A Two-level Cooperative Game-based Approach for Joint Relay Selection and Distributed Resource Allocation in MIMO-OfDM-Based Cooperative Cognitive Radio Networks

TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerging Tel. Tech. (2014)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2784

RESEARCH ARTICLE

A two-level cooperative game-based approach for jointrelay selection and distributed resource allocationin MIMO-OFDM-based cooperative cognitiveradio networksMehdi Ghamari Adian* and Hassan Aghaeinia

Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran

ABSTRACT

We consider the downlink of a multi-input multi-output orthogonal frequency-division multiplexing-based cooperativecognitive radio network in which the secondary users (SUs) with extra resources work as relay to help the SUs withless available resources to improve their utility in terms of the throughput and fairness by forming a coalition graph. Wemodel the problem of joint relay selection and power allocation in multi-input multi-output orthogonal frequency-divisionmultiplexing-based cooperative cognitive radio network as a two-level cooperative game problem, and the objectives of thiswork are twofold. Firstly, we assign each weak SU to one of the relays (rich SUs) through solving a problem achieved bya non-transferable utility coalition graph game, and this comprises the first level of the game. Secondly, we jointly allocateavailable channels to the SUs such that no subchannel is allocated to more than one SU and simultaneously optimise thetransmit covariance matrices of nodes based on the Nash bargaining solution, which is the second level of the game. Wefurther model the network as a multi-cell scenario with small serving areas. After relaxing the problem and convexifyingthe relaxed version, we develop an optimal distributed algorithm, using dual decomposition. Afterwards, we propose adynamic distributed resource allocation algorithm for this purpose. Simulations confirm the convergence of our distributedalgorithms to the globally optimal solution of the two-level game. Copyright 2014 John Wiley & Sons, Ltd.

*CorrespondenceM. Ghamari Adian, Electrical Engineering Department, Amirkabir University of Technology, Tehran, Iran.E-mail: [email protected]

Received 5 October 2013; Revised 13 November 2013; Accepted 27 November 2013

1. INTRODUCTION

Cognitive radio (CR) and multi-input multi-output(MIMO) communications are among the most promisingsolutions to improve spectrum utilisation and efficiency.Dynamic and opportunistic spectrum access allows CRnodes to communicate on temporarily idle or under-utilisedfrequencies. MIMO systems boost spectral efficiency byhaving a multi-antenna node simultaneously transmit mul-tiple data streams. A timely issue is to embrace recentinnovations of the two technologies into a single system.

Based on the orthogonal frequency division multiplex-ing (OFDM) technique, it is possible to provide highspectral efficiency, multiuser diversity, robustness againstmultipath fading and flexibility in radio resource alloca-tion. However, to make it possible for all the users in alarge area to obtain access to the network, wide coverage isanother important objective. For this purpose, cooperativerelay was introduced into CR networks [1]. With the assis-

tance of a secondary user (SU) as a relay that has rich avail-able spectrum bands, some of channels between the SUtransmitter and the SU receiver can be bridged to exploitmore spectrum opportunities. In this case, relay selectionand dynamic resource allocation for SUs in cognitive radionetworks (CRN) become very important.

The issue of resource allocation in OFDM-based CRNswith or without deploying multiple-antennas at users wasexplored in [25]. The authors in [2] presented a lowcomplexity semi-distributed algorithm for resource allo-cation in MIMO-OFDM based CRNs, using game theoryapproach, the strong duality in convex optimisation and theprimal decomposition method. In [3], the authors extendedthe pricing concept to MIMO-OFDM-based CRNs andpresented two iterative algorithms for resource alloca-tion in such systems. For the single-antenna systems, theresource allocation problem in OFDM-based CRNs withthe objective of maximising the sum-rate of SUs wasconsidered in [4], with and without the availability of chan-nel state information (CSI) for the channel between the

Copyright 2014 John Wiley & Sons, Ltd.

M. Ghamari Adian and H. Aghaeinia

secondary transmitter and primary receiver at the sec-ondary base station. In [5], a resource allocation anddecoding strategy at the secondary system for underlay andinterweave OFDM-based cooperative CRNs (CCRNs) waspresented, where the objective is to maximise the sum-rateof both primary and secondary systems, subject to the inter-ference threshold constraint and the fact that the primaryreceiver always considers interference as noise.

Joint problems of relay selection and resource alloca-tion in CRNs have attracted extensive research interestsbecause of its more effective spectrum utilisation [611].The authors in [6] consider a CCRN in which the relays areselected among the existing SUs. Moreover, the quality ofservice of the relays should be ensured. For CCRNs withdecode-and-forward strategy, two relay selection schemes,namely, a full-channel state information (CSI)-based bestrelay selection and a partial CSI-based best relay selection(PBRS) were proposed in [7]. In order to obtain an optimalsubcarrier pairing, relay assignment and power allocationin MIMO-OFDM-based CCRNs, the dual decompositiontechnique was recruited in [8] to maximise the sum-ratesubject to the interference temperature limit of the PUs.Moreover, because of high computational complexity ofthe optimal approach, a suboptimal algorithm was fur-ther proposed in [8]. The issue of joint relay selectionand power allocation in two-way CCRN was consideredin [9]. A suboptimal approach for reducing the complex-ity of joint relay selection and power allocation in CCRNwas proposed in [10]. The network coding opportunitiesexisting in cooperative communications that can furtherincrease the capacity was exploited in [11]. Furthermore,the reformulation and linearisation techniques to the orig-inal optimisation problems with nonlinear and non-convexobjective functions were applied such that the proposedalgorithms can produce high competitive solutions in atimely manner.

In non-cognitive radio systems, the issue of relay selec-tion and resource allocation in MIMO systems was con-sidered in [12], where the distributed space-time codingin amplify-and-forward mode was recruited. It was shownthat amplify-and-forward distributed space-time codingresults in an opportunistic relaying scheme, in which onlythe best relay is selected to retransmit the source space-time coded signal. The impact of optimising physicallayer security metrics on the architecture and interac-tions of the nodes in multi-hop wireless networks wasstudied in [13], where a tree formation game was for-mulated in which the players are the wireless nodes thatseek to form a network graph among themselves whileoptimising their multi-hop secrecy rates or the path quali-fication probabilities, depending on their knowledge of theeavesdroppers channels.

In this work, we study the joint problems of relay selec-tion and distributed resource allocation for MIMO-OFDM-based CCRNs, which has not been explored yet, to the bestof our knowledge. The main contribution is to model theproblem of joint relay selection and distributed resourceallocation as a two-level cooperative game. In the first

level, we model the relay selection and assignment of SUsto relays as coalition graph game with non-transferableutility. The resource allocation problem in the resultantCCRN with the coalition graph game-based formation isformulated based on the Nash bargaining (NB) approach.

In the second level, we design both centralised anddistributed algorithms that allow nodes to cooperate in allo-cating channels and designing optimal power allocation tomaximise user fairness. NB is a type of cooperative game,which has been used to allocate resources while enforcingfairness (e.g. [14, 15]). These schemes are often centralisedor require the assistance of an arbitrator to manage the bar-gaining process. The only fully distributed NB design wasprovided in [16] but under the assumption of an infinitenumber of channels. Moreover, almost all of these schemeswere developed for single-antenna systems, with the onlyexception of [15], which was developed for MIMO down-link communications. However, the algorithm in [15] is acentralised, timesharing one and does not consider exclu-sive channel occupancy. The challenge that hinders a fullydistributed algorithm is the complexity of the joint powerand channel allocation problem.

