Energy-Efficient Resource Allocation in OFDMA Networks - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012

3767

Energy-Efficient Resource Allocation in OFDMA Networks Cong Xiong, Geoffrey Ye Li, Fellow, IEEE, Shunqing Zhang, Yan Chen, and Shugong Xu, Senior Member, IEEE

Abstract—The widespread application of multimedia wireless services and requirements of ubiquitous access have triggered rapidly booming energy consumption at both the base station side and the user equipment (UE) side. Hence, energy-efficient design in wireless networks is very important and is becoming an inevitable trend. In this paper, we study the energy-efficient resource allocation in both downlink and uplink cellular networks with orthogonal frequency division multiple access (OFDMA). For the downlink transmission, the generalized energy efficiency (EE) is maximized while for the uplink case the minimum individual EE is maximized, both under certain prescribed perUE quality-of-service (QoS) requirements. For both transmission scenarios, we first provide the optimal solution and then develop a suboptimal but low-complexity approach by exploring the inherent structure and property of the energy-efficient design. For the downlink case, by modifying the original problem, we also find a computationally efficient and numerically tractable upper bound on the EE, which indicates the performance limit and is demonstrated to be quite tight if the number of subcarriers is larger than that of UEs and motivates us to find a nearoptimal approach relying on the quasiconcave relation between the modified EE and transmit power. Simulation results show that the energy-efficient design greatly improves EE compared with the conventional spectral-efficient design and the low-complexity suboptimal approaches can achieve a promising tradeoff between performance and complexity. Index Terms—Energy efficiency (EE), green radio, orthogonal frequency division multiple access (OFDMA), spectral efficiency (SE).

I. I NTRODUCTION

W

ITH the explosive growth of high-data-rate wireless services and requirement of ubiquitous access, energy consumption of wireless devices is rapidly increasing. Highlevel energy consumption at the base station side usually results in a large operational expenditure. As reported in [1], the radio access part alone takes up more than 70% of the total energy consumption for many mobile operators. At the user equipment (UE) side, high-level energy consumption brings much inconvenience, especially for mobile terminals that are not able to connect to an external charger, due to limited

Paper approved by O. Oyman, the Editor for Cooperative and Heterogeneous Networks of the IEEE Communications Society. Manuscript received September 23, 2011; revised March 5, May 18, and July 7, 2012. C. Xiong and G. Y. Li are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (e-mail: [email protected]; [email protected]). S. Zhang, Y. Chen, and S. Xu are with the GREAT Research Team, Huawei Technologies Co., Ltd., Shanghai, China (e-mail: {sqzhang, eeyanchen, shugong}@huawei.com). This work was supported in part by the Research Gift from Huawei Technologies Co. and the NSF under Grant No. 1017192. This paper was presented in part at the IEEE Global Communications Conference, Houston, Texas, USA, 2011. Digital Object Identifier 10.1109/TCOMM.2012.082812.110639

battery capacity and slow advancement of battery technology. Therefore, green radio and energy-efficient design in wireless networks is becoming increasingly important and prompting new waves of research and standard development activities [2]. Orthogonal frequency division multiple access (OFDMA) has been extensively studied for the next generation wireless communication systems, such as WiMAX and the 3GPP LTE. Recently, more attention has been paid to energy-efficient design in OFDMA networks. As a special case of OFDMA systems, energy-efficient orthogonal frequency division multiplexing (OFDM), which maximizes the energy efficiency (EE) (i.e., bits-per-Joule) has been addressed with consideration of circuit energy consumption in [3]–[5], respectively. It is demonstrated that a unique global maximum EE exists and can be obtained by the proposed algorithms therein. In [6], EE and spectral efficiency (SE) tradeoff in the downlink OFDMA networks with the specific proportional rate constraints has been addressed. However, there is only limited work on energyefficient design in uplink and downlink OFDMA networks with general quality-of-service (QoS), priority, and fairness requirements and with frequency-selective channels. In this paper, we address the energy-efficient resource allocation, including performance limit and optimal and lowcomplexity suboptimal algorithms, in both downlink and uplink of OFDMA networks with frequency-selective fading channels and consideration of QoS and priority/fainess issues. For the downlink scenario, we model the problem as the maximization of generalized EE under QoS requirements while for the uplink we model the problem as the maximization of the minimum individual EE under QoS requirements. For both cases, we first give the optimal solution then develop a low-complexity suboptimal solution by exploring the inherent structure and property of the energy-efficient design. For the downlink case, we also find a tractable upper bound on EE, which is further demonstrated to be relatively tight and is the foundation of a near-optimal approach. The rest of the paper is organized as follows. In Section II, we describe the system model and formulate the optimization problem for both the downlink and uplink cases. In Section III, the downlink scenario is investigated. We give the optimal solution by exhaustive search for reference, find a numerically tractable upper bound on EE and accordingly propose an efficient near-optimal approach, then propose a suboptimal approach by examining the inherent structure of the EE objective function. In Section IV, we provide both the optimal and low-complexity suboptimal solutions for the uplink scenario. Then, we present numerical results in Section V. Finally, we conclude the paper in Section VI.

c 2012 IEEE 0090-6778/12$31.00

3768


II. S YSTEM D ESCRIPTION AND P ROBLEM F ORMULATION In this section, we introduce the system model and formulate the problems of energy-efficient design. A. System Description We consider a single cell OFDMA network, either downlink or uplink, with K active UEs. The total bandwidth, B, is B . divided into N subcarriers, each with a bandwidth of W = N Assume that each subcarrier is exclusively assigned to at most one UE each time to avoid interference among different UEs. Denote the transmit power and the channel frequency response of the kth UE on the nth subcarrier as pk,n and Hk,n , respectively. Then, the maximum achievable data rate of the kth UE on the nth subcarrier is accordingly pk,n |Hk,n |2 rk,n = W log2 1 + , (1) N0 W where N0 is the single-sided noise spectral density. Then, the aggregate rate for the kth UE and the overall throughput are Rk = ρk,n rk,n and R = Rk = ρk,n rk,n , n∈N

k∈K

k∈K n∈N

respectively, where ρk,n ∈ {1, 0} indicates whether or not the nth subcarrier is assigned to the kth UE, N = {1, 2, · · · , N } and K = {1, 2, · · · , K} denote the sets of all subcarriers and all UEs, respectively. Obviously, a feasible subcarrier assignment indictor matrix, ρ = [ρk,n ]K×N , should satisfy ρk,n ≤ 1, ∀ n ∈ N ; ρ ∈ [ρk,n ]K×N | k∈K (2) ρk,n ∈ {0, 1}, ∀ k ∈ K, n ∈ N .

blocks [7]. For the downlink transmission, the overall power consumption at the base station is given by [7] Ptot = ζP + Pc ,

where ζ is the reciprocal of drain efficiency of power amplifier and Pc represents the circuit power. Similarly, the overall power consumption at the kth uplink UE is modeled as Pktot = ζk Pk + Pkc ,

Sk ⊆ N and Sk

Sk = ∅, ∀ k = k ,

k=1

where Sk is the set of subcarriers assigned to the kth UE. Practically, the total transmit power of either base station or UE is nonnegative and also limited. Thus, any possible power allocation matrix, P = [pk,n ]K×N , should be subject to P ∈ P [pk,n ]K×N |pk,n ≥ 0, ∀ k ∈ K, ∀ n ∈ N ; pk,n ≤ Pmax (for downlink case); (3) k∈Kn∈N pk,n ≤ Pkmax , ∀k ∈ K (for uplink case) , n∈N

where Pmax and Pkmax represent the maximum total transmit power at base station for downlink transmission and at the kth UE for uplink transmission, respectively. The overall transmit power for the kth UE and the total transmit power are Pk = pk,n and P = Pk = pk,n . n∈N

k∈K

k∈K n∈N

Besides transmit power, the energy consumption also includes circuit energy consumption incurred by active circuit

(5)

where ζk and Pkc are the reciprocal of drain efficiency of power amplifier and circuit power, respectively. B. EE for Downlink Transmission We define the generalized EE for the downlink transmission as the weighted totally delivered bits per unit energy, i.e., ωk Rk DL ηEE k∈K , (6) ζP + Pc where the predetermined weights, ωk ’s, can potentially provide certain level of priority and/or fairness among the UEs and the generalized EE still reflects the notion of the conk∈K Rk in [3]–[5], [8], ventional EE that is defined as ζP +Pc where ω1 = ω2 = · · · = ωK = 1. To provide different service priorities and guarantee QoS for each UE, we consider the generalized EE under a series of traffic-related minimum ˇ k ’s, and the peak transmit power, P rate requirements, R max . The generalized EE optimization problem for the downlink transmission can be mathematically formulated as k∈K ωk n∈N ρk,n rk,n DL , (7a) ηÊE max ρ∈,P ∈P k∈K n∈N ζpk,n + Pc subject to

The subcarrier assignment constraint in (2) can be equivalently viewed from the following perspective K

(4)

ˇ k , ∀ k ∈ K, ρk,n rk,n ≥ R

(7b)

n∈N DL where ηÊE represents the optimal downlink EE.

