interdependent Detection of primary systems using wideband cognitive radios (AEÜ) International Journal of Electronics andCommunications

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International

Journal

of

Electronics

and

Communications

(AEÜ)

j o u r n al hom ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / a e u e

Detection

of

interdependent

primary

systems

using

wideband

cognitive

radios

Burak

Yılmaz,

Serhat

Erküc¸ük

∗

DepartmentofElectricalandElectronicsEngineering,KadirHasUniversity,Fatih,34083 ˙Istanbul,Turkey

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received14November2012 Accepted8May2013 Keywords: Cognitiveradios

Ultrawideband(UWB)systems Detect-and-avoid(DAA) Widebandspectrumsensing Energydetection

a

b

s

t

r

a

c

t

Cognitiveradios(CRs)maybesharingmultiplefrequencybandswithprimarysystemsiftheCRisa widebandoranultrawideband(UWB)system.Inthatcase,theCRshouldensureallthecoexisting primarysystemsinthesebandsaredetectedbeforeitcanstartdatatransmission.Inthiswork,westudy theprimarysystemdetectionperformanceofawidebandCRassumingthattherearemultiplecoexisting primarysystemsandthattheseprimarysystemsmaybejointlyactive.Accordingly,weconsiderthe implementationofenergydetectionschemeinmultiplebandsfollowedbytwodetectionmethods:(i)a maximum-a-posteriori(MAP)baseddetection(i.e.,jointdetection)thattakesintoaccountthestatistics ofsimultaneouslyoperatingsystemsinindependentbandsand(ii)aNeyman–Pearson(NP)testbased detectionthatoptimizesthethresholdvaluesindependentlyineachband(i.e.,independentdetection). Forasimplerimplementationoftheindependentdetection,weshowthatthethresholdvaluesobtained fromjointdetectioncanbeusedinordertoachievetheoptimumNPtestbasedindependentdetection results.Inadditiontoquantifyingthegainofjointdetectionoverindependentdetectionintermsof probabilitiesoffalsealarmanddetectionforpracticalscenarios,wealsopresenttheoperationcapability ofCRsintermsofthefractionsoftimetheCRcanaccessthechannelwithoutinterferingwiththeprimary systems.Theresultsareimportantforthepracticalimplementationofmultibanddetectionwhenthe primarysystemsareknowntobeinterdependent.

© 2013 Elsevier GmbH. All rights reserved.

1. Introduction

Asaresultofincreaseddemandfornewwireless communi-cationtechnologies,therehavebeennumerouslicensedsystems assigned todifferent frequency bandsin recent years. This has caused the spectrum become very crowded, and yet not well utilized. In the last decade cognitiveradios (CRs) [1] and ultra wideband(UWB)systems[2]havebeenproposedandinvestigated asunlicensedsystems,wheretheyhavebeenwidelyacceptedas alternativetechnologiesforbetterutilizationofthespectrum.From theperspectiveofalicensedprimarysystem,themajorconcern fortheimplementationofeitherCRsorUWBsystemsisthe possi-bleinterferencetheymaycausetoprimarysystems.Hence,many regulatoryagenciesworldwidehavemandateddetect-and-avoid (DAA)techniquesinvariousbands[3].Accordingly,CRsandUWB systemshavetoperformspectrumsensinginthesebandsbefore theycancommunicate.

Spectrumsensinghasbeenwidelyexploredinthecontextof cognitiveradios.Surveysofexistingspectrumsensingtechniques canbefoundin[4]and[5].Whilesometechniquesarebasedon

∗ Correspondingauthor.Tel.:+902125336532.

E-mailaddresses:[email protected](B.Yılmaz),[email protected] (S.Erküc¸ük).

matchedfilteringor featuredetectionof primaryusers’signals, energydetection [6]isthemostcommontechniquebecauseof its low computationaland implementationcomplexity in addi-tiontonotrequiringanyknowledgeoftheprimaryusers’signals, despitethechallengesinsignaldetectionreliabilityforalow signal-to-noiseratio (SNR)[4].Thereisa comprehensiveliteratureon energydetectioninasinglefrequencybandwithfurther improve-mentsusingcollaborationamongsecondaryusers[7–9],diversity schemes[10],multipleantennas[11]andamodelconsideringthe primaryuserappearanceprobability[12].

Ontheotherhand,theliteratureonenergydetectionin mul-tiplefrequencybandsisrathernew.Thisconceptisindeedquite importantasitismoredesiredtoassesstheavailabilityofawider spectrumforbetterutilization.Moreover,theCRsmaybe wide-bandorUWBsystems,andtherefore,theyshouldensureallthe coexistingprimarysystemsincommonbandsaredetectedbefore theycanstartdatatransmission.In[13],theeffectofnumberof primary users in differentbands onthedetection performance wasinvestigated.In[14],theaggregateopportunisticthroughput wasmaximizedovermultiplebandssubjecttosomeconstraints ontheamountofinterferencetoprimaryusers.In[15],softand hardfusiontechniqueswereconsideredtoimprovethedetection performanceinthepresenceofmultiplesecondaryusers.In[16], periodicsensingwasaddedtothesystemmodelof[14]inorderto improvethedetectionperformance.Thecommonassumptionin

1434-8411/$–seefrontmatter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.aeue.2013.05.003

thesestudieswasthattheprimarysystemsindifferentbandswere independent.However,ifthelicensedsystemsindifferentbands aredependent,thedetectionperformancecanbefurtherimproved. In[17],theprimarysystemdetectionperformancewasassessedfor M=2interdependentbandsusingamaximum-a-posteriori(MAP) baseddetectionmethodandthedetectiongainoverindependent detectionwasquantified.However,thisworkwaslimitedtoM=2 bands.

