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Energyefficiency maximization bidirectional direct and relay transmission
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 156 (2020)
Abstract
Energyefficient transmission (EET) has become a very important problem in wireless communication. Optimal power allocation (OPA) is one of the general methods to achieve EET. But when OPA is only considered in EET, it maybe cannot analyze EET problems accurately. This paper aims at analyzing energyefficient bidirectional direct and relay transmission (RT) through joint optimization of transmit power (TP) and transmit time (TT) allocation. In RT, direct links (DLs) are existed. The EET problems are given with three optimization cases: (i) maximizing sum throughput (ST) to maximize energy efficiency (EE), in such case, optimal TT (OTT), optimal relay position (ORP), and outage probability analysis are given; (ii) minimizing total energy consumption (TEC) to maximize EE, in such case, optimal TP (OTP), OTT with gradientdescent algorithm, and influences of asymmetry and asymmetry transmission tasks are given; and (iii) maximizing ST and minimizing TEC simultaneously to maximize EE, in such case, Dinkelbach’s algorithm and onebyone optimal algorithm are given. Simulation results are presented to validate theoretical analysis. Results reveal that relay technique and DLs in RT can improve system’s EE.
Introduction
With rapid development of wireless communication, traditional communication system cannot satisfy the requirement of green communication and spectrum resources shortage [1, 2]. A large amount of interests have been focused on improving energy efficiency (EE) and spectral efficiency (SE). At the same time, lots of works have been done with different transmission techniques to improve them separately or simultaneously, such as cognitive radio [3, 4], orthogonal frequency division multiplexing (OFDM) [5], multi antenna technique [6], millimeter wave technique [7], simultaneous wireless information and power transfer (SWIPT) technique [8, 9], and nonorthogonal multiple access [10, 11]. Among all of these techniques, relay technique has attracted a lot of attention for achieving spatial diversity [12]. The major existing relay protocols are amplifyandforward (AF) and decodeandforward (DF). In AF relay protocol, relay node simply amplifies and forwards the received signal without any decoding operation [13]. Also, AF relay protocol is easy for implementation since it only requires coarse synchronization [14]. Therefore, AF relay protocol is considered in this paper.
With relay technique, energyefficient transmission (EET) can be achieved. Energyefficient resource allocation has been investigated in [15], which demonstrated that system EE can be maximized by scheduling optimal numbers of relay antennas and corresponding relay transmit power (TP). Optimal power allocation (OPA) scheme in a cooperative relaying system has been investigated in [16], which suggested that optimal global EE is strictly quasiconcave with regard to TP. With relay technique, spectralefficient transmission (SET) can also be achieved. Optimal rate allocation scheme has been proposed in [17], and it found that system’s errorfree SET can be achieved. Energyandspectralefficient adaptive forwarding strategy for multihop devicetodevice communications overlaying cellular networks has been proposed in [18], and it found that the higher EE and SE can be achieved.
However, only OPA has been considered in [15, 16] and only ideal network environment has been considered in [17, 18] to achieve EET. Actually, joint optimization of transmit time (TT) and TP allocation can be more accurately to measure system’s energy consumption and more effectively to improve system’s EE. Considering the joint optimization concern, energyefficient relayassisted cellular network has been studied in [19], which aimed at minimizing total power consumption through joint optimization of TT and TP allocation, and EE comparisons among direct transmission (DT), oneway relay transmission (OWRT), and twoway relay transmission (TWRT) with consideration of optimal TT (OTT) and optimal TP (OTP) have been discussed in [20]. It should be noted that the TWRT in [20] transmitted signal with the twophase analog network coding (ANC) protocol. At the same time, in practical transmission systems, energy consumption does include not only TP, but also nonnegligible circuit powers (CPs). In addition, the power amplifier (PA) efficiency is usually not ideal. The CP consumption and nonideal PA efficiency are all for nonideal network environment. Considering the nonideal network environment concern, throughput optimal policies for energy harvesting transmitters with CP consumption have been studied in [21], and EE maximization of fullduplex (FD) twoway DF relay with PA efficiency and CP consumption has been discussed in [22]. While considering the above two concerns, EE analysis of relay systems with joint consideration of TP, TT, and CPs has been discussed in [23].
However, again, direct links (DLs) in relay transmission (RT) have not been considered in [19–23] with assumption of deeply fading channel. Actually, consideration of DLs that are existed in RT can achieve further SE performance gain, and some works have been done on it. For example, cellular communication scenario involving the coexistence of onehop DT and twohop relaying has been studied in [24], optimal source and relay design for multiuser MIMO AF relay communication system with DLs and imperfect channel state information (CSI) has been investigated in [25], EE optimization for a twohop AF relay network with DLs over Rayleigh fading channels has been considered in [26], RT considering DLs and aiming at maximizing EE has been discussed in [27], and the optimization for energyefficient FD transmissions with DLs has been studied in our recent work [28]. As for the comparisons between the halfduplex and FD technique of the related works, they have been given in [28]. It has been shown that the FD technique can increase the SE and EE of the system, but only the minimizing total energy consumption (TEC) case has been discussed in [27, 28].
At the same time, optimal relay position (ORP) problem is important in RT for it influences the system performance. Although it has been showed that the ORP in a general communication system is the relay node located at the middle of two source nodes, it still can be a open problem. To the best of our knowledge, there are fewer works that investigate the ORP problem from the perspective of maximizing EE, i.e., the problem of joint optimization of relay station positions and relay stations serving range for maximizing EE has been discussed in [29].
Considering all of the above concerns and comparing with existing work in the literatures, whose characteristics are summarized in Table 1, this paper investigates the energyefficient DT and RT problems. The main contributions of this paper can be summarized as follows:

Firstly, to achieve a more pratical EET in this paper, PA efficiency, CP consumption, DLs in RT, and joint optimization of TT and TP allocation are considered simultaneously, rather than only considering parts of them in existing work. At the same time, ORP problem is investigated from the perspective of maximizing EE for the system EE can be improved with ORP [29]. This problem also has not been considered in the most of existing work.

Secondly, to achieve a more comprehensive EE analysis in this paper, EET problems are given with three optimization cases, i.e., maximizing sum throughput (ST) to maximize EE, minimizing TEC to maximize EE, and maximizing ST and minimizing TEC simultaneously to maximize EE, rather than only considering minimizing TEC to maximize EE in [19] and [22, 23], or only considering maximizing ST to maximizing EE in [18] and [29, 30].

