Research ArticleFairness-Aware and Energy Efficiency Resource Allocation inMultiuser OFDM Relaying System

Guangjun Liang Qi Zhu An Yan Ziyu Pan Jianfang Xin and Tianjiao Zhang

Department of Telecommunication and Information Engineering Jiangsu Key Lab of Wireless CommunicationsKey Lab on Wideband Wireless Communications and Sensor Network Technology of Ministry of Education Nanjing University ofPosts and Telecommunications Nanjing Jiangsu 210003 China

Correspondence should be addressed to Qi Zhu zhuqinjupteducn

Received 11 September 2015 Revised 30 March 2016 Accepted 21 April 2016

Academic Editor Lin Gao

Copyright copy 2016 Guangjun Liang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A fairness-aware resource allocation scheme in a cooperative orthogonal frequency divisionmultiple (OFDM) network is proposedbased on jointly optimizing the subcarrier pairing power allocation and channel-user assignment Compared with traditionalOFDM relaying networks the source is permitted to retransfer the same data transmitted by it in the first time slot furtherimproving the system capacity performance The problem which maximizes the energy efficiency (EE) of the system with totalpower constraint and minimal spectral efficiency constraint is formulated into a mixed-integer nonlinear programming (MINLP)problemwhich has an intractable complexity in generalThe optimizationmodel is simplified into a typical fractional programmingproblem which is testified to be quasiconcave Thus we can adopt Dinkelbach method to deal with MINLP problem proposed toachieve the optimal solutionThe simulation results show that the joint resource allocationmethod proposed can achieve an optimalEE performance under the minimum system service rate requirement with a good global convergence

1 Introduction

Recently orthogonal frequency division multiple (OFDM)cooperative communication systems have been widely usedto overcome the limitations of the usersrsquo space constraintsbecause the relays can adopt the frequency diversity tech-nique to deal with channel fading In multiple frequencychannels subcarrier pairing was first devised independentlyfor a single-user communication environment in [1 2] bymatching the subcarriers in OFDM relaying networks Fur-ther in multiuser relaying networks [3 4] both the relay andusers are sharing all the channel Owing to the complexitiesand drastic variations in the channel conditions for differentusers appropriate channel-user assignments can potentiallyenable significant improvements in the spectral efficiency(SE) Recently energy efficiency (EE) [5ndash9] has emergedas one of the most promising solutions to resolve issuessuch as the rapidly increasing energy consumption andthe carbon emissions caused by the escalating growth ofwireless data traffic in next-generation networksThe authorsin [10ndash13] have conducted a preliminary research on EE

multiuser relaying resource allocation (RA) In such OFDMrelaying networks an optimal system performance shouldtake into account three problems jointly the subcarrierpairing channel-user assignment (the assignment of sub-carrier pairs to users) and the power allocation that havea strong correlation among them However the combina-torial solution of these three problems entails the mixed-integer linear programming problem (MILP)which generallyhas an excessive computational complexity Besides if wefurther consider the energy efficiency resource allocationproblem the optimization problem will be a mixed-integernonlinear programming (MINLP) problem that in generalis computationally undesirable because of its combinatorialnature which could adopt branch-and-bound method tosolve [14] Previous attempts to optimize the performanceof the multiple users OFDM relaying networks have usuallyconsidered only a subset of three problems regarding the SE[15ndash22] or a subset of three problems regarding the EE [5ndash12]or have adopted a suboptimal approach [13 23]

Based on single-user amplify-and-forward (AF) OFDMrelaying networks [15] shows that the subcarrier pairing

Hindawi Publishing CorporationMobile Information SystemsVolume 2016 Article ID 7364154 11 pageshttpdxdoiorg10115520167364154

2 Mobile Information Systems

method according to the instantaneous channel gain thatmatches the incoming and outgoing subcarriers is sum-rate optimal The authors in [16 17] propose a joint powerallocation and subcarrier pairing scheme in the same scenar-ios with [15] for the AF and the decode-and-forward (DF)networks respectively However the works mentioned aboveonly address the maximization of the end-to-end SE withoutjointly considering the direct passing path For theAF and theDF respectively [18 19] consider power allocation and sub-carrier pairing scheme jointly which could be obtained by theLagrange means in the single-user OFDM relaying networkwhen the passing path directly is available An equal powerallocation policy in [23] exploits an optimal subcarrier-userallocation method which maximizes the system throughputFor DF relaying networks [21] provides an asymptoticallyoptimal scheme to jointly allocate power and assign subcar-rier userThe same inDF relaying networks [20] considers allthe three problems with the total power constraint Howeverthis approach is suboptimal because of separately applyingthe power allocation subcarrier pairing and subcarrier-user allocation The authors in [22] address a maximizedthroughput problem by joint considering power allocationsubcarrier pairing and subcarrier pair-user allocation Basedon usual relaying strategies the optimal resource allocationproblem can be obtained by Lagrange means

For a downlink OFDM system [5] presents a bandwidthallocation scheme based on energy efficiency tomaximize thenumber of bits per Joule energy consumption Reference [6]offers a global optimal energy efficiency scheme to solve theoptimal energy efficiency of the system that decomposes thejoint optimization problem into three subproblemsHowever[5 6] do not consider the EE of the OFDM relaying net-works which are traditional OFDM networks without relaysConsidering a sum-rate constraint for a cooperative OFDMDF relaying network [7] develops a sum-power minimizedRA algorithm without the direct source-destination linkIn [8] based on AF relaying networks with a frequencyselective channel a suboptimal two-step power allocationand subcarrier pairing strategy is raised to maximize theenergy efficiency when the direct source-destination linkis unavailable In a downlink cooperative multiuser OFDMrelaying network the authors in [9] present a joint subcarrierpairing and power allocation scheme to maximize the EE ofthe system with the proportional fairness

In [10ndash13] multiple users relaying networks are consid-ered Considering only the power allocation without consid-ering the subcarrier pairing and the channel-user assignment[10] presents a multiple users AF relaying model Reference[11] formulates energy efficiency optimization problem forpower allocation with fairness in cooperative multiple usersfading channels In [12 13] maximizing the EE problem isproposed which is formulated as the ratio of the SE overthe total power dissipation and joint power and subcarrierallocation in a multiple users OFDM relaying network In[12] the optimization problem is solved under a constraintof providing the minimum required spectral efficiency in theOFDM relaying networks

In [13] the objective function (OF) is proven to bequasiconcave and can adopt Dinkelbach method to obtain

the optimal solution by solving a sequence of subtractiveconcave problems using the dual decomposition approachHowever [12 13] jointly consider only power allocationand subcarrier pairing without considering channel-userassignment

In this work we focus on the design of a fairness-awareresource allocation scheme in a cooperative OFDM networkCompared with traditional OFDM relaying networks thesource is permitted to retransfer the same data transmitted byit in the first time slot further improving the system capacityperformance The problem which maximizes the energyefficiency (EE) of the system with total power constraintand minimal spectral efficiency constraint is formulated intoa mixed-integer nonlinear programming (MINLP) prob-lem which has an intractable complexity in general Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

The main contributions of the work are summarized asfollows

(i) We propose a new energy-efficient maximizingmethod in amultiple users relaying systemwith directpassing path and total power constraints which jointlyoptimizes the subcarrier pairing the power alloca-tion and the channel-user assignment We can alsoadopt Lagrangian method and continuity relaxationto solve the joint optimal problem But the three-dimensional assignment problem is its key subprob-lem which is nondeterministic polynomial time (NP-hard) but has time complexity

(ii) Comparedwith traditional OFDM relaying networksthe source is permitted to retransfer the same datatransmitted by it in the first time slot further improv-ing the system capacity performance Because of boththe new relay cooperation protocol and the subcarrierpairs-user three-dimensional assignment algorithmthe proposed joint resource allocation method couldachieve optimal EE performance under a minimumsystem data rate requirement with a good globalconvergence which can be proved by theoreticalderivation and simulation analysis

(iii) The optimization model is simplified into a typicalfractional programming problem which is testifiedto be quasiconcave Thus we can adopt Dinkelbachmethod to deal with MINLP problem proposed toachieve the optimal solution

The remained of this paper is organized as follows InSection 2 the system model and the resource allocationproblem are described In Section 3 the problem is formu-lated and reformulated by the convex optimization techniqueIn Section 4 a resource allocation iteration algorithm willbe adopted and solved by a dual decomposition method

Mobile Information Systems 3

Source

Relay

User 1

User 2

User k

User K

Figure 1 Model of the concerned single relay multiuser OFDMnetwork

In Section 5 both the analysis and the simulation resultsare presented to compare the performances of the differentresource allocation schemes Section 6 gives a conclusion forthis paper

2 System Model

In this section we first introduce the adopted system modeland the performance measure Then the design of theresource allocation and scheduling is formulated as an opti-mization problem

21 System Model Considering an OFDM single relay mul-tiusers system as in Figure 1 source transfers the informationto all the users with the help of one relay In this paper wefocus on the joint resource allocation problem in a scenarioincluding one source one relay and 119870 users 119873 equal-bandwidth orthogonal channels haves been divided from theavailable frequency spectrum accessible by the relay andall the users Without loss of generality we assume that allthe channels occupy the same bandwidth and experienceindependent frequency selective fading It is also assumedthat all pieces of the channel state information (CSI) areavailable at the source the relay and the users [18] Thehalf-duplex transmission process can be divided into twophases The source 119878 transmits the information data to allthe other nodes including the relay and all the users in thefirst phase The relay decodes the received signals in the firstphase and retransmits them to all the users in the secondphase Compared with traditional OFDM relaying networks[22] the source is permitted to retransfer the same datatransmitted by it in the first time slot further improving thesystem capacity performance [24]

From the discussion above the state of channel 119894 candenote that for 1 le 119894 le 119873 ℎ119904119903

119894and ℎ1199041198961

119894represent the

instantaneous channel gain from source to relay and fromthe source to the user 119896 in the first phase and ℎ119903119896

119894and ℎ1199041198962

119894

signify the instantaneous channel gain from the relay to theuser 119896 and from the source to the user 119896 in the second phaseThe AWGN (Additive Gaussian White Noise) at the 119896th

user and the relay are zero mean with variances 1205902119903and 1205902

119896

respectivelySince there are multiple users for the sake of maximizing

the system throughput when allocating the resources wemust also take the user fairness into account [9 11] Thefairness factor 120596

119896for user 119896 is expressed by

119870

sum

119896=1

119908119896= 1 (1)

It should be mentioned that the considered scenario isapplicable in practice where the source the relay and theusers may act as base station the relay station and thecellular users respectively Further this system model is anexample of a wireless sensor network transmission modelwhere the source the relay and the users are all consideredas wireless sensor nodes In our system model adopting onerelay instead of multiple relays can significantly reduce theenergy consumption and the system overhead in accordancewith green communication and environmental protection

22 Channel Assignment The relay not only forwards theincoming signals to their intended users by a special relayprocess strategy but also conducts the subcarrier pairing andthe channel-user assignments Obviously the solutions of thesubcarrier pairing are closely related to the strategy of thechannels assignments to all the users Then we refer to thejoint assignment on subcarrier pairing and subcarrier pair-user allocation as the joint channel assignment problem

Suppose that the path119875(119898 119899 119896) is selected if the first-hopchannel119898 is paired with a second-hop channel 119899 and the pairof channels (119898 119899) is assigned to user 119896 Further we define theindicator functions given by [22]

120601119898119899119896

=

1 if 119875 (119898 119899 119896) is selected

0 otherwise(2)

A user can be assigned many subcarrier pairs but asubcarrier pair must be allocated to a user exclusively Hence

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896

= 1 forall119898119898 isin 1 2 119873

119873

sum

119898=1

119870

sum

119896=1

120601119898119899119896

= 1 forall119899 119899 isin 1 2 119873

(3)

