# On the Performance of Cooperative Relaying Systems with NOMA inκ−μ Fading Environment

Authors: Sari Khatalin; Hadeel Miqdadi
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Nonorthogonal multiple access (NOMA) is considered to be a promising solution for the fifth generation (5G) wireless networks. The performance of the 5G wireless networks in terms of outage probability, achievable rate, and other important network performance metrics can be remarkably improved by the integration of NOMA with cooperative relaying transmission technique. In this paper, we investigate the performance of the cooperative relaying system integrated with NOMA in terms of outage probability and achievable rate overκ−μ fading channels. Analytical expressions for the outage probability and the achievable rate are obtained for the system mentioned above. Moreover, corresponding expressions for Rician and Rayleigh fading channels are also presented in this paper as special cases of κ−μ fading channels. Some numerical examples are presented here for illustration purposes. The analytical results obtained in this paper are all validated via simulation.


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## Body

## 1. Introduction

The new applications introduced by the fifth generation (5G) wireless networks such as the internet of things (IoT) have led to a tremendous increase in demands for better spectral efficiency, massive connectivity, and low latency. Such demands have driven the need for a new development in the multiplexing techniques, because the orthogonal multiple access (OMA) techniques currently used by the wireless communications systems to allocate resources to different users and applications are not expected to support such demands [1, 2]. Nonorthogonal multiple access (NOMA) has been shown to be a suitable multiple access technique that could support all of these demands [3, 4]. Unlike OMA techniques, NOMA is capable of supporting data transmission for multiple users simultaneously by employing the same channel resources of time, frequency, and code. However, the levels of power for the signals are different from each other [5]. At the NOMA receiver, the strong signal is decoded first; then the successive interference cancellation (SIC) scheme is utilized to decode the reminder of the signals [6].Cooperative relaying communication systems have shown the capability of providing a significant enhancement to the system performance of wireless communication networks [7, 8]. Recently, some researchers have considered cooperative relaying protocols in their studies to improve the transmission reliability for the sixth generation (6G) cognitive IoT network. For example, cooperative spectrum sharing and simultaneous wireless information and power transfer (SWIPT) are studied by the authors in [9] to improve the spectrum and energy efficiency for the 6G Cognitive IoT network.The integration of a cooperative relaying communication system with NOMA should provide further enhancement to the system performance of NOMA. Wireless communications assisted by unmanned aerial vehicles (UAVs) is an effective technique for communication performance improvement and coverage range expansion under poor channel conditions. One application for UAVs is to serve as relays in cooperative communication systems. Several studies have considered the UAVs as a potential technique for assisting NOMA to achieve better performance. For example, a UAV-assisted NOMA network was proposed by the authors in [10], in which the UAV and the base station cooperate with each other to serve ground users simultaneously. The authors in [11] proposed a UAV-aided NOMA scheme using SWIPT to achieve secure transmission for ground passive receivers in the presence of a potential eavesdropper.Other forms of cooperative communication systems were also considered to assist NOMA in achieving better performance. For instance, a cooperative communication system combined with NOMA was suggested by [12], in which users with better channel conditions are considered relays to provide assistance to those users with poor channel conditions. The authors in [13] proposed a cooperative NOMA transmission scheme with a dedicated relay to enhance the transmission reliability for users with bad channel conditions. In [14], NOMA was combined with multiple-antenna amplify-and-forward (AF) relaying network to improve the system performance. The focus of the study in [14] was on the analysis of the outage behavior for the mobile users. Furthermore, the achievable sum rate of a cooperative relaying system with NOMA (CRS-NOMA) was analyzed in [15] over Rayleigh fading channels. The authors in [16, 17] have considered similar analysis as in [15] over Rician and κ−μ shadowed fading channels, respectively. In [18], the achievable rate and the outage probability of CRS-NOMA were obtained over alpha-μ fading channels. The second time slot of the CRS-NOMA scheme used in [15–18] was only employed by the relay to transmit one decoded symbol to the destination. In [19], the authors have proposed that the second time slot is used only by the relay as well, but for transmitting two symbols instead of only one symbol. The achievable rate, outage probability, and throughput were obtained in [19] over Rayleigh fading channels for their proposed scheme.In order to improve the system performance of the CRS-NOMA scheme further, the authors in [20] proposed a new CRS-NOMA scheme, in which the second time slot of the new scheme can be utilized by both the relay and the source directly, which is different from the schemes used in [15–19]. Specifically, in the first time slot of the new scheme proposed by [20], the source transmits the same exact symbol to the relay and the destination, whereas in the second time slot the relay transmits the decoded symbol to destination and also the source transmits another symbol directly to destination. The data symbols can then be decoded at the destination by utilizing the SIC detection scheme. Moreover, each node in the new scheme presented in [20] will transmit each symbol with its maximum power to avoid the need for using the complicated power allocation method used in the other CRS-NOMA schemes as in [15–19]. The authors in [20] have studied the performance of the new proposed scheme in terms of the outage probability and the achievable rate over Rayleigh fading channels. Furthermore, the authors in [21] utilized the same scheme proposed by [20] to obtain approximated expressions for the achievable rate of the system over Rician fading channels. In [22], the outage probability of the system proposed in [20] was also analyzed over Nakagami-m fading channels.It can be noted that a very few studies have been conducted on the performance of the CRS-NOMA scheme proposed in [20]. To the best of our knowledge, none of these studies conducted over generalized fading channels such as κ−μ fading channels. The κ−μ fading distribution was proposed in [23] as a general fading model, which describes the small scale signal variations in the presence of line-of-sight components. The κ−μ general fading distribution includes Nakagami-m, Nakagami-n (Rician), Rayleigh, and one-sided Gaussian fading distributions as special cases, and therefore, it is very important to study the performance of the proposed scheme over this general distribution as it covers several other distributions as mentioned above.In this paper, we study the performance of the CRS-NOMA scheme presented in [20] over κ−μ fading channels. Specifically, the outage probability and the achievable rate of this proposed system, which are important performance metrics for the quality of service in the communication system design, are analyzed in κ−μ fading environment. The main contributions of this work can be summarized as follows: analytical expressions for the outage probability of the system considered here are derived over κ−μ fading channels for both direct and indirect links. These expressions are then utilized to obtain expressions of the outage probability for two different data symbols. Furthermore, analytical expressions for the average achievable rates of the system considered in this paper are also obtained in κ−μ fading environment for two different symbols separately. The average achievable rate of the whole system combined is also presented here. Moreover, corresponding expressions of the outage probability and the achievable rate for Rayleigh and Rician fading channels are obtained here as special cases of κ−μ fading channels and then compared with the results already published in the literature. All the analytical results obtained here are illustrated via some numerical examples and validated through simulations.The reminder of the paper is organized as follows: Section2 provides a description of the system used in this paper. Section 3 shows the mathematical analysis of the outage probability and the achievable rate. Some numerical results are offered in Section 4, while Section 5 presents the conclusion of this work.


