A Distributed Power Control Algorithm for Energy Efficiency Maximization in Wireless Cellular Networks
11 A Distributed Power Control Algorithm for Energy EfficiencyMaximization in Wireless Cellular Networks
Rojin Aslani,
Student Member, IEEE , and Mehdi Rasti,
Member, IEEE
Abstract —In this paper, we propose a distributed powercontrol algorithm for addressing the global energy efficiency(GEE) maximization problem subject to satisfying a minimumtarget SINR for all user equipments (UEs) in wireless cellularnetworks. We state the problem as a multi-objective optimizationproblem which targets minimizing total power consumption andmaximizing total throughput, simultaneously, while a minimumtarget SINR is guaranteed for all UEs. We propose an iterativescheme executed in the UEs to control their transmit power usingindividual channel state information (CSI) such that the GEE ismaximized in a distributed manner. We prove the convergenceof the proposed iterative algorithm to its corresponding uniquefixed point also shown by our numerical results. Additionally,simulation results demonstrate that our proposed scheme outper-forms other algorithms in the literature and performs like thecentralized algorithm executed in the base station and maximizesthe GEE using the global CSI.
Index Terms —Distributed power control; energy efficiency;multi-objective optimization problem; wireless cellular networks.
I. I
NTRODUCTION
Nowadays, energy efficiency (EE) is a leading concern inwireless cellular networks. In [1], three metrics for EE aredefined: 1) Minimum-EE defined as the minimum EE of userequipments (UEs) in which the EE of a UE described by theratio of the UE’s throughput and its consumed power; 2) Sum-EE defined as the summation over the EE of UEs; 3) GlobalEE (GEE) defined as the ratio between the total throughput andthe total power consumption of the system. As the GEE is themost commonly used metric for EE maximization problems inwireless cellular networks [2]-[5], we focus on it in this paper.To maximize the GEE, it is required that the total through-put is maximized while the minimum power is consumed.However, the objectives of total throughput maximization andpower consumption minimization are contradicting each other.Since, for total throughput maximization, the UEs require totransmit with high power while high power increases the totalpower consumption. Thus, the transmit power control has asignificant role in maximizing the GEE in cellular networks.There is a lot of research that focuses on designing powercontrol schemes to maximize the GEE in cellular networks[1]-[9]. In [1], two centralized power control algorithms areproposed to address the problems of the GEE maximizationand the weighted minimum-EE maximization using fractionalprogramming and sequential convex optimization framework.Employing the fractional programming and Dinkelbach al-gorithm, a joint power control scheme for the uplink and
The authors are with the Department of Computer Engineering, Amirk-abir University of Technology, Tehran 1591634311, Iran (e-mail: [email protected]; [email protected]).Digital Object Identifier 10.1109/LWC.2020.3010156 downlink of a cellular network is proposed in [2] whichaims at maximizing the GEE while the quality of servicerequirements of the UEs are guaranteed. In [3] and [4], theproblem of the GEE maximization in the downlink of cellularnetworks is studied. The authors of [5] and [6] apply themulti-objective optimization (MOO) approach to address theproblem of the GEE-spectral efficiency tradeoff. In [7] and [8],power control schemes are proposed to maximize the GEE inD2D communications underlaying cellular networks.The existing power control algorithms for the GEE maxi-mization are mostly centralized [1]-[8]. However, distributedpower control schemes are practically preferred to centralizedones due to the utilization of local information and minimalfeedback from the base station (BS). The authors of [9]propose a semi-distributed algorithm that aims at maximizingthe GEE in the downlink of a cellular network. Employinggame theory, a distributed power control scheme is presentedin [1] to maximize the sum-EE. In [10], a threshold-baseddistributed power control scheme is proposed to address thesum-EE maximization problem. The fixed-target-SIR-trackingpower control (TPC) algorithm is proposed in [11] for theaggregate transmit power minimization while a minimumtarget SINR is satisfied for all UEs. In [12] and [13], anopportunistic power control scheme to maximize the totalthroughput is proposed. The distributed dynamic target-SIRtracking power control (DTPC) algorithm is presented in[14] which maximizes the total throughput while guaranteesa minimum target SINR for all UEs. The authors of [15]propose a distributed power control algorithm with temporaryremoval and feasibility check to address the gradual removalproblem in wireless networks. In [16], two distributed powercontrol schemes are presented for the aggregate transmit powerminimization and the total throughput maximization in energyharvesting wireless networks. Most of the proposed distributedpower