23 Relaying Strategy and Power Allocation In the traditionalDF relaying networks each transmission cycle is dividedinto two time slots equally In the first time slot the sourcebroadcasts a data block in every channel to both user andrelay nodes Then in the second time slot the relay startsto decode and forward the data from source to the userMeanwhile if the channel strength of the source is sufficientin the second hop the source will be assigned power tosend the data again and then obtains a diversity gain Theuser adopts the MRC in both two time slots on the receivedsignals

4 Mobile Information Systems

Along any path119875(119898 119899 119896) the source in the first time slotthe source in the second time slot and the relay transmissionpowers are denoted by 1198751199041

119898119899119896 1198751199042119898119899119896

and 119875119903119898119899119896

respectivelyThen the total power allocated to path 119875(119898 119899 119896) is 119875

119898119899119896=

1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896 The total power 119875total constraint can be

given by

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896(1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896) le 119875total (4)

Based on DF relaying networks the maximum source-destination achievable rate on path 119875(119898 119899 119896) is given by[22]

119877 (119898 119899 119896) =1

2min log

2(1 + 119886

1198981198751199041

119898119899119896)

log2(1 + 119887

119899119896119875119903

119898119899119896+ 1198881198981198961198751199041

119898119899119896+ 1198861198991198961198751199042

119898119899119896)

(5)

where 119886119898= |ℎ119904119903

119898|2

1205902

119903 119887119899119896= |ℎ119903119896

119899|2

1205902

119896 119888119898119896= |ℎ1199041198961

119898|2

1205902

119896

119886119899119896= |ℎ1199041198962

119899|2

1205902

119896 and 12 indicates that one data transmission

is needed in two time slots

3 Problem Formation andConvex Reformulation

In this section we formulate the corresponding source allo-cation problem and reformulate the optimization problem

31 Problem Formulation Given a fixed total power 119875119898119899119896

ofpath 119875(119898 119899 119896) (5) can be converted to [24]

119877 (119898 119899 119896) =1

2log2(1 + 119886

119898119899119896119875119898119899119896) (6)

119886119898119899119896

denotes the equivalent channel gain on the path 119875(119898119899 119896) expressed as

119886119898119899119896=

119886119898119886119899119896

119886119898+ 119886119899119896minus 119888119898119896

when 119887119899119896le 119886119899119896

119886119898119887119899119896

119886119898+ 119887119899119896minus 119888119898119896

when 119887119899119896gt 119886119899119896

(7)

According to the fairness factor 120596119896for every user 119896

the total end-to-end SE and the total transmit power on allthe subcarriers by the source and the relay of the system canbe expressed by

119877total (ΦP) =1

2

119870

sum

119896=1

120596119896

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896

log2(1 + 119886

119898119899119896119875119898119899119896)

119875users (ΦP) =119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896119875119898119899119896

(8)

where P = [119875119898119899119896]119873times119873times119870

Φ = [120601119898119899119896]119873times119873times119870

and 119875119898119899119896

and120601119898119899119896

satisfy the constraints mentioned previouslyThe total power consumed can therefore be expressed as

[13]

119875total (ΦP) = 119875119862 + 120589119875users (ΦP) (9)

where 119875119862is the constant circuit power consumption which

includes the power dissipations in the transmit filter themixer the frequency synthesizer and the digital-to-analogconverter that are independent of the actual transmittedpower [12] 120589 gt 1 denotes the reciprocal of the drain effi-ciencies of the power amplifiers employed at the sourceand the relay For example an amplifier with a 50 drainefficiency has 120589 = 105 = 2 119877req specifies the minimum SErequirement as a QoS constraint and satisfies that 119877total(ΦP) ge 119877req

Finally the optimization problem to maximize the aver-age system EE is formulated into

maxΦP

119902 (ΦP) =119877total (ΦP)119875total (ΦP)

st (3) and (4)

120601119898119899119896= 0 1

119875119898119899119896ge 0

forall119898 119899 119896

119877total (ΦP) ge 119877req

(10)

32 Continuous Relaxation and Convex Reformulation Theoptimization problem addressed in (10) is MINLP problemthat in general is computationally undesirable because of itscombinatorial nature which could adopt branch-and-boundmethod to solve [14] However in the following section wewill raise a method to find an optimal strategy that growspolynomial with the numbers of subcarriers available in thesystem

First ignoring the physical meaning of formula (2) werelax the constraints by allowing 120601

119898119899119896to take any value in the

interval [0 1] where

120601119898119899119896isin [0 1] forall119898 119899 119896 (11)

Hence we introduce a new variable

119878119898119899119896= 120601119898119899119896119875119898119899119896 (12)

Second the relaxed optimization problem of (10) accord-ing to Φ = [120601

119898119899119896]119873times119873times119870

and S = [119878119898119899119896]119873times119873times119870

can be refor-mulated as

Mobile Information Systems 5

maxΦS

119902 (Φ S) =(12)sum

119870

119896=1sum119873

119898=1sum119873

119899=1120596119896120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896))

119875119862+ 120589sum119873

119898=1sum119873

119899=1sum119870

119896=1119878119898119899119896

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(13)

Proposition 1 The channel-user assignments and the powerallocation matrices (Φlowast Slowast) found in the previous subsectionare globally optimal solutions to the original problem of (10)

Proof Applying the Dinkelbach approach in [25] the objec-tive functions of (10) and (13) are given as

119865 (119902) = maxΦP[119877total (ΦP) minus 119902119875total (ΦP)]

119865 (119902)

= maxΦS[1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896]

(14)

The optimization problem of (10) is not concave since 120601119898119899119896=

0 1 The optimization problem of (13) is the slack form ofoptimization problem (10) by allowing 120601

119898119899119896to take any value

in the interval [0 1]The objective function (13) is concave in (Φ S) since

120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896)) are perspectives of the con-

cave function log2(1 + 119886

119898119899119896119878119898119899119896) Further since there are

feasible points and all constraints are affine obviously Slaterrsquoscondition can be satisfied [17] Hence we can suggest thata globally optimal solution can be got in the Lagrange dualdomain because convex optimization problem (13) has a zeroduality gap Therefore relaxation problem (13) and originalproblem (10) have the same optimal solution

Note that adopting traditional convex optimization soft-ware packages cannot get the optimal solution in (10) becauseΦlowast is not ensured to be a binary In next section we will

present a three-dimensional joint subcarrier pairing andchannel-user assignment scheme that can provide an optimalsolution with the polynomial time

4 Joint Resource Allocation Scheme forMultiuser OFDM Relaying System

Since solving the nonconvex problem in (13) is not standardapproach in general in this section we transform the opti-mization problem with the Dinkelbach method and handlean iterative algorithm to obtain it

41 Objective Function Transformation Without loss of gen-erality we denote the optimal EE and the set of feasiblesolutions for the optimization proposed in (13) as 119902lowast and Tthen we have

119902lowast

= maxΦS

119877total (Φ S)119875total (Φ S)

=119877total (Φ

lowast

Slowast)119875total (Φ

lowast

Slowast)

forall Φ S isin T(15)

We define

119865 (119902) = maxΦS[119877total (Φ S) minus 119902119875total (Φ S)] (16)

The optimal energy efficiency of the optimization prob-lem proposed in (15) 119902lowast is achieved if and only if

119865 (119902lowast

) = maxΦS[119877total (Φ S) minus 119902

lowast

119875total (Φ S)]

= 119877total (Φlowast

Slowast) minus 119902lowast119875total (Φlowast

Slowast) = 0

forall Φ S isin T

(17)

We can prove (17) by following the Dinkelbach approachas in [25] In other words the optimal resource allocationpolicies Φlowast Slowast for the equivalent objective function arealso the optimal resource allocation policies for the originalobjective function

In summary the optimization of the original objectivefunction and the optimization of the equivalent objectivefunction result in the same resource allocation policiesEquation (17) reveals that for an optimization problem with

6 Mobile Information Systems

(1) Initial maximum number of iterations 119871max and maximum tolerance 120576(2) Set maximum energy efficiency 119902 = 0 and iteration index 119899 = 0(3) repeat Main Loop(4) Solve the inner loop problem in (19) for a given 119902 and channel

assignment and power allocation policy (Φ1015840 S1015840)(5) if 119877total(Φ

1015840

S1015840) minus 119902119875total(Φ1015840 S1015840) lt 120576 then(6) Convergence = true(7) return (Φlowast Slowast) = (Φ1015840 S1015840) and 119902lowast = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840)(8) else(9) Set 119902 = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840) and 119899 = 119899 + 1(10) Convergence = false(11) end if(12) until Convergence = true or 119899 = 119871max

Algorithm 1 Resource allocation iteration algorithm for EE maximization

an objective function in the fractional form there existsan equivalent objective function in a subtractive form forexample 119877total(Φ

lowast

Slowast) minus 119902lowast119875total(Φlowast Slowast) in the consideredcase As a result we can focus on the equivalent objectivefunction further in this paper

42 Resource Allocation Iteration Algorithm for EE Maxi-mization The iterative algorithm for EE maximization isdescribed in Algorithm 1

As shown in Algorithm 1 for each iteration in the mainloop that is lines (3)ndash(12) we solve the following optimiza-tion problem for a given parameter 119902 that can be expressedas

maxΦS119865 (119902) = [119877total (Φ S) minus 119902119875total (Φ S)] (18)

In combination with formula (13) (18) can be furtherconverted to

maxΦS

119865 (119902)

=1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902(119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(19)

According to (19) and setting 119902 = 0 the optimizationproblem by solving 119865(0) turns out to be an SE-oriented

resource allocation scheme 119877total(Φ S) is concave and 119878119898119899119896is affine The optimization problem of OP(119902) is concave withrespect to (120601

119898119899119896 119878119898119899119896) We assume that 119877req is achievable

under the constraints of (19) that indicates the existence ofinterior points Then Slaterrsquos condition is satisfied [26]

Hence it can be shown that the strong duality holds andthe optimization problem (19) has a zero duality In otherwords solving the dual problem is equivalent to solving theprimal problem and a globally optimal solution can be foundin the Lagrange dual domain Asmentioned in Section 32 byrelaxing the constraints and introducing a continuous 120601

119898119899119896isin

[0 1] solving the dual problem will always provide an upperbound for the original optimization problem

Specifically the optimal solution for the dual problemdoes not always result in a binary 120601

119898119899119896isin 0 1 that is

desired as a necessary constraint in the original optimizationproblem However we will prove that a globally optimalsolution always exists for the dual problem with a binary120601119898119899119896

isin 0 1 making the solution obtained available andoptimal for the optimization problem in (10)

43 Dual Problem Formulation and Decomposition SolutionThe Lagrangian of (19) is given by

119871 (Φ S 120582 120583 119902) = 12

sdot

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902(119875119862

+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896) minus 120582(

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896minus 119875119905)

minus 120583(119877req

minus1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

))

(20)

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Computer Networks and Communications

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httpwwwhindawicom Volume 2014

Advances in

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ArtificialNeural Systems

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Mobile Information Systems

method according to the instantaneous channel gain thatmatches the incoming and outgoing subcarriers is sum-rate optimal The authors in [16 17] propose a joint powerallocation and subcarrier pairing scheme in the same scenar-ios with [15] for the AF and the decode-and-forward (DF)networks respectively However the works mentioned aboveonly address the maximization of the end-to-end SE withoutjointly considering the direct passing path For theAF and theDF respectively [18 19] consider power allocation and sub-carrier pairing scheme jointly which could be obtained by theLagrange means in the single-user OFDM relaying networkwhen the passing path directly is available An equal powerallocation policy in [23] exploits an optimal subcarrier-userallocation method which maximizes the system throughputFor DF relaying networks [21] provides an asymptoticallyoptimal scheme to jointly allocate power and assign subcar-rier userThe same inDF relaying networks [20] considers allthe three problems with the total power constraint Howeverthis approach is suboptimal because of separately applyingthe power allocation subcarrier pairing and subcarrier-user allocation The authors in [22] address a maximizedthroughput problem by joint considering power allocationsubcarrier pairing and subcarrier pair-user allocation Basedon usual relaying strategies the optimal resource allocationproblem can be obtained by Lagrange means

