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Characteristics of Sequential Detection in Cognitive Radio Networks

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Characteristics of Sequential Detection in Cognitive Radio Networks

    Characteristics of Sequential Detection in

    Cognitive Radio Networks

JournalofCommunicationandComputer8(2011)587595

    ;CharacteristicsofSequentialDetectioninCognitive

    ;RadioNetworks

    ;aLl?HIN

    ;OscarFilioRodriguez,ValeriKontorovich2andSergueiPrimak

    ;,DepartmentofElectricalandComputerEngineering,TheUniversityofWesternOntario,London,Ontario,N6A5B9,Canada

    ;2ElectricalEngineeringDepartment,TelecommunicationsSection.CINVESTAV-IPNAv.IPN2508,Co1.SanPedroZacatencoC.P

    ;07360MexicoCity,Mexico

    ;Received:October29,2011/Accepted:December16,2011/Published:Mrdy31,2012 ;Abstract:SequentialAnalysisisaneffectivedetectionprocedureforspectrumsensinginCognitiveRadio(CR)Networks.Onaverage,

    ;givenPFA(probabilityoffalsealarm)andPMDandlowSNRregime,itrequireslessindependentsamplesfortheprimaryusers(PU)

    ;detectioncomparingtotheNeyman.Pearson(NP)test.Thispaperdealswiththeevaluationofthecumulantsofthedistribution(PDF)

    ;ofrandomtimeofthesequentialanalysis.however.forsimplicityweconsideronlyGaussianapproximation.ItisassumedthatthePU

    ;andsecondaryusers(SU1aresharingthesamefrequencybandwidthandforspectrumsensingSU’sapplyincoherentdiversity

    ;combiningofdiversitybrancheswithfadingdescribedbytheGeneralizedGaussian(GG1mode1. ;Keywords:CognitiveRadio,SequentialAnalysis,CumulantAnalysis

    ;1.Introduction

    ;CognitiveRadio(CR)hasattractedasignificant

    ;researcheffortduringthepastdecade.OperationofCR

    ;assumessomesortofcoexistenceoflicensed(primary)

    ;users(PU)andsecondary(unlicensed)user(SU)inthe

    ;sametime-spacefrequencyresource16].Inone

    ;modalitySUcannottransmitatallifPUisactive.

    ;Alternatively,SUcantransmitevenifPUisactive,

    ;however.1evelofitsradiatedmustbelimitedbelow

    ;prescribedlevel1,2.Inbothcasesonehasdetermine

    ;presenceorabsenceofPUandidentifyingspectrum

    ;ValerjKontorovich.Ph.D..professor.researchfields:

    ;electromagneticcompatibilityofradiocommunicationsystems,

    ;interferenceanalysis,channe1modelingcognitiveradio ;networks.E-mail:valeri@cinvestav.mx.

    ;SergueiPrimak,Ph.D.,associatedprofessor,researchfields: ;non-gaussianrandomprocesses,wirelesscommunications, ;MIMOchannelsmodeling,Statisticalelectromagnetic. ;CognitiveRadioNetworks.E-mail:sprimak@eng.uwo.ca. ;C0rrespondingauthor:0scarFilio,Ph.D.candidate, ;researchfields:cognitiveradionetworks.statisticaltheoryof ;communications.Email:ofilioro@uwo.ca.

    ;opportunitiesfortransmission.Spectrumhole ;detectionisquantifiedbyspecifyingprobabilities, ;and,minimumdetectablesignalleveland/or ;minimumSNRforsensing.Atthesametime

    ;complexityandcomputationaleffortaswellaspower ;consumptionmustbeminimized.Whilemostofthe ;effortsinanalysisofdetectionarefocusedonthe ;Neyman-Pearsondetectors[7,itiswellknown[8that

    ;sequentialdetectionprovidessubstantiallyfaster ;operationonaverage.Theexpressionforaverage ;detectiontimecanbeeasilyobtainedforthecaseof ;lowSNR[8.