## 2. System Description

In our study, we consider the CRS-NOMA scheme illustrated in Figure1(b), which consists of three nodes: a source (S), a destination (D), and a relay (R). It is assumed that R is a decode-and-forward (DF) relay node and operates in a fixed half-duplex mode. The source S can communicate with D directly and via R [7]. It is also assumed that all links between the three nodes are available and ready for usage.Figure 1
System models of two CRS-NOMA schemes.
(a)
Conventional CRS-NOMA scheme(b)
Proposed CRS-NOMA schemeThe channel coefficients of the links between the system nodesS-D, S-R, and R-D are defined as hSD, hSR, and hRD, respectively, where those channel coefficients are subject to κ−μ fading with average powers of ΩSD,ΩSR,andΩRD, respectively.Under the conventional CRS-NOMA scheme illustrated in Figure1(a), which is used in [15–18], a complicated power allocation method is utilized in the transmission process of the first time slot. Furthermore, the destination receives only one symbol during the second time slot, which is transmitted from R.In contrast with the conventional scheme mentioned above, in the proposed CRS-NOMA scheme illustrated in Figure1(b), which is the system considered here, each symbol is transmitted with its maximum power; therefore, the complicated power allocation method used in the conventional CRS-NOMA is avoided. Moreover, the destination can acquire two data symbols during the second time slot. As a result, this scheme outperforms the conventional CRS-NOMA scheme in terms of achievable rate. Specifically, the transmission process under the scheme considered here consists of two time slots as shown in Figure 1(b). In the first time slot, the symbol s1 is transmitted by the source S with power Pt to the nodes R and D. Therefore, the received signals in the first time slot at R and D can be, respectively, presented as [20–22]
(1)XSR=PthSRs1+nSR,(2)XSD=PthSDs1+nSD,where nSR and nSD denote the additive white Gaussian noise (AWGN) with zero mean and variance σ2 and Pt represents the total transmit power for each symbol with normalized power Es12=1 noting that E. represents expectation. The received signal to noise ratio (SNR) for s1 at nodes R and D can, respectively, be written as
(3)γSR,s1=hSR2Ptσ2,(4)γSD,s1=hSD2Ptσ2.In the second time slot, nodeR forwards the decoded symbol s1 to node D with power Pr, and also node S transmits symbol s2 with power Pt to node D. Therefore, the received signal at node D can be written as [20–22]
(5)rD=hRDPrs1+hSDPts2+n,where n denotes the AWGN with zero mean and variance σ2. In accordance with the protocol of NOMA technology, initially, s1 will be decoded by D while s2 is treated as noise. After that s1 is cancelled from (5) by utilizing SIC to decode s2. Assuming that the powers at the relay and the source are equal (i.e., Pt=PR=P), where P refers to the total transmit power. Hence, the SNR for the symbols s1 and s2 at D can, respectively, be given as
(6)γD,s1=hRD2P/σ2hSD2P/σ2+1,(7)γSD,s2=hSD2Pσ2.


## 3. Performance Analysis

In this section, we analyze the outage probability and the achievable rate for the CRS-NOMA system presented in Figure1(b) over κ−μ fading channels.


