Energy Efficiency Fairness for Multi-Pair Wireless-Powered Relaying Systems
Kien-Giang Nguyen, Quang-Doanh Vu, Le-Nam Tran, Markku Juntti
aa r X i v : . [ c s . I T ] O c t Energy Efficiency Fairness for Multi-PairWireless-Powered Relaying Systems
Kien-Giang Nguyen,
Student Member, IEEE , Quang-Doanh Vu,
Member, IEEE ,Le-Nam Tran,
Senior Member, IEEE and Markku Juntti,
Senior Member, IEEE
Abstract
We consider a multi-pair amplify-and-forward relay network where the energy-constrained relaysadopting time-switching protocol harvest energy from the radio frequency signals transmitted by the usersfor assisting user data transmission. Both one-way and two-way relaying techniques are investigated.Aiming at energy efficiency (EE) fairness among the user pairs, we construct an energy consumptionmodel incorporating rate-dependent signal processing power, the dependence on output power levelof power amplifiers’ efficiency, and nonlinear energy harvesting (EH) circuits. Then we formulatethe max-min EE fairness problems in which the data rates, users’ transmit power, relays’ processingcoefficient, and EH time are jointly optimized under the constraints on the quality of service and users’maximum transmit power. To achieve efficient suboptimal solutions to these nonconvex problems, wedevise monotonic descent algorithms based on the inner approximation (IA) framework, which solve asecond-order-cone program in each iteration. To further simplify the designs, we propose an approachcombining IA and zero-forcing beamforming, which eliminates inter-pair interference and reduces thenumbers of variables and required iterations. Finally, extensive numerical results are presented to validatethe proposed approaches. More specifically, the results demonstrate that ignoring the realistic aspectsof power consumption might degrade the performance remarkably, and jointly designing parametersinvolved could significantly enhance the energy efficiency.
Manuscript received March 19, 2018; revised July 6, 2018; accepted September 6, 2018. This work was supported in part bythe Academy of Finland under the projects “Wireless Connectivity for Internet of Everything–Energy Efficient Transceiver andSystem Design (WiConIE)” under Grant 297803, “Flexible Uplink-Downlink Resource Management for Energy and SpectralEfficiency Enhancing in Future Wireless Net- works (FURMESFuN)” under Grant 31089, and “6Genesis Flagship” under Grant318927. This publication has emanated from research supported in part by a Grant from Science Foundation Ireland under Grantnumber 17/CDA/4786. The work of K.-G Nguyen was supported by HPY Research Foundation, Nokia Foundation, WalterAhlström Foundation, Finnish Foundation for Technology Promotion, Tauno Tönning Foundation.Kien-Giang Nguyen, Quang-Doanh Vu, and Markku Juntti are with Centre for Wireless Communications, University of Oulu,FI-90014, Finland. Email: {giang.nguyen, doanh.vu, markku.juntti}@oulu.fi.L.-N. Tran is with School of Electrical and Electronic Engineering, University College Dublin, Ireland. Email:[email protected]).
Index Terms
Multi-pair relay networks, energy efficiency, nonlinear energy harvesting, non-ideal power amplifier,distributed beamforming, inner approximation.
I. I
NTRODUCTION
Relay-assisted cooperative communications can improve spectral and energy efficiency, and,more importantly, extend the range of coverage [1], [2]. As such, relay-assisted cooperativecommunications has been standardized in current mobile networks, e.g., 3GPP Long-Term Evo-lution (LTE) [3]. In addition, it is expected to be a major means to implement device-to-device communications in the upcoming mobile networks [4]. Various relay strategies havebeen proposed including amplify-and-forward (AF), decode-and-forward (DF), and compress-and-forward [5]. Among them, AF has attracted significant interest due to its simplicity and lowlatency [2].Relaying can be either one-way or two-way. The former refers to one-directional transmissionfrom one network node to another, which is applied to the scenario that only one node has datatransmitted to another such as the downlink transmission from an access point to a mobile phone.The latter comprises a system in which both nodes send messages to each other, introduced forimproving spectral efficiency [2]. Two-way relaying was developed based on the self-interferencecancellation employed at the destinations to extract the desired signals [6].Early works on relay systems focused on single user pair, and for improving spectral efficiency,a more general relay system including multiple pairs of users was proposed [7]. Here, therelays simultaneously assist the transmission of multiple user pairs forming an interferencechannel. Linear precoding at the relays can be used to manage the radio resource and control theinterference [8]. In resource constrained networks such as wireless sensor networks, the nodesare low-cost, i.e., each one is equipped with a single-antenna. The benefits of MIMO techniquescan be exploited if a pool of relays collaboratively operate to perform the so called distributedrelay beamforming [9].In a relay-based system where low-cost relays are equipped with limited batteries, i.e. do nothave sustainable power supplies, such as sensors or mobile devices, one of the main implemen-tation challenges is to recharge the limited batteries for keeping the network alive [10]. To thisend, simultaneous wireless information and power transfer technique is a promising solution [10]–[13]. The technique allows the relays to harvest energy from the radio-frequency (RF)signals, and thus the batteries can be wirelessly empowered.Energy efficiency (EE) has become an important performance measure in wireless networks[14]. By definition, the consumed energy plays a vital role on EE objectives. Thus, the accuracyof the power consumption model is crucial for designing practical systems. For example, signalprocessing power and the efficiency of power amplifiers (PAs) are commonly assumed to be fixed[15]–[18]. However, signal processing power is often rate-dependent [19] and PAs’ efficiencydepends on output power level [20]. It has been demonstrated that such aspects may havesignificant impacts on the network level EE performances [21], [22].
Related Works
Multi-pair one-way and two-way relaying have been investigated in many prior works. [7]considered a one-way relay network with the aim of minimizing the total transmit power atrelays. Therein, distributed relay beamforming was designed using the semidefinite relaxation(SDR). This work was generalized in [23] where transmit power at users and the distributed relaybeamforming were designed for minimizing the total transmit power at users and relays. Theconstrained concave convex procedure was used to tackle the nonconvex problem. Similarly, in[24], the users’ power and relay beamforming were jointly designed for maximizing the secrecyrate. On the other hand, [6] focused on two-way relaying where the inter-pair interference iseliminated via zero-forcing (ZF) relay beamforming. [8] considered a system where a two-wayrelay is equipped with multiple antennas. The processing matrix at the relay is designed based onZF and minimum mean-square-error criteria for achieving fairness among users and maximizingsystem signal-to-noise ratio. [25] aimed at achieving the max-min rate fairness among users.Therein, the relay’s processing matrix was designed by using the SDR and ZF. In general, designproblems for multiuser AF relay networks are nonconvex. Consequently, the related works havemainly focused on suboptimal low-complexity designs.Cooperative systems with EH relays have received considerable attention. In particular, [12]proposed time-switching and power-splitting protocols for single user pair networks where arelay harvests energy from the user’s RF signal. To further improve the network performance,the authors proposed dynamic EH time in [26]. A more general system with multiple user pairswas considered in [11]. Assuming user pairs use orthogonal channels, the work analyzed theimpacts of different power allocation strategies on the network performance. [10] considered a network where both users and the relay harvest energy and focused on user and relay powerallocation for throughput maximization under the EH constraints. [27] considered multiple-inputmultiple-output (MIMO) AF system where a relay simultaneously harvests energy transmittedfrom a destination and receives information from a source. A system with a single user pair andan EH two-way relay was studied in [28]. A more general system with multiple EH relays anda single user pair was recently studied in [29] for one-way relaying, and in [30] for two-wayrelaying. While [29] optimized the EH relays’ power splitting ratio in order to maximize thetransmit data rate, the work in [30] jointly designed EH time allocation and distributed relaybeamforming for three objectives including sum-rate maximization, total power consumptionminimization at relays, and EH time minimization.EE for relay-assisted cooperative communications has recently been studied. [15] considereda one-way MIMO AF system with one user pair and one relay. The work jointly optimizedthe user and relay precoding matrices for different channel state information assumptions. EEmaximization for the similar system model, but with a two-way relay, was studied in [17]. Morerecently, [18] solved the EE maximization for a two-way relay network with multiple user pairsand multiple relays by jointly designing user transmit power and relay matrices. [31] considereda multiple user pair one-way MIMO DF system. [32] focused on a network with one user pairand one EH two-way relay, and devised power allocation for maximizing EE performance. In theaforementioned works, signal processing power and PAs’ efficiency were assumed to be constant.In a few recent publications [33], [34], the impacts of non-ideal PA efficiency and rate-dependentsignal processing power on the EE performance were studied for two-way systems with one relayand one user pair. The EE problems for the network with multiple user pairs and multiple EHrelays have remained relatively open in the literature.
Contributions
Motivated by the above discussion and literature review, in this work, we study the one-wayand two-way multiuser AF relay networks where the low-cost relays receive energy from the usersfor assisting data transmission. The goal is to manage the EE fairness between the user pairs,which is inspired from the fact that the users in a pair might have to consume a lot of their ownenergy to charge the relays, while the transmit data rate of the pair is small. Towards a relativelyrealistic energy consumption model, we take into account the data-rate signal processing power,the dependence of PAs’ efficiency on the output power level, and consider a practical model of
EH circuit introduced in [35]. Consequently, the parameters including transmit data rate, users’transmit power, relays’ processing coefficient, and EH time, are mutually dependent, and shouldbe jointly designed. Hence, we formulate the problems of max-min EE fairness for both one-wayand two-way relay systems in which the mentioned parameters are optimization variables. Theseproblems inherit the numerical difficulties encountered in multiuser AF relay networks, and thus,are nonconvex. We then develop the low-complexity iterative algorithms based on the efficientdescent optimization framework, namely, inner approximation (IA) [37], [38]. The convergenceproofs for the algorithms are also provided. For efficient practical implementations, we transformthe convex approximate problems into the second-order-cone programs (SOCPs), which is donebased on a concave lower bound of the logarithmic function. In addition, for lower complexitydesigns, we develop solutions based on the combination of IA and ZF beamforming which havesmaller problem sizes, and thus require fewer numbers of iterations to converge. Finally, weprovide extensive numerical results which confirm that our proposed approaches are efficient interms of the EE fairness. Specifically, the main results indicate that realistic aspects of powerconsumption should be taken into consideration in the EE designs, and much better performancecan be yielded by jointly optimizing parameters involved.
Organization:
The rest of the paper is organized as follows. Section II describes the systemmodels and formulates the problems. Section III presents the iterative algorithms developedbased on IA. The designs based on the combination of IA and ZF are provided in Section IV.Section V discusses the computational complexity of the proposed solutions. Numerical resultsand discussion are provided in Section VI. Finally, Section VII concludes the paper.
