2011 IEEE 22nd Interntionl Symposium on Personl, Indoor nd Mobile Rdio Communictions Primry-User Mobility Impct on Spectrum Sensing in Cognitive Rdio Networks ngel Sr Cccipuoti,InF.kyildiz, Fellow, IEEE Brodbnd Wireless Networking Lbortory, School of Electricl nd Computer Engineering Georgi Institute of Technology, tlnt, US Emil: {ngel.cccipuoti, in}@ece.gtech.edu Luigi Pur Deprtment of Biomedicl, Electronics nd Telecommunictions Engineering, University of Nples Federico II, Itly Emil: {ngelsr.cccipuoti, pur}@unin.it. bstrct In this pper, the effects of the primry-user () mobility on spectrum sensing in Cognitive Rdio () networks re studied. To this im, first, the spectrum sensing problem is reformulted to ccount for the mobility. Then, the effects of the mobility re studied with the objective to determine the prmeters tht ffect the spectrum sensing functionlity. For this, two performnce metrics re nlyticlly derived: i) the detection cpbility, whichmesuresthemobilityimpcton the user detection probbility; ii) the mobility-enbled sensing cpcity, newmetricthtmesurestheexpectedtrnsmission cpcity chievble by user in the presence of mobility. The mthemticl nlysis is crried out in different scenrios, by using mobility nd spectrum occupncy models. The results show tht the detection cpbility is ffected by five prmeters: the protection rnge, the network region size, the mobility model, the sptil distribution, nd the number of s tht use the sme spectrum bnd. Moreover, it is shown tht the sensing cpcity cn significntly increse in the presence of mobility if the protection rnge is smller thn the network region size. The mthemticl results re derived by considering the dynmic trffic, nd vlidted through simultions. I. INTRODUCTION Spectrum Sensing is key functionlity in Cognitive Rdio () Networks [1]. Through spectrum sensing, unlicensed users ( users) cn recognize nd dynmiclly exploit portions of the rdio spectrum whenever they re vcted by licensed users, referred to s Primry Users (s). The interference on the trnsmissions depends on the ccurcy of the spectrum sensing, which cn be ffected by the wireless chnnel impirments, such s multipth fding nd/or shdowing. Thus, recently, the reserch efforts re devoted to improve the ccurcy nd efficiency of sensing techniques [1], [2]. Despite these efforts, new chllenges rise in the spectrum sensing functionlity in the presence of mobility. Mobility chnges dynmiclly the mutul distnces mong the s nd the users nd, s consequence, the connectivity between them vries in time. For this, even if t certin time n rbitrry user is inside the protection rnge 1 of mobile This work ws supported by the project HrBour trffic optimiztion system (HBITT), founded by the itlin ntionl progrm Pino Opertivo Nzionle Ricerc e Competitività 2007-2013, ndbytheu.s.ntionl Science Foundtion under wrd ECCS-0900930. 1 To void hrmful interference ginst the s, the users should be ble to detect ctive s within rnge, referred to s protection rnge, determined by the trnsmission rnge nd by the interference rnge [3]., fter the movement, the user cn be outside of it, thus becoming unble to sense the possible trnsmissions. Spectrum sensing should be wre of these topology chnges, mking necessry to revisit the current formultion of the sensing problem. In this pper, the effects of the mobility on spectrum sensing re studied, with the objective to determine the prmeters tht ffect the sensing performnce nd design. To the best of our knowledge, this is the first work tht ddresses this problem. Specificlly, first, the spectrum sensing problem is reformulted to ccount for the mobility. Then two performnce metrics re nlyticlly derived: i) the detection