control algorithms in the literature for cellular networksfocused on objectives of total throughput maximization, totalpower consumption minimization, and sum-EE maximization.In this work, we propose a distributed power control schemefor GEE maximization by simultaneously maximizing the totalthroughput and minimizing the total power consumption whilesatisfying a minimum SINR for UEs. To do this, we applyMOO framework which makes a balance between the compet-ing objectives. The contributions of this work are as follows: • We study the power control problem to maximize GEEsubject to the minimum target SINR for the UEs in theuplink of cellular networks (in contrast to [9] which con-siders downlink communications). This is in contrast withother existing literature such as [1], [10]-[16], in whichthe objective functions are sum-EE maximization, total (cid:13) a r X i v : . [ ee ss . SP ] S e p throughput maximization, or total power minimization. • We propose a distributed power control scheme to max-imize GEE. This is in contrast to [1]-[8] which pro-pose centralized schemes. For this purpose, since thestated problem is non-convex, we first reformulate itas a MOO problem which targets minimizing the totalpower consumption and maximizing total throughput,simultaneously. We then employ (cid:15) -constraint method [5]to address the MOO problem. Finally, we propose aniterative scheme that is distributed and executed in theUEs to control their transmit power using individualchannel state information (CSI). • We provide the system feasibility condition. We provethat the proposed iterative algorithm converges to itsunique fixed point also shown by the numerical results.Additionally, simulation results demonstrate that our dis-tributed algorithm which uses the local CSI performs likethe centralized algorithm proposed in [2] in which theglobal CSI is assumed to be presented in the BS.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a single cell in a wireless CDMA network withone BS and K UEs, whose set is denoted by K = { , · · · , K } .Let h i denote the channel gain from i th UE to the BS. Also, p i denotes the transmit power of the i th UE and ≤ p i ≤ p i ,where p i is the maximum transmit power for the i th UE.We assume the presence of additive white Gaussian noise(AWGN) with power σ at the BS.Given the transmit power vector p , γ i ( p ) denotes the SINRof the i th UE at the BS receiver, given by: γ i ( p ) = p i h i I i ( p ) , (1)where, I i ( p ) is the total interference caused to the i th UEand given by I i ( p ) = (cid:80) j ∈K ,j (cid:54) = i p j h j + σ . The effectiveinterference for the i th UE, denoted by φ i ( p ) , is defined as: φ i ( p ) = I i ( p ) h i . (2)The value of φ i ( p ) shows the quality of the i th UE’s channel,lower interference and higher channel gain lead to lower φ i ( p ) , implying a good channel, compared with higher inter-ference and lower channel gain, which lead to higher φ i ( p ) ,implying a poor channel [14]. According to Shannon formula,the throughput for the i th UE (bps/Hz) is given by: T i ( p ) = log(1 + γ i ( p )) . (3)The total throughput is obtained by T ( p ) = (cid:80) Ki =1 T i ( p ) . Thetotal power consumption is formed as: P T ( p ) = P CBS + K (cid:88) i =1 P C i + K (cid:88) i =1 µ i p i , (4)where, P CBS and P C i are the power consumed by the circuit ofthe BS and i th UE, respectively, and µ i is the power amplifierinefficiency of i th UE [2]. We define the GEE as the ratio oftotal throughput and total power consumption, given by: EE ( p ) = T ( p ) P T ( p ) . (5) The problem of power control for maximizing the GEEsubject to the minimum target SINR requirements for UEs andthe feasibility of transmit power for UEs is formulated as: maximize p EE ( p ) (6) s . t . C : γ i ( p ) ≥ (cid:98) γ i , ∀ i ∈ K C : 0 ≤ p i ≤ p i , ∀ i ∈ K where C corresponds to the target SINR requirement for UEs.The feasibility of transmit power for UEs is met by C .The problem (6) is non-convex due to the fractional objec-tive function which is an obstacle to address it in a distributedmanner. In the next section, we address (6) by reformulating itas an equivalent MOO problem to minimize aggregate transmitpower and maximize total throughput, simultaneously.III. T HE P ROPOSED D ISTRIBUTED P OWER C ONTROL S CHEME AND I TS A NALYSIS
As given in (5), the GEE is the ratio of the total throughputto the total power consumption. Thus, maximizing the GEE isequivalent to minimizing the total power consumption whilemaximizing the total throughput, simultaneously [6]. On theother hand, as the circuit power consumption of BS and UEs,i.e., P CBS and P C i , respectively, as well as the power amplifierinefficiency of UEs, i.e., µ i , are fixed, we can conclude thatminimizing the total power consumption is equivalent to min-imizing the aggregate transmit power. Accordingly, we refor-mulate (6) as an equivalent MOO problem, that is: f : minimize p K (cid:88) i =1 p i (7) f : maximize p T ( p )s . t . C − C , where, f is the aggregate transmit power minimization and f is the total throughput maximization. Proposition 1.