Notation:
Bold lower and upper case letters represent vectors and matrices, respectively. k·k represents the ℓ norm. |·| represents the absolute value. R m × n and C m × n represent the spaceof real and complex matrices of dimensions given in superscript, respectively. I n denotes the n × n identity matrix. CN (0 , c I ) denotes a complex Gaussian random vector with zero meanand variance matrix c I . ℜ ( · ) represents real part of the argument. A H and A T are Hermitian andnormal transpose of A , respectively. diag ( a ) represents diagonal matrix constructed from element We formulate the problems based on the EE definition, in which the objective functions contain fractional functions. Anotherapproach for achieving EE in wireless communications is to minimize the power consumption. However, as shown in manyworks (e.g. [36]), EE performances obtained by this approach are far from optimal. Another common suboptimal technique used for overcoming intractable fractional EE problems is developed based onparametric fractional programming, e.g. [39]. However, this technique may not be guaranteed to converge [40, Section 4.1]. of a . Notation ⊙ stands for Schur-Hadamard (element-wise) multiplication of two matrices. e l , [0 , . . . , | {z } l − , , , . . . , . [ a ] + denotes max(0 , a ) . h a , b i , a T b when a and b are real vectors,and h a , b i , ℜ ( a H b ) when a and b are complex vectors. Other notations are defined at theirfirst appearance. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT
In this section, we first describe the system model of multi-pair relaying. Then the transmissionprotocol and energy consumption model of one-way relaying are presented, following by thoseof two-way relaying. Finally, the EE fairness problems are formulated.We consider a multi-pair relay system consisting of a set of K user pairs, denoted by K , { , . . . , K } , and a set of L nonregenerative relays, denoted by L , { , . . . , L } , as shown inFigure 1. Let us denote by U k and U k the two users of pair k , and by R l the relay l . Supposethat there is no direct link between U k and U k for any k ∈ K , and a user intends to communicatewithin its own pair with the help of the relays. All nodes operate in a half-duplex mode and arelow-cost, i.e., each of the nodes is equipped with a single-antenna.The channels are supposed to be flat block-fading with block time T , and without loss ofgenerality, let T = 1 for notational simplicity. Let f ikl denote the complex channel coefficientbetween U ik and R l , and f ik , [ f ik , ..., f ikL ] T . The channel reciprocity holds for all links.Following [7], [23], [24] we suppose that perfect channel state information (CSI) is known at acentral node, where system optimization is performed.We further assume that the transmit user nodes are non energy-constrained while the relaysare energy-constrained. Therefore, for assisting the data transmission, the relays follow the time-switching protocol to harvest energy from the RF signal transmitted from the users [12]. Inparticular, a transmission block is divided into two portions: the first portion of duration τ , τ ∈ (0 , , is a fraction of block time used for charging the relays, referred to as EH phase. Thesecond portion is for the two-hop AF communications, referred to as information transmission Because we consider also two-way relaying, both nodes of each communicating user pair play the role of source anddestination. Therefore, we index them as 1 and 2. In the one-way relay channel, 1 is the source and 2 is the destination, whilein the two-way relaying both send and receive. Compared to power-splitting protocol, time-switching protocol requires simpler hardware implementation (i.e., simpleswitchers) [13], thus it is more suitable for low-cost nodes. U U k U U K U K R R L U k f Fig. 1. A diagram of multiple user pair AF relay systems with K pairs of users and L relays. (IT) phase. In this work, we consider both one-way and two-way relay systems. Communicationprotocol for each of the systems is detailed below. A. One-Way Relay System
In a one-way relay system, only one user in each pair transmits data to the other. Withoutloss of generality and for notational convenience, let us assume that U k is the transmitter and U k is the receiver, for all k ∈ K .
1) EH Phase (One-Way):
During EH phase, the relays harvest energy from the RF signaltransmitted by the transmitters. Particularly, the RF power at the input of the EH circuit of R l is [12] P RF,OW l ( p ) , X k ∈K p k | f kl | (1)where p ik , ( i = { , } ) is the transmit power at U ik and p , [ p , ..., p K ] T . The EH power circuitconverts P RF,OW l ( p ) to DC power used during the IT phase. Here, we consider a realistic RF-DCpower converter, whose conversion efficiency is not a constant, introduced in [35]. Specifically,the harvested energy at R l is E EH,OW l ( τ, p ) = τ ¯ P DC l − β l
11 + exp (cid:0) − c l ( P RF,OW l ( p ) − d l ) (cid:1) − β l ! (2) This scheme is for the scenario where it is inconvenient for the receivers transmitting energy to the relays. For example, thereceivers are mobile phones with low batteries, and to transmit energy to the relays could make the batteries run out quickly. where ¯ P DC l is the maximum power that can be harvested, c l and d l are parameters depending onthe circuit specifications, and β l = (1 + exp( c l d l )) − .
2) IT Phase (One-Way):
During IT phase, the remaining (1 − τ ) fraction of block time isdivided into two equal-length time slots. In the first time slot, the transmitters send data to therelays. Let x k denote the normalized complex symbol transmitted by U k . The received signalat R l is ˜ y OW l = X k ∈K √ p k f kl x k + ˜ n l (3)where ˜ n l is the additive white Gaussian noise (AWGN), i.e., ˜ n ∼ CN (0 , ˜ σ I L ) with ˜ n , [˜ n , ..., ˜ n L ] T . In the second time slot, the relays transmit the processed signal to the receivers. Wedenote by w l ∈ C the complex weight coefficient used at R l , and let w , [ w , ..., w L ] T ∈ C L × .The received signal at U k is y OW k = X l ∈L f kl w l ˜ y OW l + n k = √ p k f T k Wf k x k | {z } desired signal + X j ∈K\{ k } √ p j f T k Wf j x j | {z } interference + f T k W ˜ n + n k | {z } noise (4)where W , diag ( w ) , and n k denotes the additive noise with n k ∼ CN (0 , σ ) . The signal-to-interference-plus-noise ratio (SINR) at U k is γ OW k ( w , p ) = p k | f T k Wf k | P j ∈K\{ k } p j | f T k Wf j | + ˜ σ || f T k W || + σ = p k w H H kk w P j ∈K\{ k } p j w H H kj w + w H G k w + σ (5)where h kj , ( f k ⊙ f j ) T , H kj , h H kj h kj , and G ik , ˜ σ diag (cid:0) f H ik ⊙ f T ik ) . Let r ik be the real transmit data rate at U ik , i.e., the effective information rate is − τ r ik . For feasible transmission,the constraint r k ≤ log(1 + γ OW k ( w , p )) , ∀ k ∈ K (6)should hold. The purpose of introducing { r k } Kk =1 is to determine the rate-dependent signalprocessing energy, which is discussed in detail next.
3) Energy Consumption Model (One-Way):
We consider herein a relatively realistic energyconsumption model which takes into account the dependence of PAs’ efficiency on the outputpower level [20], [41] as well as the dependence of signal processing operators on the transmitdata rate [19]. In addition, for saving energy, a node can be idle (i.e., sleep mode) if it is neitherreceiving nor transmitting [16], [33]. In this spirit, let us first focus on the energy consumed by the users of pair k . Let P idle ik denote the consumed power of U ik in idle mode, which is assumedto be constant [16], [33]. Then, the energy consumed in this mode is E idle k ( τ ) = 1 − τ P idle k + 1 + τ P idle k . (7)On the other hand, for the clarity of description, we divide the power consumed in the activemode into three components: power consumed by the operating circuits, the amplifiers and signalprocessing. The first part includes the power consumed, e.g., by filters, mixers, etc, denoted by P act,cir ik for U ik . It is modeled as a constant [42]. For the power consumed on the amplifiers, weconsider a realistic model whose efficiency is given by [20, Eq. (2)] ˜ ǫ ik = ǫ ik r p ik ¯ P ik (8)where ǫ ik ∈ (0 , is the maximum PA’s efficiency and ¯ P ik is the maximum transmit power of U ik . From (8), the power consumed on the PA is P amp ik = p ik ˜ ǫ ik = ε ik √ p ik (9)where ε ik = p ¯ P ik /ǫ ik . Finally, the power for signal processing is modeled as a linear functionof data rate given by P sp k = ( ρ en k + ρ de k ) r k where ρ en k and ρ de k represent power for encoder at U k and decoder at U k , respectively. Their units are in W/(Gnats/s). In summary, the total energyconsumed by pair k during a block time is E OW k ( τ, p , r ) = E idle k ( τ ) + 1 + τ P amp k + P act,cir k ) + 1 − τ P sp k + P act,cir k )= 1 − τ ρ sp k r k + P ′ k ) + 1 + τ ε k √ p k + P ′′ k ) (10)where ρ sp k = ρ en k + ρ de k , P ′ k = P idle k + P act,cir k , and P ′′ k = P idle k + P act,cir k , which are constant; r , [ r , ..., r K ] T .We now describe the energy consumed by the relays. The radiated power at R l is P rad l ( p , w ) , w ∗ l (cid:16)P k ∈K p k | f kl | + ˜ σ (cid:17) w l = w H A l w + P k ∈K p k w H B kl w where A l , ˜ σ diag ( e l ) and B kl , | f kl | diag ( e l ) . Then, the total energy consumed at R l is given by E R,OW l ( τ, p , w ) , − τ p ¯ P R l P rad l ( p , w ) ǫ R l + E R,const l (11)where ǫ R l ∈ (0 , is the maximum PA’s efficiency and ¯ P R l is the maximum transmit powerof R l . In (11), the first term is the energy consumed by the PA, and E R,const l is the consumedenergy for activating the basic functions which is constant [42]. Since the relays do not encode or decode data, the rate-dependent signal processing energy does not exist. Clearly, for successfullyassisting the data transmission, the energy consumption cannot exceed harvesting or E R,OW l ( τ, p , w ) ≤ E EH,OW l ( τ, p ) , ∀ l ∈ L . (12) B. Two-Way Relay System
In a two-way system, the relays assist the bi-directional communication of all pairs, i.e., bothof the two users of each pair transmit and receive data.
