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Non-orthogonal multiple access in full-duplex-based coordinated direct and relay transmission (CDRT) system: performance analysis and optimization

Abstract

This paper considers non-orthogonal multiple access (NOMA)-based coordinated direct and relay transmission (CDRT) system, where the base station (BS) directly communicates with the nearby users while it requires the help of a dedicated relay to communicate with the cell-edge users. We derive exact closed form expression for the outage probabilities experienced by the downlink users and the system outage probability of the considered CDRT network with full-duplex relaying (FDR) technique. Further, we derive approximate closed form expressions for the ergodic rates achieved by the users. The channel of all the links experience Nakagami fading distribution and the analysis takes into account the residual interference generated due to the imperfect successive interference cancelation (I-SIC) technique. We provide numerical and simulation results to identify the impact of key system parameters on the outage and ergodic rate performance of the users and the system outage performance. The outage and ergodic rate performance of users in the considered FDR-based NOMA-CDRT system has been observed to be significantly improved compared to a FDR-based OMA (orthogonal multiple access)-CDRT system. It is observed that random selection of NOMA power allocation coefficients at the BS leads to higher outage for the near users compared to the far users. We determine numerical results for the NOMA power allocation coefficient that leads to equal outage performance for both the users. Finally, we derive analytical expression for the optimal power allocation (OPA) coefficient at the BS that minimizes the system outage probability. Through extensive numerical and simulation studies, we establish that OPA can lead to significant reduction of system outage probability compared to random selection of power allocation coefficients at the BS.

Introduction

Recently non-orthogonal multiple access (NOMA) has been identified as an effective multiple access technique to improve the spectrum efficiency in the fifth generation (5G) wireless networks. NOMA technique allows multiple users to coexist and share the same time-frequency resource block via power domain multiplexing mechanism [1, 2]. In this case, a NOMA enabled BS will apply the superposition coding technique to combine multiple user’s signals with distinct power levels, while the receivers use the successive interference cancelation (SIC) technique to decode the message. In a cooperative NOMA system, strong users (i.e., users experiencing better channel conditions) act as relays for delivering the message to weak users in the system [3]. In NOMA-based CDRT systems, the BS directly communicates with the nearby users while it requires the help of a dedicated relay to communicate with the far users [4]. The spectral efficiency of the system can be improved by operating the relay in the full-duplex mode that enables it to carry out simultaneous reception and transmission in the same frequency resource. However, in-band FDR technique leads to the generation of self-interference (SI) at the relay node, which is induced from its transmitter to the receive section [5]. Even though, the effect of SI can be mitigated by sophisticated interference suppression techniques, the relay node will still be affected by residual self-interference (RSI), which proportionally grows with the used transmit power [6].

The focus of the current work is on the performance analysis of NOMA-based CDRT system that employs FDR technique. The performance of cooperative NOMA system with half duplex relaying (HDR) has been analyzed thoroughly in the literature (e.g., [714] and references therein). The performance of cooperative NOMA-based CDRT has also been analyzed in the literature [4, 1517]. In [4], the authors have considered the application of NOMA in two-user HDR based CDRT system and analyzed the outage and ergodic sum rate performance. In [15], the authors have considered the application of uplink NOMA in HDR-based CDRT system. The ergodic sum capacity of the system has been analyzed under both perfect and imperfect SIC conditions. In [16], the authors have analyzed the outage and ergodic rate performance of the near and the far users in HDR-based NOMA-CDRT system, where an energy harvesting relay has been employed to assist the BS for delivering message to the far user. The authors of [17] have considered HDR-based NOMA-CDRT system with two cell-center users (CCUs) and a cell-edge user (CEU), which is assisted by a relay. Notice that all the above papers consider the performance evaluation of HDR-based NOMA-CDRT system.

In downlink cooperative NOMA system, the strong users will first decode the signal corresponding to the weak users from the received NOMA signal. They will then implement the SIC technique to cancel the signal corresponding to the weak users, before decoding their own symbol [18]. Imperfect SIC (I-SIC) will lead to the generation of residual interference at the near users. Recently, the performance of FDR-based cooperative NOMA system has been investigated extensively in the literature [1826]. In [18, 19], the authors have considered a cooperative NOMA system with half/full duplex relaying and evaluated the performance of the system in terms of outage probability, ergodic rate, and energy efficiency, under perfect SIC (P-SIC) conditions. The impact of relay selection strategy on the performance of cooperative NOMA system has been analyzed in [20], assuming that the relays can operate in either full-duplex (FD) or half-duplex (HD) mode. In [21], the authors have analyzed the outage probability performance of downlink cooperative NOMA system where the relay harvests energy from source, assuming P-SIC condition. In [22], the authors have analyzed the outage and ergodic rate performance of FDR-based NOMA-CDRT system in Rayleigh fading channels, assuming P-SIC. The outage and ergodic sum rate performance of FDR-based NOMA-CDRT system has been analyzed in [23], in the presence of Nakagamai fading channels under P-SIC condition. Further, to simplify the analysis, the RSI at the relay node has been modeled as a Gaussian random variable in [23]. The outage performance of FD-based NOMA-CDRT system has been analyzed in [24, 25] as well, assuming Nakagami fading channels under I-SIC; however, the evaluation of ergodic rates and ergodic sum rate have been ignored. In [26], the authors have considered a cooperative NOMA system, where a CCU is paired with multiple CEUs on a time sharing basis to improve the spectrum efficiency. The authors have established that the proposed system can achieve significant increase of ergodic sum rate as compared to existing bench mark schemes.

In this paper, we derive closed form expressions for the outage probabilities experienced by the users and the system outage probability of FDR-based NOMA-CDRT system assuming the links to experience Nakagami fading, under the realistic assumption of I-SIC. Further, we derive approximate closed form expressions for the ergodic rates achieved by the users in the system under I-SIC. A single-cell downlink NOMA-CDRT system is considered consisting of a BS, a cell-centric (i.e., near) user, a cell-edge (i.e., far) user, and a relay which operates in the FD mode and assists the BS to deliver information to the far user. The major contributions of this paper are outlined as follows: ∙ Closed-form expressions are derived for the outage probabilities experienced by the users in FDR-based NOMA-CDRT system, under the realistic assumption of I-SIC. An analytical expression for the system outage probability is also presented. Approximate closed-form expressions for the ergodic rates achieved by the users are derived, assuming I-SIC. Numerical results for the outage and ergodic rate performance of the users are presented. Further, numerical results for the system outage and ergodic sum rate performance of the considered FDR-based NOMA-CDRT network are also presented. Analytical results are corroborated by Monte Carlo-based extensive simulation studies. ∙ In the considered FDR/HDR-based NOMA-CDRT system, random selection of NOMA power allocation coefficient at the BS leads to poor outage performance for the near user compared to the far user. We present insights on selection of NOMA power allocation coefficient that provides equal outage performance for both the near and the far users. ∙ The performance of the users in FDR-based NOMA-CDRT system has been compared against that is perceived in a CDRT system which use conventional OMA technique based on time division multiple access (TDMA) scheme. ∙ We derive analytical expression for the OPA factor at the BS that minimizes the system outage probability of FDR-NOMA-CDRT system. We evaluate the percentage improvement in system outage under the OPA compared to random (i.e., non-optimal) power allocation (RPA) at the BS. With the help of numerical and simulation investigations, we establish that the system outage improves significantly under the OPA strategy.

Notice that the system model (i.e., the network and signal model) considered in our paper is similar to that considered in [2225]. In [22, 23], the authors have assumed P-SIC, while the works reported in [24, 25] have considered I-SIC condition. Further, the authors of [22] have assumed Rayleigh fading while Nakagami fading model was used in [23], where the RSI was approximated as a Gaussian random variable. Even though I-SIC condition was assumed in [24, 25], evaluation of ergodic rates of the users and the ergodic sum rate of the system under the realistic assumption of I-SIC have not appeared in these papers. Moreover, according to our best knowledge, the evaluation of the system outage probability of the considered FD-based NOMA-CDRT network and the investigation of OPA that minimizes the system outage probability have not appeared in the literature so far. The rest of the paper is organized as follows. The system model/experimental used in the paper are summarized in Section 2. Section 3 describes the system model while the derivation of outage probability and ergodic rates are presented in Section 4. Section 5 considers minimization of system outage probability and derives analytical expressions for the OPA to meet the desired objective. The numerical and simulation results are described in Section 6. Finally, the paper is concluded in Section 7.

Methods/experimental

In this work, we have considered a CDRT system where the BS delivers message to downlink users using NOMA technique. From a practical perspective, the considered system resembles the conventional relay-based cellular wireless communication scenario. To improve the spectral efficiency of the NOMA-CDRT system, the relay node is assumed to operate in FD mode. The reliability of downlink communication system is analyzed in terms of outage probabilities. Further, the achievable ergodic rates of downlink users are also analyzed. We have used realistic channel model, basics of digital communication principles, and probability theory to analyze the performance of the considered FDR-based NOMA-CDRT system. Furthermore, OPA factor at the BS for improving the system outage performance is investigated. Experimental investigations are carried out using Monte Carlo techniques to validate the analytical findings.

System model and preliminary details

The downlink cooperative FDR-based NOMA-CDRT system shown in Fig. 1 is considered where user 1 (U1) happens to be the near user and user 2 (U2) is the far user. The BS has direct communication link to U1 while it is assumed that, due to heavy shadow fading, the direct communication link between BS and U2 is absent. Accordingly, the BS employs a dedicated FD-based relay (R) to deliver the messages to U2. We consider that R can operate as a DF relay. Let {hij,i(s,r),j(r,1,2)} be the channel coefficients corresponding to the links between nodes i and j. We assume the links to experience independent non-identically distributed (i.n.i.d.) Nakagami fading with shape parameter mij and mean power \(\mathbb {E}[|h_{ij}|^{2}] = \pi _{ij}\). Accordingly, |hij|2 have Gamma distribution with shape parameter mij and scale parameter βij=πij/mij. The probability density function (PDF) of |hij| is given by [27]:

Fig. 1
figure1

A sample example of FDR-based NOMA-CDRT system

$$ f_{|h_{ij}|}(x) = \left(\frac{m_{ij}}{\pi_{ij}}\right)^{m_{ij}} \frac{(2x)^{m_{ij}-1}}{\Gamma(m_{ij})}e^{-\frac{m_{ij}}{\pi_{ij}}x^{2}} $$
(1)

where Γ(.) is the Gamma function. The CDF and PDF of |hij|2 are given by [27]:

$$ F_{|h_{ij}|^{2}}(x) = 1-e^{-\frac{x}{\beta_{ij}}} \sum_{j=0}^{m_{ij}-1} \frac{(\frac{x}{\beta_{ij}})^{j}}{j!} $$
(2a)
$$ f_{|h_{ij}|^{2}}(y) = (\beta_{ij})^{-m_{ij}}\frac{y^{m_{ij}-1}}{\Gamma({m_{ij}})}e^{-\frac{y}{\beta_{ij}}} $$
(2b)

Notice that (2a) assumes mij to take integer values only. We assume \(\pi _{ij} = (\frac {d_{ij}}{d_{0}})^{-n}\) where n is the path loss exponent; dij is the distance between nodes i and j; i(s,r),j(1,r,2); and d0 is the reference distance (in the far-field region of the transmitting node). Further, it is assumed that all the links experience frequency flat block fading. Furthermore, all the nodes in the network experience additive white Gaussian noise (AWGN) of equal variance σ2.

When R operates in the FD mode, it suffers from strong SI which is induced from its transmitter to the receiver side. It is assumed that all the nodes in the network shown in Fig. 1 (except R) use single antenna while R uses two directional antennas. This enables R to perform simultaneous transmission and reception in the same frequency band. Use of directional antennas at R reduces the effect of SI to a great extent [5, 6]. According to the recent literature on state-of-art methods for SI suppression/cancelation, multiple techniques have to be successively applied to the SI signal on top of one another to make the SI cancelation effective. Recently, many techniques have been reported for the mitigation of SI present in FDR systems such as (i) physical isolation, (ii) analog cancelation, and (iii) digital cancelation. Physical isolation techniques attempt to physically prevent the transmitted signal from reaching the receiver-end of the FD node by employing different approaches [2830] such as (i) placing shielding plates between the transmitter and receiver sections, (ii) using directional transmit antennas with nulls spatially projected at the receive antennas, and (iii) using orthogonally polarized transmit and receive antennas.

