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Location-Based Joint Relay Selection and Channel Allocation for Cognitive Radio Networks

In cognitive radio networks (CRNs), dynamic spectrum access has been demonstrated as an effective way to improve the spectrum utilization. Spectrum holes can be exploited not only in certain time slots or frequency bands, but also at particular
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  Location-based Joint Relay Selection andChannel Allocation for Cognitive Radio Networks ∗ Fangyong Li, Bo Bai, Jun Zhang and Khaled Ben Letaief,  Fellow, IEEE  ECE Department, Hong Kong University of Science and Technology, Kowloon, Hong KongEmail: {eeli, eebob, eejzhang, eekhaled}   Abstract —In cognitive radio networks (CRNs), dynamic spec-trum access has been demonstrated as an effective way toimprove the spectrum utilization. Spectrum holes can be exploitednot only in certain time slots or frequency bands, but alsoat particular locations. In relay assisted CRNs, one relay ata certain location can help to identify and provide differentspectrum holes over multiple channels. In this paper, a multi-dimensional combinatorial optimization problem is formulatedfor joint relay selection and channel allocation. We proposea weighted bipartite graph model and a minimum weightedassignment approach to efficiently get the optimal solution of theconsidered problem. Simulation results show that by applyingthis approach, spectrum efficiency, relay selection diversity andpower efficiency can be improved simultaneously for the cognitiveusers. Besides, only the statistical channel state information isneeded and the allocation results can be computed efficiently byusing the proposed approach. I. I NTRODUCTION C OGNITIVE radio has been considered as a key enablingsolution to the spectrum scarcity problem by applyingdynamic spectrum sharing [1]. It allows cognitive users toshare the spectrum owned by the licensed users in the primarynetwork (PN) without introducing much interference. In cog-nitive radio networks (CRNs), the key challenge lies in howto improve the spectrum efficiency by taking advantage of themulti-dimensional spectrum holes. Most of the previous workson cognitive radio focused on the sensing and utilization of spectrum holes in the frequency or time domain, while how toimprove the utilization of spectrum holes by using the locationinformation of the primary and cognitive users has not beenstudied in a systematic way. With path loss, the location playsa key role to leverage the interference between primary andcognitive users. Location awareness can help find multiplelocal spectrum holes and a cognitive user may be encouragedto access the frequency band owned by distant primary users.The location information can be obtained by the positioningsystem GPS or other localization techniques [2].There are only a few related works on location basedspectrum access. Zheng and Peng considered the differencesin the spectrum environment for cognitive users in differentlocations [3]. A graph coloring method was then proposedto deal with the spectrum allocation in that system. Bai etal. proposed a location-based cluster and list-coloring methodfor spectrum sharing [4]. A distributed coordination schemefor dynamic spectrum allocation that takes users’ location into ∗ This work is supported by the Hong Kong Research Grant Council undergrant # 610309 and Qatar National Research Fund. account was proposed in [5]. Xie et al. introduced a geometricrouting architecture to improve the spectrum efficiency [6].In [7], an infrastructure based CRN was proposed whichuses cooperative relays to improve the spectrum efficiency.A heuristic greedy algorithm was developed for the relayselection and channel allocation.In the CRNs, cognitive relays can help to reduce theinterference and acquire more spectrum holes. Cognitive userscan usually work over multiple channels and the channels maybe