1) EH Phase (Two-Way):
The relays receive energy from the both two users of each pair.Hence, the RF power at the input of EH circuit of R l is P RF,TW l (˜ p ) , X k ∈K X i =1 p ik | f ikl | (13)where ˜ p , [ p , p , ..., p K , p K ] T . Accordingly, the harvested energy at R l is E EH,TW l ( τ, ˜ p ) = τ ¯ P DC l − β l
11 + exp (cid:0) − c l ( P RF,TW l (˜ p ) − d l ) (cid:1) − β l ! . (14)
2) IT Phase (Two-Way):
In the first time slot of IT phase, all the users transmit their signalsto the relays using the same frequency band. Particularly, the received signal at R l is ˜ y TW l = X k ∈K X i =1 √ p ik f ikl x ik + ˜ n l . (15)During the second time slot, the relays broadcast the processed signals to all the users. Thereceived signal at U ik is expressed as y rec ik = X l ∈L f ikl w l ˜ y TW l + n ik = X j ∈K X ˆ i =1 p p ˆ ij f T ik Wf ˆ ij x ˆ ij + f T ik W ˜ n + n ik . (16)As with most of the related works (see [6], [8], [18], [25] and the references therein), we supposethat the self-interference can be completely canceled at the users (with the known CSI). Thenthe signal for decoding at U ik reduces to y TW ik = √ p ¯ ik f T ik Wf ¯ ik x ¯ ik | {z } desired signal + X j ∈K\{ k } X ˆ i =1 p p ˆ ij f T ik Wf ˆ ij x ˆ ij | {z } interference + f T ik W ˜ n + n ik | {z } noise (17)where ¯ i = { , } \ { i } . Thus the SINR at U ik can be written as γ TW ik ( w , ˜ p ) = p ¯ ik | f T ik Wf ¯ ik | P j ∈K\{ k } P i =1 p ˆ ij | f T ik Wf ˆ ij | + ˜ σ || f T ik W || + σ = p ¯ ik w H H kk w P j ∈K\{ k } P i =1 p ˆ ij w H ˜ H ik ˆ ij w + w H G ik w + σ (18)where ˜ h ik ˆ ij , ( f ik ⊙ f ˆ ij ) T and ˜ H ik ˆ ij , ˜ h H ik ˆ ij ˜ h ik ˆ ij . We note that ˜ H ik ¯ ik = ˜ H ¯ ikik = H kk . Similar tothe one-way system, we need the following set of constraints for successful transmissions r ik ≤ log(1 + γ TW ¯ ik ( w , ˜ p )) , ∀ k ∈ K , i = { , } . (19)
3) Energy Consumption Model (Two-Way):
Different from the one-way relay system, the usersin the two-way relay system are always active since each of them either transmits or receivesduring block time. In addition, the energy for the power amplifiers accounts on the both usersof a pair, and the rate-dependent signal processing energy for pair k is calculated based on therate transmitted from U k and U k . Thus the energy consumed by pair k can be expressed as E TW k ( τ, ˜ p , ˜ r ) , X i =1 P act,cir ik !| {z } energy for circuits + 1 + τ X i =1 P amp ik !| {z } energy for PAs + 1 − τ X i =1 P sp ik !| {z } energy for signal processing = E TW,cir k + 1 + τ X i =1 ε ik √ p ik ! + 1 − τ X i =1 ρ sp ik r ik ! (20)where E TW,cir k = 1( P i =1 P act,cir ik ) is a constant, ρ sp ik = ρ en ik + ρ de ¯ ik , and ˜ r , [ r , r , ..., r K , r K ] T .For two-way relay R l , the radiated power is P rad,TW l (˜ p , w ) , w ∗ l (cid:16)P k ∈K P i =1 p ik | f ikl | +˜ σ (cid:17) w l = w H A l w + P k ∈K P i =1 p ik w H B ikl w . Then, the total consumed energy at R l is E R,TW l ( τ, ˜ p , w ) , − τ q ¯ P R l P rad,TW l ( w , ˜ p ) ǫ R l + E R,const l . (21)Again, the following set of constraints on the harvested and consumed energy is required forsuccessful relaying E R,TW l ( τ, ˜ p , w ) ≤ E EH,TW l ( τ, ˜ p ) , ∀ l ∈ L . (22) C. Energy Efficiency Fairness Problems
We focus on the max-min EE. Here, the shared relays use energy contributed by the usersfor assisting data transmission, when each user exchanges information with the one in the samepair only. Hence, it is relevant to maintain the EE fairness (EEF) between the user pairs.
1) EEF for One-Way Relay System:
With the model specified in Section II-A and by definition,the individual EE of pair k is given by f EE,OW k ( τ, p , r ) , − τ r k E OW k ( τ, p , r ) , k ∈ K (23)Thereby the problem of max-min EEF can be mathematically formulated asmaximize p , w , r ,τ min ≤ k ≤ K f EE,OW k ( τ, p , r ) (24a)subject to − τ r k ≥ Q k , ∀ k ∈ K (24b)EEF-OW , < p k ≤ ¯ P k , ∀ k ∈ K (24c) P rad l ( p , w ) ≤ ¯ P R l , ∀ l ∈ L (24d)(6) , (12) . (24e)Constraint (24b) guarantees the quality of service (QoS) for each user pair, where Q k > is apredefined threshold. (24c) and (24d) represent the transmit power constraints at the transmittersand the relays, respectively.
2) EEF for Two-Way Relay System:
Similarly, we obtain the problem of max-min EEF forthe two-way system as maximize ˜ p , w , ˜ r ,τ min ≤ k ≤ K f EE,TW k ( τ, ˜ p , ˜ r ) , − τ P i =1 r ik E TW k ( τ, ˜ p , ˜ r ) (25a)subject to − τ r ik ≥ Q ik , ∀ k ∈ K , i = 1 , (25b)EEF-TW , < p ik ≤ ¯ P ik , ∀ k ∈ K , i = 1 , (25c) P rad,TW l (˜ p , w ) ≤ ¯ P R l , ∀ l ∈ L (25d)(19) , (22) . (25e)In this work, we assume that the feasible sets of EEF-OW and EEF-TW are nonempty. Theobjectives in EEF-OW and EEF-TW are nonsmooth nonconvex—the numerators of the fractionsare linear, but the denominators are nonconvex. Also, the feasible sets are nonconvex. Hencethe problems are intractable and it is impossible to transform the problems into the equivalentconvex ones. Like many studies on wireless communication designs [18], [24], [27], [30], weaim at finding approximate, but efficient, solutions to these problems. III. T HE P ROPOSED A LGORITHMS FOR S OLVING
EEF-OW
AND
EEF-TWIn this section, we propose algorithms for solving EEF-OW and EEF-TW based on theinner approximation (IA) framework [37], [38], which is an efficient approach widely usedfor dealing with nonconvex programs. First, the general principles of the IA and the usefulapproximation functions are provided. Then, the IA-based algorithms solving EEF-OW andEEF-TW are presented, followed by the convergence discussion. Finally, the approach arrivingat the SOCP approximations is provided.
A. Useful Approximate Formulations
For exposition purpose, we first provide some approximate formulations which are used todevise proposed solutions. Generally the basic idea of IA is to successively approximate anonconvex set to inner convex ones. Specifically, let h ( x ) ≤ be a nonconvex constraint where h ( x ) : C n → R and h ( x ) is continuously differentiable. An inner approximation is obtainedby replacing h ( x ) by a convex upper bound ˜ h ( x ; g ( x ′ )) , i.e., h ( x ) ≤ ˜ h ( x ; g ( x ′ )) , where g ( x ) : C n → C m is a parameter vector and x ′ is some feasible point. Function ˜ h ( x ; g ( x ′ )) must satisfythe following conditions h ( x ) = ˜ h ( x ; g ( x )) , ∇ x ∗ h ( x ) = ∇ x ∗ ˜ h ( x ; g ( x )) (26)where ∇ x ∗ h () denotes the gradient of h () with respect to the complex conjugate of x . If x is areal vector, then ∇ x ∗ h () is simply replaced by ∇ x h () . The approximations presented next followthese principles.
1) Approximation for Bilinear Function:
Consider nonconvex constraint x x ≤ y where ( x , x , y ) ∈ R . An approximation of bilinear function x x is given by [37, Lem. 3.5] x x ≤ h bi ( x , x ; λ ) , . (cid:18) λx + x λ (cid:19) (27)where λ = x ′ x ′ . We remark that the bilinear function can be rewritten as difference-of-convexones, e.g., x x = 0 . x + x ) − . x − x ) = 0 . x + x ) − . x + x ) = 0 . x + x ) − . x − x ) . Then the approximates can be obtained by using the first order Taylor seriesapproximation of the nonconvex parts. Herein, we use (27) for problems EEF-OW and EEF-TW,since we numerically observe that with (27), the iterative procedures require fewer number ofiterations for convergence (see Fig. 3(a) for the numerical example).
2) Approximation for Fractional-Linear Function:
Consider nonconvex constraint x x ≤ y where ( x , x , y ) ∈ R . In light of (27), an approximation of fractional-linear function x x canbe obtained as x x ≤ h frac ( x , x ; λ ) , . (cid:18) λx + 1 λx (cid:19) (28)where λ = x ′ x ′ . We note that constraint h frac ( x , x ; λ ) ≤ y can be expressed by the followingtwo second-order cone (SOC) ones . (cid:18) λx + z λ (cid:19) ≤ y, ≤ x z. (29)
3) Approximation for Quadratic-over-Linear Function:
Consider concave function h ( x , z ; A ) , − x H Ax z where x ∈ C n , z ∈ R ++ , and A (cid:23) . We can use the first order Taylor series to obtaina convex upper bound of h ( x , z ; A ) given as h qol ( x , z ; x ′ , z ′ ; A ) , h ( x ′ , z ′ ; A ) + (cid:10) [ ∇ x ∗ h ( x ′ , z ′ ; A ) , ∇ z h ( x ′ , z ′ ; A )] T , [ x − x ′ , z − z ′ ] T (cid:11) = ( x ′ ) H Ax ′ ( z ′ ) z − ℜ (( x ′ ) H Ax ) z ′ . (30)
4) Approximation for Logarithmic Function:
Consider logarithmic function h ( x ) , log( x ) where x ∈ R ++ . An approximated function of h ( x ) is given by h ( x ) ≤ h log ( x ; x ′ ) , log( x ′ ) − xx ′ . (31)
5) Approximation for Power Function:
Consider power function h ( x ; m ) , − x m where x ∈ R ++ . Here, we only focus on the cases m < or m > where h ( x ; m ) is concave. Its convexapproximation is given by h ( x ; m ) ≤ h po ( x ; x ′ ; m ) , ( m − x ′ ) m − m ( x ′ ) m − x. (32) B. Solution for EEF-OW
Directly applying IA to (24) is difficult, since the nonconvex parts here are not explicitlyexposed. As a necessary step, we translate (24) into an equivalent, but more tractable, formulation.We first introduce variable η > and arrive at the epigraph form of (24) given asminimize p , w , r ,τ,η η (33a)subject to f EE,OW k ( τ, p , r ) ≥ η − , ∀ k ∈ K (33b)(6) , (12) , (24b) , (24c) , (24d) . (33c)Here the nonconvex parts include (6), (12), (24d), and (33b).
1) Changes of Variables:
We now make some changes of variables. Specifically, we willdenote q ik = p ik , ∀ k ∈ K , i = 1 , , and turn the nonconvex products of linear and quadraticfunctions, e.g., p k w H H kk w , into the quadratic-over-linear functions. We also define ˜ τ = τ − τ ,i.e., τ = ˜ τ − τ +1 . It is important to note that these changes of variables still preserve the convexityin (24b) and (24c) as well as turn nonconvex constraint (24d) into a convex one. In addition,they make (6), (12), and (33b) become more convenient to handle, as shown next.
