Joint Tx/Rx Energy-Efficient Scheduling in Multi-Radio Networs: A Divide-and-Conquer Aroach Qingqing Wu, Meixia Tao, and Wen Chen Deartment of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, China. Emails: {wu.qq,mxtao,wenchen}@sjtu.edu.cn. arxiv:152.52v1 [cs.it] 31 Jan 215 Abstract Most of the existing wors on energy-efficient wireless communication systems only consider the transmitter (Tx or the receiver (Rx side ower consumtion but not both. Moreover, they often assume the static circuit ower consumtion. To be more ractical, this aer considers the joint Tx and Rx ower consumtion in multile-access radio networs, where the ower model taes both the transmission ower and the dynamic circuit ower into account. We formulate the joint Tx and Rx energy efficiency (EE imization roblem which is a combinatorial-tye one due to the indicator function for scheduling users and activating radio lins. The lin EE and the user EE are then introduced which have the similar structure as the system EE. Their hierarchical relationshis are exloited to tacle the roblem using a divide-and-conquer aroach, which is only of linear comlexity. We further reveal that the static receiving ower lays a critical role in the user scheduling. Finally, comrehensive numerical results are rovided to validate the theoretical findings and demonstrate the effectiveness of the roosed algorithm for imroving the system EE. I. INTRODUCTION The increasing number of new wireless access devices and various services lead to a significant increase in the demand for higher user data rate. While the higher energy consumtion is a great concern as well for future wireless communication systems. Recently, there has been an usurge of interest in the energy efficiency (EE otimization field. Basic concets of energy-efficient communications are introduced in [1] and several advanced hysical layer techniques for EE are studied in [2] [6]. However, all the above wors only consider one side ower consumtion, i.e., either the transmitter (Tx or the receiver (Rx side. In fact, the exectation of limiting electric exenditure and reducing carbon emissions requires the base station to erform in an energy-efficient manner [1], while minimizing the user side energy consumtion also deserves more efforts due to caacity limited batteries and user exerience requirements [7], [8]. Moreover, according to [9], the techniques adoted to imrove the EE of one end of the communication system may adversely affect the EE of the This wor is suorted by the National 973 Project #212CB31616, by NSF China #6132212, #6116113529, and #6132811, by the STCSM Science and Technology Innovation Program #13517112, by the SEU National Key Lab on Mobile Communications #213D11. Wen Chen is also with the School of Electronic Engineering and Automation, Guilin University of Electronic Technology. other end. Therefore, it is necessary to consider the joint Tx and Rx EE otimization, which shall rovide more flexibility for the energy saving at the side interested or both. For EE oriented research, one of the most imortant tass is to quantify the ower consumtion of the communication system [9]. Most of the existing wors only consider a constant circuit ower so as to simlify the system analysis and mae the roblem more tractable. However, it has been reorted in [1], [1] that a rough modeling for the ower consumtion can not reflect the true behaviour of wireless devices and thus might rovide misleading conclusions. Therefore, the ower consumtion modeling should not only cature the ey system comonents but also characterize the reality [1]. The main contributions of this aer are summarized as follows: 1 We formulate the joint Tx and Rx EE imization roblem in which the lin deendent signal rocessing ower, the static circuit ower as well as the transmission ower are considered based on a comrehensive study [9], while the ower model of existing wors [2] [5], [11], [12] are basically secial cases. 