C. EE for Uplink Transmission For the uplink scenario, we optimize the minimum individual EE, which guarantees satisfying EE as much as possible even for the worst UE. The individual EE of the kth UE for the uplink transmission is conventionally defined as UL ηEE,k

Rk , ζk Pk + Pkc

(8)

like that in [9]. Mathematically, the EE optimization problem for the uplink transmission can be expressed as

n∈N ρk,n rk,n UL min (9a) ηÊE max c ρ∈,P ∈P k∈K n∈N ζk pk,n + Pk subject to

ˇ k , ∀ k ∈ K. ρk,n rk,n ≥ R

(9b)

n∈N

III. D OWNLINK T RANSMISSION In this section, we will develop the optimal and lowcomplexity suboptimal approaches for the energy-efficient

XIONG et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN OFDMA NETWORKS

resource allocation in the downlink transmission and study the performance limit of the downlink energy-efficient design.

3769

filling equations pˇk,n =

A. Optimal Solution

Theorem 1. For any fixed subcarrier assignment indicator matrix ρ ∈ and its corresponding subcarrier assignment sets Sk ’s (∀k ∈ K), the maximum achievable EE at a certain total transmit power, P , namely, (ρ)

Rw (P ) pk,n ≥0 ζP + Pc k∈K ωk n∈Sk rk,n max , pk,n ≥0 ζP + Pc

ηÊE (P ) max

subject to

(10a)

ˇ k , ∀ k ∈ K, rk,n ≥ R

(10b)

n∈Sk

pk,n = P,

(10c)

k∈Kn∈Sk

has the following properties: (ρ)

(i) ηÊE (P ) is continuously differentiable and strictly quasiconcave in P , (ρ) (ii) ηÊE (P ) either strictly decreases or first strictly increases decreases with P starting from

and then strictly ˇ k ,1 P0 = k∈K Rk−1 Sk , R ⎧ ˆ (ρ) (P ) (ρ) dR ⎪ > 0 if ηÊE (P ) < ζ1 wdP ⎪ ⎨ (ρ) ˆ (ρ) (P ) dˆ η (P ) (ρ) dR (iii) EE , = 0 if ηÊE (P ) = ζ1 wdP dP ⎪ ⎪ (ρ) ˆ ⎩ (ρ) (P ) d R < 0 if ηÊE (P ) > ζ1 wdP (ρ) (ρ) ˆw where R maxpk,n ≥0 Rw (P ) = (P ) maxpk,n ≥0 k∈K ωk n∈Sk rk,n under constraints (10b) and (10c) is the maximum Weighted Sum Rate (WSR). And its derivative satisfies (ρ)

ˆ w (P ) dR ≡ dP

max

k∈K,n∈Sk

≡ |H

(12a)

ˇk . W log2 (μk gk,n ) = R

(12b)

n∈{n∈Sk |pˇk,n >0}

Problem (7) is in general NP-hard for the optimal solution. To obtain insight on the problem, we first investigate the properties of the case with a given subcarrier assignment, which are summarized in the following theorem and proved in Appendix A.

(ρ)

+ 1 μk − , ∀ n ∈ Sk , gk,n

ωk W gk,n log2 e 1 + pˆk,n gk,n

ωk W log2 e μk k∈K ωk W log2 e μ

max

if P = P0

,

(11)

pˆk,n = pˇk,n + μ −

1 gk,n

+ − pˇk,n

,

(13a)

1 − pˇk,n = P − pˇk,n , μ− gk,n

k∈K n∈{n∈Sk |pˆk,n >pˇk,n}

k∈Kn∈Sk

(13b) +

where (x) represents max(x, 0) and μk and μ are intermediate variables. For any strictly quasiconcave function, there is always a unique global maximum. Thus, Property (i) guarantees the existence and uniqueness of the global maximum and reveals (ρ) the differentiability of ηÊE (P ). Property (ii) further indicates that the maximum is always achieved at a finite transmit power. Property (iii) connects the sign of the first derivative with the relative size of the EE and the scaled reciprocal of the water-filling level. From Theorem 1, once the subcarrier assignment is fixed, the corresponding optimal power allocation strategy for (8) can be easily obtained by a derivative-assisted bisection method that is based on the single-UE water-filling in (12), and the multilevel water-filling in (13). The algorithm is named bisection-based power adaptation (BPA) and is sketched in Table I.2 The basic idea of the BPA algorithm (ρ) is to search bidirectionally for the maximum of ηÊE (P ) by utilizing Properties (i), (ii), and (iii). It starts by doing the (ρ) single-user water-filling to check the monotonicity of ηÊE (P ) (ρ) at P0 , which is shown from Line 1 to Line 5. If ηÊE (P ) is decreasing at P0 , the optimal EE is achieved at P0 and the BPA algorithm terminates; otherwise, it has to find another (ρ) feasible point at which ηÊE (P ) is decreasing or it finds out (ρ) that ηÊE (P ) is even increasing at Pmax , which is described from Line 6 to Line 26. Then, it searches for the optimal total transmit power (the corresponding EE is optimal) between the two boundary points (P (1) and P (2) ), which is illustrated from Line 27 to Line 37. The optimal solution to (7) can be obtained by applying the BPA algorithm to every feasible subcarrier assignment ρ ∈ and then choose the one with the maximum EE. However, the complexity is extremely high and makes it prohibitive for practical scenarios.

if P > P0

|2

is the channel-gain-to-noise ratio (CNR) where gk,n Nk,n 0W of the kth UE on the nth subcarrier and pˆk,n (n ∈ Sk ) is the (ρ) ˆw optimal power on the nth subcarrier for achieving R (P ). The optimal power can be calculated from the following water1 R (X , P ) and R−1 (X , R ) denote the maximum aggregate data rate k k k k achieved by optimally allocating total power Pk over subcarrier set X and the minimum transmit power required for realizing aggregate rate Rk over subcarrier set X for the kth UE, respectively.

B. Near-Optimal Solution To facilitate practical application of the optimal energyefficient design, we will first exploit and prove the quasicon(ρ) cave relation between an upper bound on maxρ∈ ηÊE (P ) and total transmit power, P . 2 Note that in Table I the variables with superscripts of (1) and (2) correspond to the two boundary points for the bisection search, respectively.

3770


TABLE I B ISECTION - BASED P OWER A DAPTATION (BPA) A LGORITHM .

Algorithm BPA ˇk , ∀ k ∈ K Input: ρ = [ρk,n ]K×N , Pc , Pmax ; ωk , R Output: P = [pk,n ]K×N 1. 2. 3. 4.

FOR each UE k ∈ K Do single-user water-filling using (12) to get pˇk,n and μk ; END (1) (1) pk,n ]K×N ; P (1) ← pk,n ; P (1) [pk,n ]K×N ← [ˇ

5.

ηÊE

(ρ),(1)

←

(ρ),(1)

ˇ

k∈K ωk Rk ζP (1) +Pc

k∈Kn∈Sk ωk W log 2 e ; μk k∈K

; d(1) ← max

6. IF ηÊE ≥ ζ1 d(1) 7. P [pk,n ]K×N ← P (1) ; 8. RETURN; 9. ELSE (2) ]K×N ← P (1); P (2) ← P (1) ; 10. P (2) [pk,n 11.

P (1) ← min κP (1) , Pmax , where κ > 1, e.g., κ ← 1.5;

12. 13.

Do multilevel water-filling with total transmit power P (1) using (13) to get P (1) ←[ˆ pk,n ]K×Nand μ; (ρ),(1)

←

k∈K

ωk

(1) n∈Sk log2 1+pk,n gk,n ζP (1) +Pc

ηÊE

15. 16. 17.

WHILE ηÊE < ζ1 d(1) && P (1) < Pmax (2) (1) P ← P ; P (2) ← P (1); (1) ← min κP (1) , Pmax ; P

18. 19.

