Energy- and Spectral-Efficiency Tradeoff in Downlink OFDMA Networks

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Energy- and Spectral-Efficiency Tradeoff in Downlink OFDMA Networks∗ Cong Xiong† , Geoffrey Ye Li† , Shunqing Zhang‡ , Yan Chen‡ , and Shugong Xu‡ † School of ECE, Georgia Institute of Technology, Atlanta, GA, USA ‡ GREAT Research Team, Huawei Technologies Co., Ltd., Shanghai, China

Abstract—Conventional design of wireless networks mainly focuses on system capacity and spectral efficiency (SE). As green radio (GR) becomes an inevitable trend, energy-efficient design in wireless networks is becoming more and more important. In this paper, the fundamental relation between energy efficiency (EE) and SE in downlink orthogonal frequency division multiple access (OFDMA) networks is addressed. We first set up a general EE-SE tradeoff framework, where the overall EE, SE and peruser quality-of-service (QoS) are all considered, and prove that EE is strictly quasiconcave in SE. We also find a tight upper bound and a tight lower bound on the EE-SE curve for general scenarios, which reflect the actual EE-SE relation. We then focus on a special case that priority and fairness are considered and develop a low-complexity but near-optimal resource allocation algorithm for practical application of the EE-SE tradeoff. Numerical results corroborate the theoretical findings and demonstrate the effectiveness of the proposed resource allocation scheme for achieving a flexible and desirable tradeoff between EE and SE. Index Terms—Energy efficiency (EE), green radio (GR), orthogonal frequency division multiple access (OFDMA), spectral efficiency (SE)

I. I NTRODUCTION In recent years, the widespread application of high-data-rate wireless services and requirement of ubiquitous access have triggered rapidly booming energy consumption. Meanwhile, the escalation of energy consumption in wireless networks leads to large amount of greenhouse gas emission and high operation expenditure. Green radio (GR) [1], which emphasizes on energy efficiency (EE) besides spectral efficiency (SE), has been proposed as an effective solution and becomes an inevitable trend for future wireless network design. Unfortunately, EE and SE do not always coincides and sometimes may even conflict [1]. Hence, how to balance EE and SE is well worth studying. Orthogonal frequency division multiple access (OFDMA) has been extensively studied from the SE perspective and proposed for next generation wireless communication systems, such as WiMAX and the 3GPP LTE. While OFDMA can provide high throughput and SE, its EE is previously not much concerned. To keep pace with GR, it is necessary for OFDMA to guarantee a certain level of EE at the same time. Recently, more attention has been paid to energy efficient design in OFDMA networks. For uplink OFDMA transmission with flat fading channels, it is shown that EE-oriented design always ∗ The work was supported in part by the Research Gift from Huawei Technologies Co. and the NSF under Grant No. 1017192. Corresponding author. Email: [email protected].

consumes less energy than the traditional fixed power schemes [2]. Meanwhile, we notice that there is only limited work on joint design of EE and SE for downlink OFDMA networks. In this paper, we address the EE-SE relation in downlink OFDMA networks. We build a general EE-SE tradeoff framework, prove that EE is quasiconcave in SE, then bound the EESE curve for general scenarios by a double-side approximation process, which relies on a tight upper bound and a lower bound obtained by Lagrange dual decomposition [3], [4]. When priority and fairness are considered, we propose a computationally efficient algorithm for resource allocation to facilitate application of EE-SE tradeoff. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION In this section, we introduce the system model of downlink OFDMA and formulate the problem of EE-SE tradeoff. A. System Model We consider the downlink of a single cell OFDMA network consisting of K active users. The total bandwidth B is equally divided into N subcarriers, each with a bandwidth of W = B/N . Let K = {1, 2, · · · , K} and N = {1, 2, · · · , N } denote the sets of all users and all subcarriers, respectively. Assume that each subcarrier is exclusively assigned to at most one user each time to avoid interference among different users. Denote the transmit power of user k on subcarrier n as pk,n . Then, the achievable data rate of user k on subcarrier n is accordingly pk,n gk,n , rk,n = W log2 1 + σ2 where gk,n = |hk,n |2 is the channel power gain of user k on subcarrier n, hk,n is the corresponding frequency response and is assumed to be accurately known at the transmitter, and σ 2 is the noise power, which is, without loss of generality, assumed to be the same for the all users on all subcarriers. B. Problem Formulation For a downlink OFDMA network, EE and SE are, respectively, defined as ηEE =