    ;Obtainingotherstatisticalcharacteristics,suchas ;completeprobabilitydensityofthedecisiontime,is ;ratherdifficult.ApproximationbytheWaldPDForby ;firstfewcumulantsweresuggestedonlyincaseof ;significantlydifferentthresholdsofdetection,i.e.,for ;<<.ThelatterisnotalwaysapplicableforCR ;network.Inaddition,modemreceiverstendtouse ;588CharacteristicsofSequentialDetectioninCognitiveRadioNetworks

    ;multipleantennasanditisimpo~anttounderstand ;effectsofcombiningonperformanceofthesequential ;detector.Weconsidersituationofnoncoherent

    ;combiningoftwobrunches,eachsufferingfrom ;GeneralizedGaussianfadingasdescribedinRe~[911].

    ;Thepaperisorganizedasfollows.Section2is ;dedicaedtodescribetheGeneralizedGaussian ;ChannelMode1.Basicsofsequentialanalysisare ;presentedatSection3.Section4isdedicatedtothe ;cumulatesofthePDFofthetimeofsequentialanalysis ;AtSection5generalresultsofthesequentialanalysis ;oftheincoherentdiversitydetectionforgeneral ;Gaussianchannelmodelarepresented.Section6is ;devotedtocone1usionremarks.

    ;2.GeneralizedGaussianChannelModel

    ;Majorityofexistingfadingchannelmodelsare ;basedonastatisticalrepresentationofthemagnitude ;andphase(orinphaseandquadraturecomponents)of ;acomplexchanneItransferfunction.Often,such ;modelingcouldbeaccomplishedbypostulating ;GaussianPDFforboththeinphaseandquadrature

    ;componentsxand9,i.e.,theirjointPDF

    ;p(x,Y)isgivenby:

    ;p(x,)=2

    ;~ro-x

    ;_gfourparameters:{x,y}and

    ;{:,o:}[9-lO].Therefore,PDFgivenby(2)is ;knownas”four—parametricdistribution”[10].The

    ;followingseriesrepresentationof()areoften ;usefulinpracticalcalculations:

    ;p(r)=

    ;where

    ;+ym

    ;2or2

    ;ey

    ;o-x0

    ;(壶一ymy)

    ;(o;?:)

    ;Here/0(z)isamodifiedBesselfunctionofzero ;orderand()areHermitianpolynomials12].

    ;Letusintroducethefollowingnormalized ;parameters,whichareusefulinfurthercalculations: ;g::;z:,

    ;:arctan

    ;+(,-:()-

    ;<>:++o-~;y2=+y2.(4)<.>:+:+y.(4) ;Itisoftenbeneficialtoapproximatefull ;fourparametricdistributionwiththelessgeneralbut ;yetveryflexibleNakagamidistribution[9.The

    ;equivalentparametermofNakagamidistributionis ;givenby:

    ;:

    ;(1+f12)2

    ;(1+q2)2

    ;211+a4..~2q2(1+/32)(/32cos~~+sin2~o)].(5),:——’J

    ;BeckmandistributionfollowsfromEq.(3)when ;m=0,70=Iml;whileHoytPDFappearswhen ;?()-,o-0==y=0.Rayleighand

    ;truncatedGaussianfollowswheno’00and

;;0,m=m=0respectively.Itisworthto

    ;mentionthatNakagamidistributionisonlyan ;approximationtothefour-parametercase,butit ;adequatelyrepresentsofthevariationofthefunctional ;four.parameterPDFform.

    ;一一

    ;?

    ;CharacteristicsofSequentialDetectioninCognitiveRadioNetworks

    ;3.SequentialAnalysisofA.Wald

    ;IncontrasttoNPtestwherethelogarithmofthe ;MaximumLikelihoodRatio(MLR,iscomparedtoa ;singlethresholdAoatapredefinedandfixed ;observationintervalT,thesequentialtestofA.Wald ;[8]comparesMLRtotwothresholdsA1and2

    ;sequentially,untilcertainconditionisnotsatisfied.In ;bothcasesparametersofthetestarecalculatedbased ;onrequiredandasdescribedinRefi8,

    ;13].However,inNPtestdurationoftestingischosen ;basedondesiredprobabilitiesoferrors,whileinthe ;Waldtestthedecisiontimeisallowedtofluctuate, ;resultinginthefastest(onaverage)detector.Asa ;results,abetteraveragedecisiontimecouldbe ;achieved.Thethresholdinquestioncanbecalculated ;(upperbounded)asRef.[8].

