Performance Analysis of Two-Way Relay NOMA Systems with Hardware Impairments and Channel Estimation Errors

Abstract

In this paper, we consider a two-way relay non-orthogonal multiple access (TWR-NOMA) system with residual hardware impairments (RHIs) and channel estimation errors (CEEs), where two group users exchange their information via the decode-and-forward (DF) relay by using NOMA protocol. To evaluate the performance of the considered system, exact analytical expressions for the outage probability of the two groups users are derived in closed-form. Moreover, the asymptotic outage behavior in the high signal-to-noise ratio (SNR) regime is examined and the diversity order is derived and discussed. Numerical simulation results verify the accuracy of theoretical analyses, and show that: i) RHIs and CEEs have a deleterious effects on the outage probabilities; ii) CEEs have significant effects on the performance of the near user; iii) Due to the RHIs, CEEs, inter-group interference and intra-group interference, there exists error floors for the outage probability.

1. Introduction

 Non-orthogonal multiple access (NOMA) has been identified as a promising technique for the fifth generation (5G) mobile communication network since it has the advantages of low latency, massive connectivity and high spectral efficiency [1-3]. In general, NOMA can be classified into two categories: power domain NOMA [4] and code domain NOMA [5]. For power domain NOMA, multi-users can be simultaneously served by the same base station using the same resource, while for code domain NOMA, the signals of different users are spread by using different codes and then multiplexed over the same time-frequency resources [6]. In this paper, we consider the power domain NOMA. It is worth noting that the power domain NOMA mentioned through this paper will be replaced by NOMA. For NOMA systems, in order to ensure a trade-off between throughput and user fairness, more powers are allocated to the users with poorer channel conditions. At the transmitter, the superposition signals are sent by power multiplexing, while the signals can be separately decoded at receivers by successive interference cancellation (SIC) [7].

 Cooperative communication is another effective way to improve spectral efficiency, reduce transmit power and broaden the network coverage [8]. To further improve the system performance, two-way relay (TWR) was proposed for its ability to exchange information with the aid of a common relay with bidirectional information-transmission [9]. For combat channel fading and improve transmission reliability, the diversity behavior of the generalized MIMO TWR networks was studied in [10], where a two relay antenna selection (RAS) scheme was proposed. In [11], the outage behavior of a TWR network subject to a nonlinear transmission at the relay was studied, where fixed-gain and variable-gain amplify-and-forward (AF) relay were taken into account. Considering mixed asymmetric line-of-sight (LoS)/non-LoS (NLoS) fading scenario, the authors in [12] investigated the outage probability and channel capacity of the TWR networks. In [13], a distributed robust beamforming scheme was designed to minimize the total transmit power of the cognitive TWR networks. To further enhance the performance, the multi-antenna technique was introduced to the TWR networks [14], in which an optimal linear beamforming scheme was proposed to minimize the weighted mean squared error. As a further advance, massive multiple-input multiple-output (MIMO) inspired TWR network were involved in [15], where the impact of residual hardware impairments (RHIs) was investigated by considering Rician fading channels.

 Recently, the combination of TWR and NOMA, known as TWR-NOMA, has drawn a considerable research attention, since it exploits advantages of both TWR and NOMA [16–19]. In [16], the outage probability and ergodic rates of a TWR-NOMA system were investigated, in which perfect and imperfect SIC were analyzed. Sparked by full-duplex, the authors in [17] proposed a full duplex cooperative NOMA system, the outage probability and ergodic capacity of the proposed system were derived. For increase the system throughput and reduce the signaling overhead, the multi-pair TWR-NOMA network were developed in [18], in which an optimal group decoding scheduling scheme joint fair rate allocation was proposed for uplink and downlink. With the emphasis on secure transmission, different decoding schemes of the NOMA-based full-duplex TWR networks were proposed in [19] based on SIC for the legitimate users, relay and eavesdroppers.

 Although the above-mentioned works provide a solid foundation on the TWR-NOMA systems, one of the limitations of the above works is that the perfect hardware is assumed. In practice, due to deploying low-cost and low power efficiency RF components, the transceivers are prone to hardware imperfections and impaired by some types of hardware impairments, such as in-phase/quadrature-phase (I/Q) imbalance, amplifier non-linearities, and phase noise [20–22]. Although some signal processing algorithms can be used to compensate for the above imperfections, there still remains some RHIs due to estimation errors, inaccurate calibration and different types of noise [23], [24]. As stated in [20] [22] [25], the RHIs have significant effects on the system performance. The authors in [20] quantified the aggregated impact of RHIs on dual-hop relaying systems. The joint impact of RHIs and imperfect CSI on the multi-relay NOMA system was investigated in [22]. The authors in [25] derived the expressions of outage probability and approximate ergodic sum rate for analysis the deleterious effect of RHIs on the dual-hop NOMA network. The performance of wireless communication systems in the presence of RHIs has been extensively studied, e.g., see [26–28] and the references therein. The authors in [26] analyzed the effect of RHIs on the ergodic channel and ergodic sum rates of optimal and linear minimum mean-square-error (MMSE) receivers of MIMO systems. In [27], authors investigated the lower bound for the achievable sum rate of regular and large-scale MIMO systems with zero-forcing receivers in the presence of RHIs. In [28], exact closed-form expressions of outage probability and asymptotic expressions were derived in a TWR cooperative network with opportunistic relay selection, and the allocation of the fixed hardware impairments are analyzed as well. Recently, there are some research works dealing with the performance analysis of related topics of NOMA with RHIs, e.g., [29–31]. In [29], the authors investigated the performance of the one way cooperative NOMA network with RHIs by deriving a closed-form expression for the outage probability. Considering RHIs at the relay, [30] derived analytical expressions for the outage probability and symbol error rate of a TWR network. In [31], a one way relay dual-hop NOMA network with RHIs at source, relay and destination was considered, in which the outage probability, asymptotic individual ergodic rate (IER) and ergodic sum rate (ESR) were obtained. Nevertheless, the above works assume that all nodes have perfect channel state information (CSI). In fact, the performance of wireless communication system is greatly affected by wireless channel, such as shadow fading and frequency selective fading, which makes the path between transmitter and receiver very complicated. Due to randomness nature of wireless channels, it is a great challenge to obtain perfect channel knowledge. The common way of doing this is to use some estimation algorithms to obtain the estimated. However, the perfect estimation is not available due to channel estimation errors (CEEs). Therefore, it is of significance to investigate the joint impact of RHIs and CEEs on the TWR-NOMA system.

 Motivation and Contribution: Motivated by the above discussion, different from the existing works on TWR-NOMA systems, we investigate the performance of TWR-NOMA systems in the presence of RHIs and CCEs, where two groups NOMA users exchange their information with the aid of decode-and-forward (DF) relay. The contributions of this paper are summarized as follows:

 • Contrary to the existing research works on the performance of TWR-NOMA, we consider two practical factors, namely, RHIS and CEE. We aim at investigating the joint effects of the two factors on TWR-NOMA networks, which is a valuable problem for practical system analysis and design.

 • We derived exact closed-form analytical expressions for the outage probability of the far users and near users. In addition, we also derived exact analytical expressions for the outage probability of the considered network under the condition of ideal RF components and CSI.

 • We examine the asymptotic outage behavior of the users and discuss the diversity order in the high signal-to-noise ratio (SNR) region. It reveals that RHIs and CEEs can cause the outage performance to deteriorate. Moreover, the results show that there exists an error floor for the outage probability due to inter/intra group interference and CEEs, and the diversity order is zero. Additionally, CEEs have a deleterious effect on the outage probability on the near users.

 Organization: The rest of this paper is organized as follows. In section II, we present the TWR-NOMA system model in the presence of RHIs and CEEs. The exact closed-form analytical and asymptotic expressions for the outage probability and diversity order are derived and discussed in section III. In section IV, we present some numerical and simulation results to verify the accuracy of our analysis. A brief summary of this paper is concluded in section V.

 Notations: In this paper, the main notations are shown as follows: E{·}  denotes the expectation operation, Pr{·}  is the probability,  \(\prod(\cdot)\) denotes the continuous multiplication operation, \(\sum(\cdot)\)  is the summation.

 

2. System Model

 A TWR-NOMA system is considered, which consists of one two-antenna relay R   and two groups of NOMA users \(G_{1}=\left\{U_{1}, U_{2}\right\}\)  and  \(G_{2}=\left\{U_{3}, U_{1}\right\}\) as illustrated in Fig. 1. For NOMA users, we assume that two types of users are deployed: U₁  and  U₃ are the near users, and U₂  and U₄  are the far users. The users of same type aim to exchange their informations via a two-antenna relay R , namely R₁  and R₂ . In addition, there is no direct link between two groups of users due to heavy shadowing. We also assume that all users are equipped with a single antenna and operate in a half-duplex mode. h_i, \(i \in\{1,2,3,4\}\), denotes the channel fading coefficients between U_i  and R , where h_i  follows independent Rayleigh random variables with \(\left|h_{i}\right|^{2} \sim C N\left(0, \delta_{i}^{2}\right)\) .

