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v1 [

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Spectrum Leasing and Cooperative Resource Allocation inCognitive OFDMA Networks

Meixia Tao and Yuan Liu

Abstract: This paper considers a cooperative OFDMA-based cog-nitive radio network where the primary system leases some ofitssubchannels to the secondary system for a fraction of time inex-change for the secondary users (SUs) assisting the transmission ofprimary users (PUs) as relays. Our aim is to determine the cooper-ation strategies among the primary and secondary systems soas tomaximize the sum-rate of SUs while maintaining quality-of-service(QoS) requirements of PUs. We formulate a joint optimizationproblem of PU transmission mode selection, SU (or relay) selection,subcarrier assignment, power control, and time allocation. By ap-plying dual method, this mixed integer programming problem isdecomposed into parallel per-subcarrier subproblems, with eachdetermining the cooperation strategy between one PU and oneSU.We show that, on each leased subcarrier, the optimal strategy is tolet a SU exclusively act as a relay or transmit for itself. This resultis fundamentally different from the conventional spectrumleasingin single-channel systems where a SU must transmit a fraction oftime for itself if it helps the PUs transmission. We then propose asubgradient-based algorithm to find the asymptotically optimal so-lution to the primal problem in polynomial time. Simulation resultsdemonstrate that the proposed algorithm can significantly enhancethe network performance.

Index Terms: Cooperative communications, cognitive radio net-works, orthogonal frequency-division multiple-access (OFDMA),resource allocation, two-way relaying.

I. Introduction

Cognitive radio (CR), with its ability to sense unused fre-quency bands and adaptively adjust transmission parameters,has recently attracted considerable interest for solving the spec-trum scarcity problem [1, 2]. A key concept in cognitive radionetworks (CRNs) is opportunistic or dynamic spectrum access,which allows secondary users (SUs) to opportunistically accessthe bands licensed to primary users (PUs). Most of the workson dynamic spectrum access regard the secondary transmissionas harmful interference and hence the SUs do not participateinthe primary transmission. Recently, a new cooperation strategybetween the primary system and the secondary system was pro-posed in [3] and further investigated in [4]. Therein, the PUlinkleases its channel to the SUs for a fraction of time to transmitsecondary traffic in exchange for the SUs acting as relays to as-sist the transmission of primary traffic. The spectrum-leasingbased cooperation can improve the performance of both the pri-

Manuscript received March 6, 2012; revised June 30, 2012; accepted Septem-ber 2, 2012; approved for publication by Prof. Young-June Choi.

This work is supported by the Innovation Program of ShanghaiMunicipalEducation Commission under grant 11ZZ19, and the Program for New CenturyExcellent Talents in University (NCET) under grant NCET-11-0331.

The authors are with the Department of Electronic Engineering at ShanghaiJiao Tong University, Shanghai, 200240, P. R. China. Emails: {mxtao, yuan-liu}@sjtu.edu.cn.

mary and secondary systems and result in a win-win situation.The early works [3] and [4] on spectrum leasing only inves-

tigated the time slot allocation in the single-PU, multi-SU, andsingle-channel scenario. More specifically, in [3], one PU tar-gets at maximizing its rate while multiple SUs compete witheach other to access the single channel. However, this schememay result in an extreme case that the PU is aggressive and theSUs have no opportunity to access the channel. Recall that inCRNs, PUs are willing to share the spectrum resource with SUsif their quality-of-service (QoS) requirements are satisfied [1,2].In [4], the PU maximizes its utility in terms of rate and revenuewhile the SUs competitively make decisions based on their ratesand payments. Nevertheless, the virtual payment and revenuemay lead to another extreme case that the PU provides all of thetransmission time to the SUs on the single channel, which is notpractical in CRNs.

In this paper, we consider the general spectrum leasingand resource allocation problem in multi-channel multi-userCRN based on orthogonal frequency-division multiple-access(OFDMA). The motivation of using OFDMA is two-fold. First,OFDMA is not only adopted in many current and next gener-ation wireless standards but also a strong candidate for CRNs[5]. The second is that OFDMA-based systems can flexibly in-corporate dynamic resource allocations in CRNs (e.g., [68]).The primary system consists of multiple user pairs conductingbidirectional communication. The secondary system is a cellu-lar network consisting of a base station (BS) and a set of SUs.The two systems operate in a cognitive and cooperative man-ner by allowing the SUs to occupy certain subcarriers given thatthe QoS of the PUs are satisfied with the assistance of SUs ascooperative relays.