Even though these techniques significantly reduce the SI, additional mitigation is usually required due to the overwhelming strength of the interfering signal. Although originally believed to be impractical, FD wireless operation has been recently shown to be feasible through the use of novel techniques for SI isolation and cancelation [3136]. In spite of the advancements made on the design of SI cancelation methods, it has been well established that SI cannot be canceled completely, and thus, residual self-interference (RSI) would always be present at the FDR nodes. Experimental studies reported in [31, 37] have suggested that the RSI channel can be modeled as a fading channel. However, the probability distribution of the RSI fading channel differs according to the isolation/cancelation technique employed [31], where the PDF has been modeled as Rician with appropriate K (i.e., Rician factor) values. According to the results in [37], when a strong passive suppression is employed, the line-of-sight (LOS) component of SI is sufficiently suppressed and the PDF becomes Rayleigh. Research work reported in [38, 39] have used Nakagami fading model for the RSI channel. Since Nakagami fading model can represent both Rayleigh and Rician cases, we assume the RSI channel to undergo Nakagami fading. Let hrr be the channel coefficient corresponding to the RSI channel; we assume |hrr| to follow Nakagami-m fading with parameters mrr and mean RSI power = k2πrr, where k2 (0≤k2≤1) represents the extent of SI cancelation; k2=0 means RSI is absent in the system.

In the considered CDRT system, BS generates the NOMA signal by superposition coding and transmits x(t) as

$$ x(t) = \sqrt{P_{s} a_{1}}x_{1}(t)+\sqrt{P_{s} a_{2}}x_{2}(t) $$
(3)

where x1(t) and x2(t) are the information symbols for U1 and U2 respectively; a1 and a2 are the power allocation coefficients such that a1+a2=1,a1<a2; and Ps is the source power. Thus, the far user is allocated higher power as compared to the near user. According to the NOMA protocol employed, both R and U1 will receive the NOMA signal. Now R will try to recover the symbol x2 by treating signal corresponding to U1 as interference. Under DF relaying, R will forward a clean copy of the re-encoded symbol x2 to U2. Since it operates in the FD node, there will be RSI present at the receiver of R. Accordingly the received signal at R is

$$ {}y_{r}(t) \,=\, \sqrt{P_{s} a_{1}}h_{sr} x_{1}(t)\,+\,\sqrt{P_{s} a_{2}}h_{sr} x_{2}(t)\,+\,\sqrt{P_{r}}{h}_{rr} x_{2}(t\,-\,\tau)\,+\,n_{r}(t) $$
(4)

In (4), the third term represents the RSI present at the relay node, Pr is the transmit power of R, τ is the processing delay, and nr(t) is the AWGN component at R. The relay tries to decode x2 in the presence of signal corresponding to x1 and RSI. The corresponding SINR is given by:

$$ \Gamma_{r2} = \frac{|h_{sr}|^{2} \rho_{s} a_{2}}{|h_{sr}|^{2} \rho_{s} a_{1}+|{h}_{rr}|^{2} \rho_{r} +1} $$
(5)

where \(\rho _{s}=\frac {P_{s}}{\sigma ^{2}}\) and \(\rho _{r}=\frac {P_{r}}{\sigma ^{2}}\). The corresponding achievable rate is given by:

$$ C_{r2} = log_{2}(1+\Gamma_{r2}) $$
(6)

Once R forwards the re-encoded symbol x2, the received signal at U2 is given by:

$$ y_{2}(t) = h_{r2} x_{2}(t-\tau) +n_{2}(t) $$
(7)

where n2(t) is the AWGN component at U2. From the received signal, U2 tries to recover the symbol x2 and the corresponding SNR is

$$\begin{array}{*{20}l} \Gamma_{22} = \rho_{r} |h_{r2}|^{2} \end{array} $$
(8)

The achievable rate for R- U2 link is

$$ C_{22} = log_{2}(1+\Gamma_{22}) $$
(9)

Meanwhile, the received signal at U1 is given by

$$ \begin{aligned} y_{1}(t) \!&=\! \sqrt{P_{s} a_{1}}h_{s1} x_{1}(t)\,+\,\sqrt{P_{s} a_{2}}h_{s1} x_{2}(t)\!\\&+\!\sqrt{P_{r}}h_{r1} x_{2}(t-\tau)+n_{1}(t) \end{aligned} $$
(10)

Here, the third term represents the interference at U1 arising due to transmissions from R. According to the NOMA principle, U1 can decode the far user’s symbol x2; thus, x2(tτ) is known at U1 apriori. Thus, U1 can cancel the third term in (10) completely. However, we assume that perfect cancelation of the third term is not possible at U1; thus, (10) is written as follows:

$$ {\begin{aligned} y_{1}(t) \!&=\! \sqrt{P_{s} a_{1}}h_{s1} x_{1}(t)\,+\,\sqrt{P_{s} a_{2}}h_{s1} x_{2}(t)\\&+\!\sqrt{P_{r}} \hat{h}_{r1} x_{2}(t\,-\,\tau)\,+\,n_{1}(t) \end{aligned}} $$
(11)

In (11), \(|\hat {h}_{r1}|\) is assumed to have Nakgami PDF, and thus, \(|\hat {h}_{r1}|^{2}\) has Gamma PDF with mean k1πr1 where k1(0≤k1≤1) represents the level of residual interference created at U1 due to incomplete cancelation of interference from R. The SINR corresponding to the decoding of x2 at U1 is

$$ \Gamma_{12} = \frac{|h_{s1}|^{2} \rho_{s} a_{2}}{|h_{s1}|^{2} \rho_{s} a_{1} + |\hat{h}_{r1}|^{2} \rho_{r} +1} $$
(12)

The corresponding achievable rate is given by

$$ C_{12} = log_{2}(1+\Gamma_{12}) $$
(13)

After decoding x2 successfully, U1 will decode x1 by performing SIC. In this case, the decoded symbol x2 must be subtracted from y1(t) before the decoding of x1 is carried out. If x2 is decoded successfully, it can be completely subtracted from the composite received signal, i.e., SIC will be perfect. Otherwise, the decoding of x1 will be carried out in the presence of residual interference due to I-SIC. Thus, SINR corresponding to the decoding of x1 at U1 in the presence of I-SIC is given by

$$ \Gamma_{11} = \frac{|h_{s1}|^{2} \rho_{s} a_{1}}{|h_{s1}|^{2} \rho_{s} \beta a_{2} + |\hat{h}_{r1}|^{2} \rho_{r} +1} $$
(14)

where 0≤β<1; i.e., β=0 means P-SIC and 0<β≤1 implies I-SIC.

The achievable rate corresponding to the decoding of x1 at U1 is given by

$$ C_{11} = log_{2}(1+\Gamma_{11}) $$
(15)

Under DF relaying, the maximum achievable rate for U2 is given by

$$ C_{2} = log_{2}[1+min(\Gamma_{12},\Gamma_{r2},\Gamma_{22})] $$
(16)

Performance analysis

In this section, we present analytical models for finding the outage probabilities and ergodic rates of U1 and U2 in the considered FDR-based NOMA-CDRT system. We derive closed form expressions for the system outage probability as well, under imperfect SIC condition.

Outage probability analysis

Assume that R1 and R2 (expressed in bits per channel use, i.e., bpcu) are the target rates for the successful decoding of symbols x1 and x2, respectively, in the considered FDR-based NOMA-CDRT system. Let \(u_{1}^{FD} = 2^{R_{1}}-1\) and \(u_{2}^{FD} =2^{R_{2}}-1 \) be the corresponding SINR threshold values. If HDR technique is considered instead of FDR technique, the system requires two distinct time slots to complete the transmission of symbols x1 and x2. Thus, achievable rate for HD system is halved. For a fair comparison, we set the target rates for the equivalent HDR-based NOMA-CDRT system to be the same as that of the FDR-based system. Accordingly, the SINR threshold values are given by \(u_{1}^{HD} = 2^{2R_{1}}-1\) and \(u_{2}^{HD} = 2^{2R_{2}}-1\) for U1 and U2, respectively.

Outage probability experienced by U 1 in FDR-based NOMA-CDRT system

Notice that the near user U1 would not experience outage if both x1 and x2 are decoded successfully at U1. Thus, the outage probability of U1 is given by

$$ P_{out,1}^{FD} = 1-Pr\{\Gamma_{12} \geq u_{2}^{FD};\Gamma_{11} \geq u_{1}^{FD}\} $$
(17)

Proposition 1: Assuming \(u_{2}^{FD}<\frac {a_{2}}{a_{1}}\) and \(u_{1}^{FD} < \frac {a_{1}}{a_{2} \beta }, P_{out,1}^{FD}\) is given by the following equation:

$$ {\begin{aligned} P_{out,1}^{FD} =& 1-\left[e^{-\frac{1}{\phi \rho_{s} \beta_{s1}}}\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1}\frac{1}{j!}\left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}\right)^{j} \sum_{k=0}^{j} C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}\right.\\ &\left. (m_{r1}+k-1)!\times \left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}+\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k}\right] \end{aligned}} $$
(18)

where \(\phi = min\big (\frac {a_{2}-u_{2}^{FD} a_{1}}{u_{2}^{FD}}, \frac {a_{1}-\beta a_{2} u_{1}^{FD}}{u_{1}^{FD}}\big)\). Further, when either \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}\) or when \(u_{1}^{FD} \geq \frac {a_{1}}{\beta a_{2}}, P_{out,1}^{FD}\) becomes unity.

Proof: Refer Appendix Appendix A.

Outage probability experienced by U 2 in FDR-based NOMA-CDRT system

Notice that the far user U2 would not suffer from outage if x2 is decoded successfully at R and U2. Thus, the outage probability experienced by U2 is given by

$$ P_{out,2}^{FD} = 1-Pr\{\Gamma_{r2} \geq u_{2}^{FD},\Gamma_{22} \geq u_{1}^{FD}\}a $$
(19)

Proposition 2: Assuming that \(u_{2}^{FD}<\frac {a_{2}}{a_{1}}, P_{out,2}^{FD}\) is given by the following expression

$$ {\begin{aligned} P_{out,2}^{FD} =& 1-\left[e^{-\frac{1}{\psi \rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \sum_{j=0}^{m_{sr}-1}\right.\\ & \frac{1}{j!} \left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}\right)^{j} \sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} (m_{rr}+k-1)!\\ &\left. \left(\frac{\rho_{r} }{\psi \rho_{s} \beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)^{-m_{rr}-k} e^{-\left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)} \sum_{i=0}^{m_{r2}-1} \frac{1}{i!} \left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)^{i}\right] \end{aligned}} $$
(20)

where \(\psi = \frac {a_{2}-u_{2}^{FD} a_{1}}{u_{2}^{FD}}\) and \({~}^{j}C_{k} = \frac {j!}{k!(n-k)!}\). Further, when \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}, P_{out,2}^{FD}\) becomes unity.

Proof: Refer Appendix Appendix B.

System outage probability derivation

System outage probability is the probability of the event that either one user or both the users in the considered FDR -based NOMA-CDRT network suffer outage conditions. Thus, we determine the system outage probability as follows:

$$\begin{array}{*{20}l} {}P_{out,sys}^{FD} &\,=\, 1\,-\,P_{r}\!\left\{\!\Gamma_{12}\!\geq\! u_{2}^{FD},\Gamma_{11}\!\geq\! u_{1}^{FD},\Gamma_{r2}\!\geq\! u_{2}^{FD},\Gamma_{22}\!\geq\! u_{2}^{FD}\!\right\} \end{array} $$
(21a)

To find the closed form expression for the system outage probability, we substitute the expressions for Γ12,Γ11,Γr2, and Γ22 in (21a). Accordingly, \(P_{out,sys}^{FD}\) becomes:

$$\begin{array}{*{20}l} P_{out,sys}^{FD} &=1-P_{r} \left\{\frac{|h_{s1}|^{2} \rho_{s} a_{2}}{|h_{s1}|^{2} \rho_{s} a_{1} + |\hat{h}_{r1}|^{2} \rho_{r} +1}\right.\\ &\geq u_{2}^{FD}, \frac{|h_{s1}|^{2} \rho_{s} a_{1}}{|\hat{h}_{r1}|^{2} \rho_{r} +1} \geq u_{1}^{FD}, \\ & \qquad \frac{|h_{sr}|^{2} \rho_{s} a_{2}}{|h_{sr}|^{2} \rho_{s} a_{1}+|{h}_{rr}|^{2} \rho_{r} +1} \\ & \left.\geq u_{2}^{FD}, \rho_{r} |h_{r2}|^{2} \geq u_{2}^{FD} \right\} \end{array} $$
(21b)
$$\begin{array}{*{20}l} &= 1-P_{r} \left\{|h_{s1}|^{2} \rho_{s}\geq \frac{u_{2}^{FD}(|\hat{h}_{r1}|^{2} \rho_{r} +1)}{(a_{2} -u_{2}^{FD} a_{1})}; |h_{s1}|^{2}\right.\\ & \rho_{s}\!\geq\! \frac{u_{1}^{FD}(|\hat{h}_{r1}|^{2} \rho_{r} +1)}{(a_{1} -u_{1}^{FD} a_{1} \beta)}; \end{array} $$
(21c)
$$\begin{array}{*{20}l} & \left. |h_{sr}|^{2} \rho_{s} \geq \frac{u_{2}^{FD}(|h_{rr}|^{2} \rho_{r}+1)}{(a_{2} - u_{2}^{FD}a_{1})}; |h_{r2}|^{2}\rho_{r}\geq u_{2}^{FD} \right\} \\ &= 1-Pr\left\{|h_{s1}|^{2} \rho_{s}\geq \frac{1}{\phi}(|\hat{h}_{r1}|^{2} \rho_{r} +1); |h_{sr}|^{2} \rho_{s}\right. \\&\left. \geq \frac{(|h_{rr}|^{2} \rho_{r}+1)}{\psi}; |h_{r2}|^{2}\rho_{r}\geq u_{2}^{FD} \right\} \end{array} $$
(21d)