occupied by the primary users at different positions. Inaddition, one cognitive relay at a certain location may finddifferent spectrum holes over different channels so that it canhelp multiple cognitive users. Therefore, for cognitive users,relay selection and channel allocation can be jointly optimizedto achieve high spectrum efficiency and relay selection diver-sity. It should be noticed that power control is a key factorfor the joint optimization, because it not only controls theinterference between the primary and cognitive users, but alsocan improve the power efficiency.In this paper, we consider a cognitive radio system withmultiple source-destination pairs who want to share multiplechannels licensed to the primary pairs. Multiple cognitiverelays are deployed to assist the cognitive pairs to achievesuccessful communications. We consider the cross-layer de-sign for joint relay selection and channel allocation with twoobjectives: 1) to maximize the number of active cognitivepairs, and 2) to minimize the sum transmission power of theCRN. A two-stage two-objective combinatorial optimizationproblem is formulated for joint resource allocation. Sucha problem is difficult and of high computation complexity.Observing the special structure of our problem, we propose aweighted bipartite graph model to reformulate it. Based uponthe weighted bipartite graph, a minimum weighted assignmentmethod is developed to perform the joint relay selection andchannel allocation, with the two objectives achieved simul-taneously. Compared with the heuristic greedy algorithm, thesimulation results show that more cognitive pairs can be activewhen employing the proposed method, which implies a higherspectrum efficiency.The rest of the paper is organized as follows. The systemmodel is given in Section II and then the problem formulationis presented in Section III. A weighted bipartite assignmentmethod is proposed for the joint relay selection and channelallocation in Section IV. Some numerical results which il-lustrate the potential of the proposed approach are shown inSection V. Finally, Section VI concludes the paper.  󰁃󰁒 󰀳 󰁐󰁓 󰀱 󰁐󰁓 󰀲 󰁐󰁓 󰁍 󰀮󰀮󰀮 󰁐󰁄 󰀱 󰁐󰁄 󰁍 󰁐󰁄 󰀲 󰁃󰁨 󰀱 󰁃󰁨 󰁍 󰁃󰁨 󰀲 󰁃󰁓 󰀱 󰁃󰁓 󰀲 󰁃󰁓 󰁎 󰁃󰁄 󰀱 󰁃󰁄 󰁎 󰁃󰁄 󰀲 󰁃 󰁨 󰀱  󰁃 󰁨  󰀱 󰁃 󰁨  󰀲  󰁃 󰁨 󰀲 󰁐󰁲󰁩󰁭󰁡󰁲󰁹 󰁎󰁥󰁴󰁷󰁯󰁲󰁫󰁃󰁯󰁧󰁮󰁩󰁴󰁩󰁶󰁥 󰁒󰁡󰁤󰁩󰁯 󰁎󰁥󰁴󰁷󰁯󰁲󰁫 󰁋 󰁃󰁒󰁳 󰁃󰁒 󰀴 󰀮󰀮󰀮󰀮󰀮󰀮󰀮󰀮󰀮 Figure 1. System model of the relay-assisted CRN coexisting with thePN:  M   primary sources-destination (PS,PD),  N   cognitive source-destination(CS,CD),  K   cognitive relays (CR) and  M   channels (Ch). II. S YSTEM  M ODEL Consider a multi-channel spectrum sharing CRN, as shownin Fig. 1, which is composed of   N   source-destination pairs.The direct link is not considered, since a longer transmissionrange introduces higher interference to primary users.  K  cognitive relays are deployed to assist the communicationsof the CRN. In the same region, there are  M   primarysource-destination pairs occupying  M   orthogonal channels.The cognitive users can access the channels only if they donot cause high interference to primary users. We assume thatthe locations of all the users in the CRN and PN are known tothe cognitive users by applying some positioning algorithms[2].As shown in Fig. 1,  N   cognitive pairs are denoted as { ( CS n , CD n ) } n =1 , 2 ,...,N   and  K   cognitive relays are denotedas  { CR k } k =1 , 2 ,...,K  .  M   licensed channels  { Ch m } m =1 , 2 ,...,M  are used by  M   primary pairs { ( PS m , PD m ) } m =1 , 2 ,...,M   respec-tively. Let  d A,B  denote the distance between the two subscriptnodes.The channels between any transmission/interference sourceand receiver are assumed to be flat Rayleigh fading andthe outage probability is mainly affected by the signal-to-interference-plus-noise-ratio (SINR) with large-scale fading.Due to the hardware limitation, the cognitive relays are as-sumed to work in a half-duplex and decode-and-forward mode.All the channels are orthogonal and occupied by the primarypairs