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 Open Access
Optimal power allocation and throughput performance of fullduplex DF relaying networks with wireless power transferaware channel
 XuanXinh Nguyen^{1} and
 DinhThuan Do^{1}Email author
https://doi.org/10.1186/s136380170936x
© The Author(s) 2017
 Received: 20 January 2017
 Accepted: 23 August 2017
 Published: 7 September 2017
Abstract
In terms of modern applications of wireless sensor networks in smart cities, relay terminals can be employed to simultaneously deliver both information and energy to a designated receiver by harvesting power via radio frequency (RF). In this paper, we propose time switching aware channel (TSAC) protocol and consider a dualhop fullduplex (FD) relaying system, where the energy constrained relay node is powered by RF signals from the source using decodeandforward (DF) relaying protocols. In order to evaluate system performance, we provide an analytical expression of the achievable throughput of two different communication modes, including instantaneous transmission and delayconstrained transmission. In addition, the optimal harvested power allocation policies are studied for these transmission modes. Most importantly, we propose a novel energy harvesting (EH) policy based on FD relaying which can substantially boost the system throughput compared to the conventional halfduplex (HD) relaying architecture in other transmission modes. Numerical results illustrate that our proposed protocol outperforms the conventional protocol under the optimal received power for energy harvesting at relay. Our numerical findings verify the correctness of our derivations and also prove the importance of FD transmission mode.
Keywords
 Energy harvesting
 Fullduplex
 Decodeandforward
 Outage capacity
 Delayconstrained
1 Introduction
It is undoubted that wireless communication systems have attracted much research interest in recent years. In particular, energyaware radio access solutions can be implemented to deal with the massive increase in the consumption of energy in telecommunication networks and the efficient use of power is important for energy optimization. In addition, applications based on internet of things networks have become increasingly popular, so they require novel approaches for energy saving applied in lowpower devices. Energy harvesting is the amount of energy available at the transceiver node powered by surrounding energy sources such as solar, wave, vibration, and radio frequency.
Since energy harvesting plays an important role in relaying regardless of power optimization at relays which are assumed to be powered by ideal power sources. In sensor networks and cellular networks, wireless devices using rechargeable or replaceable batteries are often out of order in a short period of time, as the battery powered devices in such wireless networks usually suffer from limited operating times. Unlike portable devices, the maintenance cost of sensor nodes is often higher in case they are replaced or recharged. Additionally, it is noted that it may be dangerous to replace batteries in toxic environments and powering medical sensors implanted inside human bodies is also challenging. To supply a perpetual power in such networks, energy harvesting is considered as a potential method to prolong lifetime of wireless devices [1, 2]. To take advantage of information transfer over wireless channels, receivers can scavenge power from the transmitted signal. Since ambient radio signals can carry energy and information, energy harvesting brings more major advantages compared to the conventional grid power supply [3].
Since radio frequency (RF) signals are capable of carrying both information and energy, a new concept in green wireless communications was put forward, namely simultaneous wireless information and power transfer (SWIPT). To take advantage of SWIPT, more practical receiver architectures have been developed with two separated circuits to carry out energy harvesting and information decoding [4, 5]. There are two major schemes in the receiver, including time switching and power splitting. The authors in [4] presented performance of system under capability of energy harvesting applied in a simple singleinput singleoutput scenario while multipleinput multipleoutput (MIMO) broadcasting scenarios was introduced in [6].
In order to obtain practical insights in term of optimal time and power allocation. The work in [7] focused on time allocation policy for the two transmitters in case the efficiency of energy transfer is maximized by an energy beamformer under the impact of Channel State Information (CSI) received in the uplink by the energy transmitter. Furthermore, in [8], the time fraction in TSR impacts on the optimal throughput and such parameter can be found in a numerical method.
We next consider several system models regarding existing cooperative networks with capability of energy harvesting. Firstly, the employed relay in SWIPT networks [9, 10] or the source terminal [11] can harvest energy from the radiated signal of the source terminal or the employed relay. Secondly, in multihop networks, energy is transferred to remote terminals via multihop [12, 13]. In multihop systems, the high path loss of the energybearing signal can be eliminated [12]. Unlike [12], the authors in [13] investigated a multiantenna relay adopting two separate terminals with capability of information processing and power transfer, respectively, and the expressions of the transmission rate and outage probability were presented under the impact of remote energy transfer. In addition, relay selection is considered as solution to determine a tradeoff between the efficiency for the information transmission and the amount of energy forwarded to the energy receivers [14–16].
