IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 3, MARCH 2013 541

On the Energy Efficiency Optimality ofOFDMA for SISO-OFDM Downlink System

Qingjiang Shi, Weiqiang Xu, Dapeng Li, Yaming Wang, Xiaowei Gu, and Wenshu Li

Abstract—This letter considers energy–efficient resource allo-cation for a general SISO-OFDM downlink system. A fundamen-tal result is proven that OFDMA is optimal for achieving themaximum energy efficiency measured by bits per joule. Usingthe OFDMA solution structure, we turn the nonconvex energy–efficient power allocation problem into a quasi–convex optimiza-tion problem and propose an efficient power allocation algorithmwhich requires solving at most two simple subproblems. Theachieved energy efficiency performance of the proposed powerallocation algorithm is verified via simulations.

Index Terms—Energy–efficient resource allocation, SISO–OFDM downlink system, OFDMA.

I. INTRODUCTION

OVER past decades, most efforts are made to improve thesystem throughput or spectral efficiency of wireless net-

works, and various resource allocation methods are developedto obtain higher QoS data transmission service in wirelesscommunication. Generally, high network throughput implieslarge energy consumption. As a result, the current informationand communication technology contributes more and moregreenhouse gas emissions, which poses a challenge in cellularsystem design. As a viable solution, energy–efficient systemdesign is becoming an increasingly important topic in wirelesscommunication and recently has gained a lot of attention inboth industry and academia [1].

So far, there are two main classes of energy efficientsystem design methods. The first class is network planing.For example, it is shown in [2] that, reducing cell size canincrease energy efficiency which is measured by the numberof bits transmitted per unit energy. In this letter, we focus onthe second class—energy-efficient resource allocation [1], [3]–[7]. As opposed to the traditional spectral-efficient resourceallocation, dynamic resource allocation is designed to achievethe maximum energy efficiency. Miao et. al. [3], [4] have con-sidered energy efficiency optimization for an uplink OFDMAsystem and showed that energy efficiency is improved byrate adaption or adjusting the distribution of subchannelsamong users. In [5], the authors assume flat–fading across

Manuscript received December 27, 2012. The associate editor coordinatingthe review of this letter and approving it for publication was J. Armstrong.

Q. Shi, W. Xu, Y. Wang, X. Gu, and W. Li are with the School of Info.Sci. & Tech., Zhejiang Sci-Tech University (ZSTU), Hangzhou 310018, China(e-mail: qing.j.shi@gmail.com).

D. Li is with Nanjing University of Posts and Telecommunications, Nanjing210003, China.

This work is supported by the National Nature Science Foundation ofChina under grant 61272311, 61070063, 60702081, 61201162, the KeyProject of Chinese Ministry of Education under grant 212066, the ZhejiangProvincial Science Foundation under grants LY12F02042, LQ12F01009, andthe Science Foundation of Zhejiang Sci-Tech University (ZSTU) under GrantNo. 1203805Y.

Digital Object Identifier 10.1109/LCOMM.2013.020413.122898

subcarriers and have considered energy-efficient bandwidth al-location for a downlink OFDMA system. Accounting for QoSconstraints, [6] has studied the problem of energy-efficientjoint subcarrier allocation and rate/power allocations. Notethat, all the above works assume OFDMA system where nointer-user interference exists. For interference-limited multi-cell system/interference channel, [7] has studied distributedenergy-efficient power optimization based on a noncooperativegame method.

In this letter, we consider energy-efficient power optimiza-tion for a general single cell SISO–OFDM downlink systemwhere subcarriers are allowed to be shared among users.[8] has shown a fundamental result on the optimality ofOFDMA for spectral efficient resource allocation in SISO-OFDM system. As a counterpart, an open question is whetherOFDMA is also optimal for energy-efficiency optimization.This letter first answers this question by mathematical proof.Then the nonconvex energy–efficient power allocation prob-lem is simplified to a quasi-convex optimization problemby using the OFDMA solution structure. Finally, a simpleenergy-efficient power allocation algorithm is proposed whichrequires only solving at most two simple subproblems.

The rest of the paper is organized as follows. In SectionII, we describe the system model and give the problem for-mulation. In Section III, we prove the optimality of OFDMAfor energy efficiency. A simple power allocation algorithm isproposed in Section IV. Section V demonstrates simulationresults while Section VI concludes the paper.