For a downlink OFDM system [5] presents a bandwidthallocation scheme based on energy efficiency tomaximize thenumber of bits per Joule energy consumption Reference [6]offers a global optimal energy efficiency scheme to solve theoptimal energy efficiency of the system that decomposes thejoint optimization problem into three subproblemsHowever[5 6] do not consider the EE of the OFDM relaying net-works which are traditional OFDM networks without relaysConsidering a sum-rate constraint for a cooperative OFDMDF relaying network [7] develops a sum-power minimizedRA algorithm without the direct source-destination linkIn [8] based on AF relaying networks with a frequencyselective channel a suboptimal two-step power allocationand subcarrier pairing strategy is raised to maximize theenergy efficiency when the direct source-destination linkis unavailable In a downlink cooperative multiuser OFDMrelaying network the authors in [9] present a joint subcarrierpairing and power allocation scheme to maximize the EE ofthe system with the proportional fairness

In [10ndash13] multiple users relaying networks are consid-ered Considering only the power allocation without consid-ering the subcarrier pairing and the channel-user assignment[10] presents a multiple users AF relaying model Reference[11] formulates energy efficiency optimization problem forpower allocation with fairness in cooperative multiple usersfading channels In [12 13] maximizing the EE problem isproposed which is formulated as the ratio of the SE overthe total power dissipation and joint power and subcarrierallocation in a multiple users OFDM relaying network In[12] the optimization problem is solved under a constraintof providing the minimum required spectral efficiency in theOFDM relaying networks

In [13] the objective function (OF) is proven to bequasiconcave and can adopt Dinkelbach method to obtain

the optimal solution by solving a sequence of subtractiveconcave problems using the dual decomposition approachHowever [12 13] jointly consider only power allocationand subcarrier pairing without considering channel-userassignment

In this work we focus on the design of a fairness-awareresource allocation scheme in a cooperative OFDM networkCompared with traditional OFDM relaying networks thesource is permitted to retransfer the same data transmitted byit in the first time slot further improving the system capacityperformance The problem which maximizes the energyefficiency (EE) of the system with total power constraintand minimal spectral efficiency constraint is formulated intoa mixed-integer nonlinear programming (MINLP) prob-lem which has an intractable complexity in general Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

The main contributions of the work are summarized asfollows

(i) We propose a new energy-efficient maximizingmethod in amultiple users relaying systemwith directpassing path and total power constraints which jointlyoptimizes the subcarrier pairing the power alloca-tion and the channel-user assignment We can alsoadopt Lagrangian method and continuity relaxationto solve the joint optimal problem But the three-dimensional assignment problem is its key subprob-lem which is nondeterministic polynomial time (NP-hard) but has time complexity

(ii) Comparedwith traditional OFDM relaying networksthe source is permitted to retransfer the same datatransmitted by it in the first time slot further improv-ing the system capacity performance Because of boththe new relay cooperation protocol and the subcarrierpairs-user three-dimensional assignment algorithmthe proposed joint resource allocation method couldachieve optimal EE performance under a minimumsystem data rate requirement with a good globalconvergence which can be proved by theoreticalderivation and simulation analysis

(iii) The optimization model is simplified into a typicalfractional programming problem which is testifiedto be quasiconcave Thus we can adopt Dinkelbachmethod to deal with MINLP problem proposed toachieve the optimal solution

The remained of this paper is organized as follows InSection 2 the system model and the resource allocationproblem are described In Section 3 the problem is formu-lated and reformulated by the convex optimization techniqueIn Section 4 a resource allocation iteration algorithm willbe adopted and solved by a dual decomposition method

Mobile Information Systems 3

Source

Relay

User 1

User 2

User k

User K

Figure 1 Model of the concerned single relay multiuser OFDMnetwork

In Section 5 both the analysis and the simulation resultsare presented to compare the performances of the differentresource allocation schemes Section 6 gives a conclusion forthis paper

2 System Model

In this section we first introduce the adopted system modeland the performance measure Then the design of theresource allocation and scheduling is formulated as an opti-mization problem

21 System Model Considering an OFDM single relay mul-tiusers system as in Figure 1 source transfers the informationto all the users with the help of one relay In this paper wefocus on the joint resource allocation problem in a scenarioincluding one source one relay and 119870 users 119873 equal-bandwidth orthogonal channels haves been divided from theavailable frequency spectrum accessible by the relay andall the users Without loss of generality we assume that allthe channels occupy the same bandwidth and experienceindependent frequency selective fading It is also assumedthat all pieces of the channel state information (CSI) areavailable at the source the relay and the users [18] Thehalf-duplex transmission process can be divided into twophases The source 119878 transmits the information data to allthe other nodes including the relay and all the users in thefirst phase The relay decodes the received signals in the firstphase and retransmits them to all the users in the secondphase Compared with traditional OFDM relaying networks[22] the source is permitted to retransfer the same datatransmitted by it in the first time slot further improving thesystem capacity performance [24]

From the discussion above the state of channel 119894 candenote that for 1 le 119894 le 119873 ℎ119904119903

119894and ℎ1199041198961

119894represent the

instantaneous channel gain from source to relay and fromthe source to the user 119896 in the first phase and ℎ119903119896

119894and ℎ1199041198962

119894

signify the instantaneous channel gain from the relay to theuser 119896 and from the source to the user 119896 in the second phaseThe AWGN (Additive Gaussian White Noise) at the 119896th

user and the relay are zero mean with variances 1205902119903and 1205902

119896

respectivelySince there are multiple users for the sake of maximizing

the system throughput when allocating the resources wemust also take the user fairness into account [9 11] Thefairness factor 120596

119896for user 119896 is expressed by

119870

sum

119896=1

119908119896= 1 (1)

It should be mentioned that the considered scenario isapplicable in practice where the source the relay and theusers may act as base station the relay station and thecellular users respectively Further this system model is anexample of a wireless sensor network transmission modelwhere the source the relay and the users are all consideredas wireless sensor nodes In our system model adopting onerelay instead of multiple relays can significantly reduce theenergy consumption and the system overhead in accordancewith green communication and environmental protection

22 Channel Assignment The relay not only forwards theincoming signals to their intended users by a special relayprocess strategy but also conducts the subcarrier pairing andthe channel-user assignments Obviously the solutions of thesubcarrier pairing are closely related to the strategy of thechannels assignments to all the users Then we refer to thejoint assignment on subcarrier pairing and subcarrier pair-user allocation as the joint channel assignment problem

Suppose that the path119875(119898 119899 119896) is selected if the first-hopchannel119898 is paired with a second-hop channel 119899 and the pairof channels (119898 119899) is assigned to user 119896 Further we define theindicator functions given by [22]

120601119898119899119896

=

1 if 119875 (119898 119899 119896) is selected

0 otherwise(2)

A user can be assigned many subcarrier pairs but asubcarrier pair must be allocated to a user exclusively Hence

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896

= 1 forall119898119898 isin 1 2 119873

119873

sum

119898=1

119870

sum

119896=1

120601119898119899119896

= 1 forall119899 119899 isin 1 2 119873

(3)

23 Relaying Strategy and Power Allocation In the traditionalDF relaying networks each transmission cycle is dividedinto two time slots equally In the first time slot the sourcebroadcasts a data block in every channel to both user andrelay nodes Then in the second time slot the relay startsto decode and forward the data from source to the userMeanwhile if the channel strength of the source is sufficientin the second hop the source will be assigned power tosend the data again and then obtains a diversity gain Theuser adopts the MRC in both two time slots on the receivedsignals

4 Mobile Information Systems

Along any path119875(119898 119899 119896) the source in the first time slotthe source in the second time slot and the relay transmissionpowers are denoted by 1198751199041

119898119899119896 1198751199042119898119899119896

and 119875119903119898119899119896

respectivelyThen the total power allocated to path 119875(119898 119899 119896) is 119875

119898119899119896=

1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896 The total power 119875total constraint can be

given by

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896(1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896) le 119875total (4)

Based on DF relaying networks the maximum source-destination achievable rate on path 119875(119898 119899 119896) is given by[22]

119877 (119898 119899 119896) =1

2min log

2(1 + 119886

1198981198751199041

119898119899119896)

log2(1 + 119887

119899119896119875119903

119898119899119896+ 1198881198981198961198751199041

119898119899119896+ 1198861198991198961198751199042

119898119899119896)

(5)

where 119886119898= |ℎ119904119903

119898|2

1205902

119903 119887119899119896= |ℎ119903119896

119899|2

1205902

119896 119888119898119896= |ℎ1199041198961

119898|2

1205902

119896

119886119899119896= |ℎ1199041198962

119899|2

1205902

119896 and 12 indicates that one data transmission

is needed in two time slots

3 Problem Formation andConvex Reformulation

In this section we formulate the corresponding source allo-cation problem and reformulate the optimization problem

31 Problem Formulation Given a fixed total power 119875119898119899119896

ofpath 119875(119898 119899 119896) (5) can be converted to [24]

119877 (119898 119899 119896) =1

2log2(1 + 119886

119898119899119896119875119898119899119896) (6)

119886119898119899119896

denotes the equivalent channel gain on the path 119875(119898119899 119896) expressed as

119886119898119899119896=

119886119898119886119899119896

119886119898+ 119886119899119896minus 119888119898119896

when 119887119899119896le 119886119899119896

119886119898119887119899119896

119886119898+ 119887119899119896minus 119888119898119896

when 119887119899119896gt 119886119899119896

(7)

According to the fairness factor 120596119896for every user 119896

the total end-to-end SE and the total transmit power on allthe subcarriers by the source and the relay of the system canbe expressed by

119877total (ΦP) =1

2

119870

sum

119896=1

120596119896

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896

log2(1 + 119886

119898119899119896119875119898119899119896)

119875users (ΦP) =119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896119875119898119899119896

(8)

where P = [119875119898119899119896]119873times119873times119870

Φ = [120601119898119899119896]119873times119873times119870

and 119875119898119899119896

and120601119898119899119896

satisfy the constraints mentioned previouslyThe total power consumed can therefore be expressed as

[13]

119875total (ΦP) = 119875119862 + 120589119875users (ΦP) (9)

where 119875119862is the constant circuit power consumption which

includes the power dissipations in the transmit filter themixer the frequency synthesizer and the digital-to-analogconverter that are independent of the actual transmittedpower [12] 120589 gt 1 denotes the reciprocal of the drain effi-ciencies of the power amplifiers employed at the sourceand the relay For example an amplifier with a 50 drainefficiency has 120589 = 105 = 2 119877req specifies the minimum SErequirement as a QoS constraint and satisfies that 119877total(ΦP) ge 119877req

Finally the optimization problem to maximize the aver-age system EE is formulated into

maxΦP

119902 (ΦP) =119877total (ΦP)119875total (ΦP)

st (3) and (4)

120601119898119899119896= 0 1

119875119898119899119896ge 0

forall119898 119899 119896

119877total (ΦP) ge 119877req

(10)

32 Continuous Relaxation and Convex Reformulation Theoptimization problem addressed in (10) is MINLP problemthat in general is computationally undesirable because of itscombinatorial nature which could adopt branch-and-boundmethod to solve [14] However in the following section wewill raise a method to find an optimal strategy that growspolynomial with the numbers of subcarriers available in thesystem

First ignoring the physical meaning of formula (2) werelax the constraints by allowing 120601

119898119899119896to take any value in the

interval [0 1] where

120601119898119899119896isin [0 1] forall119898 119899 119896 (11)

Hence we introduce a new variable

119878119898119899119896= 120601119898119899119896119875119898119899119896 (12)

