedwithAlgorithm 4 isthesameasthetargetRSD.Recallthat,inAlgorithm 2 ,theRSDisdecomposedwithanapproximatemethod,thussomedistributiondifferenceappearsathighdegreeorders.Hence,withthesametargetRSD,thedecomposedh-DLTcodeisexpectedtoperformbetterthantheSD-DLTcode. 63

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Algorithm4:HybridRSDDecomposition Input: ThetargetdegreeRSDdistribution(x)anddesiredmoderatiod Result: ThedecomposedDDPs(x)and!(x) Initialization: Constructasmoothdistribution~(x)=((x)+~(x))=with~(x)=Pk=Ri=1(R=ik)xi; Chooseatentativeratio~; repeat Compute1(x)=~~(x);Determinethehybriddecomposition1(x),2(x),!(x)andusingAlgorithm 3 ;Calculatethemoderatio=!();Increase~=~+.until=d; Computethedecomposeddistribution(x)=1(x)+(1)]TJ /F6 11.955 Tf 11.96 0 Td[()2(x). 4.2.6DLTCodePerformance ToverifytheDLTcodeperformance,wesimulatethedecodingprobability,andcalculateitscomputationcost.Theresultsarecomparedwiththenon-decomposedLTcodesandthedistributedLTcodes. 4.2.6.1Degreedistributions ForatargetRSD(x)withparametersk=1000,c=0.08and=0.05,weapplytheSD-DLTcodedistributions(x)and!(x)withAlgorithm 2 ,andtheh-DLTcodeswithAlgorithm 4 fordifferentmoderatios=0.8,0.6,0.4.Theresultantdistribution^(x)iscalculatedaccordingtoEquation( 4 )andEquation( 4 ).ThedistributionofthedistributedLTcodeisalsoobtainedaccordingto[ 67 ].ToquantifythedifferencebetweentheresultantdistributionofthedecomposedLTcodesandtheRSD(x),theKullback-Leibler(KL)divergenceiscomputedwithresultsofDSD)]TJ /F8 7.97 Tf 6.59 0 Td[(DLT=0.0176bits,DDistLT=0.0011bitsandDh)]TJ /F8 7.97 Tf 6.59 0 Td[(DLT=0forallvalues,ThisindicatesthattheSD-DLTanddistributedLTcodesbothhavesomedistributionvariationwhilethehybriddecompositionalgorithmgeneratesexactlythesameresultantdistributionastheRSD.Morespecically,asshowninFigure 4-5 ,theresultantSD-DLTdistributionresemblestheRSDwithexactlythesamevaluesfortherstD!=40degrees.Thedifference 64

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showsupatthetailpart.Thusitisofinteresttoinvestigatetheeffectofthedistributiondifferenceondecodingperformance. 4.2.6.2Decodingperformance IntheLTcodes,thereceiversuccessfullyrecoversallkinputsymbolswithhighprobabilityforareceptionofk(1+)codedsymbols,whereisthedecodingoverhead.Thus,toillustratethedecodingperformance,wesimulatetheprobabilityofsuccessfuldecodingwithrespecttodifferentoverheadsfortheSD-DLT,distributedLTandprimitiveLTcodesobtainedabove.TheresultsareplottedinFigure 4-6 .ObservethatbothdecomposedLTcodesachievesimilarperformanceastheprimitiveLTcode.ThusthedifferencebetweentheSD-DLTdistributionandtheRSDinhighdegreeordersdoesnothavemuchimpactontheperformance.Thisobservationalsoveriestheproofin[ 66 ]thatthedecomposedLTcodeachievesthesameperformanceasthenon-decomposedLTcodeasymptotically. Toillustratethedecodingperformance,wesimulatetheprobabilityofsuccessfuldecodingwithrespecttodifferentoverheadsfortheobtainedh-DLTcodes.Forpresentationclarity,onlytheresultfor=0.4isincludedinFigure 4-6 .Observethattheh-DLTcodeachievessimilarperformanceasthenon-decomposedLTcodeandotherdecomposedLTcodes.Toillustratethedetailedperformancedifference,weplotthecomplementarycumulativedistributionfunctions(CDF)oftherequiredoverhead()forsuccessfuldecoding.AsseenfromFigure 4-6 ,atleast1)]TJ /F6 11.955 Tf 11.96 0 Td[(packetsaresuccessfullyrecoveredwithanoverheadlargerthan0.55.Figure 4-7 depictstheoverheadCDFfortherangelargerthan0.55.Asexpected,theh-DLTcodeperformsthebestamongalldecomposedLTcodes,becausetheresultantdegreeoftheh-DLTcodeisidenticaltotheRSD,whiletherearesomedifferencesfortheothertwodecomposedLTcodes.Inaddition,comparedtotheprimitiveLTcode,someperformancedegradationisalsoobservedfortheh-DLTcodes.ThedegradationcomesfromthedegreereductioninducedbypacketcollisionwhentheselectedDLT-1packetscontainthesameraw 65

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packetsatthesecondlayer.However,thesmalldifferenceconrmsthattheprobabilityofsucheventsisveryslim.Inaddition,theSD-DLTcodeandthedistributedLTcodehavesimilarperformancewithmoredegradationcomparedtotheLTcode,whichisduetotheirdistributiondifferencestotheRSDandthedegreereductioneffect. 4.2.6.3Computationcost ThedecomposedLTcodesaredesignedtoreducethecomputationcomplexityateachencoder.Toverifythis,wecomputetheaverageencodingdegreesforallcodesobtainedusingEquation( 4 ),whicharelistedinTable 4-1 .IntheconcatenatedLTcode,fullencodingisperformedateachlayer,thuslargeencodingcostof11.64isneededatbothlayers.ThesmallC1andC2valuesofboththeSD-DLTanddistributedLTcodesindicatethatmuchlesscomputationcostisrequiredatbothencoders.Inaddition,foragivenLTcode,thecorrespondingdecomposedSD-DLTanddistributedLTcodeshavexedcomputationcostsatbothlayers.Whiletheh-DLTcodeshaveexiblecomputationallocation.Asexpected,whenthemoderatiodecreases,moreencodingcostshiftsfromtherelaytothesource. 4.3DecomposedRaptorCodes Inthissection,thedecompositionofRaptorcodeswillbeexplored.SincethecoreofRaptorcodesistheinnerLTcodes,thedecompositionmethoddevelopedintheprecedingsectioncanbeappliedtoconstructDRC.However,underadifferentencodingsetup,moreefcientdecompositionschemecanbedesigned.Inaddition,linearprogrammingbasedapproachcanalsobedevisedforcodeconstruction. 4.3.1ProblemFormulation Insomeapplications,thesourcepacketsexhibitgroupfeatures.Forexample,inzone-basedsensornetworks[ 76 77 ],thesourcepacketsarelocatedingeographicallyseparatedzones.Thustherst-layerrandomencodingisconstrainedtobewithineachzone.Abstractedfromthistypeofsystem,thefollowinggroup-wisetwo-layerrandom 66

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encodingschemeisconsidered.Atotalofkpacketsisdividedintog=k=mgroupswithmpacketsineachgroup.TheDRCencodingconsistsofthreephases: 1. mpacketsineachgroupareencodedwithatraditionalerasurecodesofrater,whichresultsinm=rprecodedpackets; 2. Theprecodedpacketsgeneratedineachgrouparegatheredtogether,andtherstlayerLTencodingisperformedonthemwithDDP(x)=PDi=1ixi; 3. Allrst-layerLTcodedpacketsfromggroupsaregatheredtogetherandfurtherencodedamonggroupswithanotherDDP!(x)=PD!i=1!ixi. SimilartoDLTcodes,wecandecomposetheRaptorDDP(x)intotwovalidDDPs,butnontrivialsolutionsdonotexistforexactdecompositionofRaptordistributions[ 69 ]asLemma 1 .ThusweneedtoexploreapproximatesolutionsasProblem 2 ,whichisrestatedforRaptordistributionsinthefollowing. ProblemStatement3(ApproximateDecomposition). ForatargetRaptordistribution(x)withmaximumdegreeD,wewanttodetermineanapproximatedecompositionof(x)and!(x)withmaximumdegreeDandD,respectively,suchthat 1. (1)=1andi0,i2f1,2,,Dg;and 2. thefollowingequationhasnonnegativesolution!0, ~!=~, (4) where~=[1,2,,D!],~!=[!1,!2,,!D!],and~isaD!D!lower-triangularmatrixoffull-rank, ~=2666666666641000221003212310...............D!1377777777775. (4) 67

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4.3.2EfcientDecomposition ThealgorithmdevelopedinAlg. 1 canbeappliedtotheRaptorcodesdecomposition.However,therecursivemethodrendershighcomputationcomplexity.Inthefollowing,wewilldevelopamoreefcientmethodutilizingthegroupedencodingfeature. 4.3.2.1Validchoiceof(x) FromEquation( 4 ),weobservethateachtermintherstcolumnof~matrix,!1i,representstheprobabilitythatanaloutputpacketofadegree-ihasalltheisourcepacketscomingfromthesamegroup.InprimitiveRaptorcodes,thisprobabilitycanalsobecomputedfromthedistribution(x),whichisg1)]TJ /F8 7.97 Tf 6.59 0 Td[(ii.Thuswecanintentionallyenforcethesetwovaluestobethesame: !1i=g1)]TJ /F8 7.97 Tf 6.59 0 Td[(ii,i2f1,,Dg. (4) BynormalizingPDi=1i=1,wecanobtain!1andconsequentlyalli. However,furtherinvestigationrevealsthatthissolutionof(x)willpossiblyleadtonegativesolutionof!inEquation( 4 ),andthenegativevaluestartsat!3.Mathematically,!3canbecomputedas:!3=g)]TJ /F3 11.955 Tf 11.96 0 Td[(1 g2 31g+1 g3 2)]TJ /F3 11.955 Tf 11.95 0 Td[(22 1=g)]TJ /F3 11.955 Tf 11.96 0 Td[(1 g2 31g+1 g3 2)]TJ /F3 11.955 Tf 13.39 8.09 Td[(2 g2 1. UndertheconstraintinEquation( 4 ).Thevalueof!3isnegativeifthenumberofgroupsisg<222 13)]TJ /F3 11.955 Tf 12.6 0 Td[(1==3)]TJ /F3 11.955 Tf 12.6 0 Td[(1.ByexaminingtheRaptordistributioninEquation( 4 ),wenoticethatthecoefcientsfollowamonotonicdecayingfeaturesinsteadofdegree1andD+1terms.Thesmallvalueof1induceslargeratioof2=1,whichconsequentlyrendersnegative!3.Intuitively,wecanincreasetheweightof1suchthatthenonnegativityofthesolutionisguaranteed.Therefore,thepotentialsolutionof(x)ismodiedbyaddingaweightingfactorw: i=1 !11 (wg)i)]TJ /F10 7.97 Tf 6.59 0 Td[(1i,i2f2,,Dg, (4) 68