As CRNs are typically hierarchical and heterogeneous, it isintuitive that if SUs can aggressively help PUs transmission,then less subcarriers will be needed by the PUs to satisfy theirQoS requirements, and as a result the SUs can access more sub-carriers for maximizing their own data rates. Meanwhile, asthecommunication in the primary system is bidirectional, the coop-eration of SU as relays can also bring network coding gain inthe form of two-way relaying1. Thus, more subcarriers can beleased to the SUs. The increased spectral efficiency is in turntransformed into cooperation opportunities. Optimizing suchcooperative CRN has unique attractiveness and challenges asfollows.

Firstly, for the primary system, when relaying is necessary, ithas to decide which cooperative transmission modes (one-wayrelaying and two-way relaying) to select and which set of SUsto

1In two-way relay systems, a pair of nodes exchange information with thehelp of a relay node using physical layer network coding [911]. Two-wayrelaying can achieve much higher spectral efficiency than the traditional one-way relaying.

http://arxiv.org/abs/1209.3105v1

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choose, since it has higher priority in a CRN. Secondly, for thesecondary system, it needs to schedule appropriate SUs to uti-lize the leased subcarriers for maximizing its total throughput.Moreover, for those SUs that not only be selected as relays butalso be scheduled to transmit for themselves, the secondarysys-tem needs to balance their resource utilization. Thirdly, from thecommon perspective of the primary and secondary systems, itiscrucial to determine which set of subcarriers to cooperate on to-gether with how much power and time slots to transmit signals,in order to satisfy the QoS requirements of the primary system.

The main contributions of this paper are summarized as fol-lows:1. We propose an optimization framework for joint bidirec-tional transmission mode selection, SU selection, subcarrier as-signment, power control, and time slot allocation in the coop-erative CRNs. The objective is to maximize the sum-rate of allSUs while satisfying the individual rate requirement for each ofthe PUs. There are three distinct features in our optimizationframework.First, by subcarrier assignment and allocating timeslot between PUs and SUs in cooperation sessions, multiuserdiversity can be achieved in both frequency domain and timedomain.Second, as the communication in the primary system isbidirectional, we can exploit network coding gain in the form oftwo-way relaying to improve spectral efficiency via the SUsas-sistance.Third, using the OFDMA-based relaying architecture,each PU pair can conduct the bidirectional communication bymultiple transmission modes, namely direct transmission,one-and two-way relaying, each of them can take place on a differentset of subcarriers.2. We show that in the multi-channel cooperative CRNs, the op-timal strategy is to let a SU exclusively act as a relay for a PUor transmit data for itself on a cooperated channel. This resultfundamentally differs from the conventional cooperation in thesingle-channel scenario where a SU must transmit a fractionoftime for itself if it forwards the PUs transmission on the chan-nel.3. Using the Lagrange dual decomposition method, the joint op-timization problem is decomposed into parallel per-subcarrier-based subproblems. An efficient algorithm is proposed to findthe asymptotically optimal solution in polynomial time.

The remainder of this paper is organized as follows. Sec-tion II introduces the optimization framework, including systemmodel and problem formulation. Section III presents the de-tails of the Lagrange dual decomposition method for the jointresource-allocation problem. Section IV provides the simula-tion results. Finally, we conclude this paper in Section V.

II. Optimization Framework

We consider an OFDMA-based CRN where the primary sys-tem coexists with the secondary system as shown in Fig. 1. Theprimary system is an ad hoc network, consisting of multiple userpairs with each user pair conducting bidirectional communica-tions. The secondary system of interest is the uplink of a single-cell network where a BS communicates a set of SUs. Notethat the downlink can be analyzed in the same way. The pro-posed model can be justified in the IEEE 802.22 standard, wherethe CR systems are based on cellular basis. By taking advan-tage of the parallel OFDMA-based relaying architecture, each

BS

PU

SU

Fig. 1. System architecture of the CRN.

frequency

accesssecond hopfirst hop

timePUSU SUPU SUBS

(1 ) / 2t (1 ) / 2t t

SU

PU

Fig. 2. Time slot allocation between a PU and a SU. PU and SU can

transmit directly (on different subcarriers) or by a cooperation manner.