R P + Pc

and

ηSE =

R , B

where R, P and Pc denote the system overall throughput, total transmit power, and circuit power, respectively. To obtain a

978-1-61284-231-8/11/$26.00 ©2011 IEEE


high EE as well as a desirable SE and guarantee quality-ofservice (QoS) for each user, it is reasonable to maximize EE under a satisfying minimum overall throughput requirement, ˇ a series of (minimum) rate requirements, R ˇ k ’s, depending R, on the traffic of the corresponding user, and the peak transmit power, PT . Since the capability of providing differentiate services is an important feature for future wireless networks, a heterogeneous traffic model includes both real-time and nonreal-time traffic is considered in our work. For users with real-time services [5], such as video conferencing and online ˇ i ’s are required. For users with nongaming, fixed rates R real-time services [5], such as file transfers and online video, ˇ j ’s are demanded. Let only minimum rate requirements R K1 = {1, 2, · · · , K0 − 1} and K2 = {K0 , K0 + 1, · · · , K} denote the sets of real-time users and non-real-time users, respectively. Accordingly, the optimization problem can be formulated as max ηEE ρ,P

subject to K

ρ,P

if there is a sufficiently large number of subcarriers. Moreover, ˇ ˆ R R ∗ and ηSE = B , ηEE (ηSE ) is in the region between ηSE = B (i) strictly decreasing with ηSE and maximized at ηSE = if ∗ dηEE (ηSE ) ≤ 0, ˇ dηSE ηSE = R

ˇ R B

B

(ii) strictly increasing with ηSE and maximized at ηSE = if ∗ dηEE (ηSE ) >0 ˇ dηSE R ηSE = B ∗ dηEE (ηSE ) and ≥ 0, ˆ dηSE ηSE = R

ˆ R B

B

ρk,n k=1 N K

(iii) first strictly increasing and then strictly decreasing with R if ηSE and maximized at ηSE = EE,max B ∗ dηEE (ηSE ) >0 ˇ dηSE R ηSE = B ∗ dηEE (ηSE ) and < 0, ˆ dηSE ηSE = R

≤ 1, ∀ n, ρk,n ∈ {1, 0}, ∀ k ∈ K, n,

ˇ ρk,n rk,n = R ≥ R,

k=1 n=1 N

ˇ k , ∀ k ∈ K1 , ρk,n rk,n = R

n=1 N

ˇ

R , achieved with Theorem 1. For any given SE, ηSE ≥ B subcarrier allocation matrix, ρ, and power allocation matrix, P , that satisfy all constraints but not necessarily including the peak transmit power one in (1), the maximum EE, ∗ (ηSE ) = max ηEE (ηSE ), is strictly quasiconcave in ηSE ηEE

B

(1)

ˆ is the maximum throughput under all constraints in where R (1), which is obviously achieved by transmitting at the peak transmit power, PT , and REE,max is the throughput which max , under all constraints achieves the global maximum EE, ηEE except the peak transmit power one in (1).

where ρ = [ρk,n ]K×N and P = [pk,n ]K×N are the subcarrier allocation indicator matrix and transmit power matrix, respectively. For convenience, some other notations used in this paper are listed in advanced as follows. set of subcarriers assigned to user k, Sk number of assigned subcarriers of user k, mk set of users without assigning any subcarrier KE (with empty Sk ’s) overall transmit power for user k, and Pk = Pk N ρ n=1 k,n pk,n , Rk overall data rate for user k, and Rk = N ρ n=1 k,n rk,n , f (Rk , Sk ) power needed by water-filling to fulfill rate Rk over subcarrier set Sk .

