Noncooperative Multicell Resource Allocation of FBMC-Based Cognitive Radio Systems

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012 799

Noncooperative Multicell Resource Allocation ofFBMC-Based Cognitive Radio Systems

Haijian Zhang, Didier Le Ruyet, Senior Member, IEEE, Daniel Roviras, Senior Member, IEEE, andHong Sun, Member, IEEE

Abstract—Cognitive radio (CR) has been proposed to improvespectral efficiency while avoiding interference with licensed users.In this paper, we propose a resource allocation (RA) algorithm toperform uplink frequency allocation and power allocation amongnoncooperative multicells with multiuser per cell in CR systems.The maximization of the total information rate of multiple usersin one cell is considered for the Rayleigh channel with path losssubject to the power constraint on each user. Since the opti-mization formulation for rate maximization of multiple users ineach cell is an integer optimization problem, the multiple accesschannel (MAC) technique is proposed. With the aid of MAC,the original integer optimization problem is transformed intoa concave optimization problem, thereby establishing a distrib-uted game model in which the base station of each cell, whichtries to maximize the sum rate of its own users, is a player.The proposed game-theoretic algorithm for distributed multiuserpower allocation is viewed as an extension of the iterative water-filling algorithm, which is applied to distributed single-userpower allocation. From the perspective of effectiveness, the pro-posed game-theoretic algorithm based on the MAC techniqueis compared with the traditional algorithm based on frequency-division multiplexing access (FDMA). Final numerical resultsshow that the MAC-based RA algorithm can achieve more in-formation rate and better convergence performance than theFDMA-based RA algorithm.

Index Terms—Cognitive radio (CR), filter-bank-based multicar-rier (FBMC), game theory (GT), multiple access channel (MAC),orthogonal frequency-division multiplexing (OFDM), resource al-location (RA).

I. INTRODUCTION

COGNITIVE radio (CR) has been proposed as a possiblesolution to improve spectrum utilization via dynamic

spectrum access over the past few years [1]–[4]. The licensedspectrum bands to primary users (PUs) could be shared bysecondary users (SUs) by sensing the existence of spectrumholes. Therefore, spectrum sensing is necessary to ensure thatSUs would not interfere with PUs [5], [6]. When imperfect

Manuscript received June 21, 2011; revised October 31, 2011; acceptedDecember 3, 2011. Date of publication December 21, 2011; date of currentversion February 21, 2012. This work was supported in part by PHYDYASUE under Project FP7-ICT-2007-1-211887. The review of this paper wascoordinated by Dr. E. D. S. Au.

H. Zhang, D. Le Ruyet, and D. Roviras are with the LAETITIA/CEDRICLaboratory, Conservatoire National des Arts et Métiers, 75003 Paris, France(e-mail: [email protected]; [email protected]; [email protected]).

H. Sun is with the Signal Processing Laboratory, Electronic InformationSchool, Wuhan University, Wuhan 430079, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2011.2180743

spectrum sensing is not considered in the resource allocation(RA) algorithm, the interference introduced into PUs is exces-sive, and the service for PUs cannot be guaranteed. The problemof joint spectrum sensing and RA was investigated in [7] and[8], where the sensing error due to the imperfect sensing methodwas considered in CR networks. In this paper, we investigate theRA problem for noncooperative multicell in CR systems at theassumption that spectrum sensing has been well implemented.That is, the perfect spectrum sensing information about theactivities of PUs is assumed available at the base station (BS)of the secondary CR network.

Conventional orthogonal frequency-division multiplexing(OFDM) is regarded as a technology that is well matched forCR physical layer. However, interference avoidance is regardedas an important issue in CR systems. Due to the significantspectral leakage of OFDM, an accurate time synchronization isneeded to avoid the cross-channel interference with the licensedsystem. A lot of works in the literature put their focus on theRA considering networks with the coexistence of the primarysystem and the secondary system through OFDM-based airinterface [9]–[11]. The interference constraint between PUs andSUs was taken into account because of the significant out-of-band emission. Another alternative is to use filter-bank-basedmulticarrier (FBMC) modulation [12]–[14]. As demonstratedin [15] and [16], in an FBMC-based CR system, no timesynchronization or only a coarse time synchronization wasadequate to suppress the cross-channel interference because ofits low spectral leakage property. FBMC strongly relaxes theinterference constraint on the licensed system. As a result, theRA strategy without involving interference constraint for anFBMC-based secondary system can be feasible.

In [16], we proposed an RA algorithm by taking the impactof intercell interference (ICI) resulting from timing offset intoaccount. The maximization of total information rates was for-mulated under an uplink scenario with path loss and Rayleighfading subject to maximum power constraint as well as mutualinterference constraint between PUs and SUs. However, theproposed algorithm only considered the study of a scenariowith one secondary cell and one primary cell. This paper,which is an extension of the previous work, will investigatethe RA algorithm in multiple unsynchronized CR cells withmultiple users per cell, where CR users in different cells reusethe same spectrum resource to enhance the spectral efficiency.Therefore, CR users assigned with the same spectrum bandwill interfere with each other, i.e., ICI exists among differentCR cells. Because of the high communication overheads forcentralized RA, our focus is casted on the distributed RA withno cooperation between CR cells.

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800 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2, FEBRUARY 2012

Relevant research focusing on the noncooperative RA al-gorithm among multiple independent secondary cells can befound in the literature. In [17], the original distributed powercontrol for frequency-selective multiuser interference channelwas modeled as a noncooperative game and implemented bymeans of iterative water-filling algorithm (IWFA) in the contextof digital subscriber line systems, where each user water-filled its power to different subchannels regarding the power ofother users as interference. A distributed noncooperative gameto perform subchannel assignment, adaptive modulation, andpower control for multicell OFDM networks with one userper cell was proposed in [18]. To achieve Pareto improve-ment compared with the solution in [18], a pricing policyto the users’ transmit power by adding a penalty price wasproposed in [19]. However, the authors in [18] and [19] have notprovided provable uniqueness of Nash equilibrium (NE) andglobal convergence to an NE. In [20], the optimization problemmaximizing the information rate of each link for Rayleighfrequency-selective interference channel was formulated as astatic noncooperative game, and an asynchronous IWFA wasproposed to reach the NE of the game. In this asynchronousalgorithm, each user updated its power spectrum density in acompletely distributed and asynchronous way. Moreover, theauthors provided conditions that ensure global convergence ofthe asynchronous IWFA to the unique NE point. In [21], adistributed power allocation algorithm based on a new class ofgames called potential game was proposed. Convergence ruleand steady-state characterization were analyzed using potentialgame theory (GT). The proposed potential game algorithmwas shown to achieve higher energy efficiency in comparisonwith pure IWFA. In [22], a full distributed RA in multicellwith multiple users per cell was first presented by adoptinga game-theoretic approach. The unique NE point was provedto exist in some constrained environment. However, at eachiteration, subchannel assignment and power control were sepa-rately implemented by the player (BS), which required iterativecalculations of the subchannel assignment matrix and the powervector. Furthermore, a heuristic RA approach was presentedin [23], in which a selfish and a good neighbor decentralizeddynamic spectrum access strategies were proposed. However,dynamic RA for noncooperative multicell with multiple usersper cell in the context of CR is still an open topic.