PU pair can conduct the bidirectional communication throughthree transmission modes, namely direct transmission, one- andtwo-way relaying, on different sets of subcarriers. As shownin Fig. 2, the PUs can transmit directly and the SUs can accessthe PUs residual subcarriers, or they transmit by a cooperationmanner. On each cooperated subcarrier, a SU can assist a PU(or PU pair) using one- or two-way relaying. This setup canfully explore available diversities of the network, including user,channel, and transmission mode.

We model the wireless fading environment by large-scale pathloss and shadowing, along with small-scale frequency-selectivefading. The channels between different links experience inde-pendent fading and the network operates in slow fading environ-ment, so that channel estimation is perfect. We assume that thetwo-hop transmission uses the same subcarrier for both links,i.e., the sourcerelay link and the relaydestination link. Thetime slot allocation between a PU and a SU on a cooperatedsubcarrier is illustrated in Fig. 2, in which we further assumethat the two hops of the cooperative transmission use equal timeslots. This is true for amplify-and-forward (AF) relaying strat-egy because AF needs equal time allocation, but more flexi-bility can be provided if the two hops pursue time adaptationfor decode-and-forward (DF). Nevertheless, we still adoptequaltime slot allocation between the two hops for simplicity.

Let N = {1, 2, , N} denote the set of subcarriers andK = {1, , k, ,K} denote the set of users, with the first

3

KP being the PU pairs and the remainingKS = K KPbeing SUs. Herek represents PU pair index if1 k KP and represents SU index ifKP + 1 k K. De-note k1 and k2 as the two users in thek-th PU pair, 1 k KP . DenoteP n = [P1,n, , Pk,n, , PK,n]T andRn = [R1,n, , Rk,n, , RK,n]T as the power and achiev-able rate vectors on subcarriern, respectively. If1 k KP ,Pk,n = [Pk1,n, Pk2,n]

T andRk,n = [Rk1,n, Rk2,n]T . For in-

terference avoidance, at most one PU (or PU pair) and one SUare active on each subcarrier. This exclusive subcarrier assign-ment and best relay selection can be implicitly involved inP nandRn. Let Pmax = [Pmax1 , , Pmaxk , , PmaxK ]T denotethe peak power constraints vector. Again, if1 k KP ,Pmaxk = [P

maxk1

, Pmaxk2 ]T . Let r = [r1, , rk, , rKP ]T (with

rk = [rk1 , rk2 ]T ) be the rate requirements of the PUs. Denote

tn = [tKP+1,n, , tk,n, , tK,n]T whose element0 tk,n 1is the duration that SUk transmits on subcarriern. Note thatas aforementioned there is at most one SU active on a subcar-rier, thus at most one non-zero element intn. Without loss ofgenerality, we assume that additive white noises at all nodesare independent circular symmetric complex Gaussian randomvariables, each of them has zero mean and unit variance. As-suming channel reciprocity in time-division duplex, we then use|hpk1,k2,n|2, |hsk,BS,n|2, and|h

psk1,k,n

|2 (|hpsk2,k,n|2) to representthe effective channel gains between PUk1 andk2 of PU pairk,SU k and BS, and PUk1 (k2) and SUk, respectively, on sub-carriern. For brevity, we denote all of them as a vectorHn. APU can cooperate with multiple SUs and a SU can assist multi-ple PUs. Thanks to the use of OFDMA, the inter-user interfer-ence can be avoided. In addition, the intra-pair interference forthe PU pairs will be treated as back-propagated self-interferenceand canceled perfectly after two-way relaying. Finally, weletP = [P 1, ,P n, ,PN ]T , R = [R1, ,Rn, ,RN ]T ,andt = [t1, , tn, , tN ]T be the power, achievable rate, andtime slot allocation matrices, respectively.

In this paper, our objective is not only to optimally allocatepower, subcarriers, and time slot but also to choose best trans-mission modes and relays for the PUs so as to maximize thesum-rate of all SUs while satisfying the individual rate require-ment for each of the PUs. Mathematically, the optimizationproblem can be formulated as

maxP ,R,t

K

k=KP+1

N

n=1

Rk,n (1a)

s.t.