2 We exlore the fractional structure of the system EE and introduce the concet of the individual EE, i.e., the lin EE and the user EE. Based on these, an otimal aroach of linear comlexity is roosed to solve the non-convex EE imization roblem. Moreover, this aroach can also be used to otimally solve the roblem in [3] where only a quadratic comlexity method is roosed. 3 We reveal that the static receiving ower has an imlicit interretation of the otimal number of scheduled users. In the extreme case when the static receiving ower is negligible, time division multilexing access (TDMA is otimal for the energy-efficient transmission. II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION A. System Model Consider a multi-user multi-radio networ, where K users are communicating with one access oint (AP over M orthogonal radio lins simultaneously. It is assumed that each user, for = 1,...,K, is assigned rior with a fixed subset of radio lins, denoted as M, and that the radio lins of different users do not overla with each other so as to void interference, i.e., M Mm = Ø. The multile radio lins can be formed by orthogonal multilexing techniques, such as frequency division multilexing. The channel between the
AP and each user is assumed to be quasi-static fading and they are all equied with one antenna. It is assumed that the erfect and global channel state information (CSI of all users is available for the AP, which allows us to do energyefficient scheduling. The channel gain of user over liniand the corresonding ower allocation of this lin are denoted as g,i and,i, resectively. The receiver noise is modelled as a circularly symmetric comlex Gaussian random variable with zero mean and variance σ 2 for all lins. Then the data rate of user over lin i, denoted as r,i, can be exressed as ( r,i = Blog 2 1+,ig,i Γσ 2, (1 where B is the bandwidth of each radio lin and Γ characterizes the ga between actual achievable rate and channel caacity due to a ractical modulation and coding design [3]. Consequently, the overall system data rate can be exressed as K K R tot = ω R = ω r,i, (2 i M where R is the data rate of user and ω which is rovided by uer layers, reresents the riority of user. B. Joint Tx/Rx Power Consumtion Model In this wor, we adot the ower consumtion model from [9] ublished by Energy Aware Radio and networ technologies (EARTH roject, which rovides a comrehensive characterization of the ower consumtion for each comonent involved in the communication. At the user side, the ower dissiation consists of two arts, i.e., the transmission ower and the circuit ower. Denote P T as the overall transmission ower of user and it is given by i M P T =,i, (3 where (,1] is a constant which accounts for the efficiency of the ower amlifier. DenoteP C as the circuit ower of user. According to [9], the circuit ower of each device contains a dynamic art for the signal rocessing which linearly scales with the number of active lins, and a static art indeendent of lins for other circuit blocs, i.e., P C (n o = no P dyn, +I(n o P sta,, (4 where n o is the number of active lins and can be exressed as n o = i M I(,i. Here, the indicator function I(x is defined as { 1, if x >, I(x = (5, otherwise. Secifically, if,i >, then I(,i = 1 means that lin i is active, and if n o >, then I(no = 1 means that user is scheduled. In (4, P dyn, and P sta, are dynamic and static comonents of the circuit ower for user, resectively. Considering different tyes of terminals in ractical systems, P dyn, and P sta, can be different for different user. Now, the overall ower consumtion of user, denoted as P, is P = P T +P C (n o. (6 At the AP side, the receiving circuit ower consumtion also consists of two similar arts as the user device [7], [9]. Denote P dyn, and P sta, as the dynamic and static receiving circuit ower, resectively. Then the overall ower consumtion at the AP side can be exressed as K P = n o P dyn, +P sta,. (7 Finally, the overall ower consumtion of the system can be