1 Introduction

Spectral efficiency, energy efficiency, low latency processing, and wider connectivity are the key performance pillars of fifth-generation (5G) wide spreading wireless networks’ deployment and implementation [1]. 5G not only aims to take forward some prior technologies like relaying, small-cells, heterogeneous networks, massive antenna systems [2] etc. but also welcomes cutting-edge techniques like multi-user multiple-in-multiple-out (MIMO), device-to-device communication [3, 4], energy harvesting and non-orthogonal multiple access (NOMA) to foster the aim of future wireless communication [5, 6]. The apparent aim of relaying is to offload traffic of the base station (BS) with the help of intermediate nodes so that communication is realised between remotely located devices and BS or between two devices. Relaying plays a vital role in ever growing dense networks to serve cell-edge users for improved throughput and better quality of service (QoS). For prolonged communication, the relays can be powered with radio frequency energy harvesting (RFEH) refraining them to utilize their own energy for relaying purposes. RFEH is on boom these days as it exploits available radio signals to recharge devices enabling simultaneous wireless information and power transfer (SWIPT) so that the devices self sustain without counting on any extra energy sources within the network [7].

1.1 Related works and motivation

With expected sharp rise in future wireless network devices, cell phones, sensors etc., NOMA has gained substantial attention to improve spectral efficiency and reduce latency [8]. In general NOMA is classified into two brackets, power-domain NOMA and code-domain NOMA [6]. Specifically, power-domain NOMA, recently proposed to 3GPP LTE [9] multiplexes users on account of power i.e., the information of two users, hereafter referred as NOMA pair, is clubbed in a single frequency/time block with varying power levels. A comprehensive study on NOMA for 5G is presented in [9] with insights on real world deployment. So far, NOMA has been examined in uplink (UL) and downlink (DL) communication from various points of views in [10, 11]. In recent research works, concept of cooperative NOMA (CNOMA) is proposed in which one of the user in NOMA pair serves as a relay to forward the information to other user [12]. In a conventional CNOMA scheme, only one user from the NOMA pair contributes as a relay to increase signal diversity, improve reception reliability and extend coverage of the network [12]. In DL communication, a coordinated two-point system based on superposition coding (SC) was investigated in [13] exploiting the use of NOMA in cooperative communications. The authors in [14] used decode-and-forward (DF) relaying while the authors in [15] applied amplify and forward (AF) relaying with NOMA over different fading channels and analysed outage probability and sum rates of the system. Lately, two cooperative user relaying scenarios with near user in half duplex (HD) or full duplex (FD) mode are studied in [16] to compare the performance gain of HD or FD relaying in CNOMA. It is important to realise that both of the NOMA users can also act as relays to forward the information to some devices which are not in the proper coverage range of the source. Such situations can occur practically in internet of things (IoT) or machine-to-machine (M2M) communications. For example, in smart automotive networks, two cars moving away from the source towards a low connectivity area may require some assistance or emergency signals from the source. Therefore, it is possible to exploit two users in a NOMA pair to act as relays and assist two or more out-of-coverage devices. Albeit the use of relay nodes helps to improve coverage, reception reliability and QoS of a network, but at the cost of relay’s own energy, hence the need of sufficient energy supply to relay information cannot be ignored. It has been recognised already that SWIPT allows relays to scavenge from the wireless signals carrying information [17, 18] with the help of two key relaying protocols: (1) time switching (TS) and (2) power splitting (PS) [19]. SWIPT has been adopted in HD as well as FD relaying schemes [20, 21]. A research work in [22] used PS relaying scheme to harvest energy at near user in CNOMA for improved DL communication of far user. Most recently, a user selection scheme with the use of TS and PS relaying protocol is proposed for CNOMA netwoks using SWIPT in DL transmission only [23]. While aforementioned research works have already used CNOMA in DL relaying with or without SWIPT for enhanced signal diversity, the use of NOMA based relaying for extension of coverage in two-way communication is in infancy. A novel work to foster this concept is introduced in [24] where, the authors utilized NOMA pair as relays with TS protocol to serve two disconnected users simultaneously for DL communication. However, only outage performance of end users in DL communication was analysed. To the best of our knowledge, the use of NOMA pair based relaying in both DL and UL communication considering energy incentive for relaying information has not been investigated yet. Not quite the same as past works, we aim to derive relaying benefits from a NOMA pair with SWIPT for extending coverage of a network in DL and UL transmissions. More specifically, we explore the potential capacity of NOMA pair in user relaying in a NOMA–OMA based network with identifying the following key impact factors.

  • Will two users in a NOMA pair be able to act as relays for remote users for two-way communication solely on the basis of harvested energy? If yes, then what would be the impact on the signal outages and what would be the systems performance in delay-limited and delay tolerant modes in terms of throughput?

  • Will NOMA based relaying contributes towards improved system throughput than conventional OMA based network along with the use of SWIPT?

  • What are the significant factors that may impact the system performance in this hybrid NOMA–OMA network?

1.2 Contributions

We propose a concept of intermediate wireless powered NOMA relays (WPNR) in a NOMA based network where a source utilizes both users in a NOMA pair to act as relays for remotely located cell-edge users. Based on the proposed system, the main contributions of this paper are summarised as follows:

  • We investigate two-way hybrid WPNR based communication scheme for UL and DL transmission enabling the source to establish bi-directional communication with users which may not be accessed via single relay.

  • To highlight the behaviour of proposed system, we derive analytical expressions for the outage probabilities of UL and DL information symbols and system throughput in delay-limited transmission mode based on these outages. Furthermore we obtain closed form expressions for ergodic sum rate to extract key insights on the performance.

  • Our findings reveal that compared to the conventional OMA based network with SWIPT, the proposed system provides improved performance in terms of overall system throughput.

