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Nonorthogonal multiple access in fullduplexbased coordinated direct and relay transmission (CDRT) system: performance analysis and optimization
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 24 (2020)
Abstract
This paper considers nonorthogonal 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 celledge 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 fullduplex 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 (ISIC) 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 FDRbased NOMACDRT system has been observed to be significantly improved compared to a FDRbased 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 nonorthogonal 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 timefrequency 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 NOMAbased 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 fullduplex mode that enables it to carry out simultaneous reception and transmission in the same frequency resource. However, inband FDR technique leads to the generation of selfinterference (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 selfinterference (RSI), which proportionally grows with the used transmit power [6].
The focus of the current work is on the performance analysis of NOMAbased 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., [7–14] and references therein). The performance of cooperative NOMAbased CDRT has also been analyzed in the literature [4, 15–17]. In [4], the authors have considered the application of NOMA in twouser 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 HDRbased 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 HDRbased NOMACDRT 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 HDRbased NOMACDRT system with two cellcenter users (CCUs) and a celledge user (CEU), which is assisted by a relay. Notice that all the above papers consider the performance evaluation of HDRbased NOMACDRT 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 (ISIC) will lead to the generation of residual interference at the near users. Recently, the performance of FDRbased cooperative NOMA system has been investigated extensively in the literature [18–26]. 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 (PSIC) 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 fullduplex (FD) or halfduplex (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 PSIC condition. In [22], the authors have analyzed the outage and ergodic rate performance of FDRbased NOMACDRT system in Rayleigh fading channels, assuming PSIC. The outage and ergodic sum rate performance of FDRbased NOMACDRT system has been analyzed in [23], in the presence of Nakagamai fading channels under PSIC 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 FDbased NOMACDRT system has been analyzed in [24, 25] as well, assuming Nakagami fading channels under ISIC; 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 FDRbased NOMACDRT system assuming the links to experience Nakagami fading, under the realistic assumption of ISIC. Further, we derive approximate closed form expressions for the ergodic rates achieved by the users in the system under ISIC. A singlecell downlink NOMACDRT system is considered consisting of a BS, a cellcentric (i.e., near) user, a celledge (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: ∙ Closedform expressions are derived for the outage probabilities experienced by the users in FDRbased NOMACDRT system, under the realistic assumption of ISIC. An analytical expression for the system outage probability is also presented. Approximate closedform expressions for the ergodic rates achieved by the users are derived, assuming ISIC. 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 FDRbased NOMACDRT network are also presented. Analytical results are corroborated by Monte Carlobased extensive simulation studies. ∙ In the considered FDR/HDRbased NOMACDRT 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 FDRbased NOMACDRT 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 FDRNOMACDRT system. We evaluate the percentage improvement in system outage under the OPA compared to random (i.e., nonoptimal) 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 [22–25]. In [22, 23], the authors have assumed PSIC, while the works reported in [24, 25] have considered ISIC 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 ISIC 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 ISIC have not appeared in these papers. Moreover, according to our best knowledge, the evaluation of the system outage probability of the considered FDbased NOMACDRT 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 relaybased cellular wireless communication scenario. To improve the spectral efficiency of the NOMACDRT 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 FDRbased NOMACDRT 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 FDRbased NOMACDRT system shown in Fig. 1 is considered where user 1 (U_{1}) happens to be the near user and user 2 (U_{2}) is the far user. The BS has direct communication link to U_{1} while it is assumed that, due to heavy shadow fading, the direct communication link between BS and U_{2} is absent. Accordingly, the BS employs a dedicated FDbased relay (R) to deliver the messages to U_{2}. We consider that R can operate as a DF relay. Let {h_{ij},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 nonidentically distributed (i.n.i.d.) Nakagami fading with shape parameter m_{ij} and mean power \(\mathbb {E}[h_{ij}^{2}] = \pi _{ij}\). Accordingly, h_{ij}^{2} have Gamma distribution with shape parameter m_{ij} and scale parameter β_{ij}=π_{ij}/m_{ij}. The probability density function (PDF) of h_{ij} is given by [27]:
where Γ(.) is the Gamma function. The CDF and PDF of h_{ij}^{2} are given by [27]:
Notice that (2a) assumes m_{ij} to take integer values only. We assume \(\pi _{ij} = (\frac {d_{ij}}{d_{0}})^{n}\) where n is the path loss exponent; d_{ij} is the distance between nodes i and j; i∈(s,r),j∈(1,r,2); and d_{0} is the reference distance (in the farfield 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 σ^{2}.