In (21d), the constants ϕ and ψ were defined in propositions 1 and 2, respectively. Now, the channel power gains, |hsr|2,|hs1|2, and |hr2|2 are independent since they correspond to distinct communication links in the network. Thus, \(P_{out,sys}^{FD}\) is given by

$$\begin{array}{*{20}l} P_{out,sys}^{FD} =& 1- \left[P_{r} \left\{|h_{s1}|^{2} \rho_{s}\geq \frac{1}{\phi}(|\hat{h}_{r1}|^{2} \rho_{r} +1)\right\}\right. \\ &\times P_{r} \left\{|h_{sr}|^{2} \rho_{s} \geq \frac{(|h_{rr}|^{2} \rho_{r}+1)}{\psi}\right\} \\& \left. \times P_{r} \left\{|h_{r2}|^{2}\rho_{r}\geq u_{2}^{FD}\right\}\right] \\ \end{array} $$
(21e)

Proposition 3: Assuming that \(u_{2}^{FD} < \frac {a_{2}}{a_{1}}\) and \(u_{1}^{FD} <\frac {a_{1}}{\beta a_{2}}\), the closed from expression for \(P_{out,sys}^{FD}\) is given as follows:

$$\begin{array}{*{20}l} P_{out,sys}^{FD} &= 1-\left[e^{-\frac{1}{\phi \rho_{s} \beta_{s1}}}\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})}\right. \\ &\sum_{j=0}^{m_{s1}-1}\frac{1}{j!}\left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}\right)^{j} \sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}\\ & (m_{r1}+k-1)!\times \left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}+\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k}\\ & \times e^{-\frac{1}{\psi \rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \end{array} $$
$$\begin{array}{*{20}l} & \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}\right)^{l} \sum_{q=0}^{l} {~}^{l}C_{q} \left(\frac{1}{\rho_{r}}\right)^{l-q} \!\!\!\!\!(m_{rr}\,+\,q\,-\,1)! \\ & \left(\frac{\rho_{r} }{\psi \rho_{s} \beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)^{-m_{rr}-q} e^{-\left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)}\\ & \left. \sum_{i=0}^{m_{r2}-1} \frac{1}{i!} \left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)^{i} \right] \end{array} $$
(21f)

When either \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}\) or \(u_{1}^{FD} \geq \frac {a_{1}}{\beta a_{2}}, P_{out,sys}^{FD}\) becomes unity.

Proof: Appendix Appendix C.

Ergodic rate analysis

In this section, we analyze the ergodic rates achieved by U1 and U2 in the presence of I-SIC condition.

Ergodic rate from BS to U 1 in FDR-based NOMA-CDRT system

The ergodic rate achieved by the near user U1 (\(E[_{R_{1}}^{FD}] \)) is determined as follows:

$$\begin{array}{*{20}l} E[_{R_{1}}^{FD}] &= \mathbb{E}[log_{2}(1+\Gamma_{11})]\\ &= \int_{0}^{\infty} log_{2}(1+x) f_{\Gamma_{11}}(x) dx \\ &=\frac{1}{ln2} \int_{0}^{\infty} \frac{1-F_{\Gamma_{11}}(x)}{1+x} dx \end{array} $$
(22a)

where \(F_{\Gamma _{11}}(x)\) and \(F_{\Gamma _{11}}(x)\) are the CDF and PDF of Γ11, respectively.Proposition 4: An approximate closed form expression for \(E[R_{1}^{FD}]\) is given as follows.

$$\begin{array}{*{20}l} {}E[_{R_{1}}^{FD}] & \!\cong\! \frac{1}{ln2}\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!}\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}\\&(m_{r1}+k-1)! \times \frac{\pi}{N} \sum_{n=0}^{N} \frac{ a_{n} b_{n}}{d_{n}} e^{-c_{n}} \end{array} $$
(22b)

where \(a_{n} = \left (\frac {a_{1} \rho _{r} (1+\phi _{n})}{\beta a_{2} \rho _{s} a_{1} (1-\phi _{n})}\right)^{j} \sqrt {1-\phi _{n}^{2}}, b_{n} = \left (\frac {a_{1} \rho _{r} (1+\phi _{n})}{\beta a_{2} \rho _{s} a_{1} (1-\phi _{n})} + \frac {1}{k_{1}\beta _{r1}}\right)^{m_{r1}-k}, c_{n} = \frac {a_{1} (1+\phi _)}{\beta a_{2} \rho _{s} a_{1} (1-\phi _{n})}, d_{n} = \frac {2\beta a_{2} + a_{1}(1+\phi _{n})}{2 \beta a_{2}}\) and \(\phi _{n} = cos\left (\frac {(2n-1)\pi }{2N}\right)\). Notice that (22b) is obtained by using the Gaussian-Chebyshev quadrature formula, which is described in Appendix Appendix D. Here, Nis the complexity accuracy trade-off parameter in this approximation.

Proof: Refer Appendix Appendix D.

Ergodic rate from BS to U 2 in FDR-based NOMA-CDRT system

The ergodic rate achieved by the far user U2 (\(E[R_{2}^{FD}]\)) is determined as follows:

$$\begin{array}{*{20}l} E[R_{2}^{FD}] &= \mathbb{E}[log_{2}(1+min\{\Gamma_{12},\Gamma_{r2},\Gamma_{22}\})]\\ &= \mathbb{E}[log_{2}(1+Y)]\\& =\frac{1}{ln2} \int_{0}^{\infty} \frac{1-F_{Y}(y)}{1+y} dy \end{array} $$
(23a)

where y=min{Γ12,Γr2,Γ22} and FY(y) is the CDF of Y.Proposition 5: An approximate closed form expression for \(E[R_{2}^{FD}]\) is given as follows.

$$\begin{array}{*{20}l} E[_{R_{2}}^{FD}] &\cong \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} \\&\times(m_{r1}+k-1)! \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})}\\& \times \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \sum_{p=0}^{l} {~}^{l}C_{p} \left(\frac{1}{\rho_{r}}\right)^{l-p} (m_{rr}+p-1)! \\&\times \sum_{q=0}^{m_{r2}-1} \frac{1}{q!} \times \frac{\pi}{N ln2} \\& \times \sum_{n=0}^{N}\frac{e_{n} f_{n} g_{n} h_{n} r_{n}}{w_{n}} e^{-s_{n}}e^{-t_{n}}e^{-v_{n}} \end{array} $$
(23b)

where \(e_{n} = \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1-\phi _{n}) \beta _{s1}} + \frac {1}{\beta _{r1} k_{1}}\right)^{-m_{r1}-k} \sqrt {1-\phi _{n}^{2}}, f_{n} = \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1-\phi _{n}) \beta _{s1}} \right)^{j}, g_{n} = \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1-\phi _{n}) \beta _{sr}} \right)^{l}, h_{n} \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1-\phi _{n}) \beta _{s1}} + \frac {1}{\beta _{rr} k_{2}}\right)^{-m_{rr}-p}, r_{n} = \left (\frac {a_{2} (1+\phi _{n})}{2 a_{1} \rho _{r} \beta _{r2}}\right)^{q}, s_{n} = \frac {2 a_{1}}{2a_{1} + a_{2}(1+\phi _{n})}, t_{n} = \frac {(1+\phi _{n})}{a_{1} \rho _{s} (1-\phi _{n}) \beta _{s1}}, v_{n} = \frac {(1+\phi _{n})}{a_{1} \rho _{s} (1-\phi _{n}) \beta _{sr}}, w_{n} = \frac {a_{2} (1+\phi _{n})}{2 a_{1} \rho _{r} \beta _{r2}}, \phi _{n} = cos\left (\frac {(2n-1)\pi }{2N}\right)\) and N is the complexity accuracy trade-off parameter, relating to the Gaussian-Chebyshev quadrature method.

Proof: Appendix Appendix E.

Optimal power allocation (OPA) for minimizing system outage probability

In this section, our aim is to find the OPA factor at the BS, i.e., a1,opt that minimizes the system outage probability in FDR-based NOMA-CDRT system. The outage minimization problem can be formulated as

$$\begin{array}{*{20}l} &\underset{a_{1}}{min} \qquad P_{out,sys}^{FD} \\ & s.t. \qquad a_{1}+a_{2} = 1 \end{array} $$
(24)

Proposition 5: For the considered FDR-based NOMA-CDRT system, the OPA coefficient a1,opt that minimizes the system outage probability is given by

$$ a_{1,opt} = \frac{u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}} $$
(25)

Proof:

Recall the expression for the system outage probability given in section 4, i.e.,

$$ P_{out,sys}^{FD} = 1-(C_{0} \times A_{0} \times B_{0}) $$
(26)

Here, A0 and B0 are given by (42) and (43), respectively, and \(C_{0} = 1-P_{out,1}^{FD}\), where \(P_{out,1}^{FD}\) is given by (18). These are reproduced below:

$$\begin{array}{*{20}l} {}A_{0} =& e^{-\frac{1}{\psi \rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}\right)^{l}\\ &\quad \sum_{q=0}^{l} {~}^{l}C_{q} \left(\!\frac{1}{\rho_{r}}\!\right)^{l-q} (m_{rr}+q-1)! \\& \times \left(\frac{\rho_{r} }{\psi \rho_{s} \beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)^{-m_{rr}-q} \end{array} $$
(27)
$$\begin{array}{*{20}l} B_{0} &= Pr\{|h_{r2}|^{2}\rho_{r} \geq u_{2}^{FD}\}= P_{r}\left\{|h_{r2}|^{2} \geq \frac{u_{2}^{FD}}{\rho_{r}}\right\} \\& = e^{-\left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)} \sum_{i=0}^{m_{r2}-1} \frac{1}{i!} \left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)^{i} \end{array} $$
(28)
$$\begin{array}{*{20}l} C_{0} & \,=\, e^{-\frac{1}{\phi \rho_{s} \beta_{s1}}}\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1}\frac{1}{j!}\left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}\right)^{j} \\ &\quad\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}\\& (m_{r1}+k-1)!\times \left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}+\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k} \end{array} $$
(29)

It is assumed that SIC is perfect, i.e., β=0 in this section. Notice that in the expression for A0,ψ must be grater than 0 which implies \(u_{2}^{FD}<\frac {a_{2}}{a_{1}}\). Since a2=1−a1, this implies that a1 must satisfy the condition \(0 < a_{1} < \frac {1}{1+u_{2}^{FD}}\). Now, consider \(C_{0} = 1-P_{out,1}^{FD}\) where \(\phi = min\left (\frac {a_{2} - u_{2}^{FD}a_{1}}{u_{2}^{FD}},\frac {a_{1}}{u_{1}^{FD}}\right)\), which will give rise to two distinct cases as given below:

Case(I): \(\frac {a_{2} - u_{2}^{FD}a_{1}}{u_{2}^{FD}} < \frac {a_{1}}{u_{1}^{FD}} \)

Since a2=1−a1, the above implies that a1 must satisfy the following condition: \(\frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}} < a_{1} < \frac {1}{1+u_{2}^{FD}}\). In this case, the system outage can be written as follows:

$$ P_{out,sys}^{FD} = 1-(A_{0}(a_{1}) \times B_{0} \times C_{0}(a_{1})) $$
(30)

where A0(a1) and C0(a1) are given by

$$\begin{array}{*{20}l} A_{0}(a_{1}) =& e^{-\frac{u_{2}^{FD}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma({m_{rr})}} \sum_{l=0}^{m_{sr}-1}\\&\quad \frac{1}{l!} \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}}\right)^{l} \\&\quad \times \sum_{q = 0}^{l} {~}^{l}C_{q} \left(\frac{1}{\rho_{r}}\right)^{l-q} (m_{rr}+q-1)! \\&\quad\left(\frac{u_{2}^{FD}\rho_{r}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}} +\frac{1}{k_{2} \beta_{rr}} \right)^{-m_{rr}-q} \end{array} $$
(31a)
$$\begin{array}{*{20}l} C_{0}(a_{1}) =& e^{-\frac{u_{2}^{FD}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}} \sum_{j=0}^{m_{s1}-1}\\& \frac{1}{j!} \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}}\right)^{j} \\& \times \sum_{k = 0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} (m_{r1}+k-1)!\\& \left(\frac{u_{2}^{FD}\rho_{r}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}} +\frac{1}{k_{1} \beta_{r1}} \right)^{-m_{r1}-k} \end{array} $$
(31b)