Furthermore, fullduplex (FD) mode was evaluated, in which it allows transmitting and receiving signals at the same frequency band at the same time slot. Various theoretical analysis and practical designs have been conducted in terms of FD networks like in [17–21]. Thanks to the use of FD mode, the resources are utilized more efficiently and it can double the spectral efficiency compared to halfduplex (HD) mode. However, due to practical constraints, the performance of FD communication can be affected by the selfinterference (SI) stemming from FD node transmission.
In addition, energy harvesting along with throughput optimization has been mentioned in previous works. In [22] and [23], throughput optimization with constraints was studied under a static channel condition for obtaining best efficiency of energy harvesting transmitters.
Moreover, energy harvesting based on power control policies for wireless powered transmission over fading channels suffer from several problems, i.e., the randomness of RF energy source, wireless fading channels and the maximum power constraints. To address this situation, several existing works have considered offline optimal power control designs for fading channels [24, 25]. The work in [25] considered that the offline optimization in an efficient optimal solution was presented to achieve optimal energy efficiency. Although the authors in [26] investigated the offline scheduling and formulated the corresponding performance optimization problems of twoway relay networks, the statistics of energy and fading channels are assumed to be parameters at the transmitter, and optimal function can be derived in a numerical manner under high computation.
The authors in [27] considered that the optimal time splitting coefficients for the fullduplex dualhop relaying lead to enhance the system throughput in comparison with the traditional halfduplex relaying scheme for all kinds of modes including instantaneous transmission, delayconstrained transmission, and delaytolerant transmission. The energyconstrained FD relay node can be applied in the multipleinput singleoutput (MISO) system; the optimal power allocation and beamforming design are investigated in [28]. In another line of research, the FD decodeandforward system using the timeswitching protocol is embedded in the multiple antennaassisted relay to obtain more energy from the source and transfer signal to the destination as in [29, 30]. Interestingly, in order to achieve the maximal throughput performance, the optimal time switching coefficient is adaptively selected based on channel state information (CSI), accumulated energy, and threshold signaltonoise ratio (SNR) [31, 32] and [33].
Motivated from the previous works [27], we focus on optimal throughput of a twohop case, where RF energy harvesting powers the relay. In this paper, the impact of FD transmission is investigated in terms of the system throughput to determine the performance of RF energy harvesting relaying system. The twoantenna configuration is proposed in the FD mode, where the relay is equipped with two antennas, one for transmitting signal and the other one for receiving signal simultaneously. In this paper, two different transmission modes are investigated, namely instantaneous transmission and delayconstrained transmission. Furthermore, we examine the throughput of DF relaying protocols and characterize the fundamental tradeoff between energy harvesting and system throughput. In order to compare with the effect of the FD relaying architecture, the HD relaying architecture is also investigated. The main aim of this paper is that FD relaying is an attractive and promising solution to enhance the throughput of RF energy harvestingbased relaying systems.
Comparing to [27] and [28], although this work consider a same system model with those works, our investigation also design a novel wireless power transfer strategy to improve the system performance. The authors in [27] consider a conventional time splitting protocol, in which relay switches from energy harvesting to information transfer with fixed EH fraction time allocated. While [28] design a selfenergy recycling strategy for energy harvesting at relay, this EH manner can collect transmitted signal itself and scavenge to energy. Contrarily, our work redesigns the conventional time switching protocol as in [27] to a novel EH schedule. The proposed EH will acquire the channel state information (CSI) to allocate amount of EH time. In particular, the relay transmission power can be preset level. The magnitude of self interference at relay thus can be controlled based on relay transmitted power, which can improve the system performance.

A new protocol for wireless power transfer called time switching aware channel protocol (TSAC) is proposed in FD DF relaying networks and further analysis is presented as well.