Inthispaper,motivatedbyquantifyingthedetection perfor-mance gain when there are M>2 interdependent systems, we generalizetheworkin[17]tomultiplebands.Here,M>2could beanexampleofMsystemsinindependentfrequencybandswith knownactivitystatistics.Forexample,thestatisticsmightindicate thattwoofthethreesystemsarejointlyactive40%ofthetime, whileallthreesystemsarejointlyactive50%ofthetimeand pas-sive10%ofthetime.Forthatweconsidertheimplementationof energydetectionschemeinmultiplebandsfollowedbyeithera MAPbaseddetection(i.e.,jointdetection)oraNeyman–Pearson (NP)testbaseddetectionthatoptimizesthethresholdvalues inde-pendentlyineachband(i.e.,independentdetection).Differentfrom [17],thecontributionofthecurrentstudyisfourfold.Accordingly, we

1generalizetheprobabilityoffalsealarmanddetection expres-sionsforM>2forbothjointandindependentdetection, 2usethe thresholdvalues obtainedfromjoint detection soas

toachievetheoptimumNPtest basedindependentdetection resultswithasimplerimplementation,

3providepracticalexamplestoquantifytheperformancegainof jointdetectionoverindependentdetectionforvariousscenarios, and

4presenttheoperationcapabilityofCRsasanadditional perfor-mancemeasureintermsofthefractionsoftimetheCRcanaccess thechannelwithoutinterferingwiththeprimarysystems.

Inthiswork,inadditiontogeneralizingtheprobabilityoffalse alarmand detectionexpressions [18],theaccuratemodeling of the decision variable with 2 _{distribution is discussed. This is}

importantasmoststudiesconsidertheapproximateGaussian dis-tributionintheirmodels.Usingthe2_{distributioninMAPbased}

detection,jointdetectionanalyticalexpressionsandtheassociated thresholdvaluesarederived andpresented indetail. Whilethe commonlyusedperformancemeasuresareprobabilitiesoffalse alarmanddetectionintheliterature,theoperationcapabilityof CRsis introducedasanadditionalperformancemeasurein this study.ThisisanimportantperformancemeasureasCRswith sim-ilardetectionperformancesmayindeedutilizethecommonband quitedifferentlydependingonsystemactivityvalues.Accordingly, themodelsandtheresultspresentedinthisstudyareimportantfor thepracticalimplementationofmultibanddetectionwhenthere aremultipleprimarysystemsthatareknowntobeinterdependent. Therestofthepaperisorganizedasfollows.InSection2,the receivermodeloftheCRispresented.InSection3, implementa-tionsofjointdetectionandindependentdetectionarepresented. InSection4,numericalandsimulationresultsareprovidedforthe comparisonoftheconsidereddetectionmethodsunderdifferent scenarios.ConcludingremarksaregiveninSection5.

2. Receivermodel

We assume that there are M primary systems operating in orthogonalfrequencybandsandcoexistingwithawidebandCR. Eachprimarysystemisassumedtocommunicatebytransmitting aprimarysignal,sm(t),whereeachsystemhasabandwidthofWm,

m={1,2,...,M}.Thesesystemsmaybeactiveorpassivedepending

onthetimeoftheday.Thereceivedsignalsarefilteredusingideal zonalbandpassfilterswithbandwidthsWmateachorthogonal

fre-quencybandtoeliminatetheout-of-bandnoise[6,10].Accordingly, thetwohypothesescorrespondingtotheabsenceandpresenceof thefilteredsignalreceivedfromthemthsystem,respectively,are

H0,m: rm(t)=nm(t) (1)

H1,m: rm(t)=Amejmsm(t−m)+nm(t), (2)

whereeach primarysignalsm(t)passes throughachannel with

amplitudeAmandphasemuniformlydistributedover[0,2),m

isthetimingoffsetbetweenthetwosystems,nm(t)isband-limited

additivewhiteGaussiannoise(AWGN)withvariance2

nm=N0Wm andN0/2isthetwo-sidednoisepowerspectraldensity.Notethat

thechannelsareassumedtobefrequencynonselectiveforthemth system,however,eachsystemmayhavedifferentattenuation val-uesineachfrequencyband,whichisthemaincriticalassumption inoursystemmodel(i.e.,differentSNRlevelsindifferentbands).