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where!1=1+PDi=21 (wg)i)]TJ /F26 5.978 Tf 5.76 0 Td[(1i.Toguaranteethenonnegativityof!3,wneedstosatisfythatwwmin=3=(g))]TJ /F3 11.955 Tf 12.33 0 Td[(1=g.Forexample,g=100and=0.015,thenwmin=1.99.Noticethatwming=3=)]TJ /F3 11.955 Tf 11.03 0 Td[(1,whichisnotdependentongorm.Thus(x)isdeterminedbyDandwonly.Inaddition,thefollowinglemmaindicatesthatthechoiceof(x)inEquation( 4 )rendersavalidsolutiontoProblem 3 Lemma4. Forsufcientlylargenumberofgroupsorwithappropriatelychosenweight-ingfactorw,thesolutionof!toEquation( 4 )isnonnegative. Proof. SeeAppendix C 4.3.2.2Decompositionsolutionof!(x) Afterobtaining(x),weneedtodeterminethesolutionof!(x).Besidesnonnegativity,the!icoefcientsalsoneedtobenormalized.BysolvingEquation( 4 ),wehavetherstD!coefcientsoftheresultantDDP^(x)=!((x))perfectlymatchthoseoftheRaptorDDP.Thefollowinglemmashowsthepropertyofthesolvedcoefcients!i. Lemma5. ThesummationoftheD!coefcients!iobtainedinEquation( 4 )islessthan1. Proof. ForiD!,^i=PD!j=1~i+1,j!j.SimilarlytoLemma 4 ,wecanprovethat,i+1)]TJ /F3 11.955 Tf 13.35 2.65 Td[(^i+1=i+1)]TJ /F8 7.97 Tf 14.74 14.94 Td[(D!Xj=1~i+1,j!j0. PutallD+1inequalitiestogether,wecanexpressedtheminmatrixform:)]TJ /F16 11.955 Tf 12.19 0 Td[(A0oralternativelyinpolynomialform:(x))]TJ /F6 11.955 Tf 11.96 0 Td[(!((x))0.Hence,(1))]TJ /F6 11.955 Tf 11.96 0 Td[(!((1))=1)]TJ /F8 7.97 Tf 14.75 14.94 Td[(D!Xi=1!i0. Thenormalizationcansimplybedonebydividingeverycoefcientbythesumofall!i.However,byexaminingthecodeperformanceinequalityinEquation( 4 ),wenoticethatabetterchoiceistointroduceahigherorderterm!D!+1toincludealltheremaining 69

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probability,insteadofmagnifyingtherstD!coefcientsof!(x).ThisstrategyisbenecialtothedecodingperformanceofobtainedDRC.RecallthatagoodDDPneedstosatisfytheconstraintinEquation( 4 )withsmall.Investigationsshowthatasxapproaches1,thelargeritermixi)]TJ /F10 7.97 Tf 6.58 0 Td[(1hasmorecontributiontothevalueof0(x).Thuslargerhigh-ordercoefcientswillenlargetherangethatsatisescondition,whichcorrespondstosmaller.KnownfromLemma 5 that^ii,iD!.Thusweneedtoboostthehighordercoefcients^itopreservethedecodingperformanceofDRC.Therefore,theadditionof!D!+1termisbenecialintwofolds.First,thepreservationofthecoefcientmatchoftherstD!termswillguaranteethattheresultant^(x)satisesEquation( 4 )forsmallandmediumx.Second,theD!+1termenlargestherangeofx,whichmaketheresultantDRCapproachtheperformanceoftheRaptordistribution(x). Withalltherationalepresentedabove,wecannowsummarizethealgorithmforRaptorcodedecompositioninAlgorithm 5 .TheobtainedDRCistermedasDRC-ANA. Algorithm5:DRCDecomposition Input: ThetargetRaptordistribution(x) Result: ThedecomposedDDPs(x)and!(x) Compute(x)as:1=1 !1,where!1=1+PDi=2i (wg)1)]TJ /F14 5.978 Tf 5.75 0 Td[(iandw>wmin;i=1 (wg)1)]TJ /F14 5.978 Tf 5.76 0 Td[(ii !1,i2f2,3,,Dg; CalculatetherstD!coefcients!ifromEquation( 4 ); Determinetheextraterm!D!+1=1)]TJ /F13 11.955 Tf 11.95 8.97 Td[(PD!i=1!i. 4.3.3FiniteLengthDRC Inpracticalsystemswithnitenumberofpackets,acorrespondingnite-lengthRaptordistributioncanbeobtainedthroughlinearprogrammingbyminimizingtheaverageencodingdegree[ 69 ].SimilartechniquescanbedevelopedtoobtainoptimalDRCdistributions.Theoptimizednite-lengthDRCcanobtainedasfollows:theDDP(x)and!1arepredeterminedasinEquation( 4 ),thentheoptimal!i,i2values 70

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arecomputedbyminimizing!0(1): minPD!i=2i!i (4) s.t.)]TJ /F13 11.955 Tf 11.29 8.96 Td[(PDi=2i[(x)]i)]TJ /F10 7.97 Tf 6.58 0 Td[(1!iln(1)]TJ /F8 7.97 Tf 6.59 0 Td[(x)]TJ /F8 7.97 Tf 6.59 0 Td[(cp 1)]TJ /F14 5.978 Tf 5.75 0 Td[(x Nk) (1+)0(x)+!1,x2[0,1)]TJ /F6 11.955 Tf 11.96 0 Td[(]PD!i=2!i=1)]TJ /F6 11.955 Tf 11.96 0 Td[(!10!i1,i2f2,,D!g. Fordiscretevaluesofx,theoptimal!(x)canbefoundnumerically.TheDRCcomputedwiththismethodisnamedDRC-LP. 4.3.4PerformanceofDRC ToillustratetheperformanceoftheproposedDRC,wewillsimulatethedecodingprobabilityanddeterminetheircomputationcost.In[ 66 ],exhaustivesearchisusedtoobtainthedecomposeddistributionswiththeminimumencodingdegree!0(1).TheDRCdeterminedbythismethodisnamedasDRC-MIN.ComparisonsareshownamongtheDRCobtainedwithdifferentmethodsandwiththeprimitiveRaptorcodes.Inaddition,affectingfactorsontheDRCperformancewillalsobeevaluated. 4.3.4.1DRCdistributions TheresultantdegreedistributionofDRC-ANAshouldbeexactlythesameastheprimitiveRaptordistributionfortherstD!degrees,whilethedifferenceexistsathigherdegreeorders.TheDRCcomputedthroughlinearprogrammingwillbeverydifferent.Torevealthedifference,weobtaintheDRCdistributionsforacodesystemwithtotalg=100groupsandmaximumrstlayerencodingdegreeDk=5withdifferentmethods.ThecoefcientsofresultantDDPs!((x))areplottedinFigure 4-8 .Forreadability,theresultantdistributionofDRC-MINisomitted.ResultsshowthatthedistributionofDRC-ANAmatcheswellwith(x)exceptabumpatlargedegreesaroundd=20.Thisisinducedbytheextradegreetermatd=D!+1=18.TheresultantdistributionofDRC-LPfollowssimilartrendas(x)withthepeakdistributionatdegreed=2,butithasmoredistributioninthemiddlerange. 71

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4.3.4.2Decodingperformance TorevealtheperformanceofdifferentDRC,wesimulatetheirrecoveryratioinasystemwithg=100groupsand20packetspergroup.Inaddition,theperformanceoftheprimitiveRaptorcodesforatotal2000packetsissimulated.TheaveragedecodingratiocurvesaredisplayedinFigure 4-9 .ThegureshowsthatDRC-ANAhasverysimilarperformanceastheprimitiveRaptorcodes,whileDRC-LPhassomeperformancedegradation,about5%moreoverhead.However,DRC-MINdoesnotperformwellforthissystemsetup,only70%recoverywith50%overhead.ExtendedsimulationshowsthatDRC-MINneedsabout200%overheadtoachievefullrecovery.Thereasonsforthisobservationareasfollows.ForDRC-MIN,inordertominimize!0(1),moreencodingweightsareshiftedtotherstlayer(x).Thustherstlayerhasheavierencoding,whichresultsinsmall1.Forasystemwithsmallgroupsize,inordertohaveenoughdegree-1packettofacilitatedecoding,alargeoverheadisrequired.Otherwise,thedecodingofthepacketsfromthisgroupwillincurarecoveryfailure.Onthecontrary,ourproposedmethodobtains(x)byintentionallyassigningmoreprobabilitytothedegree-1packet.Thusthedecodingsuccessfulrateishigh. Forthesimulatedsystem,thesecondlayeraverageencodingdegreesofdifferentcodesare!0ANA(1)=3.93,!0LP(1)=3.61and!0MIN(1)=1.36,respectively.Wecanalsocomputethetotalencodingdegree.Withtherstlayeraverageencodingdegree0(1)=1.14and0MIN(1)=3.15,thetotalaverageencodingdegreesare4.49forDRC-ANA,4.12forDRC-LPand4.28forDRC-MIN.ThusthetotalencodingweightsaresimilarforallthreeDRC,andsmallerthan0(1)=5.89fortheprimitiveRaptorcode.SinceDRC-MINneedsamuchhigheroverhead=2forfullrecovery,thustheoverallencodingenergycostismuchhigher.Fromanotherpointofview,tofacilitatethesecond-layerLTencoding,everygroupneedstoprovideonaverage(1+)!0(1)krst-layerLT-codedpackets.Thisvalueis4.24kforDRC-ANA,4.04kforDRC-LP,and4.08kforDRC-MIN,whicharecomparableforallthreecodes. 72