exressed as K P tot = P +P. (8 C. Problem Formulation Energy efficiency is commonly defined by the ratio of the overall system rate R tot over the overall system ower consumtion P tot [2], [3], [5]. Our goal is to jointly otimize the user scheduling, the lin activation and the ower control to imize the EE of the considered system. Mathematically, we can formulate the EE otimization roblem as (P1 K ω ( i M Blog 2 1+,i g,i s.t. K n o = Γσ 2 ( i M,i +P C (n o +no P dyn, +P sta, i M I(,i, 1 K,i M,,i P,i, 1 K,i M, (9 where {,i = 1,2,...K;i M }. For ractical consideration, we assume that each radio lin i of user has a imum allowed transmit ower P.,i Note that the authors in [3] consider a similar roblem formulation to (9 but without ower constraint which is thereby a secial case of this aer. The existence of the two layered indicator functions, i.e., I(,i and I(n o = I( i M I(,i maes the objective function discontinuous and hence non-differentiable. The global otimal solution of (9 is generally difficult to be obtained with an efficient comlexity. In the following section, we exlore the articular structure of system EE and show that the global otimal solution can actually be obtained using a divide-and-conquer aroach with low comlexity. III. ENERGY-EFFICIENT SCHEDULING In this section, we solve the system EE imization roblem directly from a fractional-form ersective. This idea results from the connection of the EE from three levels, namely, the lin Energy Efficiency, the user Energy Efficiency, and the system Energy Efficiency. A. Lin Energy Efficiency and User Energy Efficiency Definition 1 (Lin Energy Efficiency: The EE of lin i of user, for i M, = 1,..,K, is defined as the ratio of the weighted achievable rate of the user on this lin over the consumed ower associated with this lin, i.e., ee,i = ω ( Blog 2 1+,ig,i Γσ 2,i, (1 +P dyn, +P dyn,
where the lin-level ower consumtion counts the transmission ower of the user over the lin, er-lin dynamic circuit ower of the user and the AP, resectively. It is easy to rove that this fractional tye function have the stationary oint which is also the otimal oint [13]. By setting the derivative of ee,i with resect to,i to zero, we obtain that the otimal ower value,i and the otimal lin EE under ea ower constraint satisfies,i = [ Bω Γσ2 ln2 ee,i g,i ] P,i,,i M, (11 where [x] a Bω b min{{x,b},a}. Note that ee Γσ2,i ln2 > g,i, i.e.,,i > always holds for ee,i, since otherwise ee,i would be zero. Based on (1 and (11, the numerical values of ee,i and,i can be easily obtained by the bisection method. Definition 2 (User Energy Efficiency: The EE of user, for = 1,...,K, is defined as the ratio of the weighted total achievable rate of the user on all its reassigned radio lins over the total ower consumtion associated with this user, i.e., EE = ( ω i M Blog 2 1+,i g,i Γσ 2 i M,i +n o (P dyn, +P dyn, +P sta,, (12 where the user-level ower consumtion counts the total transmission ower of the user, the overall circuit ower of the user and the dynamic rocessing ower of the AP related to this user. Now, we find the otimal ower control to imize the user EE. The roblem is formulated as {,i } s.t. EE n o = i M I(,i, 1 K,i M,,i P,i, 1 K,i M,,i, 1 K,i M. (13 Define Φ as the set of active lins for user and then n o is the cardinality of Φ. Given any Φ, it is easy to rove that EE is strictly quasiconcave in,i. Thus, similar to the lin EE, the otimal ower allocation under set Φ satisfies [ Bω,i = EE ln2 Γσ2 g,i ] P,i,,i Φ. (14 Note that if,i =, it suggests that this lin should not be active in the otimal solution, but its corresonding circuit ower P dyn, +P dyn, has already been accounted in calculating the total ower consumtion in (13. Therefore, we have to obtain the set Φ in which all radio lins are allocated with strictly ositive owers in imizing EE. Let EE Φ denote the otimal intermediate user EE of user when its current set of active lins is Φ, and then the value of EE Φ can be obtained by (13 and (14. The next theorem rovides a general condition for determining whether an arbitrary lin should be scheduled. Theorem 1: For any lin i / Φ, if EEΦ ee,i, then there must be EEΦ EEΦ {(,i} ee,i, and the lin i should be activated and added to Φ ; else if EEΦ > ee,i, then there must be EEΦ > EEΦ {(,i} > ee,i, and the lin i should not be activated and added to Φ. Proof: Please see Aendix A. The interretation is also obvious: the new lin i should have a better utilization of the ower than its user. In what follows, we introduce how to obtain the otimal user EE based on the lin EE, and the details of this rocedure are summarized in line 1-14 of Algorithm 1. Sort all radio lins of user according to their lin EE ee,i in descending order, i.e., ee,1 ee,2... ee,n, and set the initial Φ = Ø. Then we successively tae one lin from the order and judge whether it should be added to Φ. Until some lin is determined not to be activated or all lins are activated, then based on the current Φ, we can obtain the otimal user EE. Remar 1: The otimality of the roosed rocedure for imizing the user EE is ensured by the ordering of the lin EE as well as the conclusion of Theorem 1. This idea oens u a new way to address the fractional-form EE imization roblem. B. User Scheduling and Lin Adatation In this subsection, we show how to solve the original roblem (9 based on the lin EE and the user EE. For the exlanation convenience, we first introduce two auxiliary sets. Denote Φ as the set of active lins of all users, i.e., Φ = {(,i,i >, i,}, with its otimum denoted as Φ. DenoteU as the set of scheduled users which have at least one active lin belonging to set Φ, i.e., U = { (,i Φ,,i}. Aarently, U can be sufficiently determined by Φ. Given the set of overall active lins Φ, and accordingly the set of scheduled users U, then n o can be readily calculated and roblem (9 is simlified into the following roblem U ω i Φ Blog ( 2 1+,i g,i Γσ 2 s.t. U ( i Φ,i +P C (n o +no P dyn, +P sta, <,i P,i, 1 K,i Φ. (15 Obviously, roblem (15 can be verified as a standard quasiconcave otimization roblem and thereby can be readily solved as (13. Then our tas is transformed to find the scheduled users and its corresonding active lins. Recall that in obtaining the otimal user EE, some lins may not be activated and for all inactive lins, use (,i to denote them. Then we define each inactive lin, say lin i of user as a virtual user l just lie the real users in the system, and let {(,i } = Φ l. Therefore, the EE of this virtual user l is exactly the EE of lin i of user, i.e., EEΦ l = ee,i. In the rest, unless secified otherwise, term user refers to both real users and virtual users. The difference between the real user and the virtual user is that each real user may contain several lins and its circuit ower includes the static user scheduling ower P sta, as well as the lin-deendent ower
Algorithm 1 Energy-Efficient Scheduling Algorithm 1: for = 1 : K 2: Comute ee,i for all i M, by (1 and (11; 3: Sort all lins of user in descending order ofee,i,xxxx xxi.e., ee,1 ee,2... ee,n ; 4: Set Φ = Ø and EE Φ = ; 5: for i = 1 : n 6: if EE Φ ee,i do 7: Φ = Φ {(,i} ; 8: Comute,i and EE Φ by (13 and (14; 9: else EEΦ > ee,i 1: Φ = Φ ; 11: EEΦ = EE Φ ; return 12: end 13: end 14: end 15: Sort all users (include both real users and virtual users in descending order of EEΦ i.e., EE, Φ EE 1 Φ,..., 2 EEΦ L; 16: Set Φ = Ø, U = Ø, and EEΦ = ; 17: for = 1 : L 18: if EEΦ EE Φ do 19: Φ = Φ Φ and U = U {} ; 2: Obtain,i and EE Φ by solving roblem (15; 21: else EEΦ > EE Φ 22: Φ = Φ; 23: EEΦ = EE Φ ; return 24: end 25: end P dyn, +P dyn,, while each virtual user only contain one lin and its circuit ower thereby is given by P dyn, +P dyn,. We first sort all users in descending order according to the user EE EEΦ, i.e., EE Φ EE 1 Φ,..., EE 2 Φ where L, L is the overall number of real users and virtual users. Then, we have the following lemma to characterize a roerty of the order. Lemma 1: Assume that the virtual