For any continuous and strictly quasiconcave function, there is always a unique global maximum over a finite domain [6, Ch. 8]. Thus, according to Theorem 1, a unique globally optimal EE of (1) always exits. More importantly, as a result of this quasiconcavity, (1) can be decomposed into two layers and solved iteratively, (i) Inner layer: For a given SE, ηSE , ∗find the maximum EE, dη (ηSE ) ∗ ηEE (ηSE ), and its derivative, EE . dηSE opt (ii) Outer layer: Find the optimal EE, ηEE , by bisection search like the GABS algorithm in [7]. The corresponding joint inner- and outer-layer optimization (JIOO) algorithm is listed in Table I. Then the key of the JIOO algorithm lies in the inner-layer algorithm that finds dη ∗ (ηSE ) ∗ (ηSE ) and EE and will be studied in detail in ηEE dηSE the following sections.

ρk,n rk,n n=1 K N

ˇ k , ∀ k ∈ K2 , ≥R

ρk,n pk,n = P ≤ PT , pk,n ≥ 0, ∀ k ∈ K, n,

k=1 n=1

III. EE-SE R ELATION In this section, we will study the EE-SE tradeoff. A. Fundamentals for EE-SE Tradeoff Relation The following theorem demonstrates the quasiconcavity of EE, ηEE , in the SE, ηSE , and is proved in the Appendix.

B. Bounds on the EE-SE Tradeoff As indicated before, the solution of (1) now relies on finding dη ∗ (ηSE ) ∗ ηEE (ηSE ) and EE . Since the exact solution is too dηSE complicated to obtain in reality, we will use Lagrange dual decomposition to approximately solve it, which has been used in [3], [4] for similar problems and demonstrated to be quite


TABLE I J OINT I NNER - AND O UTER - LAYER O PTIMIZATION (JIOO) A LGORITHM

Algorithm JIOO ˇ R B

(0)

Input: initial value of SE: ηSE =

Output: optimal subcarrier and power allocation matrices, ρopt and 1. 2. 3. 4.

opt P opt , which result in the optimal EE, ηEE ∗ dη (ηSE ) (1) (0) ηSE = ηSE , d1 ← EE (1) dηSE (1)

opt then ηSE ← ηSE (2)

(1)

(1)

(1)

ηSE =ηSE

(1) BηSE (1) ∗ ηEE ηSE

=

− Pc and α > 1, i.e., α = 2

while d1 > 0 && P1 < PT (2)

(1)

(1)

(1)

(1)

do ηSE ← ηSE , ηSE ← αηSE , P1 ← P ∗ (ηSE ), dη ∗ (ηSE ) d1 ← EE (1) dηSE

6.

8.

(1)

else ηSE ← ηSE , ηSE ← αηSE , P1 ← P ∗ (ηSE ), ∗ dηEE (ηSE ) (1) d1 ← , where P ∗ (ηSE ) (1) dηSE

5.

7.

ηSE =ηSE

if d1 ≤ 0

ηSE =ηSE

while no convergence (∗ bisection search ∗) opt do ηSE ←

dopt ←

(1)

(2)

ηSE +ηSE (opt) ∗ ,P 2 opt ← P (ηSE ), opt ∗ dηEE (ηSE ) dηSE (1) ηSE =ηSE

if dopt > 0 && Popt < PT

9.

(2)

10.

opt then ηSE ← ηSE

11.

opt else ηSE ← ηSE

(1)

12. return ρopt , P opt

accurate with affordable computational complexity. For any ˇ the Lagrange dual problem of the given throughput, R ≥ R, equivalent problem of (1), which is to minimize the transmit power, P , can be expressed as K N ρk,n pk,n max min λ1 ,λ2 ,λ3 ρ,P

+

+

k=1 n=1 N

λ1,n

n=1 K

λ2,k

K

+ λ3 R −

ρk,n − 1

k=1

ˇk − R

N

ρk,n rk,n

n=1

k=1

K 0 −1 k=1

ˇk − R

K

N

ρk,n rk,n

k=K0 n=1

subject to ρk,n ∈ {1, 0}, pk,n ≥ 0, ∀ k ∈ K, n, λ1,n ≥ 0, ∀ n, λ2,k ≥ 0, ∀ k ∈ K, λ3 ≥ 0,

(2)

where λ1,n , λ2,k , and λ3 are the introduced Lagrange multipliers, and λ1 = [λ1,1 , λ1,2 , · · · , λ1,N ]T and λ2 = [λ2,1 , λ2,2 , · · · , λ2,K ]T , respectively. Note that we drop the peak power constraint in (1) here since it will be imposed by the outer-layer processing as shown in the JIOO algorithm. Such a problem as (2) can be solved with dual decomposition as suggested in [3].

