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Energy efficiency maximization of fullduplex twoway relayassisted devicetodevice communications underlaying cellular networks
 Yiliang Chang^{1},
 Hongbin Chen^{1, 2}Email authorView ORCID ID profile and
 Feng Zhao^{1}
https://doi.org/10.1186/s1363801607212
© The Author(s). 2016
 Received: 3 July 2016
 Accepted: 6 September 2016
 Published: 17 September 2016
Abstract
With the substantial progress in selfinterference (SI) cancelation, the fullduplex (FD) technique, which allows the communication user to transmit and receive signals over the same frequency band simultaneously, enables a significant enhancement of spectral efficiency (SE) in comparison with the traditional halfduplex (HD) technique. Recently, relayassisted devicetodevice (D2D) communications underlaying cellular networks have aroused a great deal of research interests due to its high SE. For the new meaningful paradigm of the combination of the FD and the amplifyandforward (AF) relayassisted D2D communications, analyzing the SE and energy efficiency (EE) is crucial, which have not been investigated in the existing works. In this paper, we focus on the EE of the FD twoway (FDTW) relayassisted D2D communications with uplink channel reuse by considering the residual SI at the D2D users and compare it with the HD counterpart. Our goal is to find the optimal transmission powers and amplification gain to maximize the system EE while guaranteeing SE requirements and maximum transmission power constraints. A new twotier alternative iteration optimization algorithm is proposed to solve the optimization problem. Simulation results show that (1) the results obtained by the proposed algorithm is very close to those obtained by the exclusive searching method, (2) smaller residual power of SI leads to better performance of EE and SE, (3) the SE obtained by FDTW relayassisted D2D networks is higher than the SE obtained by the HD counterpart, and (4) the EE comparison of FDTW relayassisted D2D networks and its HD counterpart depends on the residual power of SI. The EE obtained by FDTW relayassisted D2D is higher than the EE obtained by HD counterpart only when the residual power of SI is sufficiently small.
Keywords
 Relayassisted devicetodevice (D2D) communications
 Fullduplex (FD)
 Selfinterference (SI)
 Energy efficiency (EE)
 Spectral efficiency (SE)
1 Introduction
It is expected that mobile data traffic will increase a thousandfold over the next decade which will be driven by the expected 50 billion mobile devices connected to the cloud, anywhere and anytime. With a rapid increase in the number of connected devices, some challenges for next 5G networks appear which will be responded by increasing capacity and by improving energy efficiency (EE), coverage, spectrum utilization, and so on [1]. As the key techniques to address the requirements for 5G networks, direct devicetodevice (D2D) communication and inband fullduplex (FD) have attracted a great deal of research interests [2–4].
D2D communications allow proximate cellular users to communicate with each other directly under the control of base station (BS) with lower transmit power requirements. The high channel quality of shortrange D2D links facilitate high data rates for local services, prolong users’ battery lives, and offload heavy traffic of BS [5]. In addition, D2D links can underlay cellular links by reusing the same time and frequency resources; thus, the spectral efficiency (SE) of cellular networks can be improved obviously. However, the introduction of D2D communications into cellular networks poses new challenge in the resource allocation design due to cochannel interference caused by spectrum reuse, which has marked impacts on the performance of communication reliability and network throughput. Thus, effective power allocation is required to alleviate the interference problem. Besides, another efficient way to reduce the impact of interference is providing a relay user between D2D pair [6]. We refer to this as relayassisted D2D communications which can be an efficient approach to provide a better Quality of Service (QoS) and lower transmission powers for communication between distant D2D users.
Relay communication, in which the relay forwards the signal received by a source to a destination, has been envisaged as a spectral and energyefficient technology for cellular networks [7]. Based on the directions of signal transmission, relaying systems can be classified into three transmission types (i.e., oneway, twoway, and multiway). Compared with oneway relaying, twoway relaying can achieve higher SE, which allows a relay user to simultaneously communicate with two end users. Existing studies also show that the twoway relayassisted D2D communications have significant improvement on system performance (e.g., outage probability, system sum rate) [8, 9]. Thus, with the assistance of twoway relaying, the performance advantage of D2D communications can be further improved. Multiway relaying, in which multiple users exchange information with the assistance of a relay, can achieve better performance and flexibility than twoway relaying. However, its protocol and encoder design are rather complicated, making it a potential choice for relayassisted D2D communications in the future.