We provide analytical expressions in instantaneous transmission mode for both cases, i.e., one antenna and dual antennas at relay node in EHbased networks.

The outage probability and average throughput for delayconstrained transmission mode are derived in closedform expressions for tractable computation. The optimal values can be achieved in various simulation results.

The advantages of the proposed protocol are also compared with the previous works. The most critical performance metric (i.e., optimal throughput efficiency) is thoroughly analyzed and systematically validated via comparative simulations.

It can be seen that optimal power allocation leads to enhance throughput and resulting in system performance in comparison with nonpower allocation solution in the literature.
The remainder of the paper is organized as follows. In Section 2, the energy harvesting cooperative scenario with one source node equipped two antennas in FD mode is considered and two different strategies for single antenna or dual antennas for energy harvesting are investigated. In Section 3, the power allocation in instantaneous transmission is evaluated while outage probability and optimal throughput in delayconstraint are given in Section 4 for performance evaluation. Section 5 examines HD mode for performance comparison with FD relay. Numerical results and useful insights are provided in Section 6. Eventually, Section 7 draws a conclusion for our paper.
2 System model and energy harvesting protocol
2.1 System model
2.2 Channel model
We assume that the \({\mathcal {S}} \to {\mathcal {R}}\) and \({\mathcal {R}} \to {\mathcal {D}}\) channel links include both largescale path loss and statistically independent smallscale Rayleigh fading. We denote d _{1} is distance between source and relay node and d _{2} is distance between relay and destination node. We also denote the main channels such as h and k are the links from the source to first antenna and second antenna at relay, respectively, and g is the channel from relay to destination node. It is also assumed that the main channels experience Rayleigh fading and remain constant over the block time T and varies independently and identically from one block to the other.
In this paper, we also use character f to represent the SI link at relay node as normal manner. Due to the short distance of this link, the lineofsight (LoS) path is likely to represent to SI channel; hence, it can be shown that the Rician distribution can handle such SI channel as [18]. However, due to the complicated Rician fading probability density function (PDF), the analytical expressions become extremely difficult. Fortunately, the alternative model of the Nakagamim fading distribution provides a very good approximation to the Rician distribution. Motivated by this, and to simplify in the analysis, we adopt the Nakagamim fading with fading severity factor m _{ f } and mean λ _{ f } to model the loop interference channel in this paper.
So that h^{2} and k^{2} are independent and identically distributed (i.i.d.), exponential random variables with mean λ _{ h } and λ _{ k }, where λ _{ h }=λ _{ k }, g^{2} is exponentially distributed with mean λ _{ g }. The selfinterference power link at relay, i.e., f^{2}, is a Gamma random variable distributed as Γ(m _{ f },λ _{ f }/m _{ f }), in this paper we also assume m _{ f } is an integer number.
The transmit power of source and relay are represented by \( P_{\mathcal {S}} \) and \( P_{\mathcal {R}} \), respectively. Due to the shadowing effect, the direct transmission between source node and destination node does not exist [2, 5, 8, 20, 27, 29, 32]. The FD mode causes selfinterference, which can be addressed by the novel methods in the literature as [17, 19] and these algorithms are beyond the scope of our paper. Unfortunately, the residual selfinterference still exists after interference suppression in the practical receiver architecture and it also impairs the performance of FD relay networks as [17–21, 27]. In this paper, we mainly focus on the impact of the residual selfinterference on system performance in terms of the harvested power.
2.3 TSAC protocol description
In this subsection, the energy harvesting protocol is presented. In spirit of [31–33] suggesting adaptive time switching strategies for SWIPT system, we redesigned TS protocol related to CSI for FD relay transmission mode, named TSAC protocol. The detail of modified TSAC protocol is described as below. The harvested energy is stored in a rechargeable battery and then totally used to feed power circuits and transmit information to the destination node. Particularly in TSAC policy, each communication block time is slit into two slots, including wireless power transfer (WPT) slot and wireless information transfer (WIT) slot as mentioned in [4–7, 11, 12, 27, 31–33]. In each block time, WPT slot represents the first α _{ i } T of block time while the WIT slot stands for the rest of (1−α _{ i })T of block time. During WIT phase, the source transmits its symbol toward the intermediate relay simultaneously, the cooperative relay retransmits its decoded symbol to destination at the same time and bandwidth. The relay thus suffers from loop interference (LI).
