1 Introduction

Green wireless communication has been fascinated great interest in the investigation of energy harvesting (EH) and information relaying in recent years. With pioneering growth of data services, the power demand for wireless device is constantly increasing that can lead to battery depletion problem. To overcome the battery depletion problem, EH is a promising green technology to provide effective power supply to wireless devices. The EH components and power management circuits have been studied [1], where energy is harvested with the help of RF signal, motion and vibration, temperature difference, and photovoltaic cells. In some practical applications such as cellular communications and wireless sensor network, the EH technique is used to improve the efficiency and the lifetime of entire communication system.

With RF signal, the EH is state-of-the-art in green wireless communication to provide energy efficient wireless networks. It can be classified into three categories. In first approach, ideal receiver design is able to tradeoff transmitting energy and transmitting information simultaneously using a capacity-energy function [2]. The information rate has been maximized with minimum energy constraints for a single point-to-point channel [2, 3] for a set of parallel point-to-point channels. In [4], authors have considered two-way communication system for EH and information transmission. In second approach, the authors have studied three nodes multi-input–multi-output (MIMO) broadcast wireless system to investigate optimal transmission strategy to achieve tradeoff of information rate and energy transfer, where one receiver harvests energy and another receiver decodes information separately from signals sent by a common transmitter [5]. In this scenario, the EH receiver circuit cannot decode the information signal directly. To overcome this issues, they have further designed the practical receivers such as power splitting (PS) receiver and time-switching (TS) receiver. In power-splitting case, the PS receiver splits received signal into two separate streams of signal with different power at each antenna, one of signal stream sends to EH receiver and other is directed to information decoding (ID) receiver, whereas in time-switching case, each receiving antenna switches periodically between EH receiver and ID receiver.

In third approach, several authors have proposed wireless relaying networks [6–29], where wireless devices exchange information and transfer energy via relay/relays. The energy efficiency and energy-efficient network are interesting topics in wireless networks which have been studied in [30–37]. The authors have introduced network coding based energy efficient multicast routing [30] and topology control based cooperative multicast algorithm [31] to achieve better performance of multi-hop wireless networks. Moreover, an energy efficient cooperative multicast routing in multi-hop wireless networks for smart medical applications has been presented in [32]. Further, they have studied energy-efficient problem with quality of service (QoS) [33] and a robust energy efficiency routing algorithm by combining sleeping redundant link algorithm with minimum critical routing algorithm [34] to reduce the energy consumption and improve energy efficiency in large-scale cloud computing networks. In the previous work, it is difficult to achieve optimal network selection. In [35], authors have introduced the multi-criteria access selection algorithm to select suitable network in heterogeneous wireless vehicular networks to achieve efficient energy of networks. The spectrum allocations and the inter-cell interference (ICI) are challenging issues in multi-cell networks. To overcome such issues, the authors have been applied soft frequency reuse (SFR) based energy efficiency algorithm to allocate spectrum resources and reduce the ICIs in [36]. In [37], the energy-efficient multi-constraint routing algorithm with load balancing has been proposed to improve the energy-efficiency of networks for smart city applications. In [38], the authors have studied fine time granularity from sampled traffic traces in high speed backbone networks for software defined networks applications to recover better end-to-end network traffic.

Motivated by advancement of green wireless communication, we consider FD two-way AF relaying network, where energy constrained relay and user harvest energy from high power access point (denoted as H) and information transmits only from user to H via relay, where the relay also harvests energy with self-energy recycling. With application of FD operation, nodes transfer wireless information and power over same frequency band and improve the spectral efficiency of wireless relaying system. The foremost contributions can be summarized as follows.

  1. 1.

    Considering two-way AF relaying system, the FD-based PS and TS protocols are proposed to improve the spectral efficiency of relaying network.

  2. 2.

    We investigate the closed-form expressions of optimal PS ratio and end-to-end ergodic capacity for PS protocol. Further, we consider TS protocol to derive the TS ratio and end-to-end ergodic capacity.

  3. 3.

    To obtain the PS ratio and the TS ratio results, we first express the end-to-end signal-to-interference-plus-noise ratio (SINR). Then, we formulate the PS and TS ratios so that the end-to-end channel capacity should be maximized. After that the quasi-convex/concave optimization problem is solved to achieve the PS and TS ratios.

  4. 4.

    The numerical results show that our proposed FD-based EH schemes outperform the HD-based EH schemes (considered as the benchmark schemes).

The remainder of this paper is organized as follows. The related works and problem statement are introduced in Sect. 2. The Sect. 3 describes the system model and evaluates the ergodic capacity. In this section, we also investigate the PS and TS protocols and derive optimal PS ratio and optimal TS ratio and end-to-end ergodic capacity for Rayleigh fading. The numerical results and discussions are presented for better understanding of validation of our proposed FD-based EH schemes in Sect. 4. Finally, Sect. 5 concludes this paper and provides future works.

2 Related works

Practically wireless device cannot distinguish EH and ID from RF signal at same time. To overcome this challenge, TS protocol and PS protocol are studied for three nodes relay network in [6] to support EH and information processing (IP) at relay. In [7, 39], authors have considered wireless powered communication network (WPCN), where users harvest energy from hybrid access point (HAP) in downlink (DL) channel and then users transmit information to HAP in uplink (UL) channel using harvest-then-transmit (HTT) protocol by time-division-multiple-access (TDMA) [7] and for harvest-then-cooperate (HTC) protocol with TS protocol [39], where the HTC protocol is superior as compare to HTT protocol. The [39] shows that the WPCN experiences double near-far problem. To overcome double near-far problem, they have investigated to allot shorter/longer time to near/far users in UL channel. However, user unfairness and cooperation in WPCN for double near-far problem have been still known. Further, in [40], they have considered WPCN to maximize the weighted sum rate of two users with jointly optimizing time and power allocation (PA) in the network for wireless energy transfer (WET) in the DL and WIT in the UL and at the same time user unfairness is achieved by overcoming the doubly near-far problem.

Impacts of very fast data rate in the network reduce transmission efficiency and sometime it causes battery depletion problem. In the existing work, design a protocol for extra accumulation energy is still an open problem. In [8], authors have presented two-hop amplify-and-forward (AF) and decode-and-forward (DF) relaying strategies to harvest extra energy at relay for future use. However, co-channel interference (CCI) signal based EH for wireless system has not been studied. Further, RF-based two-hop DF relaying network has been presented in [9], where relay harvests energy from the superposition of received information signal and CCI signals using TS protocol and then using harvested energy to forward the decoded signal to destination. In EH relaying system, coherent communication consumes more power for accurate estimation (or, tracking) of instantaneous channel state information (ICSI). To overcome this limitation, authors have investigated non-coherent framework for simultaneous wireless information and power transfer (SWIPT) and studied non-coherent EH protocols to reduce the power consumption of ICSI [10]. In [11], SWIPT for non-generative MIMO orthogonal frequency-division multiplexing (MIMO-OFDM) technique has been considered to achieve end-to-end information rate using TS and PS protocols. In previous work, multicarrier EH with signal redistribution at relay has not been included. Further, authors have studied two-hop multicarrier DF relaying network [12] with TS protocol and PA protocol to maximize the end-to-end rate, where harvested energy at relay can be reorganized among all the subcarrier. The benefits of partial relaying in EH networks have been implemented to achieve throughput of source by providing stable region of network [13]. Unlike [10–13], in [14], authors have considered RF chain for RF-EH device to obtain throughput for orthogonal relay channel with joint source and relay PA strategy. To reduce the difficulty of channel state information (CSI) overhead, the adaptive TS-based EH protocols for AF and DF relaying have been investigated to achieve maximum throughput efficiency [15].