Inpractice,timingoffsetisanimportantdegradationfactoron thedetectionperformance.Therearetwomajorcausesfortiming offset:(i)asynchronismbetweenusersand(ii)asynchronismwith thereceivedsignal.Thefirstoneoccurswhentherearemultiple primaryusersinthesamefrequencyband.Iftheprimarysignals communicatewithtimingmisalignment,thismaycausedecreased spectrum opportunities [19]. The second one occurs when the receiverisnotsynchronizedwiththereceivedsignal,wherethe sig-nalofinterestmayarriveattheCRreceiverwhiletheCRisalready sensingthespectrum[20].Inthatcase,theobservedsignalmay beacombinationofonly-noiseandsignal-plus-noisecomponents, andthedetectionperformancemaybedegraded.Inthisstudy,we assumethatthereisasingleuserineachfrequencybandandthat theprimaryusersignalhasarrivedattheCRreceiver,beforetheCR startssensingthespectrum(i.e.,theprimaryusersignaliseither presentorabsentthroughoutthesensingdurationoftheCR).This isawidelyusedassumptionintheliterature,andisavalid assump-tionforthecurrentstudyastherelativedetectionperformanceof jointandindependentdetectionmethodsisofmaininterest.Next, modelingthedecisionvariableusingenergydetectionisexplained.

2.1. Modelingthedecisionvariable

Consideringthereceivedsignalsin(1)and(2),anenergy detec-tion schemecan be used[6].Using a square-law detector and normalizingtheoutputwiththetwo-sidednoisepowerspectral density N0/2, the decision variable for the mth systemcan be

obtainedas dm= 2 N0



Tm 0



_rm(t)



2 dt, (3)

whereTmistheintegrationtimeforthemthsystemand|·|isthe

absolutevalueoperator.Adoptingthesamplingtheorem approx-imationused forbandpasssignals in [6] and [10],the decision variablecanbeapproximatedas

dm≈ 1 N0Wm T



mWm i=1



AIs(i)I −AQs(i)Q +n (i) I



2 +



AIs(i)_Q +AQs(i)I +n (i) Q



2



, (4)

where s(i)_I and n(i)_I (s_Q(i) and n(i)_Q) denote the ith samples of the low-pass equivalent in-phase (quadrature) components of sm(t−m)andnm(t),respectively,sampledattheNyquistrateWm,

10−4 10−3 10−2 10−1 100 10−4 10−3 10−2 10−1 100 P f P md simulation χ2 _modelling Gaussian approximation SNR=5dB, N=20 SNR=5dB, N=12 SNR=5dB, N=4 SNR=10dB, N=16

Fig.1.ComplimentaryROCcurvesinthepresenceofaprimarysystemforvariousSNRandintegrationtime-bandwidthproductvalues.

H0,m,onlythenoiseterms,n_I(i)andn(i)_Q,willdeterminethe

deci-sionvariabledmin(4).Assumingthatthesamplesoftheprimary

signal,s(i)_I ands(i)_Q,givenin(4) arezero-meanGaussianrandom variables,1_{thedecisionvariabled}

mwillconsistonlyofzero-mean

Gaussianrandomvariablesineitherhypothesis.Hence,underH0,m

it canbeshown that dm can bemodeled using2 distribution

withNm=2TmWmdegreesoffreedom,wherethevariancetermis

2

m=(2nm)/(N0Wm)=1[17].Similarly,underH1,m,itcanbeshown thatdmcanbemodeledusing2 distributionwithNm=2TmWm

degreesoffreedom,wherethevariancetermis2

m=m+1with

SNRdefinedasm=(A2m2s)/(N0Wm)ands2beingthevarianceof

theprimarysignalsamples.Accordingly,theprobabilitydensity function(pdf)ofdmforeitherhypothesiscanbeexpressedas

fDm(dm)= 1 Nm m 2Nm/2(Nm/2) dNm/2−1 m e−dm/2 2 m_, (5)

where (a,b)=

_b∞e−tta−1_{dt is the upper incomplete Gamma}

functionand(a)=(a,0)istheGammafunction[22].In[8]and [14], assuming largeNm,dm was assumed to be normally

dis-tributedwithdm∼N(Nmm2,2Nmm4), wherem2 =1forH0,m and

2

m=(m+1)for H1,m. In the nextsubsection, we will discuss

whetherthe2_{distributionorthenormaldistributionismore}

suit-ablefor modelingthedecisionvariable.Next,thedetectionofa singlesystemispresented.

1_{Forexample,thesamplesofanorthogonalfrequency-divisionmultiplexing} (OFDM)basedprimarysignalsampledattheNyquistratecanbewell-approximated asindependentandidenticallydistributed(i.i.d.)zero-meanGaussianrandom vari-ablesbasedonthecentrallimittheorem[21].

2.2. Detectionofasinglesystem

Inconventionaldetection,thedecisionvariabledmiscompared

toapre-selectedthresholdvalue minordertomakeadecision

forthemthsystem.Theperformancemeasures,probabilityoffalse alarmandprobabilityofdetection,canberespectivelyexpressed as

Pf,m=Pr[dm> m|H0,m] (6)

Pd,m=Pr[dm> m|H1,m], (7)

where(6)and(7)canbesimplifiedto

Px,m=Q

Nm 2 , m 22 m



= 



(Nm/2),( m/2m2)

(Nm/2) , x∈



f,d



(8) withthecorresponding2

m valuesforH0,mandH1,m,andQ(a,b)

istheregularizedupperincompleteGammafunction[22].Ifdm

wasassumedtobenormallydistributedasin[8]and[14],then(8) simplifiesto Px,m=Q



m−Nmm2 2 m



2Nm



, x∈



f,d



(9)

withthecorresponding2

m valuesfor H0,m andH1,m,andQ(·)is

theGaussianQ-functiongivenasQ (x)=(1/√2)

_x∞e−t2/2_dt[22].