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4.3.4.3Effectofgroupsizemontheperformance AsdiscussedinSection 4.3.2.1 ,theDRCobtainedwiththeproposedmethodisnotdependentongroupsizeornumberofgroups.InFigure 4-10 ,weplottheminimumoverheadrequiredtorecoverallpacketsforsystemswithtotalmg=10000packetsbutdifferentgroupsizem.ResultsverifythattheperformancesofDRC-ANAandDRC-LParenotaffectedbygroupsize.However,forDRC-MIN,theoverheadisstronglydependentonthegroupsize.Forsystemswithsmallgroupsizes,cangoupto5.Asthegroupsizeincreasestoover140,theperformanceapproachesthoseoftheDRC-ANAandDRC-LP.Thisconrmstheproofin[ 66 ]thatDRC-MINcanachieveasymptoticallygoodperformanceasgroupsizegoestoinnity. 4.3.4.4EffectofwandDontheDRC-ANA RecallfromSection 4.3 thattheDDPsofDRC-ANAaredependentontheweightingfactorwandthemaximumencodingdegreeDorD!.WewillevaluatetheeffectsofthesetwofactorsontheDRC-ANAintermsofand!0(1). ForRaptordistributionwith=0.05,werstcomputeand!0(1)valuesofDRC-ANAwithxedD=5butvariousweightfactorswasmultiplesofwmintoevaluatetheeffectofw.Besides,foraxedw=2,theand!0(1)valuesarecalculatedfordifferentD.TheresultsareshowninFigures 4-11 and 4-12 ,respectively.InFigure 4-11 ,weobservethatincreasesforlargerwandD.ButtheeffectofDismuchmoreprominent.Figure 4-12 showsthattheaverageencodingdegree!0(1)increasesslightlyasweightingfactorw,butdecreasesfastwithD.Therefore,DhasmuchmoreimpactonDRC-ANAwhilewhasnegligibleeffects,andsmallerDispreferableforDRC-ANA. Theseresultsarereasonable.ForthesameRaptordistributionwithxedD,asDincreases,D!becomessmaller.KnownfromEquation( 4 )thattheresultantDRCdistribution^(x)willhavelesstermsthatmatch(x).Consequentlymorehigherordertermswillhavelessvaluethan(x)(Lemma 5 ).Henceweobserveanincreaseof,butadecreaseofaverageencodingdegree.Additionally,recallfromEquation 73

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( 4 )thatwithlargerw,wehavesmaller!1,thus1islargerbuti,i2aresmaller.Numericalresultsshowthatthesetwoeffectsenlargesome!icoefcientsbutshrinksomeothers.Thusthetotaleffectisnotnotable. 4.4Summary Inthischapter,weinvestigatedthedecomposedfountaincodeswithmulti-layerencoding.ExtensiveanalyseswerecarriedouttodevelopvalidencodingdistributiondecompositionmethodsforDLTcodeconstruction.H-DLTcodes,whichenableexiblecomputationcostallocationbetweentwoencoders,werealsodevelopedfordecompositionoftheRSDs.SimulationresultsandcomparisonsshowthatallDLTcodeshavesimilarresultantdistributionandperformanceasthecorrespondingLTcodes.AmongthesedecomposedLTcodes,theh-DLTcodeoutperformsthedistributedLTcodeandtheSD-DLTcode.Inaddition,thecomputationcostisgreatlyreducedatbothlayersandcanbeexiblyallocatedintheh-DLTcodes.Furthermore,weinvestigatedthedecompositionoftheRaptorcodeswithgroup-wiseencoding.Amoreenergy-efcientmethodisdeveloped.SimulationsshowthatDRC-ANAcanachievesimilarperformanceastheprimitiveRaptorcodes,whiletheperformanceofDRC-LPhassomedegradation.ThoughtheproposedschemesneedhigheraverageencodingdegreethanDRC-MIN,theyoutperformDRC-MINespeciallyinsystemswithsmallgroupsize.Inaddition,theanalysisofaffectingfactorsrevealsthatDismoreinuentialthantheweightingfactor,andsuggestssmallerDforbetterperformance. 74

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Table4-1. Theaverageencodingdegree CodeType1stlayer(C1)2ndlayer(C2) SD-DLT2.275.02h-DLT(=0.8)2.853.47h-DLT(=0.6)2.943.06h-DLT(=0.4)3.582DistributedLT5.901.86ConcatenatedLT11.6411.64 Figure4-1. TheencodingdiagramforaDLTcodewithtwoencodingDDPsof(x)and!(x). Figure4-2. TheencodingdiagramoftherstencoderoftheSD-DLTcodes. Figure4-3. TheencodingdiagramofthesecondencoderoftheSD-DLT/h-DLTcodes. 75

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Figure4-4. Theencodingdiagramoftherstencoderoftheh-DLTcodes. Figure4-5. ThecomparisonoftheresultantdegreedistributionoftheSD-DLTcodeandtheRSD.Thesystemparametersare:k=1000,c=0.08and=0.05. 76

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Figure4-6. TheaveragerecoveryratiowithrespecttodifferentoverheadfortheSD-DLT,distributedLT,h-DLTandprimitiveLTcodes.Thesystemparametersare:k=1000,c=0.08and=0.05. Figure4-7. Thecomplementarycumulativedistributionfunctionofrequiredoverhead()forsuccessfuldecodingfortheSD-DLT,distributedLT,h-DLTandprimitiveLTcodes.Thesystemparametersare:k=1000,c=0.08and=0.05. 77

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Figure4-8. ThecomparisonoftheresultantdistributionsofDRCsandtheRaptordistribution.Theparametersare=0.05,g=100,Dk=5andw=2.ForDRC-LP,=0.02andc=0.05. Figure4-9. ThecomparisonofaveragerecoveryratioforbothDRCsandtheRaptorcode.Theparametersare=0.05,m=21,g=100andw=2.ForDRC-LPandDRC-MIN,=0.02andc=0.05. 78

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Figure4-10. Theeffectsofgroupsizemontheoverheadrequiredforfullrecovery.Theparametersare=0.05,mg=10000,w=2andDk=5.ForDRC-LPandDRC-MIN,=0.02andc=0.05. Figure4-11. TheeffectsofwandDontheDRCperformance.Theparametersare=0.05,m=100,g=100andD=84. 79

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Figure4-12. TheeffectsofwandDontheaverageencodingratio!0(1).Theparametersare=0.05,m=100,g=100andD=84. 80

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CHAPTER5UNDERWATERAPPLICATIONSOFDECOMPOSEDFOUNTAINCODES Decomposedfountaincodesarewell-suitedtomulti-levelreliablesystems.Inthischapter,wewillexploretheirapplicationsinunderwateracousticnetworkstoenhancethedatacommunicationandstoragereliability. 5.1ReliableUnderwaterCooperativeRelayCommunications Cooperativerelaycommunicationshavebeenextensivelystudiedintheliteraturetoimprovecommunicationreliability.Tofurtherensureend-to-endcooperativecommunicationreliability,link-layercontrolisnecessary.Inthissection,wewilldesignaDLTbasedreliablecooperativecommunicationscheme. 5.1.1RelatedWorks Theinherentdual-hopwirelesstransmissionnatureofcooperativerelaycommunicationsinducessomeuniquerequirementsonthecooperativeprotocoldesign.First,duetotheindependentchannelfadingonsource-relayandrelay-destinationlinks,thetransmissionreliabilityneedstobeassuredforbothhopsfromthelink-layerpointofview[ 53 54 ].Secondly,theend-to-enddatadeliverylatencyneedstobereduced,especiallyfordelay-sensitiveapplications,suchasvideostreaming[ 55 ].Thirdly,heterogeneousnodeenergyisacriticalfactorthatlimitsthenetworklifetime.Intheliterature,residualenergy-awarerelayselectionprotocolsaredesignedtocopewiththisproblem[ 56 58 ],whilethehybridautomaticretransmissionrequest(ARQ)schemeiscommonlyadoptedtoaddressthersttwoissues[ 59 60 ].Fountaincodes[ 62 ]areveryattractive,andvariousprotocolshavebeendevelopedandanalyzedintheliterature. In[ 15 53 63 ],independentfountainencodingisadoptedateachhoptoensurethedual-hoptransmissionreliability.Intheseschemes,thesourceencodeseverypacketwithafountaincode.Therelaysneedtodecodeandre-encodeeachpacket,andsendacknowledgementsbacktothesourcetoconrmcorrectreceptionofallpackets.Clearly,highcomputationcostisrequiredattherelaysandlargetransmission 81

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latencyisinducedbythefrequentfeedbackmessages.Toaddresstheseissues,concatenatedencodingisadoptedin[ 9 54 64 65 ],wheretherelaynodessimplyapplyasecond-layerofcodingtothefountain-codedsourcedatawithoutdecoding.Apparently,complexdecodingisneededatthedestinationtorecoverthesourcepackets.In[ 64 ],severalrelayencodingschemesaredesignedandcompared.Thexed-ratesystematiccodesrequiretherelay-destinationchannelerasureinformationfeedback,whichintroducesadditionallatency;andthegreedyrandomcodeshavelowlatencybutimposehighdecodingcomplexityatthedestination.In[ 9 ],amodiedfountainencodingschemeisdevelopedfortherelay.Bystoringalargefountainencodingmatrixandasequencepermutationmap,therelaycanencodethepacketsaccordingtotheirarrivalorder.Thustherelayforwardingdelayissignicantlyreduced.However,largestoragecapacityisrequiredattherelay.In[ 54 ],thesourceadoptsrandomlinearfountaincodesandeachrelayperformsrandomXORoperations.Aspectral-efcientrelayingprotocolisalsodesignedforatwo-relaycooperativesystem.Thisworkisextendedtomulti-relaysystemsin[ 65 ],wherearandomly-select-and-forwardrelaycooperationprotocolisproposed.Nevertheless,allcooperativetransmissionprotocolsbasedonconcatenatedcodingrequiresignicantdecodingcomplexityatthedestination. Therefore,energy-efcientdual-layerreliablecooperativerelaycommunicationschemeisfavorable.Wewilldesignanh-DLTcodesassistedtransmissionschemetofulllthistask.Severalbenetscanbeachieved.First,randomencodingatboththesourceandtherelaysensurescommunicationreliabilityonbothlinks.Secondly,therateadaptationonbothlinksenablesreducedtransmissionlatencyandcommunicationcost.Thirdly,lowcomputationcomplexitycanbeachievedatallnodesduetothedecomposedencodingandthesingle-layerdecoding.Finally,bycarefullychoosingtheh-DLTmoderatioaccordingtotherelativeresidualenergyofthecooperativenodes,thecomputationcostcanbebalancedamongthetransmittingnodes. 82