Second the relaxed optimization problem of (10) accord-ing to Φ = [120601

119898119899119896]119873times119873times119870

and S = [119878119898119899119896]119873times119873times119870

can be refor-mulated as

Mobile Information Systems 5

maxΦS

119902 (Φ S) =(12)sum

119870

119896=1sum119873

119898=1sum119873

119899=1120596119896120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896))

119875119862+ 120589sum119873

119898=1sum119873

119899=1sum119870

119896=1119878119898119899119896

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(13)

Proposition 1 The channel-user assignments and the powerallocation matrices (Φlowast Slowast) found in the previous subsectionare globally optimal solutions to the original problem of (10)

Proof Applying the Dinkelbach approach in [25] the objec-tive functions of (10) and (13) are given as

119865 (119902) = maxΦP[119877total (ΦP) minus 119902119875total (ΦP)]

119865 (119902)

= maxΦS[1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896]

(14)

The optimization problem of (10) is not concave since 120601119898119899119896=

0 1 The optimization problem of (13) is the slack form ofoptimization problem (10) by allowing 120601

119898119899119896to take any value

in the interval [0 1]The objective function (13) is concave in (Φ S) since

120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896)) are perspectives of the con-

cave function log2(1 + 119886

119898119899119896119878119898119899119896) Further since there are

feasible points and all constraints are affine obviously Slaterrsquoscondition can be satisfied [17] Hence we can suggest thata globally optimal solution can be got in the Lagrange dualdomain because convex optimization problem (13) has a zeroduality gap Therefore relaxation problem (13) and originalproblem (10) have the same optimal solution

Note that adopting traditional convex optimization soft-ware packages cannot get the optimal solution in (10) becauseΦlowast is not ensured to be a binary In next section we will

present a three-dimensional joint subcarrier pairing andchannel-user assignment scheme that can provide an optimalsolution with the polynomial time

4 Joint Resource Allocation Scheme forMultiuser OFDM Relaying System

Since solving the nonconvex problem in (13) is not standardapproach in general in this section we transform the opti-mization problem with the Dinkelbach method and handlean iterative algorithm to obtain it

41 Objective Function Transformation Without loss of gen-erality we denote the optimal EE and the set of feasiblesolutions for the optimization proposed in (13) as 119902lowast and Tthen we have

119902lowast

= maxΦS

119877total (Φ S)119875total (Φ S)

=119877total (Φ

lowast

Slowast)119875total (Φ

lowast

Slowast)

forall Φ S isin T(15)

We define

119865 (119902) = maxΦS[119877total (Φ S) minus 119902119875total (Φ S)] (16)

The optimal energy efficiency of the optimization prob-lem proposed in (15) 119902lowast is achieved if and only if

119865 (119902lowast

) = maxΦS[119877total (Φ S) minus 119902

lowast

119875total (Φ S)]

= 119877total (Φlowast

Slowast) minus 119902lowast119875total (Φlowast

Slowast) = 0

forall Φ S isin T

(17)

We can prove (17) by following the Dinkelbach approachas in [25] In other words the optimal resource allocationpolicies Φlowast Slowast for the equivalent objective function arealso the optimal resource allocation policies for the originalobjective function

In summary the optimization of the original objectivefunction and the optimization of the equivalent objectivefunction result in the same resource allocation policiesEquation (17) reveals that for an optimization problem with

6 Mobile Information Systems

(1) Initial maximum number of iterations 119871max and maximum tolerance 120576(2) Set maximum energy efficiency 119902 = 0 and iteration index 119899 = 0(3) repeat Main Loop(4) Solve the inner loop problem in (19) for a given 119902 and channel

assignment and power allocation policy (Φ1015840 S1015840)(5) if 119877total(Φ

1015840

S1015840) minus 119902119875total(Φ1015840 S1015840) lt 120576 then(6) Convergence = true(7) return (Φlowast Slowast) = (Φ1015840 S1015840) and 119902lowast = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840)(8) else(9) Set 119902 = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840) and 119899 = 119899 + 1(10) Convergence = false(11) end if(12) until Convergence = true or 119899 = 119871max

Algorithm 1 Resource allocation iteration algorithm for EE maximization

an objective function in the fractional form there existsan equivalent objective function in a subtractive form forexample 119877total(Φ

lowast

Slowast) minus 119902lowast119875total(Φlowast Slowast) in the consideredcase As a result we can focus on the equivalent objectivefunction further in this paper

42 Resource Allocation Iteration Algorithm for EE Maxi-mization The iterative algorithm for EE maximization isdescribed in Algorithm 1

As shown in Algorithm 1 for each iteration in the mainloop that is lines (3)ndash(12) we solve the following optimiza-tion problem for a given parameter 119902 that can be expressedas

maxΦS119865 (119902) = [119877total (Φ S) minus 119902119875total (Φ S)] (18)

In combination with formula (13) (18) can be furtherconverted to

maxΦS

119865 (119902)

=1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902(119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(19)

According to (19) and setting 119902 = 0 the optimizationproblem by solving 119865(0) turns out to be an SE-oriented

resource allocation scheme 119877total(Φ S) is concave and 119878119898119899119896is affine The optimization problem of OP(119902) is concave withrespect to (120601

119898119899119896 119878119898119899119896) We assume that 119877req is achievable

under the constraints of (19) that indicates the existence ofinterior points Then Slaterrsquos condition is satisfied [26]

Hence it can be shown that the strong duality holds andthe optimization problem (19) has a zero duality In otherwords solving the dual problem is equivalent to solving theprimal problem and a globally optimal solution can be foundin the Lagrange dual domain Asmentioned in Section 32 byrelaxing the constraints and introducing a continuous 120601

119898119899119896isin

[0 1] solving the dual problem will always provide an upperbound for the original optimization problem

Specifically the optimal solution for the dual problemdoes not always result in a binary 120601

119898119899119896isin 0 1 that is

desired as a necessary constraint in the original optimizationproblem However we will prove that a globally optimalsolution always exists for the dual problem with a binary120601119898119899119896

isin 0 1 making the solution obtained available andoptimal for the optimization problem in (10)

43 Dual Problem Formulation and Decomposition SolutionThe Lagrangian of (19) is given by

119871 (Φ S 120582 120583 119902) = 12

sdot

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902(119875119862

+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896) minus 120582(

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896minus 119875119905)

minus 120583(119877req

minus1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

))

(20)

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

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ArtificialNeural Systems

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RoboticsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 3

Source

Relay

User 1

User 2

User k

User K

Figure 1 Model of the concerned single relay multiuser OFDMnetwork

In Section 5 both the analysis and the simulation resultsare presented to compare the performances of the differentresource allocation schemes Section 6 gives a conclusion forthis paper

2 System Model

In this section we first introduce the adopted system modeland the performance measure Then the design of theresource allocation and scheduling is formulated as an opti-mization problem

21 System Model Considering an OFDM single relay mul-tiusers system as in Figure 1 source transfers the informationto all the users with the help of one relay In this paper wefocus on the joint resource allocation problem in a scenarioincluding one source one relay and 119870 users 119873 equal-bandwidth orthogonal channels haves been divided from theavailable frequency spectrum accessible by the relay andall the users Without loss of generality we assume that allthe channels occupy the same bandwidth and experienceindependent frequency selective fading It is also assumedthat all pieces of the channel state information (CSI) areavailable at the source the relay and the users [18] Thehalf-duplex transmission process can be divided into twophases The source 119878 transmits the information data to allthe other nodes including the relay and all the users in thefirst phase The relay decodes the received signals in the firstphase and retransmits them to all the users in the secondphase Compared with traditional OFDM relaying networks[22] the source is permitted to retransfer the same datatransmitted by it in the first time slot further improving thesystem capacity performance [24]

From the discussion above the state of channel 119894 candenote that for 1 le 119894 le 119873 ℎ119904119903

119894and ℎ1199041198961

119894represent the

instantaneous channel gain from source to relay and fromthe source to the user 119896 in the first phase and ℎ119903119896

119894and ℎ1199041198962

119894

signify the instantaneous channel gain from the relay to theuser 119896 and from the source to the user 119896 in the second phaseThe AWGN (Additive Gaussian White Noise) at the 119896th

user and the relay are zero mean with variances 1205902119903and 1205902

119896

respectivelySince there are multiple users for the sake of maximizing

the system throughput when allocating the resources wemust also take the user fairness into account [9 11] Thefairness factor 120596

119896for user 119896 is expressed by

119870

sum

119896=1

119908119896= 1 (1)

It should be mentioned that the considered scenario isapplicable in practice where the source the relay and theusers may act as base station the relay station and thecellular users respectively Further this system model is anexample of a wireless sensor network transmission modelwhere the source the relay and the users are all consideredas wireless sensor nodes In our system model adopting onerelay instead of multiple relays can significantly reduce theenergy consumption and the system overhead in accordancewith green communication and environmental protection

22 Channel Assignment The relay not only forwards theincoming signals to their intended users by a special relayprocess strategy but also conducts the subcarrier pairing andthe channel-user assignments Obviously the solutions of thesubcarrier pairing are closely related to the strategy of thechannels assignments to all the users Then we refer to thejoint assignment on subcarrier pairing and subcarrier pair-user allocation as the joint channel assignment problem

Suppose that the path119875(119898 119899 119896) is selected if the first-hopchannel119898 is paired with a second-hop channel 119899 and the pairof channels (119898 119899) is assigned to user 119896 Further we define theindicator functions given by [22]

120601119898119899119896

=

1 if 119875 (119898 119899 119896) is selected

0 otherwise(2)

A user can be assigned many subcarrier pairs but asubcarrier pair must be allocated to a user exclusively Hence

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896

= 1 forall119898119898 isin 1 2 119873

119873

sum

119898=1

119870

sum

119896=1

120601119898119899119896

= 1 forall119899 119899 isin 1 2 119873

(3)

23 Relaying Strategy and Power Allocation In the traditionalDF relaying networks each transmission cycle is dividedinto two time slots equally In the first time slot the sourcebroadcasts a data block in every channel to both user andrelay nodes Then in the second time slot the relay startsto decode and forward the data from source to the userMeanwhile if the channel strength of the source is sufficientin the second hop the source will be assigned power tosend the data again and then obtains a diversity gain Theuser adopts the MRC in both two time slots on the receivedsignals

4 Mobile Information Systems

Along any path119875(119898 119899 119896) the source in the first time slotthe source in the second time slot and the relay transmissionpowers are denoted by 1198751199041

119898119899119896 1198751199042119898119899119896

and 119875119903119898119899119896

respectivelyThen the total power allocated to path 119875(119898 119899 119896) is 119875

119898119899119896=

1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896 The total power 119875total constraint can be

given by

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896(1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896) le 119875total (4)

Based on DF relaying networks the maximum source-destination achievable rate on path 119875(119898 119899 119896) is given by[22]

119877 (119898 119899 119896) =1

2min log

2(1 + 119886

1198981198751199041

119898119899119896)

log2(1 + 119887

119899119896119875119903

119898119899119896+ 1198881198981198961198751199041

119898119899119896+ 1198861198991198961198751199042

119898119899119896)

(5)

where 119886119898= |ℎ119904119903

119898|2

1205902

119903 119887119899119896= |ℎ119903119896

119899|2

1205902

119896 119888119898119896= |ℎ1199041198961

119898|2

1205902

119896

119886119899119896= |ℎ1199041198962

119899|2

1205902

119896 and 12 indicates that one data transmission

is needed in two time slots

3 Problem Formation andConvex Reformulation

In this section we formulate the corresponding source allo-cation problem and reformulate the optimization problem

31 Problem Formulation Given a fixed total power 119875119898119899119896

ofpath 119875(119898 119899 119896) (5) can be converted to [24]

119877 (119898 119899 119896) =1

2log2(1 + 119886

119898119899119896119875119898119899119896) (6)

119886119898119899119896

denotes the equivalent channel gain on the path 119875(119898119899 119896) expressed as