  • We further investigate the impact of some key factors for e.g, relay’s distance, power splitting factor and orientation of relay on the system throughput and symbol outage. The results reflect the fact that WPNR based relaying is beneficial in providing energy incentive to intermediate relays and in terms of spectral efficiency.

In the end, we also verified our analytical model with the help of numerical simulations to validate the mathematical derivations of outage probabilities and ergodic sum rates of the proposed system. The approach followed in this paper provides following benefits: (1) the use of SWIPT in a NOMA pair eliminates doubly near-far effect [25], (2) two disconnected users which are far from each other are simultaneously served by the source in UL and DL communication, (3) relays do not need to utilize their own energy in transferring the information to the end users.

1.3 Organisation and notations

The rest of the paper is organised as follows: In Sect. 2, system model and important notations are outlined. The detailed explanation of operation modes is in Sect. 3. In Sect. 4, performance analysis is reported with remarks on important results. Numerical explanation and verification of analytical results is done in Sect. 5, while the conclusion is given in Sect. 6 with future research directions.

The notations used in the paper are as follows: \({\mathbb {E}}[.]\) represents expectation operator and \(\Pr\)(.) shows the probability of an event; \(f_X(x)\) and \(F_X(x)\) denote the probability density function (PDF) and the cumulative density function (CDF) of a random variable (RV) X respectively; \({\mathcal {C}}{\mathcal {N}}(\mu ,\sigma ^2)\) denotes a circularly symmetric complex Gaussian RV with mean \(\mu\) and variance \(\sigma ^2\).

2 The system model

2.1 Wireless powered NOMA relays (WPNR)

We assume a scenario as illustrated in Fig. 1, where a source S communicates with two users \(U_1\) and \(U_2\) that are far from each other and are located in a poorly covered area. As in [24], there is no direct link between each of these users and the source due to heavy shadowing. Since these users lie far part from each other, one common relay is not able to establish individual communication links between the users and source at the same time. Hence, source exploits two relays \(R_1\) and \(R_2\) in the form of a NOMA pair to establish simultaneous communication paths to the users. The users in NOMA pair, referred as WPNR, employ PS based SWIPT protocol and scavenge energy from the received wireless signals for relaying information. The following transmission protocol and channel model further explain the proposed system in detail.

Fig. 1
figure 1

Proposed system model for hybrid NOMA with SWIPT

Full size image

2.2 Channel model description

A dual-hop wireless network with a source S, user \(U_i\) and WPNR \(R_j\) is assumed, where \(i=j\epsilon \{1,2\}, \forall i=j\), shown in Fig. 1. All nodes are assumed to be single antenna nodes [26] and relays employ DF protocol. Euclidean distance between users and relays is denoted by \(d_{U_iR_i}\) while \(d_{U_i}\), \(d_{R_j}\) represents distance between S and users, and S and relays respectively, where \(d_{R_1} < d_{R_2}\). Assuming that S is located at the center of the cell, \(\angle R_jSU_i\) represents the angle \(U_i, S\) and \(R_j\) thus \(d_{U_iR_j}=\sqrt{d^2_{U_i}+d^2_{R_j}-2d_{U_i}d_{R_j}\cos (\angle R_jSU_i)}\), where \(-\pi \le \angle R_jSU_i\le +\pi\). For a realistic communication model, we assume that the communication links exhibit large-scale path loss effects and experience small scale fading. Under the premise of perfect channel state information (CSI) at all nodes [27], all channels are quasi-static Rayleigh fading channels which means that each element of the channel coefficients is circularly symmetric complex Gaussian variable and channel power gains are exponentially distributed. The complex channel coefficients for the links \(S \longleftrightarrow R_j\), \(S\longleftrightarrow U_i\), \(U_i \longleftrightarrow R_j\) are denoted as \(h_{R_j}\sim {\mathcal {C}}{\mathcal {N}}(0,1)\), \(h_{U_i}\sim {\mathcal {C}}{\mathcal {N}}(0,1)\) and \(h_{U_iR_j}\sim {\mathcal {C}}{\mathcal {N}}(0,1)\) respectively.

Table 1 Communication slots
Full size table

3 Operation modes

Table 1 shows that the UL and DL communication between S and the two users \(U_i\) via relays \(R_j\) respectively happens in four time slots due to HD constraint. During \(t_{1,3}\), the received power by \(R_j\) is split by a power-splitting factor called \(\rho\) based on PS architecture for decoding information and the remaining power is utilized in harvesting energy [19]. Energy harvested in these time slots is utilized by \(R_j\) for relaying DL/UL data between S and \(U_i\). Table 2 lists the important notations used hereafter in this paper.

Table 2 Important symbols
Full size table

3.1 Downlink transmission

During \(t_1\), S transmits DL NOMA message \(\sum _{i=j=1}^{2}x_{U_i}\sqrt{p_{R_j}}\) to relay \(R_j\) with fixed power allocation coefficient \(p_{R_j}\) for message \(x_{U_i}\), where \(x_{U_i}\) is normalised unit power signal with \({\mathbb {E}}\{|x_{U_i}|^2\}=1\). To state fairness among relays \(p_{R_1}<p_{R_{2}}\) and \(\sum _{j=1}^{2} p_{R_j}=1\) [10]. Power corresponding to harvested energy by \(R_j\) is expressed as

$$\begin{aligned} P^{H}_{R_j,t_1}=\dfrac{\mathcal {P}_S(1-\rho )\eta _{e,R_j}\zeta _{e,R_j} |h_{R_j}|^2}{(\varepsilon +{d_{R_j}^{\alpha }})}, \end{aligned}$$
(1)