List of important symbols
Symbol  Definition 

\(P_{\mathcal {S}}\)  The fixed transmission power, preset at source node. 
\(P_{\mathcal {R}}\)  The fixed transmission power, preset at relay node. 
\({\mathcal {E}}_{i}\)  The relay transmission energy during time slot ith corresponding to preset power, \(P_{{\mathcal {R}}}\). 
\(P_{{\mathcal {R}},i}^{EH}\)  The harvested power from EH at relay node at time slot ith. 
\({\mathcal {E}}^{EH}_{i}\)  The relay transmission energy during time slot ith corresponding to harvested power \(P_{{\mathcal {R}},i}^{EH}\). 
α _{ i }  The duration time allocated to EH in time slot ith. 
This process can be done as steps below. Because relay only harvests sufficient amount of energy, \({\mathcal {E}}_{i}^{EH} = f \left (\alpha _{i} \right)\) (where f(α _{ i }) means that function of α _{ i }), one can exactly determine the EH time by equaling amount of harvested energy and that of preset energy, i.e., \({\mathcal {E}}_{i}^{EH} = {\mathcal {E}}_{i}\). Finally, the EH time duration, α _{ i }, is derived. It is noted that the suggested TSAC protocol does not require more additional time slot since the total frame for communication is the same as [27] (see Remark 1), and the preset relay transmit power is a constant value (\(P_{\mathcal {R}}\)) while the EH time is a function of the random variable WPT channel gain(s) as [31–33].
As aforementioned, to exactly determine this duration of energy harvesting time, channel gain(s) of WPT link is an important parameter(s). Therefore, we assume that the channel state information (CSI) during the first hop is available at source and relay node, which can be obtained by using novel estimation algorithm. Interestingly, the TSAC protocol adjusts the WPT time duration, α _{ i }, in each time slot to satisfy amount of installed power \(P_{\mathcal {R}}\). In contrast, the fixedtime allocation protocol in [5, 27], the harvested power, \(P_{\mathcal {R}}\), vary in each block based on channel gain.
2.4 Signal model
where i is the block time index, x _{ i } and \(\hat {x}_{i} \) are message symbol at \( {\mathcal {S}} \) and decoded symbol at \( {\mathcal {R}} \) with unit power and zero average, respectively. We also assume that the relay decodes when \( {\mathcal {S}}{\mathcal {R}} \) link does not suffer from outage. It is assumed that \(n_{{\mathcal {R}},i}\) and \(n_{{\mathcal {D}},i} \) are additive white Gaussian noise (AWGN) at \( {\mathcal {S}} \) and \( {\mathcal {D}} \) in block time ith, respectively.
where noise terms at \( {\mathcal {R}} \) and \( {\mathcal {D}} \) are zero mean and variances of \({\sigma _{{\mathcal {R}},i}^{2} }\), \({\sigma _{{\mathcal {D}},i}^{2} }\), respectively.
In WPT time slot of SWIPT, we investigate the performance of two schemes based on the number of antennas in EH phase at relay as [27], where (i) only one antenna is responsible for receiving signals and harvesting RF signals in EH phase or (ii) either two antennas equipped for FD communication can be used during the EH stage.
2.4.1 Single antenna for energy harvesting
where 0≤η≤1 is the energy conversion efficiency that depends on the rectification process and the energy harvesting circuitry. It is noted that we ignore the harvested energy from noise. So that the harvested power can be determined during (1−α _{ i })T as \(P_{{\mathcal {P}},i}^{EH}= {\mathcal {E}}_{i}^{EH} / (1  \alpha _{i})T\).
2.4.2 Dual antennas for energy harvesting
Remarks 1
In (9) and (12), the duration of time allocated in energy harvesting phase is a function of some variables, including channel gains, optimal power at relay node, the distance between \( {\mathcal {S}} \) and \( {\mathcal {R}} \), i.e. d _{1}, energy harvesting coefficient and power transmission at source, i.e. \(P_{\mathcal {S}} \). Specifically, this time fraction is always less than one, i.e., α _{ i }<1, which implies the allocated time which is required for simultaneous wireless information and power transfer.