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 stateofart 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 receiverend of the FD node by employing different approaches [28–30] 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 [31–36]. 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 selfinterference (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 lineofsight (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 h_{rr} be the channel coefficient corresponding to the RSI channel; we assume h_{rr} to follow Nakagamim fading with parameters m_{rr} and mean RSI power = k_{2}π_{rr}, where k_{2} (0≤k_{2}≤1) represents the extent of SI cancelation; k_{2}=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
where x_{1}(t) and x_{2}(t) are the information symbols for U_{1} and U_{2} respectively; a_{1} and a_{2} are the power allocation coefficients such that a_{1}+a_{2}=1,a_{1}<a_{2}; and P_{s} 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 U_{1} will receive the NOMA signal. Now R will try to recover the symbol x_{2} by treating signal corresponding to U_{1} as interference. Under DF relaying, R will forward a clean copy of the reencoded symbol x_{2} to U_{2}. Since it operates in the FD node, there will be RSI present at the receiver of R. Accordingly the received signal at R is
In (4), the third term represents the RSI present at the relay node, P_{r} is the transmit power of R, τ is the processing delay, and n_{r}(t) is the AWGN component at R. The relay tries to decode x_{2} in the presence of signal corresponding to x_{1} and RSI. The corresponding SINR is given by:
where \(\rho _{s}=\frac {P_{s}}{\sigma ^{2}}\) and \(\rho _{r}=\frac {P_{r}}{\sigma ^{2}}\). The corresponding achievable rate is given by:
Once R forwards the reencoded symbol x_{2}, the received signal at U_{2} is given by:
where n_{2}(t) is the AWGN component at U_{2}. From the received signal, U_{2} tries to recover the symbol x_{2} and the corresponding SNR is
The achievable rate for R U_{2} link is
Meanwhile, the received signal at U_{1} is given by
Here, the third term represents the interference at U_{1} arising due to transmissions from R. According to the NOMA principle, U_{1} can decode the far user’s symbol x_{2}; thus, x_{2}(t−τ) is known at U_{1} apriori. Thus, U_{1} can cancel the third term in (10) completely. However, we assume that perfect cancelation of the third term is not possible at U_{1}; thus, (10) is written as follows:
In (11), \(\hat {h}_{r1}\) is assumed to have Nakgami PDF, and thus, \(\hat {h}_{r1}^{2}\) has Gamma PDF with mean k_{1}π_{r1} where k_{1}(0≤k_{1}≤1) represents the level of residual interference created at U_{1} due to incomplete cancelation of interference from R. The SINR corresponding to the decoding of x_{2} at U_{1} is
The corresponding achievable rate is given by
After decoding x_{2} successfully, U_{1} will decode x_{1} by performing SIC. In this case, the decoded symbol x_{2} must be subtracted from y_{1}(t) before the decoding of x_{1} is carried out. If x_{2} is decoded successfully, it can be completely subtracted from the composite received signal, i.e., SIC will be perfect. Otherwise, the decoding of x_{1} will be carried out in the presence of residual interference due to ISIC. Thus, SINR corresponding to the decoding of x_{1} at U_{1} in the presence of ISIC is given by
where 0≤β<1; i.e., β=0 means PSIC and 0<β≤1 implies ISIC.