Do multilevel water-filling with P (1) using (13) to get pk,n ]K×N and μ; P (1) ← [ˆ (ρ),(1)

k∈Kωk

log2 1+p

(1)

gk,n

k,n k ; d(1)← ωkWμlog2 e ; 20. ηÊE ← ζP (1) +Pc 21. END (ρ),(1) ≤ 1ζ d(1) 22. IF ηÊE 23. P [pk,n ]K×N ← P (1) ; 24. RETURN; 25. END 26. END 27. WHILE no convergence (1) (2) ; 28. P ← P +P 2 29. Do multilevel water-filling with P using (13) to get ← [ˆ pk,n ]K×N and μ; 30. P [pk,n ]K×N (ρ) k∈K ωk n∈Sk log2 (1+pk,n gk,n ) 31. ηÊE ← ; d ← ωk Wμlog2 e ; ζP +Pc (ρ) 32. IF ηÊE < 1ζ d 33. P (2) ← P ; P (2) ← P ; 34. ELSE 35. P (1) ← P ; P (1) ← P ; 36. END 37. END n∈S

subject to

; d(1)← ωkWμlog2 e ;

14.

(ρ),(1)

to be zero (although r˜k,n goes to infinity), which agrees with that the nth subcarriers is nearly not assigned to the kth UE. When ρ˜k,n is close to one, ρ˜k,n r˜k,n is close to ρk,n rk,n , which indicates that the nth subcarrier is almost entirely assigned to the kth UE. Therefore, when ρ˜k,n is close to zero or one, the approximation of ρk,n rk,n by ρ˜k,n r˜k,n becomes precise. Moreover, ρ˜k,n r˜k,n is easier to deal with mathematically compared to ρk,n rk,n . As a result of its nice tractability and acceptable accuracy, such approximation is widely used in literature on OFDMA resource allocation [10]– [13]. We will use ρ˜k,n r˜k,n instead of ρk,n rk,n to represent the rate of the kth UE on the nth subcarrier in the following modified EE optimization problem. We then formulate a modified EE optimization problem of (7) as follows ˜k,n r˜k,n k∈K ωk n∈N ρ UB , (16a) ηÊE max ρ∈ ˜ ,P ˜ ∈P ζp k,n + Pc k∈K n∈N ˇ k , ∀ k ∈ K. ρ˜k,n r˜k,n ≥ R

(16b)

n∈N

In (16), such fractional ρ˜k,n ’s can be either interpreted as frequency domain sharing of subcarriers [10], [11] or regarded as time domain sharing of subcarriers [12]. Clearly, by relaxing ρk,n ∈ {0, 1} to ρ˜k,n ∈ [0, 1] and replacing rk,n with r˜k,n , problem (16) always yields an upper bound on the EE of (7), UB DL ≥ ηÊE , although it does not necessarily guarantee a i.e., ηÊE solution where ρ˜k,n is either 0 or 1 and thus the satisfaction of (16b) cannot ensure the feasibility of (7b). The properties of the EE upper bound are summarized in Theorem 2 and proved in Appendix B. Theorem 2. The upper bound on the maximum achievable (ρ) EE, maxρ∈ ηÊE (P ), at a certain total transmit power, P , namely, Rw (ρ, ˜ P) ζP + Pc ˜k,n r˜k,n k∈K ωk n∈N ρ max , ˜ ∈,p ρ ˜ k,n ≥0 ζP + Pc

UB ηÊE (P )

subject to

max

˜ ∈,p ρ ˜ k,n ≥0

n∈N

ˇ k , ∀ k ∈ K, ρ˜k,n r˜k,n ≥ R

pk,n = P,

(17a)

(17b) (17c)

k∈K n∈N

has the following properties Define ρ, ˜ ˜, ρ˜k,n , and r˜k,n as follows ρ˜k,n ≤ 1, ∀ n ∈ N ; ρ˜ ∈ ˜ [˜ ρk,n ]K×N | k∈K

r˜k,n

(14) ρ˜k,n ∈ [0, 1], ∀ k ∈ K, n ∈ N . pk,n |Hk,n |2 = W log2 1 + . (15) ρ˜k,n N0 W

Note that when ρ˜k,n approaches zero, ρ˜k,n r˜k,n also tends

UB (i) ηÊE (P ) is continuously differentiable and strictly quasiconcave in P , UB (P ) either strictly decreases or first strictly in(ii) ηÊE creases and P, ⎧ then strictly decreases with ˆ w (P ) d R 1 UB ⎪ > 0 if ηÊE (P ) < , ⎪ ζ dP ⎨

(iii)

UB dˆ ηEE (P ) dP

UB (P ) = = 0 if ηÊE ⎪ ⎪ ⎩ < 0 if ηÛB (P ) > EE

ˆ w (P ) 1 dR , ζ dP ˆ w (P ) 1 dR , ζ dP

ˆ w(P ) where R ˜ P) = ˜ k,n ≥0 Rw (ρ, maxρ˜ ∈,p maxρ˜ ∈,p ω ρ ˜ r ˜ under constraints ˜ k,n ≥0 k∈K k n∈N k,n k,n


(17b) and (17c) is the maximum Modified Weighted Sum Rate (MWSR). And its derivative satisfies ˆ w (P ) ωk W ρ˜∗k,n gk,n log2 e dR ≡ max , k∈K,n∈N 1 + ρ dP ˜∗k,n gk,n p∗k,n

(18)

where ρ˜∗ = [˜ ρ∗k,n ]K×N and P ∗ = [p∗k,n ]K×N are the optimal subcarrier and power allocation matrices for achieving ˆ w(P ). R UB (P ), Theorem 2 demonstrates the quasiconcavity of ηÊE which is an upper bound on the EE that is defined as (ρ) DL ηÊE (P ) maxρ∈ ηÊE (P ) under constraints (7b) and (17c), in the transmit power, P , and implies the existence UB . More and the uniqueness of the global maximum, i.e., ηÊE importantly, as a result of the quasiconcavity, problem (16) can be decomposed into two layers and solved iteratively by the joint inner- and outer-layer optimization as follows

(i) Inner layer: For a given transmit power, P ≤ Pmax , UB find the maximum EE, ηÊE (P ), and (the sign of) its UB dˆ ηEE (P ) derivative, . dP (ii) Outer layer: Search for the transmit power that results in UB , by bisection power search like the the maximum, ηÊE bisection power search in the BPA algorithm. The bisection search in the outer-layer is clear and easy. UB The key lies in the inner-layer algorithm that finds ηÊE (P ) UB dˆ ηEE (P ) . For a given total transmit power, and (the sign of) dP UB (P ) is equivalent to the inner-layer subproblem to find ηÊE maximizing the constrained MWSR, which means solving ˆ w (P ). Since Rw (ρ, ˜ P) ω ˜k,n r˜k,n for R k k∈K n∈N ρ is proved to be strictly and jointly concave in ρ˜k,n and pk,n [10], and the constraint set is convex from Appendix B, the constrained MWSR maximization is in the standard form of a convex programming problem that can be solved by standard numerical methods such as the interior-point method [14]. When the optimal subcarrier assignment matrix, ρ ˜∗ , and power allocation matrix, P ∗ , for the constrained MWSR maximization problem are obtained, the sign of the dˆ η U B (P ) , can be readily determined following first derivative, EE dP Property (iii) in Theorem 2. Hence, problem (16) can be successfully solved by the aforementioned joint inner- and outer-layer optimization. UB is found, the corresponding optimal ρõpt When ηÊE k,n ’s are not ensured to be either 0 or 1. To get a feasible solution to the original downlink EE maximization problem (7), we need to round the possibly fractional ρõpt k,n ’s to 0 or 1 and then perform the BPA algorithm to get the maximum EE for the round-off ρõpt k,n ’s. Such manipulations may not result in the optimal solution to (7). Luckily, this is rarely a problem when the number of subcarriers is large compared with the number UB DL is quite close to ηÊE . In fact, of UEs and in this case ηÊE the optimal ρ˜k,n ’s for the constrained MWSR maximization problem mostly tend to be either 0 or 1 when K N [10], [11]. On the other hand, such fine tightness of the EE upper bound and the fact that the optimal ρ˜k,n ’s are almost either 0 or 1 implicitly enable the use of the original WSR, k∈K ωk n∈N ρk,n rk,n , instead of the constrained MWSR for maximization in the inner-layer optimization with an expectation of good performance. This enables us to