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5.1.2H-DLT-assistedCooperativeRelayCommunications Consideragenericcooperativerelaycommunicationsystemwithonesource,multiplerelaysandonedestination[ 54 65 ],asshowninFigure 5-1 .Theh-DLTbasedcooperativecommunication(DLT-CC)schemeisdescribedhere.Withoutlossofgenerality,timedomainmultipleaccess(TDMA)isassumed. IntheDLT-CCscheme,thesourceandtherelay(s)willperformtherstandsecond-layerencodingoftheh-DLTcode,respectively.Inordertochooseanh-DLTcodewithappropriatecomputationallocationratioforthecooperativenetwork,thesourcewillrstbroadcasttherequestforrelayingwiththedestinationinformation.Theavailableintermediatenodeswillreplywiththeirresidualenergy,distancetothedestinationandrelatedinformation.Basedonthefeedback,thesourcewillrstchooseasetofrelaynodesaccordingtosomeperformancecriteria(e.g.,[ 56 58 ]).Inaddition,theoptimalmoderatioiscomputednumericallywithAlgorithm 4 suchthattheencodingcostratio(C1=C2)matchestheresidualenergyratiobetweenthesourceandthechosenrelays.Atthesametime,theencodingdegreedistributions(x)and!(x)areobtained.TheDLT-CCprotocolconsistsofthreeparts. 5.1.2.1Sourceencodingandbroadcast Withkrawpacketstobetransmitted,thesourcewillgeneratecodedpacketsusingtherst-layerDDPoftheh-DLTcode(x).OneIDbitisaddedtoeachcodedpackettoindicateitsencodingmode:LT(ID=0)orDLT(ID=1).Thepacketsarecontinuouslygeneratedandbroadcasttotherelays/destination.Thesourcewillceaseitstransmissionuntilacknowledgements(ACKs)arereceivedfromtherelays. 5.1.2.2Relayencodingandcooperativeforwarding Attherelay,eachreceivedpacketatthephysicallayerrstundergoesanerrordetection(e.g.CRC)process.IftheCRCchecksucceeds,thepacketisfurtherprocessed.TherelayreadsthereceivedpacketID.IfID=0,thepacketisreadytobeforwardedasanh-DLTpacket;whereasforID=1,therelayencoderwillstorethe 83

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packetinthememoryandgenerateanh-DLTpacketusingencodingdegreedistribution!(x)untilD!DLT-1packetsareaccumulated.However,ifareceivedpacketfailstheCRCcheck,thepacketisdropped,andanewh-DLTpacketisgeneratedfromthestoreddata.Inthenexttimeslot,eachrelaywillforwardtheh-DLTpackettothedestination.Eachrelaykeepsforwardingh-DLTpacketsuntilanACKisreceivedfromthedestination,andtheACKisrelayedtothesource. 5.1.2.3Destinationdecoding Eachreceivedh-DLTpacketpassingtheCRCcheckisforwardedtotheBPdecodertorecoverthesourcedata.Aslongasthereceiverrecoversallksourcepackets,anACKissentbacktotherelaystonotifysuccessfuldatatransmission. 5.1.3PerformanceofDLT-CC ToevaluatetheperformanceoftheproposedDLT-CCprotocol,weanalyzetheend-to-endcommunicationlatency,totalcommunicationcostandcomputationcomplexityofthesystem.TheresultsarecomparedwithexistingschemesintheliteraturetoverifythebenetsoftheDLT-CCscheme. 5.1.4End-to-endLatency Theend-to-endlatencyofthecooperativerelaycommunicationconsistsoftwoparts:thetotaldatapackettransmissiontimeandround-tripcontroltime.Consideratotalofksourcepackets,wecancomputethelatencyTLasfollows. TL(p1,p2,k)=tpNC(p1,p2,k)+tRTTNR(p1,p2,k), (5) wheretpandtRTTarethetransmissiontimeofeachpacketandend-to-endround-triptime(RTT).NCandNRarethetotalnumberofpackettransmissionandretransmissionrequests.p1andp2aretheaveragepacketerasureratesonthesource-relayandrelay-destinationlinks. IntheDLT-CCscheme,thetransmitterscanadapttothechannelerasureratebycontinuouslytransmittingcodedpacketsuntilafeedbackmessageisreceived.Thus 84

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onlyoneroundtripisneeded,NRDLT=1.Inaddition,theaveragenumberoftotaltransmissionsis: NCDLT(p1,p2,k)=2k(1+) 1)]TJ /F5 11.955 Tf 9.96 0 Td[(p2, (5) ToillustratethebenetsofadoptingadecomposedLTcode,wecomparetheresultswithtwoothertraditionalschemes:ARQ-basedcooperativecommunication(ARQ-CC)andLT-basedhybridARQscheme(LT-CC).IntheARQ-CCscheme,nodataencodingisadoptedandtherelay(s)simplyforwardthedatatothedestination.ThedestinationwillsendbackNACKmessagestothesourcetorequestthelostpackets;IntheLT-CCscheme,thesourcedataisencodedwithanLTcode.ThesourcewillkeepsendingcodeddatatothedestinationwiththerelayforwardinguntilthedestinationsendsbackanACK. InclassicARQbasedtransmissions,duetotheerror-pronewirelesschannel,thepacketerasurewillincurmanyretransmissions.Foratransmissionwindowofsizek,thetotalnumberofretransmissionscanbecomputedrecursivelyas: NRARQ(p1,p2,k)=1 1)]TJ /F5 11.955 Tf 9.96 0 Td[(pk"1+k)]TJ /F10 7.97 Tf 5.17 0 Td[(1Xi=1kipi(1)]TJ /F5 11.955 Tf 9.96 0 Td[(p)k)]TJ /F8 7.97 Tf 5.17 0 Td[(iNRARQ(p,i)#. Andtheend-to-endtotalnumberoftransmissionsis: NCARQ(p1,p2,k)=2 1)]TJ /F5 11.955 Tf 9.97 0 Td[(pk"k+k)]TJ /F10 7.97 Tf 5.18 0 Td[(1Xi=1kipi(1)]TJ /F5 11.955 Tf 9.96 0 Td[(p)k)]TJ /F8 7.97 Tf 5.18 0 Td[(iNCARQ(p,i)#, (5) wherep=p1+p2)]TJ /F5 11.955 Tf 9.96 0 Td[(p1p2. FortheLT-CCscheme,thesourcewillgenerateallredundantpacketstocountererasuresonbothlinks,thusthetotalnumberofpackettransmissionsis: NCLT(p1,p2,k)=k(1+) (1)]TJ /F5 11.955 Tf 9.96 0 Td[(p1)(1)]TJ /F5 11.955 Tf 9.96 0 Td[(p2)+k(1+) 1)]TJ /F5 11.955 Tf 9.96 0 Td[(p2=k(1+)(2)]TJ /F5 11.955 Tf 9.97 0 Td[(p1) (1)]TJ /F5 11.955 Tf 9.96 0 Td[(p1)(1)]TJ /F5 11.955 Tf 9.96 0 Td[(p2), (5) andthenumberofretransmissionsisNRLT=1. 85

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WithEquation( 5 ),wecomputetheaveragelatencyperpacketforallthreesystemsandfordifferenttRTT=tpratios.Thelatencyvalueisshownintermsoftp,andtheresultsareplottedinFigure 5-2 .NoticethattheDLT-CCschemeentailsthesmallestlatency,anditisnotsensitivetothetRTT=tpratio.IntheARQ-CCsystem,thelatencyincreasessignicantlywiththetRTT=tpratio.ThisindicatesthatourDLT-CCschemeismorebenecialforcommunicationsystemswithlargetRTT=tpratio,suchasunderwateracousticcommunicationsandsatellitecommunications.Inaddition,theDLT-CCsystemalsooutperformstheLT-CCschemewithreducedlatency,sincetherelayswillgenerateredundantpacketstoovercometherelay-destinationerasureintheDLT-CCsystem,insteadofthesourceintheLT-CCscheme. 5.1.4.1Communicationcost Thetotalenergyconsumptionofacooperativerelaycommunicationsystemconsistsoftwoparts:thedatacommunicationenergycostandthenodecomputationcost.Here,wewillrstevaluatethecommunicationenergycostintermsofthetotalnumberofpackettransmissions. WiththeanalyticalresultsinEquations( 5 ),( 5 )and( 5 ),wecancomputetheaveragenumberoftransmissionsperpacketforallthreeschemeswithrespecttodifferentpacketerasurerates.AsshowninFigure 5-3 ,theDLT-CCschemerequiresthesmallestcommunicationcost.Astheerasurerateincreases,morecommunicationenergycanbesavedbytheDLT-CCschemecomparedtotheARQ-CCandLT-CCschemes. TocomparetheDLT-CCschemewithotherdecomposedLTcodesbasedschemes,suchasthedistributedLTcodes[ 67 ],wecancalculatethecommunicationcostusingthesameformulaEquation( 5 ).Thus,withthesamepacketnumberkanderasureratep,wecancomparetheaverageoverheadoftheseschemestoindicatethecommunicationcostdifference.TheoverheadcomparisonresultsobtainedinFigure 86

