WITH the ever increasing demand for higher reliable

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1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE Energy-Efficient Resource Allocation in Downlink OFDM Wireless Systems With Proportional Rate Constraints Zhanyang Ren, Shanzhi Chen, Senior Member, IEEE, Bo Hu, and Weiguo Ma Abstract Orthogonal frequency-division multiplexing OFDM, with its own advantages of spectral efficiency SE enhancement and flexible resource allocation, is a promising basic technique for fulfilling the ever increasing demand for high-data-rate transmission and various service type support for mobile multimedia communications. In the meantime, energy efficiency EE has now become a critical metric for green system design. In this paper, energy-efficient and fair resource allocation is investigated in a downlink OFDM-based mobile communication system. Given a subcarrier assignment, the bisection-based optimal power allocation BOPA is proposed at first, which achieves the maximum EE and guarantees proportional data rates for users. Then, a two-step subcarrier assignment is designed to avoid unaffordable computational complexity of exhaustive search. In the first step, the estimated energy-efficient transmit power is found via the assumptions on flat fading and subcarrier sharing. In the second step, the traditional spectral-efficient subcarrier assignment SESA is introduced to complete the bandwidth resource allocation among users. Although the two-step subcarrier assignment is suboptimal due to the fact that the optimization is done independently in two separate steps, numerical results demonstrate that its performance is very close to the optimum. This paper also studies the difference between the energy-efficient solution and the traditional spectral-efficient policy and observes that they are similar with each other in the low channel-gain-to-noise ratio CNR regime. This observation is helpful and could enable that the energy-efficient design can be turned into the relatively simpler spectral-efficient policy when the CNR is low. Index Terms Energy efficiency EE, mobile multimedia communications, orthogonal frequency-division multiplexing OFDM, proportional rate constraints, spectral efficiency SE. Manuscript received July 8, 013; revised January 18, 014; accepted February 7, 014. Date of publication March 11, 014; date of current version June 1, 014. This work was supported in part by the Major National Science and Technology Special Project under Grant 013ZX , by the National High-Technology Program 863 Program of China under Grant 014AA01A701, and by the China Next Generation Internet Project under Grant CNGI The review of this paper was coordinated by the Guest Editors for the Special Section on Green Mobile Multimedia Communications. Z. Ren and B. Hu are with State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing , China childyoung@gmail.com; hubo@bupt.edu.cn. S. Chen and W. Ma are with the State Key Laboratory of Wireless Mobile Communications, China Academy of Telecommunications Technology CATT, Beijing , China chensz@datanggroup.cn; mawg@datanggroup.cn. Digital Object Identifier /TVT I. INTRODUCTION WITH the ever increasing demand for higher reliable data transmission and various quality-of-service requirements for emerging mobile services and applications such as content sharing and video streaming [1], mobile multimedia communications are expected to be provided in a more effective and efficient way. From the wireless communications perspective, channel variation and interference also challenge system robustness. In third-generation wireless systems, directsequence code-division multiple access DS-CDMA has been chosen as the basic technique for providing such multimedia communications []. Orthogonal frequency-division multiplexing OFDM [3] and CDMA-based technologies, such as multicarrier CDMA MC-CDMA, are viewed as the evolution path toward higher network performance [4]. Nowadays, OFDM has been adopted in the next-generation mobile broadband systems, such as Long Term Evolution LTE and Worldwide Interoperability for Microwave Access WiMAX. As a basic broadband technique, OFDM provides not only a good physical layer with higher and more stable performance but also benefits service diversity due to its persubcarrier resource-allocation capability with finer granularity. To this end, in addition to the signal transmission point of view, the OFDM-based system can be also regarded as a highly potential candidate for mobile multimedia communications from resource-allocation perspective. User data rates in mobile multimedia communications vary and may change dynamically due to the existing and emerging multimedia applications. To support such dynamic user data rates, adaptive resource allocation in OFDM systems has been well investigated in the literature [5] [9]. Generally speaking, in addition to user rate consideration, rate-adaptive schemes [5], [6] aim to maximize system throughput with given transmit power constraints, whereas the marginal-adaptive policies [7] [9] try to minimize the power consumption with a target transmit rate requirement. Both approaches can be viewed as spectral efficiency SE enhancement policies. However, with the rapidly growing number of communication devices and the demand for environment protection, energy efficiency EE is adopted to be the key performance in green radio system design [10]. Different from the traditional SE, EE design aims to maximize the throughput with each unit of power consumption bits/joule, i.e., given the amount of data needed to be delivered, the objective of EE design is to consume minimal power resources IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 140 