As explained earlier, we consider the NB-based formu-lation as the second level of the game to ensure fairnessamong various nodes. NB-based formulation drives net-work nodes to cooperate and bargain in the process ofassigning subchannels and adjusting their transmit covari-ance matrices. Using dual decomposition [17], a dis-tributed algorithm for the relaxed timesharing problem isdeveloped and proved to drive the problem to the glob-ally optimal solution. The distributed bargaining algorithmunder timesharing allows us to gauge preferences of dif-ferent SUs on a give channel. Using these preferences, aheuristic distributed algorithm is derived.

The remainder of the paper is as follows. The systemis modelled, and the problem is formulated in Section 2.The coalition heads and their subscribers are determinedin Section 3. Distributed channel assignment and powerallocation is discussed in Section 4. The performance ofthe proposed algorithms is evaluated in Section 5, andSection 6 concludes the paper.

Notation: Boldface uppercase is used for matrices andboldface lowercase for vectors. j.j, tr../ and ../H denote thedeterminant, the trace and the conjugate transpose opera-tors, respectively. Diag../ gives the diagonal elements of amatrix. The N N identity matrix is denoted by IN . Thecardinality of a set S is denoted by jSj.

2. SYSTEM MODEL ANDPROBLEM FORMULATION

2.1. System model

We consider a single cell MIMO-OFDM-based coopera-tive CRN. The secondary network, consisting of a cogni-tive base station (CBS) and NSU SUs with cognitive radiocapabilities, coexists with a primary network. The CBS isin charge of the downlink transmission to each SU within

Trans. Emerging Tel. Tech. (2014) 2014 John Wiley & Sons, Ltd.DOI: 10.1002/ett


the cell. We assume that the PUs and SUs, are uniformlydistributed in the cell. It is further assumed that there areNF orthogonal subchannels in the system, which are alsoreferred to as channel for brevity throughout the paper. Allnodes, including the CBS and the SUs are assumed to beequipped with multiple antennas, and for ease of exposi-tion and without loss of generality, it is assumed that allnodes are equipped with M antennas.

We have two observations in the SU system. Firstly, theentire available spectrum of some SUs may not be neededbecause of the low traffic demand of these SUs. Sec-ondly, the data rate in the direct link between the CBS andsome SUs may be less than their minimum requested ratebecause of long distance or there may be blockage betweenthe CBS and those SUs. If we can utilise the rich SUs ashelpers to relay the CBSs transmission data to other SUswith their otherwise wasted spectrum, we can significantlyimprove the system throughput. Therefore, the CBS servesdirectly some SUs, those rich SUs, which are also referredto as relays from now on, and other SUs (weak SUs)areserved cooperatively using the rich SUs.

Each channel can be used exclusively by the CBS or therelays in one of the groups of the links, that is, CBS-to-relays (to serve the rich SUs), CBS-to-relays (to supportother SUs) and relays-to-SUs (to support other SUs). Thechannel allocation coefficients in the CBS-to-SUs, CBS-to-relays and relays-to-SUs links are denoted by aBk,i, a

Brk,i

and ar.j/k,i , respectively, where amk,i D 1 (m 2 fB, Br, r.j/g)

if the k-th channel is allocated to node m, which serves thei-th SU; otherwise, amk,i D 0. The channels in all links areassumed time variant and frequency selective but constantduring one time slot.

The set of channels, all SUs, SUs served by the CBS andfinally SUs served cooperatively, are denoted by SF , SSU ,SBSU and SrSU , respectively. Some notational conventionsused throughout this paper are gathered in Table I.

In this work, we model the problem of joint relay selec-tion and power allocation in MIMO-OFDM-based CCRNas a two-level cooperative game problem. The relay selec-tion and assignment of weak SUs to each of relays com-prise the first level and is accomplished through solvinga problem achieved by a non-transferable utility coalitiongraph game. The transmit power allocation in each subslotof the SU transmissions and channel assignment to all SUlinks based on the NB solution (NBS) consist of the secondlevel of the game.

Therefore, in the first level of the system, it must bedetermined which SUs are eligible to play the role of arelay to weaker SUs and which weak SUs are efficient tobe served by each of relays. This scenario is well mod-elled by coalitional game theory. The rich SUs can thenbe called coalition heads and are responsible for receivingdata from CBS and relaying to their supported SUs, alsoreferred to as coalition members. Inside a coalition S, thecoalition graph formation between the coalition head andthe coalition members is a top-down tree with a top manand jSj 1 bottom men. Here, top man and bottom menmap to coalition head and coalition members.

Table I. Some key notations.

NSU Number of SUsNF Number of subchannelsM Number of antennas deployed at all usersaBk,i Coefficient of allocating channel k to SU i (in the

CBS-to-rich SUs link)aBrk,i Coefficient of allocating channel k to SU i (in the

CBS-to-relays link to serve weak SUs)ar.i/k,i Coefficient of allocating channel k to SU i (in the

relays-to-SUs links)SCi i-th coalitionNR Number of relays (coalitions)RBk,i Data rate in the direct link from CBS to the i-th relay

in the k-th channelHBk,i Channel from CBS to the i-th relay in the k-th

channelQBk,i Transmit covariance matrix of CBS intended for

relay i in the k-th channelRBrk,i Data rate in the link from CBS to the r-th relay which

serves SU i in the k-th channelHBrk,i Channel from CBS to the r-th relay which serves SU

i in the k-th channelQBrk,i Transmit covariance matrix of CBS intended for

relay i which serves SU i in the k-th subchannelRr.i/k,i Data rate in the link from relay r to SU i in the k-th

channelHr.i/k,i Channel from relay r to SU i in the k-th channelQr.i/k,i Transmit covariance matrix of relay r which serves

SU i in the k-th channelPT Maximum transmit power for serving the SUsPI,k Maximum tolerable amount of interference by the

PUs over the k-th channelbi Minimum rate requirement of SU iNminm,i Number of non-zero singular values of H

mk,i

Pmk,i,j Power allocated to stream j on channel k, intendedfor SU i

SF Set of channelsSSU Set of all SUsSBSU Set of SUs served by the CBS directly (rich SUs)SrSU Set of SUs served cooperatively

SU, secondary user; CBS, cognitive base station.

A coalition consists of several SUs denoted by SCi DnSCi,0, : : : , S

Ci,Ni

o, where SCi,0 is the coalition head of the i-th

coalition and SCi,js, 1 6 j 6 Ni, are coalition members.Clearly, Ni < NSU number of SUs are in the coalition i.The number of coalitions is denoted by NR and as a result,SNR

iD1 SCi D SSU andNRP

iD1Ni D NSU . A collection of coali-

tions, C, is called a coalitional structure and is a partition ofSSU , that is, for n 6 NR, S D

SC1 , : : : ,SCn SSU , suchthat SCi \ SCj D ; for all i j.