Notice that (31a) and (31b) are obtained by substituting \(\psi =\phi =\frac {1-a_{1}-u_{2}^{FD}a_{1}}{u_{2}^{FD}}\) in (42) and (18), respectively. The derivative of \(P_{out,sys}^{FD}(a_{1})\) with respect to a1 can be written as

$$ [P_{out,sys}^{FD}(a_{1})]' = -B_{0}[A_{0}(a_{1})C'_{0}(a_{1}) +A'_{0}(a_{1})C_{0}(a_{1})] $$
(32)

where the first derivatives \(A^{\prime }_{0}(a_{1})\) and \(C^{\prime }_{0}(a_{1})\) can be determined by differentiating (31a) and (31b), respectively, with respect to a1. Thus, we write \(A^{\prime }_{0}(a_{1}) \triangleq x(a_{1})+y(a_{1})+z(a_{1})\) where x(a1),y(a1) and z(a1) are given as follows:

$$\begin{array}{*{20}l} x(a_{1}) =& \frac{-\frac{u_{2}^{FD}}{\rho_{s} \beta_{sr}}(u_{2}^{FD}+1)e^{-\frac{u_{2}^{FD}}{(1-a_{1}-a_{1}u_{2}^{FD})\rho_{s} \beta_{sr}}}}{(1-a_{1}-a_{1}u_{2}^{FD})^{2}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma({m_{rr})}}\\& \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}}\right)^{l} \end{array} $$
$$\begin{array}{*{20}l} & \times \sum_{q = 0}^{l} {~}^{l}C_{q} \left(\frac{1}{\rho_{r}}\right)^{l-q} (m_{rr}+q-1)!\\& \left(\frac{u_{2}^{FD}\rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}} +\frac{1}{k_{2} \beta_{rr}} \right)^{-m_{rr}-q} \end{array} $$
(33a)
$$\begin{array}{*{20}l} {}y(a_{1}) =& e^{-\frac{u_{2}^{FD}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma({m_{rr})}}\\& \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \frac{-\frac{u_{2}^{FD} \rho_{r}}{\rho_{s} \beta_{s1}}l (-u_{2}^{FD}-1) \left(\frac{u_{2}^{FD} \rho_{r}}{\rho_{s} \beta_{s1} (1-a_{1}-a_{1}u_{2}^{FD})}\right)^{l-1}}{(1-a_{1}-a_{1}u_{2}^{FD})^{2}} \\& \times \sum_{q = 0}^{l} {~}^{l}C_{q} \left(\frac{1}{\rho_{r}}\right)^{l-q} (m_{rr}+q-1)!\\& \left(\frac{u_{2}^{FD}\rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}} +\frac{1}{k_{2} \beta_{rr}} \right)^{-m_{rr}-q} \end{array} $$
(33b)
$$\begin{array}{*{20}l} z(a_{1}) =& e^{-\frac{u_{2}^{FD}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma({m_{rr})}}\\& \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{sr}}\right)^{l} \\& \times \sum_{q = 0}^{l} {~}^{l}C_{q} \left(\frac{1}{\rho_{r}}\right)^{l-q} (m_{rr}+q-1)!\\& \frac{\frac{u_{2} \rho_{r}}{\rho_{s} \beta_{s1}}(m_{rr}+q)(-u_{2}^{FD}-1)}{(1-a_{1}-a_{1}u_{2}^{FD})^{2}} \\& \times \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}-a_{1}u_{2}^{FD})\rho_{s} \beta_{sr}}+\frac{1}{k_{2} \beta_{rr}}\right)^{-m_{rr}-q-1} \end{array} $$
(33c)

In a similar way, we write \(C^{\prime }_{0}(a_{1}) = f(a_{1})+g(a_{1})+h(a_{1})\) where f(a1),g(a1) and h(a1) are given as follows:

$$\begin{array}{*{20}l} f(a_{1}) =& \frac{-\frac{u_{2}^{FD}}{\rho_{s} \beta_{s1}}(u_{2}^{FD}+1)e^{-\frac{u_{2}^{FD}}{(1-a_{1}-a_{1}u_{2}^{FD})\rho_{s} \beta_{s1}}}}{(1-a_{1}-a_{1}u_{2}^{FD})^{2}} \frac{(k_{2} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}}\\&\times \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}}\right)^{j} \\ &\times \sum_{k = 0}^{j} {~}^{j}C_{k} \big(\frac{1}{\rho_{r}}\big)^{j-k} (m_{r1}+k-1)! \end{array} $$
$$\begin{array}{*{20}l} &\times \left(\frac{u_{2}^{FD}\rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}} +\frac{1}{k_{2} \beta_{r1}} \right)^{-m_{r1}-k} \end{array} $$
(34a)
$$\begin{array}{*{20}l} {}g(a_{1}) =&\, e^{-\frac{u_{2}^{FD}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}}} \frac{(k_{2} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}}\\[-2pt]\times& \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \frac{-\frac{u_{2}^{FD} \rho_{r}}{\rho_{s} \beta_{s1}}j (-u_{2}^{FD}-1) \left(\frac{u_{2}^{FD} \rho_{r}}{\rho_{s} \beta_{s1} (1-a_{1}-a_{1}u_{2}^{FD})}\right)^{j-1}}{(1-a_{1}-a_{1}u_{2}^{FD})^{2}} \\[-2pt]\times& \sum_{k = 0}^{j} {~}^{j}C_{k} \big(\frac{1}{\rho_{r}}\big)^{j-k} (m_{r1}+k-1)!\\[-2pt]\times& \left(\frac{u_{2}^{FD}\rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}} +\frac{1}{k_{2} \beta_{r1}} \right)^{-m_{r1}-k} \end{array} $$
(34b)
$$\begin{array}{*{20}l} {}h(a_{1}) =& e^{-\frac{u_{2}^{FD}}{(1-a_{1}-a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}}} \frac{(k_{2} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}}\\\times& \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}- a_{1} u_{2}^{FD})\rho_{s} \beta_{s1}}\right)^{j} \\\times& \sum_{k = 0}^{j} {~}^{j}C_{k} \big(\frac{1}{\rho_{r}}\big)^{j-k} (m_{r1}+k-1)!\\\times& \frac{\frac{u_{2} \rho_{r}}{\rho_{s} \beta_{s1}}(m_{r1}+k)(-u_{2}^{FD}-1)}{(1-a_{1}-a_{1}u_{2}^{FD})^{2}} \\\times& \left(\frac{u_{2}^{FD} \rho_{r}}{(1-a_{1}-a_{1}u_{2}^{FD})\rho_{s} \beta_{s1}}+\frac{1}{k_{2} \beta_{r1}}\right)^{-m_{r1}-k-1} \end{array} $$
(34c)

Through numerical investigations, we observe that \([P_{out,sys}^{FD}]'>0\), for the range of a1 considered. Thus, we conclude that \(P_{out,sys}^{FD}\) is a monotonically increasing function of a1 for \(\frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}} < a_{1} < \frac {1}{1+u_{2}^{FD}}\).

Case II: \(\frac {a_{2}-u_{2}^{FD}a_{1}}{u_{2}^{FD}}>\frac {a_{1}}{u_{1}^{FD}}\)

Since a2=1−a1, the above condition implies that a1 must satisfy \(0< a_{1}<\frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}}\). In this case, the system outage probability can be written as \(P_{out,sys}^{FD}(a_{1})= 1-[A_{0}(a_{1}) \times B_{0} \times C_{1}(a_{1})]\) where A0(a1) is given by (31a) and C1(a1) is obtained by substituting \(\phi = \frac {a_{1}}{u_{1}^{FD}}\) in the expression for \(C_{0} = 1-P_{out,1}^{FD}\), with \(P_{out,1}^{FD}\) given by (18). Thus, C1(a1) is given by

$$\begin{array}{*{20}l} C_{1}(a_{1}) =& e^{-\frac{u_{1}^{FD}}{a_{1}\rho_{s} \beta_{s1}}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \left(\frac{u_{1}^{FD} \rho_{r}}{a_{1}\rho_{s} \beta_{s1}}\right)^{j} \end{array} $$
$$\begin{array}{*{20}l} &\times \sum_{k = 0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} (m_{r1}+k-1)! \\& \times \left(\frac{u_{1}^{FD}\rho_{r}}{a_{1}\rho_{s} \beta_{s1}} +\frac{1}{k_{1} \beta_{r1}} \right)^{-m_{r1}-k} \end{array} $$
(35)

The first-order derivative of \(P_{out,sys}^{FD}\) is computed as \([P_{out,sys}^{FD}(a_{1})]' = -B_{0}[A_{0}(a_{1}) C'_{1}(a_{1}) + A'_{0}(a_{1}) C_{1}(a_{1}) ]\). Notice that \(A^{\prime }_{0}(a_{1})\), which is the first order derivative of A0(a1), can be determined by combining (33a)-(33c) as in the previous case I. The first derivative of C1(a1), i.e., C1′(a1) is given by \(C^{\prime }_{1}(a_{1}) = u(a_{1})+v(a_{1})+w(a_{1})\) where u(a1),v(a), and w(a1) are given by

$$\begin{array}{*{20}l} u(a_{1}) =& \frac{-\frac{u_{1}^{FD}}{\rho_{s} \beta_{s1}} e^{-\frac{u_{1}^{FD}}{a_{1}\rho_{s} \beta_{s1}}} }{a_{1}^{2}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}} \sum_{j=0}^{m_{sr}-1}\\&\times \frac{1}{j!} \left(\frac{u_{1}^{FD} \rho_{r}}{a_{1}\rho_{s} \beta_{s1}}\right)^{j} \sum_{k = 0}^{j} {~}^{j}C_{k} \big(\frac{1}{\rho_{r}}\big)^{j-k} \\& \times (m_{r1}+k-1)! \left(\frac{u_{1}^{FD}\rho_{r}}{a_{1}\rho_{s} \beta_{s1}} +\frac{1}{k_{1} \beta_{r1}} \right)^{-m_{r1}-k} \end{array} $$
(36a)
$$\begin{array}{*{20}l} v(a_{1}) =& e^{-\frac{u_{1}^{FD}}{a_{1}\rho_{s} \beta_{s1}}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \frac{-j\left(\frac{u_{1}^{FD} \rho_{r}}{a_{1}\rho_{s} \beta_{s1}}\right)^{j}}{a_{1}}\\&\times \sum_{k = 0}^{j} {~}^{j}C_{k} \big(\frac{1}{\rho_{r}}\big)^{j-k} (m_{r1}+k-1)! \\& \times \left(\frac{u_{1}^{FD}\rho_{r}}{a_{1}\rho_{s} \beta_{s1}} +\frac{1}{k_{1} \beta_{r1}} \right)^{-m_{r1}-k} \end{array} $$
(36b)
$$\begin{array}{*{20}l} w(a_{1}) =& e^{-\frac{u_{1}^{FD}}{a_{1}\rho_{s} \beta_{s1}}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma({m_{r1})}} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \left(\frac{u_{1}^{FD} \rho_{r}}{a_{1}\rho_{s} \beta_{s1}}\right)^{j}\\&\times \sum_{k = 0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} (m_{r1}+k-1)! \\&\times \frac{\frac{u_{1}^{FD}\rho_{r}}{\rho_{s} \beta_{s}1}(m_{r1}+k) \left(\frac{u_{1}^{FD}\rho_{r}}{a_{1}\rho_{s} \beta_{s1}} +\frac{1}{k_{1} \beta_{r1}} \right)^{-m_{r1}-k-1}}{a_{1}^{2}} \end{array} $$
(36c)

Through numerical investigations, we find that the first derivative \([P_{out,sys}^{FD}]'<0\) for the range of a1 considered. Hence, we conclude that \(P_{out,sys}^{FD}\) is a decreasing function of a1 if \(0< a_{1} < \frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}}\).

Thus, we observe that \(P_{out,sys}^{FD}\) is a monotonically decreasing function of a1 for \(0< a_{1}<\frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}}\) and monotonically increasing function of a1 for \(\frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}} < a_{1} < \frac {1}{1+u_{2}^{FD}}\). Thus, the optimal value of the power allocation coefficient a1 that minimizes the system outage is obtained as \(a_{1,opt} = \frac {u_{1}^{FD}}{u_{1}^{FD}+u_{2}^{FD}+u_{1}^{FD}u_{2}^{FD}}\). This completes the proof.