all the time. For channel allocation, we assume that the1st and 2nd hops for one cognitive pair use the same channel.One channel can be used by only one pair so that there is nointerference between the cognitive pairs. For relay selection,one relay may be used by multiple pairs as needed. Theinterference exists between and only between the cognitiveand primary pairs who share the same channel, at both the 1stand 2nd hops.Taking advantage of the location information, we firstconsider to achieve the objective of maximizing the number of the active cognitive pairs by joint relay selection and channelallocation. We also set minimizing the total transmissionpower of the CRN as the second objective, which can savepower consumption and reduce the interference between theCRN and other networks.III. J OINT  R ELAY  S ELECTION AND C HANNEL  A LLOCATION In the considered system, there are multiple relays andmultiple channels that may be used by the cognitive pairs.To achieve the designed objectives, the relay selection andchannel allocation are needed and they are coupled. For this joint relay selection and channel allocation problem, we definea joint allocation matrix  Φ = [ φ mnk ] M  × N  × K   as φ mnk  =   1 ,  Ch m ,  CR k  are allocated to  ( CS n , CD n ) , 0 ,  otherwise . (1)  A. Outage Probability and Power Control The quality-of-service (QoS) of both the primary users andcognitive users should be guaranteed so that they can work properly. In the Rayleigh fading scenario, power control isused to guarantee the QoS in terms of the outage probability(the source-destination pair is in outage when the pre-definedtransmit rate  R  is not achieved). The mutual information  I   isgiven by  I   = log(1 +  SINR ) . The outage probability,  ξ  out ,based on the statistical channel state information in Rayleighfading network is given by [8]: ξ  out =  Pr { I < R }  = exp  − Θ σ 2 n /σ 20 P  t d − αt          µ SNR ×  11 + Θ P  i d − αi P  t d − αt             µ SIR , where  P  t  and  P  i  are the transmission power of the transmis-sion and interference source respectively,  σ 20  is the varianceof the channel without large-scale fading,  σ 2 n  is the noisepower,  d t  and  d i  are the distances between the transmission,interference sources and the receiver respectively,  α  denotesthe path-loss exponent and  Θ = 2 R −  1 . We see thatthe signal-to-noise-ratio (SNR) and signal-to-interference-ratio(SIR) effects are independent, and the transmission power anddistance are the main impact factors.The transmission power of the cognitive users should not betoo large to interfere with the primary users. The transmissionpower of the primary sources, which is denoted by  P  PS , areassumed to be the same to simplify the formulation. Thus, µ SNR for the primary pairs is a constant. When  φ mnk  = 1 ,to make sure that the outage probability of the primary pair ( PS m , PD m )  is lower than a pre-defined threshold  ǫ th , thetransmission power of the specific cognitive source and relay,denoted as  P  CS mnk  and  P  CR mnk , should satisfy [6]  P  CS mnk  ≤  µ SNR m  − (1 − ǫ th )Θ(1 − ǫ th )  ×  d α CS n, PD m d α PS m, PD m P  PS  P  CSmax mnk  ,P  CR mnk  ≤  µ SNR m  − (1 − ǫ th )Θ(1 − ǫ th )  ×  d α CR k, PD m d α PS m, PD m P  PS  P  CRmax mnk  . (2)Meanwhile, the transmission power of the cognitive usersshould not be too small to have their own transmission in  outage either. Proposition 1 is given to help the QoS analysisfor the cognitive users. Proposition 1.  