The achievable rate corresponding to the decoding of x_{1} at U_{1} is given by
Under DF relaying, the maximum achievable rate for U_{2} is given by
Performance analysis
In this section, we present analytical models for finding the outage probabilities and ergodic rates of U_{1} and U_{2} in the considered FDRbased NOMACDRT system. We derive closed form expressions for the system outage probability as well, under imperfect SIC condition.
Outage probability analysis
Assume that R_{1} and R_{2} (expressed in bits per channel use, i.e., bpcu) are the target rates for the successful decoding of symbols x_{1} and x_{2}, respectively, in the considered FDRbased NOMACDRT 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 x_{1} and x_{2}. Thus, achievable rate for HD system is halved. For a fair comparison, we set the target rates for the equivalent HDRbased NOMACDRT system to be the same as that of the FDRbased 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 U_{1} and U_{2}, respectively.
Outage probability experienced by U _{1} in FDRbased NOMACDRT system
Notice that the near user U_{1} would not experience outage if both x_{1} and x_{2} are decoded successfully at U_{1}. Thus, the outage probability of U_{1} is given by
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:
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 FDRbased NOMACDRT system
Notice that the far user U_{2} would not suffer from outage if x_{2} is decoded successfully at R and U_{2}. Thus, the outage probability experienced by U_{2} is given by
Proposition 2: Assuming that \(u_{2}^{FD}<\frac {a_{2}}{a_{1}}, P_{out,2}^{FD}\) is given by the following expression
where \(\psi = \frac {a_{2}u_{2}^{FD} a_{1}}{u_{2}^{FD}}\) and \({~}^{j}C_{k} = \frac {j!}{k!(nk)!}\). 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 NOMACDRT network suffer outage conditions. Thus, we determine the system outage probability as follows:
To find the closed form expression for the system outage probability, we substitute the expressions for Γ_{12},Γ_{11},Γ_{r2}, and Γ_{22} in (21a). Accordingly, \(P_{out,sys}^{FD}\) becomes:
In (21d), the constants ϕ and ψ were defined in propositions 1 and 2, respectively. Now, the channel power gains, h_{sr}^{2},h_{s1}^{2}, and h_{r2}^{2} are independent since they correspond to distinct communication links in the network. Thus, \(P_{out,sys}^{FD}\) is given by
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:
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 U_{1} and U_{2} in the presence of ISIC condition.
Ergodic rate from BS to U _{1} in FDRbased NOMACDRT system
The ergodic rate achieved by the near user U_{1} (\(E[_{R_{1}}^{FD}] \)) is determined as follows:
where \(F_{\Gamma _{11}}(x)\) and \(F_{\Gamma _{11}}(x)\) are the CDF and PDF of Γ_{11}, respectively.Proposition 4: An approximate closed form expression for \(E[R_{1}^{FD}]\) is given as follows.
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 \(\phi _{n} = cos\left (\frac {(2n1)\pi }{2N}\right)\). Notice that (22b) is obtained by using the GaussianChebyshev quadrature formula, which is described in Appendix Appendix D. Here, Nis the complexity accuracy tradeoff parameter in this approximation.
Proof: Refer Appendix Appendix D.
Ergodic rate from BS to U _{2} in FDRbased NOMACDRT system
The ergodic rate achieved by the far user U_{2} (\(E[R_{2}^{FD}]\)) is determined as follows:
where y=min{Γ_{12},Γ_{r2},Γ_{22}} and F_{Y}(y) is the CDF of Y.Proposition 5: An approximate closed form expression for \(E[R_{2}^{FD}]\) is given as follows.