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4-7 revealthattheDLT-CCschemebasedontheh-DLTcodesoutperformstheoneusingthedistributedLTcode. 5.1.4.2Computationcomplexity TheDLT-CCschemeisdesignedbasedontheh-DLTcodestoenableexiblecomputationallocationbetweenthesourceandtherelay(s).Toverifythisdesign,wecomputetheaverageencodingdegreeateachnodefortheDLT-CCschemesusingtheh-DLTcodesobtainedinSection 4.2.6 withdifferentmoderatios.AccordingtotheanalysisinEquation( 4 ),thecomputationcostsarecomputedandlistedinTable 4-1 .Observethat,asthemoderatiodecreases,moreencodingcostshiftsfromtherelaytothesourceasexpected.ComparedtotheSD-DLTanddistributedLTcodes,theh-DLTcodeshavemuchmoreexibilityincomputationcomplexityallocation.ThisconrmsthattheDLT-CCschemecansolvetheheterogeneousnodeenergyproblembyintelligentlychoosingthemoderatio. Inaddition,ourDLT-CRCschemeisbenecialforlesscomputationcostcomparedwithotherfountaincodebasedcommunicationprotocols.Inthedecode-and-reencodebasedschemes[ 15 53 63 ],thesourceneedstoperformfullfountainencoding,andtheencodingcostisCs=0(1)k.Attherelay,thetotalcomputationisCr=O(klnk)+CswithO(klnk)beingthedecodingcomplexityoftheBPdecoder.ThedestinationdecodingcostisCd=O(klnk).Fortheconcatenatedfountaincodingschemes[ 9 64 ],theencodingcostsofthesourceandrelay(s)arebothCs,butthedestinationdecodingcostis2Cd.Intheschemesusingrandomlinearfountaincodes[ 54 65 ],theencodingcostsareO(k2)forboththesourceandrelay(s),andthedestinationdecodingcostisashighasO(k3).Inaddition,inourDLT-CCscheme,thecostsatthesource,relay(s)anddestinationareCDs=C1k,CDr=C2k(1+)andCDd=Cd,respectively.ThecomputationcostsforalldifferentschemesarelistedinTable 5-1 .Clearly,theDLT-CCschemerequirestheleastcomputationcostforallnodes.Forexample,usingtheresultshowninTable 4-1 ,for=0.4,thesourceandtherelay(s)cansaveabout69%and 87

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82%ofthecomputationcomparedtotheconcatenatedencodingscheme.Besides,thedestinationrequiresonlyhalfofthedecodingcost.Therefore,theDLT-CCprotocolisexpectedtoprolongthecooperativenetworklifetimewithreducedcomputationandadaptiveenergyallocation. 5.2ReliableUnderwaterData-CentricStorage Theessentialtaskofsensornetworksistocollectenvironmentalinformationandprovidetheaccessoftheinformationdatatooutsideusers.Insevereunderwaterenvironments,real-timedatadeliverytothegrounddatacenterisverychallengingduetotheunavailabilityofpermanentroutes,longpropagationdelay,lowthroughputandscarcesensornodeenergy.Therefore,in-networkdatastorageisfavorableforunderwateracousticsensornetworks(UASN)[ 25 34 ].Data-centricstorage(DCS)isappealingforthedatawithfrequentqueryinlarge-scaleUASN,becauseofitsfastdataretrievalandin-networkdataprocessing.However,theadverseunderwaterenvironmentchallengesthereliabilityofDCSsystemsintwoaspects.First,theunreliableunderwaterchannelrequiresmorerobustdesignofdatatransportfromsensingnodestostoragelocation;Secondly,highnodefailureratedemandsbetterprotectionofstoreddatainformation.Thus,themulti-layerprotectionprovidedbyDFCsarebenecialinunderwaterDCS.Inthissection,wewilldesignaDRC-assistedDCS(DCS-DRC)protocolforreliableandenergy-efcientunderwaterin-networkdatastorage.AnalysesandsimulationsareprovidedtoillustratethebenetsoftheDCS-DRCprotocol. 5.2.1RelatedWorks 5.2.1.1Data-centricstorage InDCS,alldataofthesametypearestoredinonenode/region.ThedatatypetostoragelocationmappingisimplementedwithGeographicHashingTable(GHT).Inthisscheme,toretrieveonetypeofdata,thequerieronlyneedstosendtherequesttothemappedstoragelocation.Thisdirectedretrievaldesignisbothtimeandenergyefcientcomparedtolocalstoragescheme.However,reliabledesignsarecriticalin 88

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DCSduetotheconcentratedstorageandmulti-hopdatatransportfromsensorstostoragenodes.Mostliteratureworksfocusonenhancingthereliabilityofeitherdatastorageorinformationtransport. TherstDCSprotocolpresentedin[ 42 ]hashesonedatacategorytoonestoragenode.Toavoidsinglestoragenodefailureandstorage/communicationhotspot,StructuredReplication(SR)aroundthestoragenodewasdesignedin[ 42 ].Besides,severaldata-to-zonemappingbasedschemeswereproposed,[ 76 77 ].Intheseprotocols,eachdatatypeishashedtomultiplenodesinonegeographiczoneinsteadofonenode.Betternodefailureresilienceisachievedandthecommunication/storagehotspotproblemisalleviated.Replicationstorageisalsoadoptedtoenhancethedatareliability.Clearly,thestorageoverheadisextremelyhighinalltheseschemes. Forreliablecommunications,GreedyPerimeterStatelessRouting(GPSR)protocol[ 79 ]wascommonlyadoptedastheunderlyingroutingschemeinDCS.Thisstatelessroutingprotocolovercomesthenetworkroutechangedynamicsbyadaptivelyforwardingtothenexthopthatisnearesttothedestination,accordingtothegeographiclocationsoftheneighbors.Toimproveroutingreliability,BidirectionalPerimeterRouting(BPR)wasintroducedin[ 75 ].ButnoefcienttransportlayerprotocolisdesignedforDCS. 5.2.1.2Fountaincodesbaseddistributedstorage Inanotherstreamwork,fountaincodeshavebeenappliedtowirelesssensornetworkstoenhancedatastoragereliability[ 80 81 ].Intheseschemes,fountaincodesareimplementedinadistributedmanner,wherealldatapacketsrandomlywalkintheentirenetworkandeachsensornodeindependentlyencodesandstoresthepacketsatrandom.Improvedstoragereliabilityisachievedwithsmallstoragespaceoverhead.However,thecommunicationcostisextremelyhighduetothedatarandomwalkintheentirenetwork.Thedataqueryisalsoenergycostly. 89

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Therefore,wewilldesignareliableunderwaterzone-basedDCSschemewiththeassistanceofDRCtoachievebothcommunicationandstoragereliability,aswellasenergyefciency. 5.2.2DRC-basedUnderwaterDCS TheDRCconsistsofonelayeroftraditionalerasureprecodingandtwolayersofLTencodingwithdecomposedRaptorDDPs.Azone-basedDCSprotocolconsistsofthreeparts:sensingzonedatasharing,inter-zonedatatransportandstoragezonedataprocessing.ByappropriatelyapplytheDRCintoeachpartofDCS,wecandesignareliableDCSprotocoltailoredforsensornetworksinharshseaenvironments. 5.2.2.1Protocoldescription ConsiderasensornetworkwithNnodesdeployedunderwaterasdepictedinFigure 5-4 .Withzone-basedtopology,theentirenetworkisvirtuallydividedintomultiplegeographiczones,whichareidentiedbytheirzoneID.Inaddition,thefollowinginformationisassumedknowntoeachsensornode:onenode'sowngeographiclocationandallzoneboundaries,whichallowsthenodetodetermineitshomezoneID;theneighboringnodeswithinthenode'scommunicationrange;thegeographichashingfunctionusedforcomputingthestoragezoneIDforeachdatatype.Tofacilitatethedescriptionofourprotocol,thefollowingnotationsareused:thenumberofzonesinthenetworkisNz,thezonesizeisNn,andN=NzNn. 5.2.2.2Homezoneencodingandooding Inasensornetwork,eachsensorkeepssensingtheenvironmentandrecordsthemeasuredinformationintodatapackets.Eachdataisidentiedbyfourvalues:sensingtime,location,sensingzoneIDanddatatype.Themeasureddataaredeliveredtothestoragezoneperiodically.Toenabledatatransportreliability,alldataarerstsharedinthehomezoneforencoding. Whenthestorageperiodcomes,eachsensornodeaccumulatesmnewdatapacketsP0ofonetype.Thesedataareoodedinthehomezone.Toenhancethe 90

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spreadingreliability,allP0packetsareencodedwithtraditionalerasurecodeswithratertoobtainm=rcodedpacketsP1,whicharereadytobesenttoallnodesinthehomezone.InP1packetooding,eachhomezonenodekeepstrackofthepacketsfromallothernodesinthezone.ARQ-basedcontrolisadoptedtoensurereliability.Morespecically,ateachnode, atimerissetupuponthereceptionoftherstpacketfromallothernodesinthesamezone; everynewlyarrivedpacketfromallothernodeswillbetemporarilystored; afterthetimeout,thecompletenessofreceptionfromeachothernodeischecked.Ifenoughpacketsarereceivedfromonenode,anACKissentbacktothecorrespondnode.Otherwise,aNACKmessageistransmittedwithneededpacketinformation. WhenP1packetsarecorrectlyreceived,allnodesinthesensingzonewillcollectivelybehaveasadatafountain.Eachnodeperformstherst-layerLTencodingonallreceivedP1packetswithDDP(x),andkeepsgeneratingcodedpacketsP2untilanACKisreceivedfromthestoragezone. 5.2.2.3Inter-zonedatatransport ForeachP2packet,thecorrespondingstoragezoneIDiscomputedbytheGHTmoduleaccordingtothedatatype,andthenitisgreedilyroutedtothestoragezonewithGPSR.Ateveryhop,thepacketisforwardedtooneofitsneighborsclosesttothedestination.Whenthepacketarrivesattherstnodeofthestoragezone,thepacketwillrandomlywalkinthestoragezone. Duetotheunreliableunderwateracousticlinks,theend-to-enddatadeliveryreliabilityneedstobebetterprotected.Furthermore,thelongpropagationdelayhinderstheimplementationoftraditionalsmallwindowARQ-basedcontrolmechanism.Toguaranteetheinter-zonedatatransportreliability,afountaincodebasedtransmissioncontrolisdesigned.Foreachnodeinthestoragezone,thereareseveralactionsitneedstotakeafterreceivingapacket. 91