119886119898119899119896=

119886119898119886119899119896

119886119898+ 119886119899119896minus 119888119898119896

when 119887119899119896le 119886119899119896

119886119898119887119899119896

119886119898+ 119887119899119896minus 119888119898119896

when 119887119899119896gt 119886119899119896

(7)

According to the fairness factor 120596119896for every user 119896

the total end-to-end SE and the total transmit power on allthe subcarriers by the source and the relay of the system canbe expressed by

119877total (ΦP) =1

2

119870

sum

119896=1

120596119896

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896

log2(1 + 119886

119898119899119896119875119898119899119896)

119875users (ΦP) =119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896119875119898119899119896

(8)

where P = [119875119898119899119896]119873times119873times119870

Φ = [120601119898119899119896]119873times119873times119870

and 119875119898119899119896

and120601119898119899119896

satisfy the constraints mentioned previouslyThe total power consumed can therefore be expressed as

[13]

119875total (ΦP) = 119875119862 + 120589119875users (ΦP) (9)

where 119875119862is the constant circuit power consumption which

includes the power dissipations in the transmit filter themixer the frequency synthesizer and the digital-to-analogconverter that are independent of the actual transmittedpower [12] 120589 gt 1 denotes the reciprocal of the drain effi-ciencies of the power amplifiers employed at the sourceand the relay For example an amplifier with a 50 drainefficiency has 120589 = 105 = 2 119877req specifies the minimum SErequirement as a QoS constraint and satisfies that 119877total(ΦP) ge 119877req

Finally the optimization problem to maximize the aver-age system EE is formulated into

maxΦP

119902 (ΦP) =119877total (ΦP)119875total (ΦP)

st (3) and (4)

120601119898119899119896= 0 1

119875119898119899119896ge 0

forall119898 119899 119896

119877total (ΦP) ge 119877req

(10)

32 Continuous Relaxation and Convex Reformulation Theoptimization problem addressed in (10) is MINLP problemthat in general is computationally undesirable because of itscombinatorial nature which could adopt branch-and-boundmethod to solve [14] However in the following section wewill raise a method to find an optimal strategy that growspolynomial with the numbers of subcarriers available in thesystem

First ignoring the physical meaning of formula (2) werelax the constraints by allowing 120601

119898119899119896to take any value in the

interval [0 1] where

120601119898119899119896isin [0 1] forall119898 119899 119896 (11)

Hence we introduce a new variable

119878119898119899119896= 120601119898119899119896119875119898119899119896 (12)

Second the relaxed optimization problem of (10) accord-ing to Φ = [120601

119898119899119896]119873times119873times119870

and S = [119878119898119899119896]119873times119873times119870

can be refor-mulated as

Mobile Information Systems 5

maxΦS

119902 (Φ S) =(12)sum

119870

119896=1sum119873

119898=1sum119873

119899=1120596119896120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896))

119875119862+ 120589sum119873

119898=1sum119873

119899=1sum119870

119896=1119878119898119899119896

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(13)

Proposition 1 The channel-user assignments and the powerallocation matrices (Φlowast Slowast) found in the previous subsectionare globally optimal solutions to the original problem of (10)

Proof Applying the Dinkelbach approach in [25] the objec-tive functions of (10) and (13) are given as

119865 (119902) = maxΦP[119877total (ΦP) minus 119902119875total (ΦP)]

119865 (119902)

= maxΦS[1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896]

(14)

The optimization problem of (10) is not concave since 120601119898119899119896=

0 1 The optimization problem of (13) is the slack form ofoptimization problem (10) by allowing 120601

119898119899119896to take any value

in the interval [0 1]The objective function (13) is concave in (Φ S) since

120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896)) are perspectives of the con-

cave function log2(1 + 119886

119898119899119896119878119898119899119896) Further since there are

feasible points and all constraints are affine obviously Slaterrsquoscondition can be satisfied [17] Hence we can suggest thata globally optimal solution can be got in the Lagrange dualdomain because convex optimization problem (13) has a zeroduality gap Therefore relaxation problem (13) and originalproblem (10) have the same optimal solution

Note that adopting traditional convex optimization soft-ware packages cannot get the optimal solution in (10) becauseΦlowast is not ensured to be a binary In next section we will

present a three-dimensional joint subcarrier pairing andchannel-user assignment scheme that can provide an optimalsolution with the polynomial time

4 Joint Resource Allocation Scheme forMultiuser OFDM Relaying System

Since solving the nonconvex problem in (13) is not standardapproach in general in this section we transform the opti-mization problem with the Dinkelbach method and handlean iterative algorithm to obtain it

41 Objective Function Transformation Without loss of gen-erality we denote the optimal EE and the set of feasiblesolutions for the optimization proposed in (13) as 119902lowast and Tthen we have

119902lowast

= maxΦS

119877total (Φ S)119875total (Φ S)

=119877total (Φ

lowast

Slowast)119875total (Φ

lowast

Slowast)

forall Φ S isin T(15)

We define

119865 (119902) = maxΦS[119877total (Φ S) minus 119902119875total (Φ S)] (16)

The optimal energy efficiency of the optimization prob-lem proposed in (15) 119902lowast is achieved if and only if

119865 (119902lowast

) = maxΦS[119877total (Φ S) minus 119902

lowast

119875total (Φ S)]

= 119877total (Φlowast

Slowast) minus 119902lowast119875total (Φlowast

Slowast) = 0

forall Φ S isin T

(17)

We can prove (17) by following the Dinkelbach approachas in [25] In other words the optimal resource allocationpolicies Φlowast Slowast for the equivalent objective function arealso the optimal resource allocation policies for the originalobjective function

In summary the optimization of the original objectivefunction and the optimization of the equivalent objectivefunction result in the same resource allocation policiesEquation (17) reveals that for an optimization problem with

6 Mobile Information Systems

(1) Initial maximum number of iterations 119871max and maximum tolerance 120576(2) Set maximum energy efficiency 119902 = 0 and iteration index 119899 = 0(3) repeat Main Loop(4) Solve the inner loop problem in (19) for a given 119902 and channel

assignment and power allocation policy (Φ1015840 S1015840)(5) if 119877total(Φ

1015840

S1015840) minus 119902119875total(Φ1015840 S1015840) lt 120576 then(6) Convergence = true(7) return (Φlowast Slowast) = (Φ1015840 S1015840) and 119902lowast = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840)(8) else(9) Set 119902 = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840) and 119899 = 119899 + 1(10) Convergence = false(11) end if(12) until Convergence = true or 119899 = 119871max

Algorithm 1 Resource allocation iteration algorithm for EE maximization

an objective function in the fractional form there existsan equivalent objective function in a subtractive form forexample 119877total(Φ

lowast

Slowast) minus 119902lowast119875total(Φlowast Slowast) in the consideredcase As a result we can focus on the equivalent objectivefunction further in this paper

42 Resource Allocation Iteration Algorithm for EE Maxi-mization The iterative algorithm for EE maximization isdescribed in Algorithm 1

As shown in Algorithm 1 for each iteration in the mainloop that is lines (3)ndash(12) we solve the following optimiza-tion problem for a given parameter 119902 that can be expressedas

maxΦS119865 (119902) = [119877total (Φ S) minus 119902119875total (Φ S)] (18)

In combination with formula (13) (18) can be furtherconverted to

maxΦS

119865 (119902)

=1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902(119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(19)

According to (19) and setting 119902 = 0 the optimizationproblem by solving 119865(0) turns out to be an SE-oriented

resource allocation scheme 119877total(Φ S) is concave and 119878119898119899119896is affine The optimization problem of OP(119902) is concave withrespect to (120601

119898119899119896 119878119898119899119896) We assume that 119877req is achievable

under the constraints of (19) that indicates the existence ofinterior points Then Slaterrsquos condition is satisfied [26]

Hence it can be shown that the strong duality holds andthe optimization problem (19) has a zero duality In otherwords solving the dual problem is equivalent to solving theprimal problem and a globally optimal solution can be foundin the Lagrange dual domain Asmentioned in Section 32 byrelaxing the constraints and introducing a continuous 120601

119898119899119896isin

[0 1] solving the dual problem will always provide an upperbound for the original optimization problem

Specifically the optimal solution for the dual problemdoes not always result in a binary 120601

119898119899119896isin 0 1 that is

desired as a necessary constraint in the original optimizationproblem However we will prove that a globally optimalsolution always exists for the dual problem with a binary120601119898119899119896

isin 0 1 making the solution obtained available andoptimal for the optimization problem in (10)

43 Dual Problem Formulation and Decomposition SolutionThe Lagrangian of (19) is given by

119871 (Φ S 120582 120583 119902) = 12

sdot

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902(119875119862

+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896) minus 120582(

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896minus 119875119905)

minus 120583(119877req

minus1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

))

(20)

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

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ReconfigurableComputing

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Mobile Information Systems

Along any path119875(119898 119899 119896) the source in the first time slotthe source in the second time slot and the relay transmissionpowers are denoted by 1198751199041

119898119899119896 1198751199042119898119899119896

and 119875119903119898119899119896

respectivelyThen the total power allocated to path 119875(119898 119899 119896) is 119875

119898119899119896=

1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896 The total power 119875total constraint can be

given by

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896(1198751199041

119898119899119896+ 1198751199042

119898119899119896+ 119875119903

119898119899119896) le 119875total (4)

Based on DF relaying networks the maximum source-destination achievable rate on path 119875(119898 119899 119896) is given by[22]

119877 (119898 119899 119896) =1

2min log

2(1 + 119886

1198981198751199041

119898119899119896)

log2(1 + 119887

119899119896119875119903

119898119899119896+ 1198881198981198961198751199041

119898119899119896+ 1198861198991198961198751199042

119898119899119896)

(5)

where 119886119898= |ℎ119904119903

119898|2

1205902

119903 119887119899119896= |ℎ119903119896

119899|2

1205902

119896 119888119898119896= |ℎ1199041198961

119898|2

1205902

119896

119886119899119896= |ℎ1199041198962

119899|2

1205902

119896 and 12 indicates that one data transmission

is needed in two time slots

3 Problem Formation andConvex Reformulation

In this section we formulate the corresponding source allo-cation problem and reformulate the optimization problem

31 Problem Formulation Given a fixed total power 119875119898119899119896

ofpath 119875(119898 119899 119896) (5) can be converted to [24]

119877 (119898 119899 119896) =1

2log2(1 + 119886

119898119899119896119875119898119899119896) (6)

119886119898119899119896

denotes the equivalent channel gain on the path 119875(119898119899 119896) expressed as

119886119898119899119896=

119886119898119886119899119896

119886119898+ 119886119899119896minus 119888119898119896

when 119887119899119896le 119886119899119896

119886119898119887119899119896

119886119898+ 119887119899119896minus 119888119898119896

when 119887119899119896gt 119886119899119896

(7)

According to the fairness factor 120596119896for every user 119896

the total end-to-end SE and the total transmit power on allthe subcarriers by the source and the relay of the system canbe expressed by

119877total (ΦP) =1

2

119870

sum

119896=1

120596119896

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896

log2(1 + 119886

119898119899119896119875119898119899119896)

119875users (ΦP) =119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

120601119898119899119896119875119898119899119896

(8)

where P = [119875119898119899119896]119873times119873times119870

Φ = [120601119898119899119896]119873times119873times119870

and 119875119898119899119896

and120601119898119899119896

satisfy the constraints mentioned previouslyThe total power consumed can therefore be expressed as

[13]

119875total (ΦP) = 119875119862 + 120589119875users (ΦP) (9)

where 119875119862is the constant circuit power consumption which

includes the power dissipations in the transmit filter themixer the frequency synthesizer and the digital-to-analogconverter that are independent of the actual transmittedpower [12] 120589 gt 1 denotes the reciprocal of the drain effi-ciencies of the power amplifiers employed at the sourceand the relay For example an amplifier with a 50 drainefficiency has 120589 = 105 = 2 119877req specifies the minimum SErequirement as a QoS constraint and satisfies that 119877total(ΦP) ge 119877req