2.2. Problem formulation

The interference introduced by the PU signal into the bandof the k-th channel used by the SU system can be mod-



elled as additive white Gaussian noise by applying the lawof large numbers as described in [18] or by assuming thatthe primary and cognitive systems are using an indepen-dent and random Gaussian codewords [19]. Without losingthe generality, it is assumed that the amount of interferenceintroduced by the PU system to all nodes of the SU sys-tem are equal. It is further assumed that the noise power atall SU nodes are equal. Therefore, the achievable data ratein the direct link from the CBS to the i-th SU in the k-thsubchannel can be written as

RBk,i D log det

IM C HBk,iQBk,iHBH

k,i

(1)

where HBk,i represents the direct channel from CBS to thei-th SU in the k-th channel and QBk,i is the transmit covari-ance matrix of CBS intended for SU i in the k-th subchan-nel. The achievable data rate in the link from CBS to ther.j/-th relay that serves the SU j, RBr.j/k,j and the link fromrelay r.j/ to the j-th SU, Rr.j/k,j , both in the k-th subchannel,can be defined similarly, according to the following

RBrk,j D log det

IM C HBrk,jQBrk,jHBrH

k,j

(2)

Rr.j/k,j D log det

IM C Hr.j/k,j Qr.j/k,j Hr.j/H

k,j

(3)

where QBrk,j is the transmit covariance matrix of CBSintended for the r-th relay, which serves SU j in the k-thchannel and Qr.j/k,j represents the transmit covariance matrixof the r-th relay, which serves SU j in the k-th channel. Thechannel matrices HBrk,j and H

r.j/k,j can be defined similar to

(1). The total data rate over all channels assigned to SU ican be written as

Ri DX

k2SFaBk,iR

Bk,i C

Xk2SF

aBrk,iRBrk,i C

Xk2SF

ar.i/k,i R

r.i/k,i (4)

Before formulating the resource allocation problem as acooperative game (which will comprise the second level ofthe game), we introduce the NBS briefly. A special typeof cooperative games are bargaining games, where playersnegotiate/bargain their actions/strategies to reach an agree-ment with guaranteed minimum pay-offs. The agreementis associated with a utility vector u D u1, : : : , uN , whereui is the utility of player i, and there are N players in thegame. Let di and Di denote the action and action space forplayer i, respectively (di 2 Di). The utility ui is a functionof the action vector d D d1, : : : , dN . The utility space Uis the set of all possible pay-off allocations u, which resultfrom all possible action vectors d. It is also possible that noagreement is reached after bargaining, a situation referredto as a disagreement point. A disagreement point is asso-ciated with a utility vector u0, which consists of minimumpay-offs that players insist on having.

Given the variety of outcomes, Nash [14] suggestedto, instead of study all possible outcomes, specify char-acteristics or axioms of one or several outcomes that weexpect and find how to drive the bargaining process to that

agreement point. Nash proposed the following axioms thatdescribe an NBS [14], denoted by S .u, u0/:

An NBS is Pareto optimal. At the NBS, all players are guaranteed their minimum

pay-offs that they insist on at the beginning of thebargaining process.

An NBS is symmetric, meaning that all players havethe same priority.

Given the aforementioned properties, the problem iswhether a unique NBS exists and how to find such a uniqueNBS. Nash proved the following theorem that answersthese three key questions [14]:

Theorem 1. If the utility space U is upper-bounded,closed and convex, then there exists a unique NBS, whichis the solution of the following problem:

u D arg maxfu2Ug

NYiD1

ui u0,i

(5)It was pointed out in [20] that an NBS-based resource

allocation mechanism is a generalised proportionally fairone. An NBS reduces to a proportional fair alloca-tion if the minimum utility pay-offs are all zero. TheNBS-based resource allocation is one that first allocatesresource to meet players minimum requirements thenallots remained/leftover resources to all players in a pro-portionally fair manner.

Note that although the utility function Ri of playeri is concave with respect to (w.r.t.) the set of transmitcovariance matrices, Ri is not concave w.r.t. to both thechannel allocation indicator and the set of transmit covari-ance matrices. This fact prevents us from directly applyingTheorem 1. In the following, we propose the following for-mulation to jointly allocate spectrum and optimise transmitcovariance matrices for MIMO-OFDM-based CCRNs:

max(aBk,i, a

Brk,i , a

r.i/k,i

QBk,i, QBrk,i , Qr.i/k,i

) RNB

s.t. C1 :X

k2SF

0@X

i2SBSUtrQBk,iC X

i2SrSU

trQBrk,i

Ctr

Qr.i/k,i1A 6 PT

C2 : trQBk,iC tr QBrk,iC tr Qr.i/k,i 6 PI,k,

8i 2 SSU , 8k 2 SFC3 :

Xk2SF

aBk,iRBk,i > bi,

Xk2SF

aBrk,iRBrk,i > bi,X

k2SFa

r.i/k,i R

r.i/k,i > bi, 8i 2 SSU

C4 :X

i2SBSUaBk,i C

Xi2SrSU

aBrk,i C ar.i/k,i

6 1, 8k 2 SF

C5 : aBk,i D f0, 1g , aBrk,i D f0, 1g , ar.i/k,i D f0, 1g ,8i 2 SSU , 8k 2 SF (6)



Figure 1. Similarity between the system model and a multi-cell network.

where RNB D Pi2SBSU

log P

k2SFaBk,iR

Bk,i bi

!C P

i2SrSUlog

Pk2SF

aBrk,iRBrk,i bi

!C P

i2SrSUlog

Pk2SF

ar.i/k,i R

r.i/k,i bi

!

and bi is the minimum rate requirement of SU i. (C1) is themaximum transmit power constraint in the system. (C2) isto ensure that PU reception is protected from CR transmis-sions. (C3) guarantees the minimum rate requirement forall CR links. (C4) and (C5) convey the exclusive-channeloccupancy policy.

We consider the SUs served by the relays, SrSU , as vir-tual users that are directly connected to the CBS. In otherwords, we interpret the links between the CBS and therelays as the direct links between the CBS and some vir-tual SUs. As shown in Figure 1, this perspective helps usto divide the serving area (single cell) into smaller areas(multi cells) served by NR C 1 nodes, where node 1 is theCBS, while nodes m, m D 2, : : : , NR, are the coalitions,constituting a coalition head and coalition members. Wedenote the set of SUs served by node m with Sm; in par-ticular, S1 D SBSU . Clearly, Sm , 2 6 m 6 NR C 1, is thesame as SCm1. Therefore, we use the following notationsfor each user:

Rmk,i D

8 bi, m D 1, : : : , NRC1, 8i 2 Sm

C4 :NRC1XmD1

Xi2Sm

amk,i 6 1, 8k 2 SF

C5 : amk,i D f0, 1g , m D 1, : : : , NRC1,8i 2 Sm, 8k 2 SF (8)

2.3. MIMO transmit andreceive beamforming

The best strategy for the transmitter and receiver of a givenMIMO link is to design their beamformers so that their datastreams do not interfere with each other. These beamform-ers can be derived from the CSI matrices of the links usingsingular-value decomposition, as follows [21]:

Hmk,i D Smk,imk,iVmH

k,i (9)

for all m 2 f1, : : : , NR C 1g, i 2 Sm and k 2 SFS,where Smk,i and V

mk,i are unitary matrices and

mk,i is a

diagonal matrix formed from the singular values mk,i,j,j D 1, : : : , Nminm,i , of the channel gain matrix Hmk,i, wherethe number of non-zero singular values of Hmk,i is denotedby Nminm,i and Nminm,i 6 M. We set the transmit covariancematrix of the m-th node intended for SU i in the k-th chan-nel, Qmk,i, to Vmk,iPmk,iVm

H

k,i , where Pmk,i is a diagonal matrix

whose j-th diagonal element Pmk,i,j is the power allocatedto stream/antenna j on channel k , intended for SU i. Thereceived signal at SU i, served by the m-th node, in the k-thchannel is multiplied with SmHk,i leading to



Qymk,i D SmH

k,i ymk,i

D SmHk,i

Smk,imk,iV

mH

k,i Vmk,i

Pmk,i

1=2xmk,i C nmk

D mk,i

Pmk,i1=2

xmk,i C Qnmk (10)

where xmk,i is a column vector of M information symbols,sent directly from the m-th node to the i-th SU in the k-thchannel and nmk 2 CM is a complex Gaussian vector withidentity covariance matrix IM , representing the floor noiseplus normalised interference from PUs on channel k.