cpbility, i.e., the probbility of user being inside the protection rnge of, which mesures the mobility impct on the detection probbility; ii) the mobility-enbled sensing cpcity,newmetricthtmesurestheexpectedtrnsmission cpcity chievble by user in the presence of mobility. The mthemticl nlysis is crried out by utilizing two populr mobility models [4], i.e., Rdom Wlk mobility Model with reflection (RWM) nd Rndom WyPoint mobility Model (RWPM). Moreover, we consider two different spectrum occupncy models. In the first model clled Single for Bnd (SPB), the s roming within the network region use different bnds. In the second model clled Multiple s for Bnd (MPB), different mobile s cn use the sme bnd. For both mobility nd spectrum occupncy models, we derive closed-form expressions for both the detection cpbility nd the mobility-enbled sensing cpcity. The detection cpbility results show tht the detection cpbility is ffected by five different prmeters. For the SPB scenrio, the detection cpbility depends on the protection rnge, the extension of the network region, the mobility model, nd the sptil distribution. For the MPB scenrio, the detection cpbility depends on ll the bove prmeters, but lso on the number of s tht use the sme bnd. Hence, the MPB nlysis revels tht, from user perspective, the totl number of s roming within the network region is not importnt but the number of s tht use the sme bnd. The derived mobility-enbled sensing cpcity shows tht, when the protection rnge is not comprble with the 978-1-4577-1348-4/11/$26.00 2011 IEEE 451
network region size, the mobility increses the sensing cpcity chievble by the users. ll results re derived by considering the dynmic trffic. The rest of the pper is orgnized s follows. In Section II we explin the reserch problem. In Section III we derive the detection cpbility nd the mobility-enbled sensing cpcity for the SPB model, while in Section IV we derive these performnce mesures for the MPB model. We vlidte the nlyticl results by simultion in Section V. In Section VI we conclude the pper. In the ppendix we provide the proofs. II. PROBLEM STTEMENT Here we present the models cpturing the mobility, the trffic nd the network. We then formulte the spectrum sensing problem under mobility.. Models nd ssumptions Mobility Models: Both the RWM (Rndom Wlk Model) nd the RWPM (Rndom WyPoint Model) initilly plce the s rndomly ccording to uniform distribution in network region, whichisssumedveryoftens line or s squre. Under the RWM, ech rndomly chooses direction ccording to uniform distribution in the rnge [0, 2π] nd speed in the rnge [v min,v mx ] m/s. t the end of ech movement period, new direction nd speed re clculted. When the edge of the network region is reched, is bounced bck to the region. Thismodel produces uniform stedy-stte sptil distribution regrdless of the verge speed [5]. Under the RWPM, ech rndomly chooses destintion point inside ccording to uniformdistribution,nditmovestowrdsthisdestintion with velocity chosen uniformly t rndom in the intervl [v min,v mx ] m/s. When reches its destintion, it remins fixed for certin puse time ( think time ), nd then it strts moving gin ccording to the sme rule. This model produces non-uniformstedy-sttesptildistribution[6].f X (x ) denotes the probbility density function (pdf) 2 of the stedystte sptil distribution ccording to the dopted mobility model, nd R is the protection rdius. Trffic Model: Thetrfficismodeledstwostte birth-deth process [2], with deth rte α nd birth rte β. In the on sttetheisctive,ndinthe off stteitis inctive. The probbilities of the on nd off sttesre: P on = β α + β, P off = α (1) α + β Users Network: Theusersressumedstticnd uniformly distributed