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 014 It is well known that orthogonal frequency-division multiple access OFDMA is optimal for SE design in OFDM networks [11]; likewise, each subcarrier assigned to only one user is demonstrated optimal also for EE optimization [1]. In [13] and [14], energy-efficient power allocation for OFDM systems is studied. Fixed circuit power consumption is considered in [13], whereas in [14], a linear sum rate dependence is also taken into account in addition to the constant part. Other than power allocation, more complete resource allocation with subcarrier assignment included can be found in [15] and [16] for uplink OFDM, where the flat-fading and frequencyselective fading are considered, respectively. Furthermore, proportional fairness [17] is also taken into account for the long-term performance. In the downlink, resources in a base station, e.g., subcarrier, power and antenna, are allocated among users to achieve the maximum EE [18] []. In [18], lowcomplexity subcarrier assignment and optimal power allocation are developed, respectively, to maximize EE and meet user rate requirements as well. Tradeoff between SE and EE is investigated in [19]; in addition, tradeoffs between deployment efficiency and EE, bandwidth and power, and delay and power in energy-efficient networks are also introduced in [3]. Other than user rate requirements, a maximum tolerable channel outage probability is also considered in [0] and [1], where the base station equipped with a large number of antennas and secure transmission are addressed, respectively. Given limited backhaul capacity, the work in [] carried out energyefficient resource allocation and scheduling in a multicell scenario. However, most existing work does not focus on the system fairness issue in energy-efficient design for OFDM systems, except for the uplink long-term fairness consideration in [15] and [16]. However, downlink fairness is critical for supporting various multimedia applications in future mobile communication systems. With such a motivation, we propose energyefficient downlink resource allocation and take instantaneous fairness into account in this paper. To this end, proportional rate constraints [4] are considered, which can be viewed as a fairness-level configuration to ensure that each user would obtain a predetermined proportion of the system throughput in each resource-allocation determination. The bisection-based optimal power allocation BOPA is first given under a fixed subcarrier assignment. Then, low-complexity energy-efficient subcarrier assignment is proposed via two steps: 1 energyefficient transmit power estimation EETPE, and spectralefficient subcarrier assignment SESA. The proposed algorithm enhances EE significantly with the performance approaching the optimal solution, which can be only obtained by full enumeration. The remainder of this paper is organized as follows. In Section II, the system model and overall description of the proposed algorithm are given. In Section III, the optimal power allocation under a fixed subcarrier assignment is proposed. In Section IV, a suboptimal subcarrier assignment is developed through two steps: 1 the EETPE, and SESA. The complexity of the proposed EE solution is further analyzed in Section V. Numerical results are given and discussed in Section VI, and finally, this paper is concluded in Section VII. II. SYSTEM MODEL AND OVERALL DESCRIPTION OF THE PROPOSED ALGORITHM A. System Model In this paper, a downlink multiuser scenario in OFDM wireless systems is considered, and perfect instantaneous channel information is assumed available at both the base station and the user side. Based on the channel state information, the resourceallocation algorithm assigns subcarriers set N = 1,,..., N} to users set K = 1,,...,K}, with the total transmit power constraint P trans. In addition, constant circuit power is also considered to calculate downlink total power consumption, i.e., the same as [5] P total = ζp trans + P C 1 where ζ denotes the reciprocal of drain efficiency of the power amplifier, and P C is the constant circuit power. Let B be the system bandwidth and subcarrier bandwidth can be expressed as B/N accordingly. Let h n, k and N 0 be the channel gain and noise power spectral density of additive white Gaussian noise, respectively. Further, let μ k, n = 1 denote if subcarrier n is assigned to user k, and μ k, n = 0, otherwise; then, we can express SE in terms of bits/s/hz based on the Shannon capacity formula as SEμ, p Δ = n N μ k, n N log 1 + p k, nh k, n N 0 B N where μ = μ k, n },n N is the subcarrier assignment indicator set, and p = p k, n },n N is the indicator set for power allocation. For simplicity, g n, k is used to denote the channelgain-to-noise-ratio CNR h n, k /N 0B/N in the following. Accordingly, EE can be expressed in terms of bits/hz/joule as EEμ, p Δ = μ k, n n N N log 1 + p k, n ζ n N p. 3 k, n + P C Hence, the EE optimization problem can be formulated mathematically as maximize EEμ, p subject to p k,n P trans n N p k,n 0 for all k, n μ k,n = 0, 1}, for all k, n μ k,n = 1, for all n R 1 : R :...: R K = α 1 : α...: α K 4 where α 1,...,α K } is the proportional rate constraints set for fairness consideration [4]. It denotes that the data rate among users would follow a predetermined proportion, or in other words, it quantizes the user priority. For user k, the proportion of system throughput distributed would be α k / α k.the third and the fourth constraints show that one subcarrier can be only assigned to one user. User rate R k can be expressed as R k = n N μ k, n N log 1 + p k, n. 5