Remarks 2
Consider energy harvesting time in (9) and (12), to convenience in represent we use two characters \(\alpha _{i}^{single}\) and \(\alpha _{i}^{dual}\) to denote α _{ i } in (9) and (12), respectively. We can observe that, since k _{ i }^{2}≥0 and other parameters are invariable then \(\alpha _{i}^{single} \ge \alpha _{i}^{dual}\). This implies that the time EH with dual antennas is less than that with single antenna. In other words, the amount of WIT time with dual antennas EH case is greater than that with single antenna EH case.
Remarks 3
The proposed modifying TSAC protocol for FD cooperative relaying system is not more require additional time slot since the total frame for communication is as same as [27] (see Remark 1), and the prior installed relay transmit power is a constant value (\(P_{\mathcal {R}}\)) while the EH time is a function of the random variable WPT channel gain(s) as [31–33].
The previous relevant works and their contributions
Reference  EH protocol  Transmission mode  Relay mode  Works area 

Nasir et al. [5]  TS, PS  Half duplex  AF  Throughput analysis 
Nasir et al. [33]  Continues and discrete TS  Half duplex  AF, DF  Throughput analysis 
Chen et al. [17]  Non EH  Full duplex  DF  Optimal power allocation 
Zeng and Zhang [28]  TS with energy recycling  Full duplex  AF  Optimizing relay transmission power 
Zhong et al. [27]  TS  Full duplex  AF, DF  Optimizing throughput 
Ding et al. [32]  Adaptive TS  Half duplex  AF, DF  Optimizing throughput 
Ding et al. [15]  PS with storage  Half duplex  DF  Outage performance analysis 
Krikidis [16]  PS with storage  Half duplex  DF  Outage performance analysis 
Our work  TS aware channel  Full duplex  DF  Optimizing transmit power at relay 
Throughput analysis. 
CSI requirement for proposed TSAC protocol schemes
EH type  Transmission mode  CSI requirements  Channel information 

Single antennas  Instantaneous  High  h _{ i }^{2}, g _{ i }^{2}, f _{ i }^{2} 
Single antennas  Delayconstrained  Medium  h _{ i }^{2}, and distributions of g _{ i }^{2}, f _{ i }^{2} 
Dual antennas  Instantaneous  High  h _{ i }^{2},k _{ i }^{2}, g _{ i }^{2}, f _{ i }^{2} 
Dual antennas  Delayconstrained  Medium  h _{ i }^{2},k _{ i }^{2}, and distributions of g _{ i }^{2}, f _{ i }^{2} 
3 Optimal power allocation for maximized instantaneous throughput
Energy harvesting is an effective method to enhance the performance of the powerconstrained relay with available RF signals, in which the transmission power at relay is allocated to maximize the instantaneous throughput. In this paper, the optimal transmitted power in energy harvesting scheme is proposed for FD relay networks.
3.1 Energy harvested by single antenna
Theorem 1
where \(\pi _{0} = \frac {{d_{1}^{m} \sigma _{\mathcal {R}}^{2} }}{{P_{\mathcal {S}} \left  h \right ^{2} }}\), \(\pi _{1} = \frac {{d_{1}^{m} \left  f \right ^{2} }}{{P_{\mathcal {S}} \left  h \right ^{2} }}\), \(\pi _{2} = \frac {{\left  g \right ^{2} }}{{d_{2}^{m} \sigma _{\mathcal {D}}^{2} }}\), \(\pi _{3} = \frac {{\eta P_{\mathcal {S}} \left  h \right ^{2} }}{{d_{1}^{m} }}\), \(\Delta = \pi _{0}^{2} + \frac {{4\pi _{1} }}{{\pi _{2} }}\), and \({\mathcal {W}}\left (x \right)\) is the Lambert function in [34], where \({\mathcal {W}}\left (x \right)\) can be found due to the problem solving of \({\mathcal {W}}\exp \left ({\mathcal {W}} \right) = x\).