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Ifthisistherstpacketreceivedbythisnode,thenthenodewillcomputethenumberofpacketsforstorageksandsetupatimerfortherandomwalkofthisP2packet; Ifthisisnottherstpacket,thenthepacketisusedforencoding. Eachstoragenodekeepsreceivingdatapacketsfromonehomezoneuntilonepacketfromthishomezonetimesoutwithoutbeingacceptedbyanystoragenode.ThenthestoragenodewillsendanACKmessagetothecorrespondinghomezonetoceasethetransmission. 5.2.2.4Storagezoneencoding Inthestoragezone,distributedLTencodingisusedateachnodetogeneratestoredpacketsand!(x)isthecorrespondingencodingDDP.Eachstoragenodewillstoreks=d(1+s)rmNzecodedpackets,wheresistheencodingredundancyratio.Inthestorageperiod, ksstorageslotsarerstallocatedateachnode.Foreachstoragememoryslot,anencodingdegreedisrandomlychosenfromthedistribution!(x)anddzoneIDsareuniformlyselectedtoformtheencodingsetSz. ForeachP2packetarrival,thestoragenodewillrstfetchthepacketzoneID,andcompareitwiththeencodingsetSzateachstorageslotconsecutively: IfoneslotmatchesthezoneID,thepacketisacceptedandthecorrespondingslotwillupdateitsstoragebyXORingitwiththeexistingdataintheslot:x+=x)]TJ /F2 11.955 Tf 9.85 -4.33 Td[(xc,wherex)]TJ /F1 11.955 Tf 10.4 -4.33 Td[(istheexistingdata,xcistheincomingdataandx+isthenewdata.Inaddition,thezoneIDisremovedfromSz,andtheencodingdegreeisdecreasedby1; IfthezoneIDisnotacquiredbyanystorageslotatthisnode,thenthepacketisrandomlyforwardedtooneneighbor,andthehopcountisupdatedbyinitializingthevalueordecreasedby1.Ifthehopcountreacheszero,thenthepacketisdeletedandatimeoutprocessisswitchedonforreliabletransportcontrol. 5.2.2.5Dataretrieval ToretrievethestoreddatafromtheUASN,anunderwatervehiclerstneedstondthestoragezoneoftheintendeddatatypebyGHT.Thenitsendsaquerytothestoragezonefromitscurrentlocation,ordrivestothestoragezoneinstead.Theneach 92

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storagenodereceivingtherequestwilltransfertheDRCcodeddatabacktothevehicle.Afterenoughcodedpacketsarereceivedfordecoding,aSTOPmessageissentouttoceasefurthertransmissions. 5.2.3PerformanceAnalysis 5.2.3.1Inter-zonedatatransport Ratelesscodesbasedreliabledatatransportcansignicantlyreducetheend-to-endtransmissiondelay.ForDRC-assistedinter-zonedatatransport,thenumberofretransmissionsisTDRC=1.IntraditionalARQwithselectiverepeat(ARQ-SR)schemes,ForawindowsizeofL,theaveragenumberretransmissionscanbecomputedinarecursivemanner: TARQ,L=1 1)]TJ /F5 11.955 Tf 11.96 0 Td[(pLr"1+L)]TJ /F10 7.97 Tf 6.58 0 Td[(1Xi=1Lipir(1)]TJ /F5 11.955 Tf 11.95 0 Td[(pr)L)]TJ /F8 7.97 Tf 6.58 0 Td[(iTARQ,i#, (5) wherepristheroutepacketfailurerate.Forpr=0.1andL=1000,theaverageretransmissionnumberisTARQ,1000=3.8.Apparently,theend-to-endlatencyismuchlesswithDRCbaseddatatransport. 5.2.3.2Networkcommunicationcost Energycostisacriticalfactorinwirelessnetworks.WewillanalyzetheenergycostoftheDCS-DRCschemeintwoparts:encodingcomputationcostanddatacommunicationcost,whichcanberepresentedbytotalnumberofgeneratedcodedpacketsandtotalnumberofpacketstransmimission,respectively. Thetotalnumberofsensing-zoneencodedandstorage-zoneencodedRaptorpacketsarecomputedas C1=!0(1)(1+s)mN=r(1)]TJ /F5 11.955 Tf 11.95 0 Td[(pr), (5) andC2=(1+s)mN=r,respectively.ThusthetotalnumberofXORoperationsis,CComp=0(1) 1)]TJ /F5 11.955 Tf 11.95 0 Td[(pr+1!0(1)(1+s)mN=r. 93

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Assumetheaverageroutefailurerateispr=0.1,forthethreeDRCsobtainedinSection 4.3.4 ,theencodingcostsare9.62(1+s)mN=rforDRC-ANA,9.17(1+s)mN=rforDRC-LPand18.36(1+s)mN=rforDRC-MIN,respectively.TheDRC-ANAandDRC-LPcodesrequireonlyhalfofthecostofDRC-MIN. InDCS-DRC,thecommunicationscostcanbeestimatedasthesumofthetransmissionsinthehome-zoneooding(O(Nn)),theinter-zonetransport(O(p N))andthestorage-zonerandom-walkO(Nnln(Nn)): CComm=O(NNn)+O(RNp N)+O(RNNnlnNn), (5) whereR=(1+s)!0(1)m=r.Forthesamenetworksetup,theRaptorcodesbaseddistributedstorageschemein[ 81 ]requiresacommunicationcostofO(N2lnN).ThustheDCS-DRCismuchmoreenergyefcient.ForalargenetworkwithNnodesandxedzonesizeNnandD,thecommunicationcostvarieswithNasCCommO(Np N).WithxednetworksizeN,thecommunicationcostwillchangewiththezonesizeasymptoticallyasCCommO(!0(1)NzlnNz). 5.2.4Simulationresults Inthissection,wesimulateourDCS-DRCprotocolwithNS2toillustrateitsperformanceintermsofenergycost.ThenumberofRaptorpacketsgeneratedbythesensornodesandthenumberofpacketstransmittedinthenetworkarerecordedtoverifytheanalysis.Inaddition,theeffectsofnetworksizeandzonesizearerevealed. 5.2.4.1Simulationsetup ToobtaintheenergyperformanceoftheDCS-DRCprotocolintrueunderwaterenvironment,wesimulateoursystemwithNSMiracle,anextensionofNS2[ 82 ],wheretheunderwaterPHYlayermoduleisprovided.Besides,thedetailedcommunicationprotocolsareimplementedinallotherlayers.Inthewholenetworkprotocolsuite,fountaincodesbasedtransportcontrolschemeisusedinthetransportlayer;GPSRisimplementedastheinter-zoneroutingprotocol;AlohawithACKisadoptedintheMAC 94

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layer,assuggestedin[ 83 ],duetotheslowpropagationofacousticsignals.Considerasparseunderwatersensornetworkwithgridtopology,wheretheinter-nodedistanceis1000m,andthecommunicationrangeis1500m.Thestoragezoneislocatedattheleftupcorner.Thesystemparametersare[s,r,m]=[0.1,4=7,10]. 5.2.4.2Networksize ForaxedzonesizeofNn=4,wesimulatenetworkswithN=50,100,200,and300nodes.ThesimulationishaltedwhenthestoragezonehassuccessfullyreceivedRNcodedpackets.ThenumberofallRaptorpacketsgeneratedandtransmittedwithrespecttodifferentnetworksizeisplottedinFigure 5-5 .Asnetworksizeincreases,thetotalnumberofRaptorpacketsincreaseslinearly.Forthesamezonesize,sensing-zoneoodingcreatesuniformbackgroundtrafc,thusroutefailureprissimilarforallscenarios.AccordingtoEquation( 5 ),theRaptorpacketnumbervarieslinearlyasN.Inaddition,thetotalcommunicationcostchangesnonlinearlywithnetworksize.AccordingtotheresultinEquation( 5 ),thecommunicationcostisO(Np N)forlargenetworksize.Thesimulationresultsverifythenonlinearity. 5.2.4.3Zonesize FornetworkswithN=100nodes,weevaluatetheeffectofzonesizeontheenergyperformance.InFigure 5-6 ,thepacketnumberfordifferentzonesizesisdisplayed.Weobservethatthecommunicationcostincreasesnonlinearlywithzonesize.Thisisduetotheincrementaltransmissionofsensing-zoneoodingandstorage-zonerandomwalk.AccordingtotheanalysesinEquation( 5 ),thecommunicationcostwillvarieswithNnasO(!0(1)NnlnNn)).Interestingly,thenumberofRaptorpacketgenerationshowsaconcavefeaturewithrespecttozonesize.KnownfromEquation( 5 ),theRaptorpacketnumberisaffectedby!0(1)andpr.Asthezonesizeenlarges,!0(1)decreases,butthesensing-zoneoodinggrows,whichworsenstheroutequalitypr.Thesetwooppositeeffectsresultinthisspecialfeature.Therefore,fromthecommunicationcost 95

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pointofview,smallerzonesizeisbenecialforDCS-DRC,whichisalsoapreferablenetworksetupforUASN. 5.3Summary Inthischapter,weinvestigatedtheapplicationsofDFCsinunderwaternetworks.Morespecically,forcooperativerelaycommunications,wedesignedanhDLT-assistedcooperativecommunicationscheme,whichseamlesslyincorporatestheh-DLTcodesintothedatatransmission.Thisschemealsotakesintoaccounttheheterogeneousnoderesidualenergytoprolongthenetworklifetime.Analysesrevealthattheproposedschemesignicantlyreducesthetransmissionlatencyandenergyconsumption.Inaddition,weproposedareliableunderwaterDCSsystemwiththeassistanceofDRC.TheDCS-DRCprotocolistailoredforunderwaternetworkswithreliabledatatransportandstorage.Analysesshowthatdatareliabilityisenhancedwithreducedenergycost.NS2simulationsshowthatsmallerzonesizeismorefavorableforlesscommunicationcost. 96