II. PROBLEM FORMULATION

Consider a single-cell SISO downlink communication sys-tem where a base station (BS) sends signals to K usersover a frequency-selective channel. Using N -point FFT basedOFDM, the frequency-selective channel can be equivalentlyviewed as N flat fading channel over N subcarriers [9]. Thesystem model can be described as

yk,n = hk,n

K∑k=1

√pk,nxk,n + zk,n,

n = 1, 2, . . .N, k = 1, 2, . . . ,K.

(1)

where yk,n is the received signal by user k on subcarrier n,hk,n denotes the channel coefficient between the user k and BSon subcarrier n, xk,n ∼ CN (0, 1) is the transmitted signal foruser k on subcarrier n, pk,n is the transmission power used forthe user k’s signal, zk,n ∼ CN (0, σ2

k,n) is the additive whiteGassuan noise (AWGN).

By assuming that each user uses single-user detection (i.e.,view other users’ signal as noise), the sum rate in nats/Hz of

1089-7798/13$31.00 c© 2013 IEEE

542 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 3, MARCH 2013

the system is given by

R =1

N

N∑n=1

K∑k=1

log

(1 +

pk,n|hk,n|2∑j �=k pj,n|hk,n|2 + σ2

k,n

). (2)

As in [10], we adopt the following power consumption modelfor the base station:

PBS = a

N∑n=1

K∑k=1

pk,n + Poff

where the coefficient a accounts for the power consumptionthat scales with the transmission (radiated) power due toamplifier and feeder losses as well as cooling of sites. Theterm Poff denotes a power offset which is independent of thetransmission power and induced by signal processing, batterybackup, as well as site cooling.

Then the energy efficiency can be expressed as [3]

E =R

a∑N

n=1

∑Kk=1 pk,n + Poff

. (3)

We are here interested in maximizing the system’s energyefficiency, which is mathematically equivalent to solving

max{pk,n≥0}

∑Nn=1

∑Kk=1 log

(1 +

pk,n|hk,n|2∑j �=k pj,n|hk,n|2+σ2

k,n

)a∑N

n=1

∑Kk=1 pk,n + Poff

s.t.K∑

k=1

pk,n ≤ Pn, n = 1, 2, . . . , N,

N∑n=1

K∑k=1

pk,n ≤ Ptot.

(4)

where Pn denotes the spectral mask constraint on the subcar-rier n and Ptot denotes the total power constraint.

In our resource allocation scheme, subcarriers are allowedto be shared among users. Hence, there may be inter-userinterference in the system. A suboptimal but interference-freeresource allocation scheme is to employ OFDMA where eachsubcarrier is allocated only to one user in order to ensure nointer-user interference. However, finding the best subcarrierallocation scheme requires solving an integer optimizationproblem which is not easy. In the next section, we show thatthe OFDMA scheme is optimal for achieving the best energyefficiency.

III. OPTIMALITY OF OFDMA

In this section, we demonstrate our main result.Lemma 1: Let {p∗k,n} denotes the optimal solution of the

energy efficiency optimization (EEO) problem (4) and defineq∗n �

∑Kk=1 p

∗k,n. Then {p∗k,n} is also the optimal solution of

the following problem

max{pk,n≥0,∀k,n}

∑Nn=1

∑Kk=1 log

(1 +

pk,n|hk,n|2∑j �=k pj,n|hk,n|2+σ2

k,n

)a∑N

n=1 q∗n + Poff

s.t.

K∑k=1

pk,n = q∗n, n = 1, 2, . . . , N,

(5)

Proof: Let E1 be the optimal value of the EEO problemand E2 be the optimal value of problem (5). Because {p∗k,n}is the optimal solution of the EEO problem, we have q∗n ≤Pn, ∀n, and

∑Nn=1 q

∗n ≤ Ptot. Hence, the optimal solution of

problem (5) must be a feasible solution of the EEO problem,implying E2 ≤ E1. Since p∗k,n is feasible to problem (5),we have E1 ≤ E2. Hence, we conclude E2 = E1, implyingthat {p∗k,n} is also the optimal solution of problem (5). Thiscompletes the proof.

Based on the result of Lemma 1, we have the followingproposition.

Proposition 1: Let {p∗k,n} denotes the optimal solution of

the EEO problem (4). Define q∗n �∑K

k=1 p∗k,n and kn =

maxk|hk,n|2σ2k,n

. The optimal solution of the EEO problem must

have an OFDMA structure, i.e.,

p∗kn,n = q∗n, p∗k,n = 0, ∀k �= kn. (6)

Proof: By noting that problem (5) is separable across nand

∑Kk=1 pk,n = q∗n, problem (5) reduces to N subproblems.

The n-th subproblem is

max{pk,n}

K∑k=1

log

(1 +

pk,n(q∗n − pk,n) + σ2

k,n/|hk,n|2)

s.t.