Thirdly, with the three optimization cases: (1) OTT, ORP, and outage probability analysis are given with maximizing ST to maximize EE; (2) OTP, OTT with gradientdescent algorithm, and influences of asymmetry and asymmetry transmission tasks are given with minimizing TEC to maximize EE; (3) Dinkelbach’s algorithm and onebyone optimal algorithm are given with maximizing ST and minimizing TEC simultaneously to maximize EE.
The rest of this paper is organized as follows. Section 2 describes the system model. Section 3 provides the throughput analysis, energy consumption model, and problem formulation. Sections 4, 5, and 6 presents the three optimization cases to maximize EE. Section 7 gives simulation results and followed by conclusions in Section 8.
System model
In this section, the models of bidirectional DT and RT are explained. In DT, there are two source nodes S_{1} and S_{2}. In RT, there is also a relay node R except for two source nodes S_{1} and S_{2}. At the same time, the relay node R lies between two source nodes S_{1} and S_{2}. To achieve a bidirectional transmission, it needs two time slots in DT, it needs four time slots in OWRT, and it needs three time slots in TWRT. The relay protocol is AF, and nodes’ work mode is halfduplex. The links experience independent block Rayleigh fading and remain unchanged during one block. The following optimization and analysis are based on one block duration T_{t}. The transmission signal for the node S_{1} is x_{1} with variance \(E\{{x_{1}}^{2}\}\,=\,1\), and the transmission signal for the node S_{2} is x_{2} with variance \(E\{{x_{2}}^{2}\}\,=\,1\). The TP for the node S_{1} is P_{1}, and the TP for the node S_{2} is P_{2}. The channel gain between the nodes S_{1} and S_{2} is h_{3}, the channel gain between the nodes S_{1} and R is h_{1}, and the channel gain between the nodes S_{2} and R is h_{2}. At the same time, the channel gains between the same two nodes are all reciprocal, and the nodes have the full CSI. The noise at the node S_{1} is n_{1}, the noise at the node S_{2} is n_{2}, and the noise at the node R is n_{r}. Meanwhile, the DLs between two source nodes S_{1} and S_{2} in RT are existed and they can be exploited to convey information [25]. The noises are zeromean symmetric complex Gaussian vector with variance 1. The system bandwidth is W. The receive signals are combined at two nodes S_{1} and S_{2} by maximum ratio combining (MRC) technique.
The DT model is shown in Fig. 1a. In the first time slot, source node S_{1} transmits signal x_{1} to node S_{2}, and the receive signal at node S_{2} is \(y_{2}^{d}=\sqrt {P_{1}}h_{3}x_{1}+n_{2}\). In the second time slot, source node S_{2} transmits signal x_{2} to node S_{1}, and the receive signal at node S_{1} is \(y_{1}^{d}=\sqrt {P_{2}}h_{3}x_{2}+n_{1}\).
The OWRT model is shown in Fig. 1b. In the first time slot, source node S_{1} transmits signal x_{1} to node R and S_{2}, and the receive signals at node R and S_{2} are respectively \(y_{1r}^{o}=\sqrt {P_{1}}h_{1}x_{1}+n_{r}\) and \(y_{2}^{o}=\sqrt {P_{1}}h_{3}x_{1}+n_{2}\). In the second time slot, relay node R amplifies and forwards the receive signals \(y_{1r}^{o}\) to node S_{2}, and the receive signal at node S_{2} is \(y_{r{2}}^{o}=\sqrt {P_{r{2}}}h_{2}x_{1r}+n_{2}\), where P_{r2} is relay node’s TP in the second time slot of OWRT, x_{1r} is node R amplifies and forwards signal in the second time slot of OWRT with \(x_{1r}=A_{1}y_{1r}^{o}\), A_{1} is amplify coefficients and \(A_{1}\approx \frac {1}{\sqrt {{P_{1}{h_{1}}^{2}}}}\) [31]. In the third time slot, source node S_{2} transmits signal x_{2} to node R and S_{1}, and the receive signals at relay node R and S_{1} are respectively \(y_{2r}^{o}=\sqrt {P_{2}}h_{2}x_{2}+n_{r}\) and \(y_{1}^{o}=\sqrt {P_{2}}h_{3}x_{2}+n_{1}\). In the fourth time slot, relay node R amplifies and forwards the receive signals \(y_{2r}^{o}\) to node S_{1}, and the receive signal at node S_{1} is \(y_{r{1}}^{o}=\sqrt {P_{r{1}}}h_{1}x_{2r}+n_{1}\), where P_{r1} is relay node’s TP in the fourth time slot of OWRT, x_{2r} is node R amplifies and forwards signal in the fourth time slot of OWRT with \(x_{2r}=A_{2}y_{2r}^{o}\), A_{2} is amplify coefficients, and \(A_{2}\approx \frac {1}{\sqrt {{P_{2}{h_{2}}^{2}}}}\) [31].
The TWRT model is shown in Fig. 1c, and it transmits signal with the threephase ANC protocol. The threephase ANC protocol is also called time division broadcast channel (TDBC) protocol as [31] suggested. In the first time slot, source node S_{1} transmits signal x_{1} to node R and S_{2}, and the receive signals at node R and S_{2} are respectively \(y_{1r}^{t}=y_{1r}^{o}\) and \(y_{2}^{t}=y_{2}^{o}\). In the second time slot, source node S_{2} transmits signal x_{2} to node R and S_{1}, and the receive signals at relay node R and S_{1} are respectively \(y_{2r}^{t}=y_{2r}^{o}\) and \(y_{1}^{t}=y_{1}^{o}\). In the third time slot, relay node R broadcasts x_{r} to nodes S_{1} and S_{1}, where \(x_{r}=\zeta _{1}y_{1r}^{t}+\zeta _{2}y_{2r}^{t}\) is node R amplifies and forwards signal of TWRT. ζ_{1} and ζ_{2} are forward coefficients, where \(\zeta _{1}\approx \sqrt {\frac {o_{1}}{P_{1}{h_{1}}^{2}}}\) and \(\zeta _{2}\approx \sqrt {\frac {o_{2}}{P_{2}{h_{2}}^{2}}}\). o_{1} and o_{2} can be seemed as the signal combining factors, and they determine how relay node R combines the two signals \(y_{1r}^{t}\) and \(y_{2r}^{t}\), where 0≤{o_{1},o_{2}}≤1 and o_{1}+o_{2}=1 [31]. The original receive signals of TWRT \({y_{r1}^{t}}'\) and \({y_{r2}^{t}}'\) can be expressed as \({y_{r1}^{t}}'=\sqrt {P_{r}}h_{1}x_{r}+n_{1}\) and \({y_{r2}^{t}}'=\sqrt {P_{r}}h_{2}x_{r}+n_{2}\), where P_{r} is relay node’s TP in TWRT. Since each of source node receives a copy of its own transmitted signal as interference, the signal transmitted from the other source node can be decoded after selfinterference cancellation (SIC). Finally, the receive signals at the nodes S_{1} and S_{2} can be respectively expressed as \(y_{r1}^{t}\,=\,\sqrt {P_{r}}h_{1}\left (\zeta _{2}\sqrt {P_{2}}h_{2}x_{2}+\zeta _{1}n_{r}+\zeta _{2}n_{r}\right)+n_{1}\) and \(y_{r2}^{t}\,=\,\sqrt {P_{r}}h_{2}\left (\zeta _{1}\sqrt {P_{1}}h_{1}x_{1}+\zeta _{1}n_{r}+\zeta _{2}n_{r}\right)+n_{2}\).
Throughput analysis, energy consumption model, and problem formulation
In this section, throughput analysis is given first, followed by the energy consumption model and problem formulation.
Throughput analysis
In this subsection, throughput analysis is given. As we have suggested in the system model, one block duration is T_{t} and within which a round of bidirectional transmission is accomplished [22].
In DT and OWRT, T_{1} and T_{2} can be used to represent TT from node S_{1} to S_{2} and from node S_{2} to S_{1}, respectively. With throughput definition in [22], \(y_{1}^{d}\), and \(y_{2}^{d}\), throughput of DT can be given by \(\phantom {\dot {i}\!}C_{d}=C_{{d}_{1}}+C_{{d}_{2}}\), where C_{d} is ST of DT, \(\phantom {\dot {i}\!}C_{{d}_{1}}=T_{1}W\log _{2}\left (1+\gamma _{{d}_{1}}\right)\) is throughput at node \(S_{2}\phantom {\dot {i}\!}\) of DT, and \(\phantom {\dot {i}\!}C_{{d}_{2}}=T_{2}W\log _{2}\left (1+\gamma _{{d}_{2}}\right)\) is throughput at node \(S_{1}\phantom {\dot {i}\!}\) of DT. \(\phantom {\dot {i}\!}\gamma _{{d}_{1}}\) and \(\phantom {\dot {i}\!}\gamma _{{d}_{2}}\) are signaltonoise ratios (SNRs) at nodes \(\phantom {\dot {i}\!}S_{2}\) and \(\phantom {\dot {i}\!}S_{1}\) of DT, respectively. With noise variance is 1, they can be given by \(\gamma _{{d}_{1}}=P_{1}h_{3}^{2}\phantom {\dot {i}\!}\) and \(\gamma _{{d}_{2}}=P_{2}h_{3}^{2}\phantom {\dot {i}\!}\).
With \(y_{2}^{o}\phantom {\dot {i}\!}\), \(y_{r2}^{o}\phantom {\dot {i}\!}\), \(\phantom {\dot {i}\!}y_{1}^{o}\), and \(\phantom {\dot {i}\!}y_{r1}^{o}\), throughput of OWRT can be given by \(\phantom {\dot {i}\!}C_{o}=C_{{o}_{1}}+C_{{o}_{2}}\), where \(C_{o}\phantom {\dot {i}\!}\) is ST of OWRT, \(\phantom {\dot {i}\!}C_{{o}_{1}}=\frac {T_{1}}{2}W\log _{2}\left (1+\gamma _{{o}_{1}}\right)\) is throughput at node \(S_{2}\phantom {\dot {i}\!}\) of OWRT, and \(\phantom {\dot {i}\!}C_{{o}_{2}}=\frac {T_{2}}{2}W\log _{2}\left (1+\gamma _{{o}_{2}}\right)\) is throughput at node \(S_{1}\phantom {\dot {i}\!}\) of OWRT. The 1/2 is due to two time slots in completing a transmission process in each direction. \(\gamma _{{o}_{1}}\phantom {\dot {i}\!}\) and \(\phantom {\dot {i}\!}\gamma _{{o}_{2}}\) are respectively SNRs at nodes \(\phantom {\dot {i}\!}S_{2}\) and S_{1} of OWRT, and they can be given by \(\phantom {\dot {i}\!}\gamma _{{o}_{1}}=P_{1}h_{3}^{2}+\frac {A_{1}^{2}P_{1}P_{r{2}}h_{1}^{2}h_{2}^{2}}{A_{1}^{2}P_{r{2}}h_{2}^{2}+1}\) and \(\phantom {\dot {i}\!}\gamma _{{o}_{2}}=P_{2}h_{3}^{2}+\frac {A_{2}^{2}P_{2}P_{r{1}}h_{2}^{2}h_{1}^{2}}{A_{2}^{2}P_{r{1}}h_{1}^{2}+1}\).