On the other hand, the inband FD transmission, which allows transmitting and receiving at the same frequency and the same time, enables an enormous enhancement of SE than halfduplex (HD) [10–12]. The main limitation impacting FD transmission is the strong selfinterference (SI) signal imposed by the transmit antenna on the receive antenna within the same transceiver. Both academia and industry reached consensus that SI cancelation would play the most pivotal role in implementing FD communication systems. There are usually two main categories: passive suppression (PS) and active cancelation [13]. Therein, the active cancelation also includes analog cancelation (AC) and digital cancelation (DC). Since none of the individual cancelation techniques is capable of satisfying the system requirements in terms of the attainable SI cancelation capability, a highcapability cancelation scheme by combining the active and passive techniques is necessarily developed. Among all the SI cancelation techniques, the PS has an important position. The primary advantage to performing PS in the propagation domain is that the downstream receiver hardware can process signals more accurately with a large dynamic range [10]. In addition, DC techniques can be performed in the digital domain, which are the lowestcomplexity active SI cancelation techniques in FD systems [10, 14]. As the design of AC usually requires a complex and largesize hardware circuits with respect to mobile devices [11, 14, 15], a combination of the latest PS and DC (referred to as PSDC) techniques will be an appropriate way to implement the mobile FD system [13, 14].
Hence, it can be practically beneficial applying the FD techniques into twoway relayassisted D2D communications with certain SI cancelation where the relay user and D2D users can transmit and receive signals simultaneously, and thus, the SE gets further improved. Since FD transmission possesses strong SI, it generally consumes more power than HD. Therefore, there is an urgent demand for maintaining high system throughput while limiting energy consumption. EE, defined as the ratio of throughput to power consumption, is an important measure of green communication solutions [7]. It is urgent and interesting to investigate the EE of FD twoway (FDTW) relayassisted D2D communications underlaying cellular networks.
1.1 Related work
In the existing literature, many works have been done on the EE of D2D communications underlaying cellular networks. For example, the EE of D2D communications without channel reuse was investigated in [16]. The authors showed the EE improvement with the deployment of D2D communications in heterogeneous networks (HetNets) compared with the full smallcell deployment, thus provided a greener alternative to cellular network deployment. Correspondingly, [17] and [18] presented the analysis of the EE of D2D systems with cellular channel reuse. Therein, [17] revealed the tradeoff between EE and delay in D2D communications, where stochastic traffic arrivals and timevarying channel conditions were both considered. Given the SE requirement and maximum transmission power constraints, the authors in [18] investigated the EE and SE, in which each user is selfinterested and wants to maximize its own EE. Inspired by the performance gain of a combination of relay and D2D communications, the authors in [19] extended radio resource management algorithms into oneway relayassisted D2D communications to balance SE and EE while considering mode selection and resource allocation constraints. In [6], the maximum achievable transmission capacity was studied in oneway relayassisted D2D communications while guaranteeing the outage probability of both cellular and D2D links. For a higher performance improvement, twoway relaying with network coding technique which can improve transmission efficiency will be a promising choice in D2D communications. The authors in [20] demonstrated the performance gain of D2D communications assisted by twoway relaying over Rayleigh fading channels. The achievable rate of different twoway transmission schemes of relayassisted D2D communications were presented in [9], while considering the interference due to spectrum sharing. Compared with the traditional cellular communications through BS, the EE and SE of twoway relayassisted D2D communications were studies in [21]. The optimal user equipment (UE) transmission powers to maximize the EE were also derived. However, there is little work addressing the EE in FDTW relayassisted D2D communications underlaying cellular networks, which is the focus of this paper.
1.2 Summary

To the best of our knowledge, this is the first work to investigate the EE of FDTW relayassisted D2D communications underlaying cellular networks. Moreover, the comparison between the FDTW relayassisted D2D network and the HD twoway (HDTW) relayassisted D2D network in terms of SE and EE is made. Finally, we demonstrate that the proposed twotier alternative iteration optimization algorithm can converge very quickly.

From the perspective of practical implementation, SI cannot be fully canceled due to technical challenges. Therefore, we consider remaining SI after cancelation at the FD users in a more reasonable and relatively tractable way to study EE and SE in the FDTW relayassisted D2D communications.