in the network region. f X (x ) denotes the pdf of the user sptil distribution. B. Spectrum Sensing Problem Definition under Mobility Consider typicl sensing scenrio in which user monitors certin spectrum bnd. In sttic Networks (Ns), the users re ssumed lwys inside the 2 Throughout the pper, rndom vribles re denoted with upper cse letters; specific outcomes of these vribles re denoted with lower cse. Protection Rnge (PrR). Hence, the locl sensing for signl detection is formulted s binry hypothesis problem [1], [2]: { v(t) H 0 x(t) = (2) g(t) s(t)+v(t) H 1 where s(t) is the signl, g(t) is the sensing chnnel gin, nd v(t) is the dditive white Gussin noise. H 0 nd H 1 denote, respectively, the hypotheses of no signl nd signl trnsmitted. In the presence of mobility, the binry problem (2) must be modified, since the common ssumption of user being lwys inside the PrR does not hold nymore, becuse of the dynmic chnge of the N topology. To ddress the spectrum sensing definition problem under the mobility, we introduce the following definitions: Definition 1: I denotes the event: n rbitrry user is inside the protection rnge. Definition 2: O denotes the event: n rbitrry user is out of the protection rnge. If the event O occurs, user cnnot listen to the trnsmission; insted, if I occurs, user cn sense the possible trnsmissions. By using the previous considertions, it is possible to reformulte the locl sensing for detection, by distinguishing between the events I nd O: Spectrum Sensing Problem under the event I: { v(t) H 0 x(t) = (3) g(t) s(t)+v(t) H 1 Spectrum Sensing Problem under the event O: { x(t) = v(t) H 0 (4) The detection performnce of n rbitrry user re evluted through the detection P d nd flse-lrm P f probbilities. From the sensing problem definitions (3) nd (4) in the presence of mobility, these cn be expressed s: P d = P (Y > γ H1 )=P(Y > γ H 1, I) P (I)+ }{{} P d I =0 {}}{ + P (Y > γ H 1, O) P (O) =P d I P (I) (5) P f = P (Y >γ H0 )=P(Y > γ H 0, I) P (I)+ }{{} P f I + P (Y > γ H 0, O) P (O) =P }{{} f I = P f O (6) P f O where Y is the decision vrible, which depends on the dopted sensing strtegy, γ is the decision threshold, nd P (I) nd P (O) re the probbilities of the events I nd O, respectively. P d I denotes the detection probbility conditioned to the event I nd it depends on the dopted sensing strtegy. In (6) the lst two equlities re justified by the symmetry of the hypothesis H 0 in both the events I nd O. From (5), it results tht the detection probbility P d 452
depends on P (I), referredtosdetection cpbility, which depends on the mobility (see Section III). Specificlly, since the normlized detection probbility, i.e., P d /P d I,coincides with the detection cpbility P (I), itisrequiredto study this term for ddressing the impct of the mobility on P d.thisissueisdevelopedinsectioniii. III. SINGLE FOR BND SCENRIO In this section, we study the impct of the mobility on the spectrum sensing for the SPB scenrio. Definition 3: userisinsidetheprotectionrnger of if the Eucliden distnce between them is not greter thn R, i.e., = X X R. By ccounting for Definition 1 nd Definition 3, it results tht: R P (I) =P [ R] = 0 f S (s) ds (7) where f S (s) is the pdf of the rndom vrible S.. Detection Cpbility for the Rndom Wlk Model In this subsection, we derive P (I) for one-dimensionl (Theorem 1) nd bi-dimensionl (Theorem 2) network regions. Proposition 1: For one-dimensionl network region, i.e., = [0,], thepdfoftherndomvrible representing the Eucliden distnce between user nd moving ccording to the RWM is given by: f 1D-RWM S 2 ( 1 s ) rect ( ) s /2 where rect(s) denotes the rectngulr window. Theorem 1: The probbility P (I) tht user is inside