The MOO problem in (7) is equivalent to theGEE maximization problem in (6).
Proof.
Suppose D is the set of feasible solutions to (6)spanned by constraints C − C . We denote q ∗ as the optimalGEE in (6) represented by q ∗ = T ( p ∗ ) P T ( p ∗ ) = maximize p ∈D T ( p ) P T ( p ) ,where p ∗ is the optimal transmit power vector for (6). It wasproved in [17] that for the fractional objective function in (6),there is an equivalent subtractive form as follows: maximize p ∈D T ( p ) − q ∗ P T ( p ) , (8)which shares the same optimal solution, i.e., p ∗ . The objectivefunction in (8) represents a linear scalarization for a multi-objective function in the form of maximize p ∈D { T ( p ) , − P T ( p ) } with the weights of and q ∗ , respectively. The objective of maximize p ∈D − P T ( p ) can be rewritten as minimize p ∈D P T ( p ) .Considering the variable part of total power consumption P T ( p ) , the problem (8) can be rewritten as the problem (7).This completes the proof. Now, we find a solution for MOO problem (7) by employing (cid:15) -constraint method [5]. To do this, we keep f as the primaryobjective function and redefine f as a constraint. Accordingly,the new problem is stated as: f : minimize p K (cid:88) i =1 p i (9) s . t . C : T ( p ) ≥ (cid:15) C − C , where, C ensures that the total throughput is greater than (cid:15) .The feasibility of (9) and the closeness between its solutionand the solution of (6) highly depend on the value of (cid:15) . Toanalyze the significance of the parameter (cid:15) on the solution of(9), we consider three following cases in a similar way to [5]: Case 1: If (cid:15) = 0 , then (9) would turn into the problem ofthe aggregate transmit power minimization subject to theminimum target SINR constraints for all UEs which is aconventional power control problem in cellular networksand addressed by the TPC algorithm proposed in [11]. Case 2: If (cid:15) = T max , where T max is the maximum totalthroughput, then (9) would turn into the problem of thetotal throghput maximization subject to the minimum targetSINR constraints for all UEs which is a conventional powercontrol problem in cellular networks and addressed by theDTPC algorithm proposed in [14]. Case 3: If (cid:15) ≥ T max , then (9) would be infeasible.With this analysis, we conclude that the solution of (9) highlydepends on the value of (cid:15) . More specifically, the parameter (cid:15) makes a trade-off between the total power consumption andthe total throughput. Therefore, we require to find a value for (cid:15) corresponding to the maximum ratio of the total throughputand the total power consumption, i.e., the GEE.From the above cases, it is derived that the minimum valueof (cid:15) is and the maximum value that (cid:15) can take withoutmaking the problem (9) infeasible is the value of T max obtainedby applying the DTPC algorithm. Thus, we have ≤ (cid:15) ≤ T max .Let define (cid:15) as (cid:15) = δT max , where δ ∈ [0 , . The value ofthe GEE modifies depends on the value of δ ; however, themaximum GEE is achieved for a specific value of δ . To find thevalue of δ , by replacing T ( p ) and (cid:15) , we