Researchers have studied two-way relaying channels (TWRCs) including multi-access broadcast (MABC) and time-division broadcast (TDBC) for EH with implementation of various relaying strategies such as DF, AF, compress-and-forward (CF) and compute-and-forward (CaF) in [16–20]. Considering EH in TWRCs, authors have investigated transmission power strategies to achieve maximum sum-throughput, where energy constrained relay performs full-duplex (FD) and half-duplex (HD) operation with relaying techniques such DF, AF, CF and CaF for EH [16]. However, bidirectional wireless information and power transfer with PS and TS protocols have not been considered. In [17], authors have studied BWIP relaying to design the PS and TS protocols to achieve end-to-end throughput for EH at both the relay and the user. The bidirectional wireless information and power transfer policies for EH need to improve performance of system and stability of wireless networks. Considering TS receiver, authors have designated the EH and IP phases and studied dual-source, single-fixed-source and single-best-source power transfer (PT) protocols for EH. A comparative framework for each of the wireless PT policies has also been studied in IP phase for bidirectional relaying using MABC and TDBC [18]. However, multi-pair energy-constrained with full-powered access point has not been considered. Further, in [19], authors have presented WPCN with multi-antenna two-way AF relaying and with single-antenna multi-pair uses to study ergodic spectral and the energy efficiency with HTT protocol for unknown CSI, partially known CSI and perfectly known CSI in PT phase. Moreover, the EH by RF signal and renewable energy (RE) are designed to power the relays using PS protocol for two-way relaying networks (TWRNs) that have been studied in [20]. In the existing works, we notice that most of the authors have considered HD relaying mechanism that limit the data transmission and reception simultaneously at relay in the same frequency band.

To increase the spectral efficiency and to boost up the EH techniques, FD green wireless communication is achieved with advance technology of antenna and signal processing capability. In [21–29], authors have studied FD relaying networks to improve the spectral efficiency and EH capability. In [21], authors have suggested a two-stage iterative cancellation scheme to improve the accuracy of self-interference cancellation for wideband FD wireless communication system. Two-hop FD relaying network has been considered to investigate TS protocol first time by [22] and later extended in [23] for multi-input-and-single-output (MISO) with equally divided phases, and then PS protocol by [24] where relay harvests energy from RF signal. Recently, in [25], authors have studied the distributed switch and stay combining technique and imperfect CSI with FD relay selection strategy. In [26–29], authors have presented in band FD wireless powered TWRNs to study TS protocol [26, 27] for PS protocol, where relay harvest energy from both the sources with self-recycling. The relay selection does not consume extra power from batteries and harvests more energy to boots up the relaying system. In [28], authors have considered relay selection in TWRNs to design PS-based FD energy harvesting scheme to improve the spectral efficiency of relaying network, where relays transmit over orthogonal channels to avoid inter-relay inference. Moreover, the time division duplexing static PS protocol and FD static PS protocol have been investigated for wireless power TWRNs by considering simple relay selection strategy [29]. Best of our knowledge full-duplex wireless information and power transfer with energy constrained of two nodes for two-way relaying network via relay has not been considered still now yet.

3 System model and formulation of ergodic capacity

In this section, we introduce system model and related channel parameters. The power-splitting protocol and time-switching protocol are described for FD two-way AF relaying networks. We derive the PS ratio and ergodic channel capacity for PS protocol. Further, we investigate TS ratio and ergodic channel capacity for TS protocol.

3.1 System model and preliminaries

We consider FD two-way relaying network, where hybrid power access-point H and user u transmit energy and information respectively to u and H via relay R which is shown in Fig. 1. The solid lines represent the energy flow in forward direction (i.e., from H to u via R) while dotted lines represent information flow in reverse direction (i.e., from u to H via R). We assume that the direct link between H and u are not available due to heavy channel attenuation. The H and R use dual antenna for FD operation, where one antenna is used for transmission and other for reception. The FD operation indicates that the wireless device transmits and receives signal over same frequency band. Furthermore, we assume that the R uses AF relaying strategy for signal transmission and signal reception.

Fig. 1
figure 1

Full-duplex energy harvesting two-way relaying network consists of hybrid power access-point H, relay R and user u, where solid lines represent energy flow and dotted lines represent information flow

Full size image

Let \(h_{1}\) and \(h_{2}\) be the channel coefficients between channel H-to-R and channel u-to-R. For simplicity, we assume that the channels are reciprocal in each slot i.e., channels H-to-R and u-to-R are identical to R-to-H and R-to-u respectively. We assume that the \(h_{H}\) and \(h_{\varvec{R}}\) are the residual self-interference channel response of H and R respectively. The fading gains \(\left| {h_{1} } \right|^{2}\), \(\left| {h_{2} } \right|^{2}\) and residual self-interference (RSI) channel gain, \(\left| {h_{R} } \right|^{2}\) are faded Rayleigh channel and followed exponential distribution with means \(\mu_{1} = {\mathbb{E}}\left\{ {\left| {h_{1} } \right|^{2} } \right\}\), \(\mu_{2} = {\mathbb{E}}\left\{ {\left| {h_{2} } \right|^{2} } \right\}\) and \(\mu_{R} = {\mathbb{E}}\left\{ {\left| {h_{R} } \right|^{2} } \right\}\) respectively, where \({\mathbb{E}}\left\{. \right\}\) denotes statistical expectation. Initially, R and u are equipped with battery of sufficient amount of energy for signal transmission and signal reception. These wireless device are exclusively powered by H with the help of RF energy signal. As compared to radiation power, circuit consumes negligible power.

3.2 Formulation of ergodic capacity

In this section, we investigate the power-splitting protocol and time-switching protocol to achieve ergodic capacity with FD-based energy harvesting for two-way relaying network.