N

n=1

Pk,n Pmaxk , k (1b)

N

n=1

Rk,n rk, 1 k KP (1c)

P 0, t [0, 1] (1d)Rk,n R (P n, tn,Hn) , 1 k KP , n (1e)

Rk,n = tk,nC

(

Pk,n|hsk,BS,n|22BS

)

,KP + 1 k K, n,

(1f)

whereC(x) = log2(1 + x), 2BS is the noise variance at the BS,

andR is the set of achievable rates for the PUs, which is relatedto P n, tn, Hn, and the transmission modes. Note again thatthe exclusive subcarrier assignment and best relay selection areimplicitly involved inP n, Rn, andtn.

Remark 1: In this paper, we assume that a central controlleris available, so that the network channel state informationandsensing results can be reliably gathered for centralized process-ing. Notice that the centralized CRNs are valid in IEEE 802.22standard [12], where the cognitive systems operate on a cellu-lar basis and the central controller can be embedded with a basestation (BS). This assumption is also reasonable if a spectrumbroker exists in CRNs for managing spectrum leasing and ac-cess [13,14]. Such centralized approach is commonly used inavariety of CRNs (e.g., [68,1319]). Compared with distributedapproaches (e.g., [3,4,20]), a CRN having a central managerthatpossesses detailed information about the wireless networken-ables highly efficient network configuration and better enforce-ment of a complex set of policies [13].

Remark 2: For CRNs, there is no single figure of QoS meritto measure the performance of the primary system. In this paper,we choose the rate requirement as the QoS metric. Other QoSmetrics, like outage probability and signal-to-interference-plus-noise ratio (SINR), can be easily accommodated in the problemformulation. Moreover, the weighted sum-rate maximization forthe SUs can be taken into account for the fairness issue, whichdoes not affect the proposed algorithms in the sequel.

III. Lagrange Dual Decomposition Based Optimization

The optimization problem in (1) is a mixed integer program-ming problem. In [21], the authors show that for the nonconvexresource optimization problems in OFDMA systems, the dual-ity gap becomes zero under the time-sharing condition. It isalsoproved in [21] that the time-sharing condition is always satisfiedas the number of OFDM subcarriers goes to infinity, regardlessof the nonconvexity of the original problem. This means thatsolving the original problem and solving its dual problem areequivalent. Based on the result, the Lagrange dual decomposi-tion method is recently applied to OFDMA-based cellular andcognitive radio networks in [22] and [19], respectively. Inthissection, we shall apply the result from [21] to solve our prob-lem in (1). In particular, some valuable insights are obtained formulti-channel cooperative CRNs, which shows that the general-ization from the single-channel case [3, 4] to the multi-channelcase is nontrivial.

We first introduce two sets of dual variables, = [1, , k, , K ]T (k = [k1 , k2 ]T if 1 k KP ) and = [1, , k, , KP ]T (k = [k1 , k2 ]T ) associated withconstraints (1b) and (1c) respectively, where 0 and 0.The Lagrange of the problem in (1) can be written as

L(P ,R, t,,) =

K

k=KP+1

N

n=1

Rk,n

+

K

k=1

k

(

Pmaxk N

n=1

Pk,n

)

+

KP

k=1

k

(

N

n=1

Rk,n rk)

.(2)

4

DefineD as the set of all primary variables{P ,R, t} that sat-isfy constraints (1d)-(1f). The dual function is given by

g(,) = max{P ,R,t}D

L(P ,R, t,,), (3)

and the dual optimization problem can be expressed as

min,

g(,) (4a)

s.t. 0, 0. (4b)

The dual function (3) can be rewritten as

g(,) =

N

n=1

gn(,) +

K

k=1

kPmaxk

KP

k=1

krk, (5)

where

gn(,) = max{P ,R,t}D

[

K

k=KP +1

Rk,n +

KP

k=1

kRk,n

K

k=1

kPk,n

]

(6)

are theN independent per-subcarrier-based optimization sub-problems.

Since a dual function is always convex by definition [23],subgradient-based ellipsoid method [24] can be used to mini-mizeg(,) by updating{,} simultaneously along with ap-propriate search directions, and it is guaranteed to converge tothe optimal solution{,}.