    ;,:lnL:InCInC,

    ;A.:ln:lnD.(6),=ln=,A.=ln=ln.LOJ

    ;1P1P

    ;ThedecisionisnotmadeaslongastheMLRis

    ;betweenthethresholdsA1andA1.Decisioninfavor ;ofthehypothesis7-to(absenceofPU)ismadeifthe ;MLRbecomessmallerthanlandthehypothesis

    ;(presenceofPU)isadmittedifMLRexceeds,.

    ;SO,incontrarytoNPtest,thetimeinstantfordecision ;makingforsequentialanalysisisnotfixedanditis ;generallyarandomvariable.

    ;Itispossibletoobtainaveragenumberofsamples ;(meandecisiontime)foracceptingoneofthetwo ;hypothesis,dependingonwhichoneofthetwo ;hypothesisandiscorrect[8:

    ;?)lnnjr1(7)

    ;=lPMD+(1njHere

    ;A:M{lng(?I};B:M{lne(?f};

    ;andf?isaMLRafterVsteps.

    ;Furthermore,sincethedecisiontimeT=

;589

    ;(hereisthesamplingintervalandisanumber ;ofstepstomakeadecision)isarandomvariable,it ;couldbedescribedbyitsPDFP_,(f).Exactshapeof ;thisPDFisnotknown,althoughsomeapproximations ;aresuggestedbyWald8fortheasymptoticcases

    ;D=const.C0andC=const.D0(3:the

    ;approximatesolutionisknownastheWald’sPDF.

    ;EstimatesofthevarianceG2Narealsoknownunderthe ;assumptionthat<<.Theoriginalbook[88also

    ;containsaprocedurefordeterminingthecharacteristic ;functionandthemomentsofthePDF.(f). ;Derivationsoftheparametersofthedistribution ;Pr(t)arebasedonthenotionofthesocalledOperation

    ;Characteristic(0c)()whichisdefinedas8for

    ;thecaseofsmallvalueofa3(1owSNRcase): ;()=D()

    ;D(C(.)’(8)

    ;Hereaistheparameterofhypothesistestingsuch ;thata=a0correspondstothehypothesiswhile ;a=correspondstothehypothesis.Indetection ;infadingproblemsaissimplyproportionaltoSNR ;ofthePU,sothatao=0ifPUisabsentandequalto ;theaveragelinkSNR

    ;Furthermore,h(a)isa

    ;equmion[8]:

    ;(q)inthecaseof

    ;uniquenonzerorootofthe

    ;r…where(x)isPDFoftheobservationbasedonthe ;parameteraandP()/P()isalikelihood

    ;ratiofortwohypothesisand.Itwasshownat ;[8]thath(aI)=1andh(ao)=1.

    ;4.CumulantAnalysisofthePDFofthe

    ;RandomTimeofSequentialAnalysisor

    ;RandomSampleSize

    ;SincetheanalysistimeTandthenumberof ;samplesneededtomakedecisionarerelated ;throughT=,whereisafixedsampling

    ;590CharacteristicsofSequentialDetectioninCognitiveRadioN

    ;etwOrks

    ;interval,wewillfocusondeterminingparametersof ;distributionof.Theaveragetime(7)canbe ;expressedintermsofOCL(a)asRef.[8,l3]: ;)=,

    ;whereCandDarethresholdsdefinedinEq. ;(7)

    ;and

    ;Here

    ;Md,=(m

    ;n

    ;(x)dx,

    ;isthelog.likelihoodratioforthetwohypothesis ;and.