where \(e_{n} = \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1\phi _{n}) \beta _{s1}} + \frac {1}{\beta _{r1} k_{1}}\right)^{m_{r1}k} \sqrt {1\phi _{n}^{2}}, f_{n} = \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1\phi _{n}) \beta _{s1}} \right)^{j}, g_{n} = \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1\phi _{n}) \beta _{sr}} \right)^{l}, h_{n} \left (\frac {(1+\phi _{n})}{a_{1} \rho _{s} (1\phi _{n}) \beta _{s1}} + \frac {1}{\beta _{rr} k_{2}}\right)^{m_{rr}p}, r_{n} = \left (\frac {a_{2} (1+\phi _{n})}{2 a_{1} \rho _{r} \beta _{r2}}\right)^{q}, s_{n} = \frac {2 a_{1}}{2a_{1} + a_{2}(1+\phi _{n})}, t_{n} = \frac {(1+\phi _{n})}{a_{1} \rho _{s} (1\phi _{n}) \beta _{s1}}, v_{n} = \frac {(1+\phi _{n})}{a_{1} \rho _{s} (1\phi _{n}) \beta _{sr}}, w_{n} = \frac {a_{2} (1+\phi _{n})}{2 a_{1} \rho _{r} \beta _{r2}}, \phi _{n} = cos\left (\frac {(2n1)\pi }{2N}\right)\) and N is the complexity accuracy tradeoff parameter, relating to the GaussianChebyshev 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., a_{1,opt} that minimizes the system outage probability in FDRbased NOMACDRT system. The outage minimization problem can be formulated as
Proposition 5: For the considered FDRbased NOMACDRT system, the OPA coefficient a_{1,opt} that minimizes the system outage probability is given by
Proof:
Recall the expression for the system outage probability given in section 4, i.e.,
Here, A_{0} and B_{0} are given by (42) and (43), respectively, and \(C_{0} = 1P_{out,1}^{FD}\), where \(P_{out,1}^{FD}\) is given by (18). These are reproduced below:
It is assumed that SIC is perfect, i.e., β=0 in this section. Notice that in the expression for A_{0},ψ must be grater than 0 which implies \(u_{2}^{FD}<\frac {a_{2}}{a_{1}}\). Since a_{2}=1−a_{1}, this implies that a_{1} must satisfy the condition \(0 < a_{1} < \frac {1}{1+u_{2}^{FD}}\). Now, consider \(C_{0} = 1P_{out,1}^{FD}\) 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 a_{2}=1−a_{1}, the above implies that a_{1} 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:
where A_{0}(a_{1}) and C_{0}(a_{1}) are given by
Notice that (31a) and (31b) are obtained by substituting \(\psi =\phi =\frac {1a_{1}u_{2}^{FD}a_{1}}{u_{2}^{FD}}\) in (42) and (18), respectively. The derivative of \(P_{out,sys}^{FD}(a_{1})\) with respect to a_{1} can be written as
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 a_{1}. Thus, we write \(A^{\prime }_{0}(a_{1}) \triangleq x(a_{1})+y(a_{1})+z(a_{1})\) where x(a_{1}),y(a_{1}) and z(a_{1}) are given as follows:
In a similar way, we write \(C^{\prime }_{0}(a_{1}) = f(a_{1})+g(a_{1})+h(a_{1})\) where f(a_{1}),g(a_{1}) and h(a_{1}) are given as follows:
Through numerical investigations, we observe that \([P_{out,sys}^{FD}]'>0\), for the range of a_{1} considered. Thus, we conclude that \(P_{out,sys}^{FD}\) is a monotonically increasing function of a_{1} 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 a_{2}=1−a_{1}, the above condition implies that a_{1} 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 A_{0}(a_{1}) is given by (31a) and C_{1}(a_{1}) is obtained by substituting \(\phi = \frac {a_{1}}{u_{1}^{FD}}\) in the expression for \(C_{0} = 1P_{out,1}^{FD}\), with \(P_{out,1}^{FD}\) given by (18). Thus, C_{1}(a_{1}) is given by
The firstorder 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 A_{0}(a_{1}), can be determined by combining (33a)(33c) as in the previous case I. The first derivative of C_{1}(a_{1}), i.e., C1′(a_{1}) is given by \(C^{\prime }_{1}(a_{1}) = u(a_{1})+v(a_{1})+w(a_{1})\) where u(a_{1}),v(a_{)}, and w(a_{1}) are given by