Finally the optimization problem to maximize the aver-age system EE is formulated into

maxΦP

119902 (ΦP) =119877total (ΦP)119875total (ΦP)

st (3) and (4)

120601119898119899119896= 0 1

119875119898119899119896ge 0

forall119898 119899 119896

119877total (ΦP) ge 119877req

(10)

32 Continuous Relaxation and Convex Reformulation Theoptimization problem addressed in (10) is MINLP problemthat in general is computationally undesirable because of itscombinatorial nature which could adopt branch-and-boundmethod to solve [14] However in the following section wewill raise a method to find an optimal strategy that growspolynomial with the numbers of subcarriers available in thesystem

First ignoring the physical meaning of formula (2) werelax the constraints by allowing 120601

119898119899119896to take any value in the

interval [0 1] where

120601119898119899119896isin [0 1] forall119898 119899 119896 (11)

Hence we introduce a new variable

119878119898119899119896= 120601119898119899119896119875119898119899119896 (12)

Second the relaxed optimization problem of (10) accord-ing to Φ = [120601

119898119899119896]119873times119873times119870

and S = [119878119898119899119896]119873times119873times119870

can be refor-mulated as

Mobile Information Systems 5

maxΦS

119902 (Φ S) =(12)sum

119870

119896=1sum119873

119898=1sum119873

119899=1120596119896120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896))

119875119862+ 120589sum119873

119898=1sum119873

119899=1sum119870

119896=1119878119898119899119896

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(13)

Proposition 1 The channel-user assignments and the powerallocation matrices (Φlowast Slowast) found in the previous subsectionare globally optimal solutions to the original problem of (10)

Proof Applying the Dinkelbach approach in [25] the objec-tive functions of (10) and (13) are given as

119865 (119902) = maxΦP[119877total (ΦP) minus 119902119875total (ΦP)]

119865 (119902)

= maxΦS[1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896]

(14)

The optimization problem of (10) is not concave since 120601119898119899119896=

0 1 The optimization problem of (13) is the slack form ofoptimization problem (10) by allowing 120601

119898119899119896to take any value

in the interval [0 1]The objective function (13) is concave in (Φ S) since

120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896)) are perspectives of the con-

cave function log2(1 + 119886

119898119899119896119878119898119899119896) Further since there are

feasible points and all constraints are affine obviously Slaterrsquoscondition can be satisfied [17] Hence we can suggest thata globally optimal solution can be got in the Lagrange dualdomain because convex optimization problem (13) has a zeroduality gap Therefore relaxation problem (13) and originalproblem (10) have the same optimal solution

Note that adopting traditional convex optimization soft-ware packages cannot get the optimal solution in (10) becauseΦlowast is not ensured to be a binary In next section we will

present a three-dimensional joint subcarrier pairing andchannel-user assignment scheme that can provide an optimalsolution with the polynomial time

4 Joint Resource Allocation Scheme forMultiuser OFDM Relaying System

Since solving the nonconvex problem in (13) is not standardapproach in general in this section we transform the opti-mization problem with the Dinkelbach method and handlean iterative algorithm to obtain it

41 Objective Function Transformation Without loss of gen-erality we denote the optimal EE and the set of feasiblesolutions for the optimization proposed in (13) as 119902lowast and Tthen we have

119902lowast

= maxΦS

119877total (Φ S)119875total (Φ S)

=119877total (Φ

lowast

Slowast)119875total (Φ

lowast

Slowast)

forall Φ S isin T(15)

We define

119865 (119902) = maxΦS[119877total (Φ S) minus 119902119875total (Φ S)] (16)

The optimal energy efficiency of the optimization prob-lem proposed in (15) 119902lowast is achieved if and only if

119865 (119902lowast

) = maxΦS[119877total (Φ S) minus 119902

lowast

119875total (Φ S)]

= 119877total (Φlowast

Slowast) minus 119902lowast119875total (Φlowast

Slowast) = 0

forall Φ S isin T

(17)

We can prove (17) by following the Dinkelbach approachas in [25] In other words the optimal resource allocationpolicies Φlowast Slowast for the equivalent objective function arealso the optimal resource allocation policies for the originalobjective function

In summary the optimization of the original objectivefunction and the optimization of the equivalent objectivefunction result in the same resource allocation policiesEquation (17) reveals that for an optimization problem with

6 Mobile Information Systems

(1) Initial maximum number of iterations 119871max and maximum tolerance 120576(2) Set maximum energy efficiency 119902 = 0 and iteration index 119899 = 0(3) repeat Main Loop(4) Solve the inner loop problem in (19) for a given 119902 and channel

assignment and power allocation policy (Φ1015840 S1015840)(5) if 119877total(Φ

1015840

S1015840) minus 119902119875total(Φ1015840 S1015840) lt 120576 then(6) Convergence = true(7) return (Φlowast Slowast) = (Φ1015840 S1015840) and 119902lowast = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840)(8) else(9) Set 119902 = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840) and 119899 = 119899 + 1(10) Convergence = false(11) end if(12) until Convergence = true or 119899 = 119871max

Algorithm 1 Resource allocation iteration algorithm for EE maximization

an objective function in the fractional form there existsan equivalent objective function in a subtractive form forexample 119877total(Φ

lowast

Slowast) minus 119902lowast119875total(Φlowast Slowast) in the consideredcase As a result we can focus on the equivalent objectivefunction further in this paper

42 Resource Allocation Iteration Algorithm for EE Maxi-mization The iterative algorithm for EE maximization isdescribed in Algorithm 1

As shown in Algorithm 1 for each iteration in the mainloop that is lines (3)ndash(12) we solve the following optimiza-tion problem for a given parameter 119902 that can be expressedas

maxΦS119865 (119902) = [119877total (Φ S) minus 119902119875total (Φ S)] (18)

In combination with formula (13) (18) can be furtherconverted to

maxΦS

119865 (119902)

=1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902(119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(19)

According to (19) and setting 119902 = 0 the optimizationproblem by solving 119865(0) turns out to be an SE-oriented

resource allocation scheme 119877total(Φ S) is concave and 119878119898119899119896is affine The optimization problem of OP(119902) is concave withrespect to (120601

119898119899119896 119878119898119899119896) We assume that 119877req is achievable

under the constraints of (19) that indicates the existence ofinterior points Then Slaterrsquos condition is satisfied [26]

Hence it can be shown that the strong duality holds andthe optimization problem (19) has a zero duality In otherwords solving the dual problem is equivalent to solving theprimal problem and a globally optimal solution can be foundin the Lagrange dual domain Asmentioned in Section 32 byrelaxing the constraints and introducing a continuous 120601

119898119899119896isin

[0 1] solving the dual problem will always provide an upperbound for the original optimization problem

Specifically the optimal solution for the dual problemdoes not always result in a binary 120601

119898119899119896isin 0 1 that is

desired as a necessary constraint in the original optimizationproblem However we will prove that a globally optimalsolution always exists for the dual problem with a binary120601119898119899119896

isin 0 1 making the solution obtained available andoptimal for the optimization problem in (10)

43 Dual Problem Formulation and Decomposition SolutionThe Lagrangian of (19) is given by

119871 (Φ S 120582 120583 119902) = 12

sdot

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902(119875119862

+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896) minus 120582(

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896minus 119875119905)

minus 120583(119877req

minus1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

))

(20)

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Applied Computational Intelligence and Soft Computing

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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 5

maxΦS

119902 (Φ S) =(12)sum

119870

119896=1sum119873

119898=1sum119873

119899=1120596119896120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896))

119875119862+ 120589sum119873

119898=1sum119873

119899=1sum119870

119896=1119878119898119899119896

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(13)

Proposition 1 The channel-user assignments and the powerallocation matrices (Φlowast Slowast) found in the previous subsectionare globally optimal solutions to the original problem of (10)

Proof Applying the Dinkelbach approach in [25] the objec-tive functions of (10) and (13) are given as

119865 (119902) = maxΦP[119877total (ΦP) minus 119902119875total (ΦP)]

119865 (119902)

= maxΦS[1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896]

(14)

The optimization problem of (10) is not concave since 120601119898119899119896=

0 1 The optimization problem of (13) is the slack form ofoptimization problem (10) by allowing 120601

119898119899119896to take any value

in the interval [0 1]The objective function (13) is concave in (Φ S) since

120601119898119899119896

log2(1 + 119886

119898119899119896(119878119898119899119896120601119898119899119896)) are perspectives of the con-

cave function log2(1 + 119886

119898119899119896119878119898119899119896) Further since there are

feasible points and all constraints are affine obviously Slaterrsquoscondition can be satisfied [17] Hence we can suggest thata globally optimal solution can be got in the Lagrange dualdomain because convex optimization problem (13) has a zeroduality gap Therefore relaxation problem (13) and originalproblem (10) have the same optimal solution

Note that adopting traditional convex optimization soft-ware packages cannot get the optimal solution in (10) becauseΦlowast is not ensured to be a binary In next section we will

present a three-dimensional joint subcarrier pairing andchannel-user assignment scheme that can provide an optimalsolution with the polynomial time

4 Joint Resource Allocation Scheme forMultiuser OFDM Relaying System

Since solving the nonconvex problem in (13) is not standardapproach in general in this section we transform the opti-mization problem with the Dinkelbach method and handlean iterative algorithm to obtain it

41 Objective Function Transformation Without loss of gen-erality we denote the optimal EE and the set of feasiblesolutions for the optimization proposed in (13) as 119902lowast and Tthen we have

119902lowast

= maxΦS

119877total (Φ S)119875total (Φ S)

=119877total (Φ

lowast

Slowast)119875total (Φ

lowast

Slowast)

forall Φ S isin T(15)

We define

119865 (119902) = maxΦS[119877total (Φ S) minus 119902119875total (Φ S)] (16)

The optimal energy efficiency of the optimization prob-lem proposed in (15) 119902lowast is achieved if and only if

119865 (119902lowast

) = maxΦS[119877total (Φ S) minus 119902

lowast

119875total (Φ S)]

= 119877total (Φlowast

Slowast) minus 119902lowast119875total (Φlowast

Slowast) = 0

forall Φ S isin T

(17)

We can prove (17) by following the Dinkelbach approachas in [25] In other words the optimal resource allocationpolicies Φlowast Slowast for the equivalent objective function arealso the optimal resource allocation policies for the originalobjective function

In summary the optimization of the original objectivefunction and the optimization of the equivalent objectivefunction result in the same resource allocation policiesEquation (17) reveals that for an optimization problem with

6 Mobile Information Systems

(1) Initial maximum number of iterations 119871max and maximum tolerance 120576(2) Set maximum energy efficiency 119902 = 0 and iteration index 119899 = 0(3) repeat Main Loop(4) Solve the inner loop problem in (19) for a given 119902 and channel

assignment and power allocation policy (Φ1015840 S1015840)(5) if 119877total(Φ

1015840

S1015840) minus 119902119875total(Φ1015840 S1015840) lt 120576 then(6) Convergence = true(7) return (Φlowast Slowast) = (Φ1015840 S1015840) and 119902lowast = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840)(8) else(9) Set 119902 = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840) and 119899 = 119899 + 1(10) Convergence = false(11) end if(12) until Convergence = true or 119899 = 119871max

Algorithm 1 Resource allocation iteration algorithm for EE maximization

an objective function in the fractional form there existsan equivalent objective function in a subtractive form forexample 119877total(Φ

lowast

Slowast) minus 119902lowast119875total(Φlowast Slowast) in the consideredcase As a result we can focus on the equivalent objectivefunction further in this paper

42 Resource Allocation Iteration Algorithm for EE Maxi-mization The iterative algorithm for EE maximization isdescribed in Algorithm 1