In general, the dual decomposition solution for subcarrier allocation indicator matrix, transmit power matrix and total transmit power, (ρ∗d , P ∗d , Pd∗ ), to the dual problem (2) yields a lower bound of total transmit power on the optimal solution, ρ∗p , P ∗p , Pp∗ , to the primal problem as a result of the nonconvexity of the primal one [3]. To guarantee the solution feasible, we find an achievable upper bound on the minimum transmit power based on the subcarrier allocation strategy obtained from the dual decomposition. It is easy to verify that for the subcarrier allocation strategy, ρ = ρ∗d , the tightest and ∗ , on total transmit power can be achievable upper bound, Pub achieved in two stages: in the first stage, power is distributed individually among the subcarriers of each user, Sk , by waterfilling, to merely fulfill its own (minimum) rate requirement, ˇ k ; in the second stage, extra power is then distributed among R all subcarriers of the non-real-time users (each has an initial water level due to the first phase processing) by water-filling till the throughput, R, is achieved. Using dual decomposition, we approximate the minimum transmit power, Pp∗ , tightly from both sides, i.e., Pd∗ ≤ Pp∗ ≤ ∗ , which allows us quite an accurate EE-SE tradeoff relation. Pub R R On the other hand, using ηEE = P +P and ηSE = B , the c above bounds on total transmit power, P , correspond to inverse ∗ (ηSE ). Since it was impossible to find out bounds on EE, ηEE dη ∗ (ηSE ) ∗ (ηSE ), EE could the closed-form expression for ηEE dηSE only be calculated via the original definition of derivative, dη ∗ (ηSE ) η ∗ (η +Δη )−η ∗ (η ) = limΔηSE →0 EE SE ΔηSE EE SE . For i.e, EE dηSE SE practical implementation, we can choose a small positive ΔηSE = ΔR B . To further reduce complexity, we do not need to repeat the whole process for the new SE, ηSE + ΔηSE . ∗ dηEE (ηSE ) but not Note that we only need the sign of dηSE ∗ ∗ ηEE (ηSE +ΔηSE )−ηEE (ηSE ) = ΔηSE ∗ ∗ ηEE(ηSE+ΔηSE)−ηEE (ηSE) = sgn ΔηSE

necessarily for its value. Since

R ΔR−ΔP P +P c +Pc +ΔP , we have that PΔR ∗ sgn ΔP −ηEE (ηSE ) , where sgn (x)

B

denotes the sign of x. We can simply distribute ΔR among all subcarriers of the non-real-time users by water-filling to approximately find out ΔP based on the subcarrier allocation and power allocation dη ∗ (ηSE ) ∗ (ηSE ), and the sign of EE matrix corresponding to ηEE dηSE is obtained accordingly. C. Priority, Fairness and Low-Complexity Algorithm

For many practical scenarios, capability to provide different service priority fairness among users is important. Let K and . ˇ k , then the two rate constraints in (1) can ˜= R R − k=1 R be rewritten as N ˜ ˇ k + ωk R, ρk,n rk,n = R n=1 K

k=1

ωk = 1, ωk ≥ 0, ∀ k,

where ωk is the weight factor for user k. We can determine the weights according to user traffic types, fairness and priority requirements. For users with real-time services, ωi can be simply set to zero. For users with non-real time services, ˇ j ’s, we can which have only minimum rate requirements R prioritize them and enforce certain notations of fairness by


adjusting ωj . For example, we can determine ωj by making user rates proportional to a set of predetermined values to ensure proportional fairness among non-real-time users [8]. With certain predetermined weight vector, ω, the equivalent problem of (1) given the throughput R can be regarded as the conventional margin adaptation (MA) optimization problem [3], [9]–[11] as follows. min ρ,P

subject to

N K

IV. N UMERICAL R ESULTS AND C ONCLUSION ρk,n pk,n

k=1 n=1 K

ρk,n ≤ 1, ∀ n, ρk,n ∈ {1, 0}, ∀ k, n,

k=1

rk,n ≥ 0, ∀ k, n N

one user getting too many subcarriers since benefit of acquiring subcarriers is decreasing. Meanwhile, the MPDF algorithm needs O(N K) times of water-filling in total. dη ∗ (ηSE ) to On the other hand, to obtain the sign of EE dηSE implement the JIOO algorithm, we can follow the same way as for the general case indicated before.