InFig.1,complementaryreceiveroperatingcharacteristic(ROC) curves(i.e.,Pf vs.Pmd=1−Pd)areplottedtocomparethe

theoret-icalperformanceaccordingto(8)and(9)toasimulatedprimary systemdetectedusingasquare-lawdetectoraccordingto(1)–(3), (6)and(7).TheprimarysystemisassumedtobeaWiMAX-OFDM systemas defined in [23] withfurtherassumptions of quadra-turephase-shiftkeying(QPSK)modulation,K=256subcarriersand

a systembandwidthof W1=8MHz beingused. Theintegration

timesin(3) areselectedasT1={0.25,0.75, 1,1.25}␮sresulting

inN={4,12,16,20},andanSNRof1={5,10}dBareassumed.

Itcanbeobservedthattheresultsfora2_distributedd

1matches

thesimulationresultswellevenonalog-scale,whereasthe nor-maldistributionyieldsagoodapproximationinthelinear-scalefor lowSNRonly[8].Sincethesystemimplementationrequiresvery lowPfandPmdvalues,weneedtomonitorthechangesinthe

log-scale.Therefore,wewillbuildourmodelbasedontheaccurate2

distribution.

2.3. Detectionofmultiplesystems

If the CR is a wideband system, then it has to assess the presence of all coexisting primary systems before it can com-municate.Accordingly, thehypotheses have to beredefined as H=



[HxM,M,...,Hx2,2,Hx1,1]|xm∈{0,1}



.SincethereareM pri-marysystems,thereare2M_{possiblecombinationsofhypotheses.}

Accordingly,theCRcanonlytransmitifxm=0,

∀

m,whichcanbe

representedbyH0.Fortherest2M−1combinationsevenifasingle

primarysystemisactive,thentheCRisnotallowedto communi-cate.Thehypothesescorrespondingtohavingatleastoneactive systemcanberepresentedbyH1,i,1≤i≤2M−1,wheretheactive

andpassivesystemsineachhypothesiscanbedeterminedbythe relation

i=(xM···x2x1)2 (10)

with(·)2denotingthebase-2representationofi.Hence,the

prob-ability of false alarm and probability of detection for multiple systemscanbeexpressedas

Pf =1−Pr



M m=1(dm< m )H0



(11) Pd=1− 2

_

M₋₁ i=1 Pr



M m=1(dm< m)H1,i



Pr



H_1,i



H1



, (12) whereH1=



2M₋₁ i=1 H1,i.

Theprobabilityofdetectionexpressiongivenin(12)isdifferent from the conventional expression mainly due to the probabil-itytermbeingconditionedondifferenthypotheses,H1,i.Hence,

theprobabilitiesofthesehypothesesareimportantin determin-ing(12).Accordingly,theprobabilitythatalltheprimarysystems arepassiveisp0=Pr[H0],whereaspi=Pr[H1,i],1≤i≤2M−1,isthe

probabilitythat H1,i holds, where



2M₋₁

i=0 pi=1. For example,if

thereareM=4interdependentsystemsandp7 isclosetounity,

thatmeansthefirstthreesystemsarejointlyactivemostofthe timewhilethefourthsystemis notactive(i.e.,7=(0111)2).To

note,theprobabilities{pi}canalsobereferredtoasjointsystem

activityvalues.

InSection3,probabilityoffalsealarmanddetectionexpressions givenin(11)and(12)willbeadaptedforjointandindependent detectionmethods,andexactexpressionsfortheseprobabilities willbeobtained.

2.4. Operationcapability

Inadditiontoprobabilitiesoffalsealarmanddetection,itisalso importanttoassesshowthewidebandCRwillbeabletoutilizethe commonband.Accordingly,wedefine

Tu=p0(1−Pf) (13)

Th=(1−p0)(1−Pd) (14)

asadditionalperformancemeasures,whereTuandTharethe

frac-tionsoftimethewidebandcognitiveradioisoperatingusefully andharmfully(causinginterferencetoprimarysystems), respec-tively.Itshouldbenotedthatthefractionoftimeduringwhichthe cognitiveradioisnotoperating,Tn,isgivenbyTn=1−Tu−Th.

3. Detectionmethods

Inthefollowing,weconsidertheimplementationoftwo detec-tionmethodsforM>2primarysystemsthatareinterdependent. Forbothmethods,itisassumedthatthesystems’jointactivity val-ues{pi}andthepdfsofthedecisionvariables{dm}areknowna

priori.Thisisareasonableassumptionasthetrafficinformationof theprimarysystemsmaybeavailabletosecondaryusers,andthe SNRoftheprimarysignalscanbeestimatedatthereceiver.