user l is derived from the lin i of the real user. Following the descending order of the user EE, the order index of this virtual user l must be larger than that of its associated real user. Proof: According to the user EE, we haveeeφ > ee,i, i.e, EEΦ > EE Φ l. Therefore, when they are mixed together to secify the order, the virtual user l (inactive lin must be raned after its corresonding real user. This lemma guarantees that those virtual users (inactive lins of real user must be less liely to be active in the system EE comared with the user (the active lins in the user EE, otherwise it may lead to the case that some lin is scheduled finally in the system, but its associated real user is not scheduled, which contradicts the reality. In the following, we exlore the secial structures of the system EE, the user EE and the lin EE, and show how to obtain the otimal set Φ. In each round, we add one user to the set U following the order and add all its active lins in Φ to Φ, resectively. Then based on Φ, the otimal system EE EEΦ can be calculated as (15. By the following theorem, we obtain the imum system EE of roblem (9. Theorem 2: 1 For any Φ / Φ, If EE Φ EE Φ then, there must be EEΦ EE Φ Φ EE Φ else if EE ; Φ > EEΦ, then there must be EE Φ > EE Φ Φ > EE Φ 2 If ; any user is scheduled, then all active lins in terms of the otimal user EE will also be activated in imizing system EE. Proof: Please see Aendix B. The first statement suggests that in each round, the comarison result of EEΦ and EE Φ is necessary and sufficient to determine whether the th user can be scheduled to imrove the system EE. While the second statement guarantees the otimality of the active lins in imizing the system EE. This theorem guarantees the otimality of the roosed method which exhibits the concet of divide-and-conquer following the EE of three levels. The rocess of method is summarized in Algorithm 1 and it is easy to show that the comlexity of the divide-and conquer aroach overall has a linear comlexity of the ower control. C. Imact of Static Receiving Power on User Scheduling The next theorem reveals the relationshi between the user scheduling and the static receiving ower. Theorem 3: 1 The otimal number of scheduled users in imizing the system EE is nondecreasing with the static receiving ower P sta, ; 2 When P sta, is negligible, i.e., P sta,, TDMA is otimal for energy-efficient transmission; 3 When P sta, is sufficiently large, all users will be scheduled for energy-efficient transmission. Proof: Due to the sace limitation, we only rovide a setch of the roof here. It is easy to show that the system EE is decreasing with the static receiving owerp sta,. Then, from Theorem 2, we can show that less users would be scheduled for a higher system EE. A more detailed roof will be given in the journal version of this aer. The intuition is that when P sta, is larger, the additional ower consumtion brought from scheduling users is less dominant, which maes it more effective to achieve higher EE. If there is no additional ower consumtion for oerating systems, i.e., P sta, =, the otimal energy-efficient strategy is only to schedule the best user where the best is in terms of the user EE. It has the similar interretation as that of the throughut imization roblem in TDMA systems: only the user with the best channel gain will be scheduled. From Theorem 3, it is also interesting to note that the number of users scheduled can not be guaranteed although the weights have been imosed on users, esecially for the case with low static receiving ower. IV. NUMERICAL RESULTS In this section, we rovide simulation results to validate our theoretical findings and demonstrate the effectiveness roosed methods. There are eight equally weighted users in the system and each user is configured with twenty radio lins. Without