3 REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS 141 Fig. 1. Overall procedure of the proposed algorithm. B. Overall Description of the Proposed Algorithm In this paper, the proposed resource-allocation algorithm is completed through two consecutive steps. The first step is subcarrier assignment. To reduce the computational complexity, this step is comprised of two substeps. In the first substep, we carry out an EETPE algorithm to obtain the estimated energy-efficient transmit power, followed by another substep of running the SESA algorithm to assign subcarriers among users. Then, the second step of the proposed algorithm is power allocation, which is to execute the BOPA algorithm to complete the energy-efficient resource allocation. The overall procedure is shown in Fig. 1. In the following, we will first introduce the BOPA algorithm since this power allocation scheme is optimal and we can easily find a brutal-search-based subcarrier assignment to obtain the optimal solution. Then, the EETPE and SESA algorithms are given due to the prohibitive computational complexity of the brutal search. III. OPTIMAL POWER ALLOCATION Here, we propose the optimal power allocation with a given subcarrier assignment. Single-user power allocation is discussed at first, and then the system-level multiuser optimal power allocation under proportional rate constraints is developed. A. Single-User Discussion Suppose that the given subcarrier assignment is D k }, where D k is the subcarrier set for user k. Obviously, to make the resource allocation most energy efficient, each user should consume minimum power resource to reach a target data rate. It is a water-filling operation and can be solved by Lagrangian method as follows: } 1 ˆp k, n =max θ ln 1, 0 n N log p k, n =R k 1 N where ˆp k, n } n Dk is the power allocation among subcarriers for user k, and θ is the Lagrangian multiple, which should be chosen such that user rate R k is satisfied. Given subcarriers set D k, from results 6, we further define a single-user power consumption function of rate level R k as P k R k D k = Δ ˆp k, n, k K 7 for which we can have the following lemma. The proof is given in Appendix A. Lemma 1: The single-user power consumption function 7 is strictly convex in R k, and its first derivative is } dp k R k D k 1 +ˆpk, n = N ln min 8 dr k where ˆp k, n } n Dk is the power allocation under rate level of R k via 6. B. Multiuser Optimal Power Allocation Now, we consider all the users in system and the proportional rate constraints. We introduce a rate parameter λ, which is defined as λ = Δ R 1 = R = = R K. 9 α 1 α α K Hence, the user power can be expressed as P k λα k D k, and the total transmit power can be rewritten as P T λ Δ = P k λα k D k. 10 Since P k λα k D k is monotonously increasing in λ, the upper bound λ max satisfying P T λ max =P trans can be found using the bisection method. Through the conversion, the EE can be reformed as a function of λ, i.e., EEλ = Δ λ α k 11 ζp T λ+p C and the total transmit power constraint also has a linear form, i.e., λ λ max. 1 According to [13], we have the following. Lemma : If transmit power P T R is strictly convex in rate vector R, UR is strictly quasiconcave, where R UR =. 13 P T R+P C Apparently, P T λ is strictly convex in λ; hence, EEλ in 11 is strictly quasiconcave, and there is a unique globally optimal solution to obtain the maximum EE. Therefore, for the reformulated EE problem 11 under linear constraint 1, we have the following theorem readily. The proof is given in Appendix B. Theorem 1: EEλ is first strictly increasing and then strictly decreasing in λ. Furthermore, The unique local optimal solution ˆλ for EE problem 11 under linear constraint 1 satisfies λmax, if fλ ˆλ = max 0 λ 14, otherwise where λ is the global optimum satisfying fλ =0. fλ is defined as fλ Δ = ζp T λ+p C λζ ln min 1 +ˆpk, n } α k 15 where ˆp k, n } n Dk is the power allocation via 6 under the rate level of λα k.

4 14 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 014 TABLE I BOPA ALGORITHM According to Theorem 1, we can obtain the optimal power allocation easily. The BOPA algorithm is shown in Table I, where we set the initial points as follows to speed up the searching of λ 1 max : λ ini, low =min λ ini, high =max R k P trans K D k α k } R k P trans K D k α k } 16 where R k P trans /K D k is the maximum data rate under power level of P trans /K for user k, i.e., water-filling with P trans /K among subcarrier set D k. As shown in Table I, the BOPA algorithm can be divided into three steps. First, the upper bound of λ related to the maximum transmit power is initialized by the bisection method. Second, the derivative of EE at λ max is examined. If the derivative is positive, then λ max is the optimum according to Theorem 1, i.e., utilizing all the transmit power would reach the maximum EE. Otherwise, the optimum ˆλ lies between 0 and λ max, i.e., the energy-efficient transmit power is lower than the maximum. Hence, exploiting the monotonicity of fλ, ˆλ would be found by the bisection method in the last step. IV. LOW-COMPLEXITY SUBCARRIER ASSIGNMENT Intuitively, enumerating each possible subcarrier assignment, the optimum is the one with which applying the BOPA algorithm would obtain the maximum EE. However, the complexity is too high to be affordable in a real system. Here, we try to develop a low-complexity policy instead and to achieve a promising performance meanwhile. 