As shown in Algorithm 1 for each iteration in the mainloop that is lines (3)ndash(12) we solve the following optimiza-tion problem for a given parameter 119902 that can be expressedas

maxΦS119865 (119902) = [119877total (Φ S) minus 119902119875total (Φ S)] (18)

In combination with formula (13) (18) can be furtherconverted to

maxΦS

119865 (119902)

=1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902(119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(19)

According to (19) and setting 119902 = 0 the optimizationproblem by solving 119865(0) turns out to be an SE-oriented

resource allocation scheme 119877total(Φ S) is concave and 119878119898119899119896is affine The optimization problem of OP(119902) is concave withrespect to (120601

119898119899119896 119878119898119899119896) We assume that 119877req is achievable

under the constraints of (19) that indicates the existence ofinterior points Then Slaterrsquos condition is satisfied [26]

Hence it can be shown that the strong duality holds andthe optimization problem (19) has a zero duality In otherwords solving the dual problem is equivalent to solving theprimal problem and a globally optimal solution can be foundin the Lagrange dual domain Asmentioned in Section 32 byrelaxing the constraints and introducing a continuous 120601

119898119899119896isin

[0 1] solving the dual problem will always provide an upperbound for the original optimization problem

Specifically the optimal solution for the dual problemdoes not always result in a binary 120601

119898119899119896isin 0 1 that is

desired as a necessary constraint in the original optimizationproblem However we will prove that a globally optimalsolution always exists for the dual problem with a binary120601119898119899119896

isin 0 1 making the solution obtained available andoptimal for the optimization problem in (10)

43 Dual Problem Formulation and Decomposition SolutionThe Lagrangian of (19) is given by

119871 (Φ S 120582 120583 119902) = 12

sdot

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902(119875119862

+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896) minus 120582(

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896minus 119875119905)

minus 120583(119877req

minus1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

))

(20)

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Mobile Information Systems

(1) Initial maximum number of iterations 119871max and maximum tolerance 120576(2) Set maximum energy efficiency 119902 = 0 and iteration index 119899 = 0(3) repeat Main Loop(4) Solve the inner loop problem in (19) for a given 119902 and channel

assignment and power allocation policy (Φ1015840 S1015840)(5) if 119877total(Φ

1015840

S1015840) minus 119902119875total(Φ1015840 S1015840) lt 120576 then(6) Convergence = true(7) return (Φlowast Slowast) = (Φ1015840 S1015840) and 119902lowast = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840)(8) else(9) Set 119902 = 119877total(Φ

1015840

S1015840)119875total(Φ1015840 S1015840) and 119899 = 119899 + 1(10) Convergence = false(11) end if(12) until Convergence = true or 119899 = 119871max

Algorithm 1 Resource allocation iteration algorithm for EE maximization

an objective function in the fractional form there existsan equivalent objective function in a subtractive form forexample 119877total(Φ

lowast

Slowast) minus 119902lowast119875total(Φlowast Slowast) in the consideredcase As a result we can focus on the equivalent objectivefunction further in this paper

42 Resource Allocation Iteration Algorithm for EE Maxi-mization The iterative algorithm for EE maximization isdescribed in Algorithm 1

As shown in Algorithm 1 for each iteration in the mainloop that is lines (3)ndash(12) we solve the following optimiza-tion problem for a given parameter 119902 that can be expressedas

maxΦS119865 (119902) = [119877total (Φ S) minus 119902119875total (Φ S)] (18)

In combination with formula (13) (18) can be furtherconverted to

maxΦS

119865 (119902)

=1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119902(119875119862+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)

st (3)

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896le 119875119905

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

119877total (Φ S) ge 119877req

(19)

According to (19) and setting 119902 = 0 the optimizationproblem by solving 119865(0) turns out to be an SE-oriented

resource allocation scheme 119877total(Φ S) is concave and 119878119898119899119896is affine The optimization problem of OP(119902) is concave withrespect to (120601

119898119899119896 119878119898119899119896) We assume that 119877req is achievable

under the constraints of (19) that indicates the existence ofinterior points Then Slaterrsquos condition is satisfied [26]

Hence it can be shown that the strong duality holds andthe optimization problem (19) has a zero duality In otherwords solving the dual problem is equivalent to solving theprimal problem and a globally optimal solution can be foundin the Lagrange dual domain Asmentioned in Section 32 byrelaxing the constraints and introducing a continuous 120601

119898119899119896isin

[0 1] solving the dual problem will always provide an upperbound for the original optimization problem

Specifically the optimal solution for the dual problemdoes not always result in a binary 120601

119898119899119896isin 0 1 that is

desired as a necessary constraint in the original optimizationproblem However we will prove that a globally optimalsolution always exists for the dual problem with a binary120601119898119899119896

isin 0 1 making the solution obtained available andoptimal for the optimization problem in (10)

43 Dual Problem Formulation and Decomposition SolutionThe Lagrangian of (19) is given by

119871 (Φ S 120582 120583 119902) = 12

sdot

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902(119875119862

+ 120589

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896) minus 120582(

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896minus 119875119905)

minus 120583(119877req

minus1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

))

(20)

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Hindawi Publishing Corporation

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RoboticsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 7

Associated with the total power constraint 120582 ge 0 and120583 ge 0 denote the LagrangianmultipliersThe dual function istherefore given by

min 119892 (120582 120583) = minmaxΦS119871 (Φ S 120582 120583 119902)

st (3)

0 le 120601119898119899119896le 1

119878119898119899119896ge 0

forall119898 119899 119896

(21)

To deal with (21) we resolve the dual problem into threesubproblems and solve it iteratively

(1) Power Allocation First with a given value of 120582 and 120583 wewill have

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

(1 + 120583

2120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

) minus 119902120589119878119898119899119896minus 120582119878119898119899119896minus 120583119877req) + 120582119875119905 minus 119902119875119862

(22)

We take the partial derivative of (22) with respect to 119878119898119899119896

and then make the derivative equal to zero According tothe Karush-Kuhn-Tucker (KKT) conditions [26] the optimal119878119898119899119896

for a given value of 120582 and 120583 can be obtained in a water-filling form as

119878lowast

119898119899119896(120582 120583) = [

120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

120601119898119899119896 (23)

where [119909]+ = max(119909 0) Equation (23) is a type of multipla-nar water-filling solution [12]

For any path 119875(119898 119899 119896) we maximize the equivalentchannel gain 119886

119898119899119896according to the instantaneous channel

information and the Fibonacci method to obtain the optimal120591lowast

119898119899119896 For a given value of 120591lowast

119898119899119896 according to (23) and (12)

the source transmission power for the first time slot 1198751199041119898119899119896

thesource transmission power for the second time slot 1198751199042

119898119899119896 and

relay node transmission power 119875119903119898119899119896

can be given by

1198751199041

119898119899119896=119886119899119896(1 minus 120591

lowast

119898119899119896) + 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

1198751199042

119898119899119896=(119886119898119896minus 119888119898119896) (1 minus 120591

lowast

119898119899119896) minus 119887119899119896120591lowast

119898119899119896

119886119898+ 119886119899119896minus 119888119898119896

119875119898119899119896

119875119903

119898119899119896= 120591lowast

119898119899119896119875119898119899119896

(24)

(2) Channel-User Assignment Substituting 119878119898119899119896

in (23) into(22) 119892(120582 120583) can therefore be reformulated as

119892 (120582 120583) = maxΦS119871 (Φ S 120582 120583 119902)

=

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583) + 120582119875max minus 119902119875119862

minus 120583119877req

(25)

where 119860119898119899119896(120582 120583) can be given by

119860119898119899119896(120582 120583)

=1

2120596119896log(1 + 119886

119898119899119896[120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

)

minus (120582 + 120589119902) [120596119896(1 + 120583)

2 ln 2 (120582 + 119902120589)minus1

119886119898119899119896

]

+

(26)

Then we have

maxΦ

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120601119898119899119896119860119898119899119896(120582 120583)

st (3)

0 le 120601119898119899119896le 1 forall119898 119899 119896

(27)

To calculate Φ = [120601119898119899119896]119873times119873times119870

which turn into solving(X ymn

) the model of (27) can be given by

maxXymn

119873

sum

119898=1

119873

sum

119899=1

119909119898119899

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583) (28)

We solve 119909119898119899

and 119910119898119899119896

in (28) over two stages Firstly wemaximize the inner-sum term over 119910119898119899

119896for each subcarrier

pair (119898 119899) that is

1198601015840

119898119899(120582 120583) = max

ymn

119870

sum

119896=1

119910119898119899

119896119860119898119899119896(120582 120583)

st119870

sum

119896=1

119910119898119899

119896= 1

0 le 119910119898119899

119896le 1 forall119896

(29)

The optimal 119910119898119899119896

in (29) is readily gained by

119910119898119899

119896

lowast

=

1 if 119896 = arg max1le119897le119870

119860119898119899119897(120582 120583)

0 otherwise(30)

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Mobile Information Systems

Secondly taking 1198601015840119898119899(120582 120583) into (28) we reduce (28) to

the following linear optimization problem

maxX

119873

sum

119898=1

119873

sum

119899=1

1199091198981198991198601015840

119898119899(120582 120583)

st119873

sum

119899=1

119909119898119899= 1 forall119898

119873

sum

119898=1

119909119898119899= 1 forall119899

0 le 119909119898119899le 1 forall119898 119899

(31)

It is easy to know that a binary solution Xlowast isin 0 1119873times119873to this problem (31) will always exist Further with thebinary constraints onX this is depicted as a two-dimensionalassignment problem such as the Hungarian Algorithm [27]

Finally the optimal 120601lowast119898119899119896

in (25) is gained as the productof 119909lowast119898119899

and 119910119898119899119896

lowast Since both 119909lowast119898119899

and 119910119898119899119896

lowast are binary 120601lowast119898119899119896

isbinary

(3) Subgradient Updating The previous subsection proposesthe method to get the Lagrange function 119892(120582 120583) with a givenvalue of the Lagrange multipliers 120582 and 120583 Since the dualfunction is differentiable we use the subgradient method toupdate 120582 and 120583 The subgradient-updated equations of 120582 and120583 are

120582 (119899 + 1) = [120582 (119899) minus 120572120582(119899)(119875

119905minus

119873

sum

119898=1

119873

sum

119899=1

119870

sum

119896=1

119878119898119899119896)]

+

120583 (119899 + 1) = [120583 (119899) minus 120572120583(119899)

sdot (1

2

119870

sum

119896=1

119873

sum

119898=1

119873

sum

119899=1

120596119896120601119898119899119896

log2(1 + 119886

119898119899119896

119878119898119899119896

120601119898119899119896

)

minus 119877req)]

+

(32)

where 119899 120572120582(119899) and 120572

120583(119899) denote the iteration index and the

positive diminishing the 119899th inner iteration step sizes of thedual variables respectively The subgradient method aboveis guaranteed to converge to the optimal dual variables ifthe step sizes are chosen following the diminishing step sizepolicy [26]

5 Simulation Results and Analysis

We have obtained the optimal solution of joint sourceallocation scheme in the previous sections In this section wecompare the performances of the proposed joint EE resourceallocation schemewith the other resource allocation schemes

The multiusers OFDM channel gains are modeled ascomplex Gaussian distributions with a zero mean and thevariance is 119888 sdot 119889minusV

119894119895 indicating that ℎ

119894119895sim 119862119873(0 119888 sdot 119889

minusV119894119895) [19]

Aver

age t

rans

miss

ion

rate

(bits

sec

her

tz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

Pmax (dBm)

9

8

7

6

5

4

3302520151050

Figure 2 Average SE performance of different resource allocationschemes

We assume that there is a circle which is centered in therelay node with a radius of 500m Users as shown inFigure 1 are randomly distributed in the right half circleCorrespondingly the source is randomly distributed in theleft half circle The number of users is set to 119870 = 4 and thenumber of OFDM system subcarriers is set to 119873 = 16 Thefairness factors for every user are set to [015 015 035 035]The variances of the AWGNs are set to 1205902