where \(\varepsilon\) is the reference parameter for bounded path-loss model to ensure finite path loss and finite harvested energy [24]. With \(p_{R_2}>p_{R_1}\), \(R_1\) invokes successive interference cancellation (SIC) of \(x_{U_2}\) i.e. it subtracts \(x_{U_2}\) from the received signal first and then decodes \(x_{U_1}\) [10]. Therefore, received signal to interference and noise ratio (SINR) of \(x_{U_2}\) and \(x_{U_1}\) at \(R_1\) is given by

$$\begin{aligned} \gamma ^{R_1}_{x_{U_2}}= & {} \dfrac{\varOmega _{R_2}|h_{R_1}|^{2}}{\varOmega _{R_1} |h_{R_1}|^{2}+ \sigma ^{2}_{R_1}(\varepsilon +d^{\alpha }_{R_1})}, \end{aligned}$$
(2)
$$\begin{aligned} \gamma ^{R_1}_{x_{U_1}}= & {} \dfrac{\varOmega _{R_1}|h_{R_1}|^{2}}{\sigma ^{2}_{R_1} (\varepsilon +d^{\alpha }_{R_1})}, \end{aligned}$$
(3)

where \(\varOmega _{R_j}=\mathcal {P}_Sp_{R_j}\rho\). On the other hand, \(R_2\) is not required to perform SIC because of lower transmit power allocation to \(x_{U_1}\). Therefore received SINR of \(x_{U_2}\) at \(R_2\) is given by

$$\begin{aligned} \gamma ^{R_2}_{x_{U_2}}=\dfrac{\varOmega _{R_2}|h_{R_2}|^{2}}{\varOmega _{R_1} |h_{R_2}|^{2}+ \sigma ^{2}_{R_2}(\varepsilon +d^{\alpha }_{R_2})}, \end{aligned}$$
(4)

After decoding the messages, \(R_1\) and \(R_2\) transmit \(x_{U_1}\) and \(x_{U_2}\) to \(U_1\) and \(U_2\) respectively using the energy harvested in \(t_1\). Received SINR of \(x_{U_1}\) at \(U_1\) is expressed as

$$\begin{aligned} \gamma ^{U_1}_{x_{U_1}} =\dfrac{P^H_{R_1,t_1}|h_{U_1R_1}|^2}{\sigma ^2_{U_1}(\varepsilon +d^{\alpha }_{U_1R_1})}, \end{aligned}$$
(5)

and SINR of \(x_{U_2}\) at \(U_2\) is written as

$$\begin{aligned} \gamma ^{U_2}_{x_{U_2}} =\dfrac{P^H_{R_2,t_1}|h_{U_2R_2}|^2}{\sigma ^2_{U_2}(\varepsilon +d^{\alpha }_{U_2R_2})}, \end{aligned}$$
(6)

3.2 Uplink transmission

During \(t_3\), \(U_i\) transmits \(x'_{U_i}\) to \(R_j\) with transmit power \(\mathcal {P}_U\), where \(x'_{U_i}\) is normalised unit power signal with \({\mathbb {E}}\{|x'_{U_i}|^2\}=1\). Similar to DL approach, a portion \(\rho\) of the received power is consumed in decoding information while remaining is utilized in harvesting energy. Power corresponding to harvested energy by each relay in this time slot is expressed as

$$\begin{aligned} P^{H}_{R_j,t_3}=\dfrac{\mathcal {P}_U(1-\rho )\eta _{e,R_j} \zeta _{e,R_j}|h_{U_iR_j}|^2}{(\varepsilon +{d_{U_iR_j}^{\alpha }})}, \end{aligned}$$
(7)

As stated in Table 1, this transmission is OMA based hence the relays do not need to perform SIC. Thus, SINR at \(R_1\) to decode \(x'_{U_1}\) is given by

$$\begin{aligned} \gamma ^{R_1}_{x'_{U_1}} =\dfrac{\mathcal {P}_U\rho |h_{U_1R_1}|^2}{\sigma ^2_{U_1} (\varepsilon +d^{\alpha }_{U_1R_1})}, \end{aligned}$$
(8)

and SINR of \(x'_{U_2}\) at \(R_2\) is written as

$$\begin{aligned} \gamma ^{R_2}_{x'_{U_2}} =\dfrac{\mathcal {P}_U\rho |h_{U_2R_2}|^2}{\sigma ^2_{U_2} (\varepsilon +d^{\alpha }_{U_2R_2})}, \end{aligned}$$
(9)

After decoding UL messages, \(R_1\) and \(R_2\) send UL NOMA signal to S, which ranks relays based on their channel conditions. The geographical separation of these relays from their end users is the basis of difference in their harvested energy which in turn leads to different transmit powers in UL NOMA signal. At this moment, it can be observed that our proposed system coherently eliminates doubly-near far problem and power allocation issue in UL NOMA communication. In order to detect individual user’s message, SIC is carried out at S and messages \((x'_{U_1}\) and \(x'_{U_2})\) are decoded one by one, starting with the message received with higher power first. Received SINR at S for \(x'_{U_1}\) is given by

$$\begin{aligned} \gamma ^{S}_{x'_{U_1}}=\dfrac{|h_{R_1}|^{2}P^H_{R_1,t_3}}{(\varepsilon +d_{R_1} ^{\alpha })(|h_{R_2}|^{2}P^H_{R_2,t_3}(\varepsilon +d_{R_2} ^{\alpha })^{-1}+ \sigma ^2_S)}, \end{aligned}$$
(10)

Since \(R_1\) is closer to S as compared to \(R_2\), S subtracts higher power symbol \(x'_{U_1}\) from the received UL signal and continues to decode \(x'_{U_2}\), hence the SINR at S to decode \(x'_{U_2}\) is given as:

$$\begin{aligned} \gamma ^{S}_{x'_{U_2}}=\dfrac{|h_{R_2}|^{2}P^H_{R_2,t_3}}{\sigma ^2_S(\varepsilon +d^{\alpha }_{R_2})}, \end{aligned}$$
(11)

This completes one transmission cycle and the UL and DL information is relayed by WPNRs between S and the serviced users.