K∑k=1

pk,n = q∗n,

pk,n ≥ 0, ∀k.

(7)

From Lemma 1, it is easily known that, p∗k,n, k = 1, 2, . . . ,K ,is also the optimal solution of problem (7). On the other hand,it is shown in [8] that the solution of problem (7) must be anextreme point of the polyhedron specified by the constraints.Hence, we must have,

p∗kn,n = q∗n, p∗k,n = 0, ∀k �= kn. (8)

This completes the proof.Proposition 1 suggests that, at the optimality each subcarrier

can be only allocated to one user, i.e., OFDMA is optimal forenergy-efficient design of the SISO-OFDM system. Hence, theinter-user interference in the EEO problem (4) is null, i.e., theEEO problem can be simplified.

IV. OPTIMAL POWER ALLOCATION ALGORITHM

In this section, we first turn the EEO problem into a quasi-convex optimization problem by using the optimal solutionstructure. We then propose an efficient and optimum powerallocation algorithm which requires only solving at most twosimple subproblems.

For notation convenience, let pn � pkn,n, hn � |hkn,n|2σ2kn,n

.

Based on the result of Proposition 1, the EEO problem reducesto the following quasi-convex optimization problem

max{pn}

∑Nn=1 log(1 + pnhn)

a∑N

n=1 pn + Poff

s.t.0 ≤ pn ≤ Pn, n = 1, 2, . . . , N,N∑

n=1

pn ≤ Ptot.

(9)

SHI et al.: ON THE ENERGY EFFICIENCY OPTIMALITY OF OFDMA FOR SISO-OFDM DOWNLINK SYSTEM 543

Problem (9) is a quasi-convex optimization problem since theobjective function possesses a structure of a concave functionover a linear function. It is well-known that [11], the bisectionmethod can be used to solve quasi-convex problems. At eachiteration of the bisection method, given t = tl+tu

2 ∈ [tl tu]where tl and tu are respectively the lower bound and upperbound of the optimal value of problem (9), we solve a convexfeasibility problem or equivalently solve the following convexoptimization problem

max{pn}

N∑n=1

log(1 + pnhn)− t

(a

N∑n=1

pn + Poff

)

s.t.0 ≤ pn ≤ Pn, n = 1, 2, . . . , N,N∑

n=1

pn ≤ Ptot.

(10)

Note that, problem (10) does not admit a closed-form ora simple solution. In general, the bisection method requiressolving a number of (typically 10 ∼ 20) problem (10) fordifferent t. We here propose a more efficient algorithm tosolve problem (9) which requires only solving at most twosimple subproblems.

First, we give the following proposition which characterizesthe solution of problem (9).

Proposition 2: If the solution of the problem

max{pn}

∑Nn=1 log(1 + pnhn)

a∑N

n=1 pn + Poff

s.t.0 ≤ pn ≤ Pn, n = 1, 2, . . . , N,

(11)

satisfies the total power constraint∑N

n=1 pn ≤ Ptot, then itis also the solution of problem (9), otherwise, the solution ofproblem (9) satisfies the total power constraint with equality,and is also the solution of problem

max{pn}

N∑n=1

log(1 + pnhn)

s.t.0 ≤ pn ≤ Pn, n = 1, 2, . . . , N,N∑

n=1

pn = Ptot.

(12)

Proof: The first part of the proposition can be proven bynoting that problem (11) has the same objective function asproblem (9) but with a larger feasible set.

The second part is true, since, in this case the solution ofproblem (9) is also the solution of the problem

max{pn}

∑Nn=1 log(1 + pnhn)

a∑N

n=1 pn + Poff

s.t.0 ≤ pn ≤ Pn, n = 1, 2, . . . , N,N∑

n=1

pn = Ptot,

(13)

which is equivalent to

max{pn}

∑Nn=1 log(1 + pnhn)

aPtot + Poff

s.t.0 ≤ pn ≤ Pn, n = 1, 2, . . . , N,N∑

n=1

pn = Ptot.

(14)

This completes the proof.Remark 1: From the result of Proposition 2, it is known

that the solution of problem (9) can be obtained by solving atmost two subproblems, first (11) and then (14) if necessary.

Remark 2: Problem (14) is the well-known sum-rate max-imization problem. Hence, it can be expected that, when thetotal power Ptot is much smaller, the total power constraintof the EEO problem would be active at the solution andconsequently the throughput optimal design could achieve themaximum energy efficiency. This is validated via simulationin Section V.