Fig. 1. System model

 In practice communication systems, it is a great challenge to obtain perfect CSI due to the CEEs. Thus, the channel fading coefficient can be modeled as \(h_{i}=\hat{h}_{i}+e_{i}\) , where \(\hat{h}_{i}\)  represent the estimated channels coefficient,   are the CEEs, which can be modeled by Gaussian random variable with \(e_{i} \sim C N\left(0, \delta_{\varepsilon}^{2}\right)\) [32].

 The whole communication is completed in two phases: 1) multiple access channel (MAC) phase; 2) broadcasting (BC) phase.

 1) MAC Phase: In this phase, two pairs of users simultaneously transmit the respective information to the intended antenna of the relay, and the received information at R₁(R₂₎  suffers from interference from the users of G₂(G₁) . Thus, the received signals at R₁  and R₂  are respectively given by

\(y_{R_{1}}=h_{1} \sqrt{a_{1} P_{u}} x_{1}+h_{2} \sqrt{a_{2} P_{u}} x_{2}+\varpi_{1} I_{2}+\eta_{r}+n_{R_{1}}\)       (1)

\(y_{R_{2}}=h_{3} \sqrt{a_{3} P_{u}} x_{3}+h_{4} \sqrt{a_{4} P_{u}} x_{4}+\varpi_{1} I_{1}+\eta_{r}+n_{R_{2}}\)       (2)

where x_i  are the transmitted signals by the user U_i , with E{x_i}=1, p_u  is the transmission power. a₁,a₂,a₃ and a₄  are the corresponding power allocation coefficients, satisfying \(a_{1}>a_{2}, a_{1}+a_{2}=1\) ;  I₂ is the inter-group interference signal (IS) from R₂  with \(I_{2}=h_{3} \sqrt{a_{3} P_{u}} x_{3}+h_{4} \sqrt{a_{4} P_{u}} x_{4}\) ; \(w_{1} \in[0,1]\) is the impact level of inter-group IS at  R ; I_i   is the inter-group IS from R_i with  \(I_{1}=h_{1} \sqrt{a_{1} P_{u}} x_{1}+h_{2} \sqrt{a_{2} P_{u}} x_{2} \) ; \(n_{R_{1}}\) and \(n_{R_{2}}\)  are additive white Gaussian noise (AWGN) with zero mean and N₀  variance at R₁  and R₂  , respectively;   \(\eta_{r}\)represents the received RHIs at relay with  \(\eta_{r} \sim C N\left(0, \kappa_{r}^{2} P_{u} \sum_{i=1}^{4}\left|h_{i}\right|^{2}\right), \kappa_{r}\)   is used to characterize the aggregated level of RHIs from the relay [33].

 According to the DF protocol of NOMA, R₁  first decodes x₁ by treating signals from the users of another group as inter-group IS. Then, the received signal-to-interference plus noise rates (SINR) at R₁  to detect x₁  is given by

\(\gamma_{R \rightarrow x_{1}}=\frac{\rho_{u}\left|\hat{h}_{1}\right|^{2} a_{1}}{\rho_{u}\left|\hat{h}_{2}\right|^{2} a_{2}+\sum_{k=1}^{4} \rho_{u}\left|\hat{h}_{k}\right|^{2} \kappa_{r}^{2}+\sum_{j=3}^{4} \rho_{u}\left|\hat{h}_{j}\right|^{2} \varpi_{1} a_{j}+\vartheta_{1}}\)       (3)

where \(g_{1}=\sum_{i=1}^{2} \rho_{u} \delta_{e_{i}}^{2} a_{i}+\sum_{j=3}^{4} w_{1} \rho_{u} \delta_{e_{j}}^{2} a_{j}+\sum_{k=1}^{4} \kappa_{r}^{2} \rho_{u} \delta_{e_{i}}^{2}+1\) , and \(\rho_{u}=P_{u} / N_{0}\)  denotes the transmit SNR at the user nodes.

 Then, the SINR at relay to decode   is given

\(\gamma_{R \rightarrow x_{2}}=\frac{\rho_{u}\left|\hat{h}_{2}\right|^{2} a_{2}}{\sum_{k=1}^{4} \rho_{u}\left|\hat{h}_{k}\right|^{2} \kappa_{r}^{2}+\sum_{j=3}^{4} \rho_{u}\left|\hat{h}_{j}\right|^{2} \varpi a_{j}+\vartheta_1}\)       (4)

 2) BC Phase: In this phase, the relay decodes and forwards the received signals to all users. More specifically, the antennas R₂  and R₁  send the superposed signals  \(\left(\sqrt{b_{1} P_{r}} x_{1}+\sqrt{b_{2} P_{r}} x_{2}\right)\) and \(\left(\sqrt{b_{3} P_{r}} x_{3}+\sqrt{b_{4} P_{r}} x_{4}\right)\)  to  G₂ and  G₁ , respectively, where P_r is the transmission power at the relay, b₁,b₂,b₃  and b₄  are the power allocation coefficients. According the NOMA protocol, the far users are allocated more power than the near users b₂>b₁ , with b₁+b₂ =1  and  b₄>b₃, with  b₃+b_{4 = 1}.

 The received signals at U₃  and  U₄ denoted by y_u3   and y_u4 , can be respectively expressed as

\(y_{U_{3}}=h_{3}\left(\sum_{i=1}^{2} \sqrt{b_{i} P_{r}} x_{i}+\eta_{t}\right)+\varpi_{2} h_{3} \sum_{j=3}^{4} \sqrt{b_{j} P_{r}} x_{j}+n_{3}\)       (5)

\(y_{U_{4}}=h_{4}\left(\sum_{i=1}^{2} \sqrt{b_{i} P_{r}} x_{i}+\eta_{t}\right)+\varpi_{2} h_{4} \sum_{j=3}^{4} \sqrt{b_{j} P_{r}} x_{j}+n_{4}\)       (6)

where \(\varpi_{2} \in[0,1]\)  is the impact level of inter-group IS at the user nodes, n₃  and  n₄ are AWGN with mean power N₀  at U₃  and U₄ , respectively; \(\eta_{t}\)  represents the RHIs at the relay transmitter node, \(\eta_{t} \sim C N\left(0, \kappa_{t}^{2} P_{r}\right), \kappa_{t}\)  is used to characterize the aggregate level of impairments in the transmitter hardware.

 We assume that the signals x₂ from the users can be correctly decoded at R  in the MAC phase. U₃  decodes the desired signal  x_i after using SIC to decode and eliminate x₂ . The effective SINR at U₃  for the detection of x₂  is given by

\(\gamma_{U_{3} \rightarrow x_{2}}=\frac{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{2}}{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{1}+\rho_{r}\left|\hat{h}_{3}\right|^{2} \kappa_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{3}\right|^{2}+\vartheta_{2}}\)       (7)

where  \(\vartheta_{2}=\rho_{r} \delta_{e_{3}}^{2}\left(1+\kappa_{t}^{2}+\sigma_{2}\right)+1, \rho_{r}=P_{r} / N_{0}\)   is the transmit SNR at the relay node.

 Similarly, we assume that the signals x₁  from the users can be correctly decoded at R in the MAC phase. After SIC operations, the received SINR at U₃  to decode x₁  is given by

\(\gamma_{U_{3} \rightarrow x_{1}}=\frac{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{1}}{\rho_{r}\left|\hat{h}_{3}\right|^{2} \kappa_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{3}\right|^{2}+\vartheta_{2}}\)       (8)

 Then, the received SINR at U₄  to decode signal x₂  is given by

\(\gamma_{U_{4} \rightarrow x_{2}}=\frac{\rho_{r}\left|\hat{h}_{4}\right|^{2} b_{2}}{\rho_{r}\left|\hat{h}_{4}\right|^{2} b_{1}+\rho_{r}\left|\hat{h}_{4}\right|^{2} \kappa_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{4}\right|^{2}+\vartheta_{3}}\)       (9)

where \(\vartheta_{3}=\rho_{r} \delta_{e_{4}}^{2}\left(1+\kappa_{t}^{2}+w_{2}\right)+1\) . After these processes, the information is exchanged between the NOMA users in  G₁ and G₂ .

 

3. Outage Performance Analysis

 In this section, we first derive the exact analytical expressions for the outage probabil-ity. Then the asymptotic outage behavior at high SNR region is analyzed, and the diver-sity orders are also discussed.

 

3.1 Exact Outage Probability        

 1) Outage probability for U₁

 In the TWR-NOMA, the outage event of U₁  occurs in the following three cases: i) The information x₁  cannot be decoded by R₁ ; ii)   U₃  cannot decode x₂ successfully; iii) The information x_i  cannot be decoded by U₃ , while U₃  can first decode the infor-mation  x₂ correctly. Hence, the outage probability of  U₁  can be written as

\(\mathrm{P}_{U_{1}}^{\text {out }}=1-\operatorname{Pr}\left(\gamma_{R \rightarrow x_{1}}>\gamma_{\text {thf }}\right)\operatorname{Pr}\left(\gamma_{U_{3} \rightarrow x_{2}}>\gamma_{t h m}, \gamma_{U_{3} \rightarrow x_{1}}>\gamma_{t h f}\right)\)       (10)

where \(\gamma_{t h f}=2^{2 R_{f}}-1, \gamma_{t h m}=2^{2 R_{w}}-1\) .  R_f and R_m  are the target rates at U₃  to detect  x₁ and x₂ , respectively.

