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Power Control of CDMA-Based Cellular Communication Networks with Time-Varying Stochastic Channel Uncertainties

Permanent Link: http://ufdc.ufl.edu/UFE0024438/00001

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Title: Power Control of CDMA-Based Cellular Communication Networks with Time-Varying Stochastic Channel Uncertainties
Physical Description: 1 online resource (61 p.)
Language: english
Creator: Subramanian, Sankrith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Power control is used to ensure that each link achieves its target signal-to-interference-plus-noise ratio (SINR) to effect communication in the reverse link (uplink) of a wireless cellular communication network. In cellular systems using direct-sequence code-division multiple access (CDMA), the SINR depends inversely on the power assigned to the other users in the system, creating a nonlinear control problem. Due to the spreading of bands in CDMA based cellular communication networks, the interference in the system is mitigated. The nonlinearity now arises by the uncertain random phenomena across the radio link, causing detrimental effects to the signal power that is desired at the base station. Mobility of the terminals, along with associated random shadowing and multi path fading present in the radio link, results in uncertainty in the channel parameters. To quantify these effects, a nonlinear MIMO discrete differential equation is built with the SINR of the radio-link as the state to analyze the behavior of the network. Controllers are designed based on analysis of this networked system, and power updates are obtained from the control law. Analysis is also provided to examine how mobility and the desired SINR regulation range affects the choice of channel update times. Realistic wireless network mobility models are used for simulation and the power control algorithm formulated from the control development is verified on this mobility model for acceptable communication.
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 Sankrith Subramanian.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.
Local: Co-adviser: Shea, John M.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024438:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024438/00001

Material Information

Title: Power Control of CDMA-Based Cellular Communication Networks with Time-Varying Stochastic Channel Uncertainties
Physical Description: 1 online resource (61 p.)
Language: english
Creator: Subramanian, Sankrith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Power control is used to ensure that each link achieves its target signal-to-interference-plus-noise ratio (SINR) to effect communication in the reverse link (uplink) of a wireless cellular communication network. In cellular systems using direct-sequence code-division multiple access (CDMA), the SINR depends inversely on the power assigned to the other users in the system, creating a nonlinear control problem. Due to the spreading of bands in CDMA based cellular communication networks, the interference in the system is mitigated. The nonlinearity now arises by the uncertain random phenomena across the radio link, causing detrimental effects to the signal power that is desired at the base station. Mobility of the terminals, along with associated random shadowing and multi path fading present in the radio link, results in uncertainty in the channel parameters. To quantify these effects, a nonlinear MIMO discrete differential equation is built with the SINR of the radio-link as the state to analyze the behavior of the network. Controllers are designed based on analysis of this networked system, and power updates are obtained from the control law. Analysis is also provided to examine how mobility and the desired SINR regulation range affects the choice of channel update times. Realistic wireless network mobility models are used for simulation and the power control algorithm formulated from the control development is verified on this mobility model for acceptable communication.
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 Sankrith Subramanian.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.
Local: Co-adviser: Shea, John M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024438:00001


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POWERCONTROLOFCDMA-BASEDCELLULARCOMMUNICATION NETWORKSWITHTIME-VARYINGSTOCHASTICCHANNELUNCERTAINT IES By SANKRITHSUBRAMANIAN ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2009 1

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c r 2009SankrithSubramanian 2

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Tomyparents,P.R.SubramanianandIndhumathiSubramanian ;mysisterShilpa; andmyfriendsandfamilymembers,whoconstantlyprovidedm ewithmotivation, encouragementandjoy 3

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ACKNOWLEDGMENTS Iexpressmymostsincereappreciationtomysupervisorycom mitteechairand mentor,Dr.WarrenE.Dixon.Ithankhimfortheeducation,ad vice,andtheencouragement thathehadprovidedmewithduringthecourseofmystudyatth eUniversityofFlorida. IalsothankDr.JohnM.Sheaforlendinghisknowledgeandsup port,andproviding technicalguidance.Itisagreatpriviledgetohaveworkedw ithsuchfar-thinkingand inspirationalindividuals.AllthatIhavelearntandaccom plishedwouldnothavebeen possiblewithouttheirdedication. Ithankallofmycolleaguesforhelpingmewithmythesisrese archandcreatinga friendlyworkatmosphere.Ialsoextendmyappreciationtot hem,especiallyParagM. Patre,SiddharthaS.Mehta,andWilliamMackunis,forshari ngtheirknowledgeand encouragingsomethought-provokinganalyticaldiscussio ns. Mostimportantly,Iwouldliketoexpressmydeepestappreci ationtomyparents P.R.SubramanianandIndhumathiSubramanianandmysisterS hilpa.Theirlove, understanding,patienceandpersonalsacricemadethisdi ssertationpossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2Radio-ChannelModeling ............................... 14 3RobustPowerControlofCellularCommunicationNetworksw ithTime-Varying ChannelUncertainties ................................ 22 3.1ControlDevelopment ............................. 22 3.1.1ControlObjective ............................ 22 3.1.2Closed-LoopErrorSystem ....................... 23 3.2StabilityAnalysis ............................... 24 3.3EstimationofErroratUnsampledInstances ................. 25 3.4Simulation .................................... 28 3.4.1NetworkMobilityModel ........................ 28 3.4.2SimulationResults ........................... 32 4Prediction-BasedPowerControlofDistributedCellularC ommunicationNetworks withTime-VaryingChannelUncertainties ..................... 35 4.1NetworkModel ................................. 36 4.2LinearPredictionofFadingCoecient .................... 37 4.3ControlDevelopment .............................. 40 4.3.1ControlObjective ............................ 40 4.3.2ClosedLoopErrorSystem ....................... 41 4.4StabilityAnalysis ................................ 43 4.5SimulationResults ............................... 45 5CONCLUSION .................................... 51 5.1SummaryofResults .............................. 51 5.2RecommendationsforFutureWork ...................... 52 APPENDIX AESTIMATIONOFRANDOMPROCESSES .................... 53 5

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A-1GeneralMMSEbasedestimationtheory ................... 53 A-2GaussianCase .................................. 54 BOrthogonalityCondition ............................... 56 REFERENCES ....................................... 58 BIOGRAPHICALSKETCH ................................ 61 6

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LISTOFTABLES Table page 3-1PercentageofsampleswithinthedesiredSINRrange ............... 33 4-1PercentageofsampleswithinthedesiredSINRrange ............... 49 7

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LISTOFFIGURES Figure page 2-1Reverselink. ..................................... 15 2-2FadingduetoDopplershiftandscattering. ..................... 15 2-3Probabilitydensityfunction(PDF)ofaRayleighrandom variable. ....... 17 2-4Powerofthereceivedenvelopefora10Hzfadingchannel. ............. 18 3-1Autocorrelationfunctionforfading. ......................... 28 3-2Cellularnetworktopology-randomway-pointmobilitym odel .......... 29 3-3Errorplot:MTswithlowdopplerfrequencies. ................... 30 3-4Errorplot:MTswithhighdopplerfrequencies. .................. 31 3-5Error,channelgainandpowerplot:MTwithadopplerfreq uencyof1.98Hz. 31 3-6Error,channelgainandpowerplot:MTwithadopplerfreq uencyof34.14Hz. 32 4-1Distributedcellularnetworktopology. ........................ 45 4-2Error,channelgain,andpowerplotofaMTwithmaximumDo pplerfrequency 4.11Hz. ........................................ 46 4-3PredictionerroroftheMTwithmaximumDopplerfrequenc y4.11Hz. ..... 47 4-4Error,channelgain,andpowerplotofaMTwithmaximumDo pplerfrequency 30.9Hz. ........................................ 47 4-5PredictionerroroftheMTwithmaximumDopplerfrequenc y30.9Hz. ..... 48 4-6Comparisonofhighgainandpredictivepowercontrolalg orithms. ........ 50 8

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience POWERCONTROLOFCDMA-BASEDCELLULARCOMMUNICATION NETWORKSWITHTIME-VARYINGSTOCHASTICCHANNELUNCERTAINT IES By SankrithSubramanian May2009 Chair:WarrenE.DixonCo-Chair:JohnM.SheaMajor:ElectricalandComputerEngineering Powercontrolisusedtoensurethateachlinkachievesitsta rgetsignal-to-interferenceplus-noiseratio(SINR)toeectcommunicationintherever selink(uplink)ofawireless cellularcommunicationnetwork.Incellularsystemsusing direct-sequencecode-division multipleaccess(CDMA),theSINRdependsinverselyonthepo werassignedtothe otherusersinthesystem,creatinganonlinearcontrolprob lem.Duetothespreading ofbandsinCDMAbasedcellularcommunicationnetworks,the interferenceinthe systemismitigated.Thenonlinearitynowarisesbytheunce rtainrandomphenomena acrosstheradiolink,causingdetrimentaleectstothesig nalpowerthatisdesiredat thebasestation.Mobilityoftheterminals,alongwithasso ciatedrandomshadowing andmulti-pathfadingpresentintheradiolink,resultsinu ncertaintyinthechannel parameters.Toquantifytheseeects,anonlinearMIMOdisc retedierentialequationis builtwiththeSINRoftheradio-linkasthestatetoanalyzet hebehaviorofthenetwork. Controllersaredesignedbasedonanalysisofthisnetworke dsystem,andpowerupdates areobtainedfromthecontrollaw.Analysisisalsoprovided toexaminehowmobilityand thedesiredSINRregulationrangeaectsthechoiceofchann elupdatetimes.Realistic wirelessnetworkmobilitymodelsareusedforsimulationan dthepowercontrolalgorithm formulatedfromthecontroldevelopmentisveriedonthism obilitymodelforacceptable communication. 9