The joint optimization problem is nonconvex. To solve this issue, the primal optimization problem is first decomposed into two subproblems, and then, we solve the two subproblems alternatively. For the first subproblem, we prove it is convex and then solve it by the bisection method. For the second subproblem, we decomposed it into two subsubproblems again, and then, we solve the two subsubproblems alternatively. For the first subsubproblem, we transform it into concave function by using the nonlinear fractional programming, which can be effectively solved by the Dinkelbach method [22]. For the second subsubproblem, we first prove it is convex and then solve it by the bisection method.

A full range of power consumption sources is considered in the power consumption model. In particular, the power consumed by SI cancelation is considered. The impact of imperfect SI cancelation is also considered in the analysis.
The rest of this paper is organized as follows. Section 2 describes the system model and optimization problem for FDTW relayassisted D2D communications underlaying cellular networks. The elaborate solving process of the optimization problem is presented in Section 3. In Section 4, simulation results are presented. Finally, some concluding remarks are given in Section 5.
2 System model
D2D transmissions will cause interference to the BS receiver, and D2D receivers also suffer interference from the cellular uplink transmission. Moreover, the users in FD D2D link transmit and receive at the same time and frequency, causing SI to the receivers from its transmitters. It can be suppressed by the methods in [13, 14], which is beyond the scope of this paper. However, due to the practical constraint, the residual SI after interference suppression still exists and limits the performance of FDTW relayassisted D2D network, which becomes the main concern in this paper. Hence, in order to analyze the effect of SI on system performance, we assume the SI in the FDTW relayassisted D2D communications can be reduced but cannot be completely eliminated by SI cancelation techniques. The existence of SI will make the optimization problem formulation complicated as shown later. Let ρ denotes the average residual power of SI after cancelation at each user in D2D link. Based on the experimental results in [23], the residual SI power model general includes two cases: the complicated case that SI power increase linearly with the transmission power [24], and the other one in which the SI power is a constant and not a function of the transmission power [25]. Similar to our previous work in [26], we consider the second case that SI power is modeled as an invariable value. There are two main reasons for us to employ this model. One is that if ρ is regarded as a function of the transmission power of the D2D users, both the primal problem (16) and the transformed subproblem (P2) (defined in Sections 2 and 3.2, respectively) may become nonconvex. The second reason is that the power of SI can be controlled in a range after cancelation [27]. Thus, it is reasonable to regard ρ to be fixed. The issue that ρ may be associated with the transmission power of D2D users will be considered in our future work. The SI channel coefficients at SUE, RUE, and DUE are denoted by h _{ ss }, h _{ rr }, and h _{ dd } respectively, which are generally modeled by the zeromean circularly symmetric complex Gaussian (CSCG) random variables.
For a more concise expression, the channel gain between the transmitter i and j is denoted as G _{ ij } = h _{ ij }^{2}, where h _{ ij } is CSCG random variable with variance \( {\sigma}_{ij}={d}_{ij}^{v} \). Herein, d _{ ij } is the normalized distance between nodes i and j, and v is the pathloss exponent.
The power consumption in a general wireless node for communications usually contains two main parts: power amplifier and circuit power [18]. Power amplifier is related to the power amplifier drain efficiency and the transmission power level and is usually modeled as the ratio of the transmission power to the power amplifier drain efficiency [10]. Circuit power consumption is usually considered to be independent of the data rate and is regarded as a constant [29]. We consider both power amplifier and circuit power consumption in the transmitters. In addition, to be more practical, the circuit power consumption in the receivers is also considered. In a practical system, the circuit power consumption of BS in uplink is huge that can be supported by various efficient energy sources including renewable energy. Therefore, the EE of BS would be less critical as compared to that in the users since mobile devices are typically handheld devices with limited battery life and can quickly run out of battery. So, it is reasonable to neglect the circuit power consumption of BS in uplink in the system EE [17, 18]. In addition, in FD communications, applying PS generally does not consume additional power; however, the power consumed by active DC is nonnegligible. The power consumed by DC, P _{DC}, is regarded as a constant due to the power consumed by the involved chip components which are not related to the throughput state.
3 Energy efficiency maximization problem
In this section, we formulate the EE optimization problem for FDTW relayassisted D2D communications underlaying the cellular networks. Specifically, we will seek the optimal transmission powers P _{ s }, P _{ d }, P _{ u }, and the optimal relay amplification gain β to maximize EE while keeping SE above a threshold and guaranteeing transmission power constraints. Unfortunately, joint optimization over P _{ s }, P _{ d }, P _{ u }, and β is very hard to be found due to that η _{EE} is not concave in P _{ s }, P _{ d }, P _{ u }, and β jointly. So, it cannot be solved by the general convex optimization methods. To overcome this difficulty, a new twotier alternative iteration optimization algorithm is proposed in this section. The basic idea is to first optimize the EE of the cellular link by the bisection method and the FDTW relayassisted D2D link with iterative optimization algorithm, respectively, and then alternately iterate the cellular link and the FDTW relayassisted D2D link until convergence to produce the optimal system EE.