In a Rayleigh fading CRN, the outage proba-bility is lower than a pre-defined threshold   ς  th  if the minimumtransmission power, denoted as  P  min t  , satisfies P  min t  = Θ σ 2 n /σ 20 W    σ 2 n /σ 20 (1 − ς  th ) P  i d − αi exp  σ 2 n /σ 20 P  i d − αi  −  σ 2 n /σ 20 P  i d − αi d αt  , where  W   ( x )  is the Lambert W-function [9].Proof:  The proof is omitted due to the space limitation.Thus, when  φ mnk  = 1 , the transmission power of thespecific cognitive source and relay should satisfy  P  CS mnk  ≥  Θ ( σ 2 n /σ 20 ) d α CS n, CR k W    σ 2 n/σ 20 ( 1 − ζ th ) R 1 mk exp   σ 2 n/σ 20 R 1 mk  − σ 2 n/σ 20 R 1 mk  P  CSmin mnk  ,P  CR mnk  ≥  Θ ( σ 2 n /σ 20 ) d α CR k, CD n W    σ 2 n/σ 20 ( 1 − ζ th ) R 2 mn exp   σ 2 n/σ 20 R 2 mn  − σ 2 n/σ 20 R 2 mn  P  CRmin mnk  , (3)to achieve their own QoS, where  R 1 mk  =  P  PS d − α PS m , CR k and R 2 mn  =  P  PS d − α PS m , CD n .  B. Two-stage Two-objective Optimization Problem Since each channel can be shared by one cognitive pair andeach pair shares at most one channel, we have the constraints  n,k φ mnk  ≤  1  and  m,k φ mnk  ≤  1 . Then we get a two-stage two-objective optimization problem: with the channelallocation constraint and the power control constraint, wefirst try to find a set of feasible relay selection and channelallocation matrices  Φ ∗ , and then, the optimal solution willbe chosen from  Φ ∗ , which can minimize the total powerconsumption. Formally, this problem can be formulated as:( P ): P  CRN = min Φ ∗  m,n φ mnk  P  CS mnk  + P  CR mnk  ,  (4) Φ ∗ = argmax Φ  m,n,k φ mnk ,  (5)subject to  φ mnk P  CSmin mnk  ≤  P  CSmax mnk  ,  (6) φ mnk P  CRmin mnk  ≤  P  CRmax mnk  ,  (7)  n,k φ mnk  ≤  1 ,  m,k φ mnk  ≤  1  (8) φ mnk  =  { 0 , 1 } .  (9)Constraints (6) and (7) are the power control constraintsintegrated with the allocation matrix. Constraint (8) is thechannel allocation constraint as mentioned before. Objective(5) in ( P ) should be solved at the first stage and then Objective(4) is needed to be solved based on the result of Objective(5). Meanwhile, the maximum number of the active cognitivepairs  N  act  =  | Φ ∗ | . ( P ) is an integer linear programming witha variable of 3-dimensional matrix. There are few algorithmscan get the optimal solution to the best of our knowledge. Tosolve Objective (4) in ( P ), we need to find the argument  Φ ∗ forthe maximization, which is even tough. In the next section, wewill propose a minimum weighted bipartite assignment methodto solve two objectives at the same time efficiently.IV. W EIGHTED  B IPARTITE  A SSIGNMENT  M ETHOD The joint relay selection and channel allocation as formu-lated in ( P ) is a multi-dimensional combinatorial optimizationproblem [10]. It is known that such kind of problems are verydifficult to solve or to get the global optimal solution. Basedon the features of this problem, we will reformulate it as aweighted bipartite graph and develop a minimum weightedassignment method to perform the relay selection and channelallocation.Recalling that one relay may serve multiple cognitive pairswith different channels, cognitive relays are not one-to-onecorresponding to cognitive pairs or channels, i.e., there isno constrait  m,n φ mnk  ≤  1  for the joint allocation matrix Φ . Thus we see the opportunity to reduce the dimension of  joint allocation matrix, from a 3-dimensional space to a 2-dimensional space.A graph is bipartite if it has two classes of vertices andthe edges are only allowed between vertices of differentclasses, which is denoted by  G ( V  1  ∪V  2 , E  )  and there is anassociated value  w e  for edge  e  ∈ E   in a weighted bipartitegraph (WBG) [11]. We set  V  1  =  { ( CS n , CD n ) } n =1 , 2 ,...,N   and V  2  =  { Ch m } m =1 , 2 ,...,M  . We also define a channel allocationmatrix  Ψ  as ψ mn  =   1 ,  Ch m  is allocated to  ( CS n , CD n ) , 0 ,  otherwise ,  (10)and a set  R mn  =   k  |  Eq. (6) is satisfied   to representthe indices of all of the available cognitive relays which canhelp the cognitive transmission when  ψ mn  = 1 . For the edgeset  E  , there is an edge between  ( CS n , CD n )  and Ch m  if andonly if   R mn   =  Ø. To minimize the total transmission power,the locally "best" relay should be the one who helps the twohops consume the least power. Therefore, with Eq. (3), theweight of each edge is given by w mn  = min k ∈R mn  P  CSmin mnk  + P  CRmin mnk  .  (11)The optimal solution to Eq. (11), i.e., the locally "best" relay,is denoted as  k ∗ mn . So far, the WBG model is set and thedimension of the srcinal problem is reduced.Based on the WBG we have now, Problem ( P ) can bedirectly transformed to a general matching problem. Objective(5) corresponds to finding all the maximum cardinality match-ings based on this bipartite graph. Then, the total transmissionpower of every matching should be calculated and compared toget the minimum one, which is corresponding to Objective (4).The computation complexity is  O  √  NE   + N   · F   ( N,E  )   󰀨󰁃󰁓 󰀱 󰀬󰁃󰁄 󰀱 󰀩 󰁗 󰀳󰀳   󰀽󰀷󰀬 󰁫 󰀳󰀳   󰀽󰀲  󰁗  󰀴  󰀳   󰀽 󰀱󰀬  󰁫  󰀴  󰀳   󰀽 󰀴 󰀨󰁃󰁓 󰀲 󰀬󰁃󰁄 󰀲 󰀩󰀨󰁃󰁓 󰀳 󰀬󰁃󰁄 󰀳 󰀩󰀨󰁃󰁓 󰀴 󰀬󰁃󰁄 󰀴 󰀩  󰁗  󰀳  󰀲   󰀽 󰀲󰀬  󰁫  󰀳  󰀲   󰀽 󰀱 󰁃󰁨 󰀴 󰁃󰁨 󰀱 󰁃󰁨 󰀳 󰁃󰁨 󰀲  󰁗  󰀲 󰀳  󰀽 󰀳  󰀬 󰁫  󰀲 󰀳  󰀽 󰀳  󰁗  󰀱 󰀲  󰀽 󰀳  󰀬 󰁫  󰀱 󰀲  󰀽 󰀳  Figure 2. A sample of   G ( V  1  ∪V  2 , E  ) , where each edge occurs obeying Eq.(11) with weight  w mn  and locally "best" relay  k ∗ mn , and all the edges withweight  L  has been removed. The thick lines denote the assignment result. for enumerating all the maximum cardinality matchings [12]while O ( A )  for comparing the total transmission power, where E   =  |E|  and  F   ( N,E  )  is the number of the maximumcardinality matchings of the WBG. Both steps would haveexponentially increasing complexity.To solve the problem efficiently, we develop a minimumweighted bipartite assignment algorithm based on the aboveWBG. An  assignment   is a matching based on a completebipartite graph [11]. Thus we first complete the WBG by in-serting missing edges with weight  L , a large enough constant,i.e.,  L  ≫   m,n,k  P  CSmin mnk  + P  CRmin mnk  . Thus, for the completeWBG, the new weight is given by  w mn  =  w mn ,  R mn   =  Ø ,L,  otherwise . Then, we perform a minimum weight assignment based onthis complete WBG. In this assignment, the more   w mn  withthe smaller value is taken, the better solution is obtained,which is consistent with our two objectives. Note that inthe solution, there may be edges with weight  L , as wellas  R mn  =  Ø. However, for this case, Ch m  can not beallocated to the pair  ( CS n , CD n )  because no cognitive relaycan help the transmission. Thus, we need to delete such Algorithm 1  Minimum weighted bipartite assignment method1) Construct the complete WBG  G ( V  1  ∪V  2 , E  ) :  V  1  = { ( CS n , CD n ) } n =1 , 2 ,...,N  ,  V  2  =  { Ch m } m =1 , 2 ,...,M   and  w mn  =  w mn ,  R mn   =  Ø ,L,  otherwise ; 2) Execute the minimum weighted assignment, i.e., apply-ing the cost-scaling algorithm [11]. As a result, we getan assignment   M  with objective value   w sum ;3) Find out the edges with weight  L  ( t  edges) in theassignment   M  and remove them. The remaining subset M  is the final assignment result, in which each edgeconnects a cognitive pair  ( CS n , CD n )  and a channelCh m , and also associated the "best" cognitive relayCR k ∗ mn ;4) The maximum number of the active cognitive pairsis  N  act  = min { N,M  } −  t  and the minimum totaltransmission power is  P  CRN =  e mn ∈M w mn .−1.2−1−0.8−0.6−0.4−    Y PS 1 PD 1 PS 2 PD 2 PS 3 PD 3 PS 4 PD 4 CS 1 CD 1 CS 2 CD 2 CS 3 CS 4 CD 4   CD 3 Allocation result by assignment algorithmAllocation result by greedy algorithmCh 4 Ch 1 Ch 2 Ch 3 Ch 4 CR 8 Ch 3 Ch 2 CR 3 CR 6 Figure 3. A geometric distribution of the CRN and PN, where 4 channelsoccupied by the primary pairs and a sample of the joint relay selection andchannel allocation solutions with different algorithms is given. kind of edges from the minimum weighted assignment. Asa result, the optimal relay selection and channel allocationis achieved. Algorithm 1 shows the detailed steps of theminimum weighted assignment method. Fig. 2 is an exampleof the minimum weighted assignment, in which the resultare drawn by thick lines and all the edges with weight  L has been removed. By applying the cost-scaling algorithm,the minimum