In TWRT, T_{twr} can be used to represent TT for the whole information exchange. With \(y_{2}^{t}\), \(y_{r{2}}^{t}\phantom {\dot {i}\!}\), \(y_{1}^{t}\phantom {\dot {i}\!}\), and \(y_{r{1}}^{t}\), throughput of TWRT can be given by \(C_{t}=C_{{t}_{1}}+C_{{t}_{2}}\phantom {\dot {i}\!}\), where C_{t} is ST of TWRT, \(C_{{t}_{1}}=\frac {T_{twr}}{3}W\log _{2}\left (1+\gamma _{{t}_{1}}\right)\phantom {\dot {i}\!}\) is throughput at node S_{2} of TWRT, and \(C_{{t}_{2}}=\frac {T_{twr}}{3}W\log _{2}\left (1+\gamma _{{t}_{2}}\right)\phantom {\dot {i}\!}\) is throughput at node S_{1} of TWRT. The 1/3 is due to equal TT is consumed in three time slots. \(\gamma _{{t}_{1}}\phantom {\dot {i}\!}\) and \(\gamma _{{t}_{2}}\phantom {\dot {i}\!}\) are respectively SNRs at nodes S_{2} and S_{1} of TWRT, and they can be given by \(\phantom {\dot {i}\!}\gamma _{{t}_{1}}=P_{1}h_{3}^{2}+\frac {\zeta _{1}^{2}P_{1}P_{r}h_{2}^{2}h_{1}^{2}}{(\zeta _{1}^{2}+\zeta _{2}^{2})P_{r}h_{2}^{2}+1}\) and \(\phantom {\dot {i}\!}\gamma _{{t}_{2}}=P_{2}h_{3}^{2}+\frac {\zeta _{2}^{2}P_{2}P_{r}h_{1}^{2}h_{2}^{2}}{(\zeta _{2}^{2}+\zeta _{2}^{2})P_{r}h_{1}^{2}+1}\).
Energy consumption model
In this subsection, the TEC model is given. It should be noted that the TEC contains TPs and CPs. To reduce the TEC, the system may not use entire block duration T_{t} for transmission. In each block, each node has three states: transmission, reception, and idle [32], which corresponding to CP consumptions of three states are respectively P^{ct}, P^{cr}, and P^{ci}. For node S_{1}, S_{2}, and R all have three kinds of CP consumptions, we use subscript ı∈{1,2,r} to represent nodes S_{1}, S_{2}, and R, respectively. In such case, the CPs of three nodes with three states can be expressed as \(P_{\imath }^{ct}\), \(P_{\imath }^{cr}\), and \(P_{\imath }^{ci}\), respectively. Also, the SIC in TWRT needs to consume CPs P^{sic}. In such case, the CP consumptions for SIC of nodes S_{1} and S_{2} in TWRT can be expressed as \(P_{1}^{sic}\) and \(P_{2}^{sic}\), respectively. The nonideal PA efficiency is μ and μ≥1 [33].
From the above illustrations, the TEC is related to TPs, CPs, and PA efficiency, and TT can be known. Then, the TEC in DT is \(\phantom {\dot {i}\!}E_{d}=T_{1}(\mu P_{1}+P_{d_{c1}}P_{Dci})+T_{2}(\mu P_{2}+P_{d_{c2}}P_{Dci})+T_{t}P_{Dci}\), where \(\phantom {\dot {i}\!}P_{d_{c1}}=P_{1}^{ct}+P_{2}^{cr}\) and \(\phantom {\dot {i}\!}P_{d_{c2}}=P_{2}^{ct}+P_{1}^{cr}\) represent total CPs in the first and second time slots of DT, respectively. \(\phantom {\dot {i}\!}T_{t}=T_{1}+T_{2}+T'\) in DT and OWRT, where T^{′} is the time in idle state. \(\phantom {\dot {i}\!}P_{Dci}=P_{1}^{ci}+P_{2}^{ci}\) represents total idle CP in DT.
The TEC in OWRT is \(E_{o}=T_{1}\left (\frac {\mu (P_{1}+P_{r{2}})}{2}+P_{o_{c1}}P_{Rci}\right)+T_{2}\left (\frac {\mu (P_{2}+P_{r{1}})}{2}+P_{o_{c2}}P_{Rci}\right)+T_{t}P_{Rci}\), where \(P_{o_{c1}}=\frac {1}{2}\left (P_{1}^{ct}+P_{2}^{cr}+P_{r}^{cr}+P_{r}^{ct}+P_{2}^{cr}+P_{1}^{ci}\right)\) and \(P_{o_{c2}}=\frac {1}{2}\left (P_{2}^{ct}+P_{1}^{cr}+P_{r}^{cr}+P_{r}^{ct}+P_{1}^{cr}+P_{2}^{ci}\right)\) represent total CPs in the first and second two time slots of OWRT, respectively. \(P_{Rci}=P_{1}^{ci}+P_{2}^{ci}+P_{r}^{ci}\) represents total idle CP in RT.
The TEC in TWRT is \(E_{t}=T_{twr}\left (\frac {\mu (P_{1}+P_{2}+P_{r})}{3}+P_{t_{c1}}+P_{t_{c2}}+P_{t_{c3}}P_{Rci}\right)+T_{t}P_{Rci}\), where \(P_{t_{c1}}=\frac {1}{3}\left (P_{1}^{ct}+P_{2}^{cr}+P_{r}^{cr}\right)\), \(P_{t_{c2}}=\frac {1}{3}\left (P_{2}^{ct}+P_{1}^{cr}+P_{r}^{cr}\right)\), and \(P_{t_{c3}}=\frac {1}{3}\left (P_{r}^{ct}+P_{1}^{cr}+P_{2}^{cr}+P_{1}^{sic}+P_{2}^{sic}\right)\) represent total CPs in the first, second, and third time slot, and T_{t}=T_{twr}+T^{′} in TWRT.
Problem formulation
In this subsection, EE definition and problem formulation are given. The EE in this paper is defined the same as [34], and it can be given as \(\eta =\frac {C_{T}}{E_{T}}\), where C_{T} is ST and E_{T} is TEC. From the throughput analysis and energy consumption model, it can be known that both E_{T} and C_{T} have relations with TPs and TT, and we need to maximize it by maximizing C_{T} and minimizing E_{T} with OTP and OTT. Thus, energyefficient optimization problem can be summarized as
For a more comprehensive analysis, we divide the optimization problem into three optimization cases. Firstly, maximizing C_{T} to maximize η when TPs are equally allocated as [29, 30]. Secondly, minimizing E_{T} to maximize η when C_{T} is constant as [22, 23]. Thirdly, maximizing C_{T} and minimizing E_{T} simultaneously to maximize η. The EE analysis of these three cases will be given in the following three sections.
Maximize sum throughput
Firstly, when TPs are equally allocated, which means P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P, we try to maximize C_{T} with OTT to maximize η. TPs equally allocation is usually existed in practical system for simplicity, i.e., IS95CDMA system [35]. At the same time, the throughput is an important index that the communication systems want to pursue. Thus, the discussion about maximizing C_{T} to maximize η with TPs equally allocation maybe can effectively improve the EE of the IS95CDMA system under a certain situation.
Optimal transmit time
In this subsection, the OTT analysis is given. It should be noted that all the CPs are a fixed power cost from 0 to hundreds of mW [36]. However, although the CPs are a fixed power cost, they are concerned with TT. Then, we need to maximize C_{T} through OTT to maximize η. In such case, Propositions 1 and 2 about OTT to maximize η can be obtained.
Proposition 1
When P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P and different nodes’ CPs for transmit, receive, and idle are the same, i.e., \(P_{1}^{ct}=P_{2}^{ct}\), \(P_{1}^{cr}=P_{2}^{cr}\), and \(P_{1}^{ci}=P_{2}^{ci}\), the OTT in DT and OWRT to maximize η is T^{′}=0.
Proof:
With P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P and different nodes’ CPs are the same constants, \(\gamma _{o_{1}}=\gamma _{o_{2}}\) and \(P_{o_{c1}}=P_{o_{c2}}=P_{o_{c}}\) can be obtained. Then, \(C_{o}=\frac {T_{1}+T_{2}}{2}W\log _{2}(1+\gamma _{o_{1}})\phantom {\dot {i}\!}\) and \(E_{o}=(\mu P+P_{o_{c}})(T_{1}+T_{2})+(T_{t}T_{1}T_{2})P_{Rci}\phantom {\dot {i}\!}\) can be further obtained. In such case, EE of OWRT η_{o} can be given as
The (a) step is for dividing T_{1}+T_{2}, and (b) step is for T_{1}+T_{2}=T_{t} and T^{′}=0 to maximize η_{o}. The same conclusion of the OTT in DT is T^{′}=0 can also be obtained with P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P and \(P_{d_{c1}}=P_{d_{c2}}=P_{{d_{c}}}\) to maximize EE of DT η_{d}.
The proof is completed. □
Proposition 2
When P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P, the OTT in TWRT to maximize η is also T^{′}=0.
Proof:
With P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P, EE of TWRT η_{t} can be given as
The (a) step is for dividing \(\frac {T_{twr}}{3}\), and (b) step is for T_{twr}=T_{t} and T^{′}=0 to maximize η_{t}.
The proof is completed. □
Remark 1:
It should be noted that comparing with DT and OWRT, the OTT in TWRT to maximize η_{t} has no constraints for CPs. At the same time, the OTT in it is always T_{twr}=T_{t} when TPs are not functions of TT even with unequally power allocation.
Optimal relay position
In this subsection, ORP analysis is given. Considering the Rayleigh fading channel, when three nodes are in a line and the distances of two source nodes between relay node are normalized by the distance between two source nodes S_{1} and S_{2}, the channel gains can be expressed as h_{3}^{2}=1, \(h_{1}^{2}={d}^{\alpha }\), d∈(0,1), and h_{2}^{2}=(1−d)^{−α} [31], where d is the distance between node S_{1} and R, α is the path loss attenuation factor, and α∈[2,5] [37].