1 Calculate user rate with equal power allocation among users, i.e., R k P trans/k D k. The power level of the user that reaches the highest proportional rate R k /α k must be decreased, and meanwhile, the lowest proportional rate user must increase its power to ensure that all the users could achieve the same proportional rate level, i.e., λ max. Hence, λ max must lie between the highest level max R k /α k } and the lowest level min R k /α k }. The essential difference between EE and SE is the transmit power. For SE issue, the transmitter will consume all the power resource, whereas for EE aiming to maximize the data rate per unit of power consumption, there is an optimal transmit power that may be lower than the maximum in most cases. We term it the energy-efficient transmit power. Hence, the EE problem can be viewed as a special SE problem under a special energy-efficient power level. Motivated by such analysis, energy-efficient subcarrier assignment can be viewed as a twostep operation: 1 find the energy-efficient transmit power; and carry out subcarrier assignment to maximize SE under the power level obtained in the first step. However, it is generally very hard to find the optimal solution for both two steps. In the following, we propose the EETPE and suboptimal SESA instead, which can reduce the computational complexity and guarantee a promising performance. A. Energy-Efficient Transmit Power Estimation The energy-efficient transmit power is estimated via assumptions on flat fading and subcarrier sharing. For flat fading, we define the average CNR as = E n, k K. Since log 1 + pg is strictly concave in g, from Jensen s inequality, we know that log 1 + p k, n D k log 1 + p k, n, where D k denotes the cardinality of set D k, i.e., the number of subcarriers for user k. Furthermore, for subcarrier sharing, the number of subcarriers assigned to user is relaxed to a real number on 0,N]. Hence, the original EE is upper bounded by the new derived EE. In other words, EE in frequency-selective fading with subcarrier exclusively taking is upper bounded by a flat fading assumption with subcarrier sharing. Therefore, the estimated energy-efficient transmit power would be approximated by the optimal energy-efficient transmit power of the upper bounded EE here. From this description, the user rate can be expressed as R k = N k N log 1 + P k. 17 Moreover, as the BOPA algorithm in Section II, we also introduce a rate parameter η, which is defined as η = Δ R 1 = R = = R K. α 1 α α K 18 Hence, combining 17 and 18, user power can be formed as a function of η and, i.e., 1 P k η, =. 19 Since the EE optimization problem is converted into a power consumption minimization one with a given η, we define a minimum power consumption function of η as } ˆP η = Δ minimize k=n P k η,. 0 Apparently, P k η, is strictly decreasing with respect to.therefore, the total transmit power is minimized when = N rather than <N.

5 REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS 143 For ˆP η, we have the following property and give the proof in Appendix C. Lemma 3: The minimum power consumption function ˆP η is strictly convex in η, and its first derivative is TABLE II TPM ALGORITHM d ˆP η dη = N ln α k k 1 where ˆ } is the solution to 0 at η. Based on this description, the upper bounded EE under flat fading and subcarrier sharing can be formed as a function of η, i.e., EE ub η Δ = η α k ζ ˆP η+p C. According to Lemma 1, EE ub η is also strictly quasiconcave in η, and there is a unique solution to achieve the maximum EE. Since d ˆP η/dη > 0, ˆP η is monotonously increasing with respect to η; hence, the maximum transmit power constraint can be converted into a linear constraint as η η max 3 where ˆP η max =P trans. As Theorem 1, we readily have the following properties. The proof is similar with Appendix B and is thus omitted. Theorem : The upper bounded EE under flat fading and subcarrier sharing EE ub η is first strictly increasing and then strictly decreasing in η, and EE optimization problem under linear constraint 3 has a unique optimal solution as η ˆη =, if η η max 4 η max, otherwise where ˆP η max =P trans, and η is the global optimum satisfying δη =0. δη is defined as δη Δ = ζ ˆP η+p C ζn ln α k k 5 where ˆ } is the solution to 19 at η. Based on Theorem, the estimated energy-efficient transmit power can be obtained readily as P = ˆP η, if ˆP η P trans P trans, otherwise. 6 Before describing the whole algorithm, we should propose the solution to power minimization problem 0 at first. Obviously, 0 is a standard convex optimization. Karush Kuhn Tucker conditions can be listed as follows: 1 ln N = γ = 0 7 where γ is the Lagrangian multiplier. Define γ k as γ k = Δ 1 1 Nηα k ln 8 and we can have dγ k = ln d + N k 1 ln = ln N k Nηαk ln N k < 0. 9 Hence, with a given γ, we can use the bisection method to get }. Furthermore, we can also use the bisection method to get the optimum γ to satisfy N =. The transmission power minimization TPM solution can be regarded as a twolayer bisection method, and the detail is shown in Table II. Generally, the number of subcarriers assigned to users satisfies 1 N; hence, we set the initial points for searching γ as γ low =min 1 1 ηαk ln ηα k γ high =max } } 1 1 Nηαk ln. 30 By combining it with the TPM algorithm, we can propose the EETPE algorithm as in Table III. The initial points can be set as 3 ηlow = 0 η high =max log 1 + P trans KN / α k }. 31 In step 1, we first calculate δη high with }, which is obtained by the TPM algorithm. If δη high is nonnegative, then δη max would be positive since δη is strictly decreasing with respect to η and η high >η max. Hence, the maximum transmit power would be the estimated energy-efficient transmit power. Otherwise, if δη high is negative, the global optimum η would lie between η low and η high. Then, in step, η is found by the bisection method. Finally, the estimated energy-efficient transmit power is determined according to 6. 3 Suppose that all N subcarriers are assigned to each user and then allocate the total power P trans among users equally. The maximum proportional rate level max R k /α k } is much higher than η max that is obtained from ˆP η max =P trans under any possible subcarrier assignment satisfying = N.