Proposition 1: For the dual problem defined in (4),

k = Pmaxk N

n=1

P k,n, 1 k K, (7)

and

k =N

n=1

Rk,n rk, 1 k KP , (8)

are subgradients ofg(,), where{P k,n, Rk,n} are the optimalsolutions of (6) for given{,}.

Proof: Please see Appendix A. As mentioned earlier, there are at most one PU (or PU pair),

denoted asP , and one SU, denoted asS, active on a subcar-rier. HereP andS also represent thebest PU (or PU pair) andSU respectively, among all possible users, that maximizes (6)for a given subcarriern. This can be obtained by an exhaus-tive search. The complexity is detailed later. Specifically, itneeds to first compute the optimal powers and rates for all usersunder all transmission modes, then let one PU and/or SU un-der one transmission mode that maximizes (6) active on eachsubcarrier. Therefore, the per-subcarrier problems in (6)can bealternatively expressed as

maxPn,Rn,tn

RS,n + PRP,n PPP,n SPS,n (9a)

s.t. PP,n 0, PS,n 0, 0 tS,n 1 (9b)RP,n R (PP,n, PS,n, tS,n,Hn) (9c)

RS,n = tS,nC

(

PS,n|hsS,BS,n|22BS

)

. (9d)

In what follows, for brevity of notation, the subscriptn in (9)is omitted due to allN per-subcarrier-based subproblems hav-ing an identical structure. In addition, for direct transmissionand one-way relaying, it is observed that the per-subcarrier op-timization problem for the two links of a PU pair, i.e.,P1 P2andP2 P1 (with or without relaying), has the same structureand can be decoupled. Thus, for brevity, we only consider heretheP1 P2 link as an example. In fact, the per-subcarrier op-timization for direct transmission and one-way relaying needsto first compute the optimal values of the objective functionin(9) for both links and then let one of them that has the maximumvalue active on the subcarrier. Moreover, we let = |hpP1,P2,n|2,1 = |hpP1,S,n|2, 2 = |h

psS,P2,n

|2, ands = |hsS,BS,n|2.

A. Direct Transmission

In this transmission mode, either a PU or a SU occupies solelythe given subcarrier. The per-subcarrier optimization problem in(9) can be expressed as

maxPP10,PS0

RS + P2RP2 P1PP1 SPS (10a)

s.t. RS = C(PSs) (10b)

RP2 = C(PP1). (10c)

Since the problem in (10) is convex, by applying the Karush-Kuhn-Tucker (KKT) conditions [23], the optimal power alloca-tions can be obtained as

P P1 =

(

P2aP1

1

)+

, (11)

and

P S =

(

1

aS 1

s

)+

, (12)

wherea = ln 2 and(x)+ = max(0, x). For a given subcarrier,the direct transmission further needs to compute the optimal val-ues of the objective function in (10) over one ofP P1 andP

S with

the other being zero, and then let one of them that has maximumvalue active. (11) and (12) show that the optimal power alloca-tions are achieved by multi-level water-filling. In particular, thewater level of each PU depends explicitly on its QoS require-ment, and can differ from one another. On the other hand, thewater levels of all SUs are the same.

B. One-Way Relaying

If relaying is required on a given subcarrier, a fraction of1tS time is used by a PU to transmit the primary traffic with thehelp of a SU, while the resttS time is assigned to the SU totransmit its own data. In this paper, we focus on DF only for

5

simplicity of presentation, for both one-way relaying and two-way relaying. Other relaying strategies are readily applicable toour framework and algorithms. The detailed discussion is givenlater.

In case of DF one-way relaying, the per-subcarrier problemin (9) can be rewritten as

maxPP1 ,PS ,tS

RS + P2RP2 P1PP1 SPS (13a)

s.t. PP1 0, PS 0, 0 tS 1 (13b)RS = tSC(PSs) (13c)

RP2 =(1 tS)

2min {C(PP11), C(PP1 + PS2)} .

(13d)

In (13d), the first term in the min-operation is the achievablerate of theP1 S link, and the second term is the achievablerate by maximum ratio combining between theS P2 linkandP1 P2 link. The following proposition is established fordetermining the optimal value of the time slot allocation variabletS .

Proposition 2: For eachcooperated subcarrier, a SUexclu-sively acts as a relay for cooperative transmission or transmitstraffic for itself.