Performance evaluation results and discussion

This section describes the results for the outage and the ergodic rate performance of the two users in the considered CDRT system. The analytical results are validated by performing extensive Monte Carlo simulations considering a set of 105 channel realizations. We select ρs=ρr=ρ,mij=m, where ρ is directly related to the transmit power. Unless otherwise specified, the following parameters are chosen for the evaluation: R1=1 bpcu, R2=1 bpcu, m=2, and n=3. We consider a two dimensional topology for the network under consideration with (xi,yi) representing the coordinates of a given node i. Assume that BS is placed at (0,0) and BS, R, and U2 are placed on a straight line. Let the coordinates of R and U2 be (1.25,0) and (1.75,0), respectively. Further, we choose the coordinates of U1 as (0.625,0.5) so that the distances between these nodes are ds1=0.8d0,dsr=1.25d0,dr1=0.8d0,dr2=0.5d0. Here, d0 is the reference distance and is selected as 1 km. For numerical illustrations, we set k1=10−2,k2=0.64×10−2; however, the results can be modified for any given values of k1 and k2 (0≤k1,k2≤1). Further, we choose \(\pi _{ij} = E[|h_{ij}|^{2}] = \big (\frac {d_{ij}}{d_{0}}\big)^{-n}, i \in (s,r), j \in (r,1,2)\) with πrr set as equal to – 3 dB. For comparison purpose, we consider HDR-based NOMA-CDRT system as well, where the communication is completed in two time slots. In the first time slot, BS transmits the NOMA signal consisting of symbols x1 and x2. U1 decodes x1 by implementing SIC, while R decodes x2 and forwards the symbol in the second time slot. Finally, x2 is decoded successfully at U2. Here, we assume that U1 would not receive interference from R’s transmission since it happens in the second time slot during which U1 is silent. We modify the relevant equations of Section 4 to find the outage of U1 and U2 in HDR-NOMA-CDRT system. For a fair comparison among FDR and HDR systems, we consider the target rate of HDR system to be the same as that of FDR system. Since HDR system requires additional time slots for completing the transmission of symbols, the achievable rate is reduced as compared to FDR system. For the outage calculation, since the target rate for both HDR-CDRT and FDR-CDRT systems are assumed to be equal, it leads to higher SINR threshold requirement for the users in HDR-NOMA-CDRT system, as compared to the equivalent FDR system.

In Fig. 2, the outage probabilities experienced by U1 and U2 are drawn against ρ for FDR/HDR-based NOMA-CDRT system. Results show that, as ρ increases, the outage performance of both the users is improved. Further, the results in Fig. 2 show that U1 suffers higher outage probability than U2 for a given set of parameters. This happens because the power allocation factor at BS, i.e., a1 has been chosen arbitrarily. In addition, the decoding of symbol x1 at U1 requires a two-step procedure: successful decoding of symbol x2 by treating signal corresponding to x1 as interference, which is followed by decoding of x1 by applying SIC to cancel the known x2. In this process, U1 is affected by interference due to transmissions from R as well. However, decoding of x2 happens at U2 in the absence of interference from any source. Moreover, it is assumed that the relay forwards the re-encoded version of x2 with full power which increases the probability of successful decoding of x2 at U2. As ρ is varied, the outage performance of U1 and U2 shows distinct behavior in FDR/HDR-NOMA-CDRT systems. In the low transmit power region, the outage probabilities experienced by both U1 and U2 in FDR-NOMA-CDRT system is lower as compared to the outage experienced in HDR-NOMA-CDRT system. This happens due to the higher threshold SINR requirement for HDR system as mentioned before. However, in the high transmit power region, the mean RSI power at R (k2πrr) becomes higher in FDR-based system, triggering degradation of SINR at R. This increases the outage of U2 in FDR-NOMA-CDRT system in the high transmit power region, as compared to HDR system as can be seen in Fig. 2. At the same time, the outage performance of U1 is not affected by the RSI at R in FDR-NOMA-CDRT system, as the decoding of x2 or x1 at U1 is not affected by the SINR over BS-R link. However, the decoding at U1 is affected by the residual interference (k1πr1) created due to transmission over the R- U2 link in FDR-NOMA-CDRT system, which becomes significantly higher when transmit power is increased. Thus, the outage experienced by U1 in FDR-NOMA-CDRT system becomes significantly higher in the high transmit power region. Notice that this residual interference is absent in HDR system, and thus, U1 exhibits much improved performance in HDR-NOMA-CDRT system when transmit power is increased.

Fig. 2
figure2

Outage probability versus ρ (a1=0.05,β=0)

Figure 3 shows the impact of I-SIC factor β on the outage probability performance of U1 and U2 in FDR-NOMA-CDRT system. As β increases, \(P_{out,1}^{FD}\) increases owing to the higher amount of interference generated by I-SIC at U1. However, \(P_{out,2}^{FD}\) is not influenced by β, since U2 (being the far-user) does not implement SIC for decoding symbol x2. As β increases from 0.3 to 0.4, \(P_{out,1}^{FD}\) increases by 89% for ρ=30 dB.

Fig. 3
figure3

Outage probability versus ρ for distinct values of β: FDR-NOMA (a1=0.05)

Figure 4 shows the outage probability of U1 in FDR-NOMA-CDRT system against the mean residual interference (k1πr1) present at U1 (which is generated by the inaccurate cancelation of symbol x2 at U1). The results show that the mean residual interference has significant impact on the outage of U1. The results in Fig. 5 show that the outage probability of U2 increases and becomes significantly very high when the mean RSI power at the relay node is increased. Increase of mean RSI degrades the SINR at R, which affects the successful decoding of x2 at R. This degrades the outage performance of U2, while HD relaying does not induce RSI, and thus, the outage probability of U2 does not depend on mean RSI power in HDR-NOMA-CDRT system.

Fig. 4
figure4

Outage probability of U1 versus mean residual interference power, k1πr1 (a1=0.05,β=0)

Fig. 5
figure5

Outage probability of U2 versus mean RSI power, k2πrr (a1=0.05,β=0)

Equal outage for U 1 and U 2

Figure 6 shows the outage probability of U1 and U2 drawn against the NOMA power allocation coefficient a1. The results are shown for both HDR as well as FDR-based systems. Here, we keep ds1=0.8, dsr=1.25, dr1=0.8, and dr2 is varied along the straight line joining BS, R, and U2. Thus, U2 moves away from BS while the position of U1 is fixed. As dr2 increases, Pout of U2 becomes higher, whereas dr2 does not influence Pout of U1. As a1 increases, more power gets allocated to U1; thus, Pout of U1 decreases while that of U2 increases. The outages become very high and moves towards unity when \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}\) for FDR-based system. Further, we can see that the outage of U2 in FDR-based system becomes higher when RSI is increased while that of U1 does not depend on RSI as we have seen in Fig. 2 as well. From Fig. 6, it is clear that, for certain values of a1 (i.e., \( a_{1}^{*} \)), Pout of both the users can be made equal, irrespective of the location of U2. Table 1 depicts the details of the NOMA power allocation factor \( a_{1}^{*} \) that makes Pout,1=Pout,2 for FDR/HDR-NOMA-CDRT system as a function of dr2. As dr2 is increased, \( a_{1}^{*} \) reduces, owing to the fact that more power need to be allocated for U2 when it moves away from BS, so as to satisfy the equal outage criterion. Furthermore, it can be seen that the \(a_{1}^{*}\) that ensures equal outage for both the users is a function of mean RSI power in FDR-based system. As the mean RSI power increases, the SINR on the BS-R link reduces which increases the outage probability of U2. Moreover, results given in Table 2 implies that as the I-SIC factor β is increased, \(a_{1}^{*}\) has to be increased to satisfy the equal outage criterion. This is because, an increase of β will make \(P_{out,1}^{FD}\) higher; consequently, \(a_{1}^{*}\) shall be increased to meet the equal outage criterion. Thus, to ensure equal outage, \(a_{1}^{*}\) has to be reduced (i.e., \(a_{2}^{*}\) must be increased) so that more power gets allocated to U2’s symbol at BS when the mean RSI power is increased. Since RSI is absent in HDR system, \(a_{1}^{*}\) is independent of mean RSI power. Figure 7 plots Pout against ρ by choosing power allocation factor \( a_{1}^{*} \), according to the results given in Table 1 (\(a_{1}^{*}\) is calculated for each value of ρ). In this figure, results are plotted for two distinct values of dr2, i.e., dr2=0.5, and 1.5. Further, results are shown for HDR as well as FDR systems. The results are plotted by finding \(a_{1}^{*}\) separately for each case and for each value of ρ considered. The results show that proper selection of \( a_{1}^{*} \) can make the outage probabilities of both the users equal, over the entire range of transmit power values considered.

Fig. 6
figure6

Outage probabilities of U1 and U2 versus a1 (ρ=20dB, β=0)

Fig. 7
figure7

Outage probabilities of U1 and U2 versus ρ (\(a_{1} = a_{1}^{*}, \beta =0\))

Table 1 NOMA coefficient \(a_{1}^{*}\) that achieves Pout,1=Pout,2: impact of RSI
Table 2 NOMA coefficient \(a_{1}^{*}\) that achieves Pout,1=Pout,2: impact of I-SIC factor β

System outage probability evaluation

Figure 8 shows the system outage probability as a function of ρ. In the low transmit power region, the system outage of FDR-NOMA-CDRT network is lower as compared to the HDR counterpart owing to the higher threshold SINR requirement for the HDR system. However, in the high transmit power region, HDR system outperforms FDR system, owing to the higher amount of mean RSI power present in FDR system. Figure 9 shows the effect of β on system outage probability. It is evident that as the value of β increases, the system outage probability also increases. This happens owing to the fact that increases of β introduce residual interference at U1 due to I-SIC. Thus, U1 experience higher outage probability that increases system outage as well. Figure 10 shows the impact of RSI on the system outage of FDR-NOMA-CDRT system while Fig. 11 shows the effect of R- U2 distance dr2. The results confirm that the system outage increases with increase of RSI power owing to the fact that, as RSI power is increased, the SINR over BS-R link degrades so that the outage probability suffered by U2 increases. Further, increase of dr2 increases the outage experienced by U2 triggering the system outage to become higher. As can be seen in Figs. 10 and 11, the system outage increases either when a1 is reduced or when a1 is increased.

Fig. 8
figure8

System outage probability versus ρ: FDR-NOMA (a1=0.05,β=0)

Fig. 9
figure9

System outage probability versus ρ for distinct values of β: FDR-NOMA (a1=0.05)

Fig. 10
figure10

System outage probability versus a1 for distinct πrr: FDR-NOMA (ρ=30 dB, β=0)

Fig. 11
figure11

System outage probability versus a1 for distinct dr2: FDR-NOMA (ρ=30 dB, β=0)

When a1 is small, the outage probability of U1 becomes higher, which makes the system outage also to be higher. When a1 is increased, the outage experienced by U2 becomes higher, which degrades the system outage. Thus, proper selection of a1 can minimize the system outage probability.

Next, we find the optimal power allocation factor a1,opt and the corresponding optimal system outage probability (\(P_{out,sys}^{FD,opt}\)), based on the analysis described in Section 5 for FDR-NOMA-CDRT system. We also find the system outage for random power allocation (RPA) (i.e., non-optimal selection of a1) as well. The system outage for the optimal and non-optimal schemes are shown in Fig. 12. Results show that OPA outperforms RPA significantly. With R1=R2=0.5 bpcu and for the assumed set of parameters indicated in Fig. 12, OPA provides 79% improvement in system outage probability as compared to the RPA scheme. With R1=2R2=0.5 bpcu, OPA scheme leads to 42% improvement in system outage as compared to the RPA scheme. Thus, we conclude that proper selection of a1 can improve the system outage performance of the FDR-based NOMA-CDRT network considered in this paper. Table 3 lists the numerical values of a1,opt that minimizes the system outage as a function of target rates R1 and R2. When R1 becomes higher, higher values for a1,opt has to be chosen so as to minimize \(P_{out,sys}^{FD}\). A higher value for R2 makes a1,opt to decrease to meet the desired objective.