Sophisticated distributed RA algorithm for noncooperativemulticell with multiple users per cell is a worthy topic ofstudy. One of the important strategies to analyze this distributedproblem is to adopt GT [24], which is a well-developed math-ematical model of conflict and cooperation among intelligentand rational players, such as cooperation and competition. SUsbelonging to the same network will compete for the limitedspectrum resources with those who do not belong to the samenetwork, which denotes that the secondary network aiming atmaximizing their own benefit without cooperation with othernetwork entities can be well analyzed via game-theoreticalalgorithms.

IEEE 802.22, the first CR-based wireless communicationstandard, is generally composed of multiple BSs. The inter-ference arises when multiple IEEE 802.22 BSs operate in thevicinity. To deal with the spectrum sharing in this situation,

IEEE 802.22 issued an inter-BS coexistence mechanism en-abling distributed competing BSs to coexist effectively [25]. Inthis mechanism, each BS could be regarded as a player optimiz-ing their own resource in a self-organizing way. Therefore, it isnatural to apply GT to model the convergence of self-organizingbehaviors. In [26], this IEEE 802.22 inter-BS coexistence issuewas investigated from a game-theoretical perspective. Based onthe definition of the inter-BS coexistence mechanism from theIEEE 802.22 standard, various game-theoretical algorithms toallocate spectrum resource could be exploited.

In this paper, a noncooperative uplink game-theoretic al-gorithm for multicell with multiuser per cell (MC-MU) in adistributed way is taken into account in the context of theFBMC-based CR system. Specifically, we propose an uplinknoncooperative RA algorithm using GT and multiple accesschannel (MAC) technique [27], [28] among multiple FBMC-based CR cells with multiple users per cell. The secondary BS(SBS) in each CR cell, trying to optimize the requirement of itsown users, is a player. The optimization of the total informationrate of CR users in one cell is considered, subject to the powerconstraint on each CR user. Thanks to the property of the MACtechnique, the rate maximization problem can be formulated asa concave optimization problem. Since it is complicated to ob-tain a closed-form solution for multiuser power allocation likewater-filling algorithm for single user, Lagrangian algorithm(LA) and gradient projection method (GPM) are employedto solve the concave optimization problem, respectively. Theproposed distributed algorithm based on the GT and MAC tech-niques, which can simultaneously perform iterative subchannelassignment and power allocation for multiuser,1 is regardedas an extension of IWFA, which is conventionally applied foriterative single-user power allocation.

Numerical results show that the game-theoretic algorithmthat allows to share one and more spectrum subchannels formultiple CR users obtains higher information rate and betterconvergence performance (converge to NE with a small numberof iterations) than the traditional frequency-division multiplex-ing access (FDMA), which needs an exhaustive search at eachiteration. Since the implementation of MAC requires additionalhardware cost in practice,2 a heuristic MAC-FDMA transfor-mation algorithm is therefore proposed, i.e., we transform thefinal RA outcome in the form of MAC into FDMA structure.Compared with traditional FDMA, this transformed FDMAfrom MAC is shown to have better performance, particularlywhen the system dimension is high.

The remainder of this paper is structured as follows. Weprovide the system model and general problem formulation inSection II. In Section III, we describe the proposed noncoop-erative game-theoretic algorithm; two mathematical solutionsof the concave optimization problem are presented. Numericalresults are given in Section IV. Finally, this paper is concludedin Section V.

1The subchannel assignment can be covered by power allocation, e.g., whenthe allocated power on a subchannel is zero, the subchannel is not used and viceversa.

2In general, users belonging to the same cell are not allowed to transmit onthe same subchannel.

ZHANG et al.: NONCOOPERATIVE MULTICELL RESOURCE ALLOCATION OF FBMC-BASED CR SYSTEMS 801

Fig. 1. Multicell CR scenario with multiple CR cells and multiple users percell.

II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section, we illustrate the system model and clarify itscorresponding assumptions. Next, a general problem formula-tion is given.

A. System Model

In this section, we define the system model, where the dif-ferent primary BSs (PBSs) share the same frequency resource,and there are 48 subbands with 18 subcarriers in the eachsubband (the minimum frequency unit occupied by PUs, herewe have chosen the practical values of WIMAX 802.16 forthe size of the subband). In the context of the FBMC-basedCR system, multiple SUs, as well as an access point calledSBS, constitute a CR cell. As shown in Fig. 1, a scenarioof a CR network consisting of multiple independent CR cellswith multiple CR users in each CR cell is illustrated. Unlikewell-localized PBSs, the CR cells are assumed to be randomlydistributed and can have heterogeneous cell dimension dueto the flexible characteristic of CR systems. We consider afrequency-selective channel with flat Rayleigh fading in eachsubband, and it is assumed that the channel changes slowlyso that the channel gains will be constant during transmission.Moreover, no interference cancelation techniques are carriedout, and the ICI is treated by each SBS as additive colored noise.