3.2.1 Performance of power-splitting (PS) protocol with full-duplex (FD) based energy harvesting (EH) in two-way amplify-and-forward (AF) relaying network

In this subsection, we describe PS protocol to achieve closed-form expressions of ergodic capacity, PS ratio and EH at relay and user for TWRNs. As shown in Fig. 2, we elaborate the FD-based PS protocol. For execution of PS protocol, we divide the block of time, T into two equal phases. In the first phase of duration \(t \in \left[ {0, T/2} \right]\), the H and u transmit RF energy signal,\(x_{H} \left[ k \right]\) and information-bearing signal with circularly symmetric complex Gaussian (CSCG), \(x_{u} \left[ k \right]\sim{\mathcal{C}\mathcal{N}}\left( {0, 1} \right)\) to R respectively. Thus, the received signal at R can be written as

$$y_{R} \left[ k \right] = h_{1} x_{H} \left[ k \right] + h_{2} x_{u} \left[ k \right] + n_{R} \left[ k \right]$$
(1)

where k represents the index of symbol and \(n_{R} \left[ k \right]\) represents the antenna noise at R and distribution follows as \({\mathcal{C}\mathcal{N}}\left( {0, \sigma_{R}^{2} } \right)\). The received signal, \(y_{R} \left[ k \right]\) splits into two parts, where one is used for EH and another for IP. Thus, energy signal for EH at R is expressed as

$$y_{R,E} \left[ k \right] = \sqrt \alpha y_{R} \left[ k \right]$$
(2)

where \(\alpha\) represents the PS ratio with \(0 \le \alpha \le 1\). Let \(\xi\) be the energy conversion efficiency for EH with \(0 \le \xi \le 1\). The relay harvests energy from RF signal received in first phase of block time, T is given by

$$E_{R,PS}^{I} = \xi {\mathbb{E}}\left[ {\left| {y_{R,E} \left[ k \right]} \right|^{2} } \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2} = \xi \alpha P_{H} \left| {h_{1} } \right|^{2} {T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(3)

where \(P_{H}\) denotes the transmit power of H. The information received at R can be expressed as

$$y_{R,IF} \left[ k \right] = \sqrt {\left( {1 - \alpha } \right)} y_{R} \left[ k \right]$$
(4)

Note that the term \(\sqrt {\left( {1 - \alpha } \right)} n_{R} \left[ k \right]\) is ignored because it does not carry meaningful information.

Fig. 2
figure 2

Full-duplex based power-splitting protocol for two-way relaying network, where SER represents self-energy recycling

Full size image

In the second phase of duration \(t \in \left[ {T/2, T} \right]\), the relay amplifies the received signal in first phase and forwards energy signal and information signal simultaneously to u and H respectively. Concurrently, the relay harvests energy from H and self-interference by receive antenna. On the basis of perfect acknowledge of \(h_{1}\) and \(x_{H} \left[ k \right]\) at R, the energy-bearing signal, \(x_{H} \left[ k \right]\) is annulled at R before transmission of energy or, information. Thus, the desired information signal gets more power for transmitting the signal. The transmitted signal by R can be written as

$$x_{R} \left[ k \right] = \sqrt {\beta_{PS} } \left( {y_{R,IF} \left[ {k - 1} \right] + n_{R,P} \left[ {k - 1} \right]} \right)$$
(5)

where \(n_{R,P} \left[ k \right]\) denotes the noise during IP caused by RF-to-baseband conversion and distribution follows as \({\mathcal{C}\mathcal{N}}\left( {0, \sigma_{R}^{2} } \right)\) and \(\beta_{PS}\) represents the amplification coefficient for PS protocol. The relay receives energy signal from H and residual self-interference through receiving antenna that can be given as

$$y_{R,E}^{'} \left[ k \right] = h_{1} x{}_{H}\left[ k \right] + h_{R} x_{R,N} \left[ k \right] + n_{R} \left[ k \right]$$
(6)

where \(x_{R,N} \left[ k \right]\) is normalized relayed signal by R. Therefore, from (6), we can write EH in second phase as

$$E_{R,PS}^{II} = \xi {\mathbb{E}}\left[ {\left| {y^{'}_{R,E} \left[ k \right]} \right|^{2} } \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2} = \xi \left( {P_{H} \left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} } \right){T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(7)

where \(P_{R}\) is the transmit power of relay. The total energy harvested in the block of time T is equal to the summation of amount of energy computed in first and second phases that can be written as

$$E_{R}^{PS} = \xi \left[ {P_{H} \left( {1 + \alpha } \right)\left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} } \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(8)

The balanced condition can be defined as the amount of energy consumed at R is equal to amount of energy harvested at R during block of time, T i.e., \({\mathbb{E}}\left[ {\left| {x_{R} \left[ k \right]} \right|^{2} } \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2} = E_{R}^{PS}\) is used to obtain the amplification coefficient. Thereby, the amplification coefficient, \(\beta_{PS}\) is expressed as

$$\beta_{PS} = \frac{{\xi \left[ {P_{H} \left( {1 + \alpha } \right)\left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} } \right]}}{{P_{u}^{PS} \left( {1 - \alpha } \right)\left| {h_{2} } \right|^{2} + \sigma_{R}^{2} }}$$
(9)

where \(P_{u}^{PS}\) is the transmit power of user for PS protocol. We assume that the relay is energy constrained node. Thus, energy consumed in second phase is less than or, equal to the total amount of energy harvested at R during block of time, T i.e., \(\frac{T}{2}P_{R} \le E_{R}^{PS}\). By using this condition, we obtain \(P_{R}\) as

$$P_{R} \le \frac{{\xi P_{H} \left( {1 + \alpha } \right)\left| {h_{1} } \right|^{2} }}{{1 - \xi \left| {h_{R} } \right|^{2} }}$$
(10)

The received signal at H in second phase is expressed as

$$y_{H} \left[ k \right] = h_{1} x_{R} \left[ k \right] + h_{H} x_{H} \left[ k \right] + n_{H} \left[ k \right]$$
(11)

In (11), second term is removed by using notch filter. Since, it does not carry useful information signal for H. Substituting \(x_{R} \left[ k \right]\) from (5) into (11), we get

$$\tilde{y}_{H} \left[ k \right] = \underbrace {{\sqrt {\beta_{PS} } \sqrt {\left( {1 - \alpha } \right)} h_{1} h_{2} x_{u} \left[ {k - 1} \right]}}_{\text{Infromation signal}} + \underbrace {{\sqrt {\beta_{PS} } h_{1} n_{R,d} \left[ {k - 1} \right] + n_{H} \left[ k \right]}}_{\text{Noise signal}}$$
(12)

From (12), signal-to-interference-plus-noise ratio (SINR) from user to H is expressed as

$$\gamma_{uH}^{PS} = \frac{{P_{u}^{PS} \left( {1 - \alpha } \right)\left| {h_{1} } \right|^{2} \left| {h_{2} } \right|^{2} }}{{\left| {h_{1} } \right|^{2} \sigma_{R}^{2} + {{\sigma_{H}^{2} } \mathord{\left/ {\vphantom {{\sigma_{H}^{2} } {\beta_{PS} }}} \right. \kern-0pt} {\beta_{PS} }}}}$$
(13)

The user is also energy constrained node. Thus, for obtaining the \(P_{u}^{PS}\), we first express the signal received at u in second phase that can be written as

$$y_{u,E} \left[ k \right] = h_{2} x_{R} \left[ k \right] + n_{u} \left[ k \right]$$
(14)