 Then, the outage probability of U₁  for the TWR-NOMA with RHIs and CEEs is pro-vided in the following theorem.

 Theorem 1. For the non-ideal cases \(\left(\delta_{e_{i}}^{2} \neq 0, \kappa_{i} \neq 0, i \in\{1,2,3,4\}, l \in\{t, r\}\right)\), the closed-form expression for the outage probability of U₁  is expressed as

\(P_{U_{1}}^{n i, {out}}=1-e^{-\frac{\tau_{0}}{\hat{\delta}_{3}^{2}}-\frac{\theta_{1} \vartheta_1}{\hat{\delta}_{1}^{2}}}\prod_{i=1}^{3} \lambda_{i}\left(\frac{\Phi_{1} \hat{\delta}_{1}^{2}}{\lambda_{1} \hat{\delta}_{1}^{2}+\theta_{1}}-\frac{\Phi_{2} \hat{\delta}_{1}^{2}}{\lambda_{2} \hat{\delta}_{1}^{2}+\theta_{1}}+\frac{\Phi_{3} \hat{\delta}_{1}^{2}}{\lambda_{3} \hat{\delta}_{1}^{2}+\theta_{1}}\right)\)       (11)

where  \(\Phi_{1}=\left(\lambda_{2}-\lambda_{1}\right)^{-1}\left(\lambda_{3}-\lambda_{1}\right)^{-1}, \Phi_{2}=\left(\lambda_{3}-\lambda_{2}\right)^{-1}\left(\lambda_{2}-\lambda_{1}\right)^{-1}, \Phi_{3}=\left(\lambda_{3}-\lambda_{1}\right)^{-1}\left(\lambda_{3}-\lambda_{2}\right)^{-1}\),\(\theta_{1}=\frac{\gamma_{\text {lif}}}{\rho_{u}\left(a_{1}-\gamma_{\text {tif}} \kappa_{r}^{2}\right)}\)   with  \(a_{1}>\gamma_{t l j} \kappa_{r}^{2}, \tau_{0}=\max \left(\tau_{2}, \tau_{1}\right)\)  , where \(\tau_{1}=\frac{\gamma_{t h f} \vartheta_{2}}{\rho_{r}\left(b_{1}-\gamma_{t h f}\left(\kappa_{t}^{2}+\varpi_{2}\right)\right)}\)  with  ,   \(b_{1}>\gamma_{t i f}\left(\kappa_{t}^{2}+\sigma_{2}\right), \tau_{2}=\frac{\gamma_{\text {thm}} \vartheta_{2}}{\rho_{r}\left(b_{2}-\gamma_{\text {thm}}\left(b_{1}+\kappa_{t}^{2}+\sigma_{2}\right)\right)}\)with \(b_{2}>\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\), \(\lambda_{1}=1 /\left(\rho_{u} \hat{\delta}_{2}^{2}\left(a_{2}+\kappa_{r}^{2}\right)\right), \lambda_{2}=1 /\left(\rho_{u} \hat{\delta}_{3}^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\right) \text { and } \lambda_{3}=1 /\left(\rho_{u} \hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right)\)  and \(\lambda_{3}=1 /\left(\rho_{u} \hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right)\) .

 Proof: See Appendix A. 

 Corollary 1. For the ideal cases \(\left(\delta_{e_{i}}^{2}=0, \kappa_{l}=0\right)\) , the analytical expression for the outage probability of  U₁ is expressed as

\(\mathbf{P}_{U_{1}}^{i d, {out}}=1-e^{-\frac{\gamma_{ {thf}}}{\xi_{1}}-\frac{\tau_{0}^{\prime}}{\delta_{3}^{2}}}\prod_{i=4}^{6} \lambda_{i}\left(\frac{\Psi_{1} \xi_{1}}{\xi_{1} \lambda_{4}+\gamma_{ {thf }}}-\frac{\Psi_{2} \xi_{1}}{\xi_{1} \lambda_{5}+\gamma_{ {thf }}}+\frac{\Psi_{3} \xi_{1}}{\xi_{1} \lambda_{6}+\gamma_{ {thf }}}\right)\)       (12)

 where  \(\Psi_{1}=\left(\lambda_{5}-\lambda_{4}\right)^{-1}\left(\lambda_{6}-\lambda_{4}\right)^{-1}, \Psi_{2}=\left(\lambda_{6}-\lambda_{5}\right)^{-1}\left(\lambda_{5}-\lambda_{4}\right)^{-1}, \Psi_{3}=\left(\lambda_{6}-\lambda_{4}\right)^{-1}\left(\lambda_{6}-\lambda_{5}\right)^{-1}\) \(\tau_{0}^{\prime}=\max \left(\tau_{2}^{\prime}, \tau_{1}^{\prime}\right), \tau_{1}^{\prime}=\frac{\left.\gamma_{(t / f}\right)}{\rho_{r}\left(b_{1}-\gamma_{t j} \sigma_{2}\right)}\)   with \(b_{1}>\gamma_{t h f} \sigma_{2}\)  and \(\tau_{2}^{\prime}=\frac{\gamma_{\text {thm }}}{\rho_{r}\left(b_{2}-\gamma_{\text {thm }}\left(b_{1}+\varpi_{2}\right)\right)}\)  with  \(b_{2}>\gamma_{t h m}\left(b_{1}+\sigma_{2}\right) \quad, \quad \xi_{1}=\rho_{u} a_{1} \delta_{1}^{2} \quad, \quad \lambda_{4}=1 /\left(\rho_{u} a_{2} \delta_{2}^{2}\right) \quad, \quad \lambda_{5}=1 /\left(\varpi_{1} \rho_{u} a_{3} \delta_{3}^{2}\right)\)  and  \(\lambda_{6}=1 /\left(\varpi_{1} \rho_{u} a_{4} \delta_{4}^{2}\right)\).

 Proof: See Appendix B.

 2) Outage probability for U₂

 The outage events of U₂  include four conditions as follows: i) The information   x_i cannot be decoded correctly by relay. ii) The relay cannot decode x₂ , while the relay can first decode x₁ , successfully. iii)U₃   cannot decode x₂  correctly. iv)  x₂ cannot be decoded by U₄  successfully. Therefore, the outage probability of U₂  can be expressed as

\(P_{U_{2}}^{ {out }}=1-\operatorname{Pr}\left(\gamma_{R \rightarrow x_{2}}>\gamma_{ {thm }}, \gamma_{R \rightarrow x_{1}}>\gamma_{ {thf }}\right)\operatorname{Pr}\left(\gamma_{u_{3} \rightarrow x_{2}}>\gamma_{t h m}\right) \operatorname{Pr}\left(\gamma_{u_{4} \rightarrow x_{2}}>\gamma_{t h m}\right)\)       (13)

 Then, the outage probability of U₂  for the TWR-NOMA system with RHIs and CEEs is provided in the following theorem.

 Theorem 2. For the non-ideal cases \(\left(\delta_{e_{i}}^{2} \neq 0, \kappa_{l} \neq 0\right)\) , the exact analytical expression of U₂  in terms of outage probability is expressed as

\(\begin{aligned}P_{U2}^{ni,out}=1-e^{-\left(\frac{\theta_{2} \vartheta_{1}}{\hat{\delta}_{2}^{2}}+\frac{\theta_{1} \vartheta_{1}}{\hat{\delta}_{1}^{2}}+\frac{\tau_{2}}{\hat{\delta}_{3}^{2}}+\frac{\tau_{3}}{\hat{\delta}_{4}^{2}}\right)}\frac{\hat{\delta}_{2}^{2} \hat{\delta}_{1}^{2} \lambda_{1}^{\prime} \lambda_{2}^{\prime}}{\left(\theta_{2} k_{r}^{2} \rho_{u} \hat{\delta}_{1}^{2}+\hat{\delta}_{2}^{2}\right)\left(\theta_{1} \rho_{u} \hat{\delta}_{2}^{2}\left(a_{2}+k_{r}^{2}\right)+\hat{\delta}_{1}^{2}\right)\left(\lambda_{2}^{\prime}-\lambda_{1}^{\prime}\right)} \\\times\left(\frac{\hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}{\theta_{2} \hat{\delta}_{1}^{2}+\theta_{1} \hat{\delta}_{2}^{2}+\lambda_{1}^{\prime} \hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}-\frac{\hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}{\theta_{2} \hat{\delta}_{1}^{2}+\theta_{1} \hat{\delta}_{2}^{2}+\lambda_{2}^{\prime} \hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}\right)\end{aligned}\)       (14)

where \(\lambda_{1}^{\prime}=1 /\left(\rho_{u} \hat{\delta}_{3}^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\right)\)  and  \(\lambda_{2}=1 /\left(\rho_{u} \hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right)\).  \(\theta_{2}=\frac{\gamma_{\text {thm}}}{\rho_{u}\left(a_{2}-\gamma_{\text {thm}} \kappa_{r}^{2}\right)}\) with  \(a_{2}>\gamma_{t h m} \kappa_{r}^{2} \cdot \tau_{3}=\frac{\gamma_{t h m} \vartheta_{3}}{\rho_{r}\left(b_{2}-\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\right)}\).   with  \(b_{2}>\gamma_{t / m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\).