4 Performance analysis of the system

The performance of our proposed model is characterised by evaluating two important metrics: outage probability and ergodic rate.

4.1 Outage analysis

Outage probability is defined as the probability of effective end-to-end SINR at receiver node (\(\gamma _R\)) falls below a certain threshold (\(\gamma ^o\)), due to channel fading or interferences, and is denoted by \(P^{out}\) below. Mathematically, we can write

$$\begin{aligned} P^{out}=\Pr (\gamma _R < \gamma ^o), \end{aligned}$$
(12)

4.1.1 Outage in downlink transmission

The outage in downlink transmission of \({x_{U_i}}\) occurs if it is not decoded by \(U_i\) or by its WPNR \(R_j\).

$$\begin{aligned} P^{out}_{x_{U_i}} (R_{x_{U_i}})= & {} {\mathbb {E}} \bigg [ \Pr \big (\log _2(1+\gamma ^{U_i/R_j}_{x_{U_i}})< R^o_{x_{U_i}} \big )\bigg ] \nonumber \\= & {} {\mathbb {E}} \bigg [ \Pr \big (\gamma ^{U_i/R_j}_{x_{U_i}}<\gamma ^o_{x_{U_i}}\big ) \bigg ], \end{aligned}$$
(13)

where \(R^o_{x_{U_i}}\) is the target rate to detect \(x_{U_i}\), and

$$\begin{aligned} \gamma ^o_{x_{U_i}}=2^{R^o_{x_{U_i}}}-1. \end{aligned}$$
(14)

Proposition 1

The analytical expression of outage probability at\(U_1\)for\(x_{U_1}\)is expressed as:

$$\begin{aligned} P^{out}_{x_{U_1}}= 1-(\breve{\varrho }_{x_{U_1}} \times \varrho _{x_{U_1}}), \end{aligned}$$
(15)

where, \(\gamma ^{o}_{x_{U_1}}\) is defined in (14), \(\gamma ^o_{x_{U_2}}<\dfrac{p_{R_2}}{p_{R_1}}\), \(\breve{\mathcal {A}}_{U_1}= \max (\breve{a}_1,\breve{a}_2)\), \(\breve{a}_1=\dfrac{\gamma ^{o}_{x_{U_1}}}{\varOmega _{R_1}}\), \(\breve{a}_2=\dfrac{\gamma ^{o}_{x_{U_2}}}{\varOmega _{R_2}-\gamma ^{o}_{x_{U_2}}\varOmega _{R_1}}\), \(\varPsi _{x_{U_1}}=\dfrac{\sigma ^2_{U_1}(\varepsilon +d^{\alpha }_{U_1R_1})(\varepsilon +d^{\alpha }_{R_1})}{\mathcal {P}_S(1-\rho )\eta _{e,R_1}\zeta _{e,R_1}}\), \(\breve{\varrho }_{x_{U_1}}=\exp \big (-\breve{\mathcal {A}}_{U_1}\sigma ^2_{R_1}(\varepsilon +d^{\alpha }_{R_1})\big )\), \(\varrho _{x_{U_1}}=1-2\sqrt{\varPsi _{x_{U_1}}\gamma ^{o}_{x_{U_1}}} K_v\big (\sqrt{4\varPsi _{x_{U_1}}\gamma ^{o}_{x_{U_1}}}\big )\), \(K_v(.)\) represents the modified Bessel function of second kind with v-th order [8.432.6, [28]]. If in above proposition, \(\gamma ^o_{x_{U_2}}\ge \dfrac{p_{R_2}}{p_{R_1}}\) then \(P^{out}_{x_{U_1}}\)=1.

Proof

see “Appendix 1”. \(\square\)

Remark 1

\(R_1\) needs to decode both of the messages for successful relaying. However the successful decoding at user’s end is also dependent on the received power at \(R_1\) which in turn relies on its channel condition with S.

Proposition 2

The analytical expression of outage probability at \(U_2\) for \(x_{U_2}\) is given by

$$\begin{aligned} P^{out}_{x_{U_2}}= 1-\varrho _{x_{U_2}} \times \breve{\varrho }_{x_{U_2}}, \end{aligned}$$
(16)

where \(\gamma ^{o}_{x_{U_2}}\) is defined in (14), \(\varPsi _{x_{U_2}}=\dfrac{\sigma ^2_{U_2}(\varepsilon +d^{\alpha }_{U_2R_2})(\varepsilon +d^{\alpha }_{R_2})}{\mathcal {P}_S(1-\rho )\eta _{e,R_2}\zeta _{e,R_2}}\), \(\breve{\varrho }_{x_{U_2}}=\exp \big (-\breve{a}_2\sigma ^2_{R_2}(\varepsilon +d^{\alpha }_{R_2})\big )\), \(\varrho _{x_{U_2}}= 2\sqrt{\varPsi _{x_{U_2}}\gamma ^{o}_{x_{U_2}}} K_v\big (\sqrt{4\varPsi _{x_{U_2}}\gamma ^{o}_{x_{U_2}}}\big )\).

Proof

With the help of (4) and (6), we obtain Proposition 2 following same steps as of Proposition 1. The details are omitted here for the sake of brevity. \(\square\)

Remark 2

As can be observed that the outage in DL transmission is dependent on the power allocation factors of WPNR and on the joint channel distribution between source with WPNRs, and between WPNRs and the users.