From (14), energy harvesting at user end can be obtained as

$$E_{u}^{PS} = \xi \left| {h_{2} } \right|^{2} {\mathbb{E}}\left[ {\left| {x_{R} \left[ k \right]} \right|^{2} } \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2} = \xi^{2} \left| {h_{2} } \right|^{2} \left[ {P_{H} \left( {1 + \alpha } \right)\left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} } \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(15)

Therefore, the user transmit power, \(P_{u}^{PS}\) is expressed from (15) as

$$P_{u}^{PS} = \frac{{E_{u}^{PS} }}{{{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}}} = \xi^{2} \left| {h_{2} } \right|^{2} \left[ {P_{H} \left( {1 + \alpha } \right)\left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} } \right]$$
(16)

From (9), (10) and (16), we can rewrite (13) as

$$\gamma_{uH}^{PS} \left( \alpha \right) = \frac{{\xi^{2} P_{H} \left| {h_{1} } \right|^{4} \left| {h_{2} } \right|^{4} }}{{1 - \xi \left| {h_{R} } \right|^{2} }}\gamma \left( \alpha \right)$$
(17)

where

$$\gamma \left( \alpha \right) = \frac{1}{{\frac{{\left| {h_{1} } \right|^{2} \sigma_{R}^{2} }}{{1 - \alpha^{2} }} + \frac{{\xi \left| {h_{2} } \right|^{4} \sigma_{H}^{2} }}{1 + \alpha } + \frac{{\left( {1 - \xi \left| {h_{R} } \right|^{2} } \right)\sigma_{R}^{2} \sigma_{H}^{2} }}{{\xi P_{H} \left( {1 - \alpha^{2} } \right)\left( {1 + \alpha } \right)\left| {h_{1} } \right|^{2} }}}}$$
(18)

The end-to-end capacity from u-to-H for PS protocol in bits/s/Hz is given by

$$C_{uH}^{PS} = \frac{1}{2}\log_{2} \left( {1 + \gamma_{uH}^{PS} \left( \alpha \right)} \right)$$
(19)

The PS ratio, \(\alpha\) effects the performance of relaying system. The PS ratio is investigated so that the performance of relaying system can be improved.

Theorem 1

The power-splitting ratio, \(\alpha\)for FD-based EH for two-way AF relaying network is given by

$$\alpha = \mathop {Maximum}\limits_{0 \le \alpha \le 1} C_{uH}^{PS} \left( \alpha \right) = 1 - \frac{1}{2}\left[ {\sqrt {a\left( {a + 4} \right)} - a} \right];\;{\text{where}}\; a = \frac{{2\left| {h_{1} } \right|^{2} \sigma_{R}^{2} }}{{\xi \left| {h_{2} } \right|^{4} \sigma_{H}^{2} }}.$$
(20)

Proof: Refer to “Appendix 1”.

Remark 1

Here, PS ratio is obtained for very high transmit power of H. From the results, it is clear that the PS ratio is independent from residual self-interference channel gain.

Performance of FD TWRNs can be studied by ergodic capacity. So, we evaluate the ergodic capacity from u-to-H for PS protocol and that is defined as

$$C_{E,uH}^{PS} = {\mathbb{E}}[C_{uH}^{PS} ] = \frac{1}{2}{\mathbb{E}}\left[ {\log_{2} \left( {1 + \gamma_{uH}^{PS} } \right)} \right]$$
(21)

where the expectation is realized by channels gain and residual self-interference channel gain.

Theorem 2

The ergodic capacity with FD-based EH from u-to-H for Rayleigh fading channel is given by

$$C_{E,uH}^{PS} = \frac{1}{{\mu_{1} \mu_{R} \ln 2}}\iint {\left[ {\psi_{2} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) - \psi_{1} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right]}e^{{ - \frac{{\left| {h_{1} } \right|^{2} }}{{\mu_{1} }}}} e^{{ - \frac{{\left| {h_{R} } \right|^{2} }}{{\mu_{R} }}}} d\left| {h_{1} } \right|^{2} d\left| {h_{R} } \right|^{2}$$
(22)

where,

$$\psi_{l} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = ci\left( {\varGamma_{l} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\cos \left( {\varGamma_{l} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right) + si\left( {\varGamma_{l} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\sin \left( {\varGamma_{l} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right);\;{\text{for}}\; l \in \left( {1,2} \right)$$
$$\begin{aligned} & \Gamma_{1} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = \frac{{\sqrt {1 - b_{2} \left| {h_{R} } \right|^{2} } }}{{\mu_{2} }}\sqrt {\frac{{b_{3} \left| {h_{1} } \right|^{4} + b_{5} \left( {1 - b_{2} \left| {h_{R} } \right|^{2} } \right)}}{{\left| {h_{1} } \right|^{2} \left( {b_{1} \left| {h_{1} } \right|^{4} + b_{4} \left( {1 - b_{2} \left| {h_{R} } \right|^{2} } \right)} \right)}}} ;\;b_{1} = P_{H} \xi^{2} ;\;b_{2} = \xi ; \\ & b_{3} = \frac{{\sigma_{R}^{2} }}{{1 - \alpha^{2} }};\;\Gamma_{2} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = \frac{1}{{\mu_{2} \sqrt {b_{4} } }}\sqrt {\frac{{b_{3} \left| {h_{1} } \right|^{4} + b_{5} \left( {1 - b_{2} \left| {h_{R} } \right|^{2} } \right)}}{{\left| {h_{1} } \right|^{2} }}} ;\;b_{4} = \frac{{\xi \sigma_{H}^{2} }}{1 + \alpha } \\ & b_{5} = \frac{{\sigma_{R}^{2} \sigma_{H}^{2} }}{{\xi P_{H} \left( {1 - \alpha^{2} } \right)\left( {1 + \alpha } \right)}} \\ \end{aligned}$$
(23)

Proof: Refer to “Appendix 2”.

Remark 2

From (22), we observe that the ergodic capacity is very difficult to obtain in closed-form mathematically. In order to investigate the closed-form solution of ergodic capacity for PS protocol, we assume that the H and R are stationary and user, u is mobile. Note that \(ci\left( \omega \right) = - \mathop \smallint \limits_{1}^{\infty } \frac{{\cos \left( {\omega x} \right)}}{x}dx\) and \(si\left( \omega \right) = - \mathop \smallint \limits_{1}^{\infty } \frac{{\sin \left( {\omega x} \right)}}{x}dx\).

Lemma 1

If channel gain from H-to-R, \(\left| {h_{1} } \right|^{2}\)and residual self-interference channel gain, \(\left| {h_{R} } \right|^{2}\)are stationary and channel gain from u-to-R, \(\left| {h_{2} } \right|^{2}\)is exponentially distributed and followed by Rayleigh fading, then ergodic capacity with FD-based EH from u-to-H is given by

$$C_{E,uH}^{PS} = \frac{1}{\ln 2}\left[ {\sum\limits_{j = 1}^{2} {\left( { - 1} \right)^{j} \left\{ {ci\left( {\varGamma_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\cos \left( {\varGamma_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right) + si\left( {\varGamma_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\sin \left( {\varGamma_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)} \right\}} } \right]$$
(24)

Proof: Refer to “Appendix 2”.