Fig. 12
figure12

System outage probability versus target rate, R1: OPA versus RPA (ρ=20 dB,β=0)

Table 3 OPA factor a1,opt and \(P_{out,sys}^{FD}\) under OPA/RPA scheme (FDR-NOMA)

Evaluation of ergodic rates of U 1 and U 2

Figures 13 and 14 respectively show the ergodic rate achieved by the users and the ergodic sum rate of the network, for FDR-based NOMA-CDRT system. In Fig. 13, the ergodic rates corresponding to both the users are shown. The residual interference (k1πr1) and mean RSI (k2πrr) are chosen as variables. The ergodic rate of U1 depends on k1πr1. As k1πr1 increases, ergodic rate of U1 decreases due to higher interference at U1’s receiver. Ergodic rate of U1 does not depend on k2πrr, (the mean RSI power), as this quantity does not influence the SINR at U1. Ergodic rate of U2 decreases as k2πrr is increased, while k1πr1 does not have any influence on it. The impact of mean RSI (k2πrr) on the ergodic rate of U2 becomes predominant at higher transmit power since mean RSI power is higher in the high transmit power region. Further, we can see that the ergodic rate shows a saturation behavior in the high transmit power region owing to the higher amount of interference experienced by the receivers of the users. In the high transmit power region, both the RSI (k2πrr) as well as the residual interference (k1πr1) become predominant. Accordingly, ergodic rates no longer maintains linear relation with ρ; thus, a saturation behavior is seen. The ergodic sum rate of the system is shown in Fig. 14. Increase of residual interference at U1 (i.e., k1πr1) and mean RSI at R (i.e., k2πrr) decreases the ergodic sum rate of the network. The degradation of ergodic sum rate is more predominant in the high transmit power region owing to the higher amount of interference in the system. Initially as the transmit power increases, the ergodic sum rate increases; however, for larger transmit power values, it shows a saturation behavior owing to the fact that interference plays a key role in this region, and the system performance is limited by the interference. Figure 15 shows the effect of I-SIC factor β on ergodic rates of U1 and U2. As described earlier, β does not influence the performance of U2 since it does not have to implement SIC technique for decoding the message. However, increase of β increases the interference at U1, which degrades the achievable ergodic rate of U1.

Fig. 13
figure13

Ergodic rate versus ρ (a1=0.05,β=0)

Fig. 14
figure14

Ergodic sum rate versus ρ (a1=0.05,β=0)

Fig. 15
figure15

Ergodic rate versus ρ for distinct values of β (a1=0.05)

Comparison between NOMA and OMA

In this section, we compare the performance of FDR-NOMA-CDRT against conventional orthogonal multiple access (OMA)-based CDRT system, where communication is completed in two time slots. Here, the BS transmits the symbol x1 to U1 in the first time slot, which is subsequently decoded by U1 towards the end of the first time slot. In the second time slot, the BS transmits the symbol x2, which is decoded and forwarded by R to U2. Since R operates in FD mode, simultaneous reception and transmission happens at R so that x2 is decoded at U2 in the same time slot with certain processing delay, i.e., time division multiple access (TDMA) is considered as the OMA scheme, with the duration of the two time slots to be equal to T sec. We consider an OMA technique where power control is considered at the BS, i.e., power allocated to x1 is a1ρs and that for x2 is a2ρs as in NOMA, where a1+a2=1.

For a fair comparison of outage, we set the target rates for NOMA and OMA to be equal; the SINR thresholds are calculated based on this. Since OMA requires additional time slots for completing the transmission, the achievable rate under OMA gets reduced. Since the target rates for NOMA and OMA are set to be the same, the threshold SINR becomes higher for both U1 and U2 under OMA. Notice that when OMA is considered, the decoding of x1 at U1 happens in the absence of interference either due to x2 or due to transmissions from R (since U1 is silent during the second time slot). Figure 16 compares the outage performance of U1 under NOMA and OMA. The results show that in the absence of any residual interference (i.e., k1=0), the outage performance of U1 under NOMA remains to be significantly better than that under OMA scheme considered. However, when k1 increases, U1 is affected by interference from R’s transmission in the considered FDR-NOMA-CDRT system, which degrades the SINR at U1; thus, U1 suffers higher outage in NOMA system as compared to OMA. Figure 17 shows the corresponding results for the outage performance of U2. Notice that, as far as U2’s performance is considered, NOMA outperforms OMA-based scheme for the entire range of transmit power considered. Figure 18 shows the system outage probability under NOMA and OMA. The results show that the system outage probability of FDR-NOMA-CDRT is much smaller than that of FDR-OMA-CDRT for the entire range of transmit power considered, if the residual interference is negligible at U1 (i.e., k1=0). However, if k1 is non-zero, the system outage performance degrades significantly so that OMA will outperform NOMA system. As mentioned before, in the high transmit power region, HDR system performs significantly better than FDR system, owing to the enhanced RSI generated by FD operation. Further, the results shown in Fig. 19 implies that NOMA outperforms OMA in terms of ergodic sum rate as well.

Fig. 16
figure16

Outage probability of U1 versus ρ: NOMA and OMA schemes (a1=0.05,β=0)

Fig. 17
figure17

Outage probability of U2 versus ρ: NOMA and OMA schemes (a1=0.05,β=0)

Fig. 18
figure18

System outage probability versus ρ: NOMA and OMA schemes (a1=0.05,β=0)

Fig. 19
figure19

Ergodic sum rate versus ρ: NOMA and OMA schemes (a1=0.05,β=0)

Conclusion

This paper considered a full-duplex-based coordinated direct and relay transmission (CDRT) system that facilitates delivery of message from a base station (BS) to two geographically separated users, i.e., a near user and a far (cell edge) user. The BS was assumed to employ power domain NOMA to transmit the messages to the users. An intermediate full-duplex relay was used to assist the message delivery to the far user. Analytical expressions for the outage probability and ergodic rates of both the users and system outage probability were derived, assuming independent non-identically distributed Nakagami fading. The impact of imperfect SIC was considered for the analysis. The outage probability experienced by the near user was observed to be higher than that experienced by far user. Further, it was established that proper selection of NOMA power allocation coefficient at the BS can lead to equal outage probabilities for both the users. Finally, analytical expression for the optimal power allocation (OPA) coefficient at the BS that minimizes the system outage probability was also derived. Through extensive numerical and simulation investigations, it was established that selection of OPA coefficient according to the criterion given in the paper can significantly improve the system outage performance of the considered FDR-NOMA CDRT network, as compared to random power allocation at the BS.

Appendix A

Derivation of (18):

Consider the definition of \(P_{out,1}^{FD}\) given in (17). Substituting the expressions for Γ12 and Γ11 in (17), we get

$$\begin{array}{*{20}l} P_{out,1}^{FD} &= 1-Pr\left\{\frac{|h_{s1}|^{2} \rho_{s} a_{2}}{|h_{s1}|^{2} \rho_{s} a_{1} + |\hat{h}_{r1}|^{2} \rho_{r} +1}\right.\\&\quad \left.\geq u_{2}^{FD}, \frac{|h_{s1}|^{2} \rho_{s} a_{1}}{|h_{s1}|^{2} \rho_{s} \beta a_{2} +|\hat{h}_{r1}|^{2} \rho_{r} +1}\geq u_{1}^{FD}\right\} \end{array} $$
(37a)
$$\begin{array}{*{20}l} &= 1-P_{r}\left\{|h_{s1}|^{2} \geq \frac{u_{2}^{FD}(|\hat{h}_{r1}|^{2} \rho_{r} +1)}{(a_{2}-a_{1}u_{2}^{FD}) \rho_{s}}, |h_{s1}|^{2}\right. \\&\quad \left. \geq \frac{u_{1}^{FD}(|\hat{h}_{r1}|^{2} \rho_{r} +1)}{(a_{1}-a_{2}u_{1}^{FD}) \rho_{s}} \right\} \\& = 1-Pr\left\{|h_{s1}|^{2} \rho_{s}\geq \frac{1}{\phi}(|\hat{h}_{r1}|^{2} \rho_{r} +1)\right\} \end{array} $$
(37b)

where \(\phi = min\left (\frac {a_{2}-u_{2}^{FD} a_{1}}{u_{2}^{FD}}, \frac {a_{1} - \beta a_{2} u_{1}^{FD}}{u_{1}^{FD}}\right)\). The CDF and PDF of |hij|2 are given in (2a) and (2b), respectively. Utilizing these expressions, (37b) can be simplified as follows:

$$\begin{array}{*{20}l} P_{out,1}^{FD} & = 1-\int_{0}^{\infty} e^{-\big(\frac{y\rho_{r} +1}{\phi \rho_{s} \beta_{s1}}\big)} \sum_{j=0}^{m_{s1}-1}\frac{1}{j!} \left(\frac{y \rho_{r} +1}{\phi \rho_{s} \beta_{s1}}\right)^{j} \\&(k_{1}\beta_{r1})^{-m_{r1}}\frac{y^{m_{r1}-1}}{\Gamma(m_{r1})}e^{-\frac{y}{\beta_{r1} k_{1}}} dy \\ &= 1-e^{-\frac{1}{\phi \rho_{s} \beta_{s1}}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1}\\& \frac{1}{j!} \int_{0}^{\infty} e^{-\frac{y \rho_{r}}{\phi \rho_{s} \beta_{s1}}} e^{-\frac{y}{\beta_{r1} k_{1}}} y^{m_{r1}-1} \left(\frac{\rho_{r} y +1}{\phi \rho_{s} \beta_{s1}}\right)^{j} dy \end{array} $$
(38a)
$$\begin{array}{*{20}l} &=1- e^{-\frac{1}{\phi \rho_{s} \beta_{s1}}} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}\right)^{j} \\&\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} \\& \times \int_{0}^{\infty} e^{-\big(\frac{\rho_{r}}{\phi \rho_{s} \beta_{s1}}+\frac{1}{\beta_{r1} k_{1}}\big)y} y^{m_{r1}+k-1} dy \end{array} $$
(38b)

where \(u_{2}^{FD} < \frac {a_{2}}{a_{1}}\) or \(u_{1}^{FD} < \frac {a_{1}}{\beta a_{2}}\). Notice that (38b) is obtained from (38a) after using binomial expansion for term (y+1/ρr)j, i.e., (y+1/ρr)j= jCkyk(1/ρr)jk. Now, the integral in (38b) can be simplified by using [40] (3.351.3). Upon simplification, the final expression for \(P_{out,1}^{FD}\) can be obtained as in (18). Further, we can see that when \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}\) or when \(u_{1}^{FD} \geq \frac {a_{1}}{\beta a_{2}}\), the probability term on the RHS of (37a) will become zero so that Pout,1 tends to unity.

Appendix B

Derivation of (20):

Consider the definition of \(P_{out,2}^{FD}\) given in (19). Substituting the expressions for Γr2 and Γ22 as given in (5) and (8) in (19), we get the following:

$$\begin{array}{*{20}l} P_{out,2}^{FD} &= 1-P_{r} \left\{\frac{|h_{sr}|^{2} \rho_{s} a_{2}}{|h_{sr}|^{2} \rho_{s} a_{1}+|{h}_{rr}|^{2} \rho_{r} +1}\right.\\& \left. \geq u_{2}^{FD}, |h_{r2}|^{2} \rho_{r}\geq u_{2}^{FD}\right\} \\ &= 1-P_{r}\left\{|h_{sr}|^{2} \rho_{s} \geq \frac{u_{2}^{FD}}{a_{2}-u_{2}^{FD}a_{1}}(|h_{rr}|^{2} \rho_{r}+1),\right. \\&\left. |h_{r2}|^{2}\rho_{r}\geq u_{2}^{FD}\right\} \end{array} $$
(39a)
$$\begin{array}{*{20}l} &= 1 - P_{r}\left\{|h_{sr}|^{2} \rho_{s} \geq \frac{(|h_{rr}|^{2} \rho_{r}+1)}{\psi}\right\} \end{array} $$
(39b)
$$\begin{array}{*{20}l} &\times P_{r}\left\{|h_{r2}|^{2}\rho_{r}\geq u_{2}^{FD}\right\} \\& \triangleq 1-(A_{0} \times B_{0}) \end{array} $$
(39c)

where \(\psi = \frac {a_{2}-u_{2}^{FD}a_{1}}{u_{2}^{FD}}\). Notice that (39b) is written under the assumption that the channel power gains |hsr|2 and |hr2|2 are independent. Now, A0 and B0 can be evaluated by utilizing the expressions for the CDF and PDF of |hij|2 given in (2). Accordingly, we proceed as follows:

$$\begin{array}{*{20}l} A_{0} &= P_{r}\left\{|h_{sr}|^{2} \geq \frac{|{h}_{rr}|^{2} \rho_{r} +1}{\psi \rho_{s}}\right\} \\& = \int_{0}^{\infty} e^{-(\frac{y \rho_{r} +1}{\psi \rho_{s} \beta_{sr}})} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \left(\frac{y\rho_{r} +1}{\psi \rho_{s}\beta_{sr}}\right)^{j} (k_{2} \beta_{rr})^{-m_{rr}}\\& \frac{y^{m_{rr}-1}}{\Gamma(m_{{rr}})} e^{-\frac{y}{\beta_{rr} k_{2}}} dy \\& = e^{-\frac{1}{\psi \rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \int_{0}^{\infty}\\& e^{-\left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)y}y^{m_{rr}-1} \left(\frac{y \rho_{r}+1}{\psi \rho_{s} \beta_{sr}}\right)^{j} dy \end{array} $$
(40)