weighted assignment can be obtained withthe computation complexity of   O  √  NE  log( NL )   [11].Compared with O  √  NE   + N   · F   ( N,E  )  , the exponentiallyincreasing complexity by using the general matching method,the proposed method is much more efficient.V. S IMULATION  R ESULTS In this section, we present some simulation results todemonstrate the performance of the proposed minimumweighted assignment method for joint relay selection andchannel allocation. We will compare our method with theexhaustive search method and a greedy algorithm. By usingthe exhaustive search method, all the allocation solutions areenumerated and compared so as to get the best one, whichgives the global optimal solution with exponentially increasingcomplexity. Another intuitive method to solve the problemis the greedy algorithm as follows. By checking the powercontrol constraints, we can always find out whether there isany channel coupled with any relay that can be used by aspecific cognitive pair. We greedily choose the pair with theleast cost one by one. The greedy algorithm gives a suboptimalsolution and the computation complexity of it is  O  N  2  .In the simulation, there are  N   = 4  cognitive pairs in theCRN and  M   = 4  primary pairs in the PN. At the initial state,we randomly put  K   = 8  cognitive relays in the middle of thecognitive pairs and the geographic locations of all the usersare shown in Fig. 3. For all the transmissions, the target rageis fixed as  R  = 2 , and the path-loss factor is given by  α  =3 . For convenience, the pre-defined thresholds of the outage  probabilities of the primary and cognitive users are the sameand fixed as  ǫ th  =  ζ  th  = 0 . 1  [6].  P  PS will be increased since itis a key factor to affect the location-based outage probability.Then we will move the PN away from the CRN to see theimpacts.Figure 4 shows that the proposed algorithm can alwaysachieve the global optimal solution as the exhaustive algo-rithm, when the transmission power of the primary users P  PS increases. This verifies the optimality of our algorithm.Compared with the greedy method, we can see: 1) when  P  PS is low, as shown in Fig. 4(a), more cognitive pairs can beactive by applying the proposed algorithm than the greedyalgorithm, because the primary users can be easily interferedwhen their own transmission power is low. Taking  P  PS = 5  asan example, the proposed algorithm makes 3 cognitive pairscommunicate successfully while the greedy algorithm makes2. 2) when  P  PS is high, Fig. 4(a) shows the same number of cognitive pairs are active. Fig. 4(b), however, shows that theproposed algorithm requires much less transmission power,about  5  dBW, which implies a higher power efficiency.In Fig. 3, the locations of the PN and CRN are fixed. Nowwe move the whole PN up, away from the CRN, i.e., the dis-tance between the CRN and PN increases. Figure 5(a) showsthat we can have more cognitive pairs active by applying theproposed algorithm when the CRN is close to the PN and allthe cognitive pairs are active by any algorithm when the CRNis far away form the PN. It is also shown that when the samenumber of cognitive pairs work well, the proposed algorithmconsumes less power. This demonstrates the importance of thenode locations for the considered problem.At the left hand side of the dash dot line in both Fig. 4(b)and Fig. 5(b), it shows that the greedy algorithm gets lowertotal transmission power. If we refer to the correspondingparts in Fig. 4(a) and Fig. 5(a), more active pairs are activeby applying the proposed algorithm. Therefore, the greedyalgorithm cannot achieve the objective of maximizing thenumber of active cognitive pairs.VI. C ONCLUSIONS In this paper, we investigated the joint relay selection andchannel allocation in a location based CRN over Rayleighfading channels. It was formulated as a multi-dimensionalcombinatorial optimization problem with the objectives of maximizing