6 144 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 014 TABLE III EETPE ALGORITHM the subcarrier is assigned according to the max min policy as aforementioned. TABLE IV SESA ALGORITHM B. Suboptimal Spectral-Efficient Subcarrier Assignment After finding the estimated energy-efficient transmit power, the subcarrier assignment can be carried out with the objective of maximizing the SE. This is a spectral-efficient resourceallocation problem, which has been investigated thoroughly in traditional OFDM systems [4], [6], OFDM-based relay systems, [7], and cognitive radio networks as well [8]. The discrete stochastic optimization policy proposed in [8] is much more complex since power allocation is evolved in each subcarrier assignment attempt; although [6] could be more effective via water-filling, it is also more complex compared with that of [4]. To this end, the equal-power-distribution-based method in [4] is adopted to carry out subcarrier assignment in this paper. It is of low complexity and could reach average performance. The key idea is maximizing the minimum proportional rate R k /α k. That is, the user that has the minimum R k /α k would be assigned with the corresponding best subcarrier, and its data rate would be updated with equal power distribution P/ D k. The SESA is shown in Table IV. In step 1, each user chooses its best subcarrier to initialize user rate, and in step, V. C OMPLEXITY STUDY Based on our previous analysis, the proposed EE-suboptimal algorithm can be implemented in three steps: 1 EETPE; SESA; and 3 BOPA algorithms. In the EETPE algorithm, suppose that the number of iterations of bisection for finding η is T OL, and the number of iterations for finding γ and that for calculating } in the TPM algorithm are T ML and T IL, respectively; 4 thus, the complexity of the EETPE algorithm is OT OL T ML T IL K. For the SESA algorithm, the complexity is mainly caused by channel sorting and is thus OKN log N. LetT WF be the number of iterations of a practical algorithm for water-filling 6, such as 1-D search; the complexity of BOPA algorithm is thus OT OL T WF K. On the other hand, the optimal solution EE-optimal can be obtained by exhaustive search of any subcarrier assignment combining the BOPA algorithm. Obviously, there are K inds of subcarrier assignment; hence, the complexity of the optimal solution is OT OL T WF K N+1. The complexity of both the optimal and proposed suboptimal solutions is summarized in Table V. VI. SIMULATIONS A. Simulation Environment In simulations, we assume that each user s subcarrier signal undergoes identical Rayleigh fading independently. The total transmit power P trans and the circuit power P C are 1 and 0. W, respectively. The reciprocal of drain efficiency of power amplifier ζ is The average CNR is defined as E n for all k, where E n denotes statistical expectation of with respect to n. All the experiment data are obtained by Monte Carlo simulations over 1000 channel realizations. The EE-optimal and SE methods in [4] are considered the comparison schemes with our proposed EE-suboptimal solution. B. Computational Complexity Figs. and 3 show the average running time of EEsuboptimal. The simulation was run on a personal computer, and the software and hardware conditions are: Windows XP Home Edition, Pentium-M 1.73-GHz processor, and -GB memory. The average CNR is 5 db and each user has the same rate requirement. Fig. gives the comparison between EE-optimal and EEsuboptimal. Since EE-optimal is a brutal-force-search-based scheme, we fix the number of users as, and run simulation with a small number of subcarriers. It is clear that the EEoptimal is much more complex than EE-suboptimal. The running time of EE-optimal grows exponentially with the number of subcarriers, whereas that of EE-suboptimal only increases linearly. 4 These three iterations can be viewed as outer layer, middle layer, and inner layer search of the EETPE algorithm.