3771

precisely solve the original problem (7) by the joint inner- and outer-layer optimization framework with the constrained WSR maximization as the inner-layer subproblem. The discovery of such fact connects the research on the energy-efficient design to the previous research on the WSR maximization, such as the Lagrange dual decomposition [15] and the branch-and-bound method [16], by allowing them to be inner-layer candidate approaches. C. Low-Complexity Suboptimal Solution Although the convex programming is numerically stable, its computational complexity depends on the number of optimizing variables, which can be large if the number of subcarriers and/or the number of UEs are/is large. Each inner-layer MWSR maximization needs at least O(N K(1/δ 2 )) times of ˆ w (P ) < δ, water-filling for δ-optimality [17], i.e., Rw (P ) − R where Rw (P ) is the corresponding solution of the convex programming. And the total complexity of (16) also depends on the number of iterations in the outer-layer, NOL , and is O(NOL N K(1/δ 2 )). Here, we will first explore the inherent property of (7) and then, based on this property, a novel low-complexity suboptimal algorithm is proposed to solve problems like (7). DL , in (7) is always equal to Theorem 3. The optimal EE, ηÊE

ωk n∈N ρk,n rk,n DL min , (19a) max ηÊE ≡ ρ∈,P ∈P,α∈α k∈K n∈N ζpk,n + αk Pc

subject to

ˇ k , ∀ k ∈ K, ρk,n rk,n ≥ R

(19b)

n∈N

where α [αk ]K×1 | k∈K αk = 1; αk ∈ R . Theorem 3, which is proved in Appendix C, illustrates the structure of the optimal solution in a split form. From it, Corollary 1 is readily obtained. DL , in Corollary 1. For any fixed α ∈ α, the optimal EE, ηÊE (7), is lower bounded by

ωk n∈N ρk,n rk,n DL ηÊE ≥ max min , (20a) ρ∈,P ∈P k∈K n∈N ζpk,n + αk Pc

subject to

ˇ k , ∀ k ∈ K. ρk,n rk,n ≥ R

(20b)

n∈N

Assume αopt = [αopt k ]K×1 corresponds to the optimal EE, in (7) and (19). Then, αopt can be intuitively regarded k as the portion of static circuit power incurred individually by the kth UE when the maximum EE is achieved.3 Based on (19) and (20), instead of directly optimizing the EE, we can alternatively maximize the minimum individual EE, i.e., ρk,n rk,n ω to maximize the objective function mink∈K k n∈N ζpk,n +αk Pc , DL ηÊE ,

n∈N

3 α ’s may be negative. This is because α P is only the “virtual” static k k c circuit power caused by the kth UE, which is our intuition/imagination but not necessarily the physical fact. The actual total static circuit power is always Pc .

3772


under a certain properly chosen α ∈ α, and expect a satisfying EE. This idea enables us to split one joint and complex optimization objective in (7) into a series of relatively isolated and simple objectives in (19). It makes subcarrier assignment easier using heuristic algorithms because, for the kth UE, the ω ρk,n rk,n DL maximum of its individual EE, ηEE,k k n∈N , n∈N ζpk,n +αk Pc depends only on its own parameters and the subcarriers it will occupy but not on the power adaptation strategies of other UEs. Here, we propose a greedy subcarrier assignment approach, named, maximizing-EE-lower-bound-based downlink subcarrier assignment (MDSA) algorithm, which is motivated by the algorithms in [11], [18] and is sketched in Table II. The key idea of the MDSA algorithm is to iteratively assign the subcarriers aiming at maximizing the minimum individual EE, DL , under the QoS requirement. At the beginning, mink∈K ηEE,k each UE is only virtually assigned its worst subcarrier and the individual EE in this situation is individually optimized under the QoS requirement by the single-UE BPA algorithm and will be used as a benchmark to measure how urgent a UE needs a real subcarrier, which is depicted from Line 1 to Line 4. Then, in each iteration, the UE with the minimum individual EE takes its most favorite subcarrier among all unassigned ones into its exclusively occupied subcarrier set and maximizes its individual EE under the QoS requirement by the single-UE BPA algorithm. The above iteration process will proceed until all subcarriers have been assigned, which is described from Line 5 to Line 10. Besides, no total transmit power constraint is imposed in the MDSA algorithm but it will quite likely be spontaneously satisfied when the subcarrier assignment is finished. This is because: first, with properly chosen α, the eventually optimized individual EEs will tend to be of close values.4 If at a certain stage, one UE requires too much power to guarantee its QoS, then it is likely with a relatively low individual EE and will ask for more subcarriers later to lower its transmit power and increase its individual EE. Second, when the EE gain of the energy-efficient design over the spectral-efficient design is satisfactory, the actually used transmit power for the energy-efficient design should be much less than the maximum available transmit power. With the promising subcarrier assignment obtained by the MDSA algorithm, we can make further improvement by using the BPA algorithm to find the corresponding EE-optimal power adaptation strategy. The complexity of the MDSA algorithm for a given α is roughly O(NOL N ) times of waterfilling. We then suggest an effective way to determine the initial α for the MDSA algorithm. Let g¯k En (gk,n ) be the average CNR of the kth UE and Nk be the number of subcarriers assigned to the kth UE. We deliberately regard that each UE undergoes flat fading with a CNR of g¯k and solve the following energy-efficient resource allocation problem g ¯k Pk 1 + ω N W log k k 2 k∈K Nk DL max , (21a) η¯EE Nk ∈N,Pk >0 ζP + P k c k∈K

TABLE II M AXIMIZING -EE-L OWER -B OUND -BASED D OWNLINK S UBCARRIER A SSIGNMENT (MDSA) A LGORITHM .

Algorithm MDSA Input: ρ = [ρk,n ]K×N ← 0K×N ; Sk ← ∅, ∀ k ∈ K; α ← αini Output: ρ 1. 2. 3. 4. 5. 6. 7.

FOR each UE k ∈ K Find the subcarrier n ˇ k ← arg minn∈N gk,n and calculate Rk ({ˇ nk },Pk ) DL max ; ηEE,k ← ζPk +αk Pc ˇ ) Pk ≥R−1 {ˇ nk },R k k ( END WHILE N = ∅ DL ˇ ∈ K such that k ˇ ← arg min ηEE,k Find the UE k ; k∈K Find the subcarrier n ˆ kˇ ∈ N such that n ˆ kˇ ← arg max gk,n ˇ ; n∈N

nkˇ }; N ← N \ {ˆ nˇ }; Set ρk,ˆ ˇ n ˇ ← 1; Sk ˇ ← Sk ˇ ∪ {ˆ k R S ,P k ( ) ˇ ˇ k DL k k ; 9. Calculate ηEE, max ˇ ← ζPk k ˇ +αk ˇ Pc −1 ˇˇ) Pk ˇ ≥R ˇ (Sk ˇ , Rk k 10. END 8.

subject to

g¯k Pk ˇ k , ∀ k ∈ K, Nk W log2 1 + ≥R Nk Nk = N,

(21b) (21c)

k∈K

Pk ≤ Pmax .

(21d)

k∈K

By relaxing Nk ’s from positive integers to positive real numbers, similar quasiconcave relation can be proved and thus problem (21) can be precisely solved by the aforementioned joint inner- and outer-layer optimization framework like (16). For such a strictly concave inner-layer MWSR maximization problem, standard convex programming technique such as the ¯k interior-point method can be applied for solution [14]. Let N ¯ and Pk be the optimal solution to (21), then according to (19) the initial αini = [αini k ]K×1 is given by ¯kW log2 1 + g¯k¯P¯k ωk N ζ P¯k Nk αini − , ∀ k ∈ K. (22) k = DL P Pc η¯EE c The advantage of this kind of estimation is that (21) is much easier to solve than (7) and (16) because the number of variables is only linear in (two times of) the number of UEs but does not depend on the number of subcarriers, and αini does not need to update (in the long run) until any of g¯k ’s changes (much). The complexity of obtaining αini is not more than O(NOL K(1/δ 2 )) [14], [17]. In Table III, the complexity of the aforementioned optimal, near-optimal, and low-complexity alternative is listed for comparison. IV. U PLINK T RANSMISSION In this section, we will discuss the uplink EE optimization. A. Optimal Solution

4 From the proof of Theorem 3, αopt makes all the individual EEs equal DL is achieved. when ηÊE

Like the downlink case, the uplink problem (9) is, in general, a complicated integer programming problem. However,


3773

TABLE III C OMPLEXITY C OMPARISON FOR D OWNLINK T RANSMISSION .