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CHAPTER1 INTRODUCTION Varioustransmitterpowercontrolmethodshavebeendevelo pedtodeliveradesired qualityofservice(QoS)inwirelessnetworks[ 1 { 8 ].Earlyworkonpowercontrolusinga centralizedapproachwasinvestigatedin[ 9 ]and[ 10 ].TheconceptofSignal-to-Interference (SIR)balancingwasintroducedin[ 9 ]and[ 10 ],whereallreceiversexperiencethesameSIR levels.MaximumachievableSIRswereformulatedconsideri ngtheSIRbalancingproblem asaneigenvalueproblem.Astochasticdistributedtransmi tterpowerapproachwasalso investigatedin[ 6 { 8 ].Methodsweredevelopedtoreduceco-channelinterferenc eforagiven channelallocationusingtransmitterpowercontrolin[ 6 ]and[ 8 ].In[ 6 ],transmitterpower controlschemesaredevelopedtoreducethecochannelinter ferences,theperformance ofwhichismeasuredbydeningOutageprobabilitiesasthep robabilityofhavinga toolowSignaltoInterference(SIR)ratio.Anoptimum(inth esensethattheoutageor interferenceprobabilityisminimized)eigenvaluebasedp owercontrolschemeisemployed usingthisapproach. Theperformanceofoptimumtransmitterpowercontrolalgor ithmsisinvestigated in[ 8 ].Performanceboundsandconditionsofstabilityforallty pesfortransmitterpower controlalgorithmsarefound.Thesystemmodelisdeveloped using N cellswith M independentchannelpairs(andhencecrosstalkbetweencha nnelsisneglected).The thermalnoiseisneglected,asaninterferencelimitedsyst emisconsidered.Alinkgain matrixisintroducedin[ 8 ],eachofthecomponentsisdenedasthelinkgainfromthe basestationincell j tothemobileterminalincell i ,normalizedtothelinkgaininthe desiredpath,frombasestation i tomobileterminal i .In[ 8 ],aglobalpowercontrol algorithmisdenedasanalgorithmthathasaccesstotheent iregainmatrixinevery instant.Anoptimumpowercontrolalgorithmthatminimizes theinterferenceprobability isproposedin[ 8 ]assumingglobalpowercontrol,andtheperformancebounds arederived 10

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forthisalgorithm.AStepwiseRemovalAlgorithm(SRA)ispr oposedin[ 8 ]thatminimizes thecomputationalrequirements,sueredbytheoptimalpow ercontrolalgorithm. Toaddresstheproblemsassociatedwithmeasurementsoflin kgainsin[ 8 ],a distributedapproachwasmadein[ 7 ],whereonlySIRmeasurementsinthoselinks actuallyinusearemade.AdistributedSIRbalancingscheme isdevelopedandadiscrete timepowercontrolalgorithm(DPCA)isproposedas P ( l +1)= ZP ( l ) ; (1{1) where P isthepowervector,and isacontrolgainchosentoavoidincreasingpowers. Duetothedicultyincalculatingthisquantity,astepwise removalwithdistributed balancingschemeisdevelopedin[ 7 ].Thesealgorithmswereframedwhenonlypathloss waseectingthechanneluncertainty. AcentralizedSIRbalancingpowercontrolschemewasformul atedin[ 11 ].Fadingand noisewereignoredinthisapproach.Aneigenvalueapproach aimedatachievingthesame targetSIRforalltheradiolinksisused.Anupperlimitfort hepowerwasimposedto eachuserintheconstrainedpowercontrolalgorithmof[ 3 ],andanoptimumpowercontrol formaximizingtheminimumSIRisformulated. Asimpledistributedautonomouspowercontrolalgorithmwa sintroducedin[ 2 ] wherechannelreuseismaximized.Networkswherecertainpo wersettingsexistsare consideredandexponentialfastconvergencetosuchsettin gsisdemonstratedin[ 2 ].Local measurementsweremadein[ 2 ]tomeetthetargetSINRineachchannel.Forthispurpose, thedistributedcontrollawismanipulatedintermsofthepo wer,andSINRformobile terminalinaradiolink.Makinglocalmeasurementshelpsin [ 2 ]impliesthattheSINRfor eachradiochannelateverysamplinginstantismeasurable, theSINRbeingafunctionof notonlyainterferenceinthecurrentcell,butalsoco-chan nelinterferencesfromadjacent cells.Basedonthelinearanalysisofthesystem,andconstr ainingtheeigenvalues,the powerapproachesanoptimalpowervector.Thepoweralgorit hmformulationforthis 11

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approachisapplicabletobothuplink(mobileterminaltoba sestation)anddownlink (basestationtomobileterminal)channels. Aframeworkwhichintegratespowercontrolandbasestation assignmentwas introducedby[ 12 ].AMinimumTransmittedPower(MTP)wasformulatedfora CDMA-basedsysteminwhichthetotalpowerisminimizedsubj ecttomaitainingas individualtargetSIRforeachmobile.Synchronous(poweru pdatesaredoneatthesame samplingrateforallradiolinks)andasynchronous(poweru pdatesdoneatdierent samplingratesfordierentradiolinks)distributedalgor ithmsthatndtheoptimalpower vectorandbasestationassignmentisidentied. Ageneralizedframeworkforuplinkiterativepowercontrol isprovidedin[ 5 ],where commonpropertiesforinterferenceconstraintsareidenti ed.Theproblemofndingthe powercontrolvectorfortheradiolinkstoachieveacceptab lecommunicationisreducedto satisfyingavectorinequalityconditionstatedin[ 5 ]astheinterferencethatausermust overcometoachieveanacceptableconnection.Synchronous andasynchronouspower controlconvergetoanoptimalpowerwhichminimizesthetot altransmittedpower. Activelinkprotection(ALP)schemeswereintroducedin[ 1 ]and[ 13 ],wheretheQoS ofactivelinksismaintainedaboveathresholdlimittoprot ectthelinkquality.In[ 1 ], theFoschini-Miljanicpowercontrolalgorithmismodiedf orreducedmeasurementsand emphasiswasgivenonALPforDistributedPowerControl(DPC ). Optimalpowercontrolalgorithmwithoutage-probabilityc onstraintswasdeveloped in[ 14 ],wherethenetworkisinterference-limitedwithRayleigh fadingofboththedesired andinterferencesignals.PowercontrolwithjointMultius erDetection(MUD)schemewas developedin[ 15 ]forRayleighfadedsystemswithoutageconstraints. Recently,adistributedpowercontrolschemewassuggested in[ 16 ]inthepresence ofradiochanneluncertaintiescausedbymobilityoftheuse rterminals.Thesechannel uncertaintiesincludeexponentialpathloss,shadowing,a ndmulti-pathfading,which aremodeledasrandomvariablesintheSINRmeasurements.Th euncertaintyofthe 12

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multi-pathfadingeectsprovidedmotivationfortheresul tsin[ 16 ]and[ 17 ].Specically, apersistentlyexcitingadaptationschemeisproposedin[ 16 ]and[ 17 ].However,inthese works,thefadingprocessismodeledasslowlychangingsoth atthechannelgaincanbe accuratelyestimatedandpracticallimitationsoftransmi ssionpowerlimitationsarenot considered. Ofthechanneluncertainties,multi-pathfadinghasthemos tcriticaleectonthe designofapower-controlsystembecauseofthetimeandampl itudescales.Multi-path fadingiscausedbyrerectionsintheenvironment,whichcau semultipletime-delayed versionsofthetransmittedsignaltoaddtogetherattherec eiver.Thetimeosets causethesignalstoaddwithdierentphases,andthusmulti -pathfadingcanchange signicantlyoverdistancescalesasshortasafractionofa wavelength.Forinstance, forasystemusingthe900MHzcellularband,thechannelcohe rencetime(thetime forwhichthechannelisessentiallyinvariant)foramobile terminaltravelingat30 miles/hourisapproximately10ms.Thereisaneedtoquantif ythemulti-pathfading eectsofthechannelinthesystem.Inthisthesis,eortsar emadetounderstand thefadingphenomenaintheradiochannelofaCDMA-basedcel lularcommunication networkandquantifythemtodeveloppowercontrolalgorith ms.Themodelingofcellular communicationnetworksisbasedonanalysisofthenonlinea rnetworkedsystemand Lyapunov-basedcontrolstructuresareformulatedforsuch systemsinthisthesis.An analyticalapproachtochoosingpowerupdatesamplingtime isusedinthisthesiswhere channeluncertainties(especiallyRayleighfading)arequ antiedbasedonestimationof errorbetweenthedesiredandactualSignal-to-Interferen ceplusNoiseRatio(SINR). 13

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CHAPTER2 RADIO-CHANNELMODELING Inthischapter,thecharactericticsofthereverselinkoft heradiochannelis investigatedandmodeled. Thechannelgainofaradiolink(seeFigure 2-1 )iscomprisedofthreecomponents: Exponentialpath-loss,Log-normalshadowing,andMulti-p athfading.Thegainofthe channelisdened[ 18 ]as g ii ( l )= g d 0 d i ( l ) d 0 10 0 : 1 i ( l ) j X i ( l ) j 2 ; (2{1) wheretheterm j X i ( l ) j 2 isusedtomodelRayleighfading., g d 0 isthenear-eldgaingiven by[ 19 ] g d 0 = G t G r 2 (4 ) 2 d 20 L ;d f d 0 d i ( l ) ; (2{2) where G t isthetransmitterantennagain, G r isthereceiverantennagain, isthe wavelengthinmeters, L isthesystem-lossfactor, d 0 isthedistancebetweenthe transmitterandreceiverantenna,and d f =6 m istheFraunhoferdistance.Without lossofgenerality, G t G r ,and L areallassumedtobe1.Sincethepowerupdatesare providedatdiscreteinstancesduetobandwidthconstraint s,thesystemisanalyzedat discreteinstancesoftime( l 2 Z ).Forthisreason,thecontinuoustimechannelparameters areanalyzedandasuitablechannelsamplingtimeischoseni nChapter 3 Theterm d i ( l ) d 0 isusedtomodeltheaveragepathlossatdistance d i ( l )from mobileterminal(MT) i tothebasestation(BS),where isthepath-lossexponent,which typicallytakesvaluesbetweentwoandve.Theterm10 0 : 1 i ( l ) isusedtomodellarge-scale log-normalshadowingfrombuildings,terrain,orfoliage, where i ( l )isaGaussianrandom process(see[ 19 ]). Figure 2-2 showsthetypicalscenarioofaMTcommunicatingwithaBS.Th e receivedsignalattheBSisfadedduetothemobilityoftheMT scausingdopplershifts inthefrequencyofthereceivedwaveandmultipathpropagat ionofthewavecaused 14

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Figure2-1.Reverselink. Figure2-2.FadingduetoDopplershiftandscattering. 15