rewrite the constraintC in problem (9) as (cid:80) Ki =1 T i ( p ) ≥ (cid:80) Ki =1 δ i T max , where, δ i ∈ [0 , and (cid:80) Ki =1 δ i = δ . Indeed, δ i T max is the throughputportion of δT max for the i th UE. Thus, we can rewrite C as:C : T i ( p ) ≥ δ i T max , ∀ i ∈ K . (10)The problem of finding δ i is formulated as: maximize p ,δ i EE ( p ) (11) s . t . C : T i ( p ) ≥ δ i T max , ∀ i ∈ K C − C , C : 0 ≤ δ i ≤ , ∀ i ∈ K . The problem (11) can be addressed by applying centralizedschemes such as the proposed algorithm in [2]. Now, given δ i obtained as mentioned above, we proposean iterative algorithm to address (9). To do this, by replacing T i ( p ) from (3), we rewrite C in (10) as:C : γ i ( p ) ≥ δ i T max − , ∀ i ∈ K . (12)On the other hand, we have γ i ( p ) ≥ (cid:98) γ i , ∀ i ∈ K in C .Combining constraints C and C , we rewrite them in a newconstraints denoted by C (cid:48) and defined as:C (cid:48) : γ i ( p ) ≥ λ i , ∀ i ∈ K , (13)where, λ i = max (cid:8)(cid:98) γ i , δ i T max − (cid:9) . Thus, (9) is rewritten as: minimize p K (cid:88) i =1 p i (14) s . t . C (cid:48) − C . Inspired by the TPC algorithm, we propose an iterativedistributed transmit power control algorithm solving (14).Accordingly, each UE i updates its transmit power at iteration t based on the following power updating function: p i ( t +1) = f i ( p ( t )) = min (cid:110) p i , λ i φ i ( p ( t )) (cid:111) . (15)The proposed scheme executed in each UE i is given inAlgorithm 1. Note that in each iteration, each UE i updates itstransmit power based on the local information and individualCSI , so the proposed algorithm is distributed. More specifi-cally, by rearranging (1), we have h i = γ i ( p ) I i ( p ) p i . Now, byreplacing h i in (2), we rewrite it as: φ i ( p ) = p i γ i ( p ) . (16)Thus, to obtain φ i ( p ( t )) , UE i needs only p i ( t ) and γ i ( p ( t )) . Algorithm 1
Iterative Distributed Power Control Algorithm
Require: p i , (cid:98) γ i , δ i , h i , t = 0 , p i (0) repeat Receive the SINR γ i ( p ( t )) Calculate φ i ( p ( t )) using (16)Update p i ( t + 1) using (15)Set t = t + 1 until Convergence return p i ( t ) A. Convergence Analysis
In this section, we prove that the distributed power controlscheme in (15) converges to its unique fixed point by applyingtwo-sided scalable framework [13] presented in Definition 1.
Definition 1.
A power updating function f ( p ) = [ f ( p ) , f ( p ) , · · · , f K ( p )] T is two-sided scalable if ∀ a> , (1 /a ) p ≤ p (cid:48) ≤ a p implies (1 /a ) f i ( p ) ≤ f i ( p (cid:48) ) ≤ af i ( p ) [13]. Proposition 2.
The transmit power updating function f ( p ) in(15) is two-sided scalable. The value of δ i can be calculated at the BS and sent to the UEs. Proof.
Given a > , from (1 /a ) p ≤ p (cid:48) ≤ a p , wehave (1 /a ) λ i φ i ( p ) ≤ λ i φ i ( p (cid:48) ) ≤ aλ i φ i ( p ) . This implies (1 /a ) f i ( p ) ≤ f i ( p (cid:48) ) ≤ af i ( p ) . This completes the proof. Proposition 3.
The transmit power updating function f ( p ) in(15) has a unique fixed point and the proposed power controlalgorithm p ( t + 1) = f ( p ( t )) converges to it. Proof.