3.2.2 Performance of time-switching (TS) protocol with full-duplex based energy harvesting in two-way AF relaying network

In this section, we investigate FD-based EH TS protocol to achieve the closed-form expressions of TS ratio and end-to-end ergodic capacity for Rayleigh fading. The FD-based TS protocol is elaborated as shown in Fig. 3. To achieve the TS protocol, the transmissions strategy are classified into three phases of block time, \(T\). In the first phase of duration \(t \in \left[ {0, \delta T} \right]\), H transmits RF-signal to R for energy-harvesting, where \(\delta\) represents the TS ratio with \(0 \le \delta \le 1\). Thus, received RF-signal at R is expressed as

$$y_{R,E}^{TS} \left[ k \right] = h_{1} x_{H} \left[ k \right] + n_{R} \left[ k \right]$$
(25)
Fig. 3
figure 3

Full-duplex based time-switching protocol for two-way relaying network, where SER represents self-energy recycling

Full size image

Thus, energy-harvesting in first phase is defined as

$$E_{R,TS}^{I} = \xi {\mathbb{E}}\left[ {\left| {y_{R,E}^{TS} \left[ k \right]} \right|^{2} } \right]\delta T = \xi P_{H} \left| {h_{1} } \right|^{2} \delta T$$
(26)

In the second phase of duration \(t \in \left[ {\delta T, \delta T + \left( {1 - \delta } \right)T/2} \right]\), the user, u transmits information signal to R. Thus, decoded signal at R is written as

$$y_{R,IP}^{TS} \left[ k \right] = h_{2} x_{u} \left[ k \right] + n_{R}^{\prime } \left[ k \right]$$
(27)

where \(n_{R}^{\prime } \left[ k \right]\sim {\mathcal{C}\mathcal{N}}\left( {0, \sigma_{R}^{2} } \right)\) denotes the antenna noise at R. In the third phase of duration \(t \in \left[ {\delta T + \left( {1 - \delta } \right)T/2, T} \right]\), the R transmits amplified information signal received in first phase and energy signal simultaneously to H and u respectively. Concurrently it harvests energy from H and self-interference by receive antenna. Therefore, amplified information signal at R can be expressed as

$$x_{R}^{TS} \left[ k \right] = \sqrt {\beta_{TS} } \left( {y_{R,IP}^{TS} \left[ {k - 1} \right] + n_{R,P}^{'} \left[ {k - 1} \right]} \right)$$
(28)

where \(n_{R,P}^{\prime } \left[ k \right]\sim {\mathcal{C}\mathcal{N}}\left( {0, \sigma_{R}^{2} } \right)\) denotes the noise during information processing due to RF-to-baseband conversion. Furthermore, we can write information signal received at H as

$$y_{H}^{TS} \left[ k \right] = h_{1} x_{R}^{TS} \left[ k \right] + h_{H} x_{H} \left[ k \right] + n_{H} \left[ k \right]$$
(29)

By using notch filter, the second term of right side of (29) is cancelled, because it does not carry meaningful information. Substituting (27) and (28) into (29), the information signal at R can be written as

$$\tilde{y}_{H}^{TS} \left[ k \right] = \underbrace {{\sqrt {\beta_{TS} } h_{1} h_{2} x_{u} \left[ {k - 1} \right]}}_{\text{Infromation signal}} + \underbrace {{\sqrt {\beta_{TS} } h_{1} n_{R,P}^{\prime } \left[ {k - 1} \right] + n_{H} \left[ k \right]}}_{\text{Noise signal}}$$
(30)

The relay receives energy signal from H and residual self-interference through receive antenna that can be given by

$$\tilde{y}_{R,E}^{TS} \left[ k \right] = h_{1} x{}_{H}\left[ k \right] + h_{R} x_{R,N} \left[ k \right] + n_{R}^{'} \left[ k \right]$$
(31)

Therefore, energy harvested at R in third phases cab be computed as

$$E_{R,TS}^{III} = \xi {\mathbb{E}}\left[ {\left| {\tilde{y}_{R,E}^{TS} \left[ k \right]} \right|^{2} } \right]{{\left( {1 - \delta } \right)T} \mathord{\left/ {\vphantom {{\left( {1 - \delta } \right)T} 2}} \right. \kern-0pt} 2} = \xi \left( {P_{H} \left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} } \right)\left( {1 - \delta } \right){T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(32)

From (26) and (32), we can obtain total amount of energy harvested at R for TS protocol is given by

$$E_{R}^{TS} = \xi \left( {P_{H} \left| {h_{1} } \right|^{2} \left( {1 + \delta } \right) + P_{R} \left| {h_{R} } \right|^{2} \left( {1 - \delta } \right)} \right){T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(33)

The amplification coefficient for TS protocol is achieved with energy balanced condition. It can be defined as the amount of energy consumed at R is equal to amount of energy harvested at R during block of time, T i.e., \({\mathbb{E}}\left[ {\left| {x_{R}^{TS} \left[ k \right]} \right|^{2} } \right]{{\left( {1 - \delta } \right)T} \mathord{\left/ {\vphantom {{\left( {1 - \delta } \right)T} 2}} \right. \kern-0pt} 2} = E_{R}^{TS} .\) Thereby, the amplification coefficient, \(\beta_{TS}\) is written as

$$\beta_{TS} = \frac{{\xi \left[ {P_{H} \left| {h_{1} } \right|^{2} \left( {1 + \delta } \right) + P_{R} \left| {h_{R} } \right|^{2} \left( {1 - \delta } \right)} \right]}}{{\left( {P_{u}^{TS} \left| {h_{2} } \right|^{2} + \sigma_{R}^{2} } \right)\left( {1 - \delta } \right)}}$$
(34)

where \(P_{u}^{TS}\) represents the transmit power of user for TS protocol. For energy constrained node, total amount of energy harvested at R during block of time, T is greater than or equal to amount of energy consumed at R in third phase i.e., \(E_{R}^{TS} \ge P_{R} \left( {1 - \delta } \right)T/2\). Thereby, transmit power of R, \(P_{R}^{TS}\) can be expressed as

$$P_{R}^{TS} \le \frac{{\xi P_{H} \left( {1 + \delta } \right)\left| {h_{1} } \right|^{2} }}{{\left( {1 - \xi \left| {h_{R} } \right|^{2} } \right)\left( {1 - \delta } \right)}}$$
(35)

From (30), the SINR from u-to-H for TS protocol is defined as

$$\gamma_{uH}^{TS} = \frac{{P_{u}^{TS} \left| {h_{1} } \right|^{2} \left| {h_{2} } \right|^{2} }}{{\left| {h_{1} } \right|^{2} \sigma_{R}^{2} + {{\sigma_{H}^{2} } \mathord{\left/ {\vphantom {{\sigma_{H}^{2} } {\beta_{TS} }}} \right. \kern-0pt} {\beta_{TS} }}}}$$
(36)