In our study, we apply the FBMC modulation scheme, whichdoes not require cyclic prefix (CP) extension and shows higherrobustness to residual frequency offset than CP-OFDM by tak-ing advantage of the low spectral leakage of its modulation pro-totype filter. FBMC has already been considered as a physicallayer candidate for CR systems [29]. The essential differencebetween OFDM and FBMC lies in the spectral leakage prop-erty, as shown in Fig. 2, in which3 their frequency responses aredrawn in a comparable way in the left figure. It can be observedthat OFDM exhibits a significant frequency sidelobe, which im-poses a strict orthogonality constraint for all the subcarriers. Onthe contrary, FBMC has a negligible sidelobe in the frequencydomain. In [15], OFDM/FBMC interference powers from SUs

3The prototype filter used for comparison is the one designed by Bellangerin the PHYDYAS project [30].

to PUs were obtained when transmitting a single complexsymbol with power that is equal to “1” on one time–frequencyslot. The mean interference powers of OFDM and FBMC areshown in the right graphic of Fig. 2. When we transmit a burstof independent complex symbols, the interference incurred byone subcarrier is equal to the sum of the interference for alltime slots. It can be observed that the number of subcarriersthat induces harmful interference (interference powers that arelarger than “10−3”) to PUs from SUs using OFDM and FBMCare “8” and “1,” respectively. In [16] and [31], the spectralefficiency of FBMC in the CR context was compared withCP-OFDM and raised cosine windowed OFDM (RC-OFDM).Both theoretic analysis and experimental results showed thatCP-OFDM and RC-OFDM had to compromise between thespectral efficiency of the secondary system and the interferenceon the primary system, whereas FBMC offered a much higherspectral efficiency than OFDM and meanwhile introduced fewinterference to PUs. Consequently, the FBMC scheme highlyrelaxes the interference constraint compared with CP-OFDMand RC-OFDM, which makes it possible to implement aninterference-free RA.

The perceived spectrum holes by different CR BSs wouldprobably be different, particularly when the CR BSs are farfrom each other. In this case, there will be no interference issuebetween the CR cells. From the viewpoint of application, ourinterest is put on the scenario where the secondary cells arewell localized in a geographical region (e.g., some adjacentbuildings in Fig. 1) so that different secondary cells can achievethe same sensing result.

Given the above FBMC-based CR system model, the follow-ing assumptions are made.

1) Spectrum sensing has been well implemented, and theavailable spectrum holes (in time and frequency domain)obtained by spectrum sensing are assumed to be fixed andcommonly shared by all the CR cells, i.e., the frequencyreuse factor is one.

2) Since the random distribution characteristic of availablespectrum subbands, the channel gains (from SUs to SBSs)of the available bands are assumed as stationary indepen-dent Rayleigh distribution.

3) The SUs in each CR cell are synchronized and equalized,and each SBS has full channel state information and con-trol of their own attached SUs, but different CR cells donot share their information, i.e., there is no coordinationamong CR cells.

4) In the context of FBMC-based CR system, we deactivateone subcarrier adjacent to PUs in all the spectrum holes4;thus, the interference from CR cells to primary systems isassumed to be negligible, and vice versa.

5) A coarse time synchronization is implemented, so eachSBS can update its system resources by a sensing intervalin a sequential way5 (as shown in Fig. 3), and no colli-sion occurs when different SBSs implement their sensingintervals.

4As highlighted in [16], the number of subcarriers that induce harmfulinterference to the primary user for FBMC is “1.”

5We do not consider the addition or removal of secondary cells.


Fig. 2. (Left) Comparison of frequency responses of OFDM and FBMC. (Right) The mean ICI powers of OFDM and FBMC from SUs to PUs when transmitting“1” at the subcarrier indexed by “0.”

Fig. 3. Schematic diagram of the sequential RA in the time domain.

6) In each cell, every SU has a constant power constraintindicated by P . The number of total available subbands(implemented by spectrum sensing) is F . A flat-fadingchannel is considered, and the channel gain is constantduring each subband [16].

The main objective is to find a distributed algorithm thatrequires no cooperation among SBSs and achieve an optimalfrequency assignment and power allocation for each CR cell.

B. Problem Formulation

In the uplink context of CR cells with multiple users per cell,we generally consider the maximization of information rate forthe whole system as the objective function, subject to a constantpower constraint on each user; this optimization problem isformulated as

maxp

:N∑

n=1

M∑m=1

F∑f=1

θnmf log2

[1 + SINRnm

f

]

SINRnmf =

Gnmnf pnm

f

σ2n +

∑n′ �=n

∑Mm′=1 Gn′m′n

f pn′m′f

s.t.

∑Mm=1 θnm

f ≤ 1 ∀n, f∑Ff=1 pnm

f = P ∀n,mpnm

f ≥ 0(1)

where p ∈ P , and P ⊂ RNMF is the possible set of power

solution; N is the number of CR cells; M is the number of usersper cell; F is the number of free bands; θnm

f is the subchannelassignment indicator, i.e., θnm

f = 1 if the f th subchannel in

the nth cell is allocated to the mth CR user; Gn′m′nf is the

propagation channel gain from the m′th user of the n′th cellto the SBS of the nth cell in the f th band; pn′m′

f is the powerof the m′th user of the n′th cell in the f th band; and σ2

n is thenoise power.

To solve (1) by centralized constrained optimization algo-rithms, all the information of channel gains is indispensable,which causes significant computational complexity and a largeamount of channel estimation overheads, particularly whenthe number of CR users is large. Therefore, a distributedRA algorithm should be more appropriate in the realistic CRscenario.

III. NONCOOPERATIVE GAME-THEORETIC ALGORITHM

Distributed RA is preferred where users can make theirdecision based on local information. In this section, we showhow to transform the nonconcave optimization problem intoa concave form so that the globally optimal solution can beobtained.

Notice that the formulation in (1) is an integer optimizationproblem. To transform the foregoing optimization probleminto a concave optimization problem, we advocate the MACtechnique, which signifies that multiple CR users in the samecell can occupy one or more spectrum bands.

Simple schemes like time-division multiplexing access(TDMA) and FDMA are used in many practical situations.When the MAC technique is allowed for data transmission ina system, a larger capacity region can be obtained than thatachieved by TDMA or FDMA by using a common decoderfor all the users of this system [27], [28]. The bound ofthe MAC capacity region for M users (M ≥ 2) with powers(p1, p2, . . . , pM ) is given by

M∑i=1

Ri ≤ log2

[1 +

p1G1 + p2G2 + · · · + pMGM

N0

](2)


TABLE ISEQUENTIAL ITERATIVE ALGORITHM

where Gi is the fading channel gain of the ith user, and N0 isthe ambient noise power.

Our interest is casted on a game-theoretic RA algorithmwithout centralized control or coordination among the multipleCR cells. In our distributed game, the SBSs are the players,which react and compete with each other for the commonresource.