 Proof: See Appendix

 Corollary 2. For the ideal cases \(\left(\delta_{e_{i}}^{2}=0, \kappa_{l}=0\right)\) , the analytical expression for the outage probability with the ideal case of U₂  is expressed as

\(\mathrm{P}_{U_{2}}^{i d, { out }}=1-e^{-\left(\frac{\gamma_{t h m} \beta_{t}}{\rho_{u} a_{2}}+\frac{\gamma_{t h f}}{\rho_{u} a_{1} \delta_{1}^{2}}+\frac{\tau_{2}^{\prime}}{\delta_{3}^{2}}+\frac{\tau_{2}^{\prime}}{\delta_{4}^{2}}\right)}\frac{\lambda_{5} \lambda_{6}}{\beta_{t} \delta_{2}^{2}\left(\lambda_{6}-\lambda_{5}\right)}\sum_{k=5}^{6} \frac{(-1)^{k+1} \rho_{u} a_{2}}{\beta_{t} \gamma_{t h m}+\lambda_{k} \rho_{u} a_{2}}\)       (15)

where  \(\tau_{2}^{\prime}=\frac{\gamma_{\text {thm}}}{\rho_{r}\left(b_{2}-\gamma_{\text {thm}}\left(b_{1}+\sigma_{2}\right)\right)}\) with \(b_{2}>\gamma_{t h m}\left(b_{1}+w_{2}\right)\)  and \(\beta_{t}=\frac{\gamma_{t h f} a_{2}}{a_{1} \delta_{1}^{2}}+\frac{1}{\delta_{2}^{2}}\) .   .\(Y^{\prime}=Y_{5}+Y_{6} \text { with } \lambda_{5}=1 /\left(\varpi_{1} \rho_{u} a_{3} \delta_{3}^{2}\right), \lambda_{6}=1 /\left(\omega_{1} \rho_{u} a_{4} \delta_{4}^{2}\right)\)

 Proof: See Appendix D.

 

3.2 Asymptotic Outage Probability

 To gain more insights, the asymptotic outage behavior is studied in the high SNR region. For the TWR-NOMA system,  \(\rho_{u}=\varepsilon \rho_{r} \rightarrow \infty\) and \(\varepsilon>0\) . The asymptotic SINR of (3) and (4) can be approximately expressed as

\(\bar{\gamma}_{R \rightarrow x_{1}}=\frac{\left|\hat{h}_{1}\right|^{2} a_{1}}{\kappa_{r}^{2}\left|\hat{h}_{1}\right|^{2}+\left|\hat{h}_{2}\right|^{2}\left(a_{2}+\kappa_{r}^{2}\right)+\left|\hat{h}_{3}\right|^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)+\left|\hat{h}_{4}\right|^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)+\bar{\vartheta}_1}\)       (16)

\(\bar{\gamma}_{R \rightarrow x_{2}}=\frac{\left|\hat{h}_{2}\right|^{2} a_{2}}{\kappa_{r}^{2}\left|\hat{h}_{1}\right|^{2}+\kappa_{r}^{2}\left|\hat{h}_{2}\right|^{2}+\left|\hat{h}_{3}\right|^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)+\left|\hat{h}_{4}\right|^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)+\bar{\vartheta}}\)       (17)

where \(\bar{\vartheta}_{1}=a_{1} \delta_{e_{1}}^{2}+a_{2} \delta_{e_{2}}^{2}+\varpi_{1} a_{3} \delta_{e_{3}}^{2}+\varpi_{1} a_{4} \delta_{e_{4}}^{2}+k_{r}^{2}\left(\delta_{e_{1}}^{2}+\delta_{e_{2}}^{2}+\delta_{e_{3}}^{2}+\delta_{e_{4}}^{2}\right)\).

 The asymptotic SINR of (7), (8) and (9) can be approximately expressed as

\(\bar{\gamma}_{U_{3} \rightarrow x_{2}}=\frac{\left|\hat{h}_{3}\right|^{2} b_{2}}{\left|\hat{h}_{3}\right|^{2} b_{1}+\left|\hat{h}_{3}\right|^{2} \kappa_{t}^{2}+\varpi_{2}\left|\hat{h}_{3}\right|^{2}+\bar{\vartheta}_{2}}\)       (18)

\(\bar{\gamma}_{U_{3} \rightarrow x_{1}}=\frac{\left|\hat{h}_{3}\right|^{2} b_{1}}{\left|\hat{h}_{3}\right|^{2} \kappa_{t}^{2}+\varpi_{2}\left|\hat{h}_{3}\right|^{2}+\bar{\vartheta}_{2}}\)       (19)

\(\bar{\gamma}_{U_{4} \rightarrow {x}_{2}}=\frac{\left|\hat{h}_{4}\right|^{2} b_{2}}{\left|\hat{h}_{4}\right|^{2} b_{1}+\left|\hat{h}_{4}\right|^{2} \kappa_{t}^{2}+\varpi_{2}\left|\hat{h}_{4}\right|^{2}+\bar{\vartheta}_{3}}\)       (20)

where \(\bar{g}_{2}=\delta_{e_{3}}^{2}+\delta_{e_{e}}^{2} \kappa_{t}^{2}+\sigma_{2} \delta_{e_{3}}^{2}\)  and  \(\overline{\mathcal{G}}_{3}=\delta_{\epsilon_{4}}^{2}+\delta_{\varepsilon_{4}}^{2} \kappa_{t}^{2}+\varpi_{2} \delta_{e_{4}}^{2}\).

 Lemma 1. For the non-ideal cases  \(\left(\delta_{e_{i}}^{2} \neq 0, \kappa_{l} \neq 0\right)\), the asymptotic outage probability of  U₁ and U₂  can be derived as following

\(\mathrm{P}_{U_{1}}^{n_{i}, \infty} \approx 1-e^{-\frac{\bar{\tau}_{0}}{\hat{\delta}_{3}^{2}}-\frac{\bar{\theta}_{1} \bar\vartheta_{1}}{\hat{\delta}_{1}^{2}}}\prod_{i=1}^{3} \bar{\lambda}_{i}\left(\frac{\bar{\Phi}_{1} \hat{\delta}_{1}^{2}}{\bar{\lambda}_{1} \hat{\delta}_{1}^{2}+\bar{\theta}_{1}}-\frac{\bar{\Phi}_{2} \hat{\delta}_{1}^{2}}{\bar{\lambda}_{2} \hat{\delta}_{1}^{2}+\bar{\theta}_{1}}+\frac{\bar{\Phi}_{3} \hat{\delta}_{1}^{2}}{\bar{\lambda}_{3} \hat{\delta}_{1}^{2}+\bar{\theta}_{1}}\right)\)       (21)

where \(\bar{\Phi}_{1}=\left(\bar{\lambda}_{2}-\bar{\lambda}_{1}\right)^{-1}\left(\bar{\lambda}_{3}-\bar{\lambda}_{1}\right)^{-1}, \bar{\Phi}_{2}=\left(\bar{\lambda}_{3}-\bar{\lambda}_{2}\right)^{-1}\left(\bar{\lambda}_{2}-\bar{\lambda}_{1}\right)^{-1}, \bar{\Phi}_{3}=\left(\bar{\lambda}_{3}-\bar{\lambda}_{1}\right)^{-1}\left(\bar{\lambda}_{3}-\bar{\lambda}_{2}\right)^{-1}\)\(\bar{\lambda}_{1}=1 /\left(\hat{\delta}_{2}^{2}\left(a_{2}+\kappa_{r}^{2}\right)\right), \bar{\lambda}_{2}=1 /\left(\hat{\delta}_{3}^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\right), \bar{\lambda}_{3}=1 /\left(\hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right) \cdot \bar{\theta}_{1}=\frac{\gamma_{i t / r}}{\left(a_{1}-\gamma_{t i f} \kappa_{r}^{2}\right)}\) .with \(a_{1}>\gamma_{t h f} \kappa_{r}^{2}, \bar{\tau}_{0}=\max \left(\bar{\tau}_{2}, \bar{\tau}_{1}\right), \bar{\tau}_{1}=\frac{\gamma_{t h f} \bar{\vartheta}_{2}}{\left(b_{1}-\gamma_{t h f}\left(\kappa_{t}^{2}+\varpi_{2}\right)\right)}\)  with \(b_{1}>\gamma_{t h f}\left(\kappa_{t}^{2}+\varpi_{2}\right)\)  and \(\bar{\tau}_{2}=\frac{\gamma_{t h m} \bar{g}_{2}}{\left(b_{2}-\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\right)}\)  with \(b_{2}>\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+w_{2}\right)\) . 