We assume that the u is energy constrained node. The received signal at u in third phase is given by

$$y_{u,E}^{TS} \left[ k \right] = h_{2} x_{R}^{TS} \left[ k \right] + n_{u} \left[ k \right]$$
(37)

Further, we obtain energy harvesting from (37) at user, u as

$$E_{u}^{TS} = \xi \left| {h_{2} } \right|^{2} {\mathbb{E}}\left[ {\left| {x_{R}^{TS} \left[ k \right]} \right|^{2} } \right]{{\left( {1 - \delta } \right)T} \mathord{\left/ {\vphantom {{\left( {1 - \delta } \right)T} 2}} \right. \kern-0pt} 2} = \xi^{2} \left| {h_{2} } \right|^{2} \left[ {P_{H} \left( {1 + \delta } \right)\left| {h_{1} } \right|^{2} + P_{R} \left| {h_{R} } \right|^{2} \left( {1 - \delta } \right)} \right]{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-0pt} 2}$$
(38)

Thereby, the user transmit power, \(P_{u}^{PS}\) can be written from (38) as

$$P_{u}^{TS} = \frac{{E_{u}^{TS} }}{{{{\left( {1 - \delta } \right)T} \mathord{\left/ {\vphantom {{\left( {1 - \delta } \right)T} 2}} \right. \kern-0pt} 2}}} = \xi^{2} \left| {h_{2} } \right|^{2} \left[ {P_{H} \left| {h_{1} } \right|^{2} \frac{1 + \delta }{1 - \delta } + P_{R} \left| {h_{R} } \right|^{2} } \right]$$
(39)

By substituting (34), (35) and (39) into (36), we rewrite SINR from u-to-H for TS protocol as

$$\gamma_{uH}^{TS} \left( \delta \right) = \frac{{{{\left( {1 + \delta } \right)} \mathord{\left/ {\vphantom {{\left( {1 + \delta } \right)} {\left( {1 - \delta } \right)}}} \right. \kern-0pt} {\left( {1 - \delta } \right)}}}}{{f_{1} \left[ {f_{2} + f_{3} \frac{1 - \delta }{1 + \delta }} \right]}}$$
(40)

where

$$f_{1} = 1 - \xi \left| {h_{R} } \right|^{2} ;\;f_{2} = \frac{1}{{\xi P_{H} \left| {h_{1} } \right|^{2} }}\left[ {\tfrac{{\sigma_{H}^{2} }}{{\left| {h_{1} } \right|^{2} }} + \tfrac{{\sigma_{R}^{2} }}{{\xi \left| {h_{2} } \right|^{4} }}} \right];\;f_{3} = \frac{{\sigma_{R}^{2} \sigma_{H}^{2} f_{1} }}{{\xi^{3} P_{H}^{2} \left| {h_{1} } \right|^{6} \left| {h_{2} } \right|^{4} }}$$
(41)

We define end-to-end capacity from u-to-H for TS protocol in bits/s/Hz as

$$C_{uH}^{TS} = \frac{1}{2}\left( {1 - \delta } \right)\log_{2} \left( {1 + \gamma_{uH}^{TS} \left( \delta \right)} \right)$$
(42)

Theorem 3

On the basis of FD-based EH, the TS ratio \(\delta\)for two-way AF relaying network is given by

$$\delta = \mathop {Maximum}\limits_{0 \le \delta \le 1} C_{uH}^{TS} \left( \delta \right) = \frac{{1 - f_{0} \left( {1 + p} \right) - p}}{{1 - f_{0} \left( {1 + p} \right) + p}}$$
(43)

where \(f_{0} = f_{1} f_{2}\), \(p = W\left( \varepsilon \right)\)with \(W\left( . \right)\)represents Lambert \(W\)function and \(\varepsilon = \left( {\frac{1}{{f_{0} }} - 1} \right)e^{ - 1}\).

Proof: Refer to “Appendix 3”.

Remark 3

The Lambert \(W\) function defines as, \(\lambda = W\left( \lambda \right)e^{W\left( \lambda \right)}\). This results is obtained for high \(P_{H}\).

Performance of FD-based TS protocol in two-way AF relaying network can be deliberated with ergodic capacity. Thereby, the ergodic capacity from u-to-H for TS protocol is expressed as

$$C_{E,uH}^{TS} = {\mathbb{E}}[C_{uH}^{TS} ] = \frac{1}{2}\left( {1 - \delta } \right){\mathbb{E}}\left[ {\log_{2} \left( {1 + \gamma_{uH}^{TS} } \right)} \right]$$
(44)

where the expectation follows characteristic of channels gain and residual self-interference channel gain.

Theorem 4

The ergodic capacity with FD-based EH from u-to-H for TS protocol, where channels are exponentially distributed and followed by Rayleigh fading channel is given by

$$C_{E,uH}^{TS} = \frac{{\left( {1 - \delta } \right)}}{{\mu_{1} \mu_{R} \ln 2}}\iint {\left[ {\varphi_{2} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) - \varphi_{1} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right]}e^{{ - \frac{{\left| {h_{1} } \right|^{2} }}{{\mu_{1} }}}} e^{{ - \frac{{\left| {h_{R} } \right|^{2} }}{{\mu_{R} }}}} d\left| {h_{1} } \right|^{2} d\left| {h_{R} } \right|^{2}$$
(45)

where,

$$\varphi_{n} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = ci\left( {{\mathcal{D}}_{n} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\cos \left( {{\mathcal{D}}_{n} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right) + si\left( {{\mathcal{D}}_{n} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\sin \left( {{\mathcal{D}}_{n} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right);\;{\text{for}}\; n \in \left( {1,2} \right)$$
$$\begin{aligned} & {\mathcal{D}}_{1} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = \frac{1}{{\mu_{2} }}\sqrt {\frac{{k_{1} \left| {h_{1} } \right|^{4} + k_{3} \left( {1 - \xi \left| {h_{R} } \right|^{2} } \right)}}{{k_{2} \left| {h_{1} } \right|^{2} + B_{0} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)}}} ;\; B_{0} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = \frac{{k_{0} \left| {h_{1} } \right|^{6} }}{{1 - \xi \left| {h_{R} } \right|^{2} }};\;k_{0} = \frac{1 + \delta }{1 - \delta }; \\ & k_{1} = \frac{{\sigma_{R}^{2} }}{{\xi^{2} P_{H} }};\;{\mathcal{D}}_{2} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right) = \frac{1}{{\mu_{2} \sqrt {k_{2} } }}\sqrt {\frac{{k_{1} \left| {h_{1} } \right|^{4} + k_{3} \left( {1 - \xi \left| {h_{R} } \right|^{2} } \right)}}{{\left| {h_{1} } \right|^{2} }}} ;\;k_{2} = \frac{{\sigma_{H}^{2} }}{{\xi P_{H} }}\;{\text{and}}\;k_{3} = \frac{{\sigma_{R}^{2} \sigma_{H}^{2} }}{{\xi^{3} P_{H}^{2} k_{0} }} \\ \end{aligned}$$
(46)

Proof: Refer to “Appendix 4”.