2) Transformation of (6):
By introducing new variables { v k } Kk =1 and { s k } Kk =1 , we can equiv-alently represent (6) by the following set of constraints r k ≤ log(1 + v k ) , ∀ k ∈ K (34) X j ∈K\{ k } w H H kj w q j + w H G k w + σ ≤ s k , ∀ k ∈ K (35) s k v k ≤ w H H kk w q k , ∀ k ∈ K . (36)Here, only (36) is nonconvex which contains bilinear and quadratic-over-linear functions.
3) Transformation of (12):
We first rewrite (12) with the change variables as ˜ ε l s w H A l w + X k ∈K w H B kl w q k ≤ ˆ β l (˜ τ − α l exp (cid:0) − c l P k ∈K | f kl | q k (cid:1) − ¯ β l ˜ τ + ˇ β l , ∀ l ∈ L where ˜ ε l = p ¯ P R l /ǫ R l , ˆ β l = ¯ P DC l − β l , ¯ β l = β l ˆ β l + E R,const l , ˇ β l = ¯ β l − E R,const l and α l = exp( c l d l ) .Also, to reveal the hidden convexity in the constraint, we introduce new variables { u l } Ll =1 and { t l } Ll =1 , and equivalently rewrite (12) as X k ∈K w H B kl w q k ≤ u l , ∀ l ∈ L (37) log(˜ τ − t l − − log( α l t l ) + X k ∈K c l | f kl | q k ≥ , ∀ l ∈ L (38) ˜ ε l q w H A l w + u l ≤ ˆ β l t l − ¯ β l ˜ τ + ˇ β l , ∀ l ∈ L . (39)The nonconvex parts are in (37) and (38) including the power and the logarithmic functions.
4) Transformation of (33b):
Constraint (33b) is rewritten as ρ sp k + P ′ k r k + ˜ τr k (cid:16) ε k √ q k + P ′′ k (cid:17) ≤ η, ∀ k ∈ K , (40)which is equivalently represented as ˜ τr k ≤ z k , ∀ k ∈ K (41) ρ sp k + P ′ k r k + ε k z k √ q k + P ′′ k z k ≤ η, ∀ k ∈ K (42)where { z k } Kk =1 are newly introduced variables. We remark that function z k / √ q k is convex (seeAppendix B for the proof), and so is (42). Also, (41) can be rewritten as r k ≤ z k ˜ τ where thenonconvex part is quadratic-over-linear.With the above transformations, (33) can be reformulated asminimize q , w , r , ˜ τ,η v , s , u , t , z η (43a)subject to r k ≥ (1 + ˜ τ ) Q k , ∀ k ∈ K (43b) q k ≥ / ¯ P k , ∀ k ∈ K (43c) w H A l w + X k ∈K w H B kl w q k ≤ ¯ P R l , ∀ l ∈ L (43d)(34) , (35) , (36) , (37) , (38) , (39) , (41) , (42) (43e)where q , [ q k , ..., q K ] T , v , [ v , ..., v K ] T , s , [ s , ..., s K ] T , u , [ u , ..., u L ] T , t , [ t , ..., t L ] T ,and z , [ z , ..., z K ] T ; (43b), (43c), and (43d) are respectively the versions of (24b), (24c), and(24d) after change of variables. The equivalence here is in the sense of optimality (see the proofin Appendix A).We are now ready to use IA for solving (43). Specifically, by applying the approximateformulations provided in Section III-A to the nonconvex parts in (43), we obtain the followingconvex approximation of (43) solved at iteration n + 1 minimize ψ η (44a)subject to h bi ( s k , v k ; v ( n ) k s ( n ) k ) + h qol ( w , q k ; w ( n ) , q ( n )1 k ; H kk ) ≤ , ∀ k ∈ K (44b) X k ∈K w H B kl w q k + h po ( u l ; u ( n ) l ; 2) ≤ , ∀ l ∈ L (44c) log(˜ τ − t l − ≥ h log ( α l t l ; α l t ( n ) l ) + X k ∈K c l | f kl | h po ( q k ; q ( n )1 k ; − , ∀ l ∈ L (44d) r k + h qol ( z k , ˜ τ ; z ( n ) k , ˜ τ ( n ) ; 1) ≤ , ∀ k ∈ K (44e)(34) , (35) , (39) , (42) , (43b) , (43c) , (43d) (44f)where ψ , [ q T , w T , r T , ˜ τ , η, v T , s T , u T , t T , z T ] T and ψ ( n ) is some feasible point of (43). Algorithm 1
The Proposed Method Solving EEF-OW Initialization:
Set n := 0 , n ′ := 0 , and randomly generate a feasible point ψ (0) of (45). repeat {Finding a feasible point of (43)} Solve minimize ψ ∈S ( ψ ( n ′ ) ) η + b P k ∈K [(1 + ˜ τ ) Q k − r k ] + , denote the optimal by ψ ∗ fe . Update n ′ := n ′ + 1 , ψ ( n ′ ) := ψ ∗ fe . until P k ∈K [(1 + ˜ τ ∗ ) Q k − r ∗ k ] + = 0 . Set ψ (0) := ψ ( n ′ ) . repeat {Solving (43)} Obtain the optimal point of (44), denoted by ψ ∗ . Update n := n + 1 , ψ ( n ) := ψ ∗ . until convergence or predefined number of iterations. Output (solution for EEF-OW) : τ := ˜ τ ( n ) − τ ( n ) +1 , w := w ( n ) , p k := 1 /q ( n )1 k for all k ∈ K .
5) Finding Initial Feasible Points:
A feasible point of (43) is required for starting the IAprocedure, which is difficult to find due to the QoS constraints. Here we provide an efficientheuristic method inspired by [43], [44, Section 3.2] to overcome this issue. The idea is to allowthe QoS constraints to be violated, and the violation is penalized. Particularly, let us considerthe following modification of (43)minimize ψ ∈S η + b X k ∈K [(1 + ˜ τ ) Q k − r k ] + (45)where b > is a penalty parameter; S , { ψ | (34)–(42) , (43c) , (43d) } . Finding feasible pointsof (45) is easy as follows. We first randomly generate τ (0) ∈ (0 , , < p (0)1 k ≤ ¯ P k , and w (0) ∈ C L × , then (if necessary) scale w (0) so that (12) and (24d) are satisfied. Based on ( τ (0) , p (0)1 k , w (0) ) , r (0) , v (0) , s (0) , u (0) , t (0) , and z (0) are determined by setting (34), (35), (36), (37),(39), and (41) to be equality. With ψ (0) , we can start an iterative IA procedure for solving(45). Intuitively, the penalty term in (45) would force { (1 + ˜ τ ) Q k − r k } to decrease. Once (1 + ˜ τ ) Q k − r k ≤ for all k , i.e., the penalty term is zero, producing a feasible point of (43).In summary, we outline the proposed method for solving EEF-OW in Algorithm 1. In line 3, S ( ψ ( n ) ) , { ψ | (34) , (35) , (39) , (42) , (43c) , (43d) , (44b)–(44e) } is an approximate convex set of S corresponding to ψ ( n ) . C. Solution for EEF-TW
The procedure for finding a solution of EEF-TW is similar to the one presented in the previoussubsection. So, for the sake of brevity, only the main steps are presented. We first arrive at theepigraph form of EEF-TW given byminimize ˜ p , w , ˜ r ,τ, ˜ η ˜ η (46a)subject to f EE,TW k ( τ, ˜ p , ˜ r ) ≥ ˜ η − , ∀ k ∈ K (46b)(19) , (22) , (25b) , (25c) , (25d) . (46c)We focus on the nonconvex convexity induced by (19), (22), (25d), and (46b). Again, by usingthe change of variables in Section III-B1 and introducing additional variables, we transform (46)into the following equivalent problemminimize ˜ q , w , ˜ r , ˜ τ, ˜ η ˜ v , ˜ s , u , t , z ˜ η (47a)subject to r ik ≥ (1 + ˜ τ ) Q ik , ∀ k ∈ K , i = { , } (47b) q ik ≥ / ¯ P ik , ∀ k ∈ K , i = { , } (47c) w H A l w + X k ∈K X i =1 w H B ikl w q ik ≤ ¯ P R l , ∀ l ∈ L (47d) r ik ≤ log(1 + v ¯ ik ) , ∀ k ∈ K , i = { , } , ¯ i = { , } \ { i } (47e) X j ∈K\{ k } X ˆ i =1 w H ˜ H ik ˆ ij w q ˆ ij + w H G ik w + σ ≤ s ik , ∀ k ∈ K , i = 1 , (47f) s ik v ik ≤ w H H kk w q ¯ ik , ∀ k ∈ K , i = { , } , ¯ i = { , } \ { i } (47g) X k ∈K X i =1 w H B ikl w q ik ≤ u l , ∀ l ∈ L (47h) log(˜ τ − t l − − log( α l t l ) + c l X k ∈K X i =1 | f ikl | q ik ≥ , ∀ l ∈ L (47i) ˜ ε l q w H A l w + u l ≤ ˆ β l t l − ¯ β l ˜ τ + ˇ β l , ∀ l ∈ L (47j) ˜ τ P i =1 r ik ≤ z k , ∀ k ∈ K (47k) P i =1 r ik + z k ! E TW,cir k + X i =1 ε ik z k √ q ik ! + P i =1 ρ sp ik r ik P i =1 r ik ≤ ˜ η, ∀ k ∈ K (47l) where ˜ v , [ v , v , ..., v K , v K ] T , ˜ s , [ s , s , ..., s K , s K ] T . Similar to EEF-OW, the equiv-alence here is in the sense of optimality. In (47), the nonconvex parts include (47g), (47h),(47i), (47k), (47l), which can also be approximated using the approximate functions provided inSection III-A. By doing so, we arrive at the convex approximation problem given asminimize ˜ ψ ˜ η (48a)subject to h bi (cid:16) s ik , v ik ; v ( n ) ik /s ( n ) ik (cid:17) + h qol ( w , q ¯ ik ; w ( n ) , q ( n )¯ ik ; H kk ) ≤ , ∀ k ∈ K , i = { , } , ¯ i = { , } \ { i } (48b) X k ∈K X i =1 w H B ikl w q ik + h po ( u l ; u ( n ) l ; 2) ≤ , ∀ l ∈ L (48c) log(˜ τ − t l − ≥ h log ( α l t l ; α l t ( n ) l ) + X k ∈K X i =1 c l | f ikl | h po ( q ik ; q ( n ) ik ; − , ∀ l ∈ L (48d) P i =1 r ik + h qol ( z k , ˜ τ ; z ( n ) k , ˜ τ ( n ) ; 1) ≤ , ∀ k ∈ K (48e) P i =1 r ik + z k ! E TW,cir k + X i =1 ε ik z k √ q ik ! + h frac X i =1 ρ sp ik r ik , X i =1 r ik ; 1( P i =1 ρ sp ik r ( n ) ik )( P i =1 r ( n ) ik ) ! ≤ ˜ η, ∀ k ∈ K (48f)(47b) , (47c) , (47d) , (47e) , (47f) , (47j) (48g)where ˜ ψ , [˜ q T , w T , ˜ r T , ˜ τ , ˜ η, ˜ v T , ˜ s T , u T , t T , z ] T and ˜ ψ ( n ) is a feasible point of (47). Finally, forfinding initial feasible points of (47), we use a similar technique as that in Section III-B5.The proposed procedure for solving EEF-TW is outlined in Algorithm 2. In line 3, ˜ S ( ˜ ψ ( n ) ) , { ˜ ψ | (47b) , (47c) , (47d) , (47e) , (47f) , (47j) , (48b)–(48f) } is an inner convex approximation of ˜ S at ˜ ψ ( n ) . D. Convergence of Algorithms 1 and 2
The general convergence analysis of the IA framework has been provided in [37]. Thus, weonly need to examine the conditions posted there for justifying the convergence of Algorithms1 and 2. First, we recall that the approximate functions provided in Section III-A satisfy (26),which corresponds to [37, Property A]. In addition, the feasible set of (43) and (47) are compactand nonempty. Thus it is guaranteed that the objective sequences { η ( n ) } ∞ n =0 (Algorithm 1) and { ˜ η ( n ) } ∞ n =0 (Algorithm 2) are nonincreasing and converge [37, Corollary 2.3]. Algorithm 2
The Proposed Method Solving EEF-TW Initialization:
Set n := 0 , n ′ := 0 , and randomly generate a point ˜ ψ (0) ∈ ˜ S , { ˜ ψ | (47c)–(47l) } . repeat {Finding a feasible point of (47)} Solve minimize ˜ ψ ∈ ˜ S ( ˜ ψ ( n ) ) ˜ η + b P k ∈K P i =1 [(1 + ˜ τ ) Q ik − r ik ] + , and denote the optimal by ˜ ψ ∗ fe . Update n ′ := n ′ + 1 , ˜ ψ ( n ′ ) := ˜ ψ ∗ fe . until P k ∈K P i =1 [(1 + ˜ τ ∗ ) Q ik − r ∗ ik ] + = 0 . Set ˜ ψ (0) := ˜ ψ ( n ′ ) . repeat {Solving (47)} Obtain the optimal point of (48), denoted by ˜ ψ ∗ . Update n := n + 1 , ˜ ψ ( n ) := ˜ ψ ∗ . until convergence or predefined number of iterations. Output (solution for EEF-TW) : τ := ˜ τ ( n ) − τ ( n ) +1 , w := w ( n ) , p ik := 1 /q ( n ) ik for all k ∈ K , i = { , } .However, since objectives in (43) and (47) are not strongly convex, the iterates { ψ ( n ) } ∞ n =0 and { ˜ ψ ( n ) } ∞ n =0 might not converge. This issue can be overcome by using proximal terms, i.e.,replacing objective of (44) and (48) by η + a || ψ − ψ ( n ) || and ˜ η + a || ˜ ψ − ˜ ψ ( n ) || , respectively, withan arbitrary regularization parameter a > [45]. By doing so, the objective sequences { η ( n ) } ∞ n =0 and { ˜ η ( n ) } ∞ n =0 are strictly decreasing and || ψ ( n ) − ψ ( n +1) || → , || ˜ ψ ( n ) − ˜ ψ ( n +1) || → [37,Proposition 3.2], which come from the following relations η ( n ) − η ( n +1) ≥ a || ψ ( n +1) − ψ ( n ) || , ˜ η ( n ) − ˜ η ( n +1) ≥ a || ˜ ψ ( n +1) − ˜ ψ ( n ) || . E. Conic Formulations for Approximate Subproblems
The approximate subproblems (44) and (48) are cast as generic convex programs due to thelogarithmic functions involved. Theoretically, these problems can be efficiently solved usinga general purpose interior-point solver. However, from the practical perspective, it is morenumerically efficient if we can arrive at a more standard convex program, e.g., conic quadratic orsemidefinite program [46]. We observe from (44) and (48) that the objectives and constraints arelinear or SOC-representable, except the constraints containing the logarithmic functions. Hence,we are motivated to develop SOC-presentable approximations for these constraints. Towards thegoal, we present a concave lower bound of the logarithmic function given as log x ≥ log x ′ + 2 − √ x ′ √ x (49)which holds for all x > , x ′ > . Inequality (49) can be justified as follows. Let us define g ( x ; x ′ ) , log x − log x ′ − √ x ′ √ x for x > , x ′ > . We can easily prove that g ( x ; x ′ ) ≥ bychecking the first-order derivative of g ( x ; x ′ ) with respect to x , i.e., ∂g ( x ; x ′ ) ∂x = 1 x − √ x ′ x √ x = 1 x − √ x ′ √ x ! , which clearly indicates that ∂g ( x ; x ′ ) ∂x ≥ if x ≥ x ′ , and ∂g ( x ; x ′ ) ∂x ≤ if x ≤ x ′ . Accordingly, g ( x ; x ′ ) achieves the minimum at x = x ′ with g ( x = x ′ ; x ′ ) = 0 , and thus g ( x ; x ′ ) ≥ for all x > , x ′ > which validates (49). Since (49) is verified to fulfill the conditions in (26), we canreplace the constraint log x ≥ y by log x ′ + 2 − √ x ′ √ x ≥ y. (50)In the IA-based iterative procedure, x ′ is the value of x obtained in the preceding iteration. Wenote that (50) admits the SOC-representation, i.e.,(50) ⇔ ξ ≤ x (cid:13)(cid:13) [2 √ x ′ , log x ′ + 2 − y, ξ ] (cid:13)(cid:13) ≤ log x ′ + 2 − y + ξ . (51)In the same way, (34) can be approximated by log(1 + v ( n ) k ) + 2 − q v ( n ) k √ v k ≥ r k ∀ k ∈ K . (52)IV. D ESIGNS B ASED ON Z ERO -F ORCING B EAMFORMING
In multi-pair relay systems, ZF is commonly invoked to eliminate the inter-pair interference,and thus, reduces the design complexity [6], [8], [25]. For EEF-OW and EEF-TW, using ZFbeamforming does not lead to convex formulations due to the complexity involved. However, wecan obtain suboptimal solutions but with much lowered complexity, using the similar proceduresillustrated in Section III. In the rest of the section, we sequentially present the ZF-based designsfor EEF-OW and EEF-TW.
A. ZF-Based Design for EEF-OW
Let us define ¯ H k , [ h T k , ..., h T k ( k − , h T k ( k +1) , ..., h T kK ] ∈ C L × ( K − and ¯ H , [ ¯ H , ..., ¯ H K ] T ∈ C L × K ( K − . The ZF beamforming principles lead to h kj w = 0 , ∀ j = k, k ∈ K ⇔ ¯ Hw = . (53) Clearly, the null-space of ¯ H exists if L > K ( K − . Let Z ∈ C L × ( L − K ( K − be an orthogonalbasis of the null-space of ¯ H , then we can find w such as w = Z ¯ w where ¯ w ∈ C ( L − K ( K − × [47]. This allows us to rewrite SINR at U k as γ OW,ZF k ( ¯ w , p ) = p k ¯ w H H ZF kk ¯ w ¯ w H G ZF k ¯ w + σ (54)where H ZF kk , Z H H kk Z and G ZF k , Z H G k Z . Thus, the design problem with ZF beamforming ismaximize p , ¯ w , r ,τ min ≤ k ≤ K f EE,OW k ( τ, p , r ) (55a)subject to ¯ w H A ZF l ¯ w + X k ∈K p k ¯ w H B ZF kl ¯ w ≤ ¯ P R l , ∀ l ∈ L (55b) r k ≤ log(1 + γ OW,ZF k ( ¯ w , p )) , ∀ k ∈ K (55c) − τ ε l s ¯ w H A ZF l ¯ w + X k ∈K p k ¯ w H B ZF kl ¯ w + E R,const l ≤ E EH,OW l ( τ, p ) , ∀ l ∈ L . (55d)(24b) , (24c) . (55e)where A ZF l , Z H A l Z and B ZF kl , Z H B kl Z . B. ZF-Based Design for EEF-TW
To obtain ZF beamforming for two-way system, we first recall that ˜ h ik ˆ ij = ˜ h ˆ ijik and de-fine M k , [˜ h T k k +1) , ˜ h T k k +1) , ..., ˜ h T k K , ˜ h T k K ] and ¯ M , [ M , ..., M K − , ¯ H , ..., ¯ H K ] T ∈ C K ( K − × L . Then we can write the ZF constraint as ˜ h ik ˆ ij w = 0 , ∀ j = k, k ∈ K , i, ˆ i ∈ { , } ⇔ ¯ Mw = . (56)Let ˜ Z ∈ C L × ( L − K ( K − be an orthogonal basis of null-space of ¯ M , which requires the conditionthat L > K ( K − for existence. Again, we can find beamforming vector as w = ˜ Z ¯ w where ¯ w ∈ C ( L − K ( K − × . The SINR at U ik reduces to γ TW,ZF ik ( ¯ w , ˜ p ) = p ¯ ik ¯ w H ˜ H ZF kk ¯ w ¯ w H ˜ G ZF ik ¯ w + σ (57)where ˜ H ZF kk , ˜ Z H H kk ˜ Z and ˜ G ZF ik , ˜ Z H G ik ˜ Z . The EEF design problem based on ZF beamformingis given bymaximize ˜ p , ¯ w , ˜ r ,τ min ≤ k ≤ K f EE,TW k ( τ, ˜ p , ˜ r ) (58a)subject to ¯ w H ˜ A ZF l ¯ w + X k ∈K X i =1 p ik ¯ w H ˜ B ZF ikl ¯ w ≤ ¯ P R l , ∀ l ∈ L (58b) r ik ≤ log(1 + γ TW,ZF ik ( ¯ w , ˜ p )) , ∀ k ∈ K , i = { , } . (58c) − τ ε l vuut ¯ w H ˜ A ZF l ¯ w + X k ∈K X i =1 p ik ¯ w H ˜ B ZF ikl ¯ w + E R,const l ≤ E EH,TW l ( τ, ˜ p ) , ∀ l ∈ L (58d)(25b) , (25c) . (58e)where ˜ A ZF l , ˜ Z H A l ˜ Z and ˜ B ZF ikl , ˜ Z H B ikl ˜ Z . Remark . We note that in ZF-based designs, other parameters (transmit data rate, users’ transmitpower, and EH time) are still jointly designed with the ZF beamforming. Here, problems (55)and (58) can be solved by the similar IA procedures described in Sections III. The two problemsare optimized over ¯ w . Thus, the total numbers of variables in their convex approximate programsare smaller than those of EEF-OW and EEF-TW (as discussed in the next section). On the otherhand, since the inter-pair interference is canceled, it is expected that the numbers of iterations ofIA procedures solving (55) and (58) are smaller compared to those of EEF-OW and EEF-TW.This will be elaborated by numerical experiments provided in Subsection VI-C. For the ease ofexposition, we refer to the solutions of (55) and (58) as ‘ZF-based design (OW)’ and ‘ZF-baseddesign (TW)’, respectively.V. C OMPUTATIONAL C OMPLEXITY A NALYSIS
We now discuss on the computational complexity of solving the SOCP approximations (ineach of iterations) by a general interior point method based on the results in [46, Chapter 6].For Algorithm 1, the SOCP solved at an iteration includes (10 K + 6 L + 2) real variables and (10 K +5 L ) conic constraints. Then the worst case of computational complexity in an iteration ofthe algorithm is O (cid:0) (10 K +5 L ) . (10 K +6 L ) (cid:1) . For Algorithm 2, the SOCP solved at an iterationincludes (19 K + 6 L + 2) real variables and (16 K + 5 L ) conic constraints. Then the worst caseof computational complexity in an iteration of the algorithm is O (cid:0) (16 K + 5 L ) . (19 K + 6 L ) (cid:1) ,which is higher than that of Algorithm 1 due to the additional variables coming from the bi-directional transmission.For ZF-based design (OW), by using ZF beamforming at the relays, the number of realvariables in an SOCP approximation is (10 K +5 L +2 − K ) and the number of conic constraintsis (9 K + 5 L ) . Hence the worst case complexity estimate is O (cid:0) (9 K + 5 L ) . (10 K + 5 L − K ) (cid:1) .Similarly, for ZF-based design (TW), an SOCP approximation includes (19 K + 5 L + 2 − K ) Table IS
IMULATION P ARAMETERS P ARAMETERS V ALUE P ARAMETERS V ALUE
Bandwidth 250 kHz User circuit power P idle ik = 0 . mW, P act,cir ik = 1 mWNoise power σ = ˜ σ = − dBm Relay circuit power P R,const l = 1 mWQoS Q ik = 0 . nats/s/Hz Signal processing power [33] ρ en ik = ρ de ik = 50 mW/(Gnats/s)PA model [33] ¯ P R l = 33 dBm, EH model [35] ¯ P DC l = 24 mW, ǫ ik = ǫ Rl = 0 . c l = 150 , d l = 0 . real variables and (14 K + 5 L ) conic constraints. So, the complexity is O (cid:0) (14 K + 5 L ) . (19 K +5 L − K ) (cid:1) .From the above complexity estimates, it is expected that computational complexity in aniteration of ZF-based design (OW) and ZF-based design (TW) are lower than that of EEF-OWand EEF-TW, respectively. This point will be numerically elaborated in Table III.VI. N UMERICAL R ESULTS
In this section, we numerically evaluate the proposed methods. We consider a relay networkas depicted in Fig. 1 in which the distance between two users of each pair is 10 m. The relaysare randomly placed inside the rectangular region formed by the users { U k } Kk =1 and { U k } Kk =1 .The exponent path loss model is used with path loss exponent 3.5. All channels are Rayleighfading. Simulation parameters are taken from Table I, unless stated otherwise. The maximumtransmit power is set to be the same for all users, i.e., ¯ P ik = ¯ P , ∀ i, k , which varies in theexperiments. The number of user pairs is K = 3 . Other parameters will be specified in theexperiments. In all simulations, the iterative procedures of Algorithms 1 and 2 stop when eitherthe increase in the objective between two consecutive iterations is less than − or the numberof iterations exceeds 200. To solve convex problems, we use the MOSEK [48] and Fminconsolvers in MATLAB environment. A. Performances of Algorithm 1 (One-Way Relaying)