In [13], it is proved that for a given two-sided scalablefunction f ( p ) , if there is a , b > such that a ≤ f ( p ) ≤ b , then aunique fixed point exists which the power updating function p ( t +1) = f ( p ( t )) converges to. For the transmit power updat-ing function f ( p ) in (15), since < f i ( p ) ≤ p i , ∀ i ∈ K , thenby considering a = [ a , · · · , a K ] where a i = q min , ∀ i ∈ K , inwhich q min → + and b = [ b , · · · , b K ] where b i = p i , ∀ i ∈ K ,we have a ≤ f ( p ) ≤ b . Thus, there is a unique fixed pointand the power updating function in (15) converges to it. B. Feasibility Analysis
In this section, we provide conditions to check the feasibilityof the system. There is a one-to-one relation between atransmit power vector p = [ p , p , · · · , p K ] T and the SINRvector γ = [ γ , γ , · · · , γ K ] T [15], that is: p i = γ i h i ( γ i + 1) × σ − (cid:80) Kk =1 γ k γ k +1 , ∀ i ∈ K . (17) Definition 2.
The target SINR vector is feasible if a powervector ≤ p ≤ p exists that satisfies target SINRs of UEs,where it implies ≤ p i ≤ p i , ∀ i ∈ K . Also, the systemis feasible if the target SINR vector for all UEs is feasible,otherwise, the system is called infeasible.Given the target SINR (cid:98) γ i for each UE i and by using (17),we conclude that the target SINR vector is feasible if: ≤ (cid:98) γ i h i ( (cid:98) γ i + 1) × σ − (cid:80) Kk =1 (cid:98) γ k (cid:98) γ k +1 ≤ p i , ∀ i ∈ K . (18)The intersection of feasible ranges for all (cid:98) γ i as a result of lowerand upper bounds of (18) indicates the feasibility condition.IV. N UMERICAL R ESULTS
We consider a single square cell system with radius mwhere the BS is located at the midpoint and UEs are placeduniformly within the cell. The channel gains are assumed tobe generated using the path loss model PL i ( d ) = PL +10 θ log d i , in which PL is the constant path loss coefficient, θ is the path loss exponent, and d i is the distance between the i th UE and BS [3]. The power consumed by the circuit ofBS and UEs are set as P CBS = 30 dBm and P C i = 20 dBm,respectively (similar to [2] and [4]). We set the power amplifierinefficiency of the UEs as µ i = 5 (as in [3]). The upper boundon the transmit power for all UEs is set as p i = 23 dBm,similar to [3] and [4]. The AWGN power at BS receiver is setas σ = − dBm, similar to [14] and [16]. According to(18), the system is feasible for (cid:98) γ i ≤ − dB, ∀ i ∈ K . Thus,we consider (cid:98) γ i belong to this feasibility range in all simulationscenarios. For each (cid:98) γ i , the value of δ i is obtained by solvingproblem (11) using the scheme proposed in [2]. We obtain Iteration Numbers A v e r a g e G l ob a l EE ( b it s / J ou l e / H z ) Fig. 1. Average GEE (bits/Joule/Hz) vs. iteration number for different (cid:98) γ i . -Inf -60 -50 -40 -30 -20 -1000.511.522.533.5 A v e r a g e G l ob a l EE ( b it s / J ou l e / H z ) Fig. 2. Average GEE (bits/Joule/Hz) vs. (cid:98) γ i for different algorithms. all the numerical results by averaging over independentsnapshots with randomly generated location of UEs.We first investigate the convergence behavior of our iterativealgorithm. Fig. 1 shows average GEE vs. the iterations numberfor different values of (cid:98) γ i . As seen in Fig 1, average GEE fordifferent values of (cid:98) γ i converges to the fixed point after iterations. Thus, the performance of our proposed scheme after iterations is shown in the following case studies. Anothersignificant observation from Fig. 1 is that increasing (cid:98) γ i leadsto a decreased GEE. To gain insight into, we examine theeffect of the target SINR threshold on the GEE as follows.Fig. 2 shows average GEE vs. the value of (cid:98) γ i for differentalgorithms. As we observe from this figure, the GEE obtainedby our proposed algorithm is a monotonically non-increasingfunction of