Through numerical investigations, we find that the first derivative \([P_{out,sys}^{FD}]'<0\) for the range of a_{1} considered. Hence, we conclude that \(P_{out,sys}^{FD}\) is a decreasing function of a_{1} 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 a_{1} 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 a_{1} 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 a_{1} 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 10^{5} channel realizations. We select ρ_{s}=ρ_{r}=ρ,m_{ij}=m, where ρ is directly related to the transmit power. Unless otherwise specified, the following parameters are chosen for the evaluation: R_{1}=1 bpcu, R_{2}=1 bpcu, m=2, and n=3. We consider a two dimensional topology for the network under consideration with (x_{i},y_{i}) representing the coordinates of a given node i. Assume that BS is placed at (0,0) and BS, R, and U_{2} are placed on a straight line. Let the coordinates of R and U_{2} be (1.25,0) and (1.75,0), respectively. Further, we choose the coordinates of U_{1} as (0.625,0.5) so that the distances between these nodes are d_{s1}=0.8d_{0},d_{sr}=1.25d_{0},d_{r1}=0.8d_{0},d_{r2}=0.5d_{0}. Here, d_{0} is the reference distance and is selected as 1 km. For numerical illustrations, we set k_{1}=10^{−2},k_{2}=0.64×10^{−2}; however, the results can be modified for any given values of k_{1} and k_{2} (0≤k_{1},k_{2}≤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 π_{rr} set as equal to – 3 dB. For comparison purpose, we consider HDRbased NOMACDRT 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 x_{1} and x_{2}. U_{1} decodes x_{1} by implementing SIC, while R decodes x_{2} and forwards the symbol in the second time slot. Finally, x_{2} is decoded successfully at U_{2}. Here, we assume that U_{1} would not receive interference from R’s transmission since it happens in the second time slot during which U_{1} is silent. We modify the relevant equations of Section 4 to find the outage of U_{1} and U_{2} in HDRNOMACDRT 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 HDRCDRT and FDRCDRT systems are assumed to be equal, it leads to higher SINR threshold requirement for the users in HDRNOMACDRT system, as compared to the equivalent FDR system.
In Fig. 2, the outage probabilities experienced by U_{1} and U_{2} are drawn against ρ for FDR/HDRbased NOMACDRT system. Results show that, as ρ increases, the outage performance of both the users is improved. Further, the results in Fig. 2 show that U_{1} suffers higher outage probability than U_{2} for a given set of parameters. This happens because the power allocation factor at BS, i.e., a_{1} has been chosen arbitrarily. In addition, the decoding of symbol x_{1} at U_{1} requires a twostep procedure: successful decoding of symbol x_{2} by treating signal corresponding to x_{1} as interference, which is followed by decoding of x_{1} by applying SIC to cancel the known x_{2}. In this process, U_{1} is affected by interference due to transmissions from R as well. However, decoding of x_{2} happens at U_{2} in the absence of interference from any source. Moreover, it is assumed that the relay forwards the reencoded version of x_{2} with full power which increases the probability of successful decoding of x_{2} at U_{2}. As ρ is varied, the outage performance of U_{1} and U_{2} shows distinct behavior in FDR/HDRNOMACDRT systems. In the low transmit power region, the outage probabilities experienced by both U_{1} and U_{2} in FDRNOMACDRT system is lower as compared to the outage experienced in HDRNOMACDRT 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 (k_{2}π_{rr}) becomes higher in FDRbased system, triggering degradation of SINR at R. This increases the outage of U_{2} in FDRNOMACDRT 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 U_{1} is not affected by the RSI at R in FDRNOMACDRT system, as the decoding of x_{2} or x_{1} at U_{1} is not affected by the SINR over BSR link. However, the decoding at U_{1} is affected by the residual interference (k_{1}π_{r1}) created due to transmission over the R U_{2} link in FDRNOMACDRT system, which becomes significantly higher when transmit power is increased. Thus, the outage experienced by U_{1} in FDRNOMACDRT system becomes significantly higher in the high transmit power region. Notice that this residual interference is absent in HDR system, and thus, U_{1} exhibits much improved performance in HDRNOMACDRT system when transmit power is increased.