Hence, the data rates over Nminm,i spatial streams in thelink from the m-th node to SU i in channel k can bewritten as

Rmk,i DNminm,iXjD1

log

1 C mk,i,jPmk,i,j

(11)

We can rewrite the problem in (8) as follows:

maxnamk,i,P

mk,i,jo

NRC1XmD1

Xi2Sm

log

0@X

k2SF

Nminm,iXjD1

amk,i

log

1 C mk,i,jPmk,i,j

bi1A

s.t. C1:X

k2SF

NRC1XmD1

Xi2Sm

Nminm,iXjD1

Pmk,i,j 6 PT

C2:Nminm,iXjD1

Pmk,i,j 6 PI,k, m D 1, : : : , NRC1,

8i 2 Sm, 8k 2 SF

C3 :X

k2SFamk,i

Nminm,iXjD1

log

1 C mk,i,jPmk,i,j

> bi,

m D 1, : : : , NRC1, 8i 2 Sm

C4:NRC1XmD1

Xi2Sm

amk,i 6 1, 8k 2 SF

C5: amk,i D f0, 1g , m D 1, : : : , NRC1,8i 2 Sm, 8k 2 SF (12)

By introducing the transmit and receive beamformers,problem (12) is significantly simplified, compared withthe original one in (8). The phases of complex elementsof transmit covariance matrices are now specified by theunitary matrix Vmk,i.

In the problem (12), we have formulated the resourceallocation problem based on the NBS. However, it is stillnot determined which SUs have surplus resources to playthe role of the relay. Furthermore, the assignment of the

remaining SUs to each coalition head must be performedbefore allocating the resources to SUs. In the next section,we aim at providing an answer for these two importantquestions, making use of a heuristic approach. Finally, inSection 4, the optimal and distributed channel assignmentand power allocations approaches will be presented.

3. DETERMINING THE COALITIONHEADS AND THEIR SUBSCRIBERS

In this section, we propose a heuristic distributed algorithmto determine the potential coalition heads and then formthe coalition graph adaptively.

3.1. Potential coalition head selection

In order to select the potential coalition heads, the max-imum weight bipartite matching algorithm is applied toallocate optimal channel for each SU, and then SUs whosetraffic demands are satisfied over the allocated channelare selected as potential coalition heads. To provide morecalcifications, we initially assume that there is no cooper-ation in the SU system and find the optimal channel foreach SU.

The channel assignment problem is one of the funda-mental combinatorial optimisation problems in mathemat-ics. There is a close relationship between the maximumweight matching in a bipartite graph and the channelassignment problem. Therefore, the channel assignmentproblem can be transformed into a maximum weightedbipartite matching problem: take channel set SF Df1, : : : , NFg and SUs set SSU D f1, : : : , NSUg as thedisjoint bipartite subsets and define the weight of edgeek,i connecting vertex k of SF and vertex i of SSU asek,i D PTNSU tr

Gk,iGHk,i

, where Gk,i denotes channel matrix

between the CBS and SU i in the k-th channel. Hungar-ian method [22] can be employed to solve the bipartitematching problem.

It is noteworthy that the channel assigned to each SUin this stage is only aimed for coalition head selectionand coalition graph formation. After forming the coalitiongraph, the optimal channel assignment and power alloca-tion for the coalition graph and NBS-based MIMO-OFDMCCRNs-based must be performed and is postponed to thefourth section.

The coalition heads must have surplus resource to helpcoalition members. The SUs whose data rates are morethan twice of their minimum rate requirement are recog-nised as potential coalition heads. In other words, the SUswhich can complete their traffic demand in just one sub-slot can be considered as coalition heads. As a result, weneed to determine the optimal transmit covariance matri-ces of SUs and then form the coalition graph, solving thefollowing problem:



maxfPki ,i,jg

Xi2SSU

log

0@NminiX

jD1log

1 C ki,i,jPki,i,j

bi1A

s.t. C1:X

i2SSU

Nminm,iXjD1

Pki,i,j 6 PT

C2:NminiXjD1

Pki,i,j 6 PI,ki , 8i 2 SSU

C3 :NminiXjD1

log1 C ki,i,jPki,i,j

> bi, 8i 2 SSU (13)

where, with some abuse of notation, ki denotes the channelallocated to SU i; Pki,i,j is the CBS power allocated to thej-th spatial mode and on the ki-th channel, intended for thei-th SU; ki,i,j denotes the j-th singular value of the chan-nel matrix over the ki-th channel from the CBS to SU i andthe number of non-zero singular values is denoted by Nmini .The subgradient method can be used to solve the dual prob-lem of (13) with guaranteed convergence. The Lagrangianof (13) can be written as

L D X

i2SSUlog

0@NminiX

jD1log

1 C ki,i,jPki,i,j

bi1A

C 0@X

i2SSU

Nminm,iXjD1

Pki,i,j PT1A

CX

i2SSUi

0@NminiX

jD1Pki,i,j PI,ki

1A

CX

i2SSUi

0@bi

NminiXjD1


1A (14)

The optimal value of Pki,i,j can be obtained as

Pki,i,j Di 1 C i

1ki,i,j

(15)

After finding the optimal solution of the dual function,Pki,i,j, at a given dual points , i and i, the dual variablesat the (n + 1)-th iteration are updated as

.nC1/ D

.n/ @L@

C

[email protected]/

0@X

i2SSU

Nminm,iXjD1

Pki,i,j PT1A1A

C

(16)

.nC1/i D

.n/i

@L@i

C

[email protected]/

0@NminiX

jD1Pki,i,j PI,ki

1A1A

C(17)

.nC1/i D

.n/i

@L@i

C

[email protected]/

0@bi

NminiXjD1


1A1A

C

(18)As explained previously, the assigned channels to the

SUs and allocated portion of transmit power of the CBSto serve the SUs is only aimed to determine the coali-tion heads and form the coalitions. Having determined thecoalition heads, we form the coalition graph in the nextsubsection.

3.2. Coalition graph formation

In this subsection, we formulate the coalition graph forma-tion problem. We first sort the potential coalition heads indescending order of

Rki,i bi

to form the coalition heads

set V D fvig and sort the ordinary SUs in ascending orderofRki,j bj

to form the set W D wj. Therefore, vi has

more surplus resource than vj, and wi needs more help thanwj if i < j. The main idea of coalition graph formation isto enable the nodes with more surplus resource to help thenodes needing more help and meanwhile to minimise thetotal power consumption. Then, coalition graph formationprocess is as follows:

First round : Starting from w1, each wj 2 W sequen-tially searches for its matched coalition head vi andmatched relay channel pair. It is noteworthy that thematched relay channel pair of wj consists of the channelk.1/i,j between CBS and vi and channel k

.2/i,j between vi and

wj. At first, each vi forms a coalition SCi and is regardedas the coalition head within this coalition. The channelset allocated to wj is denoted by wj , and the achievabletransmission rate of wj is further denoted by Riwj when itjoins the i-th coalition. The link pairs CBS-vi and vi-wj areformed after wj joins the i-th coalition, SCi . The details arepresented in Table II.

During every round of coalition graph formation pro-cess, some rules must be obeyed. First of all, the selectedcoalition head cannot be matched by more than one ordi-nary node to prevent the coalition head from being over-crowded. Furthermore, channel allocation should satisfychannel availability at each node.

In the process of coalition graph formation, multiplerounds may be required until no new match appears orall channels are assigned. Our simulations show that threeor four runs are adequate for an optimal coalition graphformation (OC).