Problem (14) has a water-filling like solution:

pn �[ν − 1

hn

]Pn

0

, n = 1, 2, . . . , N, (15)

where the water-level ν is found to satisfy the total powerconstraint

∑Nn=1 pn = Ptot. Problem (11) can be solved by the

bisection method. At each iteration of the bisection method,given t = tl+tu

2 ∈ [tl tu] where tl and tu are respectively thelower bound and upper bound of the optimal value of problem(11), we solve the following optimization problem

max{pn}

N∑n=1

log(1 + pnhn)− t

(a

N∑n=1

pn + Poff

)

s.t. 0 ≤ pn ≤ Pn, n = 1, 2, . . . , N.

(16)

Different from (10), problem (11) admits a closed-form solu-tion

pn �[1

ta− 1

hn

]Pn

0

, n = 1, 2, . . . , N,

The whole algorithm for solving problem (9) is summarizedin Table I, where ε is an accuracy control parameter. Steps 1-15 in Table I solves problem (11) while the remaining steps,which is also based on the bisection method, solves problem(14). Note that, the total power

∑Nn=1 pn is an increasing

function in t. Hence, we use Steps 7-9 to quickly identify thecase with active total power constraint. The whole algorithmis based on the bisection method and thus has complexity ofO(log2(

1ε )N

). We emphasize again here that, compared to

the algorithm that directly uses the bisection method to solve(9), the proposed algorithm requires only solving at most twosimple subproblems and thus more efficient.

V. SIMULATION RESULTS

In this section, simulations are conducted to evaluate the en-ergy efficiency achieved by our energy-efficient system designscheme. For comparison, we also present the energy efficiencyof the common system design scheme—throughput optimalsystem design scheme. Table II lists the system parameters

544 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 3, MARCH 2013

TABLE ITHE OPTIMAL POWER ALLOCATION ALGORITHM

• Input: {hn, Pn}, a, Poff , Ptot, ε• Output: the power allocation {pn} and the optimal value t

1 set tl = 0, tu = maxn hna

, μl = 0, μu = 1minn hn

+maxn Pn

2 while tu − tl ≥ ε do3 t = tl+tu

2

4 pn =[

1ta

− 1hn

]Pn

0, n = 1, 2, . . . , N

5 s =∑N

n=1 pn

6 v =∑N

n=1 log(1+pnhn)

as+Poff

7 if s ≥ Ptot and v < t8 go to Step 169 end if

10 if v > t11 tl = t12 else13 tu = t;14 end if15 end while16 while μu − μl > ε17 μ = µu+µl

2

18 pn =[μ− 1

hn

]Pn

0, n = 1, 2, . . . , N

19 s =∑N

n=1 pn20 if s < Ptot

21 μl = μ22 else23 μu = μ;24 end if25 end while26 t =

∑Nn=1 log(1+pnhn)

a∑

Nn=1 pn+Poff

TABLE IISYSTEM PARAMETERS

Carrier frequency 2 GHzSubcarrier bandwidth 15 kHz

Subcarrier number 1024Propagation environment Urban Macro, NLOS [13]

Frequency selective fading model Extended Pedestrian A [14]Noise power -132 dBm

Power offset Poff 40 dBmPower scaling factor a 4

Spectral mask level 15 dBmNumber of users in the cell 3

used in our simulation. The same noise power and spectralmask level are assumed for each user and subcarrier.

Figure 1 compares the energy efficiency achieved by theenergy-efficient design (denoted by ‘EEO’ in the figure) andthat by the throughput (or sum rate) optimal design (denotedby ‘SRM’), where the data is averaged over 1000 simulations.One can see that, when the total power level is small (≤26dBm), the throughput optimal design is also energy–efficient.Otherwise, the energy-efficient design is dramatically betterthan the throughput optimal design in terms of energy effi-ciency. Note that, when the total power level is large (≥46dBm), the energy efficiency of both design schemes keepinvariant. This is because that the total power constraint isinactive at the solution of both the EEO problem and thethroughput maximization problem when the total power levelis very large.

.20 25 30 35 40 45 50 55 60

0.5

1

1.5

2

2.5

3

Ptot

(dBm)

Ene

rgy

effi

cien

cy (

Mbi

ts/J

oule

)

EEOSRM

Fig. 1. Energy efficiency Vs. the total power level.

VI. CONCLUSION

We have proven that OFDMA is optimal for energy-efficient design in SISO-OFDM downlink system. By usingthe OFDMA solution structure, we have proposed efficientand optimal power allocation algorithm. However, the obtainedresults do not apply to energy-efficient design of MIMO-OFDM downlink system, which is left as our future work.

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