3.1 EE maximization of cellular link with fixed P _{ s }, P _{ d }, and β
It is easy to prove that η _{ u,SE} is a monotonically increasing function of P _{ u } ∈ [0, + ∞), and according to the restricted conditions of η _{ u,SE} (i.e., η _{ u,SE } ≥ 0, \( {\eta}_{u,SE}\ge {\overline{\eta}}_{u,SE} \)), we have \( {P}_u\in \left\{{P}_u\left{P}_u\in \right.\left({\overset{\smile }{P}}_u,+\infty \right)\right\} \), where \( {\overset{\smile }{P}}_u={G}_2\left({2}^{{\overline{\eta}}_{u,SE}}1\right)/{G}_{ub}\left({2}^{{\overline{\eta}}_{u,SE}}1\right){G}_1 \), G _{1} = βG _{ ur } G _{ rb }, and G2 = P _{ s } G _{ sb } + P _{ d } G _{ db } + β(P _{ s } G _{ sr } G _{ rb } + P _{ d } G _{ rd } G _{ rb } + ρ _{ r } G _{ rb } G _{ rr } + G _{ rb } σ ^{2}) + σ ^{2}.
Since P _{ u } ∈ [0, P _{ u,max}] and \( {P}_u\in \left\{{P}_u\left{P}_u\in \right.\left({\overset{\smile }{P}}_u,+\infty \right)\right\} \), the feasible region of the subproblem (P1) can be rewritten as \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, \max}\right] \).
Theorem 1
Given P _{ s }, P _{ d }, β ∈ [0, + ∞), η _{ u,EE} is strictly quasiconcave in P _{ u } for P _{ u } ∈ [0, + ∞).
Proof
Refer to Appendix 1.
Case 1: η _{ u,EE} strictly increases with P _{ u } for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), if \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left{}_{P_u={P}_{u, max}}\right.\ge 0 \), where \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u} \) is given by (29) in Appendix 1. In this case, the optimal solution to the subproblem (P1) is achieved at \( {\widehat{P}}_u={P}_{u, max} \).
Case 2: η _{ u,EE} strictly decreases with P _{ u } for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), if \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left{}_{P_u={\overset{\smile }{P}}_u}\right.\le 0 \). The optimal solution to the subproblem (P1) is achieved at \( {\widehat{P}}_u={\overset{\smile }{P}}_u \).
Case 3: η _{ u,EE} first strictly increases and then strictly decreases with P _{ u } for \( {P}_u\in \left[{\overset{\smile }{P}}_u,{P}_{u, max}\right] \), if \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left{}_{P_u={P}_{u, \max }}\right.<0 \) and \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}\left{}_{P_u={\overset{\smile }{P}}_u}\right.>0 \). The optimal solution to the subproblem (P1) is achieved at \( {\widehat{P}}_u={P_u}^{*} \), where P _{ u }* is the point at which η _{ u,EE} reaches its maximum for P _{ u } ∈ [0, + ∞) and is obtained by solving the equation \( \frac{d{\eta}_{u,\mathrm{E}\mathrm{E}}}{d{P}_u}=0 \), as shown in Appendix 1. The exact expression of P _{ u }* is not easily obtained and can be found by the numerical methods, such as the bisection method.
3.2 EE maximization of FDTW relayassisted D2D link with a fixed P _{ u }
In the suboptimization problem (P2), both η _{ d,EE} and η _{ d,SE} are nonconvex with respect to P _{ s }, P _{ d }, and β, so that directly solving problem (P2) introduces enormous computational complexity. To address this issue, the iterative optimization algorithm is adopted in this section. The basic idea is to first optimize the objective function over a portion of variables when the others are fixed and then joint optimize these variables by utilizing the separate optimization results. Specifically, we first optimize the problem (P2) over P _{ s } and P _{ d } with a fixed β and then optimize the problem (P2) over β with fixed P _{ s } and P _{ d }. This process will be repeated until convergence.