Algorithm optimal: brute-force search based on BPA near-optimal: JIOO based on convex programming suboptimal: MDSA + BPA

once the subcarrier assignment is fixed, the maximization of EE for each UE can be regarded as a single-UE case of (7) and be readily solved by the BPA algorithm. By exhaustively searching all feasible subcarrier assignments, the optimal solution to (9) can be found by choosing the one with n∈N ρk,n rk,n the maximum min ζk ρk,n pk,n +Pc,k . The total complexity k∈K

n∈N

is about O(KNOL K N ) = O(NOL KN +1 ) times of waterfilling. Nevertheless, this kind of brute-force method is usually too computationally expensive to afford. B. Low-Complexity Suboptimal Solution

The formulation of (9) appears very similar to that of (19) except that the circuit power is now naturally consumed by each UE individually (α is not introduced) and the transmit power limit for each UE is now separately imposed. Therefore, joint power allocation across different users are not needed any more. Hence, problem (9) can be similarly solved by the MDSA with some modification. The modified subcarrier assignment approach, named, maximizing-minimum-EE-based uplink subcarrier assignment (MUSA) algorithm, is illustrated in Table IV. The basic idea of the MUSA algorithm is to iteratively assign the subcarriers aiming at maximizing the UL , under both the QoS minimum individual EE, mink∈K ηEE,k requirement and the transmit power constraint.5 At first, each UE is also virtually assigned its worst subcarrier for an estimate of its initial EE. For UEs that are not yet capable of supporting their QoS requirements, their initial EEs are obtained by transmitting at their respective maximum transmit power. For UEs that are able to satisfy their QoS requirements, they maximize their EEs under their respective QoS requirements and transmit power constraints by the singleUE BPA algorithm and use them as their initial EEs. Such an initialization process is depicted from Line 1 to Line 8. Then, in each iteration (till all subcarriers are assigned), only the UE with the minimum EE is assigned its most favorite subcarrier among all unassigned ones for an expected EE improvement. To ensure that each UE can eventually meet its QoS requirement, the subcarriers are firstly assigned among the set of UEs that are not yet able to meet their QoS requirements with their exclusively occupied subcarriers and affordable transmit power. When all UEs are capable of guaranteeing their QoS with their occupied subcarriers and affordable transmit power, the remaining subcarriers (if any exists) are then iteratively assigned among the set of all UEs. 5 Note

that in the MDSA algorithm, the total transmit power constraint is not considered as we have explained. However, for the uplink case with the MUSA algorithm, the individual transmit power constraints need to be considered along with the subcarrier assignment process in case that some of the eventually obtained EEs are not practically achievable due to transmit power constraint, which may cause the resultant minimum EE an overestimate.

Complexity O(NOL K N ) O( δ12 NOL N K) 1 O( δ2NOL K + NOL N + NOL )

TABLE IV M AXIMIZING - MINIMUM -EE- BASED U PLINK S UBCARRIER A SSIGNMENT (MUSA) A LGORITHM .

Algorithm MUSA Input: ρ = [ρk,n ]K×N ← 0K×N ; Sk ← ∅, ∀ k ∈ K; F ← K, where F is the set of UEs that are not yet capable of meeting their QoS requirements with their occupied subcarriers and affordable transmit power Output: ρ FOR each UE k ∈ K ˇ k ← arg min gk,n ; Find the subcarrier n ˇ k ∈ N such that n n∈N max ˇ 3. IF Rk ({ˇ nk }, Pk ) < Rk R ({ˇ nk },Pkmax ) UL 4. ηEE,k ← kζk P max ; +Pkc k 5. ELSE Rk ({ˇ nk },Pk ) UL ← max ; 6. ηEE,k ζk Pk +Pkc max ˇ R−1 {ˇ n }, R ≤P ≤P k k) k k k ( 7. END 8. END 9. WHILE N = ∅ 10. IF F = ∅ UL ˇ ∈ F such that k ˇ ← arg min ηEE,k 11. Find the UE k ; k∈F 12. Find the subcarrier n ˆ kˇ ∈ N such that n ˆ kˇ ← arg max gk,n ˇ ; 1. 2.

n∈N

13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

nkˇ }; N ← N \ {ˆ nkˇ }; Set ρk,ˆ ˇ n ˇ ← 1; Sk ˇ ← Sk ˇ ∪ {ˆ k ˇˇ IF Rkˇ Skˇ , Pkˇmax < R k max Rk ) ˇ ( Sk ˇ ,Pk ˇ UL ηEE, ˇ ← ζ ˇ P max +P c ; k ˇ ˇ k k k ELSE Rk ˇ ( Sk ˇ ,Pk ˇ) UL max ηEE, c ; ˇ ← ζk k ˇ Pk ˇ +P ˇ ˇ ˇ )≤P ˇ ≤P max k R−1 S , R ( ˇ ˇ ˇ k k k k k ˇ F ← F \ {k}; END ELSE UL ˇ ∈ K such that k ˇ ← arg min ηEE,k ; Find the UE k k∈K Find the subcarrier n ˆ kˇ ∈ N such that n ˆ kˇ ← arg max gk,n ˇ ; n∈N

nkˇ }; N ← N \ {ˆ nˇ }; Set ρk,ˆ ˇ n ˇ ← 1; Sk ˇ ← Sk ˇ ∪ {ˆ k R S k,P ˇ( k ˇ ˇ) UL k k ; 24. Calculate ηEE,kˇ ← max c ζk ˇ Pk ˇ +P ˇ ˇ ˇ )≤P ˇ ≤P max k R−1 Sk ( ˇ , Rk ˇ ˇ k k k 25. END 26. END 23.

This part is depicted from Line 9 to Line 26. In Table V, the complexity of the aforementioned optimal and low-complexity suboptimal solutions is listed for comparison. V. N UMERICAL R ESULTS In this section, we present simulation results to verify benefit of the energy-efficient design and performance of the low-complexity algorithms. In our simulation, the frequency spacing between adjacent subcarriers is 15 kHz. The

3774


TABLE V C OMPLEXITY C OMPARISON FOR UPLINK T RANSMISSION .

0.4

Complexity O(NOL KN+1 ) O(NOL N )

6 From the following Fig. 1 to Fig. 5, the average CNR in the horizontal axis represents the CNR of the two low CNR UEs. 7 Compared with the EE-optimal approach, the throughput improvement of the EE-suboptimal one is at the cost of more transmit power.

EE−optimal EE−suboptimal SE−optimal

0.3 EE (Mbits/Joule)

frequency-selective Rayleigh fading of the kth UE is modeled with the ITU Pedestrian-B model [19] with an average CNR of g¯k . The total bandwidth, 1.08 MHz, is equally divided into 72 orthogonal subcarriers. For the downlink transmission, there are four UEs each with the same minimum rate requirement of 100 kbps. The first two UEs are with the same average CNR while the other two UE are with the same 10 dB higher average CNR.6 The circuit power is 10 W and the maximum transmit power is 20 W for the base station. For the uplink transmission, there are four UEs each with the same average CNR and minimum rate requirement of 50 kbps. The circuit power is 50 mW and the maximum transmit power is 150 mW for each UE. For simplicity, we assume that the drain efficiency of power amplifier is 38% for both the downlink and uplink cases. Figures 1 and 2 evaluate the EE of the energy-efficient design that optimizes the generalized EE and the spectralefficient design that maximizes the WSR with the same constraints expect for the objectives in the downlink transmission. From them, the energy-efficient design significantly improves EE compared to the spectral-efficient design. And the suboptimal energy-efficient scheme based on the MDSA and the BPA algorithms results in an EE that is at least 90% of the optimal EE. In Fig. 2 (showing the case with the unequal UE weights), we also plot the resultant actual/conventional EE corresponding to the generalized EE of each scheme and find that the actual EE of the EE-suboptimal scheme is also close to that of the EE-optimal one. Figure 3 plots the throughput corresponding to the EE in Figures 1 and 2. From it, the throughput of the energyefficient design is, as expected, less than that of the spectralefficient design. The EE-suboptimal scheme has a slightly larger throughput than the EE-optimal one.7 Together with Figures 1 and 2, we find that the maximum EE and the maximum throughput (or SE) are not necessarily simultaneously achieved, which indicates a tradeoff relationship between EE and SE as investigated in [6]. Figures 4 and 5 compare the average individual rate of the low CNR UEs and high CNR UEs corresponding to EE in Figures 1 and 2. From Fig. 4, in such a case with equal weights, the rate of low CNR UEs is almost fixed at the minimum rate requirement, 100 kbps, while the rate of high CNR UEs increases with the CNR, which is unfair to the low CNR UEs. However, by adjusting the UE weights, as shown in Figure 5, the rate of low CNR UEs may also increase with the CNR and even become larger than that of the high CNR UEs. Hence, by properly choosing the UE weights rather than setting them equal for the energy-efficient design, certain

0.35

0.25

0.2

0.15

0.1

0.05 10

12

14 16 Average CNR (dB)

18

20

Fig. 1. Comparison of the EE for the downlink transmission schemes with 1 1 ω1 = ω2 = ω3 = ω4 = 1 and g¯1 = g¯2 = 10 g¯3 = 10 g¯4 .