Applying binomial expansion for the term (y+(1/ρr))j; we get the following equation:

$$\begin{array}{*{20}l} A_{0} =& e^{-\frac{1}{\psi \rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}\right)^{j} \\&\times\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}\\& \times \int_{0}^{\infty} e^{-\left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)y}y^{k+m_{rr}-1} dy \end{array} $$
(41)

Now, the integral in (41) can be simplified by using [40] (3.351.3). Accordingly, the final expression for A0 can be obtained as follows:

$$\begin{array}{*{20}l} A_{0} =& e^{-\frac{1}{\psi \rho_{s} \beta_{sr}}} \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \sum_{j=0}^{m_{sr}-1} \frac{1}{j!} \left(\frac{\rho_{r}}{\psi \rho_{s} \beta_{sr}}\right)^{j} \\&\times\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} (m_{rr}+k-1)! \\& \times \left(\frac{\rho_{r} }{\psi \rho_{s} \beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)^{-m_{rr}-k} \end{array} $$
(42)

Further, B0 is determined as follows:

$$\begin{array}{*{20}l} B_{0} &= Pr\{|h_{r2}|^{2}\rho_{r} \geq u_{2}^{FD}\}= P_{r}\left\{|h_{r2}|^{2} \geq \frac{u_{2}^{FD}}{\rho_{r}}\right\} \\& = e^{-\big(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\big)} \sum_{i=0}^{m_{r2}-1} \frac{1}{i!} \left(\frac{u_{2}^{FD}}{\rho_{r} \beta_{r2}}\right)^{i} \end{array} $$
(43)

The final expression in (20) can be obtained by substituting (42) and (43) in (39c). Further, when \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}, A_{0}\) will become zero so that \(P_{out,2}^{FD}\) tends to be unity. Proposition 2 is thus proved.

Appendix C

Derivation of (21f):

Consider the expression for \(P_{out,sys}^{FD}\) given in (21e).

Let \(C_{0} = 1-Pr\left \{|h_{s1}|^{2} \geq \frac {1}{\phi \rho _{s}}(|\hat {h}_{r1}|^{2} \rho _{r} +1)\right \} \); \(A_{0} = P_{r}\left \{|h_{sr}|^{2} \geq \frac {|{h}_{rr}|^{2} \rho _{r} +1}{\psi \rho _{s}}\right \}\); and \(B_{0} = P_{r}\left \{|h_{r2}|^{2} \geq \frac {u_{2}^{FD}}{\rho _{r}}\right \} \). Accordingly, referring to (21e), we write \(P_{out,sys}^{FD} \triangleq (1-[C_{0} \times A_{0} \times B_{0}])\). From Appendix A, \(C_{0} = 1-P_{out,1}^{FD}\). Further, A0 and B0 are given by (42) and (43) of Appendix B. Combining these equations, \(P_{out,sys}^{FD}\) can be obtained as given in (21f). However, notice that if either \(u_{2}^{FD} \geq a_{2}/a_{1}\) or \(u_{1}^{FD} \geq a_{1}/{\beta a_{2}}, C_{0}\) will becomes zero so that \(P_{out,sys}^{FD}\) tends to unity. Proposition 3 is thus proved.

Appendix D

Derivation of (22b):

The CDF of Γ11, i.e., \(F_{\Gamma _{11}}(x)\) is determined as follows:

$$\begin{array}{*{20}l} {}F_{\Gamma_{11}}(x) & = 1- P_{r}\left(\frac{|h_{s1}|^{2} \rho_{s} a_{1}}{|h_{s1}|^{2} \rho_{s} \beta a_{2} +|\hat{h}_{r1}|^{2} \rho_{r} +1} >x\right) \\& = 1\,-\,P_{r}\left(|h_{s1}|^{2} \!>\!\frac{x}{\rho_{s} (a_{1}\,-\,\beta a_{2} x)}(|\hat{h}_{r1}|^{2} \rho_{r} \,+\,1)\right) \\& = 1- \int_{y=0}^{\infty} e^{-\frac{(\rho_{r} y+1)x}{\rho_{s} (a_{1}-\beta a_{2} x) \beta_{s1}}} \sum_{j=0}^{m_{s1}-1} \\&\frac{1}{j!} \left(\frac{(\rho_{r}y+1)x}{\rho_{s} (a_{1}-\beta a_{2} x) \beta_{s1}}\!\right)^{j} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})}e^{\,-\,\frac{y}{\beta_{r1} k_{1}}} y^{m_{r1}\,-\,1} dy \\& =1 - \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{x}{\rho_{s}(a_{1}-\beta a_{2} x) \beta_{s1}}\right)^{j} \\&e^{-\frac{x}{\rho_{s} (a_{1}-\beta a_{2} x)}} \int_{y=0}^{\infty} e^{-\big(\frac{x\rho_{r}}{\rho_{s} (a_{1}-\beta a_{2} x) \beta_{s1}}+\frac{1}{k_{1} \beta_{r1}}\big)y} y^{m_{r1}-1} \\& \times (y\rho_{r}+1)^{j} dy \end{array} $$
(44)

Notice that (44) is obtained by utilizing the CDF and PDF expression for |hij|2 given in (2). By using binomial expansion for (y+(1/ρr))j and utilizing [40] (3.351.3), (44) can be evaluated as,

$$\begin{array}{*{20}l} {}F_{\Gamma_{11}}(x) &= 1- \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{x}{\rho_{s} (a_{1} - \beta a_{2} x) \beta_{s1}}\right)^{j}\\&\times e^{-\frac{x}{\rho_{s} (a_{1} - \beta a_{2} x)}}\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}(m_{r1}+k-1)! \\& \times \left(\frac{x \rho_{r}}{\rho_{s} (a_{1} - \beta a_{2} x) \beta_{s1}}+\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k} \end{array} $$
(45)

Substituting (45) in (22a), \(E[_{R_{1}}^{FD}]\) can be computed using the following expression:

$$\begin{array}{*{20}l} {}E[_{R_{1}}^{FD}] &= \frac{1}{ln2}\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!}\sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k}\\&\quad(m_{r1}+k-1)! \\& \times \int_{x=0}^{\frac{a_{1}}{\beta a_{2}}}\\&{} \frac{e^{-\frac{x}{\rho_{s} (a_{1} - \beta a_{2} x) \beta_{s1}}}\left(\frac{x \rho_{r}}{\rho_{s} (a_{1} - \beta a_{2} x) \beta_{s1}}\right)^{j}\left(\frac{x \rho_{r}}{\rho_{s} (a_{1} - \beta a_{2} x) \beta_{s1}}\,+\,\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k}}{1+x} dx \end{array} $$
(46)

It is difficult to find a closed form expression for the integral term in (46), and hence, we can apply the Gaussian-Chebyshev quadrature method [41]. The basic formula used in Gaussian-Chebyshev quadrature method is given as

$$\begin{array}{*{20}l} \int_{-1}^{1} \frac{f(x)}{\sqrt{(1-x^{2})}} dx & \approx \frac{\pi}{N} \sum_{n=1}^{N} f\left[ cos\left(\frac{2n-1}{2N} \pi \right) \right] \end{array} $$
(47)

where N is an accuracy-complexity trade-off parameter. To use (47), we substitute \(x=\frac {1}{2}\frac {a_{1}}{\beta a_{2}} (1+\phi _{n})\) (where \(\phi _{n} = cos \big (\frac {(2n-1)\pi }{2N}\big))\) in (46). Larger N leads to a more accurate approximation at the cost of higher computational complexity. The integral expression in (46) can be simplified to a form similar to (47). Thereafter, \(E[_{R_{1}}^{FD}] \) can be obtained as given in (22b) by utilizing (47).

Appendix E

Derivation of (23b):

To derive the expression for \(E[_{R_{2}}^{FD}]\) using (23a), we find the CDF ofY as follows:

$$\begin{array}{*{20}l} F_{Y}(y) &= P_{r} (min\{\Gamma_{12},\Gamma_{r2},\Gamma_{22}\}\leq y) \\& = 1-P_{r}(\Gamma_{12}>y)P_{r}(\Gamma_{r2}>y)P_{r}(\Gamma_{22}>y) \\& = 1-[A_{1}\times A_{2}\times A_{3}] \end{array} $$
(48)

Notice that (48) is obtained under the assumption that the links in the network experience i.n.i.d. fading. Now, A1,A2 and A3 are determined as follows by utilizing the CDF/PDF expressions for |hij|2 given in (2).

$$\begin{array}{*{20}l} A_{1} &= P_{r}(\Gamma_{12}>y) \\& = P_{r}\left(\frac{|h_{s1}|^{2} \rho_{s} a_{2}}{|h_{s1}|^{2} \rho_{s} a_{1} + |\hat{h}_{r1}|^{2} \rho_{r} +1} > y\right) \\ & = \int_{z=0}^{\infty} P_{r}\left(|h_{s1}|^{2} > \frac{y(z \rho_{r} +1)}{\rho_{s} (a_{2}-a_{1}y)}\right) f_{|\hat{h}_{r1}|^{2}}(z)dz \end{array} $$
(49)

Recall that the power gain |hij|2 have Gamma PDF. Accordingly, we use the CDF/PDF expression given in (2) for evaluating (49). Thus, we get

$$\begin{array}{*{20}l} A_{1} &=\int_{z=0}^{\infty} e^{-\frac{y(z \rho_{r} +1)}{\rho_{s} (a_{2}-a_{1}y)\beta_{s1}}} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{y(z \rho_{r} +1)}{\rho_{s} (a_{2}-a_{1}y)\beta_{s1}}\right)^{j}\\& \quad\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} z^{m_{r1}-1} e^{-\frac{z}{\beta_{r1} k_{1}}} dz \\& {}= e^{-\frac{y}{\rho_{s} (a_{2}-a_{1}y)\beta_{s1}}} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \left(\frac{y}{\rho_{s} (a_{2}-a_{1}y)\beta_{s1}}\right)^{j} \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \\& \times \int_{z=0}^{\infty} e^{-\frac{yz \rho_{r} }{\rho_{s} (a_{2}-a_{1}y)\beta_{s1}}} (z \rho_{r} +1)^{j} z^{m_{r1}-1} e^{-\frac{z}{\beta_{r1} k_{1}}} dz \end{array} $$
(50)

To simplify (50), we invoke binomial theorem and further use the result reported in [40] (3.351.3). Thus, we get

$$\begin{array}{*{20}l} A_{1} &= e^{-\frac{y}{\rho_{s} (a_{2}-a_{1}y)\beta_{s1}}}\frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})}\sum_{j=0}^{m_{s1}-1} \\&\frac{1}{j!} \left(\frac{y\rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{s1}}\right)^{j} \sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} \\& {}\times (m_{r1}+k-1)! \left(\frac{y\rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{s1}}+\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k} \end{array} $$
(51)

By following similar procedure, A2 and A3 can be determined as follows:

$$\begin{array}{*{20}l} A_{2} &= P_{r}(\Gamma_{r2} >y) \\&\ = P_{r}\left(\frac{|h_{sr}|^{2} \rho_{s} a_{2}}{|h_{sr}|^{2} \rho_{s} a_{1}+|{h}_{rr}|^{2} \rho_{r} +1}>y\right) \\& {}= e^{-\frac{y}{\rho_{s} (a_{2}-a_{1}y)\beta_{sr}}}\frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})}\sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \left(\frac{y\rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{sr}}\right)^{l} \\&\sum_{p=0}^{l} {~}^{l}C_{p} \left(\frac{1}{\rho_{r}}\right)^{l-p} \\& \times (m_{sr}\,+\,p-1)! \left(\frac{y\rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)^{-m_{sr}-p} \end{array} $$
(52)
$$\begin{array}{*{20}l} A_{3} &= P_{r}(|h_{r2}|^{2}\rho_{r}>y) \\& = P_{r}\left(|h_{r2}|^{2}>\frac{y}{\rho_{r}}\right) \\& = e^{-\frac{y}{\rho \beta_{r2}}} \sum_{q=0}^{m_{r2}-1}\frac{1}{q!}\left(\frac{y}{\rho_{r} \beta_{r2}}\right)^{q} \end{array} $$
(53)