Proof: The proposition means that the time slot allocationvariabletS is binary, i.e.,tS {0, 1}, which can be proved bycontradiction.

Assume that the optimal solution of (13) is(tS , PP1, P S) with

0 < tS < 1. Next, we show that we can always find anotherbetter solution withtS being binary.

Let us rewrite the objective function (13a) as

f(tS , PP1, P S) = tS

[

C(P Ss)

P2 min{

1

2C(P P11),

1

2C(P P1 + P

S2)

}

]

+P2 min

{

1

2C(P P11),

1

2C(P P1 + P

S2)

}

P1P P1 SP S . (14)

If C(P Ss) < P2 min{

1

2C(P P11),

1

2C(P P1 + P

S2)

}

, wehave

f(tS , PP1, P S) < P2 min

{

1

2C(P P11),

1

2C(P P1 + P

S2)

}

P1P P1 SP S= f(0, P P1 , P

S). (15)

Similarly, if C(P Ss) P22

min{

C(P P11), C(PP1 + P S2)

}

,we have

f(tS , PP1, P S) C(P Ss ) P1P P1 SP S

< C(P Ss ) SP S

= f(1, 0, P S). (16)

These results contradict the assumption. This completes theproof.

This proposition also holds for two-way relaying as discussedin the next subsection. The proof is similar and hence ignored.

Proposition 2 significantly simplifies the per-subcarrier opti-mization problem in (13)without loss of optimality by an ex-haustive search overtS . Specifically, we settS = 0 andtS = 1to compute the optimal values of (13a), respectively, then followthe one that has the maximum value.

The intuition is that, on a cooperated subcarrier (see Fig. 2),if the subcarrier condition on the SUBS link is good but onthe cooperative link is poor, it is better that the PU leases thewhole transmission time slot to the SU. Otherwise, the SU com-pletely devotes itself as a relay to the PU. In other words, ifaSU exclusively forwards a PUs traffic on a subcarrier, the PUshall lease other subcarrier(s) to the SU as remuneration. Thischannel-swap based multi-channel cooperation fundamentallydiffers from the single-channel cooperation case [3, 4] where ifa SU forwards a PUs traffic, it must benefit from the PU onthe channel. The spectral efficiency improvement brings morecooperation opportunities and leased subcarriers, and thus, thetotal throughput of the secondary system is increased. In thefollowing we considertS = 0 andtS = 1, respectively.

Case 1:tS = 0. In this case,S exclusively acts as a relay ona cooperated subcarrier. In DF one-way relaying, it is intuitivethatRP2 is maximized whenC(PP11) = C(PP1 + PS2),which leads to

PS = PP1 , (17)

where = (1 )/2. It is noted that DF occurs only if1 > . Substituting (17) to (13) and lettS = 0, the problemcan be rewritten as

maxPP10

P2RP2 (P1 + S)PP1 (18a)

s.t. RP2 =1

2C(PP11). (18b)

The above is a convex problem. By applying the KKT condi-tions, the optimal power allocation is given by

P P1 =

[

P22a(P1 + S

) 1

1

]+

, (19)

P S can be obtained according to (17). The above optimal powerallocation (19) shows that higher channel gain1, meaninghigher, results in lower water level, which is the extra fea-ture compared with the standard water-filling approach (e.g.,(11) and (12) in direct transmission). One also observes thatlower channel gain2 leads to lower water level and vice versa.

Case 2:tS = 1. In this case,S uses a cooperated subcarriersolely for its own transmission. The optimal power allocationcan be easily obtained and is the same as (12).

C. Two-Way Relaying

The two-way communication betweenP1 andP2 assisted byS takes place in two phases. Specifically, in the first phase, alsoknown as multiple-access (MAC) phase,P1 andP2 concurrentlytransmit signals to the assistingS. In the second phase, knownas broadcast (BC) phase,S broadcasts the processed signals tobothP1 andP2. Different from direct transmission and one-wayrelaying, two-way relaying must occur in pair [2530]. Thus

6

both the bidirectional links are taken into account together fortwo-way relaying. Here we only analyze the case oftS = 0, andthe case oftS = 1 is omitted since the optimal power allocationis the same as (12) iftS = 1.