the PrR (Protection Rnge) of, roming within onedimensionl network region = [0,] ccording to the RWM, is given by P RWM 1D (I) =2 ( ) R (8) ( ) 2 R (9) Proof: By substituting (8) in (7), fter some lgebric mnipultions, the eqution (9) is obtined. Proposition 2: For bi-dimensionl network region, i.e., = [0,] [0,], the pdf of the rndom vrible representing the Eucliden distnce between user nd movingccordingtotherwmisgivenby: ( ) 2 πs 2 8s2 3 + 2s3 4 rect f 2D-RWM ( s ) 2/2 2 (10) Theorem 2: The probbility P (I) tht n rbitrry user is inside the PrR of, roming within bi-dimensionl network region = [0,] [0,] ccording to the RWM, is given by P RWM 2D (I) =π ( ) 2 R 8 3 ( ) 3 R + 1 2 ( ) 4 R (11) Proof: By substituting (10) in (7), fter some lgebric mnipultions, the eqution (11) is obtined. B. Detection Cpbility for the Rndom WyPoint Model In this subsection, we derive P (I) for one-dimensionl (Theorem 3) nd bi-dimensionl (Theorem 4) network regions, by ssuming RWPM with no thinking times (RWPM- NTT). Then, we generlize the nlysis by ccounting for the Thinking Times (TTs) in Theorem 5. Proposition 3: For one-dimensionl network region, i.e., = [0,], thepdfoftherndomvrible representing the Eucliden distnce between user nd moving ccording to the RWPM-NTT is given by: ( ) ( ) 2 f 1D-RWP-NTT S + 4s3 4 6s2 s /2 3 rect (12) Theorem 3: The probbility P (I) tht user is inside the PrR of, roming within one-dimensionl network region = [0,] ccording to the RWPM-NTT, is given by ( ) ( ) 3 ( ) 4 R R R P1D RWPM-NTT (I) =2 2 + (13) Proof: By substituting (12) in (7), fter some lgebric mnipultions, the eqution (13) is obtined. Proposition 4: For bi-dimensionl network region, i.e., = [0,] [0,], the pdf of the rndom vrible representing the Eucliden distnce between user nd movingccordingtotherwpm-nttisgivenby: f 2D-RWP-NTT S 32 5 7 s6 + 4 3 8 s7 ( 2 π 2 s 6 π 4 s3 + 32 3 5 s4 + 9 π 4 6 s5 + ) ( s ) 2/2 rect (14) 2 Theorem 4: The probbility P (I) tht user is inside the PrR of, roming within bi-dimensionl network region = [0,] [0,] ccording to the RWPM-NTT is given by P RWPM-NTT 2D (I) =π + 3 π 8 ( ) 2 R 3 π ( ) 4 R + 32 ( ) 5 R + 2 15 ( ) 6 R 32 ( ) 7 R + 1 ( ) 8 R (15) 35 6 Proof: By substituting (14) in (7), fter some lgebric mnipultions, the eqution (15) is obtined. When TTs re considered, the resulting sptil distribution is given by liner combintion of the thinking component nd the mobile component. By denoting with p p the probbility tht puses t rndomly chosen time, the sptil distribution cn be written s [6] f X (x )= p p f X,T(x )+(1 p p )f X,m(x ),wheref X,T(x ) is the pdf of the thinking component. Theorem 5: The detection cpbility P (I) tht n rbitrry user is inside the PrR of roming within network 453
region ccording to the RWPM with TTs (RWPM-TT) is equl to P RWPM-TT (I) =p p P RWM (I)+(1 p p )P RWPM-NTT (16) where P RWM (I) is given by (9) nd (11) for one- nd bidimensionl network region, respectively. P RWPM-NTT is given by (13) nd (15) for one- nd bi-dimensionl network region. Proof: Since f X,T(x ) is uniform [6] s the distribution produced by the RWM, P (I) for moving ccording to the RWPM-TT is liner combintion of the results obtined for the RWM nd the RWPM without TTs. Remrk: The results derived bove for both the dopted mobility models show tht the detection cpbility depends on the normlized PrR, i.e., R/, onthemobilitymodel, nd on the sptil distribution. C. Mobility-Enbled Sensing Cpcity By using the previous nlysis, nd by following the pproch dopted in [2] for sttic Ns, we introduce the new notion of mobility-enbled sensing cpcity s follows: Definition 4: The mobility-enbled sensing cpcity Ci mob is the expected trnsmission cpcity on the spectrum