the target SINR threshold. Indeed, the GEE reachesthe highest value when (cid:98) γ i = -inf dB ( (cid:98) γ i = 0 ). The reason isthat as (cid:98) γ i is high, the UEs have to transmit with a higher powerin all channels to satisfy the SINR constraints leading to highertotal power consumption. Also, the higher transmit power byall UEs including UEs with low channel gain results in higherinterference which leads to lower total throughput. In contrast,when (cid:98) γ i is low, the UEs with high channel gain transmitwith high power to increase the total throughput and UEswith low channel gains transmit with low power to satisfy thelow SINR threshold which results in higher total throughputand lower total power consumption leading to higher GEE.This result has also been made in [2] for a different systemmodel and resource allocation problem. Another significantobservation from Fig. 2 is a demonstration of the behavior ofour proposed algorithm compared to TPC [11] and DTPC [14]algorithms. From this figure, we see that our proposed schemeoutperforms TPC and DTPC algorithms. More specifically,in the TPC algorithm, all UEs transmit in low power to -60 -50 -40 -30 -20 -100.511.522.533.5 A v e r a g e G l ob a l EE ( b it s / J ou l e / H z ) Our AlgorithmAlgorithm in [2]Algorithm in [1]
Fig. 3. Average GEE (bits/Joule/Hz) vs. (cid:98) γ i for different algorithms. satisfy the target SINR threshold which leads to lower totalthroughput and subsequently lower GEE. On the other hand,in the DTPC algorithm, some UEs transmit with high powerto increase their achieved SINR which results in higher totalpower consumption and subsequently lower GEE. However, bycontrolling the transmit power through our proposed scheme, abalance between total throughput and total power consumptionis achieved resulting in higher GEE.Finally, we provide the performance comparison betweenour proposed algorithm and other existing schemes in theliterature which aim to maximize the EE. We consider thecentralized scheme proposed in [2] to maximize the GEE, andthe distributed scheme proposed in [1] to maximize the sum-EE. Fig. 3 illustrates average GEE vs. the value of (cid:98) γ i for ourproposed algorithm, the algorithm in [2], and the algorithmin [1]. We observe that for all values of (cid:98) γ i , our proposeddistributed algorithm performs like the centralized algorithm in[2]. In the algorithm in [2] executed in the BS, the BS has theglobal CSI of all UEs and obtains the transmit power controlpolicy in a centralized manner. However, in our algorithmexecuted in the UEs, each UE controls its transmit powerin a distributed manner using its CSI only. Additionally, ouralgorithm outperforms the algorithm in [1] for all values of (cid:98) γ i .The reason is that in our proposed algorithm, each UE controlsits transmit power to maximize the GEE and the interferenceis managed accordingly. However, the goal of the algorithmproposed in [1] is to maximize the summation of the UEs’EE, so each UE targets at maximizing its EE. Therefore, thetransmit powers of UEs are not necessarily in accordance withthe GEE maximization, and so the interference among theUEs is not managed to maximize GEE. This causes higherinter-UEs interference which results in higher total powerconsumption and lower total throughput leading to lower GEEin the algorithm in [1].V. C ONCLUSION
In this paper, we studied the problem of GEE maximizationsubject to the minimum SINR of UEs in cellular networks.Since the stated problem was non-convex, we reformulatedit as a MOO problem with the aim of minimizing the totalpower consumption and maximizing the total throughput,simultaneously, while satisfying UEs’ target SINR. We em-ployed the (cid:15) -constraint method to address the problem. Finally,we proposed an iterative algorithm to control the transmitpower of UEs in a distributed manner. We proved that our proposed iterative algorithm converges to its correspondingunique fixed point. Numerical results demonstrated that ourproposed scheme rapidly converges and outperforms otheralgorithms. Simulation results also show that our distributedalgorithm which obtains the maximum GEE using the localCSI performs like the centralized algorithm which maximizesthe GEE by employing the global CSI.R