7 REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS 145 TABLE V COMPUTATIONAL COMPLEXITY COMPARISON Fig.. Average running time versus the number of subcarriers. The number of users is two, and the average CNR is 5 db for both two users. Fig. 4. EE versus average CNR. The number of subcarriers is ten and the number of users is two. All users have the same channel condition. Fig. 3. Average running time versus the number of subcarriers. Average CNR is 5 db for all users. The average running times of EE-suboptimal under different scenarios are given in Fig. 3. We can observe that even in condition of 56 subcarriers and 3 users, EE-suboptimal takes about 0.35 s to complete one resource allocation. It is still much more efficient than EE-optimal with only two users and ten subcarriers, which takes 0 s according to Fig.. C. EE and SE Performance Figs. 4 and 5 show the EE and SE, respectively. To reduce the simulation time, we set ten subcarriers and two users. Proportional rate constraints are set to 1:1 and 1:10, respectively, to evaluate the effect of the system fairness requirement. As shown Fig. 5. SE versus average CNR. The number of subcarriers is ten and the number of users is two. All users have the same channel condition. in Fig. 4, EE methods are more effective in terms of EE than the SE method in the high-snr regime. While in the low-snr regime, the SE and EE policies could reach a similar EE level. In Fig. 5, we can also see a similar trend in terms of SE, i.e., the SE method is more spectral efficient than the EE method in the high-snr regime, whereas in the low-snr regime the two schemes are similar with each other. It can be explained by the reason that the rate function of power is concave. Hence, the rate increment of each power unit is higher in the low-cnr regime than that in the high- CNR regime, and the energy-efficient transmit power would be the maximum transmit power with high possibility. In fact,

8 146 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 014 Fig. 6. EE versus average CNR of user. The number of subcarriers is ten and the number of users is two. User s average CNR is 5 db more than that of user 1. Fig. 8. EE versus average CNR. The number of subcarriers is 64 and the number of users is four. All users have the same channel condition. Fig. 7. SE versus average CNR of user. The number of subcarriers is ten and the number of users is two. User s average CNR is 5 db more than that of user 1. when the energy-efficient transmit power is the same as the maximum transmit power, the SE scheme and the EE scheme are equivalent. On the other hand, in the high-cnr regime, the energy-efficient transmit power would be lower than the maximum transmit power since the power would be less efficient as rate increases slowly. Hence, the discrepancy of the two designs is significant, i.e., the SE method would achieve more throughputs, and the EE method could be more efficient with each unit of power consumption. Mathematically, as is small, i.e., in the low-cnr regime, fλ max and δη max are more likely to be positive, which can be observed from 15 and 5. This means that the maximum transmit power is more likely to be the most energy efficient, which verifies our earlier discussion. In Figs. 4 and 5, we can also see that either EE or SE under proportion 1:1 achieves a higher performance than that of 1:10. It is reasonable since the channel conditions are similar between two users and neither has priority over the other. Fig. 9. SE versus average CNR. The number of subcarriers is 64 and the number of users is four. All users have the same channel condition. To show the connection between proportion and channel condition more clearly, we set different average CNRs for two users in Figs. 6 and 7. For simplicity, average CNR of user is 5 db more than that of user 1. The other parameters are the same as in Figs. 4 and 5. Apparently, when the proportion is 1:10, system performance reaches a higher level in terms of either EE or SE. In short, when the user with better channel condition requires more data rate, the system performance is better. Figs show a scenario of 64 subcarriers and four users. Proportional rate constraints are set to 1:1:1:1 and 1:1:1:10 in this case. In Figs. 8 and 9, all users have the same average CNR, and in Figs. 10 and 11, user 4 s average CNR is 5 db higher than that of the others. Considering a larger input case, we also give a scenario of 56 subcarriers and 16 users in Figs Four users have higher rate requirements and the proportional rate constraints are set to 1: 1:1:1:1 and 1: :10:10:10:10. In Figs. 1 and 13, all users have the same average CNRs, whereas in Figs. 14

9 REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS 147 Fig. 10. EE versus average CNR of user 4. The number of subcarriers is 64 and the number of users is four. User 4 s average CNR is 5 db more than that of the others. Fig. 1. EE versus average CNR. The number of subcarriers is 56 and the number of users is 16. All users have the same channel condition. Fig. 11. SE versus average CNR of user 4. The number of subcarriers is 64 and the number of users is 4. User 4 s average CNR is 5 db more than that of the others. and 15, the last four users average CNRs would be 5 db higher than that of the others. Similar with Figs. 4 7, we can also see the relation between SE design and EE design in these two scenarios. In the high- CNR regime, the two schemes are distinct from each other, whereas in the low-cnr regime, the discrepancy would not be significant. Moreover, the connection between system performance and proportion requirements is illustrated as well. When more data rate is distributed to the user that has a higher channel gain, the system performance would be enhanced in terms of both EE and SE. VII. CONCLUSION In this paper, we have investigated downlink energy-efficient and fair resource allocation in OFDM wireless systems to support mobile multimedia communications. Both EE and pro- Fig. 13. SE versus average CNR. The number of subcarriers is 56 and the number of users is 16. All users have the same channel condition. portional rate constraints are incorporated into the proposed resource-allocation approaches. First, the optimal power allocation with a given subcarrier assignment is carried out. Then, a two-step suboptimal method for assigning subcarriers is proposed to avoid the expense and complexity of exhaustive search. In the first step, the energy-efficient transmit power is estimated via an upper bounded EE with assumptions on flat fading and subcarrier sharing. In the second step, the traditional spectral-efficient scheme is introduced to complete the subcarrier assignment. The numerical results show that the proposed algorithm could approach the optimum and is more energy efficient than the traditional spectral-efficient method in the high-cnr regime. In the low-cnr regime, the SE design and the EE design are similar to the result of the concavity of Shannon capacity formulation. This observation simplifies the EE design to a certain extent since we can turn to a relatively less complex SE policy in this situation.