3.1. Jointdetection

Knowing{pi}andthepdfsof{dm},theMAPdecisionruleserves

asanoptimaldecisionrule.Thehypothesiscanbeestimatedby findingthemaximumoftheMAPdecisionmetricsas

ˆi= argmax

i∈{0,1,...,2M_−1}PMi ˆ

H=H0if ˆi=0; Hˆ =H1if ˆi={1,2,...,2M−1},

(15)

where the decision metrics are PM0=

b0p0fD1,D2,...,DM|H0(d1,d2,...,dM) and PMi= bipifD1,D2,...,DM|H1,i(d1,d2,...,dM), {i=1, 2,..., 2

M_{−1}. Thebias}

terms{bi|i=0,1,2, ...,2M−1}aretheintentionallyintroduced

termstoachieveadesiredtrade-offbetweentheprobabilitiesof false alarm and detection,and fD1,D2,...,DM|Hx(d1,d2,...,dM)are thejointpdfsconditionedonthehypothesisHx.Sincetheprimary

systemsare innon-overlappingfrequency bands,thejointpdfs conditionedonHxcanbeexpressedusing(5)as

fD1,D2,...,DM|Hx(d1,d2,...,dM)= M



m=1

dNm/2−1 m 2Nm/2_(Nm/2)



×



e−dm/2_m2 (2 m)Nm/2



=C M



m=1



e−dm/22_m (2 m)Nm/2



, (16) whereC=



M_m=1



(dNm/2−1 m )/(2Nm/2(Nm/2))

isacommonterm foralljointpdfsindependentofthehypotheses.Ontheotherhand, thesecondterminbracketsdependsonthehypothesisitis con-ditionedon,asthevariancetermdefinedbefore(5)is2

m=m+1

for H1,m and 2m=1 forH0,m.Accordingly,using therelationof

theindexiwith{xm|m=1,2,...,M}asgivenin(10),thedecision

metricscanbewrittenas

PMi=bipiC M



m=1 exp



(−dm)/(2(m+1)xm)

(m+1)xmNm/2 ,



i=0,1,2,...,2M−1



(17)

Considering(11),(12)and(15),theprobabilitiesoffalsealarmand detectioncanberedefinedas

Pf=1−Pr



2M₋₁ i=1 (PM0<PMi)



H0



(18)

Pd=1− 2

_

M₋₁ i=1 pi 1−p0 Pr



2 M₋₁ j=1 (PM0<PMj)



H_1,i



. (19)

Bysubstituting(17)intothecomparisonterm{PM0>PMi},(18)and

(19)canbesimplifiedto Pf=1−Pcond,H0



_M



m=1 (1−Pf,m)



(20) Pd=1− 2

_

M₋₁ i=1 pi 1−p0Pcond,H1,i



_M



m=1 (1−Pd,m)Xm(1−Pf,m)(1−xm)



, (21) wherePcond,Hx,{x=0}or{x=1,i}istheconditionalprobabilityterm obtainedas Pcond,Hx= 2

_

M₋₁ i=1 P _i|( 1)x1,( 2)x2,...,( 2m−1)xm,...,( 2M−1)xM,Hx (22) with P _i|( 1)x1,...,( 2M−1)xM,Hx =Pr



_M



m=1 xmamdm< i



x1a1d1 < 1,...,xMaMdM< 2M−1,Hx



(23) andam=m/(2(m+1)).Theresultingthresholdvaluesare

i=



_M



m=1 xmNm 2 ln(m+1)+ln



_p 0 pi



+ln



_b 0 bi



(24) fori=_{1,2,...,2M_{−1},where{x}

m}areobtainedfrom(10).Itshould

benotedthat



2m−1|m=1,2,...,M



correspondtoindependent thresholdvalues2 _{foreachband,m,whereastherestofthe{}

i}

values(i.e.,2M_{−M−1values)correspondtothejointbands.For}

example,whenM=4thethreshold 5correspondstobands1and

3(i.e.,5=(0101)2).We calculatetheprobabilitiesof falsealarm

anddetectionusing (20)and (21),where thetermsin (22)can becalculatednumericallyasexplainedinAppendixA.Byletting b1=b2=···=b2M₋₁=bin(24),andvaryingthevalueofb,a trade-offbetween(P_f,P_d)-pairscanbeobtainedwithaclose-to-optimal performance[17].

3.2. Independentdetection

Theprobabilitiesoffalsealarmanddetectionformultiplebands canbeexpressedas Pf=1− M



m=1 (1−Pf,m) (25) Pd=1− 2

_

M₋₁ i=1 pi 1−p0 M



m=1 (1−Pd,m)xm(1−Pf,m)(1−xm) (26)

ifthebandsareindependentlyprocessed.Theseequationscanalso beobtainedbylettingPcond,Hx=1in(20)and(21).

2_{NotethatthesevaluesresultingfromMAPdetectionandcorrespondingtothe} mthbandaredifferentfromtheconventionalthresholdvalues{ m|m=1,2,...,M} givenin(11)and(12).

3.2.1. NPtest

InordertoobtainthebestdetectionperformancetheNPtest canbeemployed,whichoptimizesthethresholdvaluesinorderto maximizePdforagiventargetPf=˛:

max { 2m−1|m=1,2,...,M} Pd s.t.Pf =˛. (27) Here,



2m−1|m=1,2,...,M



aretheindependentthreshold val-uestobeoptimized.ThisisequivalenttomaximizingPdoveran

M-dimensionalsearchspace.InFig.2,possible(Pf,Pmd)-pairsthat

areobtainedbyusing 1∈[0,100]and 2∈[0,100]in(25)and(26),

i.e.,thesearchspaceforindependentdetection,andthe numeri-callycalculatedminimumPmdvaluesforfixedPf={˛}areplotted

whenM=2withP:{p0=0.76,p1=0.06,p2=0.11,p3=0.07},N1=24,

N2=8,1=5dB,and2=10dB.Itcanbeobservedthat,asexpected,

thebest(Pf,Pmd)-pairsareobtainedbytheNPtestasthecurve

attainsthelowerboundofthesearchspace.ForlargervaluesofM, thecomputationcomplexityoftheNPtestincreases.