\(\begin{array}{lc}\mathrm{P}_{U_{2}}^{n i, \infty} \approx 1-e^{-\left(\frac{\bar{\theta}_{2} \bar{\vartheta}_{1}}{\hat{\delta}_{2}^{2}}+\frac{\bar{\theta}_{1} \bar{\vartheta}_{1}}{\hat{\delta}_{1}^{2}}+\frac{\bar{\tau}_{2}}{\hat{\delta}_{3}^{2}}+\frac{\bar{\tau}_{3}}{\hat{\delta}_{4}^{2}}\right)}\frac{\hat{\delta}_{2}^{2} \hat{\delta}_{1}^{2} \bar{\lambda}_{1}^{\prime} \bar{\lambda}_{2}^{\prime}}{\left(\theta_{2} \kappa_{r}^{2} \hat{\delta}_{1}^{2}+\hat{\delta}_{2}^{2}\right)\left(\theta_{1} \hat{\delta}_{2}^{2}\left(a_{2}+\kappa_{r}^{2}\right)+\hat{\delta}_{1}^{2}\right)\left(\bar{\lambda}'_{2}-\bar{\lambda}_{1}^{\prime}\right)}\\\times\left(\frac{\hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}{\bar{\theta}_{2} \hat{\delta}_{1}^{2}+\bar{\theta}_{1} \hat{\delta}_{2}^{2}+\bar{\lambda}_{1}^{\prime} \hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}-\frac{\hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}{\bar{\theta}_{2} \hat{\delta}_{1}^{2}+\bar{\theta}_{1} \hat{\delta}_{2}^{2}+\bar{\lambda}_{2}^{\prime} \hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}\right)\end{array}\)       (22)

where \(\bar{\lambda}_{1}=1 /\left(\hat{\delta}_{3}^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\right), \bar{\lambda}_{2}=1 /\left(\hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right) \cdot \bar{\theta}_{2}=\frac{\gamma_{44 m}}{a_{2}-\gamma_{4 m} \kappa_{r}^{2}}\) with \(a_{2}>\gamma_{t h m} \kappa_{r}^{2}\) .\(\bar{\tau}_{3}=\frac{\gamma_{t h m} \bar{\vartheta}_{3}}{b_{2}-\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)}\)   with \(b_{2}>\gamma_{\tan m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\)  .

 Lemma 2. Based on the derived results of (12) and (15), when \(\rho_{u}=\varepsilon \rho_{r} \rightarrow \infty\) , and \(\varepsilon>0\) , using   and L’Hospital’s rule, the ideal cases   of the outage probability of   and   can be approximated as follows

\(\mathrm{P}_{U_{1}}^{i d, \infty} \approx 1-\prod_{i=4}^{6} \bar{\lambda}_{i}\left(\frac{\bar{\Psi}_{1} a_{1} \delta_{1}^{2}}{a_{1} \delta_{1}^{2} \bar{\lambda}_{4}+\gamma_{t h f}}-\frac{\bar{\Psi}_{2} a_{1} \delta_{1}^{2}}{a_{1} \delta_{1}^{2} \bar{\lambda}_{5}+\gamma_{t i f}}+\frac{\bar{\Psi}_{3} a_{1} \delta_{1}^{2}}{a_{1} \delta_{1}^{2} \bar{\lambda}_{6}+\gamma_{t h f}}\right)\)       (23)

where \(\bar{\Psi}_{1}=\left(\bar{\lambda}_{5}-\bar{\lambda}_{4}\right)^{-1}\left(\bar{\lambda}_{6}-\bar{\lambda}_{4}\right)^{-1}, \bar{\Psi}_{2}=\left(\bar{\lambda}_{6}-\bar{\lambda}_{5}\right)^{-1}\left(\bar{\lambda}_{5}-\bar{\lambda}_{4}\right)^{-1}, \bar{\Psi}_{3}=\left(\bar{\lambda}_{6}-\bar{\lambda}_{4}\right)^{-1}\left(\bar{\lambda}_{6}-\bar{\lambda}_{5}\right)^{-1}\)  \(\bar{\lambda}_{4}=1 /\left(a_{2} \delta_{2}^{2}\right), \bar{\lambda}_{5}=1 /\left(\varpi_{1} a_{3} \delta_{3}^{2}\right)\) and \(\bar{\lambda}_{6}=1 /\left(\varpi_{1} a_{4} \delta_{4}^{2}\right)\) .

\(\mathrm{P}_{U_{2}}^{i d, \infty} \approx 1-\frac{\bar{\lambda}_{5} \bar{\lambda}_{6}}{\beta_{t} \delta_{2}^{2}\left(\bar{\lambda}_{6}-\bar{\lambda}_{5}\right)} \sum_{k=5}^{6} \frac{(-1)^{k+1} a_{2}}{\beta_{t} \gamma_{t h m}+\bar{\lambda}_{k} a_{2}}\)       (24)

where  \(\beta_{t}=\frac{\gamma_{t h f} a_{2}}{a_{1} \delta_{1}^{2}}+\frac{1}{\delta_{2}^{2}}\).

 3) Diversity Orders

 In this subsection, the diversity order is analyzed, which is defined as [34]

\(d=-\lim _{\rho \rightarrow \infty} \frac{\log \left(\mathrm{P}_{U_{n}}^{\infty}(\rho)\right)}{\log \rho}\)       (25)

where \(\mathbf{P}_{U_{n}}^{\infty}\)  denotes the asymptotic outage probability of U_n , \(n \in\{1,2\}\) .

 By using the definition in (25), the diversity orders for the non-ideal conditions \(\left(\delta_{\varepsilon_{\varepsilon}}^{2} \neq 0, \kappa_{l} \neq 0\right)\)  of both  U₁ and U₂  are obtained as

\(d_{1}^{n i}=-\lim _{\rho \rightarrow \infty} \frac{\log \left(\mathrm{P}_{U_{1}}^{n i, \infty}(\rho)\right)}{\log \rho}=0\)       (26)

\(d_{2}^{n i}=-\lim _{\rho \rightarrow \infty} \frac{\log \left(\mathrm{P}_{U_{2}}^{n i, \infty}(\rho)\right)}{\log \rho}=0\)       (27)

 The diversity orders for the ideal conditions  \(\left(\delta_{e_{i}}^{2}=0, \kappa_{l}=0\right)\) of both U₁  and  U₂ are derived as

\(d_{1}^{i d}=-\lim _{\rho \rightarrow \infty} \frac{\log \left(\mathrm{P}_{U_{1}}^{i d, \infty}(\rho)\right)}{\log \rho}=0\)       (28)

\(d_{2}^{i d}=-\lim _{\rho \rightarrow \infty} \frac{\log \left(\mathrm{P}_{U_{2}}^{i d, \infty}(\rho)\right)}{\log \rho}=0\)       (29)

 Remark: As can be observed from (21)-(24), there exists error floors for both the ideal and non-ideal conditions due to the intra-group interference. In addition, RHIs and CEEs deteriorate the outage probability since they can be recognized extra interference. As can also be seen in (26)-(29), the diversity orders for the ideal and non-ideal conditions are both zero due to the fixed outage probabilities at high SNRs. This means that TWR-NOMA can not obtain diversity gains.

 

4. Numerical Results and Discussions

 In this section, some numerical results are provided to verify the accuracy of the theo-retical analysis. We provide numerical illustration of our analytical results through Monte Carlo Simulations. Unless otherwise specified, the parameter values are provided in the Table 1. For convenience, we assume \(\delta_{e}^{2}=\delta_{e_{1}}^{2}=\delta_{e_{2}}^{2}=\delta_{e_{3}}^{2}=\delta_{e_{4}}^{2}, \kappa=\kappa_{r}=\kappa_{t}\)  .

Table 1. Table of simulation parameters for numerical results

Fig. 2. Outage probability versus the transmit SNR.

 Fig. 2 plots the outage probabilities of the two users versus transmit SNR for different values of k  and \(\delta_{e}^{2}\) . We consider three cases in this simulation. 1)k=0  , \(\delta_{e}^{2}\) =0.03; 2)k=0.05  ,\(\delta_{e}^{2}\) =0 ; 3)k=0 ,\(\delta_{e}^{2}\) =0 . The curves represent the exact and asymptotic analytical of outage probability forU₁  and U₂  of ideal and non-ideal conditions in (11), (12), (14), (15) and (21), (22), (23), (24), respectively. It is clear that analytical curves are in good agreement with Monte Carlo simulations. It can be seen from this figure that the outage performance of the system with RHIs or CEEs is worse than that of the system without RHIs and CEEs for both U₁  and U₂ , which means that RHIs and CEEs have deleterious effects on the system outage performance. Moreover, it is readily noticed from Fig. 2 that for the third case, there are error floors for U₁  and U₂ . The reason can be explained that the intra-group IS result in zero diversity orders. This conclusion is confirmed by (28) and (29).

Fig. 3. Outage probability versus the transmit SNR.

 Fig. 3 plots the outage performance versus SNR with different levels of inter-group IS from w={0,0.1,0.15}  . For the purpose of comparison, the results of [16] have been provided. It can be seen that with the improvement of inter-group IS coefficient, the outage performance decreases obviously. We can conclude that the existence of the inter-group IS makes the outage performance worse. In addition, for the case (w=0)  and (w=0.1) , it still have the error floors caused by the intra-group IS, which is consistent with the results of [16]. We can observe that RHIs has a negative effect on the outage performance by comparing the red curves with the blue curves.

Fig. 4. Outage probability versus the CEEs.

 Fig. 4 plots the outage performance of the TWR-NOMA system versus CEEs for different RHIs parameters k={0,0.15} . As observed from Fig. 4, the outage probabili-ties of the TWR-NOMA network for the two users increase as CEEs grow large. This happens because the CEEs act as interference for the desired signal, which reduces the system performance. Moreover, the growth of U₁  is larger than that of U₂ , which means that CEEs have more serious effect on the outage probabilities of the near users than the far users for this system.