In summary, let G = {N , {pn}n∈N , {un}n∈N }} denotethe noncooperative game structure, where N = {1, 2, . . . ,N} is the index set of the players (SBSs), pn = [pn1

1 , pn21 , . . . ,

pnM1 , pn1

2 , pn22 , . . . , pnM

F ] ∈ RMF is the power strategy space

of the nth player, and un is the utility function of the nth player.Each CR cell wants to maximize its own information rate

by allocating power into different bands for its own users,regardless of the other CR cells in a distributed way. For thenth cell, its information rate maximization problem with powerconstraint can be formulated as6

maxpn

: un(pn,p−n) =F∑

f=1

log2

[1 + SINRn

f

]

SINRnf =

∑Mm=1 Gnmn

f pnmf

σ2n + I−n

f

s.t. {∑Ff=1 pnm

f = P ∀n,mpnm

f ≥ 0(3)

where p−n = (p1, . . . ,pn−1,pn+1, . . . ,pN ) is the strategyprofile of all the players except for the nth player, and I−n

f =∑n′ �=n

∑Mm′=1 Gn′m′n

f pn′m′f denotes the ICI from the other

cells to the nth cell in the f th band, which is treated asadditive noise. Compared with the problem formulation forthe multiuser case in [16], the interference constraint is notconsidered herein because the FBMC technique is assumed tobe used by the CR cells.

The proposed iterative noncooperative game is implementedin a sequential way, i.e., each player independently updates itsstrategy according to a fixed updating order. This sequentialalgorithm is stated in mathematical terms in Table I.7

6The power constraint is generally satisfied with equality because the wholepower tends to be used by the CR users to maximize the information rate.In some cases where the intercell channel impact is very severe, this powerconstraint with equality will fail, and the final result will not converge or willconverge to an undesirable NE point.

7Where Tit denotes the prescribed iteration times.

The outcome of the proposed game involving N playersbased on the utility function un is expected to achieve NE,which is defined as in Definition 1.

Definition 1: A strategy profile p∗ is NE if no unilateraldeviation in strategy by any single player is profitable for thatplayer, that is

un

(p∗

n,p∗−n

) ≥ un

(pn,p∗

−n

) ∀n,∀pn ∈ Pn (4)

where Pn is the set of admissible strategies defined in theconstraint conditions of (3).

A. Existence of NE

Theorem 1: For a utility function un(pn,p−n) with a sup-port domain pn ∈ Pn, at least a pure strategy NE point existsfor any set of channel realizations and power constraint, if∀n, Pn is a nonempty convex set, and un is continuous andquasiconvex or quasiconcave.

Proof: Theorem 1 given in [24] can be applied to provethe existence of NE point for the proposed game-theoreticalgorithm. Specifically, two conditions should be satisfied:1) the support domain Pn is a nonempty compact convexsubset of a Euclidean space, and 2) the payoff function un

is continuous and quasiconcave in pn. The proof of thesetwo conditions for the proposed game-theoretical algorithm inTable I is provided in the Appendix. Theorem 1 is similar toNash’s theorem [24], which is a special case of this theorem.The continuous utility function implies nonempty closed-graphreaction correspondences. In other words, if the utility functionis not continuous, the reaction correspondences may fail to havea closed graph or fail to be nonempty because discontinuousfunctions need not attain a maximum. By contrast, quasi-concavity is generally harder to satisfy than the continuity ina game. The quasiconcave property implies that the reactioncorrespondences are convex valued. It is interesting to mentionthat NE can also exist even when these two conditions are notsatisfied, as these conditions are sufficient but not necessary.

B. Solutions for Concave Optimization Problem

The next problem is how to achieve the optimal solution pn

in (3), which maximizes the utility function un. When M = 1,the optimal problem in (3) is degraded to the multicell withsingle user per cell; in this special case, the known closed-form solution of WFA [17] can be used for solving the single-user power optimization problem. Nevertheless, in the generalcase of M > 1, traditional WFA does not work for the powerallocation of the multiuser case. A closed-form solution forthe MC-MU case seems to be difficult to achieve. A varietyof mathematic methods known in the literature can be usedto efficiently solve the convex optimization in (3); herein, LAand GPM are employed to solve the optimization problem ofMC-MU.

LA: The optimization problem in (3) is a multiuser non-linear optimization problem with equality and inequality con-straints. Herein, the method of Lagrange multipliers and theKarush–Kuhn–Tucker (KKT) conditions [32] are provided as a


strategy to find the maximum of the utility function un subjectto constraints. Define the Lagrangian function of the problemin (3) as

L(pn, λ, µ) = un(pn,p−n)

+M∑i=1

λi

(P − pni

1 − pni2 · · · − pni

F

)+

M∑i=1

F∑k=1

µikpni

k . (5)

The KKT conditions are necessary for a solution in nonlinearprogramming to be optimal, provided some regularity con-ditions are satisfied. It is a generalization of the method ofLagrange multipliers to inequality constraints.

Suppose that the objective function to be maximized isf , and the equality and inequality constraint functions arehi(i = 1, 2, . . . ,m) and gj(j = 1, 2, . . . , n), respectively. Fur-ther, suppose that they are continuously differentiable at apoint x∗. If x∗ is a local minimum that satisfies some regularityconditions, then there exist constants λi(i = 1, 2, . . . ,m) andµj(i = 1, 2, . . . , n) such that

∇x∗f(x∗) +∑m

i=1 λi∇x∗hi(x∗)+

∑nj=1 µj∇x∗gj(x∗) = 0

hi(x∗) = 0, gj(x∗) ≥ 0λi ≥ 0, µj ≥ 0µjgj(x∗) = 0 (∀i, j)

(6)

where ∇ is the gradient operator. These foregoing formulasare the well-known KKT conditions, which are necessary fora solution to be optimal.

In some cases, the necessary KKT conditions are sufficientfor global optimality. This is the case when the utility functionand the inequality constraints are continuously differentiableconcave functions and the equality constraints are affine func-tions. Since we already proved the concavity of the utilityfunction un in (3), the KKT condition is sufficient for theuniquely optimal solution of (3).