4.1.2 Outage in uplink transmission

The outage in uplink transmission of \({x'_{U_i}}\) occurs if the message sent by \(U_i\) is either not decoded by its WPNR \(R_j\) or by S.

$$\begin{aligned} P^{out}_{x'_{U_i}} (R_{x'_{U_i}})= & {} {\mathbb {E}} \bigg [ \Pr \big (\log _2(1+ \gamma ^{R_j/S}_{x'_{U_i}})< R^o_{x'_{U_i}} \big ) \bigg ] \nonumber \\= & {} {\mathbb {E}} \bigg [ \Pr \big (\gamma ^{R_j/S}_{x'_{U_i}} < \gamma ^o_{x'_{U_i}} \big )\bigg ], \end{aligned}$$
(17)

where \(R^o_{x'_{U_i}}\) is the threshold rate to detect \(x'_{U_i}\) and is expressed as

$$\begin{aligned} \gamma ^o_{x'_{U_i}}=2^{R^o_{x'_{U_i}}}-1, \end{aligned}$$
(18)

Proposition 3

The outage at S for \(x'_{U_1}\) is derived as

$$\begin{aligned} P^{out}_{x'_{U_1}}\approxeq 1- \varrho _{x'_{U_1}} \times {\tilde{\varrho }}_{x'_{U_1}}, \end{aligned}$$
(19)

where \(\varrho _{x'_{U_1}}=\exp \bigg (-\dfrac{\tilde{\mathcal {A}}_S(\varepsilon +d^{\alpha }_{U_1R_1})}{\mathcal {P}_U} \bigg )\), \(\tilde{\mathcal {A}}_S=\dfrac{\gamma ^{o}_{x'_{U_1}}\sigma ^2_{U_1}}{\rho }\), \(\delta _n=\cos \Big (\dfrac{2n-1}{2\mathcal {N}}\pi \Big )\), \({\tilde{\varrho }}_{x'_{U_1}}\approxeq \sum \limits ^\mathcal {N}_{n=1} \sum \limits ^\mathcal {M}_{m=1} \chi _n \chi _m \sqrt{\gamma ^{o}_{x'_{U_1}}\tilde{\chi _{mn}}} K_v\Big ( 2\sqrt{\gamma ^{o}_{x'_{U_1}}\tilde{\chi _{mn}}}\Big )\), \(\omega _n=\tan \bigg (\dfrac{\pi }{4}(\delta _n+1)\bigg )\), \(\tilde{\delta _n}=\dfrac{\pi ^2}{2\mathcal {N}}\sqrt{1-\delta ^2_n}\sec ^2(\arctan \omega _n)\), \(\xi _1=\dfrac{(\varepsilon +d^{\alpha }_{R_1})(\varepsilon +d^{\alpha }_{U_1R_1})}{(\varepsilon +d^{\alpha }_{R_2})(\varepsilon +d^{\alpha }_{U_2R_2})}\), \(\tilde{\chi _{mn}}=\xi _1\omega _n\omega _m+\xi _2\), \(\xi _2=\dfrac{\sigma _S^2(\varepsilon +d^{\alpha }_{R_1})(\varepsilon +d^{\alpha }_{U_1R_1})}{\mathcal {P}_U\eta _{e,{R_1}}\zeta _{e,{R_1}}(1-\rho )}\), \(\delta _m=\cos \Big (\dfrac{2m-1}{2\mathcal {M}}\pi \Big )\), \(\omega _m=\tan \bigg (\dfrac{\pi }{4}(\delta _m+1)\bigg )\), \(\chi _n=\tilde{\delta _n}e^{-(\omega _n+\omega _m)}\), \(\chi _m=\dfrac{\pi ^2}{4\mathcal {M}}\sqrt{1-\delta ^2_m}\sec ^2(\arctan \omega _m)\), and \(\mathcal {N}\) and \(\mathcal {M}\) are the Gauss-Cheybshev approximation parameters.

Proof

Appendix “2”. \(\square\)

Remark 3

Outage of UL transmission is computationally complex because of the fact that transmitted power of WPNR is dependent on the harvested energy from received power sent by the user and WPNR uses NOMA in relaying UL message as compared to conventional OMA in relaying DL message.

Proposition 4

The outage at S for \(x'_{U_2}\) is expressed

$$\begin{aligned} P^{out}_{x'_{U_2}}\approx 1- \varrho _{x'_{U_2}} \times {\tilde{\varrho }}_{x'_{U_2}} \times {\tilde{\varrho }}_{x'_{U_1}}, \end{aligned}$$
(20)

where \(\varrho _{x'_{U_2}}=\exp \bigg (-\dfrac{\tilde{\mathcal {B}}_S(\varepsilon +d^{\alpha }_{U_2R_2})}{\mathcal {P}_U} \bigg )\), \({\tilde{\varrho }}_{x'_{U_2}}=1 - 2\sqrt{\varpi \gamma ^{o}_{x'_{U_2}}} K_v\big (\sqrt{4\varpi \gamma ^{o}_{x'_{U_2}}}\big )\), \(\tilde{\mathcal {B}}_S=\dfrac{\gamma ^{o}_{x'_{U_2}}\sigma ^2_{U_2}}{\rho }\), \(\varpi =\dfrac{\sigma ^2_S(\varepsilon +d_{R_2} ^{\alpha })(\varepsilon +d_{U_2R_2} ^{\alpha }) }{\eta _{e,{R_2}}\zeta _{e,{R_2}} \mathcal {P}_U(1-\rho )}\), and \({\tilde{\varrho }}_{x'_{U_1}}\) is defined in Proposition 1.

Proof

Steps follows Appendix “2” and are omitted here to avoid longevity. \(\square\)

Remark 4

The analytical form of the outage probability of \(x'_{U_2}\) has extra product term which corresponds to the successful decoding of \(x'_{U_1}\) at S. One can observe that the decoding of \(x'_{U_2}\) by S is dependent on the joint channel distribution of \(R_2\) and \(U_2\) and on the successful decoding of \(x'_{U_2}\) at \(R_2\).