10 148 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 014 ˆr k, 1 n }, ˆr k, n }, and ˆr k, 3 n } be the rate distribution accordingly; thus, we can have ɛp k R 1 k D k +1 ɛpk R k D k = ɛ ˆp 1 k, n +1 ɛ ˆp k, n = ɛ N ˆr 1 k, n 1 +1 ɛ N ˆr k, n 1 n D k = ɛ N ˆr1 k, n 1 +1 ɛ N ˆr k, n 1 a ɛn ˆr 1 k, n +1 ɛn ˆr k, n 1 = P k ɛr 1 k +1 ɛr k D k b N ˆr 3 k, n 1 33 Fig. 14. EE versus average CNR of the last four users. The number of subcarriers is 56 and the number of users is 16. The last four users average CNR is 5 db more than that of the others. where a utilizes the convexity of x 1, and b utilizes the definition of P k R k D k. This shows the convexity. Let ˆr k, n +Δr k, n } n Dk be the rate distribution under user rate level R k +ΔR k, where ΔR k = Δp k, n ; then, we can have 3, shown at the bottom of the page. This gives the derivative. APPENDIX B PROOF OF THEOREM 1 Proof: Based on the derivative of P k λα k D k given in Lemma 1, the derivative of P T λ can be obtained as P T λ = dp T λ dλ = N ln min = dp k λα k D k α k dλα k 1 +ˆpk, n } α k 36 Fig. 15. SE versus average CNR of the last four users. The number of subcarriers is 56 and the number of users is 16. The last four users average CNR is 5 db more than that of the others. APPENDIX A PROOF OF LEMMA 1 Proof: Let ˆp 1 k, n }, ˆp k, n }, and ˆp 3 k, n } be the power allocation under user rate Rk 1, R k, and ɛr1 k + 1 ɛrk based on 6, respectively, where 0 ɛ 1. Let where ˆp k, n } n Dk is the single-user power allocation under rate level of λα k. Obviously, P T λ is monotonously increasing in λ; hence, let P T λ be the second derivative of P T λ,wehavep T λ > 0. For simplicity, define α = Δ K k=1 α k; we can also have deeλ = α ζp T λ+p C λαζp T λ dλ ζp T λ+p C = ζp T λ λζp T λ+p C α ζp T λ+p C fλ α = ζp T λ+p C 37 } dp k R k D k P k R k +ΔR k D k P k R k D k min N ˆr k, n +NΔr k, n 1 N ˆr k, n 1 dr k ΔR k 0 ΔR k ΔR k 0 ΔR k } min N ˆr k, n +NΔr k, n N ˆr k, n } min N ˆr k, n NΔr k, n 1 ΔR k 0 ΔR k ΔR k 0 } ΔR k min N ˆr k, n n D Δr k, n N ln } k 1 +ˆpk, n ΔR k 0 ΔR k = N ln min 3

11 REN et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION IN DOWNLINK OFDM WIRELESS SYSTEMS 149 where fλ Δ = ζp T λ λζp T λ+p C. The derivative of fλ can be given as f λ =ζp T λ λζp T λ+p C = ζp T λ ζp T λ λζp T λ = λζp T λ < Hence, fλ is strictly decreasing in λ. Since f0 =P C > 0, EEλ is either strictly increasing, or first strictly increasing and then strictly decreasing in λ. On the other hand, we know that EE0 =0, and λα α lim EEλ λ λ ζp T λ+p C λ ζp T = 0. λ 39 Therefore, from the given discussion, it can be derived that EEλ is first strictly increasing and then strictly decreasing in λ. From 37, it can be further derived that the unique global optimum λ satisfies fλ =0. Local optimum ˆλ for the constrained EE can be also obtained readily, as expressed in 14. This completes the proof. APPENDIX C PROOF OF LEMMA 3 Proof: First, we prove that P k η, is convex. The Jacobian matrix can be obtained as [ P k η, = Nα k ln 1 ln 1 ]/ 40 and then the Hessian matrix can be derived as [ Nk ] / P k η, = η 1 Nαk ln η ḡ 1 k. 41 η Since the Hessian matrix is negative semidefinite, P k η, is convex. We further proof the convexity of ˆP η in the following. Let 1 k }, k }, and 3 k } be the subcarrier assignments to reach the minimum power consumption under rate parameter η 1, η, and ɛη 1 +1 ɛη, respectively, where 0 ɛ 1. Therefore, we can have ɛ ˆP η 1 +1 ɛ ˆP η = ɛ P k η 1, ˆ 1 +1 ɛ P k η, ˆ 1 ɛp k η 1, k 1 +1 ɛp k η, k = a P k ɛη 1 +1 ɛη,ɛˆ 1 +1 ɛ k b P k ɛη 1 +1 ɛη, k 3 = ˆP ɛη 1 +1 ɛη 4 where we use the convexity of P k η, at a and the definition of ˆP η at b. Now, we calculate the first derivative of ˆP η. Let ˆ } and ˆ +Δˆ } be the subcarrier assignments under rate parameter η and η +Δη, respectively. From the definition of ˆP η, we can have ˆ+ Δ ˆ 1 k +Δˆ ˆP η 43 Nη+Δηα k ˆP η +Δη k 1 ˆ. 44 Accordingly, we can further have 34 and 35, shown at the bottom of the page. d ˆP η dη d ˆP η dη Δη 0 Δη 0 Δη 0 Δη 0 ˆP η +Δη ˆP η Δη ˆ Δη lim Δη 0 NΔηα k ˆ 1 ˆ ˆP η +Δη ˆP η Δη ˆ +Δ ˆ Δη lim Δη 0 = NΔηα k ˆ +Δ ˆ 1 ˆ = Nη+Δηα k ˆ 1 ˆ k k Δη ˆ 1 ˆ NΔηα k k 1 lim = N ln k α k Δη 0 Δη Nη+Δηα k ˆ +Δ ˆ 1 ˆ +Δ ˆ Δη ˆ +Δ ˆ ˆ +Δˆ 34 ˆ +Δ ˆ 1 ˆ +Δ ˆ NΔηα k k +Δ ˆ 1 lim Δη 0 Δη = N ln ˆ α k 35