From (2), we can see that only \(\gamma _{{o}_{1}}\phantom {\dot {i}\!}\) is needed to be maximized with ORP to maximize η_{o} when T^{′}=0. Then, Proposition 3 about ORP in OWRT can be obtained.
Proposition 3
When P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P and T^{′}=0, the ORP in OWRT to maximize η_{o} is d=0.5.
Proof:
With P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P, T^{′}=0, A_{1}, and channel gains, \(\gamma _{{o}_{1}}\phantom {\dot {i}\!}\) can be reformulated as
With (4), when d=0.5, \(\gamma _{{o}_{1}}\phantom {\dot {i}\!}\) can achieve the maximum for \(\frac {{d}^{\alpha }(1d)^{\alpha }}{(1d)^{\alpha }+{d}^{\alpha }}\leq \frac {\sqrt {{d}^{\alpha }(1d)^{\alpha }}}{2}\leq \frac {(0.5)^{\alpha }}{2}\). This means that η_{o} is the maximum when d=0.5 for in such case \(\gamma _{{o_{1}}}\) is the maximum.
The proof is completed. □
From (3), we can also know that only the sum SNR of TWRT \(\phantom {\dot {i}\!}\gamma _{t}=\gamma _{{t}_{1}}+\gamma _{{t}_{2}}+\gamma _{{t}_{1}}\gamma _{{t}_{2}}\) is needed to be maximized with ORP to maximize η_{t} when T^{′}=0. However, it also should be noted that comparing with OWRT, the o_{1} and o_{2} will influence \(\gamma _{{t}_{1}}\phantom {\dot {i}\!}\) and \(\gamma _{{t}_{2}}\phantom {\dot {i}\!}\). Then, Proposition 4 about optimal o_{1}, o_{2}, and ORP in TWRT can be obtained.
Proposition 4
When P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P and T^{′}=0, the optimal o_{1}, o_{2}, and ORP in TWRT to maximize η_{t} are o_{1}=o_{2}=0.5 and d is smaller.
Proof:
Please see the Appendix: Proof of Proposition 4.
The proof is completed. □
Outage probability analysis
In this subsection, outage probability analysis is given for it is an important criterion to measure the performance of a link. With C_{th} to avoid disruption for a given throughput, then C_{(·)}<C_{th} denotes the outage event, and Pr(C_{(·)}<C_{th}) denotes the outage probability.
Let γ=P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P, C_{d} can be reformulated as
It should be noted that bidirectional transmission system is in outage if any endtoend transmission is in outage. The outage event of DT is given by \(\min \{C_{d_{1}},C_{d_{2}}\}<C_{th}\), and it is equivalent to the event \(h_{3}^{2}<\frac {2^{\frac {C_{th}}{T_{\text {min}}W}}1}{\gamma }\), where \(T_{\text {min}}=\min \{T_{1},T_{2}\}\) in DT and OWRT. Statistically, the variances of channel gains h_{l} are \(\delta _{l}^{2}\). For Rayleigh fading, i.e., h_{l}^{2} are exponentially distribution with parameter \(\delta _{l}^{2}\) [38], where subscript l∈{1,2,3} represent different channel gains. Then, \(\lambda _{1}=\delta _{1}^{2}\), \(\lambda _{2}=\delta _{2}^{2}\), and \(\lambda _{3}=\delta _{3}^{2}\) can be obtained. Also, when x tends to 0, an equivalent infinitesimal approximation 1−e^{−x}≈x can be obtained. With them, outage probability of DT \(p_{d}^{\text {out}}\) can be calculated as
With γ, C_{o} can be reformulated as
where \(f(x,y)=\frac {xy}{x+y}\). The outage event of OWRT is given by \(\min \{C_{o_{1}},C_{o_{2}}\}<C_{th}\), and it is equivalent to the event
Assuming h_{3}^{2}=s, h_{1}^{2}=u, h_{2}^{2}=v, \(m=\frac {1}{\gamma }\), \(g(m)=\left (2^{\frac {2C_{th}}{T_{\text {min}}W}}1\right)m\), and according to [38], the following equation can be obtained as
With it, we can further obtain the following equation as
Finally, the outage probability of OWRT \(p_{o}^{\text {out}}\) can be calculated as
With γ, C_{t} can be reformulated as
where \(f_{1}(x,y)=\frac {xy}{x+y+o_{2}\gamma h_{1}^{2}}\) and \(f_{2}(x,y)=\frac {xy}{x+y+o_{1}\gamma h_{2}^{2}}\). With o_{1}=o_{2}=0.5, the outage event of TWRT is given by \(\min \{C_{t_{1}},C_{t_{2}}\}<C_{th}\), and it is equivalent to the event
where \(f_{1}(x,y)=\frac {xy}{3x+y}=\frac {1}{3}.\frac {3xy}{3x+y}=\frac {1}{3}f(3x,y)\). The above equivalent event give us an example when \(C_{t_{1}}\leq C_{t_{2}}\) to get the outage probability of TWRT and we can use the same method to get the outage probability of TWRT when \(C_{t_{1}}>C_{t_{2}}\). Assuming 3h_{1}^{2}=w, \(h(m)=\left (2^{\frac {3C_{th}}{T_{twr}W}}1\right)m\), and according to [38], the following equation can also be obtained as
With it, we can further obtain the following equation as
Finally, the outage probability of TWRT \(p_{t}^{\text {out}}\) can be calculated as
With the same method to get (16), we can know that when \(C_{t_{1}}>C_{t_{2}}\), the outage probability of TWRT \(p_{t}^{\text {out}}\) is \(p_{t}^{\text {out}}(\gamma,C_{th})=Pr(\min \{C_{t_{1}},C_{t_{2}}\}<C_{th})\approx \left (\frac {1}{6\delta _{3}^{2}}.\frac {3\delta _{1}^{2}+\delta _{2}^{2}}{\delta _{1}^{2}.\delta _{2}^{2}}\right)\left (\frac {2^{\frac {3C_{th}}{T_{twr}W}}1}{\gamma }\right)^{2}\).
Minimize total energy consumption
Secondly, when C_{T} is constant with the assumption of C_{1}=βC_{T} and C_{2}=(1−β)C_{T}, we try to minimize E_{T} with OTT and OTP to maximize η. The β∈(0,1) is the transmission task distribution factor. C_{1} and C_{2} are the minimum transmission tasks at two directions [22]. Minimizing TEC is one of the widely used methods to achieve green communication [1]. At the same time, the OPA has been widely studied in many practical wireless systems, i.e., with OPA to achieve EET of cellular system [19] and with OPA to maximize the average throughput and EE of energy harvesting system [21]. Thus, the discussion about the minimizing E_{T} to maximize η with joint optimization of TT and TP allocation maybe can effectively improve the EEs of the cellular and energy harvesting systems.
Energyefficient direct transmission
In this subsection, the problem with minimizing E_{d} to maximize η_{d} is given. With the requirements of C_{1}, C_{2}, maximum TP \(P_{t}^{\text {max}}\), and total TT T_{t}, EET in DT can be given as follows:
Through throughputs, the TPs can be expressed as functions of TT, then the minimum TECs can be obtained with only optimizing TT. For throughputs are derived from Shannon capacity formula, which means the maximum achievable throughput can be obtained under given TPs, thus, the TPs derived through throughputs are also the minimum which can support the required transmission tasks.
With \(C_{d_{1}}=C_{1}\), \(2^{\frac {C_{1}}{T_{1}W}}1=\gamma _{{d}_{1}}\) can be obtained. At the same time, with \(C_{d_{2}}=C_{2}\), \(2^{\frac {C_{2}}{T_{2}W}}1=\gamma _{{d}_{2}}\) can be obtained. Then through \(\phantom {\dot {i}\!}\gamma _{{d}_{1}}=P_{1}h_{3}^{2}\) and \(\phantom {\dot {i}\!}\gamma _{{d}_{2}}=P_{2}h_{3}^{2}\), the minimum TPs in DT can be obtained as
It can be seen that the optimal \(P_{1}^{\text {opt}}\) and \(P_{2}^{\text {opt}}\) are respectively increasing functions of \(\gamma _{d_{1}}\) and \(\gamma _{d_{2}}\); thus, they are respectively decreasing functions of T_{1} and T_{2}. As T_{1} and T_{2} decrease, the optimal \(P_{1}^{\text {opt}}\) and \(P_{2}^{\text {opt}}\) increase, and they may achieve \(P_{t}^{\text {max}}\). To simplify the analysis, we only consider all the TPs that will not achieve \(P_{t}^{\text {max}}\) situation, and the TPs that achieve \(P_{t}^{\text {max}}\) situation can be seen in our recent work [27].
With (18), the (17) can be reformulated into only optimizing TT problem and it can be given by
In (19), the term T_{t}P_{Dci} is independent of TT, and the terms \(E_{d_{1}}=T_{1}\mu \frac {\gamma _{{d}_{1}}}{h_{3}^{2}}+T_{1}(P_{d_{c1}}P_{Dci})\) and \(E_{d_{2}}=T_{2}\mu \frac {\gamma _{{d}_{2}}}{h_{3}^{2}}+T_{2}(P_{d_{c2}}P_{Dci})\) are only concerned with T_{1} and T_{2}, respectively. Then, the second order derivative (SEC) \(E_{d_{1}}''(T_{1})=\frac {2^{\frac {C_{1}}{T_{1}W}}\mu (\ln 2)^{2}C_{1}^{2}}{W^{2}h_{3}^{2}T_{1}^{3}}\) and \(E_{d_{2}}''(T_{2})=\frac {2^{\frac {C_{2}}{T_{2}W}}\mu (\ln 2)^{2}C_{2}^{2}}{W^{2}h_{3}^{2}T_{2}^{3}}\) can be obtained.
With \(E_{d_{1}}''(T_{1})\geq 0\) and \(E_{d_{2}}''(T_{2})\geq 0\), \(E_{d_{1}}\) and \(E_{d_{2}}\) that are respectively convex functions of T_{1} and T_{2} can be known. Therefore, the objective function is convex with respect to T_{1} and T_{2} for the addition of two convex functions is still a convex function [39]. The convex functions can be solved using standard solvers, such as interpoint methods, Newton method, and gradientdescent method. For the fast convergence of gradientdescent algorithm [39], we use it in this paper. For each t_{i}, the gradientdescent algorithm can be summarized as Algorithm 1.