3.2.1 Optimization over P _{ s } and P _{ d } with a fixed β
Theorem 2
For any given β, η _{ d,SE} is a concave and strictly increasing function of P _{ s } and P _{ d } for P _{ s }, P _{ d } ∈ [0, + ∞).
Proof
Refer to Appendix 2.
Theorem 3
The maximum EE \( {q}_d^{*} \) is achieved if and only if \( {\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s,{P}_d\right){q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s,{P}_d\right)={\eta}_{d,\mathrm{S}\mathrm{E}}\left({P}_s^{\ast },{P}_d^{\ast}\right){q}_d^{\ast }{P}_{d,\mathrm{total}}\left({P}_s^{\ast },{P}_d^{\ast}\right)=0 \) and for a given q _{ d }, the transformed function in subtractive form is a concave function.
Proof
Refer to Appendix 3.
Notice that the optimal transmission power terms of SUE and DUE given by (23) and (24) are similar to traditional waterfilling solutions, where the heights of the pool are defined as φ _{ s }/η and φ _{ d }/η, and the water level is partially determined by λ _{1} and q _{ d }. Since both P _{ s } and P _{ d } involve the Lagrangian multiplier λ _{1} and the KKT conditions of (22) require λ _{1} to be nonnegative, we derive λ _{1} in the following two cases.
Case 1: λ _{1} > 0
Case 2: λ _{1} = 0
3.2.2 Optimization over β with fixed P _{ s } and P _{ d }
It is easy to prove that η _{ d,SE} is monotonically increasing function of β ∈ [0, + ∞), and according to the restricted conditions of η _{ d,SE} (i.e., η _{ d,SE } ≥ 0, \( {\eta}_{d,SE}\ge {\overline{\eta}}_{d,SE} \)), we have \( \beta \in \left\{\beta \left\beta \in \left[\infty, {\overset{\smile }{\beta}}^{\prime}\right]\right.\cup \left[\overset{\smile }{\beta },+\infty \right]\right\} \), where \( {\overset{\smile }{\beta}}^{\prime }=\left(b\sqrt{b^24ac}\right)/2a \), \( \overset{\smile }{\beta }=\left(b+\sqrt{b^24ac}\right)/2a \), a = c _{2} c _{3} + c _{1} c _{4} + c _{1} c _{3} − ϖc _{2} c _{4}, b = c _{3} σ _{1} + c _{1} σ _{2} − ϖc _{2} σ _{2} − ϖc _{4} σ _{1}, and c = − ϖσ _{1} σ _{2}. Since \( \beta \in \left[0,\overline{\beta}\right] \) and \( \beta \in \left\{\beta \left\beta \in \left[\infty, {\overset{\smile }{\beta}}^{\prime}\right]\right.\cup \left[\overset{\smile }{\beta },+\infty \right]\right\} \), the feasible region of the subproblem (P22) can be rewritten as \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \).
Theorem 4
Given P _{ s }, P _{ d } ∈ [0, + ∞), η _{ d,EE} is strictly quasiconcave in β for β ∈ [0, + ∞).
Proof
Refer to Appendix 4.
From Theorem 4, similar to the optimization problem of cellular link, there are only three cases for the curve η _{ d,EE} vs. β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \).
Case 1: η _{ d,EE} strictly increases with β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \), if \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left{}_{\beta =\overline{\beta}}\right.\ge 0 \), where \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta } \) is given by (42) in Appendix 4. In this case, the optimal solution to the subproblem (P22) is achieved at \( \widehat{\beta}=\overline{\beta} \).
Case 2: η _{ d,EE} strictly decreases with β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \), if \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left{}_{\beta =\overset{\smile }{\beta }}\right.\le 0 \). The optimal solution to the subproblem (P22) is achieved at \( \widehat{\beta}=\overset{\smile }{\beta } \).
Case 3: η _{ d,EE} first strictly increases and then strictly decreases with β for \( \beta \in \left[\overset{\smile }{\beta },\overline{\beta}\right] \) if \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left{}_{\beta =\overline{\beta}}\right.<0 \) and \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta}\left{}_{\beta =\overset{\smile }{\beta }}\right.>0 \). The optimal solution to the subproblem (P22) is achieved at \( \widehat{\beta}={\beta}^{*} \), where β* is the point at which η _{ d,EE} reaches its maximum for β ∈ [0, + ∞) and is obtained by solving the equation \( \frac{d{\eta}_{d,\mathrm{E}\mathrm{E}}}{d\beta }=0 \), as shown in Appendix 4. The exact expression of β* is not easily obtained and can be found by the numerical methods, such as the bisection method.