Fig. 5. Outage probability versus the RHIs at the relay node.

 Fig. 5 illustrates the impact of the RHIs at the relay node on the outage performance of the TWR-NOMA systems. These curves represent two cases: ideal case (\(\delta_{e}^{2}\)=0)  and non-ideal case  (\(\delta_{e}^{2}\)=0.05)  . In addition, we assume a fixed transmit SNR (20dB) in this simulation. It can be observed that the outage performance becomes worse as the RHIs increase in both two cases. Moreover, when the RHIs is equal to zero (k=0) , the out-age probabilities of this two cases have different values, due to the CEEs existence in the non-ideal case.

Fig. 6. Outage probability versus the RHIs and CEEs.

 Fig. 6 illustrates the impact of the RHIs and the CEEs at the relay node on the outage performance of the TWR-NOMA systems. We use the change of color to reflect the outage performance affected. There is a chromaticity bar on the right side of the graph. As the color changes from dark to light indicates that the performance of system outage becomes degrades. There are two layers of grid in this figure, the lower one denotes the performance of U₁  and the other layer denotes the performance of U₂ . This means that the performance of U₁  is better than that of U₂ . It is apparent to see from the Fig. 6 that as RHIs increase, the color changes from dark blue to light blue, and as the CEEs increase, the color clearly changes from dark to light, which means the outage probabil-ity becomes worse, respectively.

Fig. 7. Outage probability versus 

 Fig. 7 depicts the impact of P_u  on the outage performance. In this simulation, we consider two cases: 1) P_u =-20:40  dB and P_u+P_r=30dB ; 2) P_u =-20:40dB andP_u =P_r  . For the first case, we can clearly see that there exist optimal allocation powers (about 0dB for U₁  and 6dB for U₂  ) for P_u   of U₁ and U₂ . This happens because the performance gain caused by increasing the power of users (P_u ) is larger than the per-formance loss caused by reducing the power of relay (P_r ). For the second case, we can also see that the outage probabilities for the two users always decrease as the power of users and relay increasing.

 

5. Conclusion

 This paper investigated the performance of TWR-NOMA networks with RHIs and CEEs, in which two groups of users exchange information with the aid of a DF relay. The closed-form expressions for exact outage probability of two group users were de-rived and the asymptotic behavior was discussed. Based on the derived analytical re-sults, we further evaluated the diversity orders achieved by the users. Results revealed that CEEs have more deleterious effects on the outage probabilities of the near users than the far users. Furthermore, it was shown that the TWR-NOMA networks with in-ter-group IS, intra-group IS, RHIs or CEEs cause error floors.

 

APPENDIX A

PROOF OF THEOREM 1

 It is worth noting that for the convenience of calculation, we assume  \(X_{i}=\left|h_{i}\right|^{2}\),   \(\hat{X}_{i}=\left|\hat{h}_{i}\right|^{2}\)with \(i \in\{1,2,3,4\}\) . Substituting (3), (7) and (8) into (10), the outage probabil-ity of U₁  is given by

\(\begin{array}{l}\mathrm{P}_{U_{1}}^{n i, { out}}=1-\operatorname{Pr}\left(\gamma_{R \rightarrow x_{1}}>\gamma_{ {thf }}\right) \operatorname{Pr}\left(\gamma_{U_{3} \rightarrow x_{2}}>\gamma_{ {thm }}, \gamma_{U_{3} \rightarrow x_{1}}>\gamma_{ {thf }}\right) \\\quad=1-f_{1} f_{2}\end{array}\)       (A.1)

\(f_{1}=\operatorname{Pr}\left(\frac{\rho_{u}\left|\hat{h}_{1}\right|^{2} a_{1}}{\rho_{u}\left|\hat{h}_{2}\right|^{2} a_{2}+\sum_{k=1}^{4} \rho_{u}\left|\hat{h}_{k}\right|^{2} \kappa_{r}^{2}+\sum_{j=3}^{4} \rho_{u}\left|\hat{h}_{j}\right|^{2} \varpi_{1} a_{j}+\vartheta_{1}}>\gamma_{t h f}\right)\)       (A.2)

\(\begin{aligned}f_{2}=\operatorname{Pr}\left(\frac{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{2}}{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{1}+\rho_{r}\left|\hat{h}_{3}\right|^{2} \kappa_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{3}\right|^{2}+\vartheta_{2}}>\gamma_{t h m}\right.& \\\left.\frac{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{1}}{\rho_{r}\left|\hat{h}_{3}\right|^{2} \kappa_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{3}\right|^{2}+\vartheta_{2}}>\gamma_{ {thf }}\right)\end{aligned}\)       (A.3)

 For calculating the probability of f₁  in (A.2), we set \(z=z_{1}+z_{2}+z_{3}\)  with \(z_{1}=\rho_{u}\left|\hat{h}_{2}\right|^{2}\left(a_{2}+\kappa_{r}^{2}\right), z_{2}=\rho_{u}\left|\hat{h}_{3}\right|^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right), \text { and } z_{3}=\rho_{u}\left|\hat{h}_{4}\right|^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\)and  . As we wrote earlier, \(\left|\hat{h}_{i}\right|^{2}\)  follow the exponential distribution with the means  \(\hat{\delta}_{i}^{2}, i \in\{1,2,3,4\}\)  . In addition, \(z_{1}, z_{2}\)  and  \(z_{3}\) are also independent exponentially distributed random var-iables with means \(\lambda_{1}=1 /\left(\rho_{u} \hat{\delta}_{2}^{2}\left(a_{2}+\kappa_{r}^{2}\right)\right), \lambda_{2}=1 /\left(\rho_{u} \hat{\delta}_{3}^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\right)\) and \(\lambda_{3}=1 /\left(\rho_{u} \hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right)\) , respectively. As shown in the [35], for the independent non-identical distributed fading scenario, the PDF of \(z \)  is given by

\(f_{Z}(z)=\prod_{t=1}^{3} \lambda_{t}\left(\Phi_{1} e^{-\lambda_{1} z}-\Phi_{2} e^{-\lambda_{2} z}+\Phi_{3} e^{-\lambda_{3} z}\right)\)       (A.4)

where \(\Phi_{1}=\left(\lambda_{2}-\lambda_{1}\right)^{-1}\left(\lambda_{3}-\lambda_{1}\right)^{-1}, \Phi_{2}=\left(\lambda_{3}-\lambda_{2}\right)^{-1}\left(\lambda_{2}-\lambda_{1}\right)^{-1}, \Phi_{3}=\left(\lambda_{3}-\lambda_{1}\right)^{-1}\left(\lambda_{3}-\lambda_{2}\right)^{-1}\)

 According to the above explanations,   is expressed as follows

\(\begin{aligned}f_{1} &=\operatorname{Pr}\left(\hat{X}_{1}>\theta_{1}\left(z+\vartheta_{1}\right)\right) \\&=e^{-\frac{\theta, \vartheta_1}{{\hat{\delta}}_{1}^{2}}} \prod_{i=1}^{3} \lambda_{i}\left(\frac{\Phi_{1} \hat{\delta}_{1}^{2}}{\lambda_{1} \hat{\delta}_{1}^{2}+\theta_{1}}-\frac{\Phi_{2} \hat{\delta}_{1}^{2}}{\lambda_{2} \hat{\delta}_{1}^{2}+\theta_{1}}+\frac{\Phi_{3} \hat{\delta}_{1}^{2}}{\lambda_{3} \hat{\delta}_{1}^{2}+\theta_{1}}\right)\end{aligned}\)       (A.5)

where \(\theta_{1}=\frac{\gamma_{t h f}}{\rho_{u}\left(a_{1}-\gamma_{t f f} \kappa_{r}^{2}\right)}\) , with \(a_{1}>\gamma_{t h f} \kappa_{r}^{2}\) .

   can be further calculated as follows

\(\begin{aligned}f_{2} &=\operatorname{Pr}\left(\hat{X}_{3}>\tau_{2}, \hat{X}_{3}>\tau_{1}\right) \\&=\operatorname{Pr}\left(\hat{X}_{3}>\max \left(\tau_{2}, \tau_{1}\right)=\tau_{0}\right) \\&=e^{-\frac{\tau_{0}}{\hat{\delta_{3}^{2}}}}\end{aligned}\)       (A.6)

where \(\tau_{0}=\max \left(\tau_{2}, \tau_{1}\right), \tau_{2}=\frac{\gamma_{\text {thm}} \vartheta_{2}}{\rho_{r}\left(b_{2}-\gamma_{\text {thm}}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\right)}\) with \(b_{2}>\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\)  and  \(\tau_{1}=\frac{\gamma_{t h f} \vartheta_{2}}{\rho_{r}\left(b_{1}-\gamma_{t h f}\left(\kappa_{t}^{2}+\varpi_{2}\right)\right)}\) with \(b_{1}>\gamma_{t h f}\left(\kappa_{t}^{2}+\varpi_{2}\right)\) .

 Substituting (A.5), (A.6) into (A.1), (11) can be obtained.

 The proof is completed.