119903= 1205902

119896= 1198730=

minus131 dBm and the power consumption of the constant circuitis set to 119875

119862= 30 dbm

51 Analysis of Average SE Performance The average trans-mission rate of our proposed EE scheme is comparedwith theothers schemes proposed in [13 22 23] as shown in Figure 2The resource allocation schemes proposed in [13 22 23] aredesigned to maximize the total system SE We can see fromFigure 2 that the average SE performance of our proposedscheme monotonically increases when 119875max le 18 dBm andthen saturates when 119875max gt 18 dBm When 119875max le 18 dBmbased on the analysis in previous section the proposedalgorithm by solving 119865(0) is also aimed at maximizing thetotal system SE hence the profile monotonically increaseswith respect to 119875max However when 119875max gt 18 dBmthe proposed algorithm stops consuming more power fortransmitting the radio signals to maximize the system energyefficiency In [13 22 23] the average SE of the other resourceallocation schemes monotonically increases with respect to119875max as 119865(0) by sacrificing the energy efficiency of thesystem Moreover as expected our proposed scheme canprovide a better SE performance compared to the otherresource allocation schemes when 119875max le 18 dBm it canobtain a 30 dB 20 dB and 16 dB gain approximately com-pared with schemes proposed in [13] [23] and [22] respec-tively

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 9Av

erag

e EE

(bits

Joul

ehe

rtz)

The scheme in [13]The scheme in [22]

The scheme in [23]Proposed EE scheme

65

6

55

5

45

4

35

3

25

302520151050

Pmax (dBm)

Figure 3 Average EE performance of different resource allocationschemes

In summary [23] exploited an optimal subcarrier-userassignment with an equal power assignment policy and hasthe worst performance Reference [13] jointly considers onlythe power allocation and the subcarrier pairing without con-sidering the channel-user assignment thus it has the secondworst performance Reference [22] considers the maximizedSE problem of jointly optimizing the subcarrier pairing thechannel-user assignment and the power allocation but theperformance is still not as good as our scheme because wehave adopted a new relaying strategy In this paper comparedwith traditional OFDM relaying networks [22] the sourceis permitted to retransfer the same data transmitted by itin the first time slot further improving the system capacityperformance

52 Analysis of Average EE Performance Figure 3 shows thecomparison of our proposed resource allocation method andthe other methods proposed in [13 22 23] in allusion tomaximize the total average EE performance

We can see from Figure 3 that schemes in [22 23] aimedat maximizing the total system SE monotonically increasewhen 119875max le 18 dBm and monotonically decrease when119875max gt 18 dBm But the average EE performances of ourmethod raised and [13] monotonically increase when 119875max le18 dBm and then saturate when 119875max gt 18 dBm Howeverthe average EE of our method raised remains constant while119875max gt 18 dBm after it attains an optimal EE of about56 bitJHz This is because once the maximum EE of thesystem has been obtained the source and relay will not takeany energy to do useless transmission It is also shown inFigures 2 and 3 that our proposed EE scheme provides bothbetter EE and SE performances when 119875max le 18 dBm causedby the joint channel-user assignment and the new relayingstrategy Reference [13] also considers a joint channel-user

EE (b

psJo

ule

hert

z)

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Number of out iterations

6

5

4

3

2

1

010987654321

Figure 4 Convergence performance of the outer iteration algo-rithm for EE maximization

SE (b

psh

ertz

)

Number of out iterations

13

12

11

10

9

8

7

6

5

10987654321

Rreq = 2bpshertzRreq = 4bpshertz

Rreq = 6bpshertzRreq = 8bpshertz

Figure 5 Convergence performance of SE under minimum QoSconstrain

assignment but continues to have lower EE performance thanour scheme since we adopt a new relaying strategy

53 Convergence of Iterative Algorithm Figures 4 and 5illustrate the evolution of the average system energy efficiencyof the proposed iterative algorithm for different minimumsystem service rate requirements based on the fractionalprogramming method in [25]

Figures 4 and 5 display the convergence performances ofthe total EE and the total SE under different QoS constraints

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Mobile Information Systems

respectively As shown in Figure 5 EE 119902 monotonicallyincreases with the number of outer iterations and thensaturates when the number of iterations exceeds eight On thebasis of the analysis in Section 32 119902will achieve an optimal 119902lowastwhen the outer cycle is convergent Particularly in this paperby setting119877req = 0 indicating that there is no QoS constraintwe will obtain the optimal 119902lowast

119877req=0 = 56 bitsHz and thecorresponding SE will be 119877

119877req=0 = 63 bitsHz Accordingto the above-mentioned analysis we can conclude that ourresource allocation method raised could obtain an optimalEE performance under different QoS constraints with a goodglobal astringency and a rapid convergence ability

6 Conclusions

This paper discuss energy efficiency problem for multi-ple users OFDM relaying scenario We investigate a jointresource allocation scheme including the power allocationthe subcarrier pairing and the channel assignment to theuser We maximize the overall system EE guaranteeingthe constraints of total power and a minimum SE Theoptimization model is simplified into a typical fractionalprogramming problem which is testified to be quasiconcaveThus we can adopt Dinkelbach method to deal with MINLPproblem proposed to achieve the optimal solution Thesimulation results show that the joint resource allocationmethod proposed can achieve an optimal EE performanceunder the minimum system service rate requirement with agood global convergence

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61571234) and National Basic ResearchProgram of China (973 Program (2013CB329005) and 863Program (2014AA01A705))

References

[1] Z He and M Zhai ldquoA new resource allocation algorithm insingle-user power line communication systemrdquo in Proceedingsof the International Symposium on Signals Systems and Electron-ics (ISSSE rsquo10) vol 1 pp 1ndash3 IEEE Nanjing China September2010

[2] H Zhang Y Liu and M Tao ldquoResource allocation withsubcarrier pairing in OFDMA two-way relay networksrdquo IEEEWireless Communications Letters vol 1 no 2 pp 61ndash64 2012

[3] Y Liu andW Chen ldquoAdaptive resource allocation for improvedDF aided downlink multi-user OFDM systemsrdquo IEEE WirelessCommunications Letters vol 1 no 6 pp 557ndash560 2012

[4] S Khakurel C Leung and T Le-Ngoc ldquoFast optimal energy-efficient resource allocation for downlink multi-user OFDMsystemsrdquo in Proceedings of the IEEE 24th Annual International

Symposium on Personal Indoor and Mobile Radio Communica-tions (PIMRC rsquo13) pp 1671ndash1675 IEEE LondonUK September2013

[5] A Akbari R Hoshyar and R Tafazolli ldquoEnergy-efficientresource allocation inwireless OFDMA systemsrdquo in Proceedingsof the IEEE 21st International Symposium on Personal Indoorand Mobile Radio Communications (PIMRC rsquo10) pp 1731ndash1735IEEE Instanbul Turkey September 2010

[6] G Miao N Himayat and G Y Li ldquoEnergy-efficient linkadaptation in frequency-selective channelsrdquo IEEE Transactionson Communications vol 58 no 2 pp 545ndash554 2010

[7] T Wang Y Fang and L Vandendorpe ldquoPower minimizationforOFDMtransmissionwith subcarrier-pair based opportunis-tic DF relayingrdquo IEEE Communications Letters vol 17 no 3 pp471ndash474 2013

[8] Y Chen X M Fang and Y Zhao ldquoEnergy-efficient adap-tive power allocation in orthogonal frequency division multi-plexing-based amplify-and-forward relay linkrdquo IET Communi-cations vol 7 no 15 pp 1676ndash1687 2013

[9] Y Wang J Zhang and P Zhang ldquoLow-complexity energy-efficient power and subcarrier allocation in cooperative net-worksrdquo IEEE Communications Letters vol 17 no 10 pp 1944ndash1947 2013

[10] G Y Li Z Xu C Xiong et al ldquoEnergy-efficient wireless com-munications tutorial survey and open issuesrdquo IEEE WirelessCommunications vol 18 no 6 pp 28ndash35 2011

[11] D Wang X Wang and X Cai ldquoOptimal power control formulti-user relay networks over fading channelsrdquo IEEE Trans-actions on Wireless Communications vol 10 no 1 pp 199ndash2072011

[12] K T K Cheung S Yang and L Hanzo ldquoAchieving maximumenergy-efficiency in multi-relay OFDMA cellular networksa fractional programming approachrdquo IEEE Transactions onCommunications vol 61 no 7 pp 2746ndash2757 2013

[13] E Khaledian N Movahedinia and M R KhayyambashildquoSpectral and energy efficiency in multi hop OFDMA basednetworksrdquo in Proceedings of the IEEE 7th International Sympo-sium on Telecommunications (IST rsquo14) pp 123ndash128 Tehran IranSeptember 2014

[14] S Boyd and J Mattingley Branch and Bound Methods NotesStanford University Stanford Calif USA 2003

[15] A Hottinen and T Heikkinen ldquoOptimal subchannel assign-ment in a two-hop OFDM relayrdquo in Proceedings of the IEEE 8thWorkshop on Signal Processing Advances in Wireless Communi-cations pp 1ndash5 Helsinki Finland June 2007

[16] W Wang S Yan and S Yang ldquoOptimally joint subcarriermatching and power allocation in OFDM multihop systemrdquoEURASIP Journal on Advances in Signal Processing vol 2008Article ID 241378 2008

[17] W Ying Q Xin-chunW Tong and B-L Liu ldquoPower allocationand subcarrier pairing algorithm for regenerative OFDM relaysystemrdquo in Proceedings of the IEEE 65th Vehicular TechnologyConference (VTC rsquo07) pp 2727ndash2731 IEEE Dublin IrelandApril 2007

[18] H Mu M Tao W Dang et al ldquoJoint subcarrier-relay assign-ment and power allocation for decode-and-forwardmulti-relayOFDM systemsrdquo in Proceedings of the 4th International Confer-ence on Communications and Networking in China (ChinaCOMrsquo09) pp 1ndash6 IEEE Xirsquoan China 2009

[19] C-N Hsu P-H Lin and H-J Su ldquoJoint subcarrier pairing andpower allocation for OFDM two-hop systemsrdquo in Proceedings

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 11

of the IEEE International Conference on Communications (ICCrsquo10) pp 1ndash5 IEEE Cape Town South Africa May 2010

[20] K Seong M Mohseni and J M Cioffi ldquoOptimal resourceallocation for OFDMA downlink systemsrdquo in Proceedings of theIEEE International Symposiumon InformationTheory (ISIT rsquo06)pp 1394ndash1398 Seattle Wash USA July 2006

[21] Y Cui and V K N Lau ldquoDistributive stochastic learning fordelay-optimal OFDMA power and subband allocationrdquo IEEETransactions on Signal Processing vol 58 no 9 pp 4848ndash48582010

[22] M Hajiaghayi M Dong and B Liang ldquoJointly optimal channeland power assignment for dual-hop multi-channel multi-userrelayingrdquo IEEE Journal on Selected Areas in Communicationsvol 30 no 9 pp 1806ndash1814 2012

[23] G Li and H Liu ldquoResource allocation for OFDMA relaynetworks with fairness constraintsrdquo IEEE Journal on SelectedAreas in Communications vol 24 no 11 pp 2061ndash2069 2006

[24] Z Zhou and Q Zhu ldquoJoint optimization scheme for powerallocation and subcarrier pairing in OFDM-based multi-relaynetworksrdquo IEEE Communications Letters vol 18 no 6 pp1039ndash1042 2014

[25] W Dinkelbach ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 7 no 13 pp 492ndash498 1967

[26] S P Boyd and L Vandenberghe Convex Optimization Cam-bridge University Press New York NY USA 2004

[27] H W Kuhn ldquoThe Hungarian method for the assignmentproblemrdquo Naval Research Logistics Quarterly vol 2 pp 83ndash971955

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014