0.45 0.4


0.35 EE (Mbits/Joule)

Algorithm optimal: brute-force search based on BPA suboptimal: MUSA

0.3 generalized EE 0.25 0.2 0.15 actual EE corresponding to the generalized EE 0.1 0.05 10

12


18

20

Fig. 2. Comparison of the EE for the downlink transmission schemes with 1 1 ω1 = ω2 = 2.5, ω3 = ω4 = 1, and g¯1 = g¯2 = 10 g¯3 = 10 g¯4 .

fairness/priority notions can be introduced while still reflecting the aspects of the conventional EE. Figures 6 and 7 illustrate the EE and throughput of the spectral-efficient design in [11] and the energy-efficient design based on the suboptimal MUSA algorithm, respectively. From them, the performance difference in EE and minimum individual rate between the energy-efficient design and the spectral-efficient design increase with the CNR, which also indicates the tradeoff relation between EE and SE for the uplink transmission. VI. C ONCLUSION In this paper, we have studied the energy-efficient resource allocation in both downlink and uplink OFDMA networks. For each scenario, we first find the optimal energy-efficient resource allocation approach then develop low-complexity suboptimal algorithm by exploring the inherent structure of the objective function and the feature of energy-efficient design. For the downlink case, we also obtain a computationally


3775

2

9

8


1.8 1.6 1.4

6

EE (Mbits/Joule)

7 Throughput (Mbps)

EE−suboptimal SE−optimal

ω1=ω2=ω3=ω4=1

5

1.2 1 0.8 0.6

4

0.4 3

2 10

ω =ω =2.5 1 2 ω3=ω4=1 12


0.2

18

0 15

20

Fig. 3. Comparison of the throughput for the downlink transmission schemes 1 1 g¯3 = 10 g¯4 . with g¯1 = g¯2 = 10

20

25

30

Average CNR (dB)

Fig. 6. Comparison of the EE for the uplink transmission schemes with g¯1 = g¯2 = g¯3 = g¯4 .

0.7 EE−suboptimal SE−optimal

4.5 0.6 Minimum individual rate (Mbps)

4


Invidual data rate (Mbps)

3.5 3 2.5

high CNR UEs

2 1.5 1

low CNR UEs

0.5

0.4

0.3

0.2

0.1

0.5 0 10

0 15 12


18

20

Fig. 4. Comparison of the individual data rate for the downlink transmission 1 1 g¯3 = 10 g¯4 . schemes with ω1 = ω2 = ω3 = ω4 = 1 and g¯1 = g¯2 = 10

Invidual data rate (Mbps)


2 low CNR UEs high CNR UEs

1.5

25

30

Average CNR (dB)

Fig. 7. Comparison of the minimum individual data rate for the uplink transmission schemes with g¯1 = g¯2 = g¯3 = g¯4 .

efficient and numerically tractable EE upper bound, which tends to be rather tight if the number subcarriers is large compared with that of UEs and is the foundation of a nearoptimal solution relying on the quasiconcave relation between the modified EE and transmit power. Simulation results show great EE improvement of the energy-efficient design than that of the spectral-efficient design and the proposed lowcomplexity algorithms can achieve a promising tradeoff between performance and complexity.

3

2.5

20

1

0.5

0 10

12


18

20

Fig. 5. Comparison of the individual data rate for the downlink transmission 1 g¯3 = schemes with ω1 = ω2 = 2.5, ω3 = ω4 = 1, and g¯1 = g¯2 = 10 1 g ¯ . 4 10

A PPENDIX A P ROOF OF T HEOREM 1 (ρ) ˆw Proof: We first prove that R (P ) under the constraint (10b) is strictly concave and continuously differentiable in P . With the nature of water-filling, it is easy to prove that the transmit power on each subcarrier is nondecreasing with the total transmit Then we consider the limit under the power. constraint k∈K n∈Sk Δpk,n = ΔP in (23). The existence (ρ) ˆw (P ) is continuously differof the limit indicates that R

3776

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 12, DECEMBER 2012 ˆ (ρ)(P ) dR w dP

entiable in P and

= maxk∈K,n∈Sk

(ρ)

ωk W gk,n log2 e 1+gk,n pˆk,n .

Accordingly, ηÊE (P ) is continuously differentiable in P . ω Wg log e Moreover, k1+g k,npˆk,n2 is nonincreasing with P for k ∈ K k,n

ωk W g log e and n ∈ Sk while maxk∈K,n∈Sk 1+g k,npˆk,n2 is strictly k,n ˆ (ρ)(P ) d2 R w < 0 and monotonically decreasing with P . Thus, dP 2 (ρ) ˆ Rw (P ) is strictly concave in P . (ρ) Denote the superlevel sets of ηÊE (P ) as

Sβ = {P ≥

−1 ˇ k | ηˆ(ρ) (P ) ≥ β}. Sk , R k∈K Rk EE (ρ)

According to [14], ηÊE (P ) is strictly quasiconcave in P if Sβ is strictly convex for any real number β. When β < 0, no (ρ) points exist on the counter ηÊE (P ) = β. When

β ≥ 0, Sβ is ˇ k | βζP + βPc − equivalent to Sβ = {P ≥ k∈K Rk−1 Sk , R (ρ) (ρ) ˆw ˆw R (P ) ≤ 0}. Since R (P ) is strictly concave in P , Sβ is strictly convex in P . This completes the proof of Property (i). (ρ)

Since lim

dˆ ηEE (P ) dP

=

lim

ΔP →0

ˆ (ρ) (P ) ˆ (ρ) (P +ΔP )−R R (ρ) w w −ζ η ÊE (P ) ΔP ζ(P +ΔP )+Pc

ΔP →0 (ρ) dˆ ηEE (P ) sgn dP

=

ˆ (ρ) (P +ΔP ) R w ζ(P +ΔP )+Pc

−

ˆ (ρ) (P ) R w ζP +Pc

ΔP

=

ˆ (ρ) (P ) dR (ρ) w −ζ η ÊE (P ) dP ζP +Pc

= , ˆ (ρ) dRw (P ) (ρ) − ζ ηÊE (P ) , where sgn dP

sgn (x) denotes the sign of x. Accordingly, Property (iii) is proved. On the other hand, with the strict concavity (ρ) (ρ) ˆw of R (P ), it is obvious that lim ηÊE (P ) = P →∞ o(P ) = 0. Thus, starting from ζP +Pc P →∞ (ρ) −1 ˇ k , ηˆ (P ) either strictly decreases P0 = k∈K Rk Sk , R EE (ρ) dˆ ηEE (P ) with P if | ≤ 0 or first strictly increases and P =P0 dP (ρ) dˆ η (P ) then strictly decreases with P if EE |P =P0 > 0. This dP ˆ (ρ)(P ) R w P →∞ ζP +Pc

lim

=

lim

completes the proof of Property (ii). The basic idea of the power allocation process is firstly to allocate power to make each UE merely satisfy its rate requirement (Eq. (12)) then allocate the remaining power to the subcarriers that can further maximize the WSR (Eq. (13)). It can be straightforwardly realized by the method of Lagrange multiplier and the derivations of (12) and (13) are omitted due to the spbace limit. A PPENDIX B P ROOF OF T HEOREM 2 ˆ w (P ) maxρ∈ ˜ P ) as the maxProof: Define R ˜ ˜ Rw (ρ, imum MWSR under the given power allocation matrix, P , ˜ P ) without constraint and constraint (17b). In [10], Rw (ρ, (17b) is proved to be strictly and jointly concave in ρ˜k,n and pk,n . Clearly, ˜k,n r˜k,n can be expressed in the n∈N ρ form of Rw (˜ ρ, P ) by letting ωk = 1 and ωk = 0 for ρ, P ) without constraint (17b) is strictly k = k. Since Rw (˜ and jointly concave in ρ˜k,n and pk,n for any given ωk ’s, ˇ k in (17b) naturally turns out to be a − n∈N ρ˜k,n r˜k,n ≤ −R ˜ P ) with constraint (17b) is convex constraint. Hence, Rw (ρ, also strictly and jointly concave in ρ˜k,n and pk,n . Thus, for ρ ˜1 , ρ ˜2 , and P 1 = P 2 that satisfy (17b), and θ ∈ (0, 1), we have ˜2 , θP 1 +(1−θ)P 2 ) > θRw (ρ ˜1 , P 1 )+ that Rw (θρ˜1 +(1−θ)ρ (1−θ)Rw (ρ ˜2 , P 2 ). On the other hand, for > 0, there exist