Remark 5

The results in Proposition 1–4 are consistent with the intuition that by employing WPNR in a wireless system, S can establish successful communication link with the two remote service deprived users.

4.2 System throughput

System throughput in delay-limited transmission mode [19] is dependant on the outages of the symbols \(x_{U_i}\) and \(x'_{U_i}\).

4.2.1 Downlink system throughput

In this case, S transmits \(x_{U_i}\) at a constant rate of \(R^o_{x_{U_i}}\) in a wireless fading channel, which is under the effect of outage probability. The system throughput in downlink communication is written as

$$\begin{aligned} R^{dlim}_{DL}=(1-P^{out}_{x_{U_1}})R^o_{x_{U_1}}+ (1-P^{out}_{x_{U_2}})R^o_{x_{U_2}}, \end{aligned}$$
(21)

where \(P^{out}_{x_{U_1}}\) and \(P^{out}_{x_{U_2}}\) are given in (15) and (16) respectively.

4.2.2 Uplink system throughput

For uplink communication, \(U_i\) transmits \(x'_{U_i}\) at a threshold rate of \(R^o_{x'_{U_i}}\) which is also subject to outage. We obtain uplink system throughput on the basis of outage probabilities as below

$$\begin{aligned} R^{dlim}_{UL}=(1-P^{out}_{x'_{U_1}})R^o_{x'_{U_1}}+ (1-P^{out}_{x'_{U_2}})R^o_{x'_{U_2}}, \end{aligned}$$
(22)

where \(P^{out}_{x'_{U_1}}\) and \(P^{out}_{x'_{U_2}}\) are expressed in (17) and (18) respectively.

4.3 Rate analysis

Here we analyse ergodic rate of the proposed system which is also an important performance metric if system is working under delay tolerant transmission mode [19]. Ergodic rate is the data rate achieved by a communication link averaged over all the fading states of wireless channel and is expressed with the help of effective SINR as:

$$\begin{aligned} R^{erg}=\int _{0}^{\infty } B \log _2 (1+\gamma _R) f_{\gamma _R}({\gamma }) d\gamma _R, \end{aligned}$$
(23)

where B is the received signal bandwidth.

4.3.1 Downlink ergodic rate

\(R^{erg}_{x_{U_i}}\) represents the ergodic rate achieved for \(x_{U_i}\) over the communication link \(S\rightarrow R_j\rightarrow U_i\).

Proposition 5

The closed-form expression for downlink ergodic rate is expressed as follows

$$\begin{aligned} \begin{aligned} R^{erg}_{DL}&= \dfrac{1}{\ln 2} \sum _{l=0}^{k} \sum _{q=0}^{l} \bigg (\sum _{g=0}^{\mathcal {G}}\lambda _\mathcal {G}\sqrt{1-\varTheta ^2_g}\varUpsilon _g\varPhi _g e^{-\tau _{1_g}}\\&\quad \bigg (e^{-\beta \varUpsilon _g} \varLambda (v,l,q) \cdot (\beta \varUpsilon _g)^{q-v} + \mathcal {T}_{\epsilon }\bigg ) \\&\quad +\sum _{f=0}^{\mathcal {F}} \psi _\mathcal {F} \sqrt{1-\mu ^2_f} \mathcal {W}_f \bigg (e^{-\beta \varsigma _{f}} \varLambda (v,l,q)\\&\quad \cdot (\beta \varsigma _{f})^{q-v} + \mathcal {T}_{\epsilon }\bigg ) \bigg ) \end{aligned} \end{aligned}$$
(24)

where \(\lambda _\mathcal {G}=\dfrac{\pi ^2}{4\mathcal {G}},\varTheta =\cos \bigg (\dfrac{2g-1}{2\mathcal {G}}\pi \bigg ), \varUpsilon _g=\sqrt{4\varPsi _{x_{U_1}}\tan \mathcal {X}_g}, \varPhi _g=\dfrac{\sec ^2\mathcal {X}_g}{1+\tan \mathcal {X}_g}, \tau _{1_g}=\big (-\dfrac{\tan \mathcal {X}_g(\varepsilon +d^\alpha _{R_1})\sigma ^2_{R_1}}{\varOmega _{R_1}}\big )\), \(\mathcal {X}_g=\dfrac{\pi }{4}(\varTheta _g+1)\), \(\psi _\mathcal {F}=\dfrac{\pi p_{R_2}}{2\mathcal {F}p_{R_1}}\), \(\mu _f=\cos \bigg (\dfrac{2f-1}{2\mathcal {F}}\pi \bigg )\), \(\beta =1\), \(\mathcal {W}_f=\dfrac{\sqrt{4\varPsi _{x_{U_2}}\mathcal {Y}_f}}{1+\mathcal {Y}_f}\), \(\mathcal {Y}_f=\dfrac{p_{R_2}}{2p_{R_1}}(\mu _f+1)\), \(\varsigma _{f}=\sqrt{4\varPsi _{x_{U_2}}\mathcal {Y}_f}\), q, l,  and k are accuracy parameters, and \(\mathcal {G}\) and \(\mathcal {F}\) are Gaussian–Cheybshev approximation parameters.

Proof

See Appendix “3”. \(\square\)

4.3.2 Uplink ergodic rate

\(R^{erg}_{x'_{U_i}}\) represents the ergodic rate achieved for \(x'_{U_i}\) over the communication link \(U_i\rightarrow R_j\rightarrow S\).