The per-subcarrier problem in (9) can be expressed as (recallthattS = 0)

maxPP0,PS0,RP

P1RP1 + P2RP2

P1PP1 P2PP2 SPS (20a)s.t. RP R (PP , PS , 1, 2) = CMAC

CBC,

(20b)

whereCMAC andCBC are the capacity regions for the MAC andBC phases, respectively [11,31,32]. Specifically,

CMAC ={

[RP1 RP2 ]

RP1

1

2C(PP22), RP2

1

2C(PP11),

RP1 +RP2 1

2C(PP22 + PP11)

}

, (21)

and

CBC ={

[RP1 RP2 ]

RP1

1

2C(PS1), RP2

1

2C(PS2)

}

.

(22)Note that the channel reciprocity is used in the BC phase, whichis justified by the time-division duplex mode. Since bothCMACandCBC are convex sets, and so is their intersection, the prob-lem in (20) is a convex problem and can be solved by convextechniques.

Let 1 and2 be the two dual variables associated with thetwo rate constraints in (22). We first incorporate the two rateconstraints in (22) into the objective function and rewritetheLagrange dual problem of (20) as

min1,2

max{PP ,RP }CDF

P1RP1 + P2RP2

P1PP1 P2PP2 SPS

1[

RP1 1

2C(PS1)

]

2[

RP2 1

2C(PS2)

]

(23a)

s.t. 1 0, 2 0, (23b)

whereCDF is the set of the remaining constraints in (20) that{PP , RP } must satisfy. The minimization over{1, 2} can bedone using ellipsoid method with the fact that1

2C(PS1)RP1

and 12C(PS2) RP2 are subgradients of1 and2, respec-

tively. It is observed that the optimization variables in (23) areseparable. Therefore, the maximization over{PP , RP } in (23)can be decomposed into two subproblems that can be solvedseparately. The two subproblems are

maxPP0,RP

(P1 1)RP1 + (P2 2)RP2 P1PP1 P2PP2 (24a)

s.t. {PP , RP } CMAC, (24b)

andmaxPS0

12C(PS1) +

22C(PS2) SPS . (25)

For brevity of notation, we let1 = P1 1 and 2 =P2 2. It is noted that both1 and2 must be nonnega-tive, i.e.,P1 1 andP2 2. In the following we presentthe solution to each subproblem.

The subproblem in (24) is a classic resource allocation prob-lem in the Gaussian MAC [33], where the optimal power andrate allocations can be achieved by successive decoding. Specif-ically, users signals are decoded one by one in an increasing rateweight order [33]. Without loss of generality, we assume that1 2 (here1 and2 can be regarded as the rate weightsfor P1 andP2, respectively). Based on the polymatroid struc-ture of the Gaussian MAC [33], we then incorporate the threerate constraints in (21) into the objective function of (24), thesubproblem in (24) can be expressed as

maxPP0

12C(PP22) +

22

[C(PP22 + PP11) C(PP22)]

P1PP1 P2PP2 . (26)

It is easy to validate that the objective function of (26) is jointlyconcave inPP1 andPP2 . By applying the KKT conditions, theoptimal power allocations can be obtained as

P P2 =

[

(1 2)12a(1P2 2P1)

12

]+

, (27)

and

P P1 =1

2a

[

2P1

(1 2)2

1P2 2P1

]+

. (28)

It is observed that the optimal power allocations in the MACphase have the form of water-filling. Moreover, the follow-ing proposition is provided according to the above closed-formpower allocations.

Proposition 3: For 1 2, a necessary condition for theoccurrence of two-way relaying is1P2 > 2P1 .

Proof: Please see Appendix B. The subproblem in (25) is also convex since its objective

function is concave inPS . By applying the KKT conditions,the optimal power allocation is given by

P S =

{

2+

22413

21, if S (

1 2)2/(1P2 2P1). After

some manipulations, we obtain1P2/2P1 > 1/

2. Com-

bining the condition1/2 1, we obtain1P2 > 2P1 .

This completes the proof.

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I IntroductionII Optimization FrameworkIII Lagrange Dual Decomposition Based OptimizationIII-A Direct TransmissionIII-B One-Way RelayingIII-C Two-Way Relaying

IV Simulation ResultsV ConclusionA Proof of Proposition ??B Proof of Proposition ??