bnd i tht user cn chieve in the presence of mobility: Ci mob = η i ρ i W i [(1 P (I)) + P off,i P (I)] (17) where η i, W i,ndp off,i represent the sensing efficiency, the bndwidth, nd the off stte probbility (1) of the spectrum bnd i. ρ i is the spectrl efficiency of the bnd i (bit/sec/hz) [2], nd P (I) is the detection cpbility evluted before. Remrk: (17)sttesthtusercnusecertinbndif it is outside of the PrR or if it is inside but the is in the off stte. Ci mob reflects the dynmic nture of both the topology through P (I), ndthetrfficthroughp off,i. Remrk: In sttic network, the sensing cpcity ws derived in [2] nd it is equl to Ci sttic = η i ρ i W i P off,i.by compring such n eqution with (17), it results tht C mob i >C sttic i if P (I) < 1 nd P off,i < 1 (18) = Ci sttic lim P (I) 1 Cmob i Hence, if P (I) is smll, e.g., if the normlized PrR, R/, is smll, the mobility-enbled cpcity Ci mob cn be significntly greter thn Ci sttic,withginconstitutedbythe term relted to P (O) =1 P (I). Infct,thnkstothe mobility, user hs more chnces to use the bnd, since it cn be outside of the PrR with high probbility. Clerly, if the never trnsmits, i.e., P off,i =1, Ci mob = Ci sttic. IV. MULTIPLE FOR BND SCENRIO In this section, we study the impct of the mobility on the spectrum sensing for the MPB scenrio. If in there re n mobile s tht use the sme bnd, the previous expressions re not vlid nymore, since here P (I) represents the probbility of user being inside the PrR of t lest one. By denoting with P (O) the probbility tht is not inside the PrR of ny tht use the bnd, P (I) is: P (I) =1 P (O) (19) To evlute (19), let us consider user t certin loction x. TheuserisinsidethePrRifthemobile is plced within disk C(x ) of rdius R round x.the probbility of this event is: P C (x )= f X (x )dx (20) C(x ) Since the s move independently of ech other in both the considered mobility models [5], [6], the number K of s within C(x ) obeys binomil distribution: ( n P (K = k x )= (P k) C (x )) k (1 P C (x )) n k (21) By using (21), the probbility tht user locted in x is not inside the PrR of ny s using the sme bnd is: P (O x )=P (K =0 x )=(1 P C (x )) n (22) Hence, by ccounting for (21) nd (22), the probbility tht user locted in x is inside the PrR of t lest one using the sme bnd is given by: P (I x )=1 P(O x )=1 (1 P C (x )) n = n ( n = (P k) C (x )) k (1 P C (x )) n k (23) k=1 By integrting P (I x ) over ll the loctions, we hve: P (I) = P (I x )f X (x )dx = (24) 1 (1 P C (x )) n f X (x )dx = n ( n P k) C (x ) k (1 P C (x )) n k f X (x )dx k=1 From (24), to evlute P (I) we need to derive P C (x ). Proposition 5: In one-dimensionl network region = [0,], theprobbilitythtuserloctedinx is inside PrRisequltofortheRWMndRWPM,respectively: 2R R x R P RWM C (x )= x +R 0 x R x +R R x 0 otherwise (25) PC RWPM (x )= (26) 12Rx 4 R 2 (R 2 +3x 2 3 ) R x R ( x+r) 2 [ 3 2 (x + R) ] 0 x R 1+ ( x R) 2 [ 2 (x R) 3 ] R x 0 otherwise 454
1 1 Detection Cpbility 0.3 0.2 0.1 0 Incresing Thinking Time 1D Network 2D Network RWM RWM Experimentl RWPM RWPM Experimentl RWPM TT RWPM TT Experimentl 0.1 0.2 0.3 Normlized Protection Rnge Normlized Mobility Enbled Sensing Cpcity P off = 6 RWM 1D RWM 1D Experimentl RWPM 1D RWPM 1D Experimentl P = 0.33 off RWM 2D RWM 2D Experimentl RWPM 2D RWPM 2D Experimentl Sttic 0.1 0.2 0.3 Normlized Protection Rnge Fig. 1. Detection Cpbility vs Normlized PrR Fig. 2. Normlized Mobility-Enbled Sensing Cpcity vs Normlized PrR Detection Cpbility 1 0.3 0.2 RWM RWM Experimentl RWPM RWPM Experimentl Normlized Mobility Enbled Sensing Cpcity 1 0.3 0.2 0.1 RWM RWM Experimentl RWPM RWPM Experimentl P off = 6 P off = 0.33 0.1 2 4 6 8 10 12 14 16 18 20 Number of Primry Users 0 2 