Figure 3 shows the impact of ISIC factor β on the outage probability performance of U_{1} and U_{2} in FDRNOMACDRT system. As β increases, \(P_{out,1}^{FD}\) increases owing to the higher amount of interference generated by ISIC at U_{1}. However, \(P_{out,2}^{FD}\) is not influenced by β, since U_{2} (being the faruser) does not implement SIC for decoding symbol x_{2}. As β increases from 0.3 to 0.4, \(P_{out,1}^{FD}\) increases by 89% for ρ=30 dB.
Figure 4 shows the outage probability of U_{1} in FDRNOMACDRT system against the mean residual interference (k_{1}π_{r1}) present at U_{1} (which is generated by the inaccurate cancelation of symbol x_{2} at U_{1}). The results show that the mean residual interference has significant impact on the outage of U_{1}. The results in Fig. 5 show that the outage probability of U_{2} 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 x_{2} at R. This degrades the outage performance of U_{2}, while HD relaying does not induce RSI, and thus, the outage probability of U_{2} does not depend on mean RSI power in HDRNOMACDRT system.
Equal outage for U _{1} and U _{2}
Figure 6 shows the outage probability of U_{1} and U_{2} drawn against the NOMA power allocation coefficient a_{1}. The results are shown for both HDR as well as FDRbased systems. Here, we keep d_{s1}=0.8, d_{sr}=1.25, d_{r1}=0.8, and d_{r2} is varied along the straight line joining BS, R, and U_{2}. Thus, U_{2} moves away from BS while the position of U_{1} is fixed. As d_{r2} increases, P_{out} of U_{2} becomes higher, whereas d_{r2} does not influence P_{out} of U_{1}. As a_{1} increases, more power gets allocated to U_{1}; thus, P_{out} of U_{1} decreases while that of U_{2} increases. The outages become very high and moves towards unity when \(u_{2}^{FD} \geq \frac {a_{2}}{a_{1}}\) for FDRbased system. Further, we can see that the outage of U_{2} in FDRbased system becomes higher when RSI is increased while that of U_{1} 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 a_{1} (i.e., \( a_{1}^{*} \)), P_{out} of both the users can be made equal, irrespective of the location of U_{2}. Table 1 depicts the details of the NOMA power allocation factor \( a_{1}^{*} \) that makes P_{out,1}=P_{out,2} for FDR/HDRNOMACDRT system as a function of d_{r2}. As d_{r2} is increased, \( a_{1}^{*} \) reduces, owing to the fact that more power need to be allocated for U_{2} 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 FDRbased system. As the mean RSI power increases, the SINR on the BSR link reduces which increases the outage probability of U_{2}. Moreover, results given in Table 2 implies that as the ISIC 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 U_{2}’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 P_{out} 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 d_{r2}, i.e., d_{r2}=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.
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 FDRNOMACDRT 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 U_{1} due to ISIC. Thus, U_{1} experience higher outage probability that increases system outage as well. Figure 10 shows the impact of RSI on the system outage of FDRNOMACDRT system while Fig. 11 shows the effect of R U_{2} distance d_{r2}. 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 BSR link degrades so that the outage probability suffered by U_{2} increases. Further, increase of d_{r2} increases the outage experienced by U_{2} triggering the system outage to become higher. As can be seen in Figs. 10 and 11, the system outage increases either when a_{1} is reduced or when a_{1} is increased.
When a_{1} is small, the outage probability of U_{1} becomes higher, which makes the system outage also to be higher. When a_{1} is increased, the outage experienced by U_{2} becomes higher, which degrades the system outage. Thus, proper selection of a_{1} can minimize the system outage probability.