Lemma 2

If channel gain from H-to-R, \(\left| {h_{1} } \right|^{2}\)and residual self-interference channel gain, \(\left| {h_{R} } \right|^{2}\)are static and channel gain from u-to-R, \(\left| {h_{2} } \right|^{2}\)is exponentially distributed and followed by Rayleigh fading, then ergodic capacity with FD-based EH for TS protocol from u-to-H is known as

$$C_{E,uH}^{TS} = \frac{{\left( {1 - \delta } \right)}}{\ln 2}\left[ {\sum\limits_{j = 1}^{2} {\left( { - 1} \right)^{j} \left\{ {ci\left( {{\mathcal{D}}_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\cos \left( {{\mathcal{D}}_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right) + si\left( {{\mathcal{D}}_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)\sin \left( {{\mathcal{D}}_{j} \left( {\left| {h_{1} } \right|^{2} ,\left| {h_{R} } \right|^{2} } \right)} \right)} \right\}} } \right]$$
(47)

Proof: Refer to “Appendix 4”.

Remark 4

We notice that the closed-form expression of ergodic capacity is mathematically intractable. So, the closed-form solution of ergodic capacity for TS protocol is derived when the H and R are static and user, u is mobile.

4 Numerical results and discussions

In this section, we present the numerical results to evaluate the performance of FD-based EH for two-way AF relaying network by considering Rayleigh fading channel. To achieve our goal, we propose two protocols based on FD energy-harvesting namely PS protocol and TS protocol. After that, we investigate the characteristics of our proposed FD-based EH schemes and then compare with HD-based EH schemes that is existing in literature [17]. In [17], the end-to-end ergodic capacity for HD-based EH has not been computed for relaying system. Further, we consider system model studied in [17] to investigate the end-to-end ergodic capacity.

  1. (a)

    Performance of HD-based EH relaying for PS protocol

The end-to-end ergodic capacity from user-to-access point with HD-based EH PS protocol for Rayleigh fading is given by

$$\begin{aligned} C_{E,HD}^{PS} = & \frac{1}{2}{\mathbb{E}}\left[ {\log_{2} \left( {1 + \gamma_{a} } \right)} \right] \\ = & \frac{1}{{\mu_{1} \ln 2}}\int {\sum\limits_{l = 1}^{2} {\left( { - 1} \right)^{l} \left\{ {ci\left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right)\cos \left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right) + si\left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right)\sin \left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right)} \right\}e^{{ - \frac{{\left| h \right|^{2} }}{{\mu_{1} }}}} d\left| h \right|^{2} } } \\ \end{aligned}$$
(48)

where \(\chi_{1} \left( {\left| h \right|^{2} } \right) = \frac{1}{{\mu_{2} }}\sqrt {\frac{{d_{1} \left| h \right|^{4} + d_{3} }}{{\left| h \right|^{2} \left( {\left| h \right|^{4} + d_{2} } \right)}}}\), \(\chi_{2} \left( {\left( {\left| h \right|^{2} } \right)} \right) = \frac{1}{{\mu_{{2\sqrt {d_{2} } }} }}\sqrt {\frac{{d_{1} \left| h \right|^{4} + d_{3} }}{{\left| h \right|^{2} }}}\), \(d_{1} = \frac{{\sigma_{r}^{2} }}{{\eta^{2} P_{a} \rho \left( {1 - \rho } \right)}}\), \(d_{2} = \frac{{\sigma_{a}^{2} }}{{\eta P_{a} \rho }}\) and \(d_{3} = \frac{{\sigma_{r}^{2} \sigma_{a}^{2} }}{{\eta^{3} P_{a}^{2} \rho^{2} \left( {1 - \rho } \right)}}\). Note that the symbolic presentation is given in [17]. With signal-to-noise ratio (SNR), \(\gamma_{a}\) from [17], the proof of (48) follows similar to “Appendix 2” by considering channel gains, \(\left| h \right|^{2} = \left| {h_{1} } \right|^{2}\) and \(\left| g \right|^{2} = \left| {h_{2} } \right|^{2}\). The (48) is intractable to evaluate in closed-form.


Special case 1: If channel gain \(\left| h \right|^{2}\) is static and channel gain \(\left| g \right|^{2}\) is exponentially distributed and followed by Rayleigh fading channel, then end-to-end ergodic capacity with HD-based EH for PS protocol is written as

$$C_{E,HD}^{PS} = \frac{1}{\ln 2}\left[ {\sum\limits_{l = 1}^{2} {\left( { - 1} \right)^{l} \left\{ {ci\left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right)\cos \left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right) + si\left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right)\sin \left( {\chi_{l} \left( {\left| h \right|^{2} } \right)} \right)} \right\}} } \right]$$
(49)
  1. (b)

    Performance of HD-based EH relaying for TS protocol

By considering TS protocol, the end-to-end ergodic capacity from user-to-access point with HD-based EH relaying for Rayleigh fading is expressed as

$$\begin{aligned} C_{E,HD}^{TS} = & \frac{{\left( {1 - \tau } \right)}}{2}{\mathbb{E}}\left[ {\log_{2} \left( {1 + \gamma_{a,ts} } \right)} \right] \\ = & \frac{{\left( {1 - \tau } \right)}}{{\mu_{1} \ln 2}}\int {\sum\limits_{l = 1}^{2} {\left( { - 1} \right)^{l} \left\{ {ci\left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right)\cos \left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right) + si\left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right)\sin \left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right)} \right\}e^{{ - \frac{{\left| h \right|^{2} }}{{\mu_{1} }}}} d\left| h \right|^{2} } } \\ \end{aligned}$$
(50)

where \(\chi_{1,ts} \left( {\left| h \right|^{2} } \right) = \frac{1}{{\mu_{2} }}\sqrt {\frac{{p_{1} \left| h \right|^{2} + p_{3} /\left| h \right|^{2} }}{{p_{0} \left| h \right|^{4} + p_{2} }}}\), \(\chi_{2,ts} \left( {\left| h \right|^{2} } \right) = \frac{1}{{\mu_{{2\sqrt {p_{2} } }} }}\sqrt {p_{1} \left| h \right|^{2} + p_{3} /\left| h \right|^{2} }\),\(p_{0} = \frac{\tau }{1 - \tau }\), \(p_{1} = \frac{{\sigma_{r}^{2} }}{{2\eta^{2} P_{a} }}\), \(p_{2} = \frac{{\sigma_{a}^{2} }}{{2\eta P_{a} }}\) and \(p_{3} = \frac{{\sigma_{r}^{2} \sigma_{a}^{2} }}{{4\eta^{3} P_{a}^{2} p_{0} }}\). The symbols are presented in [17]. With SNR, \(\gamma_{a,ts}\) from [17], the proof of (50) is similar to “Appendix 4”. The proof (50) is mathematically intractable to come to be in closed-form.