In the first set of experiments, we study the impact of conic formulation of (44) on thecomputational complexity of Algorithm 1. Fig. 2 shows the convergence behavior of Algorithm 1running with the generic convex program (GCP) and SOCP. Specifically, Fig. 2(a) plots theconvergence of the objective over two channel realizations, and Fig. 2(b) shows the cumulativedistribution functions (CDFs) of the required number of iterations to converge. Also, we provide . . . . Chan. real. 1 Chan. real. 2Iteration index EE O b j ec ti v e ( n a t s / s / J ) Algorithm 1-GCPAlgorithm 1-SOCP (a) Convergence over two channel realizations
20 60 100 140 18000 . . . . Number of iterations C D F Algorithm 1-GCPAlgorithm 1-SOCP (b) CDFs of the number of required iterations to converge.Fig. 2. Impact of conic formulation on the convergence behavior of Algorithm 1. We take K = 2 , L = 2 and ¯ P = 33 dBm.Table IIA VERAGE PER - ITERATION AND TOTAL SOLVER RUN TIME ( IN SECOND ) OF A LGORITHM ADOPTING
GCP
AND
SOCP. W
ETAKE K = 2 , L = 2 AND ¯ P = 33 D B M .Solver Fmincon MOSEKAlgorithm1-GCP Avg. per-iteration run time 49 N/AAvg. total run time 2.5e3Algorithm1-SOCP Avg. per-iteration run time 4.97 0.003Avg. total run time 220 0.17 the average total and per-iteration run time of the algorithm with the two formulations in TableII. We can see in the figure that with the SOCP, the algorithm converges with more iterationscompared to the GCP. However, as shown in the table, the per-iteration run time of the SOCP(solver Fmincon) is much smaller than that of GCP (solver Fmincon), resulting that the total runtime of the algorithm with the SOCP is ten times smaller than that with the GCP. In addition,the SOCP allows us to use the more efficient solver MOSEK. With this, the total run timesignificantly reduces.In Fig. 3, we illustrate the effectiveness of (27) in term of convergence. Particularly, Fig.3(a) plots the convergence of Algorithm 1 using the two approximation functions, (27) anddifference-of-convex (DoC) function, over a random channel realization with two different initialpoints generated randomly. And Fig. 3(b) shows the CDFs of number of iteration required forconvergence. The results clearly demonstrate that using DoC formulations of bilinear functionfor the considered problems is not efficient since the corresponding iterative procedure only stopsby the maximum number of iteration criteria. This confirms the use of (27).Fig. 4 depicts the averaged minimum EE performance of Algorithm 1 as a function of the . . . Initial point 1 Initial point 2Iteration index EE O b j ec ti v e ( n a t s / s / J ) Algorithm 1 (27)Algorithm 1 (DoC) (a) Convergence over a channel realization
20 60 100 140 18000 . . . . Number of iterations C D F Algorithm 1 (27)Algorithm 1 (DoC) (b) CDF of the number of requirediterations to converge.Fig. 3. Impact of approximation functions (27) and DoC on the convergence of Algorithm 1. We take K = 3 , L = 9 and ¯ P = 33 dBm.
27 30 33 36 3900 . . . . P (dBm) A c h i e v e d m i n i m u m EE ( n a t s / s / J ) Algorithm 1Baseline 1-OWBaseline 2-OWBaseline 3-OW
Fig. 4. Achieved EE versus the transmit power ¯ P with K = 3 and L = 9 . maximum user power ¯ P . For comparison purpose, we also provide the performance of threebaseline schemes: in the first scheme, namely ‘Baseline 1-OW’, the transmit power of the usersare fixed at ¯ P ; in the second scheme, namely ‘Baseline 2-OW’, EH time is fixed as τ = ; inthe third scheme, namely ‘Baseline3 -OW’, the users’ transmit power and EH time are fixed at ¯ P and τ = , respectively. For the three baseline schemes, it may happen that feasible resourceallocation cannot be obtained for some channel realizations. Thus, we set the performance ofthose infeasible channels as zero. The first observation is that the performance of Algorithm 1decreases when ¯ P increases. This result can be explained as follows. In an EE problem, thetransmit power may be smaller than the threshold, especially when the threshold is relativelylarge. For this case, increasing ¯ P brings no benefit to the optimizing of the transmit power.On the other hand, as shown in (8), both ¯ P and the optimized transmit power influence thePA efficiency. And increasing ¯ P reduces the PA efficiency, leading to more amount of energyconsumed at the PA as can be seen in (9). Another interesting observation is that the EEs
27 30 33 36 3900 . . . . P (dBm) A c h i e v e d m i n i m u m EE ( n a t s / s / J ) Algorithm 1Baseline 4Baseline 5
Fig. 5. Achieved minimum EE and versus the transmitpower ¯ P with K = 3 and L = 9 . − − − ¯ ρ (cid:0) W/(nats/s) (cid:1) A c h i e v e d m i n i m u m EE ( n a t s / s / J ) Algorithm 1Baseline 6
Fig. 6. Achieved minimum EE versus rate-dependentpower coefficient ¯ ρ with K = 3 and L = 9 . of the three baseline schemes first increase, and then decrease as ¯ P increases. The reason isthat the probabilities of infeasibility of these schemes are high when ¯ P is small. When ¯ P becomes larger, the infeasibility probabilities are smaller leading to the improved performances.When the probabilities of infeasibility are small enough, further increasing ¯ P leads to thedegraded performances due to the decrease of PA efficiency. As expected, our proposed schemeoutperforms the baseline ones.In Fig. 5, we show the impacts of PA and EH models on the minimum EE performance.For this purpose, we consider the following schemes: the first scheme, named as ‘Baseline 4’,considers linear model of PA efficiency where the efficiency is fixed at 0.35. The the secondscheme, named as ‘Baseline 5’, adopts the linear EH model with constant conversion efficiency0.8. The performances of these schemes are obtained as follows. First, the design parametersare determined by suitably modifying Algorithm 1 corresponding to the considered models.From the achieved values, the minimum EE is recalculated following the PA and EH modelsconsidered in Section II. If there is infeasibility, the corresponding minimum EE is set as zero.The figure clearly shows that PA and EH models have significant influence on the performance.Similarly to Baselines 1, 2 and 3 (in Fig. 3), the performances of Baseline 4 and Baseline 5 areinferior when ¯ P is small due to high probability of infeasibility. The performance degradationsof Baselines 4 and 5 are mainly because of the mismatch between the baseline schemes and therealistic models. The results again confirm the validity of our proposed scheme.To investigate the impacts of rate-dependent signal processing power (RSPP) on the minimumEE performance, we let the rate-dependent-power coefficients in each pair be different from thatof other pairs by simply setting as ρ en k = ρ de k = ω k ¯ ρ where ω k = k , and plot the performance asa function of ¯ ρ in Fig. 6. Here, the compared scheme, namely ‘Baseline 6’, takes ρ en ik = ρ de ik = 0 ,
30 (dBm) 33 (dBm) 36 (dBm) . . A vg . p e r- u s e r- p a i r EE ( n a t s / s / J ) User pair-1 User pair-2 User pair-3 (a) Average individual EE of user pairs. . . . . . . . Fairness index C D F ¯ P = 30 (dBm) ¯ P = 33 (dBm) ¯ P = 36 (dBm) (b) CDF of Jain’s fairness index.Fig. 7. EE fairness among the user pairs achieved by Algorithm 1 with K = 3 , and L = 12 . and its performance is obtained similarly as that of Baseline 4 and Baseline 5 (in Fig. 5). Weobserve that RSPP has insignificant influence on the performance when its coefficients are small.However, when the coefficients becomes larger, the gap between Algorithm 1 and Baseline 6 isremarkable.Fig. 7 shows the EE fairness among user pairs versus different values of ¯ P . In particular, theaverage individual EE of the user pairs is plotted in Fig. 7(a), and the CDFs of Jain’s fairnessindex [49] are shown in Fig. 7(b). It can be observed that the achieved EE is relatively balancedamong all user pairs. On the other hand, the algorithm achieves absolute fairness in more than90% of channel realizations in all considered cases of ¯ P . B. Performances of Algorithm 2 (Two-Way Relaying)
In Fig. 8, we evaluate the performances of Algorithm 2 in terms of convergence and minimumEE. Specifically, Fig. 8(a) plots the convergence behavior of the algorithm over a random channelrealization with two different initial points also generated randomly. Compared to Algorithm 1,Algorithm 2 likely requires more iterations to converge. This can be intuitively explained by theinter-pair interference in two-way relaying systems which is more difficult to manage than that inone-way relaying systems due to the bi-directional transmission. Fig. 8(b) illustrates the averageachieved minimum EE of Algorithm 2 versus the maximum transmit power ¯ P . We compareAlgorithm 2 with the three schemes, Baseline 1-TW, Baseline 2-TW and Baseline 3-TW whichare set up similarly to Baseline 1-OW, Baseline 2-OW and Baseline 3-OW in Fig. 3. Again, we According to [49], let us denote [ EE ∗ , . . . EE ∗ K ] as the individual EEs of the user pairs, then the fairness index is given as:fairness = (cid:0) P Kk =1 EE ∗ k (cid:1) K P Kk =1 ( EE ∗ k ) . Obviously, when EE ∗ = . . . = EE ∗ K , fairness = 1 which implies an absolute fairness. . . . . . Iteration index EE O b j ec ti v e ( n a t s / s / J ) Init. point 1Init. point 2 (a) Convergence of Algorithm 2 for one channel realizationwith ¯ P = 33 dBm.