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byscatteringinthepresenceofsurroundingobjects.These individualcomponentsadd upinaconstructiveordestructivemanner,dependingonran domphaseshiftsofthese componentsofthereceivedsignal. Thereceivedfadingcomponentofthesignalcanberepresent edas[ 19 ] X i ( t )= G c ( t )cos(2 f c t ) G s ( t )sin(2 f c t )(2{3) where f c isthecarrierfrequency,theGaussianrandomprocesses G c ( t )and G s ( t )are denedas G c ( t )= E 0 N X n =1 C n cos(2 f n t + n )(2{4) G s ( t )= E 0 N X n =1 C n sin(2 f n t + n ) : (2{5) Theprocesses G c ( t )and G s ( t )areuncorrelatedzero-meanGaussianrandomvariables forany t withequalvariance E 2 0 2 ,where E 0 istherealamplitudeofthelocalaverage E-eld(assumedconstant), C n istherealrandomvariablerepresentingtheamplitudeof individualwaves, n isthephaseshiftduetorerectionsoftheindividualwavesa ndisan uniformrandomvariablein[0 ; 2 ], N isthenumberofscatteredwaves,and f n ( t )isthe dopplerfrequencydenedas f n = v cos : (2{6) InEquation 2{6 v ( t )isthevelocityofmotionoftheMTand ( t )istheanglebetweenthe transmittedsignalandthedirectionofmotionoftheMT. Theenvelopeofthereceivedsignal(E-eld)is j X i ( t ) j = p G 2c ( t )+ G 2s ( t ) ; (2{7) 16

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Rayleigh Variable, Xf(X) Figure2-3.Probabilitydensityfunction(PDF)ofaRayleig hrandomvariable. where j X i ( t ) j isarandomvariablewithaRayleighdistributionwithaprob abilitydensity functionof(refertoFigure 2-3 ) p ( j X i j )= X i E 2 0 2 exp 0@ X 2 i 2 E 2 0 2 1A ; 0 X i 1 (2{8) =0 ;X i < 0 : SquaringEquation 2{7 yieldsthefadingpower,i.e., j X i ( t ) j 2 = G 2c ( t )+ G 2s ( t ) : (2{9) Thepowerofthereceivedenvelopeforafadedradiochannel( Dopplerfrequency=10Hz) isshowninFigure 2-4 Foranalyticalpurposes, X i ( t )isusuallytakentobeacomplex-valuedGaussian randomprocess,andthus j X ( t ) j isaRayleighrandomvariableforeach t when E [ X ( t )]= 0(theoperator E [ X ]isusedtorepresenttheexpectedvalueofarandomvariable X ), whichcorrespondstonoline-of-sightpathfromtheMTtothe BS.Gaussianrandom 17

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0 500 1000 1500 2000 2500 3000 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 Time (Sampling time = 1.7ms)Power (dBm) Figure2-4.Powerofthereceivedenvelopefora10Hzfadingc hannel. processesprovidegoodmodelsforthelog-normalshadowing andRayleighfadingoverthe most-probablerangeofreception.However,bothofthesepr ocessesareunbounded,which meansthatanyreceivedpowerlevelispossible.However, g ii ( l )cannottakearbitrarily largevaluesinpracticebecausethereceivedpowercannote xceedthetransmittedpower. Furthermore,acellularsystemcannotpracticallytransmi tto overfaded userswhoarein verydeepfades(i.e.,when g ii ( l )isclosetozero)becausedoingsowouldrequireextremely largepoweratthatuserandtheotherusers(becausethepowe rtransmittedtoeachuser causesinterferenceattheotherusers)[ 20 ].Hence,thesubsequentdevelopmentisbasedon theassumptionthatthefadingpower j X i ( ) j 2 isboundedandnon-zero. Thedevelopmentinthisthesisconsidersthereversechanne l(fromtheMTstothe BS)andinvestigatescontroloftheSINRsfortheMTs.TheSIN RatMT i ,denotedby x i ( l ) 2 R ,canbeexpressedas[ 21 ] x i ( l )= ag ii ( l ) P i ( l ) I i ( l ) ; (2{10) 18

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where P i ( l ) 2 R isthepowerfromtheMT i totheBS,and g ii ( l ) 2 R isthechannelgain fromtheBStotheMT i .InEquation 2{10 I i ( l ) 2 R denotestheinterference-plus-noise powerattheBSduetotransmissionsbyotherMTsinthecellul arnetwork,denedas I i ( l )= X j 6 = i g ij ( l ) P j ( l )+ a i ; (2{11) where g ij ( l ) 2 R isthechannelgainforthelinkbetweenMT j andtheBSthataects theinterferenceintheradiolinkbetweenMT i andtheBS, P j ( l ) 2 R isthepower transmittedbyMT j totheBS,and i 2 R denotesthethermalnoiseinlink i .Ina CDMAbasednetwork,eachradiolinkisforcedtosharethesam ebandwidth;hence, I i ( ) isnon-zeroandbounded.Thebandwidthspreadingfactor,or theprocessinggain[ 22 ]for thecellularsystemusingCDMAisdenotedby a denedas a = W R ; (2{12) where W isthetransmissionbandwidth,inhertz,and R isthedatarateinbits/second. Byincreasingthebandwidthspreadingfactor,theinterfer enceofthesystemcanbe reduced.Therefore,focusislaidontheeectsoffadingint heradiochannelinthis thesistodeveloppowercontrollersforradiolinksoperati nginaCDMAbasedcellular communicationnetwork. TherstdierenceoftheSINRdenedinEquation 2{10 canbedeterminedas x i ( l )= a ( I i ( l )+ I i ( l )) 1 g ii ( l ) T s P i ( l )+ a ( I i ( l )+ I i ( l )) 1 g ii ( l ) P i ( l ) T s f I i ( l )( I i ( l )+ I i ( l )) g 1 ( a X i 6 = j g ij ( l ) P j ( l ) T s g ii ( l ) P i ( l ) + a X i 6 = j g ij ( l ) P j ( l ) T s g ii ( l ) P i ( l ) ) + a ( I i ( l )+ I i ( l )) 1 : g ii ( l ) T s P i ( l ) T s a X i 6 = j g ij ( l ) P j ( l ) T 2 s : f I i ( l )( I i ( l )+ I i ( l )) g 1 g ii ( l ) P i ( l ) (2{13) 19

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whereEquation 2{10 andEquation 2{11 wereused,and T s isthepowerupdateinterval. NeglectingtheresidualtermsinsquarebracketsinEquatio n 2{13 ,approximating ( I i ( l )+ I i ( l )) I i ( l ),andusingEquation 2{10 yields x i ( l +1)= i ( l;x ) x i ( l )+ u i ( l ) ; (2{14) where i ( l;x ) 2 R isanunknown,time-varyingstate-dependentquantity,de nedas i ( l;x )= ag ii ( l +1) g 1 ii ( l ) a X i 6 = j ( g ij ( l ) P j ( l )) I 1 i ( l ) a X i 6 = j ( g ij ( l ) P j ( l )) I 1 i ( l ) = ) i ( l;x )= aI 1 i ( l ) P i ( l ) g ii ( l +1) x i ( l ) X i 6 = j ( g ij ( l ) P j ( l )) P i ( l ) + X i 6 = j ( g ij ( l ) P j ( l )) P i ( l ) # ; (2{15) and u i ( l ) 2 R isthecontrolinput,denedas u i ( l )= x i ( l ) P i ( l ) [ P i ( l +1) P i ( l )] ; (2{16) since ag ii ( l ) I i ( l ) = x i ( l ) P i ( l ) : Afterincludingmeasurementnoise i ( l;x ),theexpressioninEquation 2{14 canbe rewrittenas x i ( l +1)= i ( l;x ) x i ( l )+ u i ( l )+ i ( l;x ) : (2{17) Bydeningtheinterference I ( l ) 2 R n n asadiagonalmatrixwithentries I i ( l )expressed inEquation 2{11 g ( l ) 2 R n n asadiagonalmatrixwithentries g ii ( l ),and P ( l ) 2 R n ,then theMIMOsystemcanbedevelopedas x ( l +1)= ( l;x ) x ( l )+ u ( l )+ ( l;x ) ; (2{18) where ( l;x )=diag( i ( l;x )) 2 R n n denotestheunknown,time-varyingstate-dependent diagonalmatrix(since i ( l;x )isafunctionofthestate x i ( l )asshownintheEquation 2{15 )whichcanbeassumedtobeupperboundedbyaknownpositivec onstantfromthe precedingdiscussionontheuncertainchannelparameters, x ( l ) 2 R n isthestatevectorat 20

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instant l u ( l ) 2 R n isthecontrolinputvector, x ( l +1) 2 R n isthestatevectoratinstant l +1,and ( l;x ) 2 R n isthestochasticmeasurementnoiseboundedbyaknownconst ant. Themeasurementnoiseisassumedtobeboundedbyapositivec onstant. Here, u ( l )isexpressedintermsofthepowerupdatelawas P i ( l +1)= u i ( l ) x i ( l ) P i ( l )+ P i ( l ) : (2{19) 21

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CHAPTER3 ROBUSTPOWERCONTROLOFCELLULARCOMMUNICATIONNETWORKS WITHTIME-VARYINGCHANNELUNCERTAINTIES Inthischapter,theobjectiveistodesignandanalyzethepe rformanceofacontroller foruseinaradiochanneloperatinginaCDMAbasedcellularc ommunicationnetwork withRayleighfadingfollowingClarke'smodel[ 23 ].TheRayleighfadingprocessproduces unboundedchangesintheSINRswithnon-zeroprobability,e venforarbitrarilyshort timescales,butbyusingtheconceptofoverfadedusers[ 20 ],thechannelgainscanbe bounded.Basedonthismodel,asimpleproportionalcontrol lertominimizethesampled SINRerrorisdevelopedinthischapter.Specically,despi teuncertaintyinthemulti-path fadingeects,aLyapunov-basedanalysisisusedtodevelop anultimateboundforthe sampledSINRerrorwhichisafunctionoftheupperboundonth echanneluncertainty dividedbyanonlineardampinggainthatcanbemadearbitrar ilylargeuptosomeupper valuedictatedbythepowerupdatelaw.Theperformanceofth iscontrollerisevaluated inthischapterviasimulationunderrealisticpowerlimits andchannelchangesbasedon thestandardrandom-waypointmobilitymodel.Astatistica lanalysisoftheperformance eectsoffadingbetweenthesamplingintervalsisconsider edinthischapter,whichis usedtodiscussthechoiceofthecontrolupdaterate.Additi onalanalysisisprovided toconcludethattheexpectedvalueofthesquarednormofthe SINRerrorconverges toanultimateboundthatisafunctionofsamplingrate.Ther efore,thesamplingrate canbeadjustedtokeeptheSINRerrorwithinadesiredranget hatallowsforsignal decoding.SimulationresultsareprovidedforaRandom-Way pointmodelthatillustrates theperformanceofthedevelopedcontroller. 3.1ControlDevelopment 3.1.1ControlObjective TheSINRshouldremainbetweentwothresholdsas r min x i r max (3{1) 22