To guarantee a maximum point x∗ to be KKT, it shouldsatisfy some regularity condition, the most used is the linearindependence constraint qualification (LICQ). The constraintfunctions satisfy the LICQ if

[∇x∗gj(x∗);∇x∗hi(x∗)]T ∀i, j (7)

has full column rank, i.e., the gradients of the active inequalityconstraints and the gradients of the equality constraints arelinearly independent at x∗. For the optimization problem in (5),gi

k(pn) = pnik (∀i, k), and hi = (1 − pni

1 − pni2 · · · − pni

F ) (∀i).Assuming p∗

n is the optimal solution, according to (7), we get

Rank{[∇p∗

ngi

k(p∗n);∇p∗

nhi (p∗

n)]T

}= M × F (8)

satisfying the LICQ.Consequently, the KKT conditions in (6) can be used for

uniquely solving the global point of (3), and the Lagrangian

functions of the optimal allocation problem can be obtained as

Gnink

σ2n+I−n

k+(pn1

kGn1n

k+···+pnM

kGnMn

k )· 1log2(2)

− λi + µik = 0,

1 − pni1 − pni

2 · · · − pniF = 0,

µikpni

k = 0,λi ≥ 0, µi

k ≥ 0, pnik ≥ 0, (∀i, k)

(9)

where i = 1, 2, . . . ,M , and k = 1, 2, . . . , F . Under the as-sumption that aggregated ICI I−n

k can be measured locallyand the SBS knows all the channel information of its ownusers, the optimal frequency allocation and power control canbe iteratively computed between distributed CR cells accordingto (9).

Unfortunately, for a CR cell with M users and F freebands, the number of equations in (9) is M × (2F + 1), whichindicates a huge computational complexity in particular whenthe values of M and F are high.

GPM: Due to the computation limit of the Lagrangian opti-mization method in (9) to high-dimension network, we resort toGPM for our linearly constrained problem. Rosen’s GPM [32]is based on projecting the search direction into the subspacetangent to the active constraints. We define our constrainedoptimization problem as

max : un(pn,p−n)

s.t.

{Epn = eApn ≤ b (10)

where A is a MF × MF coefficient matrix of the inequalitylinear constraints, and the rows of E are the coefficient vectorsof the equality constraints. Comparing (10) with (3), we canobtain

A =

−1 0 · · · 00 −1 · · · 0...

.... . . 0

0 . . . 0 −1

MF×MF

,b =

0

...0

MF×1

E =

1 · · · 1 0 · · · 0 · · · · · · 00 · · · 0 1 · · · 1 0 · · · 0...

......

......

......

......

0 · · · · · · · · · · · · 0 1 · · · 1

M×MF

e =

P

...P

M×1

.

Let pn be a feasible solution, and suppose A1, A2, b1, andb2 satisfy A1pn = b1,A2pn < b2, where AT = (AT

1 ,AT2 ),

and bT = (bT1 ,bT

2 ). Supposing that MT = (AT1 ,ET ), then

the gradient projection algorithm is given as follows:1) Calculate the projection matrix P, which is given by

P = I − MT (MMT )−1M (11)

where I is the unit matrix.2) Calculate s(t) = P∇un (∇ is the operator of the

gradient).


TABLE IITRANSFORMATION ALGORITHM FROM MAC TO FDMA

3) If ‖s(t)‖ ≤ ε, terminate (ε is a threshold value).4) Determine the maximum step size

αmax = min{αk}, k = 1, 2, . . . ,MF

αk ={

ck

dkdk > 0

∞ dk ≤ 0(12)

where c = b − Apn, d = As(t). (13)

5) Solve the line-search problem to find

α = argmaxα

(un

(pn

(t) + αs(t)))

, 0 ≤ α ≤ αmax.

(14)

6) Set p(t+1)n = p(t)

n + αs(t), t = t + 1, and go to step 2.Given a proper threshold ε, GPM is an efficient way for our

optimization problem with linear constraints. To summarize,the foregoing statements show that LA is only applied for lowdimension systems due to its computation complexity. Con-versely, GPM can be used to solve the optimization problemof high-dimension systems.

To evaluate the proposed MAC GT (MAC-GT) algorithm, aFDMA-based GT (FDMA-GT) is also implemented in such away that each SBS exhaustively searches for the optimal FDMAstrategy that maximizes the whole information rate in (1)in a sequential way. The existence of the NE point cannotbe guaranteed due to the nonconcave property of the utilityfunction in (1). Moreover, when the number of CR users percell is large, a computation problem appears for the FDMA-GT algorithm because we have to try all the possible FDMAcandidates to obtain the optimal FDMA solution during eachiteration process. This exhaustive search makes FDMA-GT im-practical, and some heuristic strategies are expected to solve thehigh-dimension problem of the FDMA-GT algorithm. In thischapter, thanks to the MAC-GT algorithm, we propose a MAC-FDMA transformation algorithm, i.e., after global convergence,and the final MAC-GT result is transformed into an FDMAstructure by preferentially allocating the bands with strongpower, the detail of this MAC-FDMA algorithm is described inTable II. Since the capacity region of MAC is larger than that ofFDMA, particularly in the case of a large system dimension, theperformance of MAC-FDMA is expected to be better than theperformance of the FDMA-GT algorithm because the FDMAformat that is closest to the MAC format is chosen according toTable II.

C. Complexity Analysis

In this section, we discuss the implementation complexity ofthe proposed game-theoretical algorithm. For centralized RAalgorithms, the channel information of users in all the cells isrequired at all times to perform the optimal allocation. Thisresults in a large amount of signaling overhead. Instead, theproposed game-theoretical scheme targets on the distributedRA so that signaling channels and coordinations between CRcells are not needed.

The optimal solution to the subcarrier and power allocationfor the FDMA-GT algorithm requires an exhaustive search tofind the optimal subcarrier assignment of the F subbands, andthe complexity of this exhaustive search exponentially growsas O(2F ). As for the MAC-GT algorithm, although the interiorpoint convex optimization method can be used to solve theconcave optimization problem, its complexity is considerablyhigher than those methods that exploit the special structureof the formulation in (3). GPM is therefore applied to solvethe concave optimization problem in a low complexity. Thecomputational complexity of the gradient computation in step 2of the GPM is O(MF ). Steps 4 and 5 of the GPM determine themaximum step size and execute the line-search problem, whichhas a complexity of O(MF + K), where K is the numberof search intervals (which is dependent on the maximum stepsize). As a result, the complexity of the algorithm per iterationis;O(MF + K), which denotes that GPM has a desirable lin-ear complexity [33]. According to the theoretical analysis andthe following experimental results, the number of iterations forthe GPM usually lies in between 2 and 4, even for a big systemdimension, i.e., the convergence of GPM is very insensitive tothe increase of the number of users and available subbands. Theconvergence proof of the GPM can be found in [32].