4 6 8 10 12 14 16 18 20 Number of Primry Users Fig. 3. Detection Cpbility vs Number of s Fig. 4. Normlized Mobility-Enbled Sensing Cpcity vs Number of s By substituting (25) nd (26) in (24), nd by using f X (x ) = 1/, forx, sconsequenceof the user network model, P (I) cn be clculted. Remrk: ThepreviousequtionsshowthtP (I) depends not only on the mobility model, the sptil distribution, the normlized PrR, but lso on the number of s tht use the bnd of interest. Hence, in the MPB scenrio, from user perspective, the totl number of s roming within the network region is not importnt but the number of s tht use the sme bnd. s in Section III, to ccount for TTs in the RWPM it is enough to linerly combine the results obtined for the RWM nd RWPM-NTT. By using Definition 4 nd the previous results for P (I) in the MPB scenrio, the expression of the mobility-enbled sensing cpcity Ci mob hs to be modified. In fct, user cn use the bnd of interest i if it is not inside the PrR of ny tht cn use tht bnd, or if it is inside but no trnsmits. By ssuming tht the s ctivities re independent mong ech other, nd by denoting with Pi,l off the off stte probbility of the l-th, Ci mob cn be expressed s: n ( ) Ci mob n k = η i ρ i W i (P (I) + P k i,l off k=1 l=1 ) P C (x ) k (1 P C (x )) n k f X (x ) dx (27) From (27), it results tht when the number n of s tht use the bnd i increses, Ci mob decreses s well, pproching to zero when n +. V. VLIDTION OF THE THEORETICL RESULTS In this section we vlidte the mthemticl results with Monte Crlo simultions. We generte 10 4 topologies by plcing both the s nd the users rndomly in line/squred network region. Then,forechtopology,weletthes move ccording to the dopted mobility model (RWM or RWPM) for enough time to rech stedy-stte distribution (10 4 seconds). Then we clculte the verge P (I). SPB scenrio: InFig.1weshowthedetectioncpbility P (I) versus the normlized PrR, R/,wherethenlyticl expressions (9), (11), (13) nd (15) for both the dopted mobility models mtch well the simultion results. We observe tht, when R/ increses, P (I) increses s well, since the probbility tht user is inside the PrR increses. More specificlly, P RWPM (I) P RWM (I). In2Dnetworks,P (I) hs smller vlues with respect to 1D networks due to the higher degree of freedom given by the dditionl dimension. Moreover, in Fig. 1 we lso depict P (I) vlues when the TTs in the RWPM re considered, for two vlues of p p,i.e., p p = 0.2 nd p p =. lsointhiscsethereisvery good greement between the mthemticl nd the simultion results. Introducing TTs decreses P (I). In Fig. 2, both the normlized mobility-enbled sensing 455
cpcity C mob i = Ci mob /(η i ρ i W i ) nd the normlized sttic sensing cpcity C sttic i = P off,i re reported s functions of R/ for two vlues of P off,i.e.,p off =1/3 nd P off =2/3. For both vlues, we observe tht the mobility introduces significnt sensing cpcity gin with respect to sttic Ns for smll vlue of R/. For exmple, for P off = 6 nd R/ =0.1, C mob i is in the worst cse (1D network) t lest 44% greter thn C sttic i =6. Thisginincresesif P off =0.33, sincethemobilitycnhelptoovercomethe high trffic ctivity. The cpcity gin cn be justified by noting tht, thnks to the mobility, user hs more chnces to use the bnd of interest, since it cn be outside of the PrR. Moreover, for the RWM C mob i decreses slower thn the cpcity ssocited to the RWPM, s consequence of the results shown in Fig. 1, i.e., P RWPM (I) P RWM (I). MPB scenrio: Fig.3showsP (I) versus the number n of s tht use the sme bnd of interest, for R/ =0.1. The results vlidte the theoreticl nlysis for both the mobility models, since there is very good greement between the theoreticl nd the experimentl