### 3.1. Outage Probability

The outage probability is the probability that the instantaneous SNRγ falls below a predefined threshold γth. The probability density function (PDF) of γ for the κ−μ fading channels can be obtained from eq. 11 of [23] as
(8)fγγ=μ1+κμ+1/2κμ−1/2expμκΩμ+1/2γμ−1/2exp−μ1+κγΩIμ−12μκ1+κγΩ.γ≥0.The cumulative distribution function (CDF) ofγ for the κ−μ fading channels can be presented as [23]
(9)Fγ=1−Qμ2κμ,2μκ+1γΩ,where Ω denotes the average received SNR, κ>0 represents the ratio between the total power of the dominant components and the power of the scattered waves, μ>0 is the number of the multipath clusters, Iv. is the modified Bessel function of the first kind of order v, and Qμa,b is the generalized Marcum Q-function [24, 25].The outage probability will be analyzed for boths1 and s2. First of all, we consider the outage probability for s1, Po,S1. Outage events for s1 can be defined as follows: the symbol s1 will go into an outage if it failed to arrive at the destination, and this happens if any of the transmission links for s1 to destination (i.e., direct (S-D) or indirect (S-R-D)) experiences an outage. Now, the outage probability of s1 over the direct link, FSD,s1γth can be obtained directly from (9)as [20].
(10)FSD,s1γth=PrγSD,s1<γth=1−Qμ2μκSD,2μκSD+1γthρΩSD,where ρ=P/σ2 indicates the transmit SNR. The outage probability of s1 over the indirect link, FD,s1γth, which occurs if the communication through the relay fails due to an outage over S-R and/or R-D links, is derived in Appendix as
(11)FD,s1z=1−μκSD+1μ+1/2κSDμ−1/2expμκSDρΩSDμ+1/2exp−μkRD−zμκRD+1ρΩRD⋅∑l=0∞∑r=0μ+l−1μκRDlzμκRD+1/ρΩRDrl!r!∑i=0rriΓi+1ΓμΓi+μΓμ⋅μκSDκSD+1/ρΩSDμ−1zμκRD+1/ρΩRD+μκSD+1/ρΩSDi+μ⋅1F1i+μ;μ;μ2κSDκSD+1/ρΩSDzμκRD+1/ρΩRD+μκSD+1/ρΩSD,where 1F1a;b;x is the confluent hypergeometric function of the first kind. The end-to-end (e2e) SNR for s1 via the indirect link is γe2e,S1=minγSR,s1,γD,s1, and the e2e outage probability for s1is given as [26]
(12)Fe2e,s1γth=FD,s1γth+FSR,s1γth−FD,s1γthFSR,s1γth,where
(13)FSR,s1γth=PrγSR,s1<γth=1−Qμ2μκSR,2μκSR+1γthρΩSR.Finally, the total outage probability fors1 can be expressed as [20]
(14)Po,s1=FSD,s1γthFe2e,s1γth.The symbols2 will experience an outage if s2 cannot be decoded properly during the transmission via the S-D link. Furthermore, s2 will experience an outage when the indirect link experiences an outage situation. Based on the SIC concept, the symbol s1 will be detected first, and then it is removed from the received signal. After that, D detects the symbol s2. Therefore, if D detects s1 incorrectly, then s2 will not be detected. The outage probability for s2 is obtained as [27]
(15)Po,s2=Prγ<γth=1−PrγD,s1≥γth,γD,s2≥γth=1−PrγD,s1≥γthPrγD,s2≥γth=1−1−PrγD,s1<γth1−PrγD,s2<γth=1−1−FD,s1γth1−FSD,s2γth,where PrγSD,s2<γth can be obtained from (9) as
(16)FSD,s2γth=PrγSD,s2<γth=1−Qμ2μκSD,2μκSD+1γthρΩSDand FD,s1γth is the expression in (11) after replacing z with γth. Setting μ=1 in (14) and (15) yields the corresponding expressions for the Rician fading channels. When k=0 and μ=1, (14) and (15) reduce to the corresponding expressions for Rayleigh fading, which are the same expressions presented in [20].