Special case 2: If channel gain \(\left| h \right|^{2}\) is stationary and channel gain \(\left| g \right|^{2}\) is distributed exponentially and followed by Rayleigh fading, then end-to-end ergodic capacity with HD-based EH for TS protocol can be given by

$$C_{E,HD}^{TS} = \frac{{\left( {1 - \tau } \right)}}{\ln 2}\left[ {\sum\limits_{l = 1}^{2} {\left( { - 1} \right)^{l} \left\{ {ci\left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right)\cos \left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right) + si\left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right)\sin \left( {\chi_{l,ts} \left( {\left| h \right|^{2} } \right)} \right)} \right\}} } \right]$$
(51)

For plotting the results, we consider set of parameters as described in Table 1. The parametric range of residual self-interference channel is considered as − 85 dB to − 15 dB. The self-interference channel gain without cancellation is taken approximately as − 15 dB [23] and practical self- interference cancellation can be already suppressed interference by 70 dB or more [21]. The ergodic capacity from u-to-H is plotted with energy conversion efficiency, \(\xi\) in Fig. 4(a) and with transmit power of H, \(P_{H}\) in Fig. 4(b).

Table 1 Specification of parameters
Full size table
Fig. 4
figure 4

Variation of ergodic capacity from u-to-H with a energy conversion efficiency, \(\xi\) and b transmit power of H, \(P_{H}\)

Full size image

The performance of our proposed FD-based EH schemes are better than the HD-based EH schemes. The ergodic capacity increases with \(P_{H}\) and \(\xi\) for both the protocols. From Fig. 4(b), we observe that the PS protocol is superior to the TS protocol at high transmit power of H, \(P_{H}\) while TS protocol performs better as compared to PS protocol at lower value of \(P_{H}\). Furthermore, we notice that the ergodic capacity of proposed FD-based EH schemes lead to better results as compared to HD-based EH schemes for \(11 \le P_{H} \le 15\).

The PS ratio is varying with energy conversion efficiency, \(\xi\) in Fig. 5(a) and with transmit power of H, \(P_{H}\) in Fig. 5(b). The PS ratio for FD relaying is less than the PS ratio for HD relaying (considered as the benchmark scheme). From plotted results, it is clear that the PS ratio increases with \(\xi\) for FD-based EH while it is constant with \(\xi\) for HD-based EH. Thereby, the energy constrained node, R (or, user) harvests progressively more energy from RF signal. Since, the product of PS ratio and energy conversion efficiency (or, square of energy conversion efficiency) increases by keeping constant \(P_{H}\). Furthermore, the PS ratio is independent from \(P_{H}\).

Fig. 5
figure 5

Plot power-splitting ratio versus a energy conversion efficiency, \(\xi\) and b transmit power of H, \({\text{P}}_{\text{H}}\)

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The time-switching (TS) ratio \(\delta\) is plotted with energy conversion efficiency \(\xi\) in Fig. 6(a) and with transmit power of H \(P_{H}\) in Fig. 6(b). Our proposed FD-based TS ratio \(\delta\) is less as compared to HD-based TS ratio. The ratio \(\delta\) gradually decreases as the both the parameters \(\xi\) and \({\text{P}}_{\text{H}}\) increase. But the product of \(\delta\) and \(\xi\) increases and the relay, R gains more power. The product of \(\delta\) and \(\xi\) in our proposed scheme is less as compared to benchmark scheme. The relay harvests more energy for FD-based TS protocol than that of HD-based TS protocol. Since, the R harvests energy in two phases for FD-based TS protocol but it harvests energy in one phase for HD-based TS protocol.

Fig. 6
figure 6

Variation of time-switching ratio \(\delta\) with a energy conversion efficiency, \(\xi\) and b transmit power of H, \({\text{P}}_{\text{H}}\)

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Variation of energy-harvesting at R with transmit power of H, \(P_{H}\) is depicted in Fig. 7. The FD-based EH schemes outperform the HD-based EH schemes. Since, the R receives RF energy from H in two phases of FD relaying protocols while the R harvests energy from RF signal from access-point in one phase of HD relaying protocols. For better understanding of results, we study characteristics of EH for different range of \(P_{H}\). First we consider \(35 \le P_{H} \le 45\), the EH in case of TS protocol is superior to PS protocol. Again we assume another range \(55 \le {\text{P}}_{\text{H}} \le 65\), the FD-based EH schemes perform better as compared to HD-based EH schemes. Furthermore, we observe that the PS protocol is better than the TS protocol for FD-based EH while we focus to the HD-based EH scheme, the TS protocol gains more energy than the PS protocol. At high value of \(P_{H}\), the EH with FD and HD relaying system for PS protocol is more effective than the TS protocol. In Fig. 8, the energy harvests by u is varying with \(P_{H}\). The performance of user EH follows similar characteristics of the EH of the R. But the magnitude of energy gains by R is higher than the energy gains by u. The energy constrained node, u harvests energy from R in only one phase for both the protocols.

Fig. 7
figure 7

Performance of energy-harvesting, EH at R with transmit power of H, \({\text{P}}_{\text{H}}\)

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Fig. 8
figure 8

Energy harvesting at u versus transmit power of H, \({\text{P}}_{\text{H}}\)

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Another interesting phenomenon, we study as ergodic capacity varies with residual self-interference channel gain which is shown in Fig. 9. The ergodic capacity is constant when the lookback interference is lowered to around − 45 dB. Thereafter, it increases monotonically with RSI channel gain. From (17), (18), (40) and (41), we observe that the relation of ergodic capacity and RSI channel gain is monotonic in nature. In our proposed scheme, the RSI channel gain should be nonzero. Since, our system is benefited from loopback interference channel gain as it is also a source of energy harvesting. Moreover, we found that the PS protocol is highly preferable as compared to TS protocol.

Fig. 9
figure 9

Variation of ergodic capacity with residual self-interference channel gain

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5 Conclusions and future work

In this paper, we present full-duplex two-way AF relaying network, where energy-constrained nodes such as relay and user harvest energy from hybrid power access-point, H and relay also harvests energy with self-energy recycling. The FD-based PS and TS protocols are proposed to improve the spectral efficiency efficiently over HD relaying network. We investigate the optimal value of PS ratio for PS protocol and the optimal value of TS ratio for TS protocol. Further, we consider Rayleigh fading channel to derive the end-to-end ergodic capacity for PS and TS protocols. The numerical results are provided with effects of numerous system parameters on the performance of simultaneously wireless energy and information processing in two-way AF relay network via relay. Our proposed FD-based EH schemes outperform the benchmark schemes significantly in terms of achievable end-to-end ergodic capacity. In addition, we observe that the TS protocol is suitable for lower value of transmit power of H while PS protocol is benefited efficiently at higher value of transmit power of H. Thus, we conclude that the FD-based EH increases the life span and extends the coverage area of network in wireless cooperative communication.

In our system model, the user acts as small-size wireless device, so that the FD operation at the user is very challenging issues. Since, the propagation-domain using antennas isolation method and analog-circuit-domain are employed to achieve large portion of self-interference cancellation in FD operation which is useful for the large infrastructure system such as BS and relays. However, the small-size wireless device needs single antenna technique for FD operation to harvest energy and flow the information at user that is interesting and open research issues.