EFERENCES[1] A. Zappone, L. Sanguinetti, G. Bacci, E. Jorswieck and M. Debbah,”Energy-efficient power control: A look at 5G wireless technologies,”
IEEE Trans. on Signal Processing , vol. 64, no. 7, pp. 1668-1683, 2016.[2] R. Aslani, M. Rasti, and A. Khalili, “Energy efficiency maximization viajoint sub-carrier assignment and power control for OFDMA full duplexnetworks,” in
IEEE Trans. on Vehicular Technology , vol. 68, no. 12, pp.11859-11872, Dec. 2019.[3] J. Tang, et al., “Energy-efficient heterogeneous cellular networks withspectrum underlay and overlay access,”
IEEE Trans. on Vehicular Tech-nology , vol. 67, no. 3, pp. 2439-2453, 2018.[4] D. Ng, E. S. Lo, and R. Schober, “Wireless information and powertransfer: Energy efficiency optimization in OFDMA systems,”
IEEETrans. on Wireless Communications , vol. 12, no. 12, pp. 6352-6370, 2013.[5] A. Khalili, S. Zarandi, M. Rasti and E. Hossain, “Multi-objective op-timization for energy- and spectral-efficiency tradeoff in in-band full-duplex (IBFD) communication,”
IEEE Global Communications Confer-ence (GLOBECOM) , pp. 1-6, 2019.[6] O. Amin, E. Bedeer, M. Ahmed and O. Dobre, “Energy efficiency-spectralefficiency tradeoff: A multiobjective optimization approach,”
IEEE Trans.on Vehicular Technology , vol. 65, no. 4, pp. 1975-1981, 2016.[7] B. zbek, M. Pischella and D. Le Ruyet, “Energy efficient resourceallocation for underlaying multi-D2D enabled multiple-antennas commu-nications,”
IEEE Transactions on Vehicular Technology , Mar. 2020.[8] S. Liu, Y. Wu, L. Li, X. Liu and W. Xu, “A two-stage energy-efficientapproach for joint power control and channel allocation in D2D commu-nication,”
IEEE Access , vol. 7, pp. 16940-16951, 2019.[9] Y. Dong, H. Zhang, M. J. Hossain, J. Cheng and V. C. M. Leung,“Energy efficient resource allocation for OFDMA full duplex distributedantenna systems with energy recycling,”
IEEE Global CommunicationsConference (GLOBECOM) , pp. 1-6, 2015.[10] C. Zhang, A. Agrawal, V. S. Varma and S. Lasaulce, ”Thresholding-based distributed power control for energy-efficient interference net-works,”
IEEE 29th Annual International Symposium on Personal, Indoorand Mobile Radio Communications (PIMRC) , pp. 1-6, 2018.[11] G. J. Foschini and Z. Milijanic, ”A simple distributed autonomouspower control algorithm and its convergence,”
IEEE Trans. on VehicularTechnology , vol. 42, no. 4, pp. 641-646, 1993.[12] K. Leung and C. W. Sung, ”An opportunistic power control algorithmfor cellular network,”
IEEE/ACM Transaction on Networking , vol. 14, no.3, pp. 470-478, 2006.[13] C. W. Sung and K. Leung, ”A generalized framework for distributedpower control in wireless networks,”
IEEE Trans .on Information Theory ,vol. 51, no. 7, pp. 2625-2635, 2005.[14] M. Rasti, A. R. Sharafat and J. Zander, ”A distributed dynamic target-SIR-tracking power control algorithm for wireless cellular networks,” in
IEEE Trans. on Vehicular Technology , vol. 59, no. 2, pp. 906-916, 2010.[15] M. Rasti, A. R. Sharafat and J. Zander, “Pareto and energy-efficientdistributed power control with feasibility check in wireless networks,”
IEEE Trans. on Information Theory , vol. 57, no. 1, pp. 245-255, 2011.[16] R. Aslani, M. Rasti, “Distributed power control schemes in in-band full-duplex energy harvesting wireless networks,”
IEEE Trans. on WirelessCommunications , vol. 16, no. 8, pp. 5233-5243, 2017.[17] W. Dinkelbach, “On nonlinear fractional programming,”