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toachieveacceptablecommunicationperformanceoverthel inkwhileminimizing interferencetoadjacentcells.Thecontrolobjectivefort hefollowingdevelopmentis toregulatetheSINRtoatargetvalueforeachchannel,denot edby r 2 R n ,whileensuring thattheSINRremainsbetweenthespeciedlowerandupperli mitsforeachchannel, asdescribedinEquation 3{1 .Toquantifytheobjective,aregulationerror e ( l ) 2 R n is denedas e ( l )= x ( l ) r: (3{2) 3.1.2Closed-LoopErrorSystem Therstdierenceoftheregulationerror,denotedas e ( l ) 2 R n ,is e ( l )= e ( l +1) e ( l )= x ( l +1) x ( l )(3{3) = ( l;x ) x ( l )+ u ( l )+ ( l;x ) x ( l ) : Tofacilitatethesubsequentanalysis,theexpressioninEq uation 3{3 isrewrittenas e ( l )= ( l;x )+n( l;x )+ u ( l ) ; (3{4) where ( l;x ) 2 R n denotesanauxiliarytermdenedas ( l;x )= ( l;x ) I n 1 e ( l ) ; (3{5) andn( l;x ) 2 R n isdenedas n( l;x )= ( l;x ) I n 1 r + ( l;x ) : (3{6) MotivationforintroducingtheauxiliarytermsinEquation 3{5 andEquation 3{6 isto collecttermsthathaveacommonupperbound.Specically,u pperboundsfor ( l;x )and n( l;x )canbedevelopedas k ( l;x ) k c 1 k e ( l ) k and k n( l;x ) k c 2 ; (3{7) 23

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where c 1 ;c 2 2 R denoteknownpositiveconstants.BasedonEquation 3{4 ,Equation 3{7 andthesubsequentstabilityanalysis,aproportionalcont rollerisdesignedas u ( l ) ( c 1 + k n + k 1 ) e ( l ) ; (3{8) where c 1 isintroducedinEquation 3{7 ,and k 1 ;k n 2 R denotepositivecontrolgains. BasedonEquation 2{19 andEquation 3{8 ,thepowerupdatelawis P i ( l +1)= ( c 1 + k n + k 1 ) e i ( l ) P i ( l ) ( e i ( l )+ r ) + P i ( l )(3{9) undertheconstraintthat0


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ByusingEquation 3{7 ,theexpressioninEquation 3{12 canbeupperboundedas V c 1 k e ( l ) k 2 + c 2 k e ( l ) k ( c 1 + k n + k 1 ) k e ( l ) k 2 c 2 k e ( l ) k k n k e ( l ) k 2 k 1 k e ( l ) k 2 : (3{13) CompletingthesquaresonthersttwotermsinEquation 3{13 yieldsthefollowingupper bound V k 1 V ( e;l )+ c 22 4 k n : (3{14) Lemma13.1of[ 24 ]cannowbeinvokedtoconcludethat V ( e;l ) b l V ( e ( l 0 ) ;l 0 )+ 1 b l k 1 c 22 4 k n ; (3{15) where b =1 k 1 ; where0
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Considertheperformanceforlarge t ,suchthattheerrormagnitudesatises j e ( l ) j = j x ( l ) r j <" .Let T s denotethetimebetweensamples.Thentheerrorforthesigna lfrom MT i attime t ,where lT s
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wheretheexpectation E [ g 2 ii ( t )]iswithrespecttotherandomchangeinthefading g ii ( t ). Let R g ( )betheautocorrelationfunctionofthechannelgainproces s.Theexpected valueinEquation 3{22 canbewrittenas E [ g 2 ii ( t )]= E [ g ( t ) g ( lT s )] 2 = E [ g 2 ( t )] 2 E [ g ( t ) g ( lTs )]+ E [ g 2 ( lTs )] =2 R g (0) 2 R G ( ) ; (3{23) where = t lTs .Inmostsystems,thesamplingtimewillbefastenoughthatt he exponentialpathlossandshadowingcanbemodeledasconsta ntbetweensamplingtimes, andthustheeectsofmulti-pathfadingisonlyconsidered. Theautocorrelationfunction forthepowerinaRayleighfadingprocess(refertoFigure 3-1 )isgivenby[ 23 ] R g ( )= J 2 0 (2 f n ) ; (3{24) where J 0 isthezeroth-orderBesselfunctionoftherstkind,and f n istheDopplerspread. TheDopplerspreadisgivenby fv=c ,where f isthecarrierfrequency, v isthemobile velocity,and c isthespeedoflight. Then,themean-squarederrorisboundedby E [ e 2i ( t )] < 2 a 2 [1 J 2 0 (2 f n )] P 2 i ( t ) P j 6 = i g ij ( l ) P j ( l )+ i ( t ) # 2 + 2CT < 2 a 2 1 J 2 0 (2 f n T s ) P 2 max n 2 P 2 min + 2CT ; (3{25) wheretheweaklawoflargenumbersisappliedtothedenomina torwith E [ g 2 ij ( l )]=1. Here, P max and P min are,respectively,themaximumandminimumtransmitpowers allocatedtoanon-overfadeduser.Bytakingintoaccountth emaximumpowerratio P max =P min ,numberofusers n ,spreadinggain a ,andmaximumMTvelocity, T s canbe 27

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0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Autocorrelation functionTime Figure3-1.Autocorrelationfunctionforfading. selectedtomakethemean-squareerrorbearbitrarilyclose to CT .However,sincethe mean-squareerrorcanneverbeidentically1,itisnotpossi bletoobtainazeroerror convergenceresultforthisdiscrete-timesystem. TogiveanideaoftheimplicationofEquation 3{22 ,considertheerrorwhen CT 0. Letthecarrierfrequency f =900 Mhz ,andmaximumvelocity v =30 miles=hour .Then theDopplerspreadis40 : 2 Hz .Toachieveamaximummean-squareerrorof0 : 1 SNR max where SNR max = 2 a 2 P 2 max n 2 P 2 min ,thesamplingtimemustbeapproximately1 : 8 ms .Theability toachievethisgoaldependsonthedatarateinthesystem.Fo rexample,at100 kbps data rate,thisrequiresapowercontrolupdateevery178 bits 3.4Simulation 3.4.1NetworkMobilityModel AcellularnetworktopologywasbuiltinMATLAB,andthemobi lityoftheMTs aremodeledbyasteadystate(stationary)distributionmod el(i.e.,[ 25 ],[ 26 ]).A 28

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Figure3-2.Cellularnetworktopology-randomway-pointmo bilitymodel Random-Waypointmodelisusedtosimulatethemobilityofth eMTs.Figure 3-2 showsa typicalcellularnetworktopologybuiltusingaRandomWayPointmodel[ 26 ]. Theerrorsignalisexpressedas e i dB ( l )=10log x i ( l ) r dB; (3{26) where r =8 dB isthetargetSINRasdenedinSection 3.1.1 witharangebetween6and 10 dB .Thermalnoise, ,issetto 110 dBm .ARayleighfadedchanneliscreatedusing thechannelsamplingtimeof1 : 7 ms obtainedfromtheerroranalysis(Section 3.3 )and theDopplerfrequency,giveninEquation 2{6 ,where =0 : 33 m isthewavelengthofthe signal.Theprobabilitydensityfunctionofthevelocityis givenby[ 26 ] f i ( v )= C h v f 0 V j h ( v ) ; (3{27) 29

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0 500 1000 1500 2000 2500 3000 -30 -25 -20 -15 -10 -5 0 5 10 15 20 Iterations, lError, in dB Doppler freq.=1.98Hz Doppler freq.=4.71Hz Doppler freq.=2.49Hz Doppler freq.=9.86Hz Doppler freq.=3.25Hz Doppler freq.=6.35Hz Figure3-3.Errorplot:MTswithlowdopplerfrequencies. where f 0 V j h ( v )= 1 v max v min (3{28) = 1 48 km=hr 2 km=hr = 1 46 km=hr isaclassicalchoiceforthedensityofthevelocity,and C h =14 : 47isthenormalizing constant.Thesubscript h isusedtodenotethephaseoftheMT[ 26 ].Thevelocityfor eachoftheMTsisobtainedfromEquation 3{27 usingtheinversetransformmethodas v =exp(3 : 179 r +0 : 6931) ; (3{29) where r isuniformlydistributedbetween0and1.TheDopplerfreque ncyisobtainedfrom Equation 3{29 andbymeasuring periodically.Pathloss,withfreespacepropagation eects(near-eldeects),andlog-normalshadowingaremo deled[ 19 ]asshownin Equation 2{1 andEquation 2{2 30

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0 500 1000 1500 2000 2500 3000 -25 -20 -15 -10 -5 0 5 10 15 20 25 Iterations, lError (dB) Doppler Freq. = 34.14Hz Doppler Freq. = 18.36Hz Doppler Freq. = 18.28Hz Doppler Freq. = 10.84Hz Figure3-4.Errorplot:MTswithhighdopplerfrequencies. 500 1000 1500 2000 2500 3000 -8 -6 -4 -2 0 2 4 Iterations, lError (dB)Error Plot 0 500 1000 1500 2000 2500 3000 -140 -130 -120 -110 -100 -90 Iterations, lChannel Gain g (dB)Channel Uncertainty Plot 0 500 1000 1500 2000 2500 3000 -10 0 10 20 30 Iterations, lPower (dBm)Power Plot Figure3-5.Error,channelgainandpowerplot:MTwithadopp lerfrequencyof1.98Hz. 31

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0 500 1000 1500 2000 2500 3000 -20 -10 0 10 20 30 Iterations, lError (dB)Error Plot 0 500 1000 1500 2000 2500 3000 -130 -120 -110 -100 -90 Iterations, lChannel Gain g (dB)Channel Uncertainty Plot 0 500 1000 1500 2000 2500 3000 -30 -20 -10 0 10 20 30 Iterations, lPower (dBm)Power Plot Figure3-6.Error,channelgainandpowerplot:MTwithadopp lerfrequencyof34.14Hz. 3.4.2SimulationResults TheresultsinFigures 3-3 3-6 areobtainedwith c 1 =8 10 5 ;k 1 =5 10 5 ;k n =1 : 625 ; andthespreadingfactor a ischosenas320.Figures 3-3 and 3-4 depicttheSINRerrorsfor radiolinksoperatingatDopplerfrequenciesrangingfrom0 10 Hz andfrom10 35 Hz respectively.Figures 3-5 and 3-6 depicttheSINRerror,channeluncertainty,andpower transmissionlevelsforDopplerfrequenciesof1 : 98 Hz and34 : 14 Hz ,respectively.These plotsindicatetheintuitivenotionthattheSINRerroriswi thinthedesiredthresholdfor moresamplesatlowerDopplerfrequenciesthatatthehigher frequencies.Thesecond columnofTable3-1quantiesthepercentageofsamplesthat liewithinthedesiredSINR rangeforeachDopplerfrequency.Whensamplesexceedtheup perlimitofthedesired SINRrange(i.e., x i ( l ) r max ),theQoSfortheindividuallinkisnotcompromised. However,exceedingtheupperlimitisundesirablebecauset heinterferencetootherlinks increases,potentiallyleadingtoanoutage(i.e.,when x i ( l ) r min )[ 6 { 8 ].Anoutage 32