IV. NUMERICAL RESULTS

In the context of CR, we assume that some licensed spectrumband resources are already obtained by CR spectrum sensing.The channels of available spectrum bands are considered asstationary independent Rayleigh distribution because of therandom distributed characteristic of spectrum holes. The mainobjective of this section is to try to demonstrate more optimalsum rate and stabler convergence using MAC compared withtraditional FDMA. Simulations are divided into two parts: low-dimension systems and high-dimension systems.

A. Simulations for Low-Dimension Systems

Since low-dimension systems with small number of CR usersper secondary cell are considered, the Lagrangian solution in(9) can be used for solving the concave optimization in (3).More specific, two- and five-cell cases are considered (seeFig. 1), with two or three users uniformly distributed withina cell range (0.01–1.1 km). Assuming that the power densityof thermal noise is −174 dBm/Hz, in the case of uplink, eachuser is equipped to transmit signal with fixed P = 1 watt power.As the transmission distance increases, the attenuation also


Fig. 4. Averaged sum rate of the whole system versus distance D.

increases due to propagation path loss. The path loss of thereceived signal at a distance d (in kilometers) is [34]

P (d) = 128.1 + 37.6 · log10(d) dB. (15)

By this macropropagation model, the SNR value of the userlocated at the border of the cell (assuming the radius of cellR = 1.1 km) fluctuates around 20 dB when the Rayleigh fadingchannel is considered.

First, the numerical results for the two-cell case are dis-played. In Fig. 4, the averaged sum rate of the whole systemwith two users per cell versus the distance D between BSsis compared for FDMA-GT and MAC-GT, where the optimalcentralized algorithm8 performance for MAC and FDMA isalso given for comparison purposes. One thousand independentnetwork topologies and channel realizations are averaged fortwo and three bands, respectively. It can be seen that a largedistance can increase the sum-rate performance significantly.The reason is that there exists little interference from other cellswhen the cells are far away from each other.

In the case of two bands, the optimal centralized sum-rate performance of MAC (MAC-optimal) outperforms thatof FDMA optimal due to the larger capacity region of MAC.These dashed optimal curves in Fig. 4 can be used as perfor-mance bounds for the decentralized GT algorithm. As expected,MAC-GT always has a better sum rate performance thanFDMA-GT. However, we observe that MAC-GT has a lowsum rate compared with its optimal curve when distance Dbecomes small, which can be explained by the greedy propertyof each player in GT, in the presence of small distance (i.e.,significant interference), and the outcome of the noncooperativegame will finally converge to an inefficient NE point. The sameperformance results can happen to the case of three free bands,the only difference is that for FDMA, unlike two bands for twousers, each user can use one or two bands, e.g., for the userwho occupies two bands, power allocation will be implemented

8In this case, all the channel information among different CR cells is assumedto be known by using a centralized controller, and therefore, the optimalcentralized resource allocation can be implemented.

Fig. 5. Sum-rate cdfs of FDMA-GT and MAC-GT.

TABLE IIIITERATION SITUATION FOR FDMA-GT ALGORITHM IN

LOW-DIMENSION SYSTEMS

on these two bands with a fixed power constraint, which canimprove the performance of FDMA. This is the reason why inFig. 4 there is less performance difference between MAC andFDMA compared with the case of two bands.

For the simple case of two cells and two users per cell,both FDMA-GT and MAC-GT can converge with a rapidconvergence speed (two iteration rounds). Next, the case offive cells with three users per cell is considered. As shown inFig. 5, the sum-rate cumulative distribution functions (cdfs)of FDMA-GT and MAC-GT algorithms with three availablebands and three different distance cases (D = 0, 2.2 and4.4 km) over 200 independent channel realizations are com-pared. In the case of distance D = 0 km, these two algorithmsboth suffer from significant interferences, which causes that thefinal solution converges to some undesired NE points. In thecases of more practical distances D = 2.2 km and D = 4.4 km,the performance differences between MAC-GT and FDMA-GTare observed to be larger than the case of two cells with twousers per cell in Fig. 4. This is expected because the capacityregion disparity between MAC and FDMA increases as theusers’ number increases.

The convergence rates for FDMA-GT and MAC-GT algo-rithms in the five-cell case are presented in Tables III and IV,respectively. It can be noticed that more iterations are neededif the distance D is enough small (which means significantinterference between cells). Although FDMA-GT has a littlemore rapid convergence speed than that of MAC-GT, thereexists some nonconvergent points (NPs) resulting from somechannel situations. Conversely, MAC-GT always converges toan NE point within limited iteration times. Hence, we can


TABLE IVITERATION SITUATION FOR MAC-GT ALGORITHM IN

LOW-DIMENSION SYSTEMS

Fig. 6. Regular seven-cell CR scenario with wraparound structure.

conclude that MAC-GT can exploit more information rateand can offer better convergence stability than FDMA-GT,particularly if the number of users in the network is large.In the subsequent part, practical high-dimension systems withlarge number of CR users per cell are simulated, and it willbe demonstrated that a higher information rate is achieved forMAC-GT compared with FDMA-GT in the context of high-dimension systems.

B. Simulations for High-Dimension Systems

To generalize the application of the proposed MAC-GTalgorithm, as shown in Fig. 6, a seven-cell CR scenario (eachcell contains a large number of CR users) with a wraparoundstructure is considered, in which the seven BSs who know thepropagation channels of their own users are the players (N =7) of the game and maximize their information rate accordingto a sequential updating order. In each CR cell, multiple CRusers are uniformly distributed within the cell range (0.01–1.1 km). The transmitted power of each user is fixed at a prac-tical value P = 20 mW. The path loss model of the receivedsignal in (15) is applied.

Naturally, the computational complexity increases with theincrease of system dimension. Thus, LA is no longer efficientfor solving the optimization problem in (3). Alternatively, we

Fig. 7. Averaged sum rate of whole system versus distance.

prefer to select GPM to solve the concave optimization prob-lem, where we set the threshold parameter ε = 10−3.