Table II. Coalition graph formation algorithm.

Step 1 (Initialisations):For all i 2 SR do

Set SCi D fvig and SC DnSC1 , : : : ,SCNR

o.

For all j 2 SrSU doSet wj D ;.

End forEnd For

Step 2 (First time round robin):While SC ; doFor all j 2 SrSU doFor all i 2 SR doIf wj joins the i-th coalition (SCi ) do

Find the matched channel pair, k.1/i ,j and k.2/i ,j , which satisfies

nk.1/i ,j , k

.2/i ,j

oD argmaxn

k.1/i ,j ,k.2/i ,j

o2SF

Riwj .

End ifEnd for(1) Find the optimal coalition for SU j 2 SrSU using SCi D argmax

SCi 2SRi

wj and

the corresponding optimal channel pairnk.1/i ,j , k

.2/i ,j

oD argmaxn

k.1/i ,j ,k.2/i ,j

o2SF

Ri

wj .

(2) Let wj D wj [nk.1/i ,j , k

.2/i ,j

o, SF D SF

nk.1/i ,j , k

.2/i ,j

o, SCi D SCi [

wjand SC D SC SCi.

End forIf SC D ; do

Let SC DnSC1 , : : : ,SCNR

o.

End ifEnd while

Step 3 (After the first time round robin)While all SU j 2 SrSU have not found their coalition doRepeat Step (2) .If SF ; doFor all j 2 SrSU do

Find the matched channel pairnk.1/i ,j , k

.2/i ,j

ofor each j 2 SrSU until no new match appears or all channels are assigned.

End forEnd if

End while

SU, secondary user.

4. DISTRIBUTED CHANNELASSIGNMENT ANDPOWER ALLOCATION

In this section, we address the problem in (12) and developits distributed solution. Problem (12) is combinatorial w.r.t.to both binary variables amk,i and continuous variables P

mk,i,j.

Hence, even a centralised solution would be computation-ally expensive. For instance, one can solve this problemusing globally optimal solvers (e.g. branch and boundor exhaustive search) with exponential complexity. In theliterature, approximate solutions to binary programmingproblems can be obtained by solving them with relaxedinteger constraints (allowing amk,i to be a real number from0 to 1), followed with sequential fixing algorithms. How-ever, if we relax amk,i, (12) is still a non-convex problem, asits objective function is not concave w.r.t. amk,i and Pmk,i,j.

Moreover, even if a centralised solution to (9) isobtained, it would still be impractical for a distributed oper-ation (e.g. uplink transmission in AP (Access Point)-basedCMIMO network or peer-to-peer transmissions in an adhoc CMIMO network). In such contexts, a distributed solu-tion is more desirable. It is worth noting that almost allexisting bargaining-based resource allocation algorithms inthe literature are centralised or require the assistance ofan arbitrator (e.g. [15, 16] [23]). In our work, we developa fully distributed algorithm that allows CR links to pro-pose their requested rate demands and bargain channelassignment and power allocation with other players.

4.1. Convexification

Firstly, relaxing the binary constraint (C5), (12) becomes



maxnamk,i,P

mk,i,jo

NRC1XmD1

Xi2Sm

log

0@X

k2SF

Nminm,iXjD1

amk,i

log

1 C mk,i,jPmk,i,j

bi1A

s.t. C1, C2, C3, C4 in (12)C5: 0 6 amk,i 6 1, m D 1, : : : , NRC1, 8i 2 Sm,

8k 2 SF (19)

The problem in (19) is not convex as its objective func-tion is not concave w.r.t. amk,i and P

mk,i,j, although it is

concave w.r.t. Pmk,i,j. Let

f m

amk,i, Pmk,i,j

D

80,Pmk,i,j>0,8k2SF

Lmi

amk,i, Pmk,i,j,

mk,i, , k,

mi

(29)

Note that the local problem (29) is convex as its objec-tive function is concave over a convex feasible region,hence can be solved using standard methods like interior

point. If a central arbitrator is in place (e.g. a base stationor spectrum database/broker), after solving the local prob-lem (29), all nodes report their calculated amk,i and Pmk,i,j tothe CBS so that the dual function is computed as

Dmk,i, , k,

mi

D L

amk,i , Pmk,i,j, mk,i, , k, mi

(30)

The dual problem DP is convex [24], hence the CBScan solve it efficiently for mk,i, , k, then broadcasts thesevariables. Each node updates its local problem (29) withbroadcasted Lagrangian variables, then again solves foramk,i and Pmk,i,j. The process continues until the dual functionconverges.

We now design a fully distributed algorithm to achievethe globally optimal solution of problem (18) when nocentral controller/arbitrator is available. Because the dualproblem is convex and its objective function is differen-tiable, it can be solved with a gradient search algorithm.Specifically, variables of DP at time .t C 1/ are updated asfollows:

m,.tC1/k,i D

m,.t/k,i

@L@mk,i

!C

D0@m,.t/k,i

0@PI,k

Nminm,iXjD1

Pm,.t/k,i,j

1A1A

C

.tC1/ D

.t/ @L@

C

D0@ .t/

0@PT NRC1X

mD1

Xi2Sm

Xk2SF

Nminm,iXjD1

Pm,.t/k,i,j

1A1A

C

.tC1/k D

.t/k

@L@k

C

[email protected]/k

0@1 NRC1X

mD1

Xi2Sm

amk,i

1A1A

C

m,.tC1/i D

m,.t/i

@L@mi

C

D0@m,.t/i

0@X

k2SF

Nminm,iXjD1

amk,i

log

1 C mk,i,jPmk,i,j

bi1A

C

(31)

where > 0 is a sufficiently small step-size and ../Cdenotes the projection onto the non-negative orthant. Theaforementioned updates are iterated at an arbitrator untilconvergence. Then the arbitrator informs the nodes aboutnew channel assignment variables and transmits covariancematrices.



We have two observations. Firstly, Lagrangian vari-ables mk,i and

mi can be calculated and can be updated

using only local information at node m, which servesSU i (the fraction of time amk,i that node m, whichserves SU i, tentatively communicates on channel k andpower allocation for each data stream s on channel k,Pmk,i,j). Secondly, a node m, serving SU i, can computeLm,NBi

amk,i, P

mk,i,j,

mk,i, , k,

mi

and solve its local prob-

lem (30) using two locally computed variables mk,i and miin addition to and k, which can be obtained if othernodes n , serving SU l, broadcast their tentative time frac-tion ank,l and allocated P

nk,l,j power for each data stream,

both on channel k. Hence, we propose a fully distributedmechanism in Table III (Algorithm 1) so that nodes bar-gain their requested rates, timeshare and power allocation.The key idea in Algorithm 1 is to ignore the iterations

of updates [25] in (30), which would have been carriedover at an arbitrator. However, we can prove that this igno-rance does not affect the convergence and optimality ofAlgorithm 1, provided that nodes broadcast their tentativetimeshares mk,i and allocated powers P

mk,i,j so that other

nodes can update the instantaneous and k.

Theorem 3. For sufficiently small step-size > 0,Algorithm 1 converges the bargaining process to the glob-ally optimal solution of problem (22).

Proof. We need to show two main points. At first, weneed to prove that the proposed distributed algorithm con-verges. Then, we must show that the converged point is theglobally optimal solution of the problem (22).

Table III. Algorithm 1 (distributed algorithm using dual decomposition).