Next, we find the optimal power allocation factor a_{1,opt} and the corresponding optimal system outage probability (\(P_{out,sys}^{FD,opt}\)), based on the analysis described in Section 5 for FDRNOMACDRT system. We also find the system outage for random power allocation (RPA) (i.e., nonoptimal selection of a_{1}) as well. The system outage for the optimal and nonoptimal schemes are shown in Fig. 12. Results show that OPA outperforms RPA significantly. With R_{1}=R_{2}=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 R_{1}=2R_{2}=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 a_{1} can improve the system outage performance of the FDRbased NOMACDRT network considered in this paper. Table 3 lists the numerical values of a_{1,opt} that minimizes the system outage as a function of target rates R_{1} and R_{2}. When R_{1} becomes higher, higher values for a_{1,opt} has to be chosen so as to minimize \(P_{out,sys}^{FD}\). A higher value for R_{2} makes a_{1,opt} to decrease to meet the desired objective.
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 FDRbased NOMACDRT system. In Fig. 13, the ergodic rates corresponding to both the users are shown. The residual interference (k_{1}π_{r1}) and mean RSI (k_{2}π_{rr}) are chosen as variables. The ergodic rate of U_{1} depends on k_{1}π_{r1}. As k_{1}π_{r1} increases, ergodic rate of U_{1} decreases due to higher interference at U_{1}’s receiver. Ergodic rate of U_{1} does not depend on k_{2}π_{rr}, (the mean RSI power), as this quantity does not influence the SINR at U_{1}. Ergodic rate of U_{2} decreases as k_{2}π_{rr} is increased, while k_{1}π_{r1} does not have any influence on it. The impact of mean RSI (k_{2}π_{rr}) on the ergodic rate of U_{2} 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 (k_{2}π_{rr}) as well as the residual interference (k_{1}π_{r1}) 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 U_{1} (i.e., k_{1}π_{r1}) and mean RSI at R (i.e., k_{2}π_{rr}) 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 ISIC factor β on ergodic rates of U_{1} and U_{2}. As described earlier, β does not influence the performance of U_{2} since it does not have to implement SIC technique for decoding the message. However, increase of β increases the interference at U_{1}, which degrades the achievable ergodic rate of U_{1}.
Comparison between NOMA and OMA
In this section, we compare the performance of FDRNOMACDRT against conventional orthogonal multiple access (OMA)based CDRT system, where communication is completed in two time slots. Here, the BS transmits the symbol x_{1} to U_{1} in the first time slot, which is subsequently decoded by U_{1} towards the end of the first time slot. In the second time slot, the BS transmits the symbol x_{2}, which is decoded and forwarded by R to U_{2}. Since R operates in FD mode, simultaneous reception and transmission happens at R so that x_{2} is decoded at U_{2} 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 x_{1} is a_{1}ρ_{s} and that for x_{2} is a_{2}ρ_{s} as in NOMA, where a_{1}+a_{2}=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 U_{1} and U_{2} under OMA. Notice that when OMA is considered, the decoding of x_{1} at U_{1} happens in the absence of interference either due to x_{2} or due to transmissions from R (since U_{1} is silent during the second time slot). Figure 16 compares the outage performance of U_{1} under NOMA and OMA. The results show that in the absence of any residual interference (i.e., k_{1}=0), the outage performance of U_{1} under NOMA remains to be significantly better than that under OMA scheme considered. However, when k_{1} increases, U_{1} is affected by interference from R’s transmission in the considered FDRNOMACDRT system, which degrades the SINR at U_{1}; thus, U_{1} suffers higher outage in NOMA system as compared to OMA. Figure 17 shows the corresponding results for the outage performance of U_{2}. Notice that, as far as U_{2}’s performance is considered, NOMA outperforms OMAbased 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 FDRNOMACDRT is much smaller than that of FDROMACDRT for the entire range of transmit power considered, if the residual interference is negligible at U_{1} (i.e., k_{1}=0). However, if k_{1} is nonzero, 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.