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Table5-1. Theaveragecomputationcost SchemeSourceRelayDestinationhlineRandomlinearcode[ 54 65 ]O(k2)O(k2)O(k3)Decode-Reencode[ 53 63 ]0(1)kO(klnk)+0(1)kCBPConcatenatedLT[ 9 64 ]0(1)k0(1)k2CBPDLT-CCC1kC2kCBP Note:CBPO(klnk)isthecomputationcostofBPdecoding. Figure5-1. AcooperativerelaysystemwithLrelays. Figure5-2. TheaveragetransmissionlatencyperpacketfortheARQ,LTandDLTbasedcooperativetransmissionschemes. 97

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Figure5-3. TheaveragenumberoftransmissionsperpacketfortheARQ,LTandDLTbasedcooperativetransmissionschemes. 98

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Figure5-4. Thesensornetworktopology.ThenetworkisvirtuallydividedintoNzgeographiczones.EachzonehasNnsensornodes.Alldatasensedinthehomezonewillbedeliveredtothestoragezone. 99

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Figure5-5. ThecommunicationcostforanetworkwithxedzonesizeNz=4anddifferentnetworksizes.Eachsensornodehasm=10P0packetstotransmit. Figure5-6. ThecommunicationcostforanetworkwithxednetworksizeN=100anddifferentzonesizes.Eachsensornodehasm=10P0packetstotransmit. 100

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CHAPTER6CONCLUSIONSANDFUTUREWORKS 6.1Conclusions Inthisdissertation,weinvestigatedthereliabilityissueinunderwateracousticdatacommunicationandstorage.Duetotheuniquefeaturesofunderwateracousticsignalpropagation,weexploredtwomajortechniquestoimprovetheunderwatercommunicationnetworkreliability:cooperativerelaycommunicationsanddecomposedfountaincodes. Inunderwatercooperativerelaycommunications,westudiedthecapacitygainofrelayinganddesignedofapracticalandefcientrelayingprotocoltailoredforUAC.Basedonempiricalacousticsignalpropagationandnoisemodels,thesystemcapacityisevaluatedforadual-hoprelaysystemwitharbitraryresource(locationandpower)allocation.Theresultsarecomparedwiththetraditionaldirect-linksystem,andthecomparisonindicatesmorethan40%capacitygainwithrelaying.Besides,theaffectingfactorsoftheRA-UACsystemcapacityarealsostudied.Analyticalandnumericalresultsrevealthattherelaylocationisamorecriticalfactorindeterminingthecapacity.Aftertherelayingbenetsbeingveried,weproposeanasynchronousunderwaterrelayingprotocol,AsAP.ThenewprotocolcanaddresstheUACdifculties,suchastimesynchronization,delayvarianceandfrequency-selectivechannel.Atthesametime,ittakesadvantageofthesparsityfeatureofUACchannels,whichconvertsthespacediversityintomultipathdiversity.TheOFDMprecodingdesigniscapableofcollectingbothdiversities.Analysisandsimulationsshowthatprecodingisaneffectiveapproachandthecollectablediversityorderisboundedbytotalnumberofnon-zerotapsandtheprecodingsize. Indecomposedfountaincodes,weinvestigatedbothdecomposedLTcodesanddecomposedRaptorcodesforcooperativerelaycommunicationsandunderwaterDCSsystems,respectively.Thedecomposedcodesarewellsuitedforthedesignof 101

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multi-levelreliablesystems.FortheDLTcodes,extensiveanalyseswerecarriedouttodevelopvalidencodingdistributiondecompositionmethodsforDLTcodeconstruction.HybridDLTcodesweredevelopedfortheclassicRSD,aswellasfacilitatingexiblecomputationcostallocationbetweentwoencoders.SimulationresultsandcomparisonsshowthattheDLTcodeshavesimilarresultantdistributionandperformanceastheLTcode,andthecomputationcostisgreatlyreducedatbothlayers.FortheDRC,weexploredanefcientdecompositionmethodforRaptordistributions.Byappropriatelychoosingtherstlayerdistribution,bothanalyticalandnumericalmethodsbasedonlinearprogrammingaredesignedtoobtainthesecondlayerdistribution.Simulationsshowthatthecodeobtainedanalytically(DRC-ANA)canachievesimilarperformanceastheprimitiveRaptorcode,whiletheperformanceofthecodeobtainthroughlinearprogramming(DRC-LP)hassomedegradation.Inaddition,theanalysisofaffectingfactorsrevealsthattherst-layermaximumencodingsizeismoreinuentialandsuggestssmallerrst-layermaximumencodingsizeforbetterperformance.Inthesecondpart,weexploredtheapplicationsofthesetwocodes.Basedonh-DLTcodes,wedesignedareliableunderwatercooperativerelaycommunicationscheme,whichseamlesslyincorporatestheh-DLTcodesintothedual-hopdatatransmissions.Thisschemealsotakesintoaccountheterogeneousnoderesidualenergytoprolongnetworklifetime.Analysesrevealthattheproposedschemesignicantlyreducesthetransmissionlatencyandenergyconsumption.Inaddition,weproposedareliableunderwaterDCSsystemwiththeassistanceofDRC.ThisDCS-DRCprotocolistailoredforunderwaterenvironmentswithbothreliabledatatransportandreliablestorage.Analysisshowsthattheproposedsystemachievesbetterreliabilitywithlessenergyconsumption.NS2simulationsalsosuggestthatsmallerzonesizeisfavorableforlesscommunicationcost. 102

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6.2FutureWorks Twopromisingreliabilitytechniquesareinvestigatedinthisdissertation,however,theadaptationsofthemintounderwaterapplicationshavenotbeenthoroughlystudied.Manyinterestingproblemsstillremainopenforfurtheradvances. 1. Inthisresearch,weonlyexploredcooperativerelaysystemswithsingle-source,single-destinationandsingle-layerrelays.Tofurtherimprovethemulti-hopend-to-endcommunicationreliability,moregeneralcooperativesystemswithmulti-layerrelays[ 18 ],and/orwithmultiplesourcesanddestinations[ 84 ]canbestudied; 2. Fordecomposedfountaincodes,thedual-layerrandomcodesareconstructedthroughdecompositionofexistingsingle-layerclassicfountaincodes.Duetothemathematicalconstraintsofpolynomialdecompositions,thismethodlimitsthecodedesigncapability.SincethedesigncriterionofDFCistoenablegooddecodingperformance,theapproachofcodedesigndirectlyfromcodeperformanceperspectivecanbepromising.Inthefuture,wecanexploretheperformanceevaluationofmulti-layerrandomcodes.Thentheoptimalcodescanbeconstructedbyoptimizingcodedecodingperformance[ 87 ].Inaddition,besidetwo-layerrandomfountaincodes,generalmulti-layerrandomcodescanbeexploredformulti-layerrelaysystemsandmulti-levelhierarchicalsensornetworks. 3. Inwirelesssensornetworks,networkcodingisanattractiveschemeforreliableend-to-enddatatransmissions[ 85 86 ].Duetotherandomencodingnatureofthefountaincodes,wecaninvestigatetheapplicationoffountaincodesintounderwaternetworkcodingschemes. 103

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APPENDIXAPROOFOFNONNEGATIVESOLUTIONREQUIREMENTSOFDLTCODES Intherststep,moveinEquation( 4 )totheleftsideoftheequations.Noticethat,intherthequation,theonlynegativetermis)]TJ /F6 11.955 Tf 9.3 0 Td[(randothercoefcientsarepositive.ThusthemixednegativeandpositiveconditioninTheorem 4.1 issatised.Denef(r,0)=)]TJ /F6 11.955 Tf 9.3 0 Td[(r. ThenwecanproceedtothesecondsteptoreducetheD!equationstoD!)]TJ /F3 11.955 Tf 12.28 0 Td[(1.Inrowr=1,Pnk=1~1,k!k=1,thuswerearrangeotherrows(r>1)as:Pnk=1~r,k!k=r,andapplythemultiplicationruleonbothside, nXk=11~r,k)]TJ /F6 11.955 Tf 9.97 0 Td[(r~1,k!k=0,r>1. (A) Since~1,1=1and~1,k=0,k>1,onlythersttermineachequationinEquation( A )canpossiblybenegative.Inordertoassurethepositivesolution,thefollowingconditionmustbeguaranteed: f(r,1)=1~r,1)]TJ /F6 11.955 Tf 11.96 0 Td[(r~1,1<0. (A) UnderthenegativeconditioninEquation( A ),wecanproceedtoreducetheD!)]TJ /F3 11.955 Tf 12.41 0 Td[(1equationsby1.RearrangeEquation( A )as: nXk=21~r,k!k=)]TJ /F5 11.955 Tf 9.3 0 Td[(f(r,1)!1,r>1, (A) andmultiplyther=2equationinEquation( A )toallotherequations,wehave, nXk=2)]TJ /F5 11.955 Tf 7.3 0 Td[(f(2,1)~r,k)]TJ /F3 11.955 Tf 9.97 0 Td[(()]TJ /F5 11.955 Tf 7.3 0 Td[(f(r,1))~2,k!k=0,r>2. (A) Inthesecondrowof~matrix,~2,2=21and~2,k=0,k>2,thusthek=2termmustbenegativetoassurepositive!: f(r,2)=)]TJ /F5 11.955 Tf 9.3 0 Td[(f(2,1)~r,2)]TJ /F3 11.955 Tf 11.96 0 Td[(()]TJ /F5 11.955 Tf 9.3 0 Td[(f(r,1))~2,2<0. (A) 104