### 3.2. Achievable Rate

In this subsection, we will study the average achievable rate for the proposed CRS-NOMA scheme overκ−μ fading channels. Expressions for the average achievable rate will be derived for both symbols s1 and s2. The total achievable rate of the symbol s1 is given by [20, 21]
(17)∁s1=12minlog21+γRD,s1,log21+γSR,s1⏟∁R,s1+12log21+γSD,s1⏟∁D,s1=12log21+minλRD,λSRρ+12log21+λSDρ,where ∁R,s1 represents the achievable rate of s1 transmitted along the dual-hop (S-R and R-D) link, ∁D,s1 represents the achievable rate of s1 transmitted directly via the S-D link, λSD≜hSD2, λSR≜hSR2, λRD≜hRD2, and ρ≜Pt/σ2 is the transmit SNR. The reason for using the parameter 1/2 in (17) is because two time slots were utilized in relaying systems. The achievable rate of the symbol s2 is given by
(18)∁s2=12log21+γSD,s2⏟∁D,s2=12log21+λSDρ=∁D,s1,which is the maximum rate at which the destination can reliably decode the source message s2 in the second time slot. The total achievable rate for the whole system is the sum of the two rates in (17) and (18), which can be presented as [15–18, 21]
(19)∁sum=∁s1+∁s2=log21+minλRD,λSRρ+log21+λSDρ.Now, letγ1≜minλRD,λSR, γ2≜λSD. Then, the average achievable rate for s1 is given as [26]
(20)∁¯s1=E12log21+minλRD,λSRρ+12log21+λSDρ=12∫0∞log21+γ1ρdFγ1+12∫0∞log21+γ2ρdFγ2.The average achievable rate fors2 is given as
(21)∁¯s2=12∫0∞log21+γ2ρdFγ2.According to [16], the expressions in (20) and (21) can be presented as
(22)∁¯s1=12∙ln2Dρ+Gρ,(23)∁¯s2=12∙ln2Gρ,where
(24)Dρ=ρ∫0∞1−Fγ11+γ1ρdγ1,(25)Gρ=ρ∫0∞1−Fγ21+γ2ρdγ2.Then, the average achievable rate for the whole system (sum rate) can be presented as(26)∁¯sum=12∙ln2Dρ+2Gρ.Letz=minx,y; then the CDF of z can be presented as [16, 28]
(27)Fz=PrZ<z=1−∫x∞fxdx∫y∞fydy.Now, in order to solve the integrals inDρ and Gρ, we need to derive Fγ1 and Fγ2. From (9), we can show that for the κ−μ distribution, ∫x∞fXxdx=Qμ2κxμ,2μκx+1x/Ωx. Then, the CDF of γ1≜minλRD,λSR, Fγ1, can be obtain as
(28)Fγ1=1−Qμ2κxμ,2μκx+1γ1ΩxQμ2κyμ,2μκy+1γ1Ωy,and the CDF of γ2≜λSD, Fγ2 can be obtained as
(29)Fγ2=1−Qμ2κzμ,2μκz+1γ2Ωz,where the subscripts x, y, and z represent the links S-R, R-D, and S-D, respectively. Inserting (28) into (24), replacing Qμa,b with its infinite series representation given in (A.4) of Appendix, and evaluating the integral with the help of [29] yields
(30)Dρ=exp−κxμexp−κyμ∑n=0∞∑r=0∞κxμnn!κyμrr!∑i=0μ+n−1∑j=0μ+r−1μ1+κx/Ωxii!μ1+κy/Ωyjj!1ρi+jexpαi+j!Γ−i−j,α,where α=μ1+κx/Ωx+μ1+κy/Ωy/ρand Γ−n,x is the upper incomplete gamma function defined in ([30], p. 900). Similarly, after inserting (29) into (25), replacing Qμa,b with its infinite series representation given in (A.4), and evaluating the integral with the help of [29], we obtain
(31)Gρ=exp−κzμ∑n=0∞κzμnn!∑i=0μ+n−1μ1+κzΩzi1ρiexpβΓ−i,β,where β=μ1+κz/Ωz/ρ. Then by substituting (30) and (31) into (22) and (26), we obtain the exact expressions of the average achievable rates for s1 and the whole system, respectively. Exact expression of the average achievable rate for s2 can be obtained by inserting (31) into (23). Inserting μ=1 in the expressions represented in (28) through (31) yields the corresponding expressions for Rician fading channels. The average achievable rates for Rician fading were considered in [21]. However, the end results were given as approximated formulas, where in this paper we present the exact expressions of the average achievable rates for s1 and s2 under Rician fading.