In Algorithm 1, we give an initial point t_{i}∈[0,T_{t}]. Then, it iteratively moves toward lower values of the functions by taking steps in direction of negative gradient − ▽ f(t_{i}) with chosen step size n_{i}, and − ▽ f(t_{i}) can be obtained according to (19a), (23a), (23b), and (26a). Eventually, Algorithm 1 converges to the minimum for the functions are convex.
Energyefficient oneway relay transmission
In this subsection, with minimizing E_{o} to maximize η_{o} problem is given. The EET in OWRT can also be given under the requirements of C_{1}, C_{2}, \(P_{t}^{\text {max}}\), and T_{t}. Observing E_{o}, it can be found that the terms \(E_{o_{1}}=T_{1}\left (\frac {\mu \left (P_{1}+P_{r{2}}\right)}{2}+P_{o_{c1}}P_{Rci}\right)\) and \(E_{o_{2}}=T_{2}\left (\frac {\mu \left (P_{2}+P_{r{1}}\right)}{2}+P_{o_{c2}}P_{Rci}\right)\) are respectively only concerned with T_{1} and T_{2}, and the term T_{t}P_{Rci} is independent of TT. To make a further observation, it can be found that \(\gamma _{o_{1}}\) is not a function of P_{2} and P_{r1}, while \(\gamma _{o_{2}}\) is not a function of P_{1} and P_{r2}. Therefore, the optimization problem in OWRT can be equivalently reformulated into two subproblems as
With \(C_{o_{1}}=C_{1}\), \(2^{\frac {2C_{1}}{T_{1}W}}1=\gamma _{o_{1}}\) can be obtained. With \(\phantom {\dot {i}\!}\gamma _{{o}_{1}}=P_{1}h_{3}^{2}+\frac {A_{1}^{2}P_{1}P_{r2}h_{1}^{2}h_{2}^{2}}{A_{1}^{2}P_{r2}h_{2}^{2}+1}\), P_{r2} can be obtained as \(P_{r2}=\frac {(\gamma _{o_{1}}P_{1}h_{3}^{2})P_{1}h_{1}^{2}}{h_{2}^{2}(P_{1}h_{13}\gamma _{o_{1}})}=f(P_{1})\). With it, the TP in (20a) can be represented as P_{1}+P_{r2}=P_{1}+f(P_{1}). By setting the derivative of P_{1}+f(P_{1}) to zero and with the help of its SEC, the optimal \(P_{1}^{\text {opt}}\) and \(P_{r2}^{\text {opt}}\) with minimizing P_{1} + P_{r2} can be obtained as
where c_{1}=h_{1}^{4}, c_{2}=(h_{2}^{2}h_{3}^{2}+h_{1}^{2}h_{32}), h_{32}=h_{2}^{2}−h_{3}^{2}, h_{31}=h_{1}^{2}−h_{3}^{2}, and h_{13}=h_{1}^{2}+h_{3}^{2}.
Following the same solving procedure as (20a), the optimal \(P_{2}^{\text {opt}}\) and \(P_{r1}^{\text {opt}}\) with minimizing P_{2} + P_{r1} can be obtained as
where \(2^{\frac {2C_{2}}{T_{2}W}}1=\gamma _{o_{2}}\), c_{3}=h_{2}^{4}, c_{4}=(h_{1}^{2}h_{3}^{2}+h_{2}^{2}h_{31}), and h_{23}=h_{2}^{2}+h_{3}^{2}.
With (21) and (22), the (20) can be reformulated as
where \(P_{1}^{\text {opt}}+P_{r2}^{\text {opt}}=\frac {(\sqrt {c_{1}}+\sqrt {c_{2}})^{2}\gamma _{o_{1}}}{h_{2}^{2}h_{13}^{2}}\), and \(P_{2}^{\text {opt}}+P_{r1}^{\text {opt}}=\frac {(\sqrt {c_{3}}+\sqrt {c_{4}})^{2}\gamma _{o_{2}}}{h_{1}^{2}h_{23}^{2}}\). Then, the SEC \(E_{o_{1}}''(T_{1})=\frac {2^{\left (1+{\frac {2C_{1}}{T_{1}W}}\right)}\mu (\ln 2)^{2}C_{1}^{2}(\sqrt {c_{1}}+\sqrt {c_{2}})^{2}}{W^{2}h_{2}^{2}h_{13}^{2}T_{1}^{3}}\) and \(E_{o_{2}}''(T_{2})=\frac {2^{\left (1+{\frac {2C_{2}}{T_{2}W}}\right)}\mu (\ln 2)^{2}C_{2}^{2}(\sqrt {c_{3}}+\sqrt {c_{4}})^{2}}{W^{2}h_{1}^{2}h_{23}^{2}T_{2}^{3}}\) can be obtained.
With \(E_{o_{1}}''(T_{1})\geq 0\) and \(E_{o_{2}}''(T_{2})\geq 0\), \(E_{o_{1}}\) and \(E_{o_{2}}\) that are respectively convex functions of T_{1} and T_{2} can be known. Then, the problems can also be solved with Algorithm 1 as DT. It should be noted that the OTT in DT and OWRT should meet the requirement of \(T_{1}^{\text {opt}}+T_{2}^{\text {opt}}\leq T_{t}\). If \(T_{1}^{\text {opt}}+T_{2}^{\text {opt}}< T_{t}\), then we get \(T_{1}^{\text {opt}}\) and \(T_{2}^{\text {opt}}\) as the final OTT. Or if \(T_{1}^{\text {opt}}+T_{2}^{\text {opt}}=T_{t}\), once \(T_{1}^{\text {opt}}\) has been obtained, we can get \(T_{2}^{\text {opt}}=T_{t}T_{1}^{\text {opt}}\) through a scalar searching [23].
Energyefficient twoway relay transmission
In this subsection, with minimizing E_{t} to maximize η_{t} problem is given. The EET in TWRT can also be given under the requirements of C_{1}, C_{2}, \(P_{t}^{\text {max}}\), and T_{t}. At the same time, P_{1}=P_{2}=P in TWRT can be assumed as [31]. In such situation, the EET in TWRT can be given as follows:
Following the same solving procedure as (20a), the optimal P^{opt} and \(P_{r}^{\text {opt}}\) with minimizing 2P+P_{r} can be obtained as
where \(2^{\frac {3C_{1}}{T_{twr}W}}1=\gamma _{t_{1}}\), c_{5}=h_{1}^{2}o_{2}, c_{6}=h_{2}^{2}o_{1}, c_{7}=2h_{2}^{4}o_{1}+2h_{1}^{2}h_{3}^{2}o_{2}+2h_{2}^{2}h_{3}^{2}o_{1}−h_{2}^{2}h_{3}^{2}, c_{8}=h_{2}^{4}o_{1}+h_{1}^{2}h_{3}^{2}o_{2}+h_{2}^{2}h_{3}^{2}o_{1}, \(A=h_{2}^{2}c_{6}(c_{5}+c_{6})(c_{6}c_{7}h_{3}^{2}(c_{6}c_{7}(c_{5}+c_{6}))^{\frac {1}{2}})\), and \(B=c_{7}(c_{5}+c_{6})+h_{2}^{2}(c_{6}c_{7}(c_{5}+c_{6}))^{\frac {1}{2}}\).
With (25), the (24) can be reformulated as
where \(C=\frac {2P^{\text {opt}}+P_{r}^{\text {opt}}}{\gamma _{t_{1}}}=\frac {h_{2}^{2}c_{6}\left (2o_{1}h_{23}+h_{32}\right)+c_{5}\left (c_{7}+2o_{1}h_{2}^{2}h_{3}^{2}\right)}{c_{8}^{2}}+\frac {2h_{2}^{2}\left (c_{6}c_{7}\left (c_{5}+c_{6}\right)\right)^{\frac {1}{2}}}{c_{8}^{2}}\). Then, the SEC \(E_{t}''(T_{twr})=\frac {9\times \!2^{\frac {3C_{1}}{T_{twr}W}}\!\mu (\ln 2)^{2}C_{1}^{2}\!C}{T_{twr}^{3}W^{2}}\) can be obtained.
With Et″(T_{twr})≥0, E_{t} that is a convex function of T_{twr} can be known. Then, the problem can be solved with Algorithm 1. The complexities of Algorithm 1 are \(\mathcal {O}(\mathcal {I}_{d_{1}}+\mathcal {I}_{d_{2}})\) in DT, \(\mathcal {O}(\mathcal {I}_{d_{3}}+\mathcal {I}_{d_{4}})\) in OWRT, and \(\mathcal {O}(\mathcal {I}_{d_{5}})\) in TWRT. The \(\mathcal {I}_{d_{1}}\) and \(\mathcal {I}_{d_{2}}\) are respectively required number of iterations for T_{1} and T_{2} in DT. The \(\mathcal {I}_{d_{3}}\) and \(\mathcal {I}_{d_{4}}\) are respectively required number of iterations for T_{1} and T_{2} in OWRT, and the \(\mathcal {I}_{d_{5}}\) is the required number of iterations for T_{twr} in TWRT. In order to get the OTT, the complexity can be modeled in polynomial form in terms of the number of variables and constraints with the functions (19a), (23a), (23b), and (26a). Then, for each OTT, it mainly needs one derivation in step 3 and its complexity is \(\mathcal {O}(1)\).
Remark 2:
There is a pity that the analytic expression of OTT cannot be obtained for the existence of exponential terms of TT, and that is why we use Algorithm 1 to get the OTT in this section.
Asymmetry and asymmetry transmission tasks
In this subsection, the influences of asymmetry and asymmetry transmission tasks at two directions are given. With β to show the transmission task distribution factor at two directions, Proposition 5 can be obtained.
Proposition 5
When C_{T} is constant and different nodes’ CPs for transmit, receive, and idle are the same, β influences η_{t} and the maximum η_{t} can be obtained when β=0. However, β does not influence η_{d} and η_{o}.
Proof:
Please see the Appendix: Proof of Proposition 5.
The proof is completed. □
Maximize energy efficiency
Thirdly, we try to maximize C_{T} and minimize E_{T} simultaneously to maximize η with the proposed optimal algorithms. Maximizing EE is a more scientific method to achieve green communication, and many communication techniques have been studied to improve the EEs of different wireless communication systems, i.e., the MIMO system [1, 32]. Thus, the discussion about maximizing C_{T} and minimizing E_{T} simultaneously to maximize η maybe also can effectively improve the EE of the MIMO system.
Actually, the proposed optimal algorithms in this section can also be used in the communication systems when the multiple optimal variables cannot be obtained simultaneously [40]. The usage of the proposed optimal algorithms in the communication systems with multiple optimal variables is only to try to show its application in the existing works, and the further detailed discussions about whether the usage can improve the performance of the related systems can be studied in the further work. This is the same as Sections 4 and 5 with the application of their solutions in the related systems.
Energyefficiency maximization optimal algorithms