 

APPENDIX B

PROOF OF COROLLARY 1

 The outage probability of  U₁ for the ideal cases can be expressed as

\(\begin{array}{lc}\mathrm{P}_{U_{1}}^{i d, { out }}=1-\operatorname{Pr}\left(\gamma_{R \rightarrow x_{1}}>\gamma_{t h f}\right) \operatorname{Pr}\left(\gamma_{U_{3} \rightarrow x_{2}}>\gamma_{t h m}, \gamma_{U_{3} \rightarrow x_{1}}>\gamma_{t h f}\right) \\=1-\operatorname{Pr}\left(\frac{\rho_{u} X_{1} a_{1}}{\rho_{u} X_{2} a_{2}+\varpi_{1} \rho_{u} X_{3} a_{3}+\varpi_{1} \rho_{u} X_{4} a_{4}+1}>\gamma_{t h f}\right)\\\times \operatorname{Pr}\left(\frac{\rho_{r} X_{3} b_{2}}{\rho_{r} X_{3}\left(b_{1}+\varpi_{2}\right)+1}>\gamma_{t h m}, \frac{\rho_{r} X_{3} b_{1}}{\rho_{r} X_{3} \varpi_{2}+1}>\gamma_{t h f}\right)\\=1-\operatorname{Pr}\left(X_{1}>\frac{\gamma_{ {thf}}}{\rho_{u} a_{1}}(Y+1)\right) \operatorname{Pr}\left(X_{3}>\tau_{0}^{\prime}=\max \left(\tau_{2}^{\prime}, \tau_{1}^{\prime}\right)\right)\\=1-f_{1}^{\prime} f_{2}^{\prime}\end{array}\)       (B.1)

where \(Y=Y_{4}+Y_{5}+Y\) , with  \(Y_{4}=\rho_{u} a_{2} X_{2}, Y_{5}=\varpi_{1} \rho_{u} a_{3} X_{3}, Y_{6}=\varpi_{1} \rho_{u} a_{4} X_{4}\) . As we wrote earlierY₄,Y₅   and Y₆  are independent exponentially distributed random variables with means \(\lambda_{4}=1 /\left(\rho_{u} a_{2} \delta_{2}^{2}\right), \lambda_{5}=1 /\left(\varpi_{1} \rho_{u} a_{3} \delta_{3}^{2}\right)\) , and \(\lambda_{6}=1 /\left(\varpi_{1} \rho_{u} a_{4} \delta_{4}^{2}\right)\) .

\(f_{1}^{\prime}\)   can be further expressed as follows

\(\begin{array}{lc}f_{1}^{\prime}=\operatorname{Pr}\left(X_{1}>\frac{\gamma_{ {thf}}}{\rho_{u} a_{1}}(Y+1)\right) \\=e^{-\frac{\gamma_{ {thf}}}{\xi_{1}}} \prod_{i=1}^{3} \lambda_{i}\left(\frac{\Psi_{1} \xi_{1}}{\xi_{1} \lambda_{4}+\gamma_{ {thf }}}-\frac{\Psi_{2} \xi_{1}}{\xi_{1} \lambda_{5}+\gamma_{ {thf }}}+\frac{\Psi_{3} \xi_{1}}{\xi_{1} \lambda_{6}+\gamma_{ {thf }}}\right),\end{array}\)       (B.2)

where  \(\Psi_{1}=\left(\lambda_{5}-\lambda_{4}\right)^{-1}\left(\lambda_{6}-\lambda_{4}\right)^{-1}, \Psi_{2}=\left(\lambda_{6}-\lambda_{5}\right)^{-1}\left(\lambda_{5}-\lambda_{4}\right)^{-1}, \Psi_{3}=\left(\lambda_{6}-\lambda_{4}\right)^{-1}\left(\lambda_{6}-\lambda_{5}\right)^{-1}\) \(\xi_{1}=\rho_{u} a_{1} \delta_{1}^{2}\).

\(f_{2}^{\prime}\)  can be easily calculated as follows

\(f_{2}^{\prime}=\operatorname{Pr}\left(X_{3}>\tau_{0}^{\prime}\right)=e^{-\frac{\tau_{0}^{\prime}}{\delta_{3}^{2}}}\)       (B.3)

where  \(\tau_{0}^{\prime}=\max \left(\tau_{2}^{\prime}, \tau_{1}^{\prime}\right), \tau_{1}^{\prime}=\frac{\gamma_{\text {thf}}}{\rho_{r}\left(b_{1}-\gamma_{\text {tif}} \sigma_{2}\right)}\) with \(b_{1}>\gamma_{t i f} \varpi_{2}, \tau_{2}^{\prime}=\frac{\gamma_{t h m}}{\rho_{r}\left(b_{2}-\gamma_{t h m}\left(b_{1}+\sigma_{2}\right)\right)}\)   with  \(b_{2}>\gamma_{t h m}\left(b_{1}+\varpi_{2}\right)\).

 Combining (B.2), (B.3) into (B.1), we can obtain (12).

 The proof is completed.

APPENDIX C

PROOF OF THEOREM 2

 The outage probability of U₂  can be expressed as

\(\begin{aligned}\mathrm{P}_{U_{2}}^{n i, { out}} &=1-\operatorname{Pr}\left(\gamma_{R \rightarrow x_{2}}>\gamma_{ {thm}}, \gamma_{R \rightarrow x_{1}}>\gamma_{ {thf}}\right) \operatorname{Pr}\left(\gamma_{ {u}_{3} \rightarrow x_{2}}>\gamma_{ {thm}}\right) \operatorname{Pr}\left(\gamma_{ {u}_{4} \rightarrow x_{2}}>\gamma_{ {thm}}\right) \\&=1-f_{3} f_{4} f_{5}\end{aligned}\)       (C.1)

 Substituting (3), (4) into (C.1), \(f_{3}\)  can be further given by

\(\begin{array}{lc}f_{3}=\operatorname{Pr}\left(\gamma_{R \rightarrow x_{2}}>\gamma_{t h m}, \gamma_{R \rightarrow x_{1}}>\gamma_{t h f}\right) \\=\operatorname{Pr}\left(\frac{\rho_{u}\left|\hat{h}_{2}\right|^{2} a_{2}}{\sum_{k=1}^{4} \rho_{u}\left|\hat{h}_{k}\right|^{2} \kappa_{r}^{2}+\sum_{j=3}^{4} \rho_{u}\left|\hat{h}_{j}\right|^{2} \varpi_{1} a_{j}+\vartheta_{1}}>\gamma_{t h m}\right.,\\\left.\frac{\rho_{u}\left|\hat{h}_{1}\right|^{2} a_{1}}{\rho_{u}\left|\hat{h}_{2}\right|^{2} a_{2}+\sum_{k=1}^{4} \rho_{u}\left|\hat{h}_{k}\right|^{2} \kappa_{r}^{2}+\sum_{j=3}^{4} \rho_{u}\left|\hat{h}_{j}\right|^{2} \varpi_{1} a_{j}+\vartheta_{1}}>\gamma_{t h f}\right)\\=\operatorname{Pr}\left(\hat{X}_{2}>\theta_{2}\left(\kappa_{r}^{2} \rho_{u} \hat{X}_{1}+z^{\prime}+\vartheta_{1}\right), \hat{X}_{1}>\theta_{1}\left(\rho_{u}\left(a_{2}+\kappa_{r}^{2}\right) \hat{X}_{2}+z^{\prime}+\vartheta_1\right)\right)\\=e^{-\left(\frac{\theta_{2} \vartheta_1}{\hat{\delta}_{2}^{2}}+\frac{\theta_{1} \vartheta_1}{\hat{\delta}_{1}^{2}}\right)}\frac{\hat{\delta}_{2}^{2} \hat{\delta}_{1}^{2}}{\left(\theta_{2} k_{r}^{2} \rho_{u} \hat{\delta}_{1}^{2}+\hat{\delta}_{2}^{2}\right)\left(\theta_{1} \rho_{u} \hat{\delta}_{2}^{2}\left(a_{2}+k_{r}^{2}\right)+\hat{\delta}_{1}^{2}\right)}\int_{0}^{\infty} e^{-\left(\frac{\theta_{2}}{\hat{\delta_{2}^{2}}}+\frac{\theta_{1}}{\hat{\delta_{1}^{2}}}\right) z'} f\left(z^{\prime}\right) d z^{\prime}\end{array}\)       (C.2)

 For calculating the probability of  \(f_{3}\) in (C.1), we assume  \(z^{\prime}=z_{1}^{\prime}+z_{2}^{\prime}\) , with \(z_{1}^{\prime}=\rho_{u}\left|\hat{h}_{3}\right|^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\) , and  \(z_{2}^{\prime}=\rho_{u}\left|\hat{h}_{4}\right|^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\). \(z_{1}^{\prime}\)  and  \(z_{2}^{\prime}\) are also independent exponentially distributed random variables with means \(\lambda_{1}^{\prime}=1 /\left(\rho_{u} \hat{\delta}_{3}^{2}\left(\varpi_{1} a_{3}+\kappa_{r}^{2}\right)\right)\)  and \(\lambda_{2}^{\prime}=1 /\left(\rho_{u} \hat{\delta}_{4}^{2}\left(\varpi_{1} a_{4}+\kappa_{r}^{2}\right)\right)\) , respectively. For the independent non-identical distributed fading scenario, the PDF of \(z^{\prime}\)  is given by