27 30 33 36 390 . . . . P (dBm) A c h i e v e d m i n i m u m EE ( n a t s / s / J ) Algorithm 2Baseline 1-TWBaseline 2-TWBaseline 3-TW (b) Achieved minimum EE performance versus the transmitpower ¯ P .Fig. 8. Performances of Algorithm 2 with L = 12 .
30 (dBm) 33 (dBm) 36 (dBm) . . . . A vg . p e r- u s e r- p a i r EE ( n a t s / s / J ) User pair-1 User pair-2 User pair-3 (a) Average individual EE of user pairs. .
94 0 .
96 0 . . . . . Fairness index C D F ¯ P = 30 (dBm) ¯ P = 33 (dBm) ¯ P = 36 (dBm) (b) CDF of Jain’s fairness indexFig. 9. EE fairness among the user pairs achieved by Algorithm 2 with K = 3 and L = 12 . observe that the proposed scheme outperforms the others. On the other hand, for Algorithm 2,we can see that in the region of limited user power, the EE increases when ¯ P increases. Thisis because the effect of the gain from the additional power resource is stronger than that of thedecrease because of PA efficiency. When ¯ P is large, an increase of ¯ P has insufficient influence,and thus the performance reduces with ¯ P .In Fig. 9, we plot the individual EE performances of all user pairs (Fig. 9(a)) and the CDF offairness index (Fig. 9(b)) versus different value of ¯ P . Similar to the observation in Fig. 7, theproposed EE method for two-way relaying is able to maintain the good EE fairness among alluser pairs. C. Performance of ZF-Based Designs
In the following set of numerical experiments, we investigate the performances of ZF-baseddesigns (presented in Section IV) in terms of the minimum EE and computational complexity.
27 30 33 36 390 . . . L = 9 L = 7¯ P (dBm) A c h i e v e d m i n i m u m EE ( n a t s / s / J ) Algorithm 1ZF-based design (OW) (a) Average minimum EE of Algorithm 1 and ZF-basedscheme in the one-way relaying system.
27 30 33 36 390 . L = 13 L = 15 ¯ P (dBm) A c h i e v e d m i n i m u m EE ( n a t s / s / J ) Algorithm 2ZF-based design (TW) (b) Average minimum EE of Algorithm 2 and ZF-basedscheme in the two-way relaying system.Fig. 10. Average minimum EE performances versus ¯ P of Algorithms 1, 2, and ZF-based schemes.Table IIIS OLVER RUN TIME ( IN SECONDS ) FOR A LGORITHMS
1, 2,
AND THE
ZF-
BASED SCHEMES WITH K = 3 AND ¯ P = 33 D B M . L = 7 L = 8 L = 9 L = 13 L = 14 L = 15 Per-iteration run time Algorithm 1 0.039 0.050 0.056 Algorithm 2 0.12 0.14 0.16ZF-based (OW) 0.019 0.028 0.037 ZF-based (TW) 0.021 0.026 0.030Total run time Algorithm 1 2.24 2.53 2.40 Algorithm 2 9.97 10.16 10.36ZF-based (OW) 0.56 0.82 1.08 ZF-based (TW) 0.96 1.16 1.28
Fig. 10 shows the minimum EE performances of the considered schemes. In particular, Fig.10(a) plots the performances of Algorithm 1 and ZF-based design (OW) in the one-way relayingsystem, while Fig. 10(b) plots the performances of Algorithm 2 and ZF-based design (TW) inthe two-way relaying system. We can observe that the performances of ZF-based schemes areinferior to Algorithms 1 and 2 when L is small, and comparable, when L is sufficiently large.The results are because the ZF beamforming needs a certain number of relays to form the nullspace.To investigate the computational complexity of ZF-based schemes, we plot in Fig. 11 the CDFsof the required number of iterations for convergence of the considered schemes, and providethe corresponding solver running time in Table 2. It can be observed that the ZF-based schemesrequire smaller numbers of iterations to converge compared to Algorithms 1 and 2. In addition,the solver requires less time to solve convex subproblems in ZF-based schemes. Consequently,the total running time of the ZF-based schemes is remarkably smaller than that of Algorithms1 and 2. Combining with the results in Fig. 10, we can conclude that, when L is large, efficientsolutions can be achieved with low computational cost by using the ZF-based schemes.
20 40 60 8000 . . . . Number of iterations C D F Algorithm 1 ( L = 7 )ZF-based design (OW) ( L = 7 )Algorithm 1 ( L = 9 )ZF-based design (OW) ( L = 9 ) (a) CDF of the number of required iterations for convergenceof Algorithms 1 and ZF-based scheme in one-way relayingsystem.
20 40 60 8000 . . . . Number of iterations C D F Algorithm 2 ( L = 13 )ZF-based design (TW) ( L = 13 )Algorithm 2 ( L = 15 )ZF-based design (TW) ( L = 15 (b) CDF of the number of required iterations for convergenceof Algorithms 2 and ZF-based scheme in two-way relayingsystem.Fig. 11. The number of required iterations for convergence of Algorithms 1, 2, and ZF-based schemes. VII. C
ONCLUSION
We studied a multipair relay system where the relays harvest energy from user RF signals.We considered an energy consumption model, which accounts various realistic aspects suchas rate-dependent signal processing power, dynamic power amplifier efficiency, and nonlinearEH circuits. We have investigated the problem of max-min EE fairness among user pairs byjointly designing the transmit data rate, users’ transmit power, relays’ processing coefficient,and EH time. For both one-way and two-way relaying, we have derived iterative proceduresbased on the IA optimization framework, where each iteration only deals with an SOCP. Theproposed methods are provably convergent. In addition, for low-complexity designs, we haveproposed an approach based on a combination of ZF beamforming and IA. The effectiveness ofour approaches has been demonstrated by the numerical results.A
PPENDIX
A. Problem Equivalence
We justify the optimal equivalence between (43) and EEF-OW as follows. Let us denote ψ ∗ as the optimal solution of (43) and define ˆ k , arg max k ∈K { ˆ η k } where ˆ η k , ρ sp k + P ′ k r ∗ k + ˜ τ ∗ r ∗ k (cid:0) ε k √ q ∗ k + P ′′ k (cid:1) . We remark that constraints in (43e) hold with equality at the optimum followingthe epigraph transformation. Thus it is sufficient to show that: (i) ˆ η ˆ k is the optimal solution of(43), i.e., ˆ η ˆ k = η ∗ , and (ii) (34) and (6) with respect to user pair ˆ k hold with equality at theoptimum. In these regards, (43) and EEF-OW obtain the same optimal values of ( w ∗ , r ∗ , τ ∗ , p ∗ ) as can be seen by constraints in (43e). Thereby, we achieve f EE,OW ˆ k ( τ ∗ , p ∗ , r ∗ ) = (cid:16) ρ sp k + P ′ ˆ k r ∗ k + ˜ τ ∗ r ∗ k (cid:0) ε k √ q ∗ k + P ′′ ˆ k (cid:1)(cid:17) − = η ∗ which implies the equivalence between (43) and EEF-OW.We now show (i). It is immediately seen that (40) holds at the optimum for ˆ k and ˆ η ˆ k = η ∗ . Thisis because otherwise ˆ η ˆ k < η ∗ which means that η ∗ is not the optimum. Next, we prove (ii). Let usconsider problem (43) and suppose, to the contrary, that (34) does not hold at the optimum for ˆ k .Then, we can scale up r ∗ k by a positive-scaling factor λ > such that ˆ r k , λr ∗ k = log(1 + v ∗ k ) .And, we can easily check that new value ˆ r k is still feasible to (43). However, substituting ˆ r k to (43) results in a strictly smaller objective, i.e., ρ sp k + P ′ ˆ k ˆ r k + ˜ τ ˆ r k (cid:0) ε k √ q k + P ′′ ˆ k (cid:1) = ˆ η k < η ∗ . Thiscontradicts to the fact ˆ η k = η ∗ at the optimum. Similarly, we can argue that (6) with respectto ˆ k holds with equality at the optimum of EEF-OW. This accomplishes (ii) and completes theproof. B. Convexity of Function x / √ y We show that the function is strictly convex over x > , y > via the second-order condition.The Hessian of the function is A = h / √ y − x/y / − x/y / x / y / i . Then we have [ v v ] A [ v v ] T = 2 v √ y − xv v y / + 3 x v y / = 2 √ y ( v − xv y ) + x v y / > for all non-zero vector [ v v ] , i.e., A is positive definite.It is interesting that the constraint x / √ y ≤ t can be equivalently represented by two SOCs as x / √ y ≤ t ⇔ k [2 x, t − v ] k ≤ t + v k [2 y, v − k ≤ v + 1 . R EFERENCES [1] D. Feng, C. Jiang, G. Lim, J. Cimini, L. J., G. Feng, and G. Li, “A survey of energy-efficient wireless communication,”
IEEE Commun. Surveys Tuts. , vol. 15, no. 1, pp. 167–178, Feb. 2013.[2] L. Sanguinetti, A. A. D’Amico, and Y. Rong, “A tutorial on the optimization of amplify-and-forward MIMO relay systems,”
IEEE J. Sel. Areas Commun.