ˆ w (P i ) − for i = 1, 2. õ2 such that Rw (ρ õi , P i ) > R ρõ1 and ρ Then, we have ˆ w (θP 1 +(1−θ)P 2 ) = max Rw (ρ, R ˜ θP 1 + (1 − θ)P 2 ) ρ∈ ˜ ˜

õ2 , θP 1+(1−θ)P 2 ) ≥ Rw (θρõ1+(1−θ)ρ

> θRw (ρõ1 , P 1 )+(1−θ)Rw (ρ õ2 , P 2 ) ˆ w (P 1 )+(1−θ)R ˆ w (P 2 )−. (24) > θR

ˆ w (θP 1 + (1 − θ)P 2 ) > Since (24) holds for any > 0, R ˆ w (P 2 ). Hence, R ˆ w (P ) is strictly concave ˆ w (P 1 )+(1−θ)R θR in pk,n . ˆ w (P ) under constraints ˆ w (P ) maxp ≥0 R Define R k,n ∗ (17b) and (17c). Let P j be the transmit power matrix corˆ w (Pj ), where j = 1, 2, 3. Without loss of responding to R ∗ 2 −P3 generality, assume that P1 < P3 < P2 . Let P 3 = P P2 −P1 P 1 + ∗ ∗ ∗ P3 −P1 P2 −P3 P2 −P1 P 2 = ϑP 1 + (1 − ϑ)P 2 , where ϑ P2 −P1 . Obviously, ˆ w (P3 ) = the sum of all entries of P 3 equals P3 . Then, R ∗ ∗ ˆ ˆ ˆ ˆ Rw (P 3 ) ≥ Rw (P 3 ) > ϑRw (P 1 ) + (1 − ϑ)Rw (P ∗2 ) = ˆ w (P1 ) + (1 − ϑ)R ˆ w (P2 ). Thus, R ˆ w (P ) is strictly concave ϑR UB in P . Accordingly, the quasiconcavity of ηÊE (P ) in P can be proved in the same way as Theorem 1. ρ∗k,n ]K×N and P ∗ = [˜ ρk,n ]K×N be the matriLet ρ˜∗ = [˜ ˆ w(P ) = Rw(ρ ˆ ˜∗ , P ∗ ). ces corresponding to Rw(P ), i.e., R ∗ ∗ ∗ ˆ Similarly, let Rw(P + ΔP ) = Rw(ρ ˜ + Δρ ˜ , P + ΔP ∗ ). (ρ ˜∗ ) ∗ ˆ ˜ , P ∗ + ΔP ) and Let Rw (P + ΔP ) maxΔpk,n ≥0 Rw(ρ ∗ ∗ ( ρ ˜ +Δ ρ ˜ ) ∗ ˆw R (P ˜ + Δρ ˜∗ , P ∗ + ΔP ∗ − ) max Δpk,n ≥0 Rw(ρ ΔP ) with k∈K n∈N Δpk,n = ΔP > 0. Then we have ∗ ∗ ∗ ˆ w(P + ΔP ) and R ˆ (wρ˜ +Δρ˜ )(P ) ≤ ˆ (wρ˜ )(P + ΔP ) ≤ R that R ˆ w(P ). Thus, we have the inequalities in (25). It is obvious R when ΔP → 0, we have ΔP ∗ → 0 and Δρ ˜∗ → 0. Using the same approach as in the proof of Theorem 1, we have the (a) equalities in (26) and (27). In (27), = is obtained by removing ρ∗k,n = 0 and removing the UEs whose the terms with ρ˜∗k,n +Δ˜ rates are just equal to their minimum rate requirements since they cannot lower power on their occupied subcarriers, where K∗ denotes the set of UEs whose rates are greater than their minimum rate requirements with the subcarrier and power ˜∗ and P ∗ + ΔP ∗ . allocation matrices ρ˜∗ + Δρ ∗

ˆ (wρ˜ )(P + ΔP ) − Rw(ρ R ˜∗ , P ∗ ) lim ΔP →0 ΔP ωk W ρ˜∗k,n gk,n log2 e . = max k∈K,n∈N ρ ˜∗k,n + gk,n p∗k,n

(26)

Notice that we naturally have min ∗

ωk W ρ˜∗k,n gk,n log2 e

˜∗k,n k∈K ,n∈N ρ >0 s.t. ρ˜∗ k,n

+

gk,n p∗k,n

≤ max

ωk W ρ˜∗k,n gk,n log2 e

k∈K,n∈N

ρ˜∗k,n + gk,n p∗k,n

.

(28) Combing (25), (26), (27), and (28), we have ˆ w(P + ΔP ) − R ˆ w(P ) ωk W ρ˜∗k,n gk,n log2 e R = max . ΔP →0 k∈K,n∈N ρ ΔP ˜∗k,n + gk,n p∗k,n lim

ˆ

ωk W ρ˜∗

g

log e

2 k,n w (P ) Thus, dRdP = maxk∈K,n∈N ρ˜∗ k,n . Accordingly, +gk,n p∗ k,n k,n Property (ii) and the remaining parts of Property (iii) can be proved in the same way as Theorem 1. Moreover, we have


3777

ωk W log2 1+gk,n(ˆ pk,n +Δpk,n ) − ωk W log2 1+gk,n pˆk,n

max (ρ) (ρ) Δpk,n ≥0k∈Kn∈Sk ˆw ˆw k∈Kn∈Sk R (P +ΔP )− R (P ) = lim lim ΔP →0 ΔP →0 ΔP ΔP 1+gk,n(pˆk,n +Δpk,n ) max ωk W log2 1+gk,n pˆk,n Δpk,n ≥0 k∈Kn∈S k = lim ΔP →0 ΔP ωk W gk,n log2 e max Δpk,n Δpk,n ≥0 k∈Kn∈Sk 1+gk,n pˆk,n ωk W gk,n log2 e = lim = max . ΔP →0 k∈K,n∈Sk 1 + gk,n p ΔP ˆk,n ∗

(23)

∗

ˆ w(P + ΔP ) − R ˆ (wρ˜ )(P + ΔP ) − Rw(ρ ˆ w(P ) ˆ (wρ˜ Rw(ρ R R ˜∗ , P ∗ ) ˜∗ + Δρ ˜∗ , P ∗ + ΔP ∗ ) − R ≤ ≤ ΔP ΔP ΔP

+Δρ ˜∗ )

(P )

(25)

(ρ ˜∗ +Δρ ˜∗ )

ˆw Rw(ρ ˜∗ + Δρ ˜∗ , P ∗ + ΔP ∗ ) − R (P ) lim ΔP →0 ΔP g (p∗ +Δp∗ g (p∗ +Δp∗ k,n ) k,n−Δpk,n ) ∗ ωkW(˜ ρk,n+Δ˜ ρ∗k,n )log2 1+ k,nρ˜∗ k,n ωkW(˜ ρ∗k,n+Δ˜ ρ∗k,n )log2 1+ k,n k,n − max +Δρ˜∗ ρ˜∗ +Δρ˜∗ (a)

k,n

k∈K∗,n∈N ˜∗ s.t. ρ˜∗ k,n +Δρ k,n >0

k,n

= lim

ΔP →0

=

min

k∈K∗ ,n∈N s.t. ρ˜∗ k,n >0

ωk W ρ˜∗k,n gk,n log2 e ρ˜∗k,n + gk,n p∗k,n

Δpk,n ≥0 k∈K∗,n∈N ˜∗ s.t. ρ˜∗ k,n +Δρ k,n >0

k,n

k,n

ΔP .