the number of active cognitive pairs while mini-mizing the total transmission power. Power control was usedto reduce the interference so that more spectrum holes canbe exploited. A weighted assignment approach based on aweighted bipartite graph reformulation was proposed to solvethe optimization problem with low complexity. The resultsshowed the significant gain compared with a heuristic greedyalgorithm. The proposed approach not only provides a sys-tematic way to solve the joint relay selection and channelallocation problem, but also improves the spectrum efficiency,relay selection diversity, along with power efficiency. 456789101.522.533.5 Transmission power of primary users P PS  (dBW)    #  o   f  a  c   t   i  v  e  c  o  g  n   i   t   i  v  e  p  a   i  r  s   N   a  c   t     Exhaustive search algorithmGreedy algorithmProposed algorithm 45678910−202468101214 Transmission power of primary users P PS  (dBW)    T  o   t  a   l  p  o  w  e  r  o   f   t   h  e   C   R   N   P    C   R   N    (   d   B   W   )     Exhaustive search algorithmGreedy algorithmProposed algorithm (a) Number of active cognitive pairs (b) Total transmission power of CRNFigure 4. Joint relay selection and channel allocation when  M   =  N   =4 , K   = 8 , α  = 3 , R  = 2 , ǫ th  =  ζ  th  = 0 . 1  and CRN and PN are close. Distance between CRN and PN    #  o   f  a  c   t   i  v  e  c  o  g  n   i   t   i  v  e  p  a   i  r  s   N   a  c   t     Exhaustive search algorithmGreedy algorithmProposed algorithm Distance between CRN and PN    T  o   t  a   l  p  o  w  e  r  o   f   t   h  e   C   R   N   P    C   R   N    (   d   B   W   )   Exhaustive search algorithmGreedy algorithmProposed algorithm (a) Number of active cognitive pairs (b) Total transmission power of CRNFigure 5. Joint relay selection and channel allocation when  M   =  N   =4 , K   = 8 , α  = 3 , R  = 2 , ǫ th  =  ζ  th  = 0 . 1  and  P  PS = 10 dBW. R EFERENCES[1] I. Akyildiz, W. Lee, M. Vuran, and S. Mohanty, “Next genera-tion/dynamic spectrum access/cognitive radio wireless networks: Asurvey,”  Computer Networks , vol. 50, no. 13, pp. 2127–2159, Sep. 2006.[2] Z. Ma, W. Chen, K. Letaief, and Z. Cao, “A semi range-based iterativelocalization algorithm for cognitive radio networks,”  IEEE Trans. Veh.Tech. , vol. 59, no. 2, pp. 704 –717, 2010.[3] H. Zheng and C. Peng, “Collaboration and fairness in opportunisticspectrum access,” in  Proc. IEEE Int. Conf. Communications (ICC) ,vol. 5, May 2005, pp. 3132 – 3136.[4] B. Bai, W. Chen, and Z. Cao, “Low-complexity hierarchical spectrumsharing scheme in cognitive radio networks,”  IEEE Commun. Lett. ,vol. 13, no. 10, pp. 770–772, Oct. 2009.[5] J. Zhao, H. Zheng, and G.-H. Yang, “Distributed coordination indynamic spectrum allocation networks,” in  IEEE DySPAN 2005 , 2005,pp. 259 –268.[6] M. Xie, W. Zhang, and K.-K. Wong, “A geometric approach to improvespectrum efficiency for cognitive relay networks,”  IEEE Transactions onWireless Communications , vol. 9, no. 1, pp. 268–281, Jan. 2010.[7] Z. Qian, J. Juncheng, and Z. Jin, “Cooperative relay to improve diversityin cognitive radio networks,”  IEEE Communications Magazine , vol. 47,no. 2, pp. 111–117, Feb. 2009.[8] M. Haenggi, “On routing in random rayleigh fading networks,”  IEEE Transactions on Wireless Communications , vol. 4, no. 4, pp. 1553–1562,Jul. 2005.[9] I. M. Ryzhik, A. Jeffrey, and D. Zwillinger,  Table of integrals, seriesand products , 7th ed. Academic Press, 2007.[10] B. Bai, W. Chen, K. B. Letaief, and Z. Cao, “RBG matching basedoptimal relay selection and subchannel allocation,” in  Proc. IEEE Int.Conf. Communications (ICC) , june 2011, pp. 1 –5.[11] R. E. Burkard, M. Dell’Amico, and S. Martello,  Assignment Problems .SIAM, 2009.[12] H. Leong, H. Imai, S. Jain, and T. Uno,  Algorithms and Computation ,ser. Lecture Notes in Computer Science. Springer Berlin Heidelberg,1997, vol. 1350.
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