3.2.2. MAPtest

Alternatively,weconsiderusingthethresholdvaluesobtained fromMAPdetectioninsteadoftheNPtest.Thesethresholdvalues indeedresultfromthe{PM0>PMi}comparisons,andserve

intrinsi-callyasposterioroddsratios[24].Moreover,thesevaluesareeasier tocomputecomparedtotheNPtest.Accordingly,thethreshold valuesthatwillbeusedin(25)and(26)canbeobtainedfrom(23) and(24)bylettingtheindexof iasi=2m−1form={1,2,...,M}

independentbands,andusingtheresultingrelation

i=2m−1=(xM···xm+1xmxm−1···x1)2=(0...010...0)2, (28)

wherexmistheonlynon-zeroterm.Notethatthethresholdvalues

{ i}givenin(23)forjointdetectionarerelatedtodecisionvariables

{dm}multipliedbythetermam.Therefore,thethresholdvaluesfor

eachindependentbandcanbeobtainedas

2m−1=



Nm 2 ln(m+1)+ln

p0 p2m−1



+ln

b0 b2m−1

 

am (29)

form={1,2,...,M}.InAppendixB,thederivationof



2m−1



is pre-sentedinmoredetailbyusingthecomparisons



PM0<PM2m−1



. Similartojointdetection,bylettingb1=b2=···=b2M−1=bin

(29)andvaryingit,atradeoffbetween(Pf,Pd)-pairscanbeobtained.

4. Results

Inthissection,weinitiallyprovidesimulationandnumerical resultstoconfirmthevalidityof thejointdetection modeland theindependentdetectionmodelthatusesthethresholdvalues obtainedfromMAPdetectionwhenM>2.Wethenprovidesome numericalresultstodeterminethegainofjointdetectionover inde-pendentdetectionintermsofvarioussystemparameterssuchas theeffectsofjointsystemactivityvalues,numberof interdepen-dentsystemsandSNR.Finally,wepresenttheoperationcapability ofCRsforpracticalscenarios.Forallscenarios,itisassumedthat Pr[H0]=0.90andPr[H1]=0.10.Also,SNRandNvaluesforeachband

arefixedto10dBand8,respectively,unlessotherwiseindicated. In Fig. 3, we validate the probabilities of false alarm and detectionexpressions givenin(20)and(21)for jointdetection. Accordingly, we simulate thedecision metrics {PMi} and

eval-uate them in (18)and (19)to obtain thecomplementary ROC curves. The system activityvalues considered are P1:{p2M₋₁= 0.1,p1=p2=...=p2M₋₂=0} and P2:{randompi} for various

SNRandNvalueswhenM=3andM=4.It canbeobservedthat thesimulationresultsconfirmthevalidityofthejointdetection model.

Fig.2.The(Pf,Pmd)-pairsearchspaceandthe(Pf,Pmd)-pairsobtainedbytheNPtestwhenP:{p0=0.76,p1=0.06,p2=0.11,p3=0.07},N1=24,N2=8,SNR1=5dBand SNR2=10dB. 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 P f P md Joint Det., M=4 Joint Det., M=3 simulation P₁ N = [8,8,8] SNR dB = [8,8,8] P 2 N = [4,6,8] SNR_dB = [8,8,8] P 2 N= [8,8,8] SNR_dB = [3,5,7] P 2 N = [8,8,8,8] SNR dB = [10,5,5,10] P 1 N = [12,12,12,12] SNR dB = [8,8,8,5] P 2 N = [8,8,8,12] SNR dB = [5,5,10,10]

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 10−4 10−3 10−2 10−1 100 P f P md

Indep. Det. (MAP), P

1

Indep. Det. (MAP), P

2 Indep. Det. (NP), P 1 Indep. Det. (NP), P₂ Joint Det., P₁ Joint Det., P₂ M=4 M=3 Independent Detection Joint Detection M=3 M=4

Fig.4.ComplimentaryROCcurvesofjointandindependentdetectionforvariousPandMvalues.

In Fig. 4, we present joint and independent detection performances on the same complementary ROC plot. For that, we consider two different sets of system activity val-ues,P1:{p2M₋₁=0.07,p1=p2=p4=...=p2M−1=0,pi}andP2:

{p2M₋₁=0.08,p1=p2=p4=...=p2M−1=0,pi} for M=3 and

M=4,where pi representtheprobabilityofjointlyactivebands

andhaveequalvalues,pi=(0.10−p2M₋₁)/(2M−M−2),

∀

i.Itcan be observed that the joint detection performs better than the

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 10−4 10−3 10−2 10−1 100 P_f P md

Indep. Det. (MAP), P

1

Indep. Det. (NP), P

1

Indep. Det. (MAP), P

2 Indep. Det. (NP), P 2 M=3 M=4 SNR dB=[10,10,10] N=[8,8,12] SNR_dB=[10,10,5] N=[8,8,8,8] N=[8,12,8,12] SNR dB=[10,12,15,15]

0.020 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 5 10 15 20 25 30 p₇ Gai n P_f = 10−8 P f = 10 −₆ P f = 10 −₄ P f = 10 −₂ 2 bands jointly active All bands active

Fig.6. TheeffectofsystemactivityvaluesonthedetectionperformancewhenM=3.

independentdetectionwithMincreasingandforP2(allMbands

areactivemorefrequently)asexpected.Also,itisimportanttonote thatthethresholdvaluesobtainedbyMAPdetectionandusedin independentdetectionachievethesameperformance astheNP testresults.InFig.5,wefurthervalidatetheindependent detec-tionresultsforvariousSNRandNvalues,whentwodifferentsets ofrandomlyselectedjointsystemactivityvalues(P1forM=3and

P2 for M=4)areused.Consideringthattheperformanceresults

arecoinciding, the MAPtest based approach canbe usedwith aneasierimplementationtoreplacetheNPtestforindependent detection.