In this subsection, EE maximization optimal algorithms are given. First, maximizing η_{t} is given for we have proved in Remark 1: that the OTT in it is always T_{twr}=T_{t} when the TPs are not functions of TT. Then maximizing η_{t} problem is only concerned with TPs. With (3) and the optimization constraints of (24), maximizing η_{t} problem can be given as
where \(P_{t}^{c}=3(P_{t_{c1}}+P_{t_{c2}}+P_{t_{c3}})\). At the same time, with P_{1}=P_{2}=P, o_{1}=o_{2}=0.5 in TWRT [31], ζ_{1}, and ζ_{2}, \(\gamma _{t_{1}}=Ph_{3}^{2}+\frac {{PP}_{r}h_{1}^{2}h_{2}^{2}}{(h_{2}^{2}+h_{1}^{2})P_{r}+2Ph_{1}^{2}}\) and \(\gamma _{t_{2}}=Ph_{3}^{2}+\frac {{PP}_{r}h_{1}^{2}h_{2}^{2}}{(h_{2}^{2}+h_{1}^{2})P_{r}+2Ph_{2}^{2}}\) can be obtained.
In (27), η_{t} is nonconvex in (P,P_{r}) for the numerator of η_{t} is concave and the denominator of η_{t} is linear with respect to P and P_{r}, respectively. At the same time, η_{t} is differentiable; thus, it is pseudoconcave in terms of P and P_{r}, respectively. Since for any optimization problems, we can first optimize over some of the variables and then over the remaining ones [40]. Then, we can divide it into two suboptimization problems which optimizing P and P_{r} in alternative. The pseudoconcave properties can be proved with Hessians of them, i.e., ▽^{2}η_{t}(P)≤0 and ▽^{2}η_{t}(P_{r})≤0.
For the pseudoconcave properties of η_{t}, it can be seemed as \(\frac {f(x)}{g(x)}\), where f(x) is concave and g(x) is linear. Define the function F(ψ) as \(F(\psi)=\max _{x\in S}\{f(x)\psi g(x)\}\) with continuous and positive f, g, and compact S, then F(ψ) is convex with respect to ψ; also, it is strictly decreasing and it has a unique root ψ^{∗}. With it, the problem of finding F(ψ) can be solved with convex optimization approaches and it is shown that the problem of maximizing \(\frac {f(x)}{g(x)}\) is equivalent to finding ψ^{∗} [36]. At the same time, since the original problem has been divided into two suboptimization problems, it can be solved onebyone. For each x, Dinkelbach’s algorithm is employed for its properties in solving nonconvex fractional programming problems. Dinkelbach’s algorithm can be summarized as Algorithm 2, where the superscript (n) denotes the iteration number.
Algorithm 3 leads to global optimal values of each pseudoconcave function. In this regard, for TWRT, firstly, with P^{(n)}, Algorithm 2 is adopted in order to find \(P_{r}^{(n+1)}\). Secondly, with known \(P_{r}^{(n+1)}\), P^{(n+1)} is computed. Consequently, the onebyone optimal algorithm is required in order to optimize P and P_{r}, simultaneously. Algorithm 3 presents the onebyone optimal alternating procedure which updates the optimization parameters until convergence.
Second, maximizing η_{d} and η_{o} problems are given. With C_{d}(P_{1},P_{2},T_{1},T_{2}), E_{d}(P_{1},P_{2},T_{1},T_{2}), and the optimization constraints of (17), maximizing η_{d} problem can be given as
Observing (28), it can be found that the OTT and OTP cannot be obtained simultaneously for the TT and TPs have multiplicative terms. But we also can use Algorithms 2 and 3 to get them onebyone. However, in practice, there is an optimization order problem with onebyone optimization method especially for TT and TPs are all existed in it. To simplify the theoretical analysis, we let \(T_{1}=T_{2}=T=\frac {1}{2}T_{t}\) in DT and OWRT for it has been suggested in [23] that with high transmission tasks the system will use entail block for transmission, then (28) can be changed into
where \(\gamma _{d_{1}}=P_{1}h_{3}^{2}\), \(\gamma _{d_{2}}=P_{2}h_{3}^{2}\), and \(P_{d}^{c}=P_{d_{c1}}+P_{d_{c2}}\). Following the same analysis method as TWRT, and with ▽^{2}η_{d}(P_{1})≤0, ▽^{2}η_{d}(P_{2})≤0, (29a) which are respectively pseudoconcave in terms of P_{1} and P_{1} can be proved. Then, (29) can be solved with Algorithms 2 and 3 as TWRT.
With \(T_{1}=T_{2}=\frac {1}{2}T_{t}=T\), and the optimization constraints of (20), the maximizing η_{o} can be given as
where \(P_{o}^{c}\!=P_{o_{c1}}+P_{o_{c2}}\). At the same time, with A_{1} and A_{2}, \(\phantom {\dot {i}\!}\gamma _{{o}_{1}}=P_{1}h_{3}^{2}+\frac {P_{1}P_{r{2}}h_{1}^{2}h_{2}^{2}}{(P_{r{2}}h_{2}^{2}+P_{1}h_{1}^{2})}\) and \(\gamma _{{o}_{2}}=P_{2}h_{3}^{2}+\frac {P_{2}P_{r{1}}h_{1}^{2}h_{2}^{2}}{(P_{r{1}}h_{1}^{2}+P_{2}h_{2}^{2})}\) can be obtained. Following the same analysis method as TWRT, and with ▽^{2}η_{o}(P_{1}) ≤ 0, ▽^{2}η_{o}(P_{r2}) ≤ 0, (30a) which are respectively pseudoconcave in terms of P_{1} and P_{r2} can be proved. At the same time, with ▽^{2}η_{o}(P_{2}) ≤ 0 and ▽^{2}η_{o}(P_{r1}) ≤ 0, (30a) which are respectively pseudoconcave in terms of P_{2} and P_{r1} can be proved. Then, (30) can also be solved with Algorithms 2 and 3 as TWRT.
Complexity analysis
In this subsection, the complexity analysis of the optimal algorithms is given. To analyze the computational complexity of Algorithm 3, we should note that Algorithm 3 employs Algorithm 2. But it also should be noted that the convergence rate of Algorithm 2 is independent of the complexity of finding \(x_{\text {opt}}^{(n)}\) for its super linear convergence. As the problems of finding \(x_{\text {opt}}^{(n)}\) in Algorithm 3 are convex, their complexity can be modeled in polynomial form in terms of the number of variables and constraints. With these properties, we analyze the complexity of Algorithm 3 for DT, OWRT, and TWRT, respectively.
The complexities for step 4 and step 5 are \(\mathcal {O}(1)\), and the total complexity of one iteration of Algorithm 3 for DT is \(\mathcal {O}(\mathcal {I}_{d_{6}}+\mathcal {I}_{d_{7}})\), where \(\mathcal {I}_{d_{6}}\) and \(\mathcal {I}_{d_{7}}\) are respectively the required number of iterations for step 4 and step 5. The complexities from step 8 to step 11 are \(\mathcal {O}(1)\), and the total complexity of one iteration of Algorithm 3 for OWRT is \(\mathcal {O}(\mathcal {I}_{d_{8}}+\mathcal {I}_{d_{9}}+\mathcal {I}_{d_{10}}+\mathcal {I}_{d_{11}})\), where \(\mathcal {I}_{d_{8}}\), \(\mathcal {I}_{d_{9}}\), \(\mathcal {I}_{d_{10}}\), and \(\mathcal {I}_{d_{11}}\) are respectively the required number of iterations for step 8, step 9, step 10, and step 11. The complexities for step 14 and step 15 are \(\mathcal {O}(1)\), and the total complexity of one iteration of Algorithm 3 for TWRT is \(\mathcal {O}(\mathcal {I}_{d_{12}}+\mathcal {I}_{d_{13}})\), where \(\mathcal {I}_{d_{12}}\) and \(\mathcal {I}_{d_{13}}\) are respectively the required number of iterations for step 14 and step 15.
Simulation results
In this section, simulations are conducted to confirm the validity of theoretical analysis. The simulation parameters are given as Table 2. Considering the Rayleigh fading channel, h_{3}^{2}=1, h_{1}^{2}=d^{−α}, and h_{2}^{2}=(1−d)^{−α}, where d∈(0,1) are also used in the simulations [31]. Some of the parameters in the simulations are set to a constant, i.e., d=0.5 in RT, o_{1}=o_{2}=0.5 in TWRT, α=4, β=0.5, and T^{′}=0, unless otherwise specified.
Maximize sum throughput
In this subsection, EEs with maximizing C_{T} are given. At the same time, TPs are equally allocated. In Fig. 2, EEs with and without idle state are given which correspond to T^{′} = 1ms and T^{′} = 0, respectively. The with and without idle state corresponds to “w” and “w/o” in Fig. 2, respectively. From it, we can find that EEs when T^{′}=1 ms are lower than those when T^{′} = 0, which corresponds to Propositions 1 and 2 that the OTT both in DT and RT to maximize EE is T^{′}=0.
In Fig. 3, EEs with different o_{1} in TWRT are given. With it, two results can be found: (i) EE of TWRT when o_{1}=0.1 is the same as when o_{1}=0.9, and EE of TWRT when o_{1}=0.25 is the same as when o_{1}=0.75; (ii) EE of TWRT when o_{1}=0.5 is the maximum. These have been proved in Proposition 4 for f^{″}(o_{1})<0 and f^{′}(0.5)=0, which means the EE of TWRT is the maximum when o_{1}=0.5. In Fig. 3, EEs of DT and OWRT, and EE of relay transmission with none of DL (RTNDL) which corresponding to oneway relay system in [23] are also given. From it, we can know that EEs with DLs in RT are higher than those of RTNDL, which shows the effectiveness of DLs in RT for its SE performance gain.
In Fig. 4, EEs of RT with different d are given. In Fig. 4a, it can be found that EE of OWRT is the maximum when relay node lies in the middle of two source nodes, which has been proved in Proposition 3 for in such case \(\gamma _{o_{1}}\) is the maximum. In Fig. 4b, it can be found that the smaller the d the higher the EE of TWRT, which also has been proved in Proposition 4 for in such case γ_{t} is bigger.