Substituting (51) - (53) in (48), we get FY(y). Substituting the expression for FY(y) in (23a) and rearranging, \(E[_{R_{2}}^{FD}]\) can be obtained as:

$$\begin{array}{*{20}l} {}E[_{R_{2}}^{FD}] &= \frac{(k_{1} \beta_{r1})^{-m_{r1}}}{\Gamma(m_{r1})} \sum_{j=0}^{m_{s1}-1} \frac{1}{j!} \sum_{k=0}^{j} {~}^{j}C_{k} \left(\frac{1}{\rho_{r}}\right)^{j-k} \\&(m_{r1}+k-1)! \frac{(k_{2} \beta_{rr})^{-m_{rr}}}{\Gamma(m_{rr})} \end{array} $$
$$\begin{array}{*{20}l} &\times \sum_{l=0}^{m_{sr}-1} \frac{1}{l!} \sum_{p=0}^{l} {~}^{l}C_{p} \left(\frac{1}{\rho_{r}}\right)^{l-p} (m_{rr}+p-1)!\\& \sum_{q=0}^{m_{r2}-1} \frac{1}{q!} \times \frac{1}{ln2} \\ & {}\times \int_{y=0}^{{a_{2}/a_{1}}} e^{-\frac{y}{\rho_{s}(a_{2}-a_{1}y)\beta_{s1}}} \left(\frac{y \rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{s1}}\!\,+\,\!\frac{1}{\beta_{r1} k_{1}}\right)^{-m_{r1}-k} \\& {}\times \left(\frac{y\rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{s1}}\right)^{j} e^{-\frac{y}{\rho_{s}(a_{2}-a_{1}y)\beta_{sr}}} \left(\frac{y\rho_{r}}{\rho_{s}(a_{2}-a_{1}y)\beta_{sr}}\right)^{l} \\& \times \left(\frac{y \rho_{r}}{\rho_{s}(a_{2}\,-\,a_{1}y)\beta_{sr}}+\frac{1}{\beta_{rr} k_{2}}\right)^{-m_{rr}-p} \\&e^{-\frac{y}{\rho_{r}\beta_{r2}}}\left(\frac{y}{\rho_{r} \beta_{r2}}\right)^{q} \frac{1}{1+y}dy \end{array} $$
(54)

To evaluate integral term in (54), we make use of the Gaussian-Chebyshev quadrature method [41]. First of all, we convert the integral in (54) into a form similar to (47) by substituting \(y = \frac {1}{2} \frac {a_{2}}{a_{1}} (1+\phi _{n})\). Thereafter, \(E[_{R_{2}}^{FD}]\) can be obtained as in (23b) by utilizing (47).

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Abbreviations

AWGN:

Additive white Gaussian noise

BS:

Base station

CDF:

Cumulative distribution function

CDRT:

Coordinated direct and relay transmission

DF:

Decode and Forward

FD:

Full duplex

FDR:

Full-duplex relay

HD:

Half duplex

HDR:

Half-duplex relay

NOMA:

Non-orthogonal multiple access

OMA:

Orthogonal multiple access

OPA:

Optimal power allocation

PDF:

Probability density function

RSI:

Residual self interference

SI:

Self interference

SIC:

Self interference cancelation

SINR:

Signal to interference plus noise ratio

SNR:

Signal to interference ratio

TDMA:

Time division multiple access

References

  1. 1

    S. R. Islam, N. Avazov, O. A. Dobre, K. -S. Kwak, Power-domain non-orthogonal multiple access (noma) in 5G systems: potentials and challenges. IEEE Commun. Surv. Tutor.19(2), 721–742 (2017).

    Article  Google Scholar 

  2. 2

    Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. Yuan, V. Bhargava, A survey on non-orthogonal multiple access for 5G networks: research challenges and future trends. IEEE J. Sel. Areas Commun.35:, 10 (2017).

    Article  Google Scholar 

  3. 3

    Z. Ding, M. Peng, H. V. Poor, Cooperative non-orthogonal multiple access in 5G systems. IEEE Commun. Lett.19(8), 1462–1465 (2015).

    Article  Google Scholar 

  4. 4

    J. -B. Kim, I. -H. Lee, Non-orthogonal multiple access in coordinated direct and relay transmission. IEEE Commun. Lett.19(11), 2037–2040 (2015).

    Article  Google Scholar 

  5. 5

    Z. Zhang, K. Long, A. V. Vasilakos, L. Hanzo, Full-duplex wireless communications: challenges, solutions, and future research directions. Proc. IEEE. 104(7), 1369–1409 (2016).

    Article  Google Scholar 

  6. 6

    G. Liu, F. R. Yu, H. Ji, V. C. Leung, X. Li, In-band full-duplex relaying: a survey, research issues and challenges. IEEE Commun. Surv. Tutor.17(2), 500–524 (2015).

    Article  Google Scholar 

  7. 7

    J. -B Kim, I. -H Lee, Capacity analysis of cooperative relaying systems using non-orthogonal multiple access. IEEE Commun. Lett.19(11), 1949–1952 (2015).

    Article  Google Scholar 

  8. 8

    L. Yuanwei, D. Zhiguo, E. Maged, P. H. Vincent, Cooperative non-orthogonal multiple access with simultaneous wireless information and power transfer. IEEE J. Sel. Areas Commun.34(4), 938–953 (2016).

    Article  Google Scholar 

  9. 9

    S. Lee, D. B. Costa, T. Q. Duong, in Proceedings IEEE 27th Annual International Symposium. Outage probability of non-orthogonal multiple access schemes with partial relay selection, Personal, Indoor, and Mobile Radio Communications (PIMRC) (Valencia, 2016), pp. 1–6.

  10. 10

    S. Lee, et al., Non-orthogonal multiple access schemes with partial relay selection. IET Commun.11(6), 846–854 (2016).

    Article  Google Scholar 

  11. 11

    Z. Ding, M. Peng, H. V. Poor, Cooperative non-orthogonal multiple access in 5G systems. IEEE Commun. Lett.19(8), 1462–1465 (2015).

    Article  Google Scholar 

  12. 12

    P. Xu, et al., Optimal relay selection schemes for cooperative NOMA. IEEE Trans. Veh. Technol.67(8), 7851–7855 (2018).

    Article  Google Scholar 

  13. 13

    J. Zhao, et al., Dual relay selection for cooperative NOMA with distributed space time coding. IEEE Access. 6:, 20440–20450 (2018).

    Article  Google Scholar 

  14. 14

    Y. Zhou, V. W. S. Wong, R. Schober, Dynamic decode-and-forward based cooperative NOMA with spatially random users. IEEE Trans. Wirel. Commun.17(5), 3340–3356 (2018).

    Article  Google Scholar 

  15. 15

    Md. F. Kader, S. Y. Shin, Coordinated direct and relay transmission using uplink NOMA. IEEE Wirel. Commun. Lett.7(3), 400–403 (2018).

    Article  Google Scholar 

  16. 16

    Y. Guo, Y. Li, W. Cheng, H. Zhang, in IEEE/CIC International Conference on Communications in China (ICCC). SWIPT assisted NOMA for coordinated direct and relay transmission (Beijing, 2018), pp. 111–115.

  17. 17

    Y Xu, et al., Performance of NOMA-based coordinated direct and relay transmission using dynamic scheme. IET Commun.12(18), 2231–2242 (2018).

    Article  Google Scholar 

  18. 18

    X. Yue, et al., Exploiting full/half-duplex user relaying in NOMA systems. IEEE Trans. Commun. 66(2), 560–575 (2018).

    Article  Google Scholar 

  19. 19

    X. Yue, A. Nallanathan, Y. Liu, S. Kao, Z. Ding, in IEEE International Conference on Communications (ICC). Outage performance of full/half-duplex user relaying in NOMA systems (Paris, 2017), pp. 1–6.

  20. 20

    X. Yue, Y. Liu, L. Rongke, A. Nallanathan, Z. Ding, in IEEE Global Communications Conference. Full/half-duplex relay selection for cooperative NOMA networks (Singapore, 2017), pp. 1–6.

  21. 21

    N. T. Tan, T. M. Hoang, B. C. Nguyen, in International Conference on Engineering Research and Applications. Outage analysis of downlink NOMA full-duplex relay networks with RF energy harvesting over Nakagami-m fading channel (SpringerCham, 2018), pp. 477–487.

    Google Scholar 

  22. 22

    C. Zhong, Z. Zhang, Non-orthogonal multiple access with cooperative full-duplex relaying. IEEE Commun. Lett. 20(12), 2478–2481 (2016).

    Article  Google Scholar 

  23. 23

    T. M. C. Chu, H. J. Zepernick, Performance of a non-orthogonal multiple access system with full-duplex relaying. IEEE Commun. Lett.22(1), 2084–2087 (2018).

    Article  Google Scholar 

  24. 24

    X. -X. Nguyen, D. T. Do, System performance of cooperative NOMA with full-duplex relay over Nakagami-m fading channels. Mob. Informa. Syst (2019). https://doi.org/10.1155/2019/7547431.

    Article  Google Scholar 

  25. 25

    X. -N. Tran, in 3rd International Conference on Recent Advances in Signal Processing, Telecommunications and Computing (SigTelCom). Performance of cooperative NOMA system with a full-duplex relay over Nakagami-m fading channels (IEEE, 2019). https://doi.org/10.1109/sigtelcom.2019.8696186.

  26. 26

    M. B. Shahab, S. Y. Shin, Time shared half/full-duplex cooperative NOMA with clustered cell edge users. IEEE Commun. Lett.22(9), 1794–1797 (2018).

    Article  Google Scholar 

  27. 27

    A. Goldsmith, Wireless communications (Cambridge university press, 2005).

  28. 28

    E. Everett, et al., in IEEE Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR). Empowering full-duplex wireless communication by exploiting directional diversity, (2011), pp. 2002–2006. https://doi.org/10.1109/acssc.2011.6190376.

  29. 29

    D. Bharadia, E. McMilin, S. Katti, Full duplex radios. ACM SIGCOMM Comput. Commun. Rev.43(4), 375–386 (2013).

    Article  Google Scholar 

  30. 30

    M. A. Khojastepour, et al., in Proceedings of the 10th ACM Workshop on Hot Topics in Networks, 17. The case for antenna cancellation for scalable full-duplex wireless communications (ACM, 2011). https://doi.org/10.1145/2070562.2070579.

  31. 31

    M. Duarte, C. Dick, S. Ashutosh, Experiment-driven characterization of full-duplex wireless systems. IEEE Trans. Wirel. Commun.11:, 4296–4307 (2012).

    Article  Google Scholar 

  32. 32

    E. Everett, et al., in Proceedings 49th AACCCC. Self-interference cancellation in multi-hop full-duplex networks via structured signaling (IEEEMonticello, 2011), pp. 1619–1626.

    Google Scholar 

  33. 33

    M Jain, et al., in Proceedings of the 17th annual international conference on Mobile computing and networking. Practical, real-time, full duplex wireless (ACM, 2011), pp. 301–312. https://doi.org/10.1145/2030613.2030647.

  34. 34

    S. Hong, et al., Applications of self-interference cancellation in 5G and beyond. IEEE Commun. Mag.52(2), 114–121 (2014).

    Article  Google Scholar 

  35. 35

    Z. Zhang, et al., Full duplex techniques for 5G networks: self-interference cancellation, protocol design, and relay selection. IEEE Commun. Mag.53(5), 128–137 (2015).

    Article  Google Scholar 

  36. 36

    A. Sabharwal, et al., In-band full-duplex wireless: challenges and opportunities. IEEE J. Sel. Areas Commun.32(9), 1637–1652 (2014).

    Article  Google Scholar 

  37. 37

    E. Everett, S. Achaleshwar, S. Ashutosh, Passive self-interference suppression for full-duplex infrastructure nodes. IEEE Trans. Wirel. Commun.13(2), 680–694 (2014).

    Article  Google Scholar 

  38. 38

    Y. Wang, et al., Relay selection of full-duplex decode-and-forward relaying over Nakagami-m fading channels. IET Commun.10(2), 170–179 (2016).

    Article  Google Scholar 

  39. 39

    L. J. Rodriguez, et al., Optimal power allocation and capacity of full-duplex AF relaying under residual self-interference. IEEE Wirel. Commun. Lett.3(2), 233–236 (2014).

    MathSciNet  Article  Google Scholar 

  40. 40

    I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, 6th edn. (Academic Press, New York, 2000).

    Google Scholar 

  41. 41

    E. Hildebrand, Introduction to numerical analysis (Dover, New York, 1987).

    Google Scholar 

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V, A., A V, B. Non-orthogonal multiple access in full-duplex-based coordinated direct and relay transmission (CDRT) system: performance analysis and optimization. J Wireless Com Network 2020, 24 (2020). https://doi.org/10.1186/s13638-019-1629-4

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Keywords

  • Non-orthogonal multiple access
  • Coordinated direct and relay transmission
  • Full duplex
  • Nakagami
  • Optimal power allocation