Next,we quantify the gain of jointdetection over indepen-dentdetectionforvariousscenarios.Thegainsaredefinedasthe Pmd ratiosofindependentand jointdetectionatfixedPf values,

i.e.,Gain= (Pmd(indep.)) / (Pmd(joint)).Thisisanimportant

perfor-mancemeasurefromtheperspectiveofaprimaryuserbecause theratioindicateshowmuchmoretheprimaryuserwillbe inter-feredbytheCRifindependentdetectionisusedinsteadofthejoint detection.InFig.6,misdetectionperformancegainsarecompared atvariousPfvaluesfor2-bandjointlyactive(i.e.,only{p3,p5,p6}

arenon-zero)andall-bandactive(i.e.,_{p1,p2,···,p6}arenon-zero)

caseswhenM=3andp7varying.Theprobabilityvalue(0.10−p7)is

equallydistributedamongthenon-zeroprobabilityvaluesforboth cases.ItcanbeobservedthatthegainsincreasewithPfdecreasing

andp7increasing.Whenp7=0.1,theperformancesforbothcases

mergeatthebestgainvalue.

InFig.7,theeffectofSNRdegradationonthedetection perfor-manceisinvestigatedforvariousMvalues.Again,themisdetection gainvaluesarecalculatedatvariousPfvalues.Foreachcase,we

assumethatallsystemsarejointlyactiveallthetime,i.e.,p2M₋₁= 0.1,forM={2,3,4}.WhenM=2,gainssimilartotheonesreported in[17]areobtained.WhenMisincreased,thereisasignificant increaseingainduetoprocessingthebandsjointly.WhentheSNR

decreasesinaband,theM=4casecancompensatebetterdueto otheractivebandshavingsignificantSNRvalues.

Finally,theoperationcapabilitiesofCRsareconsidered. Accord-ingly,curveswhichshowthefractionsoftimetheCRisoperating usefullyandharmfullyareplottedusing(13)and(14)forvarious scenarios.InFig.8,theeffectofsystemactivityvaluesonthe oper-ationtimesofCRsisinvestigatedwhentheSNRofeachbandis {5,10_}dBwhenM=3.Thejointsystemactivityvaluesarenon-zero onlyforp0andp7forthreedifferentsets.SincethePfandPd

val-uesobtainedfrom(11)and(12)areconditionedonH0andH1,the

detectionperformancesofallthreecasesarethesame.However, theiroperationcapabilitiesaredifferentascanbededucedfrom (13)and(14),andasplottedinFig.8.Whenp0=0.6,p7=0.4,and

SNR=5dB,byselectingappropriatebiasvaluesforjointdetection theCRcanoperateatpointC,whereitcanusefullyoperatealmost 60%ofthetimewhileinterferingwiththeprimarysystemonly0.3% ofthetime.Ontheotherhand,usingindependentdetectionwith appropriatethresholdvalues,theCRcanoperateatpointDalmost 60%ofthetimeusefullywhileinterferingwiththeprimarysystem 1.8%ofthetime.Iftheinterferencelevelisrestrictedtoacertain amountbytheprimarysystem,thentheCRoperationpointsfor eitherdetectioncanbedeterminedaccordingly(e.g.,operatingat pointsAorBifTh=10−4isallowed).

In Fig. 9, the operation capabilities of CRs are investigated for P1:{p0=0.9, p7=0.1} (i.e., full gain), P2:{p0=0.9, p7=0.05,

p3=p5=p6=0.05/3} (i.e., joint bands active), and P3:{p0=0.9,

p7=0.05,p1=p2=···=p6=0.05/6}(i.e.,allbandsactive)whenM=3.

ThegainsofthesecasescanbeobservedinFig.6.Sincep0=0.9,the

CRcanoperateusefullyatmost90%.Therefore,particularlythe regionwheretheCRcanoperateusefullywithhighutilizationof thecommonbandiszoomedin.WhenTuis89.99%,the

interfer-ingtimeofCRduetoindependentdetectionisapproximately18 and14timestheinterferingtimeduetojointdetectionforP1and

10−8 10−7 10−6 10−5 10−4 10−3 10−2 100 101 102 103 P_f Gai n All bands 10dB 1 band 5dB 2 bands 5dB M=2 M=3 M=4

Fig.7. TheeffectofSNRvaluesonthedetectionperformanceforvariousMvalues.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−6 10−5 10−4 10−3 10−2 10−1 100

Normalized useful time for transmission (T

u)

Interfering time with primary user transmission (T

h ) SNR dB = [10,10,10]; Joint det. SNR dB = [10,10,10]; Indep. det. SNR dB = [5,5,5]; Joint det. SNR_dB = [5,5,5]; Indep. det. B C D A p₀ = 0.9 p 7 = 0.1 p 0 = 0.6 p₇ = 0.4 p 0 = 0.3 p 7 = 0.7

0.8998 0.8999 0.9 10−6 10−5 10−4 10−3 10−2 10−1 100

Normalized useful time for transmission (T

u)

Interfering time with primary user tranmission (T

h ) Joint det., P 1 Indep. det., P 1 Joint det., P₂ Indep. det., P 2 Joint det., P 3 Indep. det., P 3

Fig.9. Theeffectofsystemactivityvaluesonthefractionsoftimethecognitiveradioisoperatingusefullyandharmfully.