ˇ k + ωk R, ˜ ∀ k, ρk,n rk,n = R

(3)

n=1

Although dual decomposition can be still employed similarly as before, it typically needs O(N K 3 ) times of waterfilling to converge [3] and its complexity is still very high. Motivated by the BABS algorithm [10] for finding distribution of subcarriers in flat-fading channels, we suggest the following maximum-power-decrease-first (MPDF) algorithm. In initialization of this approach, each user is only virtually assigned its worst subcarrier and its transmit power needed in this situation will be used as a benchmark to measure how much power can be saved if the user is actually assigned a subcarrier when he has none. Then, in each iteration, each user finds its optimal subcarrier among the unassigned ones and calculates its power decrease with the additional subcarrier. Only the user with the maximum power decrease (power saving) will be assigned its favorite subcarrier in this iteration. The above iteration process will proceed until all the subcarriers have been assigned or no user needs additional subcarriers. Note that to guarantee each user is assigned at least one subcarrier at last, when the the number of unassigned subcarriers equals the number of users without any subcarriers, the subcarrier assignment should be implemented among such empty users. TABLE II M AXIMUM -P OWER -D ECREASE -F IRST (MPDF) A LGORITHM

1) Initialize: KE ← K; Sk ← ∅, mk ← 0, ∀ k ∈ K. 2) Calculate benchmarks: n ˇ k ← arg min gk,n , Pk = f (Rk , n ˇ k ), n∈N ∀ k ∈ K. nk }), ∀ k ∈ K. 3) n ˆ k ← arg max gk,n , ΔPk ← Pk − f (Rk , Sk ∪ {ˆ n∈N K ˆ ← arg max 1 + δ N − k k=1 mk −|KE | δ(mk ) ΔPk . k∈K

nkˆ }, N ← N \ {ˆ nkˆ }, mkˆ ← Assign and update: Skˆ ← Skˆ ∪ {ˆ ˆ ΔPkˆ , KE ← KE \ {k}. mkˆ + 1, Pkˆ ← Pkˆ − 4) Repeat Step 3) until K k=1 mk = N or max ΔPk = 0. k∈K

Theorem 2. The power required by water-filling with a fixed target rate is convex and non-increasing with the number of subcarriers assigned if the subcarriers are added in descending order of the power gains of subcarriers. (The proof is omitted.) From Theorem 2, the MPDF algorithm naturally prevents

In this section, we present some simulation results to verify the theoretical analysis and the effectiveness of the proposed approaches. In our simulation, the channel is modeled as a frequency-selective fading channel consisting of six independent Rayleigh multipaths with power delay profile, e−2l , l = 0, 1, · · · , 5. The total bandwidth, 1.28 MHz, is equally divided into 64 non-overlapping subcarriers. The circuit power is 2.5 W. There are two real-time users each has a fixed rate requirement of 125 kbps, and two non-real-time users each has a minimum rate requirement of 25 kbps. The fairness notion employed here for non-real-time users is the partial proportional constraint (PPC) modified from the propotional constraint in [8], where ω1 : ω2 : · · · : ωK = α1 : α2 : · · · : ˇ k log2 g¯k,n . Consequently αK . For simplicity, we let αk = R K ˇ ˇ ωk = Rk log2 g¯k,n / k =1 Rk log2 g¯k ,n . Figure 1 shows the EE-SE relation in the case that all the four users have a same average channel-gain-to-noise ratio g ¯ (CNR), σk,n 2 , of 20 dB and no specific overall throughput ˇ = 0. From the figure, the requirement is imposed here, i.e, R EE-SE relation has a bell shape curve and is also quasiconcave, since the upper bound and the lower bound derived from Lagrange dual decomposition almost perfectly match together and they are in a bell shape. ∗ , and Figure 2 demonstrates the relation between EE, ηEE ˇ R the minimum prescribed SE, ηˇSE = B . In this case, the two real-time users have the same average CNR of 20 dB. One of the two non-real-time users has an average CNR of 20 dB, while the other has an average CNR of 17 dB. Here no peak power constraint is imposed to investigate performance limit. From the figure, the performance of the MPDF-based method is quite close to that of the dual method, where the performance loss is within 3%; and it is slightly better than the DPRA [11] -based method in the high power regime. The complexity (average computational time) is evaluated using the profile in Matlab, which is shown in Fig. 3. Obviously, our MPDF-based method has relatively lower complexity and offers an attractive performance to complexity good tradeoff. A PPENDIX Proof: Let R∗1 , R∗2 and R∗3 denote the optimal rate vectors corresponding to the overall throughput R1 , R2 , and R3 , respectively, and they also satisfy all constraints but not necessarily including the peak power constraint in (1). Without loss of generality, assume that R1 < R2 < R3 . Let R2 denote the rate vector as follows. R3 − R 2 ∗ R 2 − R 1 ∗ R + R R2 = R3 − R1 1 R3 − R1 3 = λR∗1 + (1 − λ) R∗3 ,