In the first study, we hypothesize that the number of bandsis always equal to the number of users per cell9 (M = F ).We perform several simulations to corroborate the theoreticalresults. First, the simulation results with a small number ofCR users per cell are displayed. In Fig. 7, the sum rate of thewhole system with three users per cell as well as five usersper cell versus the distance D between BSs is compared forFDMA-GT and MAC-GT.10 Five hundred independent networktopologies and channel realizations are averaged for these twocases, respectively. We can see that MAC-GT always has abetter performance than FDMA-GT. In addition, due to thelarger capacity region for MAC, MAC-GT benefits more thanFDMA-GT with the increase of free bands. As shown in Fig. 7,for the case of five free bands, there is more performance gapbetween MAC-GT and FDMA-GT compared with the case ofthree free bands.

The corresponding convergence rates for FDMA-GT andMAC-GT algorithms are presented in Figs. 8 and 9, respec-tively. The same as the numerical results for low-dimensionsystems, we observe that more iterations are needed when thedistance D becomes small. FDMA-GT converges a little fasterthan MAC-GT, but there exists some NPs, the rate of whichaugments as the users’ number increases. Conversely, MAC-GT can guarantee that the noncooperative game converges toan NE point within a small iteration time.

Second, for a larger number of CR users per cell, a compu-tation problem appears for the FDMA-GT algorithm becausewe have to try all the possible FDMA candidates to obtainthe optimal FDMA solution during each iteration process. Inthis case, we apply the MAC-FDMA transformation algorithm.An illustration of this MAC-FDMA algorithm is described inFig. 10. Based on the final MAC-GT result, we first searchthe strongest power value in the MAC table. This value (0.9)

9The general case when M �= F has high simulation complexity but can stillbe applied to the proposed algorithm.

10Here, we do not give the numerical results of the optimal centralizedalgorithm because it requires a global exhaustive search, which has a highcomputational complexity for high-dimension systems.


Fig. 8. Convergence property of the case with three bands and three users percell.

Fig. 9. Convergence property of the case with five bands and five users percell.

Fig. 10. Transformation illustration from MAC to FDMA.

corresponds the second band and the first user; therefore, weallocate the second band to the first user and eliminate thesecond column and the first row from the MAC table. Then,we continue to search the strongest power value in the restof the MAC table and conduct the previous operation until

Fig. 11. Sum-rate cdfs of MAC-FDMA and MAC-GT algorithms.

TABLE VITERATION SITUATION FOR MAC-GT ALGORITHM IN

HIGH-DIMENSION SYSTEMS

all the bands are allocated. The experimental results of MAC-FDMA for three and five users are shown and compared withFDMA-GT in Fig. 7. It is interesting to find that theMAC-FDMA algorithm outperforms the exhaustive FDMA-GTalgorithm when the number of users in the system increases. Inthe rest of the simulation, we use the MAC-FDMA algorithmto replace the exhaustive FDMA-GT algorithm.

Next, the situation of a practical distance (D = 2R) withdifferent numbers of CR users per cell is considered. As shownin Fig. 11, the sum-rate cdfs and the averaged user rate ofMAC-FDMA and MAC-GT algorithms with three, five, eight,and ten CR users per cell and a distance D = 2.2 km over 200independent channel realizations are displayed, respectively.The performance of the MAC-GT algorithm is much better thanthe MAC-FDMA algorithm, and once again, the performancegaps between MAC and FDMA for high user dimension areobserved to be larger than the case of small user dimension.

The corresponding convergence properties in Fig. 11 arepresented in Table V. The higher the user dimension is, themore interference between CR cells. Thus, more iteration timesare needed for large user dimension system, which can explainthe different convergence rates in Table V.

Finally, a higher user dimension system with more than tenusers per cell is considered. In this case, the GPM with thehard threshold ε = 10−3 prevents the MAC-GT algorithm fromefficiently operating, i.e., large iteration times are needed toachieve the final convergence. To reduce the iteration times andachieve a fast convergence rate, we enlarge the threshold to


Fig. 12. Sum-rate cdfs of MAC-GT algorithm with large number of CR users.

TABLE VIITERATION SITUATION FOR MAC-GT ALGORITHM IN

HIGH-DIMENSION SYSTEMS WITH ε = 1

ε = 1. By using this much more relaxed threshold, the sum-ratecdfs of MAC-GT algorithms with eight, 10, and 20 CR users percell and a distance D = 2.2 km over 200 independent channelrealizations are displayed in Fig. 12, and its correspondingconvergence properties are presented in Table VI. In contrastto Fig. 11 and Table V, it is interesting to find that this soft-threshold MAC-GT can achieve almost the same sum-rateperformance with faster convergence rate, which indicates thatthe whole system sum rate of our MAC-GT algorithm can reacha desired performance after only two or three iteration times.Herein, we have not given out more numerical results for highuser dimension system. A higher user dimension system withmore than 20 users per cell can be implemented in the sameway by changing the threshold ε.

V. CONCLUSION

In this paper, we have proposed a noncooperative MAC-GTalgorithm to perform uplink RA with the power constraint inthe context of multicell FBMC-based CR network. A convexoptimization problem is formulated by taking advantage ofthe MAC technique. We have derived the optimization so-lutions of the proposed algorithm using LA and GPM. TheMAC-GT algorithm for iterative multiuser power allocation isan extension of IWFA, which is applied for iterative single-userpower allocation. In addition, since the high implementationcost of the MAC scheme, we propose a heuristic MAC-FDMAtransformation algorithm to avoid costly implementation ofMAC, as well as to solve the exhaustive search problem ofthe traditional FDMA strategy. Final numerical results exhibitthat the distributed MAC-GT and MAC-FDMA algorithms

deliver robust convergent behavior and achieve superior sum-rate performance than the traditional FDMA-GT algorithm,particularly for a high-dimension network with a large numberof CR users per cell.

In our CR scenario using the FBMC technique, the proposednoncooperative RA algorithm and the experimental simulationare both based on the assumption that the interference fromCR cells to primary system is neglected by deactivating onesubcarrier in the two sides of each spectrum hole. This hypoth-esis is not far from a realistic CR network without involvinginterference issue. Nevertheless, there exists significant inter-ference among the CR cells if the OFDM technique is applied.As noted in [16], to avoid interference with the primary system,“eight” subcarriers in the spectrum holes should be deactivated,which substantially decreases the capacity of the CR system.In summary, FBMC highly relaxes the interference to primarysystem and thus can be considered as a potential candidate forfuture CR physical layer data communication. Our future workwill also focus on the study of a series of GT strategies to movefrom inefficient NE toward a Pareto-efficient solution.