IEEE Commun. Mag. , vol. 52, no. 5, pp. 86–92, May 2014.[5] Q. Li, R. Q. Hu, Y. Qian, and G. Wu, “Cooperative communications for wireless networks: techniques and applicationsin LTE-advanced systems,”
IEEE Wireless Commun. , vol. 19, no. 2, April 2012. [6] B. Rankov and A. Wittneben, “Spectral efficient protocols for half-duplex fading relay channels,” IEEE J. Sel. AreasCommun. , vol. 25, no. 2, pp. 379–389, Feb. 2007.[7] S. Fazeli-Dehkordy, S. Shahbazpanahi, and S. Gazor, “Multiple peer-to-peer communications using a network of relays,”
IEEE Trans. Signal Process. , vol. 57, no. 8, pp. 3053–3062, Aug. 2009.[8] J. Joung and A. H. Sayed, “Multiuser two-way amplify-and-forward relay processing and power control methods forbeamforming systems,”
IEEE Trans. Signal Process. , vol. 58, no. 3, pp. 1833–1846, March 2010.[9] Z. Ding, W. H. Chin, and K. K. Leung, “Distributed beamforming and power allocation for cooperative networks,”
IEEETrans. Wireless Commun. , vol. 7, no. 5, pp. 1817–1822, May 2008.[10] C. Huang, R. Zhang, and S. Cui, “Throughput maximization for the Gaussian relay channel with energy harvestingconstraints,” vol. 31, no. 8, pp. 1469–1479, August 2013.[11] Z. Ding, S. M. Perlaza, I. Esnaola, and H. V. Poor, “Power allocation strategies in energy harvesting wireless cooperativenetworks,”
IEEE Trans. Wireless Commun. , vol. 13, no. 2, pp. 846–860, February 2014.[12] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and informationprocessing,”
IEEE Trans. Wireless Commun. , vol. 12, no. 7, pp. 3622–3636, July 2013.[13] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RF energy harvesting: A contemporarysurvey,”
IEEE Commun. Surveys Tuts. , vol. 17, no. 2, pp. 757–789, Secondquarter 2015.[14] Ericsson White Paper, “5G energy performance,” April 2015.[15] A. Zappone, P. Cao, and E. A. Jorswieck, “Energy efficiency optimization in relay-assisted MIMO systems with perfectand statistical CSI,”
IEEE Trans. Signal Process. , vol. 62, no. 2, pp. 443–457, Jan. 2014.[16] F. Héliot and R. Tafazolli, “Optimal energy-efficient source and relay precoder design for cooperative MIMO-AF systems,”
IEEE Trans. Signal Process. , vol. 66, no. 3, pp. 573–588, Feb. 2018.[17] J. Zhang and M. Haardt, “Energy efficient two-way non-regenerative relaying for relays with multiple antennas,”
IEEESignal Process. Lett. , vol. 22, no. 8, pp. 1079–1083, Aug. 2015.[18] Z. Sheng, H. D. Tuan, T. Q. Duong, and H. V. Poor, “Joint power allocation and beamforming for energy-efficient two-waymulti-relay communications,”
IEEE Trans. Wireless Commun. , vol. 16, no. 10, pp. 6660–6671, Oct. 2017.[19] C. Isheden and G. P. Fettweis, “Energy-efficient multi-carrier link adaptation with sum rate-dependent circuit power,” in , Dec. 2010, pp. 1–6.[20] D. Persson, T. Eriksson, and E. G. Larsson, “Amplifier-aware multiple-input multiple-output power allocation,”
IEEECommun. Lett. , vol. 17, no. 6, pp. 1112–1115, Jun. 2013.[21] O. Tervo, A. Tölli, M. Juntti, and L. N. Tran, “Energy-efficient beam coordination strategies with rate-dependent processingpower,”
IEEE Trans. Signal Process. , vol. 65, no. 22, pp. 6097–6112, Nov. 2017.[22] O. Tervo, L. N. Tran, and M. Juntti, “Energy-efficient joint transmit beamforming and subarray selection with non-linearpower amplifier efficiency,” in
IEEE GlobalSIP , Dec. 2016, pp. 763–767.[23] Y. Cheng and M. Pesavento, “Joint optimization of source power allocation and distributed relay beamforming in multiuserpeer-to-peer relay networks,”
IEEE Trans. Signal Process. , vol. 60, no. 6, pp. 2962–2973, June 2012.[24] C. Wang, H. M. Wang, D. W. K. Ng, X. G. Xia, and C. Liu, “Joint beamforming and power allocation for secrecy inpeer-to-peer relay networks,”
IEEE Trans. Wireless Commun. , vol. 14, no. 6, pp. 3280–3293, June 2015.[25] M. Tao and R. Wang, “Linear precoding for multi-pair two-way MIMO relay systems with max-min fairness,”
IEEE Trans.Signal Process. , vol. 60, no. 10, pp. 5361–5370, Oct. 2012.[26] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Wireless-powered relays in cooperative communications: Time-switching relaying protocols and throughput analysis,”
IEEE Trans. Wireless Commun. , vol. 63, no. 5, pp. 1607–1622,May 2015. [27] Y. Huang and B. Clerckx, “Relaying strategies for wireless-powered MIMO relay networks,” IEEE Trans. WirelessCommun. , vol. 15, no. 9, pp. 6033–6047, Sept. 2016.[28] Z. Chen, B. Xia, and H. Liu, “Wireless information and power transfer in two-way amplify-and-forward relaying channels,”in
IEEE GlobalSIP , Dec 2014, pp. 168–172.[29] Y. Liu, “Wireless information and power transfer for multirelay-assisted cooperative communication,”
IEEE Commun. Lett. ,vol. 20, no. 4, pp. 784–787, April 2016.[30] S. Salari, I. M. Kim, D. I. Kim, and F. Chan, “Joint EH time allocation and distributed beamforming in interference-limitedtwo-way networks with EH-Based relays,”
IEEE Trans. Wireless Commun. , vol. 16, no. 10, pp. 6395–6408, Oct. 2017.[31] F. Tan, T. Lv, and S. Yang, “Power allocation optimization for energy-efficient massive MIMO aided multi-pair decode-and-forward relay systems,” vol. 65, no. 6, pp. 2368–2381, June 2017.[32] C. Zhang, H. Du, and J. Ge, “Energy-efficient power allocation in energy harvesting two-way AF relay systems,”
IEEEAccess , vol. 5, pp. 3640–3645, March 2017.[33] Q. Cui, Y. Zhang, W. Ni, M. Valkama, and R. Jäntti, “Energy efficiency maximization of full-duplex two-way relaywith non-ideal power amplifiers and non-negligible circuit power,”
IEEE Trans. Wireless Commun. , vol. 16, no. 9, pp.6264–6278, Sept. 2017.[34] Q. Cui, T. Yuan, and W. Ni, “Energy-efficient two-way relaying under non-ideal power amplifiers,”
IEEE Trans. Veh.Technol. , vol. 66, no. 2, pp. 1257–1270, Feb. 2017.[35] E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Practical non-linear energy harvesting model and resourceallocation for SWIPT systems,”
IEEE Commun. Lett. , vol. 19, no. 12, pp. 2082–2085, Dec 2015.[36] Q.-D. Vu, L.-N. Tran, M. Juntti, and E.-K. Hong, “Energy-efficient bandwidth and power allocation for multi-homingnetworks,”
IEEE Trans. Signal Process. , vol. 63, no. 7, pp. 1684–1699, Apr. 2015.[37] A. Beck, A. Ben-Tal, and L. Tetruashvili, “A sequential parametric convex approximation method with applications tononconvex truss topology design problem,”
J. Global Optim. , vol. 47, no. 1, pp. 29–51, 2010.[38] B. R. Marks and G. P. Wright, “A general inner approximation algorithm for nonconvex mathematical programs,”
OperationsResearch , vol. 26, no. 4, pp. 681–683, Jul.-Aug. 1978.[39] S. He, Y. Huang, S. Jin, and L. Yang, “Coordinated beamforming for energy efficient transmission in multicell multiusersystems,”
IEEE Trans. Commun. , vol. 61, no. 12, pp. 4961–4971, Dec. 2013.[40] A. Zappone and E. Jorswieck, “Energy efficiency in wireless networks via fractional programming theory,”
Foundationsand Trends in Communications and Information Theory , vol. 11, no. 3-4, pp. 185–396, 2015.[41] S. Mikami, T. Takeuchi, H. Kawaguchi, C. Ohta, and M. Yoshimoto, “An efficiency degradation model of power amplifierand the impact against transmission power control for wireless sensor networks,” in , 2007, pp. 447–450.[42] G. Auer, V. Giannini, C. Desset, I. Godor, P. Skillermark, M. Olsson, M. A. Imran, D. Sabella, M. J. Gonzalez, O. Blume,and A. Fehske, “How much energy is needed to run a wireless network?”
IEEE Wireless Commun. , vol. 18, no. 5, pp.40–49, Oct. 2011.[43] T. Lipp and S. Boyd, “Variations and extension of the convex–concave procedure,”
Optimization and Engineering , vol. 17,no. 2, pp. 263–287, 2016.[44] T. P. Dinh and H. A. L. Thi, “Recent advances in DC programming and DCA,”
Transactions on Computational IntelligenceXIII , vol. 8342, pp. 1–37, April 2014.[45] T. Dinh Quoc and M. Diehl, “Sequential Convex Programming Methods for Solving Nonlinear Optimization Problemswith DC constraints,”
ArXiv e-prints , Jul. 2011. [46] A. Ben-Tal and A. Nemirovski, Lectures on modern convex optimization . Philadelphia: MPS-SIAM Series on Optimization,SIAM, 2001.[47] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMOchannels,”
IEEE Trans. Signal Process.