Energy efficiency of the system (bits/joule TABLE I SYSTEM PARAMETERS Parameter Descrition Carrier frequency 2 GHz Bandwidth of each radio lin, B 15 Hz Maximal allowed transmit ower, P,i 25 dbm Static circuit ower of the AP, P sta, 5 mw Lin deendent ower of the AP, P dyn, 45 mw Static circuit ower of user, P sta, 1 mw Lin deendent ower of user, P dyn, 5 3 mw Power density of thermal noise variance 174 dbm/hz Power amlifier efficiency,.38 Cell radius, r 1 m Path loss model Oumura-Hata Penetration loss 2 db Lognormal shadowing 8 db Fading Rayleigh flat fading Throughut of the system (bits/joule 18 16 14 12 1 8 6 4 2 x 1 4 Dinelbach method EE Otimal EE Receiver Throughut Otimal EE Transmitter 5 1 15 2 25 Transmit ower P,i (dbm 8 7 6 5 4 3 2 1 x 1 6 Fig. 1. The system EE versus the transmit ower. Dinelbach method EE Otimal EE Receiver Throughut Otimal EE Transmitter 5 1 15 2 25 Transmit ower P,i (dbm Fig. 2. The system throughut versus the transmit ower. loss of generality, P,i is assumed the same for all users and Γ is assumed as 1. Other system arameters are listed in Table I according to [11], [14] unless secified otherwise. The number of users scheduled for EE transmission Fig. 3. 8 7 6 5 4 3 2 EE Otimal 1 1 2 3 4 5 6 7 8 The static receiving ower of the AP, P sta, (W The number of scheduled users versus the static receiving ower. In Fig. 1, we comare the EE of the following methods: 1 Dinelbach method: the existing otimal method [4]; 2 EE Otimal: joint Tx and Rx otimization; 3 EE Transmitter: based on the Tx side otimization [15]; 4 EE Receiver: based on the Rx side otimization [15]; 5 Throughut Otimal: based on the throughut imization. In Fig. 1, we can first observe that our roosed method erforms the same as the Dinelbach method, which demonstrates its otimality. Moreover, as the transmit ower increases, the erformance of the EE Otimal scheme first increases and then aroaches a constant because of its energy-efficient nature, while those of the Throughut Otimal scheme and the EE Receiver scheme first increase and then decrease due to their greedy use of ower. It is also interesting to note that the EE Receiver scheme aroaches the EE Otimal scheme in the low transmit ower regime while it is more close to the Throughut Otimal scheme in the high transmit ower regime. A similar henomenon can also be found in Fig. 2 in terms of the system throughut. Moreover, the EE Transmitter scheme results in both low EE and sectral efficiency due to the fact that only one user is scheduled, which has been theoretically shown in Section III-C. Fig. 3 further demonstrates our theoretical findings in Theorem 3 which characterizes the monotonicity of the number of users scheduled with P sta,. We observe that when P sta, is negligible, the otimal energy-efficient strategy is to schedule only one user. As P sta, increases, more users are scheduled to imrove the system EE through boosting the system throughut. V. CONCLUSIONS This aer investigated the joint transmitter and receiver EE imization roblem in multi-radio networs. A holistic and dynamic ower consumtion model was established for the considered system. Then, the EE imization roblem is directly addressed from the fractional ersective, which results in an linear divide-and-conquer aroach. Moreover,
we ointed out that the static receiving ower has an imlicit interretation for the otimal number of scheduled users. In the extreme case when the static receiving ower is negligible, TDMA is the otimal scheduling strategy. In order to meet the QoS in ractice, we then extended the roose method to solve the roblem with minimal user data rate constraints, which exhibits good erformance with linear comlexity. APPENDIX A PROOF OF THEOREM 1 Denote,i as the otimal ower corresonding to ee,i by (1 and (11. Also, denote ˆ,i and ˇ,i as the otimal owers corresonding to EEΦ {(,i} and EEΦ by (15, resectively. Let S {,i,i P, i,i M, = 1,...,K} and P,i (,i = Θt,i +Θ t P dyn,. Then, we have the following EE Φ {(,i} i = ω r,l (,l S i P,i(,l +P sta, = ω r,l (ˆ,l +ω r,i (ˆ,i P,l(ˆ,l +P sta, +P,i (ˆ,i ω r,l (ˇ,l +ω r,i (,i i 1 P,l(ˇ,l +P sta, +P,i (,i { i 1 min ω r,l (ˇ,l P, ω r,i(,i },l(ˇ,l +P sta, P,i (,i = min { EEΦ,ee,i}. (16 On the other hand, = EE Φ {(,i} ω r,l (ˆ,l +ω r,i (ˆ,i P,l(ˆ,l +P sta, +P,i (ˆ,i { i 1 } ω r,l (ˆ,l P, ω r,i(ˆ,i,l(ˆ,l +P sta, P,i (ˆ,i { i 1 ω r,l (ˇ,l P, ω r,i(,i },l(ˇ,l +P sta, P,i (,i = { EEΦ,ee,i}. (17 Based on (16 and (17, we have min { EE Φ,ee,i} EE Φ {(,i} { EE Φ,ee,i}. (18 By (18, Theorem 1 can be easily roved. APPENDIX B PROOF OF THEOREM 2 The statement 1 in Theorem 2 can be similarly roved by an extension of Theorem 1, thus we omit them for brevity. We now rove 2 by contradiction. Assume that user is scheduled, but the lin i, for i Φ, is not activated in imizing the system EE, i.e., (,i / Φ. In Theorem 1, we have shown that the sufficient and necessary condition of any i Φ is that EE Φ ee,i. Thus, it follows that EE Φ ee,n o ee,i, i Φ, (19 where n o also denotes the last lin activated according to the lin EE order since user overall has n o lins activated. On the other hand, since user is scheduled, we must have EEΦ EE Φ by Theorem 2. Combining with (19, it follows that EE Φ EE Φ ee,n o ee,i = EE Φ l, i Φ, (2 where the virtual user exression is adoted, i.e.,{(, i} = Φ l. According to 1 in Theorem 2, there must be EEΦ EE Φ Φ = EE Φ l {(,i}. (21 Therefore, from (21, we can conclude that scheduling the lin i of user should be scheduled in imizing the system EE, which contradicts the assumtion that (,i / Φ. REFERENCES [1] J. Wu, S. Rangan, and H. Zhang, Green Communications: Theoretical Fundamentals, Algorithms and Alications. CRC Press, 212. [2] X. Xiao, X. Tao, and J. Lu, QoS-aware energy-efficient radio resource scheduling in multi-user OFDMA systems, IEEE Commun. Lett., vol. 17, no. 1,. 75 78, 213. [3] G. Miao, Energy-efficient ulin multi-user MIMO, IEEE Trans. Wireless Commun., vol. 12, no. 5,. 232 2313, 213. [4] D. W. K. Ng, E. S. Lo, and R. Schober, Energy-efficient resource allocation in multi-cell OFDMA systems with limited bachaul caacity, IEEE Trans. Wireless Commun., vol. 11, no. 1,. 3618 3631, 212. [5] K. Cheung, S. Yang, and L. Hanzo, Achieving imum energyefficiency in multi-relay OFDMA cellular networs: A fractional rogramming aroach, IEEE Trans. Commun., vol. 61, no. 7,. 2746 2757, 213. [6] H. Zhang, Z. Zhang, X. Chen, and R. Yin, Energy efficient joint source and channel sensing in cognitive radio sensor networs, in Proc. IEEE ICC, 211,. 1 6. [7] H. Kim and G. De Veciana, Leveraging dynamic sare caacity in wireless systems to conserve mobile terminals energy, IEEE/ACM Trans. on Networing, vol. 18, no. 3,. 82 815, 21. [8] S. Luo, R. Zhang, and T. J. Lim, Joint transmitter and receiver energy minimization in multiuser OFDM systems, arxiv rerint arxiv:1312.6743, 213. [9] G. Auer, O. Blume, V. Giannini, I. Godor, M. Imran, Y. Jading, E. Katranaras, M. Olsson, D. Sabella, P. Sillermar et al., D2. 3: Energy efficiency analysis of the reference systems, areas of imrovements and target breadown, EARTH, 21. [1] M. Gruber, O. Blume, D. Ferling, D. Zeller, M. A. Imran, and E. C. Strinati, EARTH: energy aware radio and networ technologies, in Proc. IEEE PIMRC, 29,. 1 5. [11] Z. Xu, C. Yang, G. Y. Li, S. Zhang, Y. Chen, and S. Xu, Energyefficient configuration of satial and frequency resources in MIMO- OFDMA systems, IEEE Trans. Commun., vol. 61, no. 2,. 564 575, 213. [12] J. Mao, G. Xie, J. Gao, and Y. Liu, Energy efficiency otimization for OFDM-based cognitive radio systems: A water-filling factor aided search method, IEEE Trans. Wireless Commun., vol. 12, no. 5,. 2366 2375, 213. [13] S. Boyd and L. Vandenberghe, Convex Otimization. Cambridge University Press, 24. [14] S. Zhang, Y. Chen, and S. Xu, Joint bandwidth-ower allocation for energy efficient transmission in multi-user systems, in Proc. IEEE GLOBECOM, 21,. 14 145. [15] Q. Wu, W. Chen, M. Tao, J. Li, H. Tang, and J. Wu, Resource allocation for joint transmitter and receiver energy efficiency imization in downlin OFDMA systems, IEEE Trans. Commun., 214, to aear.