APPENDIX

PROOF OF THE EXISTENCE OF THE NASH EQUILIBRIUM

In (3), the support domain Pn is a nonempty convex subsetof some Euclidean space due to the linear power constraints.Moreover, the utility function un in (3) is continuous in pn.Thus, we only need to prove that the utility function un is aquasiconcave function in pn.

First, we give the inequality of the weighted arithmetic meanand weighted geometric mean that will be helpful for our proof.

Inequality 1: Let the nonnegative numbers x1, x2, . . . , xn

and the nonnegative weights α1, α2, . . . , αn be given. Set α =α1 + α2 + · · · + αn. If α > 0, then the inequality

α1x1 + α2x2 + · · · + αnxn

α≥ α

√xα1

1 xα22 · · ·xαn

n (16)

holds with equality if and only if all the xk with αk > 0 areequal.

Thanks to the fact that the sum of concave functions is also aconcave function, we only need to prove the function

f = log2

[1 +

a1z1 + a2z2 + · · · + amzm

b

](17)

is a concave function, where a1, . . . , am, b are nonnegativenumbers, and z = (z1, z2, . . . , zm) is in a convex domain Cwith m elements. Defining θ ∈ [0, 1], x and y are two pointsin the set C. According to the definition of concave function,we have

f(θx + (1 − θ)y)

= log2

[1+

a1

b(θx1+(1 − θ)y1)+

a2

b(θx2 + (1 − θ)y2)

+ · · · + am

b(θxm + (1 − θ)ym)

]

= log2

[θ(

1 +a1

bx1 +

a2

bx2 + · · · + am

bxm

)

+ (1 − θ)(

1 +a1

by1 +

a2

by2 + · · · + am

bym

)]


θf(x) + (1 − θ)f(y)

= log2

[(1 +

a1

bx1 +

a2

bx2 + · · · + am

bxm

)θ]

+ log2

[(1+

a1

by1+

a2

by2+ · · ·+ am

bym

)1−θ]

. (18)

For simplicity, we set 1 + (a1/b)x1 + (a2/b)x2 + · · · + (am/b)xm = x′ and 1+(a1/b)y1+(a2/b)y2 + · · · + (am/b)ym =y′. Then

f (θx + (1 − θ)y) = log2 (θx′ + (1 − θ)y′)

θf(x) + (1 − θ)f(y) = log2

[(x′)θ(y′)1−θ

]. (19)

It is seen that x′ and y′ are nonnegative numbers, which isa special case of Inequality 1 when n = 2 and α = 1, α1 = θ,α2 = 1 − θ in (16). Because of the increasing property of thelog-function, f in (17) satisfies the definition of the concavefunction

f(θx′ + (1 − θ)y′) ≥ θf(x′) + (1 − θ)f(y′). (20)

Q.E.D.

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Haijian Zhang was born in 1983. He received theB.S. degree in electrical engineering from WuhanUniversity, Wuhan, China, in 2006 and the jointPh.D. degree from Conservatoire National des Artset Métiers, Paris, France, and Wuhan University,in 2010.

He is currently a Research Fellow with the Schoolof Electrical Electronic Engineering, Nanyang Tech-nological University, Singapore. His main researchinterests focus on signal detection, signal classifica-tion, dynamic spectrum allocation in the cognitive

radio domain, and time–frequency analysis.


Didier Le Ruyet (SM’11) received the Eng. andPh.D. degrees from the Conservatoire National desArts et Métiers (CNAM), Paris, France, in 1994 and2001, respectively, and the “Habilitation à dirigerdes recherches” degree from Paris XIII Universityin 2009.

From 1988 to 1996, he was a Senior Memberof the Technical Staff with SAGEM Defence andTelecommunication, France. In 1996, he was a Re-search Assistant with the Signal and Systems Lab-oratory, CNAM Paris. From 2002 to 2009, he was

an Assistant Professor with the Electronic and Communication Laboratory,CNAM Paris. Since 2010, he has been Full Professor with the CEDRICResearch Laboratory, CNAM. He has published about 70 papers in referredjournals and conferences. His main research interests lie in the areas of digitalcommunications and signal processing, including channel coding, detection andestimation algorithms, multiantenna transmission, and relaying techniques formultiuser systems.

Daniel Roviras (SM’11) was born in 1958. He re-ceived the Engineer degree from SUPELEC, Metz,France, in 1981 and the Ph.D. degree from the Na-tional Polytechnic Institute of Toulouse, Toulouse,France, in 1989.

After 7 years spent in the industry as a ResearchEngineer, he joined the Electronics Laboratory,École Nationale Supérieure d’Électrotechnique,d’Électronique, d’Informatique, et des Télécommu-nications (ENSEEIHT). He joined the EngineeringSchool ENSEEIHT in 1992 as an Assistant Professor

and has been a Full Professor since 1999. Since 2008, he has been Professorwith Conservatoire National des Arts et Métiers, Paris, France, where histeaching activities are related to radio-communication systems. He is currentlya member of CEDRIC Laboratory, CNAM. His research activity was firstcentered around transmission systems based on infrared links. Since 1992, histopics have widened to more general communication systems such as mobileand satellite communications systems, equalization and predistorsion of nonlin-ear amplifiers using neural networks, multiple-access methods (code-divisionmultiple-access, Linear Periodic Time Varying, Periodic Clock Changes, filter-bank-based multicarrier), and cooperative/cognitive radio systems.

Hong Sun (M’11) received the B.S., M.S., and Ph.D.degrees in communications and electrical systemsfrom Huazhong University of Science and Technol-ogy (HUST), Wuhan, China, in 1981, 1984, and1995, respectively.

From 1984 to 2000, she was a Professor withHUST. She was a Visiting Scholar with the Conser-vatoire National des Arts et Métiers, Paris, France,in 1997 and a Visiting Professor with the Ecole Na-tionale Supérieure des Télécommunications, Paris, in1998, 1999, 2000, 2001, and 2007. Since 2001, she

has been with the School of Electronic Information, Wuhan University, Wuhan,where she is currently a Professor and the Head of the Signal ProcessingLaboratory. Her research includes digital signal processing theory and itsapplications, including works on image interpretation, communication signalprocessing, and speech signal processing.

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Noncooperative Multicell Resource Allocation of FBMC-Based Cognitive Radio Systems