Initialisation:(1) Set an initial channel allocation coefficients amk,i D 1NRC1 , for all m D 1, : : : , NR C 1 ,i 2 Sm, k 2 SF .(2) Set initial values for allocated power for stream j of node m (which serves SU i) on subchannel k, Pmk,i,j .(3) Set initial values for Lagrangian multilpliers mk,i , ,k ,

mi for all i 2 SSU, k 2 SF , m D 1, : : : , NR C 1.

Stopping condition:Specify ".For m D 1, : : : , NR C 1 doFor all i 2 Sm doFor all k 2 SF do

(1) Compute m,i as m,i D am,.tC1/k,i am,.t/k,i .(2) Compute 'm,i as 'm,i D Pm,.tC1/k,i,j Pm,.t/k,i,j .

End forEnd for

(1) Compute m as m DP

i2Smm,i

jSmj , (jSmj is the cardinality of the set Sm)(2) Compute 'm as 'm D

Pi2SSm

'm,i

jSmj .End for

Compute DNRC1PmD1

m

NRC1and D

NRC1PmD1

'm,i

NRC1.

If < " and < " then STOP.Else CONTINUE

Iterative step : Iteration count t D 1.While Stopping Condition is not satisfied or t < tmax doFor m D 1, : : : , NR C 1 doFor all i 2 Sm doFor all k 2 SF do

(1) Compute transmit, receive beamformers Smk,i and Vmk,i and stream gains

mk,i,j using (9).

(2) Update local Lagrangian variables mk,i and mi using (31).

(3) Update and k using (31), ank,l and Pnk,l,j from nodes n m.

(4) Update Lmiamk,i , P

mk,i,j ,

mk,i , ,k ,

mi

using (28).

(5) Solve the local problem (29) for amk,i and Pmk,i,j .

(6) Broadcast amk,i and Pmk,i,j .

End forEnd for

End forn D n C 1

End while

SU, secondary user.



To show the convergence, we prove that the dual func-tion D

mk,i, , k,

mi

is non-increasing and bounded

from below. It is clear that the dual function is alwayslower-bounded by the objective function of the primalproblem, which again can be bounded from below by itsvalue at any feasible solution. Next, let us consider thedifference of the dual function between two consecutiveiterations

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

and

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

can be written as

D

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

D

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

D L

a

m,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

L

a

m,.t/k,i , P

m,.t/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

(32)

The aforementioned difference can also be expressed as

D

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

D

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

D L

a

m,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

L

a

m,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

C L

a

m,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

L

a

m,.t/k,i , P

m,.t/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

6 24 @L

@11,1, : : : ,

@L@

NRC1jSNRC1j,NF

,

@L@

,

@L@1

, : : : ,@L

@NF,

@L@11,1

, : : : ,@L

@NRC1jSNRC1j,NF

35

T

24@LNB

@11,1, : : : ,

@LNB

@NRC1jSNRC1j,NF

,

@LNB

@,

@LNB

@1, : : : ,

@LNB

@NF,

@LNB

@11,1, : : : ,

@LNB

@NRC1jSNRC1j,NF

35C 0

6 0

(33)

The inequality in the aforementioned equation fol-lows from the two facts. Firstly, L is convex w.r.t.mk,i, , k,

mi

, and for sufficiently small step-size , the

descent direction update in (31) always reduces L whilefixing amk,i and P

mk,i,j. Secondly, a

m,.t/k,i and P

m,.t/k,i,j are max-

imisers of L

am,.t/k,i , P

m,.t/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

, which

are found by solving parallel local problems in (29) whilefixing

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

.

Next, we show that the converged point is the globallyoptimal solution of the problem (22). Therefore, we needto prove that the converged point meets the Karush-Kuhn-Tucker conditions of the convex problem (22); hence, it isthe globally optimal solution. From inequality in (33), atthe converged point, the equality happens, then,

L

am,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

D L

a

m,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

(34)

and

L

am,.tC1/k,i , P

m,.tC1/k,i,j ,

m,.tC1/k,i ,

.t/,

.t/k ,

m,.t/i

D L

a

m,.t/k,i , P

m,.t/k,i,j ,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

(35)

Because L is concave w.r.t. amk,i and Pmk,i,j, while fixing

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

and convex w.r.t.

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

, while fixing amk,i and P

mk,i,j. Equations (34)

and (35) happen if and only if the gradient ofL

am,.tC1/k,i , P

m,.tC1/k,i,j ,

mk,i, , k,

mi

at

m,.tC1/k,i ,

.tC1/,

.tC1/k ,

m,.tC1/i

vanishes and am,.tC1/k,i , and P

m,.tC1/k,i,j are

also maximiser of L

amk,i, Pmk,i,j,

m,.t/k,i ,

.t/,

.t/k ,

m,.t/i

. In

other words,@L

@mk,iD @L

@D @L

@kD @L

@mk,iD @L

@amk,iD @L

@Pmk,i,jD 0

8m2f1, : : : , NRC1g , 8i2Sm, 8k2SF , 8j2n1, : : : , Nminm,i

o(36)

This is exactly the Karush-Kuhn-Tucker conditions ofthe convex problem (22).

4.3. Distributed resource allocationalgorithm

In this section, a distributed resource allocation algorithmbased on bargaining is proposed for solving the problem(22). It is worth noting that the optimal solution of problem(22) provides information on which nodes wish to accesswhich subchannels and for what fraction of time. The pro-posed distributed resource allocation algorithm is throughconsidering preferences of all other nodes on that channelwhile taking into account both user fairness and over-all system throughput. The algorithm is of great practicalinterest as it is executed in a totally distributed manner.

The gradients at the converged point of Algorithm 1must be zero if the globally optimal solution to (22) is aninterior point of the feasible region. If the solution is aboundary point, the gradient at this point must be positive



(negative) along directions outward (inward) the interiorof the feasible region [25]. The gradients of the convergedpoint of the semi-distributed algorithms can be written as

@L@Pmk,i,j

Dmk,i,j

1 C mk,i,jP

mk,i,j

amk,i

0@Pk2SF

Nminm,iPjD1

amk,i log

1 C mk,i,jP

mk,i,j

amk,i

bi

1A

mk,i Cmi

mk,i,j

1 C mk,i,jP

mk,i,j

amk,i

(37)

@L@amk,i

D

Nminm,iPjD1

0@log 1 C mk,i,jPmk,i,j

amk,i

mk,i,jPmk,i,j

amk,i1C

mk,i,jP

mk,i,j

amk,i

1A

0@ P

k2SF

Nminm,iPjD1

amk,i log

1 C mk,i,jP

mk,i,j

amk,i

bi

1A

C miNminm,iXjD1

log

1 C

mk,i,jPmk,i,j

amk,i

!

mk,i,jP

mk,i,j

amk,i1 C

mk,i,jP

mk,i,j

amk,i

1CA k (38)

Based on the aforementioned discussion, the gradientof converged point of the distributed algorithm can beexpressed as

@L@Pmk,i,j

D(

D 0 Pmk,i,j > 0< 0 Pmk,i,j D 0

(39)

@L@amk,i

D8 0 amk,i D 1< 0 amk,i D 0

(40)

For Equations (37) and (38) to be defined at amk,i D 0,we assume the timeshare amk,i D 0 if and only if Pmk,i,j D 0,j D 1, : : : , Nminm,i . This assumption is physically valid. ForPmk,i,j > 0, Equation (37) can be written as

1mi

C mi D1

mk,i,j

mk,i C

1 C

mk,i,jPmk,i,j

amk,i

!,

j D 1, : : : , Nminm,i (41)

where mi ,P

k2SF

Nminm,iPjD1

amk,i log

1 C mk,i,jP

mk,i,j

amk,i

bi is the

amount that the allocated rate of node m, which serves SUi exceeds its demand rate bi. By replacing the amount ofmi into (38) and after some manipulations, (38) can berewritten as

@L@amk,i

D

1mi

C mi Nminm,iX

jD1log

1 C

mk,i,jPmk,i,j

amk,i

!