    ;Thecumulantgenerating

    ;r(19)=ln~7,(19)where

    ;o.,(19)=Lp(x)exp(j~a),

    ;function

    ;(12)

    ;canberepresentedbyaserieswithcoefficientsf( ;beingcumulantsofPr(t)[14]

    ;)=In~r(:喜鲁((13)

    ;Given07(19)or71(19),cumulantsinquestion ;couldbef0undas

    ;=

    ;d--%,lnOr(O)l=(f(o).

    ;Inparticular

    ;

    ;71(O),K”2=?(0)=f141

    ;IfonlytwocumulantsK”1andaretakeninto

    ;account,oneobtainsGaussianapproximationofthe ;realdistribution()N(#c1,);whileiffirst ;fourcumulantsK”1?aretakenintoaccounttheSO

    ;calledcurt~sisapproximationisobtained.Thelatteris ;moreaccurateapproximationforvaluesnearthemean, ;howeveritispoorinapproximatingtailsofthe ;distribution(howeveritcouldbeusedinobtaining ;exponentiallytightbounds).Giventhecomplexityof ;calculatinghigherordercumulantswefocushereonly ;ontheGaussianapproximation.

    ;ItisshowninRef.[8thatEq.(9)leadstothe

    ;followingexpressionforthecharacteristicfunctiOn ;O,,():

    ;ov(t9):

    ;where,(t9)

    ;D2(Dl()+CJ(Cry(o)

    ;C”()Dr2(DI()C2()

    ;arerootsoftheequation:

    ;r

;g(f):JL

    ;,(15)

    ;r6,whichsatisfythefollowingconditions: ;(19)=0,f269)=().(17)

    ;UsingEq.(16)andexpandingtheexponentunder ;theintegralsignintotheTaylorseriesoneobtainsthe

    ;followingrelationshipbetweenmoments()of ;thelog.likelihoodandvaluesoftl(I9):, ;2

    ;g():(et(S)4(X)p(),(18)

    ;and

    ;Ing(l9)+f(19)+...I9.(19)

    ;ThealgorithmforevaluatinganycumulantKcan ;beaccomplishedusingthefollowingsteps: ;1.Differentiatethelogarithmof(15)i.timesand ;evaluateitwhen0.

    ;2.Differentiating(16)or(19)itimesandevaluateit

    ;whenl90takingintoaccountthattl(0)=0, ;t2(O)=h(a).

    ;3.Obtainrecurrentexpressionsfor(0)from ;.

    ;t0-])(o),f(o)….(k=1,2).

    ;4.Thiswilldefineu(0)andtherefore,K”i.

    ;AtiersomesimplebutIaboriouscalculusoneobtains ;thefollowingvaluesofderivativesoft1and ;t1(0,a1)=

    ;t2(O,a1)=

    ;M{}

    ;j

    ;f1(0,a0)=

    ;f2(O,ao)

    ;t2

    ;?

    ;.)

    ;,?

    ;J,

    ;r,.

    ;M

    ;CharacteristicsofSequentialDetectioninCognitiveRadioNetworks

    ;.

    ;(o,=

    ;=

    ;o,a.):

    ;(0,a.)=

;M3{

    ;{)

    ;Inthecaseoflthisimmediatelyresultsin ;expressions(7)forthemeanvalues(orfirstcumulants) ;Thevariance=K2canthenbefoundfrom(20) ;and(15)toproduce:

    ;广

    ;.

    ;{)l(DD+CDc)

    ;M{}

    ;+

    ;2C-2DC

    ;MsMelM{I’lq{t

    ;M2

    ;+lI

    ;+

    ;591

    ;Equationsforthespecialcases(22)and(24)arewell ;knownwhilethegeneralequations(21)and(23)are ;validforanyrelationsbetween4andP3/~)which ;tothebestofourknowledgearenotreportedinthe ;literature.

    ;Asitcanbeseenthesecondcumulantforthe ;fixedthresholdsDandCtodependtoM{)and ;fordifferenthypotheses,.Thelastone ;dependsonthedetectionscenarioPo(X)andwillbe ;consideredatSection5.