Proof
Please see in Appendix A Proof of Theorem 1. □
3.2 Energy harvested by dual antennas
The optimal relay transmission power is derived in the same way as (14).
Theorem 2
where \( \pi _{4} = \frac {{\eta P_{\mathcal {S}} \left ({\left  h \right ^{2} + \left  k \right ^{2}} \right)}}{{d_{1}^{m} }} \) and other related parameters are defined as in Theorem 1.
Proof
The Theorem can be explained by using in a similar way to Theorem 1. □
Remarks 4
In practical wireless system, only causal channel information and harvested energy are useful in calculation of power allocation. Unfortunately, the offline power allocation policy is not readily applicable in reality. Considering at a given time slot, the CSI in the second hop of the relaying network and the upcoming harvested energy are not known in advance and hence power allocation is evaluated in stochastic circumstance. However, such solution only requires low complexity when comparing online power allocation which is high computational complexity and may not be implementable in practice.
4 Analysis of outage probability and throughput in delayconstrained transmission mode
and the expected value of outage probability can be obtained by substituting (3) and (4) into (19).
As aforementioned in protocol description subsection 2.3, since the relay node will harvest a sufficient amount of energy, \({\mathcal {E}}_{i} \), before operating WIT phase, it is noted that the relay be certainly performed because the harvesting energy time is always less than 1, see Remark 1. Therefore, the relay transmit power \(P_{\mathcal {R}}\) is a constant value during the transmission duration, see Remark 3. This quantity of power is preset before (such as by technicians). It is worth noting that the \( \gamma _{i}^{\mathcal {R}},\gamma _{i}^{\mathcal {D}} \) are independent in term of \( P_{\mathcal {R}} \) as in [31–33].
where E{.} is the expectation function.
4.1 Energy harvested by single antenna
The system throughput in this case can be determined as (22) and the next theorem is proposed.
Theorem 3
Proof
The detailed explanation of the Theorem 3 is provided in Appendix B Proof of Theorem 3. □
Alternatively, we can use the following closedform when the SI channel modeled by Rayleigh fading as following corollary.
Corollary 1
Proof
The Corollary 1 can be obtained easily by substituting m _{ f }=1 into the expression in Theorem 3. □
Extracting (23) by using (24), the optimal transfer power at relay with the aim of optimizing the system throughput can be obtained. Since \(\tau _{DC} \left ({P_{\mathcal {R}}} \right)\) is a concave function of \(P_{\mathcal {R}} \), the optimal value \(P_{\mathcal {R}}^{opt} \) can be obtained by solving the equation \({{\partial \tau _{DC} \left ({P_{\mathcal {R}}} \right)}}/{{\partial P_{\mathcal {R}} }} = 0\). Although the derivation of closedform optimal expressions is challenging, the optimal value can be obtained by using numerical simulations which are presented in the next section.
4.2 Energy harvested by dual antennas
In this case, we also study the throughput in the delayconstrained mode with the following theorem.
Theorem 4
Proof
Please see in Appendix C Proof of Theorem 4. □
When the SI channel is modeled by Rayleigh fading connection, the throughput performance of system is given as corollary below.
Corollary 2
Proof
It is easy to obtain the Corollary 2 by substituting m _{ f }=1 into the expression in Theorem 4. □
Remarks 5
Regarding impact of selfinterference (SI) cancellation, the active SI suppression methods were shown experimentally to be capable of facilitating FD at short range communication. To apply energy harvesting in a real FD system, it is critical to accurately measure and suppress the SI in such FD networks. In this paper, we investigate SI in system performance through numerical simulation results.
5 Halfduplex relaying networks
In HD transmission mode, the relay node uses single antenna for energy harvesting and information processing. In order to compare with the FD relaying systems, we consider different transmission modes. Some of the results have been derived in [31–33]; however, we present here to make our work selfcontained.
5.1 Instantaneous transmission mode
The optimal allocation for power transmission at relay is similar to (14)
Theorem 5
where θ=π _{3} in case of single antenna, θ=π _{4} in case of dual antennas. And π _{0},π _{2},π _{3},π _{4} are defined as below Theorem 1.