Proposition 6

The closed-form expression for uplink ergodic rate of \(x'_{U_2}\) is expressed as follows

$$\begin{aligned} \begin{aligned} R^{erg}_{UL}&=\dfrac{1}{\ln 2}\bigg [ \sum _{l'=1}^{\mathcal {L}'}\sum _{m=1}^{\mathcal {M}} \sum _{n=1}^{\mathcal {N}} \dfrac{\pi \beth _{l'} \chi _n \chi _m}{\mathcal {L}'}\sqrt{\dfrac{1+\wp _{l'}}{1-\wp _{l'}}} \exp \bigg (\dfrac{\wp _{l'}+1}{\wp _{l'}-1}\nu \bigg )\\&\quad \bigg ( e^{-\beta 2\beth _{l'}} \sum _{n=0}^{k} \sum _{q=0}^{l} \varLambda (v,l,q) \cdot (\beta 2\beth _{l'})^{q-v} + \mathcal {T}_{\epsilon }\bigg ) \\&\quad +\sum _{z=1}^{\mathcal {Z}}\dfrac{\pi ^2\sqrt{(1-\varphi ^2_z) (\varpi \tan \mathcal {V}_z)}}{2\mathcal {Z}(1+\tan \mathcal {V}_z)} \exp (-\digamma \tan \mathcal {V}_z)\sec ^2\mathcal {V}_z \\&\quad \bigg ( e^{-2\beta \sqrt{\varpi \tan \mathcal {V}_z}} \sum _{n=0}^{k} \sum _{q=0}^{l} \varLambda (v,l,q) \cdot (2\beta \sqrt{\varpi \tan \mathcal {V}_z})^{q-v} + \mathcal {T}_{\epsilon }\bigg ) \bigg ], \end{aligned} \end{aligned}$$
(25)

where \(\beth _{l'}= \dfrac{1+\wp _{l'}}{1-\wp _{l'}}(\chi _1\omega _m\omega _n+\chi _2)\), \(\wp _{l'}=\cos \bigg (\dfrac{2l'-1}{2\mathcal {L}'}\pi \bigg )\), \(\varphi _z=\cos \bigg (\dfrac{2z-1}{2\mathcal {Z}}\pi \bigg )\), \(\nu =\dfrac{\sigma ^2_{U_1}(\varepsilon +d^{\alpha }_{U_1R_1})}{\rho \mathcal {P}_U}\), \(\digamma =\dfrac{\sigma ^2_{U_2}(\varepsilon +d^{\alpha }_{U_2R_2})}{\rho \mathcal {P}_U}\)\(\mathcal {V}_z=\dfrac{\pi }{4}(\varphi _z+1)\), and \(\mathcal {L}'\) and \(\mathcal {Z}\) are Gaussian Cheybshev approximation parameters.

5 Numerical results and discussion

This section presents the numerical results of our proposed system along with corroboration of analytical results obtained in Sect. 4. We verify our analytical results with numerical simulations as follows. We simulate a source S at the centre of a cell with users \(U_1\) and \(U_2\) located 10m away from S. \(\angle R_jSU_i\) is kept variable and is mentioned in the corresponding figures. We model the channel fading gain between S and users, and WPNR and its respective user with exponential random variable. We execute 10K channel realisations, and for each realisation in the downlink and uplink transmission, outage is evaluated. Outage occurs if the signal to noise ratio at each node in every realisation is less than the threshold signal to noise ratio. Table 3 shows the simulation parameters. The antenna and receiver conversion noise variances at all nodes are unit normalised for generality but can be altered to evaluate performance.

Table 3 Simulation parameters
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5.1 Effect of transmit power

We evaluate the impact of the transmit power of source and users on the outages of downlink and uplink messages. \(P^{out}_{x_{U_1}}\) and \(P^{out}_{x_{U_2}}\) from (15) and (16) respectively is first verified via Monte Carlo simulations and the results are shown in Fig. 2a for different value of \(R^o_{x_{U_i}}\) and \(\rho =0.5\). The precise agreement between simulated and analytical results verify our derivation of outage. The figure clearly shows that outage of \(x_{U_i}\) is inversely proportional to the SNR. And also, when the requirement of DL data rate for \(U_1\) is increased, signals reception is decreased leading to more outage at \(U_1\). The deterioration of the received signals is because of the presence of second user’s symbol at relay node which is decoded a prior by \(R_1\). Fig. 2b shows a plot of the outages of \(x_{U_1}\) and \(x_{U_2}\) vs. SNR for a varying value of \(\rho\). It is obvious from the results that an increasing the value of \(\rho\) increases the outage of the downlink symbols which is consistent with the system description mentioned in Sect. 2. For instance, \(\rho =0.8\) results in more outage than the outage achieved with \(\rho =0.4\). This demonstrates that providing less energy to energy harvesting circuitry will increase the outage in relaying link, resulting in increased outage at user’s end.

