ContentslistsavailableatSciVerseScienceDirect
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: CognitiveradiosUltrawideband(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:burak.yilmaz@khas.edu.tr(B.Yılmaz),serkucuk@khas.edu.tr (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.Usingthe2distributioninMAPbased
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 (sQ(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,nI(i)andn(i)Q,willdeterminethe
deci-sionvariabledmin(4).Assumingthatthesamplesoftheprimary
signal,s(i)I ands(i)Q,givenin(4) arezero-meanGaussianrandom variables,1thedecisionvariabled
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−1dt 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
whetherthe2distributionorthenormaldistributionismore
suit-ablefor modelingthedecisionvariable.Next,thedetectionofa singlesystemispresented.
1Forexample,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−t2/2dt[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.
Itcanbeobservedthattheresultsfora2distributedd
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,thereare2Mpossiblecombinationsofhypotheses.
Accordingly,theCRcanonlytransmitifxm=0,
∀
m,whichcanberepresentedbyH0.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− 2M−1 i=1 PrM m=1(dm< m)H1,i PrH1,iH1 , (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−1i=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/2m2 (2 m)Nm/2 =C M m=1 e−dm/22m (2 m)Nm/2 , (16) whereC=Mm=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 Pr2 M−1 j=1 (PM0<PMj) H1,i . (19)
Bysubstituting(17)intothecomparisonterm{PM0>PMi},(18)and
(19)canbesimplifiedto Pf=1−Pcond,H0
M m=1 (1−Pf,m) (20) Pd=1− 2M−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< ix1a1d1 < 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{xm}areobtainedfrom(10).Itshould
benotedthat
2m−1|m=1,2,...,Mcorrespondtoindependent 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(Pf,Pd)-pairscanbeobtainedwithaclose-to-optimal performance[17].
3.2. Independentdetection
Theprobabilitiesoffalsealarmanddetectionformultiplebands canbeexpressedas Pf=1− M
m=1 (1−Pf,m) (25) Pd=1− 2M−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).
2NotethatthesevaluesresultingfromMAPdetectionandcorrespondingtothe 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,...,Maretheindependentthreshold 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−1am (29)
form={1,2,...,M}.InAppendixB,thederivationof
2m−1is 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] SNRdB = [8,8,8] P 2 N= [8,8,8] SNRdB = [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 the10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 10−4 10−3 10−2 10−1 100 Pf 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] SNRdB=[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 Pf = 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 Pf 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. SNRdB = [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-tegrationGrantwithinthe7thEuropeanCommunityFramework
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)∗...∗fDM(dM) (31)
with∗representingtheconvolutionoperatorandeachpdffDm(dm)
obtainedfrom(5)as fDm(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,fDm(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. Sincei=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)becomesNm 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−1am (36) asgivenin(29)andam=m/2(m+1). References
[1]MitolaJ,MaguireGQ.Cognitiveradio:makingsoftwareradiosmorepersonal. IEEEPersonalCommun1999;6:13–8.
[2]WinMZ,ScholtzRA.Ultra-widebandwidthtime-hoppingspread-spectrum impulseradioforwirelessmultiple-accesscommunications.IEEETrans Com-mun2000;48:679–91.
[3]EuropeanCommission.CommissionDecisionof21April2009amending Deci-sion2007/131/EConallowingtheuseoftheradiospectrumforequipment
usingultra-widebandtechnologyinaharmonisedmannerintheCommunity. OfficialJournalofEuropeanUnion2009;L109:9–13.
[4]YücekT,ArslanH.Asurveyofspectrumsensingalgorithmsforcognitiveradio applications.IEEECommunSurveysTutorials2009;11:116–30.
[5]QuanZ,CuiS,PoorHV,SayedAH.Collaborativewidebandsensingforcognitive radios.IEEESignalProcMag2008;25:60–73.
[6]UrkowitzH.Energydetectionofunknowndeterministicsignals.IEEEProc 1967;55:523–31.
[7]GhasemiA,SousaES.Collaborativespectrumsensingforopportunisticaccess infadingenvironments.In:IEEEProcDySPAN.2005.p.131–6.
[8]MishraSM,BrodersenRW.Cognitivetechnologyforimprovingultra-wideband (UWB)coexistence.In:IEEEProcICUWB.2007.p.253–8.
[9]ZhangW,MallikRK,LetaiefKB.Optimizationofcooperativespectrumsensing withenergydetectionincognitiveradionetworks.IEEETransWireless Com-mun2009;8:5761–6.
[10]DighamFF,AlouiniMS,SimonMK.Ontheenergydetectionofunknownsignals overfadingchannels.IEEETransCommun2007;55:21–4.
[11]Pandharipande A, Linnartz JPMG. Performance analysis of primary user detectionina multipleantenna cognitiveradio.In:IEEEProcICC. 2007. p.6482–6.
[12]MaJ,ZhouX,LiGY.Probability-basedperiodicspectrumsensingduring sec-ondarycommunication.IEEETransCommun2010;58:1291–301.
[13]Erküc¸ükS,LampeL,SchoberR.AnalysisofinterferencesensingforDAAUWB-IR systems.In:IEEEProcICUWB.2008.p.17–20.
[14]Quan Z, Cui S, Sayed AH, Poor HV. Optimal multiband joint detection forspectrumsensingincognitiveradionetworks.IEEETrans SignalProc 2009;57:1128–40.
[15]KhalidL, Raahemifar K,Anpalagan A. Cooperative spectrum sensingfor widebandcognitiveOFDMradionetworks.In:IEEEProcVTC-Fall.2009.p. 1–5.
[16]Paysarvi-Hoseini P, Beaulieu NC. Optimal wideband spectrum sensing frameworkfor cognitive radiosystems. IEEE Trans SignalProc 2011;59: 1170–82.
[17]Erküc¸ükS,LampeL,SchoberR.JointdetectionofprimarysystemsusingUWB impulseradios.IEEETransWirelessCommun2011;10:419–24.
[18]YılmazB,Erküc¸ükS.Detectionofjointlyactiveprimarysystems.In:Future Network&MobileSummit.2012.p.1–8.
[19]SahinME,GuvencI,ArslanH.OpportunitydetectionforOFDMA-based cog-nitiveradiosystemswithtimingmisalignment.IEEETransWirelessCommun 2009;8:5300–13.
[20]WuJY,WangCH,WangTY.Performanceanalysisofenergydetectionbased spectrumsensingwithunknownprimarysignalarrivaltime.IEEETrans Com-mun2011;59:1779–84.
[21]Snow C, Lampe L, Schober R. Impact of WiMAX interference on MB-OFDMUWBsystems:analysisandmitigation.IEEETransCommun2009;57: 2818–27.
[22]AbramowitzM,StegunI.Handbookofmathematicalfunctions.NewYork: Dover;1964.
[23]IEEEStd802.16-2004.Part16:airinterfaceforfixedbroadbandwirelessaccess systems;2004.
[24]BergerJO.Statisticaldecisiontheory:foundations,concepts,andmethods.New York:Springer;1980.