In Fig. 5, outage probabilities are given. The simulation results of outage probabilities are obtained with (6), (11) and (16). From Fig. 5, it can be found that outage performance of TWRT is the best and DT is the worst. This is because with the given C_{th} to avoid disruption, the ST of TWRT is the best and ST of DT is the worst.
Minimize total energy consumption
In this subsection, EEs with minimizing E_{T} are given. At the same time, TPs are optimally allocated. In Fig. 6, EEs when T^{′}=1 ms and T^{′}=0 are given. The T^{′}=1 ms corresponds to the DT, OWRT, and TWRT with idle state, and it means the system can transfer into idle state when the transmission tasks have been completed. The T^{′}=0 ms corresponds to the DT, OWRT, and TWRT without idle state, and it means the system always uses the entail block duration for transmission. The with and without idle states correspond to “w” and “w/o” in the Fig. 6, respectively. From Fig. 6, the following results can be found: (i) EEs when T^{′}=0 are lower than those when T^{′}=1 ms when transmission rate is low; (ii) EEs when T^{′}=0 are equal to those when T^{′}=1 ms when transmission rate is high. This is because when the transmission task is low and it has been completed, the without idle mode still uses entail block duration T_{t} for transmission. In such case, TPs and other CPs, such as transmit and receive CPs, keep consuming. However, the with idle mode changes to the idle mode. In such case, TPs and other CPs can be saved even with idle CPs increase. However, idle CPs are smaller than TPs and other CPs. Finally, the EEs with idle mode are higher. This shows the necessity of TT optimization. While when the transmission rate is high, both with and without idle modes will use entail T_{t} for transmission, and the EEs of them are the same.
In Fig. 7, EEs with different β are given. From Fig. 7, firstly, it can be found that β does not influence the EEs of DT and OWRT, but it influences the EE of TWRT. Secondly, it can be found that the bigger the β, the lower the EE of TWRT. All of these have been proved in Proposition 5. In Fig. 7, EE of RTNDL is also given. From it, we can know that EE of RTNDL is the worst in RT even it compares with EE of TWRT when β=0.6, which also shows the effectiveness of DLs in RT.
Comparing EEs in minimizing E_{T} with maximizing C_{T}, two results can be found: (i) EEs are no more a decreasing function. This is because the OTT in maximizing C_{T} is always T^{′}=0. While in minimizing E_{T}, when the transmission tasks have been completed, the transmission models can change into idle state. Then the influences of CPs become more obvious, when SE is low, the CPs are bigger than TPs and EEs are increasing functions. While when SE is high, TPs are much bigger than CPs, the influences of CPs can be ignored, and EEs are decreasing functions; (ii) EEs of OPA are higher than those of equal power allocation, which shows the significance of OPA.
In Fig. 8, EEs with maximizing C_{T} and minimizing E_{T} simultaneously are also given. From Fig. 8, it can be found that EEs of maximizing C_{T} and minimizing E_{T} simultaneously are higher than that of only maximizing C_{T} or only minimizing E_{T}. This is because the OTPs in it need to consider C_{T} and E_{T}, simultaneously. In Fig. 8, EE of RTNDL which is the worst in RT can also be obtained.
From Figs. 2, 3, 4, 5, 6, 7, and 8, it can be found that among different transmission models, EE of TWRT is almost always the maximum and EE of DT is almost always the minimum. The TWRT has the maximum EE both for relay node’s assistance and its time delay advantage. Also, it can be found that EE of RTNDL is lower than the other RT with DLs. These two phenomenons suggest that relay technique and DLs in RT can improve system’s EE.
Conclusion
In this paper, throughput, energy consumption, and EE with consideration of nonideal PA efficiency, nonnegligible CP consumption, and DLs in RT, through joint optimization of TPs and TT allocation, have been analyzed. The EET with three optimization cases have been given, and several characteristics could be found: (i) with maximizing C_{T} to maximize EE, EEs without idle state are higher, EE of OWRT when relay node is closer to the middle of two source nodes is higher, and EE of TWRT when relay node is closer to node S_{1} and o_{1}=0.5 is higher. At the same time, outage probability of TWRT is the best; (ii) with minimizing E_{T} to maximize EE, EEs without idle state are lower when SE is low and transmission tasks ratio only influences the EEs of TWRT; (iii) with Dinkelbach’s algorithm and onebyone optimal algorithm to maximize EE, EEs are higher than that of only maximizing C_{T} or only minimizing E_{T}; (iv) relay technique and DLs in RT can improve system’s EE. All of these have been verified by the theoretical analysis and the simulations. For the energy harvesting property of SWIPT, the EEs of these three cases with SWIPT can be studied in the further work.
Appendix
Proof of Proposition 4
First, optimal o_{1} and o_{2} are analyzed to maximize γ_{t} under the constraints of o_{1}+o_{2}=1 and {o_{1},o_{2}}∈[0,1]. With P_{1}=P_{2}=P_{r1}=P_{r2}=P_{r}=P, T^{′}=0, ζ_{1}, ζ_{2}, and channel gains, \(\gamma _{{t}_{1}}\phantom {\dot {i}\!}\) and \(\gamma _{{t}_{2}}\phantom {\dot {i}\!}\), can be reformulated as
Assuming P = a, \(b(o_{1})\,=\,\frac {o_{1}{d}^{\alpha }(1d)^{\alpha }}{o_{1}(\!(1\,\,d)^{\alpha }\,\,{d}^{\alpha }\!)\,+\,2{d}^{\alpha }}\), and \(c(o_{1})\,=\,\frac {(1\,\,o_{1}){d}^{\alpha }(1\,\,d)^{\alpha }}{o_{1}(\!(1\,\,d)^{\alpha }\,\,{d}^{\alpha }\!)\,+\,{d}^{\alpha }\,+\,(1\,\,d)^{\alpha }}\), then \(\gamma _{{t}_{1}}\,=\,a\,+\,ab(o_{1})\phantom {\dot {i}\!}\), \(\phantom {\dot {i}\!}\gamma _{{t}_{2}}=a+ac(o_{1})\), and \(\phantom {\dot {i}\!}\gamma _{t}\,=\,\gamma _{{t}_{1}}+\gamma _{{t}_{2}}+\gamma _{{t}_{1}}\gamma _{{t}_{2}}=a^{2}+2a+a(a+1)(b(o_{1})+c(o_{1}))+a^{2}b(o_{1})c(o_{1})\) can be obtained. Ignore all the parts that have no relation with o_{1} in γ_{t}, the problem about optimal o_{1} to maximize γ_{t} can be given as
With f^{″}(o_{1})<0 and f^{′}(0.5)=0, η_{t} is the maximum when o_{1}=o_{2}=0.5 can be obtained for in such case γ_{t} is the maximum.
Second, ORP is analyzed to maximize γ_{t}. With o_{1}=0.5, the ORP problem in TWRT can be given as
where \(b(d)=\frac {{d}^{\alpha }(1d)^{\alpha }}{(1d)^{\alpha }+3{d}^{\alpha }}\) and \(c(d)=\frac {{d}^{\alpha }(1d)^{\alpha }}{3(1d)^{\alpha }+{d}^{\alpha }}\). With g^{′}(d)<0, γ_{t} is a decreasing function of d that can be obtained, which means when the relay node is near node S_{1}, η_{t} is higher for in such case γ_{t} is bigger.
Proof of Proposition 5
It has been shown that the optimization problem of (19) is a convex problem and it can be solved separately; then, the optimal T_{1} and T_{2} in DT should satisfy the KarushKuhnTucker (KKT) conditions. With \(P_{d_{c1}}=P_{d_{c2}}=P_{d_{c}}\), then the KKT conditions can be expressed as
where λ is the Lagrange multiplier. From (34), it can be found that the lefthand sides of (34b) and (34c) are equal to each other, then the optimal TT satisfies \(\frac {\beta C_{T}}{T_{1}}=\frac {(1\beta)C_{T}}{T_{2}}={R_{d}}\). Substituting it into the KKT conditions, it is easy to see that R_{d} is not a function of β. With it, the minimum TEC in DT can further be obtained as
which is not a function of β. It is easy to understand intuitively. With the optimal TT, the transmission tasks on each direction and each bit is transmitted with identical data rate R_{d} and identical time duration 1/R_{d}. Therefore, the energy consumed by each bit is identical no matter in which direction it is. Then the minimum TEC only depends on the C_{T}. The same conclusion can also be obtained in OWRT with the same method. All of these show that η_{d} and η_{o} have no relation with β.
In TWRT, (26a) that is a increasing function of \(\gamma _{t_{1}}=2^{\frac {{3\beta C_{T}}}{T_{twr}W}}1\) can be known. When C_{T} and T_{twr} are known, if β is increasing, then TEC is increasing. It shows that η_{t} is a decreasing function of β.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Abbreviations
 EE:

Energy efficiency
 SE:

Spectral efficiency
 AF:

Amplifyandforward
 EET:

Energyefficient transmission
 TP:

Transmit power
 OPA:

Optimal power allocation
 SET:

Spectralefficient transmission
 TT:

Transmit time
 DT:

Direct transmission
 OWRT:

Oneway relay transmission
 TWRT:

Twoway relay transmission
 OTT:

Optimal TT
 OTP:

Optimal TP
 PAs:

Power amplifiers
 CPs:

Circuit powers
 DLs:

Direct links
 RT:

Relay transmission
 CSI:

Channel state information
 ORP:

Optimal relay position
 ST:

Sum throughput
 TEC:

Total energy consumption
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This research was supported in part by China Scholarship Council and Donghua University Graduate Innovation Fund Program under Grant CUSFDHD2017086.
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The main contributions of Caixia Cai, XueQin Jiang, and Yuyang Peng were to create the main ideas, perform theoretical analysis, and execute performance evaluation by extensive simulation while Runhe Qiu worked as the advisors to discuss, create, and advise the main ideas and performance evaluations together. The authors read and approved the final manuscript.
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Cai, C., Qiu, R., Jiang, XQ. et al. Energyefficiency maximization bidirectional direct and relay transmission. J Wireless Com Network 2020, 156 (2020). https://doi.org/10.1186/s13638020017728
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DOI: https://doi.org/10.1186/s13638020017728
Keywords
 Energy efficiency
 Direct links
 Optimal transmit time
 Optimal transmit power
 Optimal relay position