(27)

ω

ρ

r

ζp

k,n k,n k,n also found that − n∈N . which requires that αk = k n∈N DL P Pc ηEE c ∗ ∗ ωk W ρ˜k,n gk,n log2 e ωk W ρ˜k,n gk,n log2 e Then it is obvious to have that = max , min

ωk ρk,n rk,n k∈K,n∈N ρ ˜∗k,n + gk,n p∗k,n ˜∗k,n + gk,n p∗k,n k∈K∗ ,n∈N ρ ∗ n∈N s.t. ρ˜k,n >0 DL DL min , ηÊE = max ηEE ≥ max ρ∈,P ∈P ρ∈,P ∈P k∈K ζpk,n + αk Pc which indicates that for all the UEs whose rates are greater n∈N than their rate requirements, their occupied subcarriers have where α ∈ α. Thus, the RHS cannot exceed the LHS for any the same “water-level”. α ∈ α. Furthermore, the equality always holds if the RHS the uses the same ρ and P as ρopt and P opt that achieve opt ωk n∈N ρopt opt k,n rk,n LHS and sets {αk } such that αk = αk − DL P η ÊE c opt A PPENDIX C n∈N ζpk,n straightforward addition with plugging in Pc . Clearly, P ROOF OF T HEOREM 3 opt opt opt ω k k∈K n∈N ρk,n rk,n DL η ˆ = concludes that αk = 1. opt opt opt opt EE ζ k∈K n∈N pk,n +Pc [pk,n ]K×N be Proof: Let ρopt [ρk,n ]K×N and P k∈K the optimal subcarrier andpower allocation matrices for (7), This completes the proof.

ρopt r opt

ωk

k∈K k,n k,n DL n∈N opt = ζ . Obviously, which indicates ηÊE k∈K n∈N pk,n +Pc for any fixed ρ ∈ and P ∈ P,

DL ηEE

ω1 ρ1,n r1,n + · · · + ωK ρK,n rK,n n∈N n∈N = ζp1,n + α1 Pc + · · · + ζpK,n + αK Pc n∈N n∈N ωk ρk,n rk,n n∈N ≥ min , k∈K ζpk,n + αk Pc n∈N

where α ∈ α. And the equality holds if and only if ω1 ρ1,n r1,n ρK,n rK,n ωK n∈N n∈N = ··· = , ζp1,n + α1 Pc ζpK,n + αK Pc n∈N

n∈N

R EFERENCES [1] T. Edler and S. Lundberg, “Energy efficiency enhancements in radio access networks,” in Erricsson Rev., 2004. [2] Y. Chen, S. Zhang, S. Xu, and G. Y. Li, “Fundamental tradeoffs on green wireless networks,” IEEE Commun. Mag., vol. 49, no. 6, pp. 30– 37, June 2011. [3] G. Miao, N. Himayat, and G. Y. Li, “Energy-efficient link adaptation in frequency-selective channels,” IEEE Trans. Commun., vol. 58, no. 2, pp. 545–554, 2010. [4] R. S. Prabhu and B. Daneshrad, “An energy-efficient water-filling alglorithm for OFDM systems,” in Proc. 2010 IEEE Int. Conf. Commun. [5] C. Isheden and G. P.Fettweis, “Energy-efficient multi-carrier link adaptation with sum rate-dependent circuit power,” in Proc. 2010 IEEE Global Telecommun. Conf. [6] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, “Energy- and spectralefficiency tradeoff in downlink OFDMA networks,” IEEE Trans. Wireless Commun., vol. 10, no. 1, pp. 3874–3886, Nov. 2011.

3778


[7] S. Cui, A. Goldsmith, and A. Bahai, “Energy-constrained modulation optimization,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2349– 2360, Sep. 2005. [8] G. Miao, N. Himayat, G. Y. Li, and A. Swami, “Cross-layer optimization for energy-efficient wireless communications: a survey,” Wiley J. Wireless Commun. Mobile Comput., vol. 9, no. 4, pp. 529–542, Apr. 2009. [9] G. Miao, N. Himayat, G. Y. Li, and D. Bormann, “Energy efficient design in wireless OFDMA,” in Proc. 2008 IEEE Int. Commun. Conf. [10] W. Yu and J. M. Coffi, “FMDA capacity of Gaussian multiple-access channel with ISI,” IEEE Trans. Commun., vol. 50, no. 1, pp. 102–111, 2002. [11] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subcarrier allocation,” in Proc. 2000 IEEE Veh. Technol. Conf. – Spring, pp. 1085–1089. [12] C. Y. Wong, R. S. Cheng, K. Lataief, and R. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747–1758, Oct. 1999. [13] H. Zhang, Y. Ma, D. Yuan, and H.-H. Chen, “Quality-of-service driven power and sub-carrier allocation policy for vehicular communication networks,” IEEE J. Sel. Areas Commun., vol. 29, no. 1, pp. 197–206, Jan. 2011. [14] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [15] K. Seong, M. Mohseni, and J. M. Cioffi, “Optimal resource allocation for OFDMA downlink systems,” in Proc. 2006 IEEE Int. Symp. Inf. Theory. [16] Z. Mao and X. Wang, “Efficient optimal and suboptimal radio resource allocation in OFDMA system,” IEEE Trans. Wireless Commun., vol. 7, no. 2, pp. 440–445, Feb. 2008. [17] N. Mokari, M. R. Javan, and K. Navaie, “Cross-layer resource allocation in OFDMA systems for heterogeneous traffic with imperfect CSI,” IEEE Trans. Veh. Technol., vol. 59, no. 2, pp. 1011–1017, Feb. 2010. [18] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2726–2737, Nov. 2005. [19] Technical specification group GSM/EDGE radio access network radio transmission and reception (release 1999), 3GPP TS 05.05 V8.15.0, Aug. 2001. Cong Xiong received his B.S.E degree from the School of Telecommunication Engineering and M.S.E degree from the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing, China, in 2007 and 2010, respectively. He is currently working toward the Ph.D. degree with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA. His research interests include MIMO, cooperative communications, energy-efficient wireless network design, and energy-efficient cross-layer optimization.

Geoffrey Ye Li received his B.S.E. and M.S.E. degrees in 1983 and 1986, respectively, from the Department of Wireless Engineering, Nanjing Institute of Technology, Nanjing, China, and his Ph.D. degree in 1994 from the Department of Electrical Engineering, Auburn University, Alabama. He was with AT&T Labs - Research at Red Bank, New Jersey and is now with Georgia Institute of Technology, Atlanta, Georgia, as a Professor. His general research interests include statistical signal processing and telecommunications, with emphasis on OFDM and MIMO techniques, cross-layer optimization, and signal processing issues in cognitive radios. In these areas, he has published over 200 papers in refereed journals or conferences and two books, 20 of which are with over 100 Google citations. He also has over 20 patents granted or filed. He has been awarded an IEEE Fellow since 2006 and won 2010 IEEE Communications Society Stephen O. Rice Prize Paper Award in the field of communications theory. Shunqing Zhang received his B.E. degree from Fudan University, Shanghai, China, in 2005 and his Ph.D. degree from HKUST in 2009, respectively. He joined the Green Radio Excellence in Architecture and Technology (GREAT) project at Huawei Technologies Co. Ltd. after the graduation. His current research interests include energy-efficient resource allocation and optimization in cellular networks, joint baseband and radio frequency optimization, fundamental tradeoffs on green wireless network design, and other green radio technologies for energy saving and emission reduction. Yan Chen was born in Hangzhou, Zhejiang, China, in 1982. She received the B.Sc. and the Ph. D degree in information and communication engineering from Zhejiang University, Hangzhou, China, in 2004 and 2009, respectively. She has been a Visiting Researcher at the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong. After graduation, she joined Huawei Technologies (Shanghai) Co., Ltd. and is currently working as a research engineer in the Central Research Department. Her current research interests include green network information theory, energyefficient network architecture and management, fundamental tradeoffs on green wireless network design, as well as the radio technologies and resource allocation optimization algorithms therein. Shugong Xu received a B.Sc. degree from Wuhan University, China, and his M.E. and Ph.D. from Huazhong University of Science and Technology (HUST), Wuhan, China, in 1990, 1993, and 1996, respectively. He is currently the Director of Access Network Technology Research Department, Principal Scientist of Huawei Corporate Research. Prior of joining Huawei, he was with Sharp Labs of American, Camas WA for seven years and a few years in academic universities including Tsinghua Univiersity and the City College of New York (CCNY). His research interests include wireless/mobile networking and communication, home networking and multimedia communications. He published more than 30 peer-reviewed research papers as lead-author in top international conferences and journals, in which the most referenced one has over 850 Google Scholar citations. He holds more than 30 granted US patents or patent applications, of which technologies have been adopted in WiFi and LTE standards. Dr. Xu is a senior member of IEEE, a Concurrent Professor at HUST, and the Technical Committee co-chair of Green Touch consortium.