\(f_{z^{\prime}}\left(z^{\prime}\right)=\prod_{t=1}^{2} \lambda_{i}^{\prime}\left(\frac{e^{-\lambda_{1} z^{\prime}}}{\lambda_{2}^{\prime}-\lambda_{1}^{\prime}}-\frac{e^{-\lambda_{2} z^{\prime}}}{\lambda_{2}^{\prime}-\lambda_{1}^{\prime}}\right)\)       (C.3)

 Substituting (C.3) into (C.2), \(f_{3}\)  can be further rewritten as

\(\begin{aligned}f_{3}=e^{-\left(\frac{\theta_{2} \vartheta_{1}}{\hat{\delta}_{2}^{2}}+\frac{\theta_{1} \vartheta_1}{\hat{\delta}_{1}^{2}}\right)}\frac{\hat{\delta}_{2}^{2} \hat{\delta}_{1}^{2} \lambda_{1}^{\prime} \lambda_{2}^{\prime}}{\left(\theta_{2} k_{r}^{2} \rho_{u} \hat{\delta}_{1}^{2}+\hat{\delta}_{2}^{2}\right)\left(\theta_{1} \rho_{u} \hat{\delta}_{2}^{2}\left(a_{2}+k_{r}^{2}\right)+\hat{\delta}_{1}^{2}\right)\left(\lambda_{2}^{\prime}-\lambda_{1}^{\prime}\right)} \\\times\left(\frac{\hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}{\theta_{2} \hat{\delta}_{1}^{2}+\theta_{1} \hat{\delta}_{2}^{2}+\lambda_{1}^{\prime} \hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}-\frac{\hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}{\theta_{2} \hat{\delta}_{1}^{2}+\theta_{1} \hat{\delta}_{2}^{2}+\lambda_{2}^{\prime} \hat{\delta}_{1}^{2} \hat{\delta}_{2}^{2}}\right)\end{aligned}\)       (C.4)

 Substituting (7), (9) into (C.1),  \(f_{4}\) and  \(f_{5}\) can be calculated as follows

\(\begin{aligned}f_{4} &=\operatorname{Pr}\left(\frac{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{2}}{\rho_{r}\left|\hat{h}_{3}\right|^{2} b_{1}+\rho_{r}\left|\hat{h}_{3}\right|^{2} k_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{3}\right|^{2}+\vartheta_{2}}>\gamma_{t h m}\right) \\&=\operatorname{Pr}\left(\hat{X}_{3}>\tau_{2}\right) \\&=e^{-\frac{\tau_{2}}{\delta_{3}^{2}}}\end{aligned}\)       (C.5)

\(\begin{aligned}f_{5} &=\operatorname{Pr}\left(\frac{\rho_{r}\left|\hat{h}_{4}\right|^{2} b_{2}}{\rho_{r}\left|\hat{h}_{4}\right|^{2} b_{1}+\rho_{r}\left|\hat{h}_{4}\right|^{2} \kappa_{t}^{2}+\varpi_{2} \rho_{r}\left|\hat{h}_{4}\right|^{2}+\vartheta_{3}}>\gamma_{t h m}\right) \\&=\operatorname{Pr}\left(\hat{X}_{4}>\tau_{3}\right) \\&=e^{-\frac{\tau_{3}}{\delta_{4}^{2}}}\end{aligned}\)       (C.6)

where \(\tau_{3}=\frac{\gamma_{t h m} \vartheta_{3}}{\rho_{r}\left(b_{2}-\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\right)}\) , with \(b_{2}>\gamma_{t h m}\left(b_{1}+\kappa_{t}^{2}+\varpi_{2}\right)\) . Finally, (14) can be obtained by combing (C.4), (C.5), and (C.6).

The proof is completed.

 

APPENDIX D

PROOF OF COROLLARY 2

 The outage probability of U₂ with ideal conditions can be expressed as

\(\begin{array}{l}\mathrm{P}_{U_{2}}^{i d, \text { out }}=1-\operatorname{Pr}\left(\gamma_{R \rightarrow x_{2}}>\gamma_{ {thm}}, \gamma_{R \rightarrow x_{1}}>\gamma_{ {thf}}\right) \operatorname{Pr}\left(\gamma_{U_{3} \rightarrow x_{2}}>\gamma_{ {thm}}\right) \operatorname{Pr}\left(\gamma_{U_{4} \rightarrow x_{2}}>\gamma_{ {thm}}\right) \\\quad=1-f_{3}^{\prime} f_{4}^{\prime} f_{5}^{\prime}\end{array}\)       (D.1)

\(f_{3}^{\prime}\)   can be further calculated as

\(\begin{aligned}f_{3}^{\prime} &=\operatorname{Pr}\left(\frac{\rho_{u}\left|h_{2}\right|^{2} a_{2}}{\rho_{u}\left|h_{3}\right|^{2} \varpi_{1} a_{3}+\rho_{u}\left|h_{4}\right|^{2} \varpi_{1} a_{4}+1}>\gamma_{ {thm}}, \frac{\rho_{u}\left|h_{1}\right|^{2} a_{1}}{\rho_{u}\left|h_{2}\right|^{2} a_{2}+\rho_{u}\left|h_{3}\right|^{2} \varpi_{1} a_{3}+\rho_{u}\left|h_{4}\right|^{2} \varpi_{1} a_{4}+1}>\gamma_{thf}\right) \\&=\operatorname{Pr}\left(X_{2}>\frac{\gamma_{ {thm}}}{\rho_{u} a_{2}}\left(Y^{\prime}+1\right), X_{1}>\frac{\gamma_{ {thf}}}{\rho_{u} a_{1}}\left(\rho_{u} a_{2} X_{2}+Y^{\prime}+1\right)\right) \\&=\frac{1}{\beta_{t} \delta_{2}^{2}} e^{-\left(\frac{\gamma_{t_{{thm}}} \beta_{t}}{\rho_{u} a_{2}} \frac{\gamma_{thf}}{\rho_{u} a_{1} \delta_{1}^{2}}\right)} \int_{0}^{\infty} e^{-\frac{\gamma_{ {thm}} \beta_{t}}{\rho_{u} a_{2}} Y^{\prime}} f\left(Y^{\prime}\right) d Y^{\prime}\\&=\frac{\lambda_{5} \lambda_{6}}{\beta_{t} \delta_{2}^{2}\left(\lambda_{6}-\lambda_{5}\right)} e^{-\left(\frac{\gamma_{t h m} \beta_{t}}{\rho_{u} a_{2}}+\frac{\gamma_{t h f}}{\rho_{u} a_{1} \delta_{1}^{2}}\right)}\sum_{k=5}^{6} \frac{(-1)^{k+1} \rho_{u} a_{2}}{\beta_{t} \gamma_{t h m}+\lambda_{k} \rho_{u} a_{2}}\end{aligned}\)       (D.2)

where Y'=Y₅+Y₆  . As we wrote earlier \(Y_{5}=w_{1} \rho_{u} a_{3} X_{3}, Y_{6}=w_{1} \rho_{u} a_{4} X_{4}\) . In addition, Y₅ ,Y₆   are independent exponentially distributed random variables with means  \(\lambda_{5}=1 /\left(\varpi_{1} \rho_{u} a_{3} \delta_{3}^{2}\right)\), \(\lambda_{6}=1 /\left(\varpi_{1} \rho_{u} a_{4} \delta_{4}^{2}\right), \beta_{t}=\frac{\gamma_{t h f} a_{2}}{a_{1} \delta_{1}^{2}}+\frac{1}{\delta_{2}^{2}}\).

  \(f_{4}^{\prime}\) and \(f_{5}^{\prime}\)  are easily obtained as follows

\(\begin{aligned}&f_{4}^{\prime}=\operatorname{Pr}\left(\frac{\rho_{r}\left|h_{3}\right|^{2} b_{2}}{\rho_{r}\left|h_{3}\right|^{2} b_{1}+\varpi_{2} \rho_{r}\left|h_{3}\right|^{2}+1}>\gamma_{t h m}\right)\\&=\operatorname{Pr}\left(X_{3}>\tau_{2}^{\prime}\right)\\&=e^{-\frac{\tau_{2}^{\prime}}{\delta_{3}^{2}}}\end{aligned}\)       (D.3)

where  \(\tau_{2}^{\prime}=\frac{\gamma_{\text {thm}}}{\rho_{r}\left(b_{2}-\gamma_{\text {thm}}\left(b_{1}+\varpi_{2}\right)\right)}\) with \(b_{2}>\gamma_{t h m}\left(b_{1}+\varpi_{2}\right)\) .

\(\begin{aligned}&f_{5}^{\prime}=\operatorname{Pr}\left(\frac{\rho_{r}\left|h_{4}\right|^{2} b_{2}}{\rho_{r}\left|h_{4}\right|^{2} b_{1}+\varpi_{2} \rho_{r}\left|h_{4}\right|^{2}+1}>\gamma_{u i f}\right)\\&=\operatorname{Pr}\left(X_{4}>\tau_{2}^{\prime}\right)\\&=e^{-\frac{\tau_{2}^{\prime}}{\delta_{4}^{2}}}\end{aligned}\)       (D.4)

 Combining (D.2), (D.3) and (D.4) into (D.1), we can obtain (15).

 The proof is completed.