where λ = (R3 − R2 ) / (R3 − R1) and 0 < λ < 1. Obviously, R2 is also in the feasible region of (1) and its sum rate is R2 . From [3], it is known that P (R) is strictly convex in R, given a sufficiently large number of subcarriers. Thus, P (R2 ) < λP (R∗1 ) + (1 − λ) P (R∗3 ). Since P (R∗2 ) is the optimal rate vector among all the rate vectors with a summation of R2 , we have P (R∗2 ) ≤ P (R2 ). Consequently, we have that P (R∗2 ) < λP (R∗1 ) + (1 − λ) P (R∗3 ). Thus, the minimum transmit power needed given the throughput R (= BηSE ), P (R) = P (R∗ ), is strictly convex in R (and ηSE ). Denote the superlevel set of ηEE (ηSE ) as

0.3 UB (dual) LB (dual−based)

0.28 0.26

EE (Mbits/Joule)

0.24 0.22 0.2 0.18 0.16 0.14 0.12

Fig. 1.

ˇ | η (ηSE ) ≥ β, β ∈ R}. Sβ = {R ≥ R EE 0

0.5

1

1.5 SE (bits/s/Hz)

2

2.5

3

EE-SE relation

UB (dual) LB (dual−based) DPRA MPDF

0.22

ηSE →∞ ηSE →∞ O(P ) BηSE = lim P = 0. Thus, starting from ηSE = lim P →∞ P P →∞ ∗ ˇ R/B, ηEE (ηSE ) is either strictly decreasing with ηSE if ∗ dηEE (ηSE ) ≤ 0, or first strictly increasing and then dηSE ˇ ηSE =R/B ∗ dηEE (ηSE ) strictly decreasing with ηSE if > 0. dηSE ˇ ηSE =R/B ˇ RPT R And the maximum EE within the SE region, B , B , is

0.2 EE (Mbits/Joule)

ˇ | βP (ηSE ) + When β ≥ 0, Sβ is equivalent to Sβ = {R ≥ R βPc − BηSE ≤ 0}, where P (ηSE ) is the minimum total ˇ As a transmit power needed for any SE ηSE ≥ R/B. result of the convexity of P (ηSE ) (i.e., P (R)) proved above, ∗ (ηSE ) is strictly Sβ is strictly convex in ηSE . Hence, ηEE quasiconcave and has a unique global maximum. ∗ SE (ηSE ) = lim PBη+P = One the other hand, lim ηEE c

0.18

0.16

straightforward as indicated in Theorem 1. 0.14

R EFERENCES 0.12

0

0.5

1 SE

min

1.5 (bits/s/Hz)

2

2.5

3

Fig. 2. Performance comparison of the MPDF-based, DPRA-based and dualbased JIOO algorithms

3

10

Normalized CPU Time

UB (dual) DPRA MPDF (proposed)

2

10

1

10

0

10

2

3

4

5 User number (K)

6

7

8

Fig. 3. Complexity comparison in the case ηˇSE = 1 bit/s/Hz, CNRk = ˇ k = 25 kbps and with PPC 20 dB, R

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