results. When n increses, P (I) increses s well. In prticulr, in the MPB scenrio, the RWPM impcts on the detection cpbility P (I) differently from the SPB scenrio: P RWPM (I) P RWM (I), forthenonuniform distribution of the s when the RWPM is dopted. In Fig. 4, we show the normlized mobility-enbled sensing cpcity C mob i versus the number n of s tht use the sme bnd, for R/ =0.1. Weobtintheseresultsbyssumingtht the s hve the sme P off =1/3 nd 2/3. Itisclergin tht the nlyticl results mtch very well with the simultion results. s expected, C mob i decreses when n increses, since the probbility tht user is outside the PrR decreses. Finlly, since P RWPM (I) P RWM (I), fortherwpmc mob i decreses slower thn C mob i ssocited to the RWM. VI. CONCLUSIONS In this pper, the effects of the primry-user () mobility on spectrum sensing in Cognitive Rdio () networks hve been studied. To this im, two performnce metrics hve been nlyticlly derived: i) the detection cpbility, which mesures the mobility impct on the user detection probbility; ii) the mobility-enbled sensing cpcity, which mesures the expected cpcity chievble by user in the presence of mobility. The mthemticl nlysis is crried out in different scenrios, by dopting two mobility nd two spectrum occupncy models. It llowed us to determine ech prmeter tht ffects the detection cpbility nd the mobilityenbled sensing cpcity. Moreover, we show tht the sensing cpcity increses significntly in the presence of mobility if the protection rnge is smller thn the network region size. The nlyticl results re vlidted through simultions. CKNOWLEDGEMENT We thnk Mssimilino Pierobon, Pu Wng nd Dr. Mrcello Cleffi for their vluble feedbck which helped us to improve the pper. PPENDIX Proof of Proposition 1: For one-dimensionl network region = [0,],byusingtherndomvribletrnsformtion Theorem [7], the pdf of the Eucliden distnce S between userndmobileisequlto: f 1D [(f X f X )(s)+(f X f X )(s)] u(s) (28) where u(s) is the step function nd (f X f X )(s) denotes the convolution between the pdfs of the sptil distributions of the users nd the s. When the RWM is dopted f X (x) is uniform [5], i.e., f X (x) =1/, for0 x, 0 otherwise, nd lso f X (x) is uniform for the dopted network model. By substituting the expressions of f X (x) nd f X (x) in (28), (8) is obtined. Proof of Proposition 2: Forbi-dimensionlnetworkregion = [0,] [0,], byusingtheindependencendidenticl distribution of the rndom vribles representing the distnce in ech dimension, the pdf of the Eucliden distnce between user nd mobile is equl to: f 2D [ f 1D ] (s) f 1D (s) u(s) (29) where f 1D (s) is given in (28). By substituting the expressions (8) of f 1D (s) for the RWM, (10) is obtined. Proof of Proposition 3: For one-dimensionl network region, by using the sme resonings used in the proof of Proposition 1, the pdf of is given by (28). For the RWPM f X (x) is nonuniform nd, with no TTs, it is equl to [6] f X (x) = 6 x 2 + 6 3 x,for0 x, 0 otherwise. f 2 X (x) is uniform. By substituting the expressions of f X (x) nd f X (x) in (28), fter lgebric mnipultions (12) is obtined. Proof of Proposition 4: Forbi-dimensionlnetworkregion, s in the proof of Proposition 2, the pdf of is given by (29). By substituting (12) in (29), fter some lgebric mnipultions (10) is obtined. Proof of Proposition 5: (25) nd (26) re derived, by substituting in (20) the expressions of the sptil distributions for the RWM nd RWPM in the one-dimensionl cse. REFERENCES [1] I. F. kyildiz, B. F. Lo, nd R. Blkrishnn, Coopertive spectrum sensing in cognitive rdio networks: survey, Physicl Communiction (Elsevier) Journl, vol.4,pp.40 62,Mr.2011. [2] W.-Y. 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