Proof
The proof is similar to that of Theorem 1. □
5.2 Delayconstrained transmission mode
In comparison with the calculation of throughput in FD mode, in (21), the factor 2 can be seen by the denominator denoting as transmission efficiency. In following calculation, we can obtain two theorem which are solved similarly as that in Theorem 4.
Theorem 6
Theorem 7
where \(\chi _{1} = \frac {{P_{\mathcal {R}} d_{1}^{m} }}{{2\eta P_{\mathcal {S}} }},\vartheta _{1} = \frac {{\gamma _{0}^{HD} d_{1}^{m} \sigma _{\mathcal {R}}^{2} }}{{P_{\mathcal {S}} }}\).
6 Numerical results
In this section, numerical simulation results are demonstrated to validate analytical expressions as concerns in the previous section, and the impact of key system parameters is investigated in detail in terms of system throughput. The energy harvesting efficiency is set η=0.8, while the path loss exponent is set m=3. The distances, d _{1} and d _{2} are normalized values which are set d _{1}=3 and d _{2}=1, respectively. The simulation environments are associated with the detailed parameters as \(\gamma _{0} = 10~\text {dB},\lambda _{f} = 0.01,\lambda _{h} = 1,\lambda _{k} = 1,\lambda _{g} = 1,P_{\mathcal {S}} = 26~\text {dB}\) and noise terms as \(\sigma _{\mathcal {D}}^{2} =  5~\text {dB},\sigma _{\mathcal {R}}^{2} =  10~\text {dB}\). In this subsection, we present simulation results to verify the analytical results. The throughput performance is calculated by averaging the throughput values over a 100,000 blocks, while fading channels for each block is perfectly independent.
7 Conclusion
In this paper, the throughput of FD and HD relaying in RF energy harvesting systems is investigated. Interestingly, the number of antennas equipped at each relay has significant influence to throughput performance due to the harvested energy at the relay node. Regarding optimal throughput, analytical expressions for the outage probability and throughput capacity of the system were derived. Therefore, the optimal time switching of energy harvesting was comprehensively evaluated. It is confirmed that by employing dual antennas at relay for energy harvesting is beneficial, and the throughput gain is significant when transmit power at source and the received power at relay are carefully calculated. In addition, in comparison with HD relaying networks, our results indicate that FD relaying can substantially boost the system throughput with optimal power allocation policy at energy harvestingenabled relay. Via mathematical and numerical analysis, the optimal throughput in both instantaneous transmission mode and delay constraint transmission mode can be obtained. More importantly, in order to compute optimal time switching fractions in energy harvesting protocol socalled TSAC, it solely relies on the channel statistics without the need of instantaneous CSI, and it has become an attractive solution to implement in future RF energy harvesting cooperative systems. Finally, for enhancing energy harvesting efficiency, future works should take into account the extra energy with multiple antenna system model (MIMO) to transmit wireless power.
8 Appendix
8.1 A Proof of Theorem 1
In case \(\pi _{2} P_{R} < \frac {1}{{\pi _{0} + \pi _{1} P_{\mathcal {R}}}}\) and combine the power at relay, e.g \(P_{\mathcal {R}} \), is the nonnegative number, we achieve \({0 \le P_{\mathcal {R}} < \frac {{\sqrt \Delta  \pi _{0} }}{{2\pi _{1} }}}\).
This is end of proof.
8.2 B Proof of Theorem 3
where F _{ X }(x) is the cumulative distribution function (cdf) function of random variable X.
where f _{ X }(x) is the pdf of random variable X, step (a) is done by f _{ i }^{2} following the gamma distribution and h _{ i }^{2} experience exponential distribution, stage (b) is revealed by using (eq. 3.381.4) in [35].
Substituting (44), (45), (46), and (47) into (43), Theorem 3 is simply derived. This ends the proof.
8.3 C Proof of Theorem 4
where the last integral can be derived by applying (Eq. 3.352.2) given in [35] To this end, pulling everything together and after some simple mathematical manipulations, Theorem 4 is derived. This is the end of the proof.
Declarations
Funding
The author declares that there is no fund of this work.
Authors’ contributions
The individual contributions of each authors to the manuscript are the same. Both authors read and approved the final manuscript.
Competing interests
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