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Table3-1.PercentageofsampleswithinthedesiredSINRran ge Doppler frequency(Hz) %ofsampleswhere r min x i r max %ofsampleswhere x i r min 1 : 9898 : 970 : 23 2 : 4998 : 670 : 27 3 : 2599 : 230 : 40 4 : 7197 : 471 : 23 6 : 3597 : 630 : 83 9 : 8694 : 901 : 97 10 : 8495 : 072 : 47 18 : 2888 : 333 : 87 18 : 3688 : 865 : 13 34 : 4175 : 4010 : 30 ofalinkdoescompromisethequalityofserviceinthesenset hatthesignalmaynot bedecodedatthatparticularsample.ThethirdcolumnofTab le3-1quantiesthe percentageofsamplesthatexperienceanoutageforeachDop plerfrequency,particularly duetofading[ 14 15 ]. Figures 3-3 through 3-6 andTable3-1indicatethatsomesamplesfalloutsideofthe desiredSINRrange(andexperienceanoutage)withincreasi ngoccurrencesathigher Dopplerfrequencies.Thesimulationmodelincludedareali sticupperlimitontheavailable power(i.e.,27 dBm (500 mW ))withaxedsamplingfrequency.Thesimulationalsoyield s rapidchangesinthechannelgains(i.e.,highfrequencycom ponentsintheuncertainty g ii ). Theserapidchangesareinruencedbyfading,whichinturnde terioratestheperformance ofthecontroller,especiallyathigherDopplerfrequencie s(i.e.,theupperbound c 2 in Equation 3{7 andEquation 3{17 hastobelargetoupperboundtheseeects).These rapidchangesareexacerbatedbyMTscomingoutofadeepfade dzone(i.e.,thechannel gainisveryclosetozero)andthechannelgainatthenextsam plecanleadtohigh valuecausinginterferencetootherusers.Increasing k n cancountertheseeects(i.e.,see Equation 3{17 ),butthemagnitudeof k n islimitedbythepowerupdatelawinEquation 3{9 andtheconstraintthat0


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coupledwithover-fading,whenthepowerofsomeMTsreachan uppersaturationlimit andthecontrollercannolongerincreasethepowertocompen sateforthefading. 34

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CHAPTER4 PREDICTION-BASEDPOWERCONTROLOFDISTRIBUTEDCELLULAR COMMUNICATIONNETWORKSWITHTIME-VARYINGCHANNEL UNCERTAINTIES FastchangingradiochannelsinaCDMAbasedcellularnetwor khavedetrimental eectsonthecontroleortsrequiredtoregulatetheSINRto thedesiredlevel,especially forchannelswithhighdopplerfrequencyMTs.Themotivatio nbehindintroducinga prediction-basedpowercontrolalgorithmistomeetthepro blemsassociatedwithrapid changesinthechannelgain(byordersofmagnitudebetweenp owerupdateintervals) inruencedbyfadingandexacerbatedbytheMTscomingoutoft he'deepfaded'zone. Forafastfadingchannel,areliablepredictionofthechann elcoecientisrequiredfor accuratecontroldesign.Forthispurpose,alinearpredict ionlterisusedinthischapter toestimatethechannelfadingparameterandthisinformati onisfedtothecontroller. AcontrollerisdevelopedinthischapterthatuseslocalSIN Rmeasurementsfromthe currentandneighboringcells([ 2 ],[ 1 ])tomaintaintheSINRsofalltheMTspresentin theacceptablecommunicationrange.ALyapunovbasedanaly sisisprovidedtoexplain theboundwhichtheSINRerrorreaches,thesizeofwhichcanb ereducedbychoosing appropriatecontrolgains.Thepowercontrolalgorithmiss imulatedonacellularnetwork withdistributedcellsandtheresultsindicatethatthecon trollerregulatestheSINRsofall theMTswithlowoutageprobability. Duetothefastfadingchannelthatapowercontrollerhastoe ncounterbefore ensuringacceptablecommunicationbetweentheMTsandtheB S,predictionofthefading powerwouldprovidethecontrollerwithusefulchannelinfo rmation.Earlierresearch conductedbyHallenonfadingpredictionfocusedonlongran geprediction[ 27 { 31 ]based onthefactthattheamplitude,frequencyandphaseofeachmu lti-pathcomponentvary muchslowerthantheactualfadingcoecient X i ( ).In[ 28 ],fadingchannelprediction iscombinedwithtransmittersignaloptimizationtomitiga tetheeectsofdeepfades. Physicalchannelmodeling,withadaptivepredictionwaspr esentedin[ 29 ].Consequently, 35

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focuswaslaidonperformanceanalysisoflongrangepredict ion,transmitterdiversityand adaptivelongrangepredictionoffastfadingchannelcoec ients[ 32 ]. Inthischapter,thebasicconceptsontheradiochannelchar acteristicsasdiscussed aboveisanalyzedandpowerofthefadingcoecientispredic ted,whichisusedinthe subsequentcontroldesign.Morespecically,foraccurate prediction,morerecentsamples areusedtoestimatethefadingcoecientatthenextinstanc e,unlikethelongrange fadingpredictionschemeswherethegoalistopredictthepa tternofthefadingenvelope. Forthispurpose,alinearMinimumMeanSquareError(MMSE)p redictorisusedto obtainareliablepredictionofthefadingcoecientatthen extinstance.Lyapunovbased analysisisperformedtoprovideanultimateboundontheSIN Rerror,thesizeofwhich canbereducedbychoosingappropriatecontrolgains.Inadd ition,variationsinother componentsoftheradiochannelsuchaspathlossandlog-nor malshadowingarealso accountedforusingthisanalysistool.Simulationresults areprovidedforanalysisand vericationofresults,andmotivationisprovidedforfutu rework. 4.1NetworkModel TheSINR x ( l ) x 1 ( l ) x 2 ( l ) ::x n ( l ) T 2 R n isdened(in dB )foreachradio link i =1 ; 2 ;:::n as x i ( l )=10log ag ii ( l ) P i ( l ) I i ( l ) (4{1) wherethefunctionlog( )denotesthebase10logarithm.Thequantitiesinsidethelo g( ) functionofEquation 4{1 isdenedinChapter 3 UnderstandinghowtheSINRchangesisbenecialforthedeve lopmentandanalysis ofthesubsequentpowercontrollaw.Takingtherstdieren ceofEquation 4{1 yields x i ( l ) T s = x i ( l +1) x i ( l ) T s = [10log( g ii ( l +1)) 10log( g ii ( l ))] T s + u i ( l ) T s [10log( I i ( l +1)) 10log( I i ( l ))] T s ; (4{2) 36

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where T s isthepowerupdateinterval,and u ( l ) u 1 ( l ) u 2 ( l ) ::u n ( l ) T 2 R n denotesanauxiliarycontrolsignaldened 8 i =1 ; 2 ;::::;n as u i ( l )=10[log( P i ( l +1)) log( P i ( l ))] ; (4{3) whichisusedtodeterminethepowerupdatelaw.Thereasonfo rdeningthestateasin Equation 4{1 istoobtainthecontrollerasEquation 4{3 thatwouldyieldapowerupdate thatismoresensitiveinoperationthanthepowerupdateEqu ation 3{9 usedinChapter 3 Afterincludingmeasurementnoise ( l;x ) 1 ( l;x 1 ) 2 ( l;x 2 ) :: n ( l;x n ) T 2 R n ,theSINRatthenextupdateinterval x ( l +1) x 1 ( l +1) x 2 ( l +1) ::x n ( l +1) T 2 R n canbeexpressedas x ( l +1)= f 1 ( x ( l ))+ f 2 ( x ( l ))+ x ( l )+ u ( l )+ ( l;x ) ; (4{4) wherethechannelgainfunctional f 1 ( x ( l )) 2 R n isdened 8 i =1 ; 2 ;::::;n as f 1 ( x i ( l ))=10log g ii ( l +1) g ii ( l ) ; (4{5) andtheinterferencefunctional f 2 ( x ( l )) 2 R n isdened 8 i =1 ; 2 ;::::;n as f 2 ( x i ( l ))=10log I i ( l ) I i ( l +1) : (4{6) 4.2LinearPredictionofFadingCoecient ThedevelopmentofapowercontrollerforradiolinksinaCDM Anetworkis challangingduetorapid,largescalechangesinthecoecie ntsofthenonlinearSINR dynamics.Afurtherchallangeisthatthepowercapacityate achMTisconstrainedas 0


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entersadeepfadedzone.Motivatedtoaddresstheseissues, thecurrentresultestimates (orpredicts) j X i ( l +1) j 2 andfeedsthisinformationtothecontrollertoaccountfors uch fastchangingchannelsandpowerlimitations. Thepredictionof j X i ( l +1) j 2 canbedenedasa OptimumMeanSquareError (MSE)problem (seeAppendix A ),i.e., 2min =min ^ X i E j X i ( l +1) j 2 ^ X i ( l ) 2 given j X i ( l ) j 2 ; j X i ( l 1) j 2 ;::; j X i ( l ( n 1 1)) j 2 ; where ^ X i ( l )istheestimateof j X i ( l +1) j 2 thatreducestheMSE.Theoptimumestimator isequaltotheconditionalmean[ 34 ] E j X i ( l +1) j 2 jj X i ( l ) j 2 ;:::::; j X i ( l ( n 1 1)) j 2 ; whichisreadastheexpectedvalueof j X i ( l +1) j 2 given j X i ( l ) j 2 ,....., j X i ( l ( n 1 1)) j 2 Obtainingconditionalestimatefornongaussianrandompro cesses(i.e., j X i ( l +1) j 2 ) isdicult;nonlinearestimatesmightbeoptimumforsuchca sesanditrequireshigher ordermomentstoobtainsuchnonlinearestimates.Forthese reasons,alinearestimatoris chosenforpredictingthefadingvariable j X i ( l +1) j 2 The LinearMMSE foranon-gaussianrandomvariable j X i ( ) j 2 canbeobtained from[ 34 ]as ^ X i ( l )= l X m = l ( n 1 1) ( m ) i n j X i ( m ) j 2 j X i j 2 o + j X i j 2 = 8>>>>>><>>>>>>: l P m = l ( n 1 1) ( m ) i j X i ( m ) j 2 + j X i j 2 1 l P m = l n 1 ( m ) i ; f n 6 =0 j X i ( l ) j 2 ; f n =0 (4{7) where j X i j 2 isthemeanoftherandomprocess j X i ( ) j 2 forall l f n isthedoppler frequencyoftheMTdenedinEquation 2{6 atinstance f ( lT p +1) ( lT p ) g .Thelinear 38