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Continuethisprocess,wecanobtainthat,inthejtheliminationstep, nXk=j)]TJ /F5 11.955 Tf 7.31 0 Td[(f(j,j)]TJ /F3 11.955 Tf 9.96 0 Td[(1)~r,k)]TJ /F3 11.955 Tf 9.96 0 Td[(()]TJ /F5 11.955 Tf 7.31 0 Td[(f(r,j)]TJ /F3 11.955 Tf 9.96 0 Td[(1))~j,k!k=0,r>j, (A) andTheorem 4.1 requires: f(r,j)=f(r,j)]TJ /F3 11.955 Tf 9.97 0 Td[(1)~j,j)]TJ /F5 11.955 Tf 9.96 0 Td[(f(j,j)]TJ /F3 11.955 Tf 9.96 0 Td[(1)~r,j<0,r>j. (A) Thelemmaisproved. 105

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APPENDIXBPROOFOFTHEVALIDRANGEOFFIRST-LAYERDISTRIBUTION AsavalidsolutiontoEquation( 4 ),thevalueof!mustsatisfy02: g(j,j)]TJ /F3 11.955 Tf 9.97 0 Td[(1)=h(2f(2,1)+(j)]TJ /F3 11.955 Tf 9.96 0 Td[(1)12)(j2)]TJ /F8 7.97 Tf 5.18 0 Td[(j)]TJ /F8 7.97 Tf 6.59 0 Td[(v4)=21ij)]TJ /F10 7.97 Tf 5.18 0 Td[(1+h(j)]TJ /F3 11.955 Tf 9.96 0 Td[(1,j)]TJ /F3 11.955 Tf 9.96 0 Td[(2). (B) Noticethatthe~elementsappearinginh(j)]TJ /F3 11.955 Tf 10.16 0 Td[(1,j)]TJ /F3 11.955 Tf 10.17 0 Td[(1)areobtainedwithk=0,kj)]TJ /F3 11.955 Tf 10.17 0 Td[(1.Thustheconstraintong(j,j)]TJ /F3 11.955 Tf 9.97 0 Td[(1)poseextralimitsonj)]TJ /F10 7.97 Tf 5.18 0 Td[(1.Bychangingtheindexfromjtoj+1,wehave, )]TJ /F5 11.955 Tf 9.96 0 Td[(h(j,j)]TJ /F3 11.955 Tf 9.97 0 Td[(1)2canbeobtained. 106

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Forj=2,00.Fromthiscubicequation,wecanobtainavalidvalueof1.ForaRSDwithparameterk=1000,=0.05,c=0.08,thepossiblerangesare:1>0.589or1<0.063.Sincethedecomposeddistributionwillhavelargevalueatdegreeoneduetothelargevalueof2,therstrangeischoseninpractice.Finally,byputtingallconstraintsforjtogether,wehavetheresultsinProposition 2 107

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APPENDIXCPROOFOFNONNEGATIVESOLUTIONOFDRC Fori=1,!1=1+PDi=21 (wg)i)]TJ /F26 5.978 Tf 5.75 0 Td[(1i>0.Fori2,thesolutionfor!icanbecomputedconsecutivelyfromEquation( 4 )as: !i=1 i1"i)]TJ /F8 7.97 Tf 13.75 14.94 Td[(i)]TJ /F10 7.97 Tf 6.59 0 Td[(1Xj=1~i,j!j#. (C) Assume!k,kiaresolvedwithnonnegativevalues.Fori+1,i+1=i)]TJ /F10 7.97 Tf 6.58 0 Td[(1 i+1i=i)]TJ /F10 7.97 Tf 6.59 0 Td[(1 i+1Pij=1~i,j!j,thus !i+1=1 i+11"i+1)]TJ /F8 7.97 Tf 19.25 14.95 Td[(iXj=1~i+1,j!j#=1 i+11iXj=1i)]TJ /F3 11.955 Tf 11.95 0 Td[(1 i+1~i,j)]TJ /F3 11.955 Tf 13.93 2.66 Td[(~i+1,j!j. (C) Observethatentriesinthe(i+1)thcolumnof~aretheconvolutionoftheithcolumnof~with.Thuseachentryinthe(i+1)thcolumn(~i+1,j)canbewrittenastheweightedsummationoftheithcolumnas:~i+1,j=i)]TJ /F8 7.97 Tf 6.59 0 Td[(j+2Xk=1k~i+1)]TJ /F8 7.97 Tf 6.59 0 Td[(k,j)]TJ /F10 7.97 Tf 6.59 0 Td[(1. ThenthedifferenceterminEquation( C )canbeexpressedas: i)]TJ /F3 11.955 Tf 11.96 0 Td[(1 i+1~i,j)]TJ /F3 11.955 Tf 13.93 2.66 Td[(~i+1,j=i)]TJ /F3 11.955 Tf 11.96 0 Td[(1 i+1i)]TJ /F8 7.97 Tf 6.59 0 Td[(j+1Xk=1k~i)]TJ /F8 7.97 Tf 6.59 0 Td[(k,j)]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F8 7.97 Tf 11.95 15.21 Td[(i)]TJ /F8 7.97 Tf 6.59 0 Td[(j+2Xk=1k~i+1)]TJ /F8 7.97 Tf 6.59 0 Td[(k,j)]TJ /F10 7.97 Tf 6.59 0 Td[(1. (C) Aftersomemanipulation,theaboveequationcanberearrangedtobe:i)]TJ /F3 11.955 Tf 9.96 0 Td[(1 i+1~i,j)]TJ /F3 11.955 Tf 11.94 2.65 Td[(~i+1,j=1i)]TJ /F3 11.955 Tf 9.97 0 Td[(1 i+1~i)]TJ /F10 7.97 Tf 5.18 0 Td[(1,j)]TJ /F10 7.97 Tf 5.17 0 Td[(1)]TJ /F3 11.955 Tf 11.94 2.65 Td[(~i,j)]TJ /F10 7.97 Tf 5.18 0 Td[(1+i)]TJ /F8 7.97 Tf 5.18 0 Td[(j+1Xk=2i)]TJ /F3 11.955 Tf 11.96 0 Td[(1 i+1k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+1~i)]TJ /F8 7.97 Tf 5.18 0 Td[(k,j)]TJ /F10 7.97 Tf 5.18 0 Td[(1)]TJ /F6 11.955 Tf 9.96 0 Td[(2~i)]TJ /F10 7.97 Tf 5.17 0 Td[(1,j)]TJ /F10 7.97 Tf 5.17 0 Td[(1. Withthetentativesolutionof(x)inEquation( 4 ),thefollowingrelationshipholds:k+1 k=1 wgk)]TJ /F10 7.97 Tf 6.59 0 Td[(1 k+11i)]TJ /F3 11.955 Tf 11.96 0 Td[(1 i+1~i)]TJ /F10 7.97 Tf 6.59 0 Td[(1,j)]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 13.93 2.66 Td[(~i,j)]TJ /F10 7.97 Tf 6.59 0 Td[(1+i)]TJ /F3 11.955 Tf 11.95 0 Td[(1 i+1)]TJ /F3 11.955 Tf 20.67 8.09 Td[(1 3wg2~i)]TJ /F10 7.97 Tf 6.59 0 Td[(2,j)]TJ /F10 7.97 Tf 6.58 0 Td[(1)]TJ /F6 11.955 Tf 11.84 0 Td[(2~i)]TJ /F10 7.97 Tf 6.58 0 Td[(1,j)]TJ /F10 7.97 Tf 6.58 0 Td[(1. 108

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Forlargenumberofgroupsgorwithappropriatelychoosingweightingfactorw,1 3wgi)]TJ /F10 7.97 Tf 6.59 0 Td[(1 i+1,thuswecanignore1 3wgwithtightapproximation.Aftersomemanipulation,wecanhavethefollowinginequality:i)]TJ /F3 11.955 Tf 9.96 0 Td[(1 i+1~i,j)]TJ /F3 11.955 Tf 11.94 2.65 Td[(~i+1,j>1(i)]TJ /F3 11.955 Tf 9.96 0 Td[(1))]TJ /F3 11.955 Tf 9.96 0 Td[(1 (i)]TJ /F3 11.955 Tf 9.96 0 Td[(1)+1~i)]TJ /F10 7.97 Tf 5.17 0 Td[(1,j)]TJ /F10 7.97 Tf 5.18 0 Td[(1)]TJ /F3 11.955 Tf 11.94 2.65 Td[(~i,j)]TJ /F10 7.97 Tf 5.17 0 Td[(1+2(i)]TJ /F3 11.955 Tf 9.97 0 Td[(2))]TJ /F3 11.955 Tf 9.96 0 Td[(1 (i)]TJ /F3 11.955 Tf 9.97 0 Td[(2)+1~i)]TJ /F10 7.97 Tf 5.17 0 Td[(2,j)]TJ /F10 7.97 Tf 5.18 0 Td[(1)]TJ /F3 11.955 Tf 11.94 2.65 Td[(~i)]TJ /F10 7.97 Tf 5.18 0 Td[(1,j)]TJ /F10 7.97 Tf 5.18 0 Td[(1. Forj=1,~i,1=i,iDand~i,1=0,i>D.Thustheinequalityi)]TJ /F10 7.97 Tf 6.59 0 Td[(1 i+1~i,j)]TJ /F3 11.955 Tf 13.98 2.66 Td[(~i+1,j0holdsforj=1.Bymathematicalinduction,itisreadilyproventhati)]TJ /F10 7.97 Tf 6.58 0 Td[(1 i+1~i,j)]TJ /F3 11.955 Tf 14.25 2.65 Td[(~i+1,j0foranyj1.PlugthisresultbackintoEquation( C ),wehave!i+10.Therefore,thelemmaisproved. 109

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BIOGRAPHICALSKETCH RuiCaowasborninJiujiang,Jiangxi,China.HereceivedhisB.S.degreeinphysicsfromNanjingUniversity,Nanjing,China,in2003,andhisM.S.degreeinphysicsfromArizonaStateUniversity,Tempe,Arizona,in2006.HereceivedhisPhDdegreeinelectricalandcomputerengineeringfromtheUniversityofFlorida,Gainesville,FL,in2011.Hiscurrentresearchinterestsincludecooperativerelaycommunications,underwateracousticcommunications,anddistributeddatastorage. 117