1amk,i

mk,i C

mi

Nminm,iXjD1

Pmk,i,j k

(42)

More precisely, @L@amk,i

can also be stated according to thefollowing:

@L@amk,i

D

Cmk,i k amk,i > 0k amk,i D 0

(43)

where

Cmk,i D

1mi

C mi Nminm,iX

jD1log

1 C

mk,i,jPmk,i,j

amk,i

!

1amk,i

mk,i C

mi

Nminm,iXjD1

Pmk,i,j (44)

It can be concluded from (44) that node m, which servesSU i, should exclusively occupy the k-th subchannel, ifCmk,i > k. Otherwise, the m-th node should timesharethe channel with other nodes or not use subchannel k ifCmk,i < k. Note that k is considered as the price of usingchannel k, which is flat for all nodes. Hence, Cmk,i can beinterpreted as the pay-off that node m, which serves SUi, gets from buying channel k. If a channel is exclusivelyallocated to no more than one node, then only the node withhighest Cmk,i should pick the channel k. This means that themost efficient/needy user (of channel k) wins the channel.Formally, we have the following rule to select the optimalnode for channel k:

amk,n D(

1 n D arg max8i2f1,:::,NRC1g

Cmk,i0 otherwise

(45)

In order to execute the rule in (45) in a distributedfashion, each node broadcasts an 1 NFvector Cmi ,hCm1,i, : : : , C

mNF ,i

i. After receiving Cmi from its neighbours,

a node m that serves SU i can autonomously decide a setSF,m of channels it should select. Notice that when compar-ing Cmk,i of different links on channel k, if a tie happens, thenwe can pick one of the nodes among those have the samepay-off Cmk,i for channel k (e.g. let the node that broadcastsits vector Cmi at last pick the channel).

The solution obtained during the timesharing assump-tion may not be optimal under the exclusive channel allo-cation according to rule (45). Moreover, under timesharing,at the converged point of Algorithm 1, all nodes are guar-anteed to obtain at least their rate demands. However, afterallocating channels according to Equation (45), some ofthe nodes may lose their timeshare on a channel that is now



Table IV. The proposed method for determining the optimal channel for all nodesin addition to calculating the value of mi .

Set Cmaxi ,hCmax1,i , : : : , C

maxNF ,i

i, where Cmaxk,i D max

nCmk,i

o, 8m 2 f1, : : : , NR C 1g.

For m D 1, : : : , NR C 1 doFor all i 2 Sm doFor all k 2 SF do

Calculatemi Dhm1,i , : : : ,

mNF .i

i, where mk,i D C

maxk,i C

mk,i

Nminm,iPjD1

log

1C

mk,i,j Pmk,i,j

amk,i

! .

End for(1) Sort Ami D Sort

mi

in an ascending order.

(2) Let A.1/m,i be the smallest positive element of Ami .

(3) Let A.2/m,i be the second smallest positive element of Ami .

(4) Set mi D A.1/m,iCA

.2/m,i

2 .(5) The optimal channel for node m, which serves SU i, is the index of A.1/m,i in

mi before sorting.End for

End for

exclusively allocated for another node. For these reasons,it is necessary to resolve the power allocation to ensure theoptimality and demand satisfaction. The power allocationfor node m, serving SU i, is now casted as follows:

maxnPmk,i,j>0,8jD1,:::,M,8k2SF,m

oX

k2SF,m

Nminm,iXjD1

log

1 Cmk,i,jP

mk,i,j

amk,i

!

s.t. C1:X

k2SF

NRC1XmD1

Xi2Sm

Nminm,iXjD1

Pmk,i,j 6PT

C2:Nminm,iXjD1

Pmk,i,j 6 PI,k, m D 1, : : : ,

NRC1, 8i 2 Sm, 8k 2 SF(46)

Problem (46) is convex, hence can be solved efficientlyusing standard solvers (e.g. interior point). With a closerobservation, (46) belongs to the class of generalised waterfilling problem with multiple water levels (one at eachband), hence can be solved more efficient with its dedicatedsolver in [26, 27].

There is also possibility of not meeting the requestedrate of node m, serving SU i, for the optimum of problem(46). If this is the case, then node m, serving SU i, shouldincrease its bargains to compete for additional channel. Letus recall pay-off vector Cmi in (44) that node m, serving SUi, uses to bid for its channels. To compete for additionalspectrum, node m, serving SU i, has to raise Cmi . Becausemi is the price of violating the minimum rate constraintC3 in (12), it is intuitive to raise mi by a sufficiently smallstep-size mi . The step-size should be picked so that node mcan win one additional channel at a time. Algorithmically,mi can be found with a binary search. However, we pro-pose an analytical method to find mi as follows. In orderto calculate mi , we first find the vector of winning pay-off

(Cmaxi ) from all channels. Afterwards, we see how far thepay-off vector Cmi of node m, serving SU i, is from Cmi andwe denote this difference as mk,i D

Cmaxk,i Cmk,iNminm,iPjD1

log

1C mk,i,jP

mk,i,j

amk,i

,

where Cmaxk,i is the k-th element of Cmi , and we have also

normalised the difference with the data rate of node m.Recalling Equation (44), if node m, serving SU i, wants towin channel k that is currently not allocated for node m,then mi must be set to be strictly greater than

mk,i. How-

ever, node m wants to request only one channel at a time.For that, we sort vector mi D

hm1,i, : : : ,

mNF .i

iin an

ascending order, then set mi to be the average of the twosmallest positive elements of mi . The proposed procedurefor obtaining the price of violating rate demand of node m(problem (12)) and channel l that node m is efficient to useis presented in Table IV.

After finding mi , node m, serving SU i, sets itsnew price of violating its rate demand, mi Dmi C mi , and recalculates its pay-off vector Cmi .Consequently, node m, serving SU i, broadcasts areallocation request message containing the channellthe index of the smallest positive element of mi

that

node m, which serves SU i, would like to possess and itsupdated Cmi . Upon hearing this message, all links recordnew Cmi (for possible future channel borrowing). Then, thecurrent owner (node n) of channel l excludes l from its setof allocated channels SF,n. Both nodes m and n resolve thepower allocation problem (46) and check if their demandsare met. The process of increasing biding price to bargainfor additional channels continues until all nodes obtaintheir requested rates.

We assume that there is enough spectrum in the networkto meet all links minimum demands (necessary conditionto apply NBS [14]), so that problem (12) is feasible. Hence,the bargaining process eventually stops. If no reallocationrequest message is heard for a given time duration, all linksstart transmitting on their selected channels. The channel



Table V. The Algorithm 3: distributed channel and power allocation.

(1) Run Algorithm 1 until convergence.(2) For m D 1, : : : , NR C 1 doFor all i 2 Sm do

(1) Compute the pay-off vector Cmi using (40).(2) Node m which serves SU i broadcasts Cmi .(3) Specify tmax (the maximum possible time for the resource allocation process).(4) t D 1.While t < tmax do

(1) After receiving Cni from other nodes (n D 1, : : : , NR C 1, n m), updatethe set of allocated channels, Sm, using (41).

(2) Solve the problem in (42) to obtain the optimal power allocation andcheck if

Pk2SF

amk,iRmk,i > bi .