Conclusion
This paper considered a fullduplexbased 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 fullduplex 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 nonidentically 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 FDRNOMA 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 Γ_{12} and Γ_{11} in (17), we get
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 h_{ij}^{2} are given in (2a) and (2b), respectively. Utilizing these expressions, (37b) can be simplified as follows:
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/ρ_{r})^{j}, i.e., (y+1/ρ_{r})^{j}= ^{j}C_{k}y^{k}(1/ρ_{r})^{j−k}. 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 P_{out,1} tends to unity.
Appendix B
Derivation of (20):
Consider the definition of \(P_{out,2}^{FD}\) given in (19). Substituting the expressions for Γ_{r2} and Γ_{22} as given in (5) and (8) in (19), we get the following:
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 h_{sr}^{2} and h_{r2}^{2} are independent. Now, A_{0} and B_{0} can be evaluated by utilizing the expressions for the CDF and PDF of h_{ij}^{2} given in (2). Accordingly, we proceed as follows:
Applying binomial expansion for the term (y+(1/ρ_{r}))^{j}; we get the following equation:
Now, the integral in (41) can be simplified by using [40] (3.351.3). Accordingly, the final expression for A_{0} can be obtained as follows:
Further, B_{0} is determined as follows:
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 \(C_{0} = 1Pr\left \{h_{s1}^{2} \geq \frac {1}{\phi \rho _{s}}(\hat {h}_{r1}^{2} \rho _{r} +1)\right \} \); \(A_{0} = P_{r}\left \{h_{sr}^{2} \geq \frac {{h}_{rr}^{2} \rho _{r} +1}{\psi \rho _{s}}\right \}\); 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, \(C_{0} = 1P_{out,1}^{FD}\). Further, A_{0} and B_{0} 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 Γ_{11}, i.e., \(F_{\Gamma _{11}}(x)\) is determined as follows:
Notice that (44) is obtained by utilizing the CDF and PDF expression for h_{ij}^{2} given in (2). By using binomial expansion for (y+(1/ρ_{r}))^{j} and utilizing [40] (3.351.3), (44) can be evaluated as,
Substituting (45) in (22a), \(E[_{R_{1}}^{FD}]\) can be computed using the following expression:
It is difficult to find a closed form expression for the integral term in (46), and hence, we can apply the GaussianChebyshev quadrature method [41]. The basic formula used in GaussianChebyshev quadrature method is given as
where N is an accuracycomplexity tradeoff parameter. To use (47), we substitute \(x=\frac {1}{2}\frac {a_{1}}{\beta a_{2}} (1+\phi _{n})\) (where \(\phi _{n} = cos \big (\frac {(2n1)\pi }{2N}\big))\) 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:
Notice that (48) is obtained under the assumption that the links in the network experience i.n.i.d. fading. Now, A_{1},A_{2} and A_{3} are determined as follows by utilizing the CDF/PDF expressions for h_{ij}^{2} given in (2).
Recall that the power gain h_{ij}^{2} have Gamma PDF. Accordingly, we use the CDF/PDF expression given in (2) for evaluating (49). Thus, we get
To simplify (50), we invoke binomial theorem and further use the result reported in [40] (3.351.3). Thus, we get
By following similar procedure, A_{2} and A_{3} can be determined as follows:
Substituting (51)  (53) in (48), we get F_{Y}(y). Substituting the expression for F_{Y}(y) in (23a) and rearranging, \(E[_{R_{2}}^{FD}]\) can be obtained as:
To evaluate integral term in (54), we make use of the GaussianChebyshev 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).
Availability of data and materials
Not applicable.
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:

Fullduplex relay
 HD:

Half duplex
 HDR:

Halfduplex relay
 NOMA:

Nonorthogonal 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
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V, A., A V, B. Nonorthogonal multiple access in fullduplexbased coordinated direct and relay transmission (CDRT) system: performance analysis and optimization. J Wireless Com Network 2020, 24 (2020). https://doi.org/10.1186/s1363801916294
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Keywords
 Nonorthogonal multiple access
 Coordinated direct and relay transmission
 Full duplex
 Nakagami
 Optimal power allocation