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estimate ^ X i ( l )cantakenon-zerovaluesifthepredictionobservationsam plingrate( T p ) canbechosenappropriately.Basedon[ 27 ],thesamplingrateischosensuchthatitis atleasttheNyquistrate,i.e.,twicethemaximumdopplerfr equencyoftheMT. The ( l 1) m 'ssatisfytheorthogonalitycondition[seeAppendix B ].Dening i h ( l ( n 1 1)) i ( l ( n 1 2)) i ..... ( l ) i i T andusingtheorthogonalityconditionyields T i = 266666664 E j X i ( l +1) j 2 j X i ( l ( n 1 1)) j 2 E j X i ( l +1) j 2 j X i ( l ( n 1 2)) j 2 : E j X i ( l +1) j 2 j X i ( l ) j 2 377777775 T Z 1 ; (4{8) where Z 2 R n 1 n 1 isdened 8 j;k =1 ; 2 ;::::;n 1 as Z jk = E j X i ( l ( n 1 j )) j 2 j X i ( l ( n 1 k )) j 2 : Theautocovariancefunctionfor j X i () j 2 isgivenby[ 23 ],[ 35 ] R j X i j 2 ( lT p )= E j X i ( l ) j 2 j X i ( l +( lT p )) j 2 J 2 0 (2 f n ( lT p )) ; (4{9) (referFigure 3-1 )where J 0 isthezeroth-orderBesselfunctionoftherstkind.Theref ore, fromEquation 4{8 T i = 266666664 J 2 0 (2 f n ( T p n 1 )) J 2 0 (2 f n ( T p ( n 1 1))) : J 2 0 (2 f n T p ) 377777775 T Z 1 ;f n 6 =0 : (4{10) wherethecomponentsof Z aredened 8 j;k =1 ; 2 ;::::;n 1 as Z jk = Z kj = 8><>: J 2 0 (2 f n T p j j k j ); j 6 = k j X i j 2 ; j = k ;f n 6 =0 : (4{11) 39

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Byorthogonality,wehavetheLMMSE[ 34 ]as 2min = E h j X i ( l +1) j 2 2 i l X m = l ( n 1 1) m i E j X i ( m ) j 2 j X i ( l +1) j 2 : Claim4-1: TheLinearpredictorinEquation 4{7 isboundedbasedonthefollowing facts.Thepowerofthefadedenvelope j X i ( ) j 2 atthereceiverisboundedsinceinaradio link,thereceivedpowercannotbegreaterthanthetransmit tedpower(refertoChapter 2 andSection 4.1 ).Therefore,themean j X i j 2 andthevariance j X i j 2 arebounded.The coecientsof i inEquation 4{10 areboundedifthecovariancematrixinEquation 4{11 isinvertible.Apredictionobservationsamplingrate( T p )equaltoorlower[ 31 ]thanthe powerupdaterate( T s )ischosen(suchthatitisatleasttheNyquistrate[ 27 ])andthe eectofadditivenoiseisincorporatedin Z [ 27 ],sothattheinverseof Z canbecomputed. Linearpredictionofthefadingprocessrequiresmeasureme ntofthe j X 1 ( ) j 2 atthe currentandpreviousinstances;theperformanceofthepred ictorcanbeimprovedby increasingthenumberofmeasurements n 1 usedtopredictthefadingprocessatinstance l +1.Practically,asmoremeasurementsareused,theperform anceofthepredictor doesnotimprovebutdegradesduetocomputationalproblems associatedwithinverting thematrix Z .NotethatthefadingpowerofindividualMTsatinstance l usedinthe predictorisalsousedinthecontrollerinthe X ( l )term. 4.3ControlDevelopment 4.3.1ControlObjective Thenetworkqualityofservicecanbequantiedbytheabilit yoftheSINRtoremain withinaspeciedoperatingrangewithupperandlowerslimi ts, r min ;r max 2 R n foreach linkdened 8 i =1 ; 2 ;::::;n as r min x i ( l ) r max : (4{12) KeepingtheSINRabovetheminimumthresholdeliminatessig naldropout,whereas remainingbelowtheupperthresholdminimizesinterferenc etoadjacentcells.Asin Chapter 3 ,thecontrolobjectiveinthischapteristoregulatetheSIN Rtoatargetvalue 40

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foreachchannel,denotedby r 2 R n ,whileensuringthattheSINRremainsbetweenthe speciedlowerandupperlimitsforeachchannel.Toquantif ythisobjective,aregulation errorisdenedas e ( l ) e 1 ( l ) e 2 ( l ) ::e n ( l ) T 2 R n where e i ( l )= x i ( l ) r; 8 i =1 ; 2 ;::::;n: (4{13) 4.3.2ClosedLoopErrorSystem BytakingtherstdierenceofEquation 4{13 ,andusingEquation 2{1 ,Equation 4{4 ,andEquation 4{5 ,theopen-looperrordynamicsforeachlinkcanbedetermine das e i ( l )=10log ( g d 0 d i ( l +1) d 0 10 0 : 1 i ( l +1) j X i ( l +1) j 2 ) 10log ( g d 0 d i ( l ) d 0 10 0 : 1 i ( l ) j X i ( l ) j 2 ) + f 2 ( x i ( l ))+ i ( l;x )+ u i ( l ) : Afterusingpropertiesofthelog( )function,theopen-looperrordynamicscanbe simpliedas e ( l )=10 d ( l;l +1)+ ( l;l +1)+10 X ( l +1) 10 X ( l )+ f 2 ( x ( l ))+ ( l;x )+ u ( l ) ; (4{14) wheretheauxiliaryfunctions d ( l;l +1) ; ( l;l +1) ; X ( ) 2 R n aredened 8 i =1 ; 2 ;::::;n as d ( l;l +1)= log d 1 ( l ) d 1 ( l +1) log d 2 ( l ) d 2 ( l +1) ... log d n ( l ) d n ( l +1) T ; (4{15) ( l;l +1)=[ 1 ( l +1) 1 ( l ) 2 ( l +1) 2 ( l )... n ( l +1) n ( l )] T ; (4{16) and ( )= log j X 1 ( ) j 2 log j X 2 ( ) j 2 ...log j X n ( ) j 2 T : (4{17) BasedonthemodeldevelopmentinChapter 2 (i.e., d i ( )and I i ( )arenon-zeroand bounded),thenormof d ( l;l +1)and f 2 ( x ( l ))canbeupperboundedbysomepositive 41

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scalarsas k d ( l;l +1) k c 1 ; (4{18) and k f 2 ( x ( l )) k c 2 : (4{19) Moroever,since ( )isazero-meangaussiandistributedvariable(in dB )withabounded standarddeviation[ 19 ],thenthenormof ( l;l +1)canbeupperboundedbysome positivescalaras k ( l;l +1) k c 3 ; (4{20) andthemeasurementnoiseisassumedtobebounded,i.e., k ( l;x ) k c 4 : (4{21) BasedonEquation 4{14 andthesubsequentstabilityanalysis,theauxiliarypower controller u ( l )isdesignedas u ( l )= ( k n + k p + k e ) e ( l ) 10 Log ^ X ( l ) +10 X ( l ) ; (4{22) wherethenotation Log ^ X ( l ) isdened 8 i =1 ; 2 ;::::;n as Log ^ X ( l ) = h log ^ X 1 ( l ) log ^ X 2 ( l ) ...log ^ X n ( l ) i T ; (4{23) and ^ X i ( ) 6 =0 ; (4{24) wherethecomponentsof Log ^ X ( l ) areobtainedfromEquation 4{7 ,andtheprediction samplingrateischosentobeatleasttheNyquistrateforEqu ation 4{24 tohold.From Equation 4{3 ,Equation 4{17 ,Equation 4{22 ,andEquation 4{23 ,thepowerupdatelaw foreachradiochannelisobtainedas P i ( l +1)=10 n i ; 8 i =1 ; 2 ;::::;n; (4{25) 42

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where n i = ( k n + k p + k e ) e i ( l ) 10 log ^ X i ( l ) +log j X i ( l ) j 2 +log( P i ( l )) : (4{26) 4.4StabilityAnalysis Theorem4-1: ThecontrollerinEquation 4{22 andEquation 4{25 ensuresthat allclosedloopsignalsarebounded,andthattheSINRregula tionerrorapproachesan ultimatebound ( k n ;k p ;l 0 ) 2 R inthesensethat k e ( l ) k! ( k n ;k p ;l 0 )as l !1 (4{27) provided k e inEquation 4{22 isselectedas 0
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AfterusingEquation 4{29 ,thefollowinginequalitycanbedeveloped V k e V + 25 rrr X ( l +1) Log ^ X ( l ) rrr 2 k p + (10 c 1 + c 2 + c 3 + c 4 ) 2 4 k n : (4{32) ProvidedthesucientconditioninEquation 4{28 issatised,Lemma13.1of[ 24 ]canbe invokedtoconcludethat V ( e;l ) (1 k e ) l V ( e ( l 0 ) ;l 0 )+ 1 (1 k e ) l k e !" (10 c 1 + c 2 + c 3 + c 4 ) 2 4 k n + 25 & k p # ; (4{33) where & = rrr X ( l +1) Log ^ X ( l ) rrr 2 isupperboundedbyapositivescalar c 5 ,i.e., & c 5 basedontheresultsfromClaim4-1,thedevelopmentinChapt er 2 ,andSection 4.2 BasedonEquation 4{33 ,anupperboundfortheSINRerrorcanbedevelopedas k e ( l ) k 2 (1 k e ) l k e ( l 0 ) k 2 + 1 (1 k e ) l k e !" (10 c 1 + c 2 + c 3 + c 4 ) 2 4 k n + 25 c 5 k p # : (4{34) Theassumptionthat X ( l ) 2L 1 ,thefactthat Log ^ X ( l ) 2L 1 fromClaim4-1and Equation 4{24 ,andthefactthat e ( l ) 2L 1 fromEquation 4{34 canbeusedtoconclude that u ( l ) 2L 1 fromEquation 4{22 ,andhence P i ( l +1) 2L 1 fromEquation 4{25 .The ultimateboundinEquation 4{34 asymptoticallyconvergesas lim l !1 k e ( l ) k 2 = (10 c 1 + c 2 + c 3 + c 4 ) 2 4 k n + 25 c 5 k p : (4{35) FromEquation 4{35 ,theultimateboundcanbedecreasedbyincreasing k n and k p ; however,themagnitudeof k n isrestrictedbytheconstraintthat0