    ;TheprobabilitythatthedurationoftheWaldtest ;+exceedssomepresetvalue

    ;4DC2C2D

    ;+

    ;(21)

    ;?+2D-2DCj

    ;

    ;IfDo..C=constoneobtainsthewell

    ;knownWald’Sformula81:

    ;Oa

    ;o

    ;{V=?

    ;Similacalculatedby

    ;Q(]./

    ;Itisoftenmorepracticaltouse1a)’insteadofthe

    ;averageM.vprovidesarealisticfigureforcomparison ;withNPtest.Moreovercanbeusedcomparing

    ;cbaracteristicsoftheclassicalsequentialanalysis ;consideredherewithnewproposalsofthesequential ;analysis(see,forexamplel5).Theresultsforthose

    ;comparisonswillbepublishedelsewhere. ;5.SequentialAnalysisoftheIncoherent ;DiversityDetectionforGeneralizedGaussian ;FadingModel

    ;5iStatisticallyIndependentFadingintheDiversity ;Branches

    ;Forthiscase,thequadraticdiversityadditionisthe ;maximumlikelihood(ML)optimumincoherent ;algorithminthecaseofknownCSIatthereceiverside ;ofthelink9,10.

    ;D?+?c,(26)=1

    ;where,arenormallydistributedstatistically ;independentsignalsofthequadraticchannels ;(quadraturecomponents),C,Darethresholds(see ;IfCSIisunknownorpartiallyknown,thesolutionis ;generalizedMLratio(seeforexample4,etc)

    ;592CharacteristicsofSequentialDetectioninCognitiveRadioNetworks

    ;above),andisaHilberttransformof.

    ;Algorithm(26)forthecaseofclosehypothesisis ;optimumforcriteriaofminimumaveragesamplesize ;(timeofanalysis)aswellbutcanbeconsideredas ;quasioptimumforanylevelofSNR.

    ;Theparametersofquadraturecomponentscanbe ;calculatedasinReL10.Inpa~icular,forhypothesis ;oneobtains:

    ;{)=?

    ;M.{I’3k}=myk2~k(27)

    ;{}=2,{}=2

    ;where

    ;2Ek22

    ;?0

    ;Ek

    ;an2=

    ;Similarly,inthecaseofthehypothesis7-t0similar ;quantitiesaregivenby:

    ;{}=

    ;{)=

    ;1

    ;2

    ;1

    ;2h~2k

;)=Do{ITk}=D0{)2h2

    ;1+2hi’

    ;(28)

    ;Inthegeneralcase(arbitrary,),thequantIty

    ;isfourparametric(seesectionII);some ;simplificati.nscanbed.nefor,hy2>>1(seefor

    ;examplefl01)-However,thisisnotsituation ;encounteredincognitiveradios.Therefore,agoodand ;simpleapproximationforlowSNRisrequiredforthe ;fourparametricdistribution.Ithasbeenshownthat[10] ;NakagamiPDFisagoodfit,especiallyforlowSNR ;scenario.Inordertoconstructsuchapproximationlet ;usfirstdefinetheparametersandqby

    ;meansofequations(27)-(28).Thenextstepistofind ;theequivalentparametermforeachhypothesis ;,,applyingEq.(5)fromRef.[5],=0,I:

    ;)_(1+)(1+g)

    ;211++2qk”(1+2)2c.s+sin.)](29)

    ;Onecanderivethat+isaFdistributed

    ;randomvariableforeachofhypothesis,with ;parameters:

    ;:

    ;

    ;ThefinalPDFforcanbefoundwiththeresults ;ofRef.16.

    ;Inthefollowingletusconsideracorrespondingto ;constantspecialacaseforfadingparameters ;independentonk.(29):

    ;…nst.(30)m’,

    ;Inthiscasep()turnsouttobeaF-distribution ;withparameters[171:

    ;?=?1m’,

    ;=

    ;?

    ;?

    ;l

    ;Q()

    ;1m()

    ;Q

    ;=.

    ;(31)

    ;m

    ;Thus,analysisofEq.(26)canbetransformedtothe ;problemofsequentialanalysisofthetwosimple

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