12 150 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 63, NO. 5, JUNE 014 Therefore, according to the squeeze theorem, we can have d ˆP η dη This completes the proof. = N ln k α k. 45 REFERENCES [1] H. Moustafa, N. Marechal, and S. Zeadally, Mobile multimedia applications: Delivery technologies, IEEE IT Prof., vol. 14, no. 5, pp. 1 1, Sep./Oct. 01. [] F. Adachi, M. Sawahashi, and H. Suda, Wideband DS-CDMA for next-generation mobile communications systems, IEEE Commun. Mag., vol. 36, no. 9, pp , Sep [3] T. Hwang, C. Yang, G. Wu, S. Li, and G. Ye Li, OFDM and its wireless applications: A survey, IEEE Trans. Veh. Technol., vol. 58, no. 4, pp , May 009. [4] F. Adachi, Evolution towards broadband wireless systems, in Proc. Int. Symp. Wireless Pers. Multimedia Commun., 00, vol. 1, pp [5] H. Yin and H. Liu, An efficient multiuser loading algorithm for OFDMbased broadband wireless systems, in Proc. IEEE Global Telecommun., 000, vol. 1, pp [6] Z. Mao and X. 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Evans, Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints, IEEE Trans. Wireless Commun., vol. 4, no. 6, pp , Nov [5] S. Cui, A. J. Goldsmith, and A. Bahai, Energy-constrained modulation optimization, IEEE Trans. Wireless. Commun., vol. 4, no. 5, pp , Sep [6] C. Mohanram and S. Bhashyam, A sub-optimal joint subcarrier and power allocation algorithm for multiuser OFDM, IEEE Commun. Lett., vol. 9, no. 8, pp , Aug [7] Y. Pan, A. Nix, and M. Beach, Distributed resource allocation for OFDMA-based relay networks, IEEE Trans. Veh. Technol.,vol.60,no.3, pp , Mar [8] R. Xie, F. R. Yu, and H. Ji, Dynamic resource allocation for heterogeneous services in cognitive radio networks with imperfect channel sensing, IEEE Trans. Veh. Technol., vol. 61, no., pp , Feb. 01. Zhanyang Ren received the B.E. and Ph.D. degrees from Beijing University of Posts and Telecommunications, Beijing, China, in 008 and 013, respectively. He then joined the Wireless Network Research Department, Huawei Technologies Company Ltd., Beijing, China. His current research interests include wireless communications theory and wireless communications systems. Shanzhi Chen SM 04 received the Ph.D. degree from Beijing University of Posts and Telecommunications, Beijing, China, in In 1994, he joined Datang Telecom Technology & Industry Group, where he has served as the Chief Technology Officer since 008. From 1999 to 011, he was a member of the Steering Expert Group on Information Technology of the 863 Program of China. He is currently the Director of the State Key Laboratory of Wireless Mobile Communications, China Academy of Telecommunications Technology, Beijing, China, and a board member of Semiconductor Manufacturing International Corporation, Shanghai, China. He has considerable contributions to time-division TD synchronous code-division multiple-access third-generation industrialization and TD Long-Term Evolution Advanced fourth-generation standardization. His current research interests include wireless mobile communications, Internet of Things, and emergency communications. Dr. Chen received the State Science and Technology Progress Award in 001 and 01. Bo Hu received the Ph.D. degree in communications and information systems from Beijing University of Posts and Telecommunications BUPT, Beijing, China, in 006. He is currently an Associate Professor with the State Key Laboratory of Networking and Switching Technology, BUPT. His current research interests include future wireless mobile communication systems and mobile-centric networking. Weiguo Ma received the B.Sc. degree in electrical engineering and the Ph.D. degree in signal and information systems from Beijing Institute of Technology, Beijing, China, in 1993 and 1998, respectively. He is currently a Professor with the State Key Laboratory of Wireless Telecommunications, China Academy of Telecommunications Technology, Beijing. He also served as the Chair to several research projects, including the National High-Technology R&D 863 Program of China as well as industrybased projects. His current research interests include prototype verification of key evolution technology of time-division Long-Term Evolution and beyond.