IfP

k2SFamk,iR

mk,i < bi do

(1) Compute mi and channel index using algorithm 2.(2) Set mi D mi C mi and update Cmi .(3) Broadcast the new Cmi , RR message.(4) Set t D 1.

End if(1) t D t C 1.(2) If a RR message is heard, set t D 1.

End whileEnd for

End for

RR, reallocation request; SU, secondary user.

and power allocation for problem (12) is summarised inTable V (Algorithm 3).

5. PERFORMANCE EVALUATION

To evaluate the system performance, we have considereda system with NF D 30 and NSU D 10 . It is alsoassumed that each relay serves only one SU. We have con-ducted extensive simulations over 1000 time slots. For thelinks from CBS or relay to users, Rayleigh channel modelis used, while the links from BS to relays are modelledwith Rician channel with factor equal to 6 dB. The cellradius is assumed to be 1000 m. We set PT D 10 W, andPI,k D 1.5 W, for all k 2 SF . The rate demands of all SUsis set to 5 bps/Hz. The path-loss exponent is 4, and the stan-dard deviation of shadowing is 6 dB. The noise power is106 W/Hz.

For better comprehending the merit of the optimalresource allocation with optimal coalition graph formation(OA-OC), we will also compare the proposed algorithmswith the approaches using OA with random coalition graphformation (OA-RC) and equal power allocation, randomchannel assignment with OC, referred to as non-OA withOC (NA-OC).

The comparison among the convergence speed ofOA-OC is presented in Figure 2. It is assumed that theCBS and all SUs are equipped with three antennas, that is,M D 3. According to Figure 2, as a result of applying theAlgorithm 1 presented in Table III (abbreviated to A1 inFigure 2), the highest convergence speed is obtained, wherea few more iterations are required for the convergence of

6 8 10 12 14 16 18 205

10

15

20

25

30

35

No. of SUs

No.

of I

tera

tions

OAOC (A1)OAOC (A3)

Figure 2. Convergence speed of optimal resource allocationwith optimal coalition formation versus number of secondary

users for different algorithms.

Algorithm 3 (Table V). This could be attributed to moreneeded iterations to reallocate channels.

Figure 3 depicts the total sum-rate of the networkunder OA-OC (with or without time sharing), OA-RC andNA-OC. The heuristic distributed algorithm (Algorithm3) is utilised for OA-OC. The sum-rate of the networkincreases with the number of SUs, for all algorithms. Thisis partially due to the higher user and frequency diversitygains. OA-OC with algorithm 3 achieves the highest sum-rate. When channels are exclusively allocated, the sum-rateas a result of OA-OC is about 13% less than that obtained



4 5 6 7 8 9 10100

150

200

250

300

350

400

450

No. of SUs (NSU)

Sum

rat

e (bp

s/Hz)

OAOC with TS (A3)OAOC without TS (A3)NAOCOARC

Figure 3. Sum-rate of the network for different algorithms withand without time sharing.

4 5 6 7 8 9 100.4

0.5

0.6

0.7

0.8

0.9

1

No. of SUs (NSU)

Jain

s fa

irnes

s in

dex

OAOC with TS (A3)OAOC without TS (A3)NAOCOARC

Figure 4. Jains fairness index for different algorithms.

by OA-OC with time sharing. It can also be concludedfrom Figure 3 that NA-OC and OA-RC algorithms performmuch poorer than OA-OC, and then it becomes inevitableto form the coalition graph and allocate resources opti-mally. Moreover, for less number of SUs, OA-RC resultsin better sum-rate where for more number of SUs, NA-OCprovides higher sum-rate. This is because at fewer numberof SUs, it is not important if the coalition graph is formedoptimally or not and the effect of OA is dominant. As thenumber of SUs grows, the effect of OC becomes evident.

In order to determine whether SUs are receiving a fairshare of system resource, Jains fairness measure as awidely used fairness measure is used. Figure 4 depictsJains fairness indices of the four algorithms, OA-OC withand without time sharing, NA-OC and OA-RC. OA-OC(with or without time sharing) achieves significantly betterfairness than those of NA-OC and OA-RC. As the num-ber of SUs increases, fairness indices under OA-RC andNA-OC decrease. However, OA-OC algorithm maintainquite stable fairness for different network sizes. This isbecause in the OA-OC algorithm, channels (or timeshare

of a channel) are allocated while accounting for the amountof extra rate. Jains index for the OA-OC with exclusivechannel allocation is about 8% (on average) less than thatunder time sharing condition.

6. CONCLUSIONS

In this paper, joint problems of relay selection and resourceallocation in MIMO-OFDM-based cooperative CRNs wereconsidered. The problem was formulated as a two-levelgame with a non-transferable utility coalition graph gameas the first level and the NB games for channel assignmentand power allocation as the second level. After determin-ing the optimal relays and assigning the SUs to the relays,we developed fully distributed algorithms to jointly allo-cate channels and optimise transmit covariance matricesfor MIMO-OFDM-based cooperative cognitive networks.The proposed algorithms in the second level allow nodesto propose their rate demands, to cooperate and bargainto obtain their channel assignment and to optimise theirtransmit covariance matrices to maximise fairness. Undertimesharing, the distributed algorithms are proved to con-verge to their network-wide globally optimal solutions.The algorithms under timesharing revealed preferences ofdifferent links on a channel that guide heuristic algorithmsto allocate channels under the exclusive channel occupancypolicy. Simulations showed that these heuristic algorithmsare very close to their optimal solutions in the second level.Furthermore, simulations confirm the convergence of ourdistributed algorithms to the globally optimal solution ofthe two level game.

APPENDIX

Proof of Theorem 2. Using Hospitals rule, it can beverified that f m

amk,i, P

mk,i,j

is continuous w.r.t. amk,i whenamk,i D 0. It is can also be shown that the feasible regionof problem (22) is an intersection of half-spaces and con-vex regions, hence convex w.r.t. amk,iand P

mk,i,j. The objective

function is a summation of NRC1 functions hm

amk,i, Pmk,i,j

where

hm

amk,i, Pmk,i,j

DXi2Sm

Xk2SF

f m

amk,i, Pmk,i,j

(A1)

We will show that hm

amk,i, Pmk,i,j

is concave w.r.t. amk,iand Pmk,i,j. Let us rewrite h

m

amk,i, Pmk,i,j

as a composite

function, hm

amk,i, Pmk,i,j

D Pi2Sm

gm

amk,i, Pmk,i,j

, where

gm

amk,i, Pmk,i,j

D Pk2SF

f m

amk,i, Pmk,i,j

. gm

amk,i, Pmk,i,j

is a concave function w.r.t. amk,i and Pmk,i,j as it is the

summation of perspective functions of concave functionslog .1 C x/ (the perspective function of a concave function



is concave [25]). Therefore, hm

amk,i, Pmk,i,j

, which is someof concave functions, is concave w.r.t. amk,i and P

mk,i,j.

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A two-level cooperative game-based approach for joint relay selection and distributed resource allocation in MIMO-OFDM-based cooperative cognitive radio networksABSTRACTIntroductionSystem Model and Problem FormulationSystem modelProblem formulationMIMO transmit and receive beamforming

Determining the Coalition heads and their subscribersPotential coalition head selectionCoalition graph formation

Distributed channel assignment and power allocationConvexificationDistributed resource allocation using dual decompositionDistributed resource allocation algorithm

Performance EvaluationConclusionsAPPENDIXREFERENCES

Documents

A Two-level Cooperative Game-based Approach for Joint Relay Selection and Distributed Resource Allocation in MIMO-OfDM-Based Cooperative Cognitive Radio Networks