Fig. 2
figure 2

Downlink outage probability

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Fig. 3
figure 3

Uplink outage probability

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Fig. 4
figure 4

System throughput-delay-limited transmission mode

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Fig. 5
figure 5

Ergodic rate- delay-tolerant transmission mode

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It is to be noted that the outage at \(U_2\) is similar to outage at \(U_1\) despite the fact that the distance between \(R_1\) and \(U_1\) is more, and that between \(R_2\) and \(U_2\) is less. The main reason behind this is that the relay located near S harvests more energy and transmits with higher power as the intended user is far. However, \(R_2\) harvests less energy because it requires less power to transmit to a closer user. This indicates the fairness for the serviced users in our proposed scheme and it also shows that the proposed model ignores doubly-near-far problem [13]. The expressions in (19) and (20) are also verified with simulations and the results are depicted in Fig. 3a where dashed/dotted lines represent analytical results while solid lines are used to show Monte Carlo simulations. Outage of \(x'_{U_i}\) symbol has an inverse relationship with transmit SNR which is consistent with the downlink transmission results. It is to be noted that the outage probability reveals an error floor in high SNR region which increases with the increase in uplink transmission rate. This is because of the interference caused by one user towards other user’s message in a single frequency/time domain which is an inherent property of uplink NOMA transmissions [29]. To verify the effect of power split ratio \(\rho\) on uplink transmissions, Fig. 3b illustrates impact of varying value of \(\rho\) on uplink outage. By comparing it with Fig. 2b, it can be demonstrated that the effect of \(\rho\) is same as explained earlier in this section for downlink transmission Figs. 2 and 3 illustrate that with the help of the WPNR pair, two remote users can be serviced adequately with nearly equal probabilities of reception at both ends. This scheme is highly useful in time-critical situations as well. For example, if an emergency signal needs to be transmitted to remote devices and a single relay is unable to communicate to both of them, then the use of a WPNR pair can serve the purpose effectively.

The results of Fig. 4a, b depict the delay-limited system throughput which is plotted with the help of expressions of the outage probabilities of the users. It is interesting to see that proposed relaying with NOMA offers superior uplink and downlink throughput as compared to OMAFootnote 1 based transmission. At \(\mathcal {P}_S=40\) dB and \(\mathcal {P}_U=30\) dB with a target threshold of unit bps, there’s a 40% percentage gain achieved with WPNRs than OMA based relaying in UL and DL transmission. Fig. 5a, b plot the ergodic rate of the proposed system which corresponds to system throughput in delay tolerant transmission mode. The dashed curves are obtained from (24) and (25) which are well matched with the solid lines representing Monte Carlo simulations. This verifies the closed form expressions of \(R^{erg}_{DL}\) and \(R^{erg}_{UL}\) derived in Propositions 5 and 6. It can be seen that ergodic rate of downlink transmission is fairly equal to ergodic rate achieved in uplink transmission, because of the fact that in uplink transmission user to relay transmission is OMA based where the relays harvest energy more as compared to downlink transmission where individual relay receives a portion of the transmit power because of NOMA.

5.2 Effect of distance and transmission angle

The graph in Fig. 6a illustrates the impact of geographical orientation of relays with respect to users and source. It is to be noted that average uplink and downlink rate of a NOMA pair degrades as the angular distance between users and relays increases. One important observation which this figure reflects is, if two relays are present at the similar distance then it is important for BS to consider the relay which lies in the line of sight of BS and relays. Another important insight is that, equal transmission rates can be achieved in the proposed model, when downlink transmit power is more than uplink transmit power. The reason behind this is the partial allocation of power to different NOMA users in downlink which leads to less harvested energy and reduced transmission rate. However, uplink NOMA assigns equal transmit power to both of the users while each user experience additional interference caused by other user’s message in the same assigned resource. To further elaborate systems performance, we plotted ergodic rate with respect to the ratio of relays’ distance from source in Fig 6b. It is witnessed that when \(d_{R_2}>>d_{R_1}\), ergodic rate in DL and UL transmission is high and for instance, it decrease by 28% and 40% respectively when the distance between the relays is kept equal with \(\mathcal {P}_S=40\) dB and \(\mathcal {P}_U=30\) dB. The explanatory reason is due to the basic concept of NOMA transmission [10]. When the two NOMA users are significantly a part, it becomes easier for the receiver to extract both messages successfully due to a difference of channel conditions.

Fig. 6
figure 6

Effect of distance and transmission angle on rate

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5.3 Uplink and downlink transmission

Figure 7a plots the downlink throughput achieved by the system vs. the data rate of both users in downlink transmission. One can observe that the system throughput is increased more when the data rate of \(R_2\) is increased as compared to the situation when data rate of \(R_1\) is increased owing to the fact that \(R_1\) decodes message of \(R_2\) first and then decodes the message intended for \(U_1\). If the data rate of \(R_1\) is increased, it leads to outage of \(x_{U_1}\) which in turn reduces the system throughput. However, for the higher data rate of \(R_2\), system throughput shows a bit sharp rise because only one message is decoded at \(R_2\)’s end. To clearly demonstrate the impact of user data rates on uplink transmission, Fig. 7b illustrates the uplink throughput versus the uplink data rate required for \(x'_{U_1}\) and \(x'_{U_2}\). Comparing Fig. 7a, b, it is apparent that the surface plot of downlink throughput has different slope than the uplink throughput. On the uplink, throughput increases sharply with increased data rate of \(R_1\) which is closer to the source. When the NOMA uplink message is sent by WPNRs, the message sent by \(R_1\) is decoded first hence its data rate contributes more towards the increased throughput. However, when data rate of \(R_2\) is increased, the outage at source increases which leads to reduction in the achieved throughput. Hence, it is important to choose the data rates appropriately to achieve the desired performance.

Fig. 7
figure 7

Ergodic rate with varying SNR and \(\rho\)

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6 Conclusion

In this paper, the concept of WPNR is applied on hybrid NOMA system to sustain communication with remotely located devices. A novel idea of using SWIPT in NOMA-based HDFootnote 2 relays is proposed which extends the coverage of the source for the isolated users. Specifically, we derived outage probabilities of the downlink and uplink transmission between the source and the remote devices. For our proposed scheme, Monte Carlo runs validated by the analytical model are executed. The system offers acceptable extension for the out-of-coverage devices with the aid of NOMA relays powered by the signals received from the source and the users. It can be concluded that with appropriate choice of uplink and downlink data rates along with careful consideration of WPNRs geographical location, this system model is adequate and satisfactory. A future extension of this study is to deploy WPNRs on a larger scale network with optimised system throughput via adequate relay selection method for multiple users. Another aspect is to employ FD WPNR to reduce the extra time slots used in this system for information transfer which in turn may improve the system throughput at the cost of extra complexity.