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Figure4-1.Distributedcellularnetworktopology. 4.5SimulationResults AcellularnetworktopologywasbuiltinMATLABandthemobil ityoftwentyMTsin thecellofinterestismodeledbyasteadystate(stationary )distributionmodel(i.e.,[ 25 ], [ 26 ]).ARandom-Waypointmodelisusedtosimulatethemobility oftheMTs(referto Section 3.4.1 ).Thesimulationiscarriedinadistributedcellularnetwo rkwithtwentyMTs operatingintheeachofthesixcellssurroundingthecellof interest.Figure 4-1 showsa distributedcellularnetworktopology. ThetargetSINR, r ischosenas8 dB witharangebetween6and10 dB ,whichis denedinSection 4.3.1 .Thermalnoise, ,issetto 110 dBm .Theinitialpowerlevelfor alltheMTsischosenas10 dBm .ARayleighfadedchanneliscreatedusingthechannel samplingtime( T s )of1 : 7 ms ,whichisobtainedbyperformingacontinuoustimeSINR 45

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200 400 600 800 1000 1200 1400 1600 -2 0 2 4 6 Iterations, lSINR Error, in dBSINR Error Plot. Maximum Doppler Frequency of MT = 4.11Hz 0 200 400 600 800 1000 1200 1400 1600 1800 -130 -120 -110 -100 -90 -80 Iterations, lChannel Gain, in dBChannel Gain Plot 0 200 400 600 800 1000 1200 1400 1600 1800 -60 -40 -20 0 20 Iterations, lPower, in dBmPower Plot Figure4-2.Error,channelgain,andpowerplotofaMTwithma ximumDoppler frequency4.11Hz. erroranalysis(seeSection 3.3 ),andtheDopplerfrequencyEquation 2{6 .Theprediction observationsamplingrate( T p )isalsochosentobe1 : 7 ms .Thevelocityforeachofthe MTsisobtainedfromEquation 3{29 .Theangle ismeasuredperiodicallyandthe DopplerfrequencyisobtainedfromEquation 3{29 andEquation 2{6 ,whichisusedto updatethecoecientsoftheLMMSEpredictor.Pathloss,wit hfreespacepropagation eects(near-eldeects),andlog-normalshadowingaremo delled[ 19 ]asshownin Equation 2{1 andEquation 2{2 TheresultsinFigures 4-2 through 4-5 areobtainedwith k n =0 : 00005 ;k p =1 : 6 ;k e =0 : 00008 ; andthespreadingfactor a ischosenas156.Thenumberofsamplesusedforprediction is5.Figure 4-2 showstheSINRerror,channelgainandpowerplotsofaMToper ating aroundamaximumdopplerfrequencyof4 : 11 Hz .ThepredictionerrorforthisMTis 46

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0 200 400 600 800 1000 1200 1400 1600 1800 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 Iterations, lPrediction Error, in dBPrediction Error Plot Figure4-3.PredictionerroroftheMTwithmaximumDopplerf requency4.11Hz. 0 200 400 600 800 1000 1200 1400 1600 1800 -30 -20 -10 0 10 20 Iterations, lSINR Error, in dBSINR Error Plot. Maximum Doppler Frequency of MT = 30.88Hz 0 200 400 600 800 1000 1200 1400 1600 1800 -160 -140 -120 -100 -80 Iterations, lChannel Gain, in dBChannel Gain Plot 0 200 400 600 800 1000 1200 1400 1600 1800 -40 -20 0 20 Iterations, lPower, in dBmPower Plot Figure4-4.Error,channelgain,andpowerplotofaMTwithma ximumDoppler frequency30.9Hz. 47

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0 200 400 600 800 1000 1200 1400 1600 1800 -25 -20 -15 -10 -5 0 5 10 15 20 Iterations, lPrediction Error, in dBPrediction Error Plot Figure4-5.PredictionerroroftheMTwithmaximumDopplerf requency30.9Hz. showninFigure 4-3 .ItcanbeseenthatthepowercontrollerregulatestheSINRo fthe MTinthedesiredrangealmostperfectly.Figure 4-4 showstheSINRerror,channelgain andpowerplotsofaMToperatingaroundamaximumdopplerfre quencyof30 : 88 Hz ThepredictionerrorforthisMTisshowninFigure 4-5 .Theinaccuratepredictionofthe linearpredictorinthedeepfadedzonescausesoutagetothe MTinthosezones(Figures 4-4 and 4-5 ).TheSINRofthisradiolinkoperatingaroundamaximumdopp lerfrequency of30 : 88 Hz isintheacceptablecommunicationrangeatallothertimes, andthepower requiredtoachievethisisintheimplementablerange,well belowthemaximumpowerof 27 dBm (500 mW )ofMTs. Table4-1showsthestatisticsoftheSINRerrorofalltheMTs inthecellof importance.ThesecondcolumnofTable4-1quantiestheper centageofsamplesthat liewithinthedesiredSINRrangeforeachDopplerfrequency .Whensamplesexceedthe upperlimitofthedesiredSINRrange(i.e., x i ( l ) r max ),thequalityofserviceforthe 48

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Table4-1.PercentageofsampleswithinthedesiredSINRran ge Max.Doppler frequency(Hz) %ofsampleswhere r min x i r max %ofsampleswhere x i r min 0 : 8099 : 780 : 05 1 : 7899 : 840 : 05 2 : 3299 : 780 : 05 3 : 9699 : 670 : 05 4 : 0099 : 560 : 16 4 : 1199 : 500 : 27 4 : 1799 : 780 : 05 6 : 1999 : 450 : 27 6 : 8299 : 730 : 16 6 : 8898 : 720 : 72 7 : 4699 : 390 : 22 8 : 0798 : 780 : 61 10 : 6199 : 450 : 22 11 : 2599 : 330 : 22 11 : 5299 : 000 : 55 14 : 6198 : 610 : 83 23 : 0398 : 500 : 83 26 : 2998 : 170 : 94 26 : 5095 : 892 : 17 30 : 8896 : 331 : 94 individuallinkisnotcompromised.However,exceedingthe upperlimitisundesirable becausetheinterferencetootherlinksincreases,potenti allyleadingtoanoutage(i.e., when x i ( l ) r min )[ 6 { 8 ].Anoutageofalinkdoescompromisethequalityofservicei nthe sensethatthesignalmaynotbedecodedatthatparticularsa mple.Thethirdcolumnof Table4-1quantiesthepercentageofsamplesthatexperien ceanoutageforeachDoppler frequency,particularlyduetofading[ 14 15 ]anditcanbeinferredfromthiscolumnthat theoutage( x i r min )foralltheMTsareregulatedatbelow3%atalltimes. Anewsetofsimulationswerecarriedoutforhigh-gain(cent ralizedpowercontrol) [ 33 ]andprediction-basedpowercontrolalgorithmssuchthatt hesealgorithmsare simulatedonthesametopologymodelwithtwentyMTsinacell toobtainaclear implicationoftheresult.Figure 4-6 showstheoutageprobabilityandmaximumpower requirementplotsplottedagainstthemaximumdopplerfreq uencyaroundwhichthe 49

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0 5 10 15 20 25 30 35 10 -4 10 -3 10 -2 10 -1 10 0 Outage Probability Vs Maximum Doppler Frequency of MTs Maximum Doppler Frequency (Hz)Outage Probability 0 5 10 15 20 25 30 35 10 15 20 25 27 30 Maximum Power Level Vs Maximum Doppler Frequency of MTs Maximum Doppler Frequency (Hz)Maximum Power (dBm) 1 Prediction Sample 3 Prediction Samples 5 Prediction Samples High-Gain (No Prediction) Figure4-6.Comparisonofhighgainandpredictivepowercon trolalgorithms. MTsoperateforthesepowercontrolalgorithms.Itcanbesee nthatprediction-based powercontrolalgorithmsareresponsiblefortheradiolink tooperatewithloweroutage probabilitiesandmuchlowermaximumpowerrequirementswh encomparedtoMTsusing high-gainpowercontrolalgorithms.Also,notethatmostof theradiolinksthatuses prediction-basedpowercontrolalgorithmsrequirepowerl esserthanthechoseninitial powerof10 dBm .ItcanalsobeinferredfromFigure 4-6 thatasthenumberofprediction observationsamples n 1 increases,theperformanceofthepowercontrollerimprove s,i.e., radiolinksoperatewithloweroutageprobabilities. 50

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CHAPTER5 CONCLUSION 5.1SummaryofResults Radiochanneluncertainties,particularlyfading,areres ponsibleforthecellular networktobecharacterizedasanonlinearsystem.Thefadin gprocessischaracterized asatime-varyingstochasticprocesswhichisresponsiblef orsignicantpowerdropsin certainregionsknownasdeepfadedregionscausingproblem sinrecoveringthesignal atthereceiver.Further,restrictionsinthemaximumpower atwhichthesignalscanbe transmittedinsuchsystemsandbandwidthavailabilityint ensiestheneedtodevelop powercontrollersforsuchradiolinks.Toaddressthesepro blems,controllersaredesigned thatusestheLyapunov-basedtoolstoanalyzethenonlinear system,andthesimulation resultsarediscussedtodemonstrateandvalidatethetheor ybehindthecontroldesign. InChapter 3 ,arobustpowercontrollerisdevelopedforawirelessCDMAbased cellularnetworksystem.Lyapunov-basedstabilityanalys isisusedtodevelopanultimate boundforthesampledSINRerrorwhichcanbedecreaseduptoa pointbyincreasinga nonlineardampinggain.Ananalysisisalsoprovidedtoillu stratehowmobilityandthe desiredSINRregulationrangeaectsthechoiceofchannelu pdatetimes.Thechoiceof theupdatetimealsoaectstheultimateboundthatthesampl edSINRerrorreachesLoweringthesamplingtimereducestheultimatebound.Simu lationsindicatethatthe SINRsofradiolinksoperatingwithlowermaximumDopplerfr equencyaremaintainedin thedesiredcommunicationrange.Radiolinksoperatingwit hahighmaximumDoppler frequencyhavehighoutageprobabilityduetothefastlytim e-varyingnatureofthe channeluncertainties,andthismotivatedtousetheconcep tofpredictiontoaddressthe issue. Chapter 4 introducesaMMSEprediction-basedpowercontrolalgorith mforawireless CDMA-baseddistributedcellularnetworkedsystem.Alinea rpredictorisusedtopredict thefadingpoweratthe l +1thinstant,andthisinformationisfedtothecontrollerf rom 51