3.2.3 Iterative optimization algorithm for the FDTW relayassisted D2D link
3.3 Twotier alternative iteration optimization algorithm
Firstly, under the given initial transmission power of CUE, to solve the EE optimization problem of D2D link, the algorithm list in Table 2 was used. Then, according to the obtained optimal power values and amplification gain value of D2D link, to solve the cellular link EE optimization problem, the bisection method was used. Repeat this process until convergence is reached, i.e., \( \left{\eta}_{\mathrm{EE}}^{\ast (n)}{\eta}_{\mathrm{EE}}^{\ast \left(n1\right)}\right \) is equal or smaller than a predefined threshold Δ.
4 Simulation results
In this section, we evaluate the performance of the proposed algorithm by using the MATLAB tool. The maximum transmission powers of SUE, DUE, RUE, and CUE are set to P _{ s,max} = 1 W, P _{ d,max} = 1 W, P _{ r,max} = 1 W and P _{ u,max} = 1 W, respectively. The minimum data rate requirements of SUE, DUE, and CUE are set to R _{ s,min} = 1 bit/s/Hz, R _{ d,min} = 1 bit/s/Hz, and R _{ u,min} = 1 bit/s/Hz, respectively. The circuit power consumption is set to P _{ c } = 0.1 W, and the power amplifier efficiency of users is set to 1/ξ = 1/0.35. The power consumed by active DC in FD mode is set to 0.1 W. The parameters of pathloss exponent and noise power are set to 4 and −50 dBm, respectively. The bandwidth W is normalized to 1, i.e., W = 1 Hz. These values of simulation parameters and channel gains are inspired by [18] and [26].
The exhaustive searching method needs to evaluate all the possible values of P _{ s }, P _{ d }, β, P _{ u }, so it needs \( O\left(\frac{P_{s, max}{P}_{d, max}{P}_{u, max}}{\varepsilon^3}\times \frac{\overline{\beta}\overset{\smile }{\beta }}{\varepsilon}\right) \) steps to obtain the optimal value, where ε is the searching accuracy. In the proposed algorithm, both the bisection method and the Dinkelbach method are adopted alternatively, which takes \( O\left({M}_2\left({M}_1\left({M}_0+{ \log}_2\left(\left(\overline{\beta}\overset{\smile }{\beta}\right)/\varepsilon \right)\right)+{ \log}_2\left({P}_{u, max}/\varepsilon \right)\right)\right) \) steps, where M _{0}, M _{1}, and M _{2} are the average iterative steps of the Dinkelbach method, firsttier and secondtier, respectively, and are both set to 10 in our simulation. Therefore, the proposed algorithm significantly reduces the computational complexity compared with the exhaustive search method.
5 Conclusions
In this paper, the EE in FDTW relayassisted D2D communications has been studied. By taking the SI at D2D users into consideration, the optimization problem in which the goal is to maximize EE while satisfying the SE requirement and the transmission power constraints was formulated. Since the optimization problem (16) is nonconcave with respect to the variables P _{ u }, P _{ s }, P _{ d }, and β, we decomposed it into two subproblems, i.e., cellular link optimization problem and D2D link optimization problem. As for the cellular link, the bisection method is used to seek the optimal solution. As for the D2D link, an iterative optimization algorithm was used to find the optimal solution. Then, a new twotier alternative iteration optimization algorithm is proposed to solve the primal optimization problem. Simulation results show that the results of proposed algorithm are consistent with exhaustive searching results. The simulation results also indicate that smaller SI leads to higher SE and EE. Furthermore, the FDTW relayassisted D2D network can achieve higher SE than the HDTW relayassisted D2D network. However, the EE obtained by FDTW relayassisted D2D underlaying cellular networks is higher than the EE obtained by HD counterpart only when the residual power of SI is sufficiently small.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (61471135, 61671165), the Guangxi Natural Science Foundation (2013GXNSFGA019004, 2015GXNSFBB139007), the Fund of Key Laboratory Cognitive Radio and Information Processing, Guilin University of Electronic Technology, China, and Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing (CRKL150104, CRKL160105), the Innovation Project of Guangxi Graduate Education (YCSZ2015144), and the Innovation Project of GUET Graduate Education (2016YJCX91).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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