P2,respectively,whereastheperformancesforP3aresimilar.These

results are also in correspondence with the gains provided in Fig.6.

WhilethecurrentstudypresentedcomplementaryROCcurves andoperationcapabilitiesofCRsforthecomparisonofjointand independentdetectionmethodsfromthedetectionperspectiveof awidebandCR,theimplementationmayrequireeachprimaryuser havingaconstraintontheinterferencelevel.Accordingly,studying theCRthroughputsubjecttosomeinterferenceconstraintsdefined foreach userasin[14] maybeaninteresting extensionofthe currentwork,however,isaresearchtopicforfuturestudy.

5. Conclusion

Inthispaper,westudiedtheprimarysystemdetection perfor-manceofawidebandCRassumingthattherearemultiple(M>2) coexistingprimarysystemsandthattheseprimarysystemsmay bejointlyactive.Accordingly,weconsideredtheimplementation ofjointandindependentdetectionmethods.Forthejointdetection, weconsideredaMAPbaseddetectionthatintrinsicallyoptimizes thethresholdvalues.Fortheindependentdetection,weconsidered theimplementationoftheoptimumNPtestbaseddetection,and asimplerimplementationthatusesthethresholdvaluesobtained fromtheMAPdetection,wherebothmethodswereshownto per-formthesame.Weconfirmedthevalidityofthejointdetection modelandtheMAPbasedindependentdetectionmodelusing sim-ulationstudies.Wethenprovidednumericalexamplestoquantify theperformancegainofjointdetectionoverindependent detec-tion.Finally,wepresentedtheoperationcapabilitiesofCRsinterms ofthefractionsoftimetheycanoperateusefullyandharmfully.The resultspresentedareimportantforthepracticalimplementation ofmultibanddetectionwhentherearemultipleprimarysystems thatareknowntobeinterdependent.

Acknowledgements

ThisworkwassupportedbyaMarieCurieInternational Rein-tegrationGrantwithinthe7th_{EuropeanCommunityFramework}

Programme.

AppendixA. Numericalevaluationoftheconditional probabilityterm

Theconditionalprobabilityterms



P _i|( 1)x1,...,( 2M−1)xM,Hx



in (22)canbenumericallycomputedconditionedonHxforgiveni

andM.Forknowniand Mvalues,initially{xi}canbeobtained

fromtherelationi=(xM...x2x1)2.Thentheexpressionin(23)can

besimplifiedas

Pr[d< i|Hx], (30)

wherethevariabledhasapdf

fD(d)=fD₁(d1)∗fD₂(d2)∗...∗fD_M(dM) (31)

with_∗representingtheconvolutionoperatorandeachpdffD_m(dm)

obtainedfrom(5)as fD_m(dm)= fDm(d m/am)/am 1−Q



(Nm/2),(( 2m−1)/(2amm2))

, 0<dm < 2m−1, (32) ifxm=1,m∈{1, 2,...,M}.Otherwise, ifxm=0,fD_m(dm)=ı(dm),

whereı(·)isa Diracdeltafunction.Forknown_{ i}values that

canbeobtainedfrom(24),theconditionalprobabilitytermcan benumericallycomputedusing(30).

AppendixB. Derivationofthethresholdvaluesfor independentdetection

ConsideringtheMAPdecision metrics{PMi},probabilitiesof

falsealarmanddetectionforthemthsystem,Pf,mandPd,m,given

in(6)and(7)canbewrittenintermsofmetriccomparisonsas Px,m=Pr[dm> 2m−1



H{0,1},m]

=Pr[PM0<PM2m−1



H{0,1},m], x∈{f,d}. (33)

The probability term in (33) can also be writ-ten as Pr



(PM0)/(PM2m−1)

<1



H{0,1},m



. Since

i=2m−1_{=(xM···xm+1xmxm−1···x1)2=(0···010···0)2 for the}

mthsystem,theratio(PM0)/(PM2m−1)<1canbesimplifiedusing

(17)as PM0 PM2m−1 =

b0 b2m−1



p0 p2m−1



e−dm/2 e−dm/2(m+1)



(m+1)Nm/2<1 (34) Takingthelnofbothsides,(34)becomes

Nm 2 ln(m+1)+ln

p0 p2m−1



+ln

b0 b2m−1



−dm



_ m 2(m+1)



<0. (35) Eq. (35) can be rearranged and used in (33) as Pr[dm>

2m−1



H{0,1},m],where 2m−1=



Nm 2 ln(m+1)+ln

p0 p2m−1



+ln

b0 b2m−1

 

am (36) asgivenin(29)andam=m/2(m+1). References

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