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whichthepowerupdatelawisobtained,whichinthiscaseisl ogarithmicinoperation unliketheproportionalcontrollerdevelopedinChapter 3 .ALyapunov-basedanalysisis usedtodevelopanultimateboundforthesampledSINRerrorw hichcanbedecreased uptoapointbyincreasinganonlineardampinggains.Simula tionsindicatethatthe SINRsofalltheradiolinksareregulatedintheregion r min x i ( ) r max witha outageprobabilityoflessthan3%.LocalSINRmeasurements areusedtosimulatethe distributedcellularnetwork.Outagesatsomeregionswere determinedtobedueto limitationsofthelinearpredictor,especiallyinthedeep fadedzones. 5.2RecommendationsforFutureWork Inordertoimprovetoperformanceofthecontrollerdesigne dinChapter 4 ,more sophisticatedpredictionandcontroldevelopmentconcept sarerequiredinsuchhighly time-varyingradiochannelsasencounteredinthefadingra diochannelsofurban environments.WienerltersandotherLyapunov-basedadap tivecontroldevelopment techniquescanbeusedtoimprovethequalityofserviceforc ellularcommunication networks.Optimalcontroldevelopmentforsuchnonlinears tochasticradiochannelscan potentiallyenhancetheradiolinkqualityofcellularcomm unicationnetworks. Themodelingandthecontroldevelopmentmethodologiesfol lowedinthiswork canbeextendedtowirelessMobileAd-HocNetworks(MANETs) ,whereinaddition totherandomtime-varyingphenomenaintheradiochannel,u npredictabletopological changes,bandwidthandpowerconstraints,multiuserinter ference,time-delays(dueto contention,back-o,etc),linkschedulingandroutingare someoftheadditionalchallenges encountered.Modelingandembeddingsuchfactorsexperien cedinMANETsinthesystem modeldevelopedinthisthesisfordevelopingpowercontrol algorithms,andoptimizingits performancebasedontheconstraintsinMANETsispotential lythenextgoalinthisline ofresearch. 52

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APPENDIXA ESTIMATIONOFRANDOMPROCESSES A-1GeneralMMSEbasedestimationtheory Let W ( l )besomerandomprocessthatneedstobeestimated.Theprobl emofnding theestimatesofthezeromeangaussianrandomvariablescan bedenedasfollows. 2min =min ^ W ( l ) E W ( l ) ^ W ( l ) 2 given W ( l 1) ;W ( l 2) ;W ( l 3) ;:: =min ^ W ( l ) E h W 2 ( l ) 2 ^ W ( l ) W ( l )+ ^ W 2 ( l ) i given W ( l 1) ;W ( l 2) ;W ( l 3) ;:: =min ^ W ( l ) E W 2 ( l ) 2 ^ W ( l ) E [ W ( l )]+ ^ W 2 ( l )given W ( l 1) ;W ( l 2) ;W ( l 3) ;:: (A-1) Tondtheminimumvalueoftheestimateof W d d ^ W ( l ) n E W 2 ( l ) 2 ^ W ( l ) E [ W ( l )] o =0given W ( l 1) ;W ( l 2) ;W ( l 3) ;:: = ) 0 2 E [ W ( l )]+2 ^ W ( l )=0given W ( l 1) ;W ( l 2) ;W ( l 3) ;:: Theestimateisgivenby[ 34 ] ^ W ( l )= E [ W ( l ) j W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ] ; (A-2) Theconditionalestimateisgivenby E [ W ( l ) j W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ] ; where W ( l ) ;W ( l 1) ;W ( l 2) ;W ( l 3) ;:: arealljointlygaussianand W ( l 1) ;W ( l 2) ;W ( l 3) ;:::: arethepastvaluesoftherandomvariable W thatareusedtoestimatethe currentvalue W ( l ). 53

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A-2GaussianCase Theconditionalprobabilitydensityfunctionisgivenby[ 36 ] f W ( l ) [ W ( l ) j W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ] = f W ( l ) ;W ( l 1) ;W ( l 2) ;::: [ W ( l ) ;W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ] f W ( l 1) ;W ( l 2) ;::: [ W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ] ; (A-3) wherethenumeratoranddenominatorarejointdensityfunct ionsofthezero-mean gaussianrandomvariables W uptoinstants l and l 1respectively.TheCovariance Matrices K n and K n 1 aredenedasfollows K n = E h [ Y l ] : [ Y l ] T i ; and K n 1 = E h [ Y l 1 ] : [ Y l 1 ] T i ; where Y l = W ( l s ) W ( l ( s 1)) ::W ( l ) T ; and Y l 1 = W ( l s ) W ( l ( s 1)) :W ( l 1) T : Sincethemeansoftherandomvariables W arezeroatany l f W ( l ) [ W ( l ) j W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ] = exp 1 2 Y T l K 1 n Y l (2 ) n 2 j K n j 1 = 2 : ( exp 1 2 Y T l 1 K 1 n 1 Y l 1 (2 ) ( n 1) 2 j K n 1 j 1 = 2 ) 1 : (A-4) Since W ( l )isazero-meangaussianrandomprocess,theMMSEestimatei salinear estimate,i.e., E [ W ( l ) j W ( l 1) ;W ( l 2) ;W ( l 3) ;:: ]canbeobtainedbymanipulating Equation A-4 .Forasimplecasewithonlyonegivenvalue,thelinearMMSEe stimationis 54

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givenby E [ W ( l ) j W ( l 1)] = W ( l ) + W ( l ) W ( l 1) W ( l ) W ( l 1) W ( l 1) W ( l 1) = W ( l ) W ( l 1) W ( l ) W ( l 1) W ( l 1) ; (A-5) where W ( l ) W ( l 1) istheautocorrelationfunction, W ( l ) and W ( l 1) arethevariances. 55

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APPENDIXB ORTHOGONALITYCONDITION Let Y X 1 X 2 X 3 ,...., X N begaussianrandomvariableswithzeromeans.The MMSEestimateistheconditionalmean,givenby E [ Y j X 1 ; X 2 ;X 3 ;::::;X N ]= N X k =1 a i X i : (B-1) Therandomvariables Y N P k =1 a i X i X 1 X 2 X 3 ,...., X N arejointlygaussian.Since thersttermisuncorrelatedwithalltherest,itcanbeinfe rredthattherandomvariable Y N P k =1 a i X i isuncorrelatedwith X 1 X 2 X 3 ,...., X N .Therefore, E Y N X k =1 a i X i j X 1 ; X 2 ;X 3 ;::::;X N # = E Y N X k =1 a i X i !# = E [ Y ] N X k =1 a i E [ X i ]=0 ; since E [ Y ]= E [ X i ]=0.Thecondition E Y N X k =1 a i X i j X 1 ; X 2 ;X 3 ;::::;X N # =0(B-2) isknownasthe OrthogonalityCondition ,whichcanalsobewrittenas Y a T X ? X; (B-3) where X = X 1 X 2 ::X N T : The a i 'scanbeobtainedfromtheorthogonalitycondition. Note: FromEquation B-2 ,weget E [ Y j X 1 ; X 2 ;X 3 ;::::;X N ] N X k =1 a i E [ X i j X ]=0 : 56

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= ) E [ Y j X 1 ; X 2 ;X 3 ;::::;X N ] N X k =1 a i X i =0 = ) E [ Y j X 1 ; X 2 ;X 3 ;::::;X N ]= N X k =1 a i X i : (B-4) Thus,theconditionalmeanofazero-meangaussianrandomva riable Y isgivenbyalinear estimateofthegivenvariables X i s. Calculationof a i 's FromtheOrthogonalityconditioninEquation B-2 ,weget[ 34 ] E Y N X k =1 a i X i j X p # =0 ; 1 p k = ) E [ YX p ]= N X k =1 a i E [ X k X p ] ; 1 p k: = ) k YX = a T K YY ; (B-5) where a a 1 a 2 ::a N T ; k YX E [ YX 1 ] E [ YX 2 ] E [ YX 3 ] ::E [ YX N ] = K YX 1 K YX 2 K YX 3 ::K YX N ; andthecovariancematrix K XX = E XX T : (B-6) FromEquation B-5 ,weget a T = k YX K T XX : (B-7) 57

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BIOGRAPHICALSKETCH SankrithSubramanianwasborninChennai,Indiain1985.Her eceivedhisbachelor's degreeininstrumentationandcontrolengineeringfromSt. Joseph'sCollegeofEngineering (AnnaUniversity),Chennai,IndiainMay2006.Hereceivedh isMasterofSciencedegree fromtheUniversityofFloridain2009.Hisinterestslieint heeldofnetworkedcontrol systems,nonlinearcontrols,wirelesscommunication,sig nalprocessingandautomation. SankrithpursuedhisMasterofSciencedegreeintheDepartm entofElectricaland ComputerEngineeringandwasagraduateresearchassistant intheNonlinearControls andRoboticsgroupintheDepartmentofMechanicalandAeros paceEngineeringatthe UniversityofFlorida,underthesupervisionofDr.WarrenE .Dixon.Hewasco-advised byDr.JohnM.SheaoftheWirelessInformationNetworkingGr oupintheDepartmentof ElectricalandComputerEngineeringattheUniversityofFl orida. TheprimaryfocusofhisresearchwastoanalyzewirelessCDM A-basedcellular communicationnetworkswithtime-varyingstochasticchan neluncertaintiesanddesign Lyapunov-basednonlinearpowercontrollers.Simulationd emonstrationswereprovided withthehelpofrealisticnetworktopologymodels.Hisfutu reinterestistoextendand applyhisresearchtowirelessad-hocnetworks. 61