## 3.1. Outage Probability

The outage probability is the probability that the instantaneous SNRγ falls below a predefined threshold γth. The probability density function (PDF) of γ for the κ−μ fading channels can be obtained from eq. 11 of [23] as
(8)fγγ=μ1+κμ+1/2κμ−1/2expμκΩμ+1/2γμ−1/2exp−μ1+κγΩIμ−12μκ1+κγΩ.γ≥0.The cumulative distribution function (CDF) ofγ for the κ−μ fading channels can be presented as [23]
(9)Fγ=1−Qμ2κμ,2μκ+1γΩ,where Ω denotes the average received SNR, κ>0 represents the ratio between the total power of the dominant components and the power of the scattered waves, μ>0 is the number of the multipath clusters, Iv. is the modified Bessel function of the first kind of order v, and Qμa,b is the generalized Marcum Q-function [24, 25].The outage probability will be analyzed for boths1 and s2. First of all, we consider the outage probability for s1, Po,S1. Outage events for s1 can be defined as follows: the symbol s1 will go into an outage if it failed to arrive at the destination, and this happens if any of the transmission links for s1 to destination (i.e., direct (S-D) or indirect (S-R-D)) experiences an outage. Now, the outage probability of s1 over the direct link, FSD,s1γth can be obtained directly from (9)as [20].
(10)FSD,s1γth=PrγSD,s1<γth=1−Qμ2μκSD,2μκSD+1γthρΩSD,where ρ=P/σ2 indicates the transmit SNR. The outage probability of s1 over the indirect link, FD,s1γth, which occurs if the communication through the relay fails due to an outage over S-R and/or R-D links, is derived in Appendix as
(11)FD,s1z=1−μκSD+1μ+1/2κSDμ−1/2expμκSDρΩSDμ+1/2exp−μkRD−zμκRD+1ρΩRD⋅∑l=0∞∑r=0μ+l−1μκRDlzμκRD+1/ρΩRDrl!r!∑i=0rriΓi+1ΓμΓi+μΓμ⋅μκSDκSD+1/ρΩSDμ−1zμκRD+1/ρΩRD+μκSD+1/ρΩSDi+μ⋅1F1i+μ;μ;μ2κSDκSD+1/ρΩSDzμκRD+1/ρΩRD+μκSD+1/ρΩSD,where 1F1a;b;x is the confluent hypergeometric function of the first kind. The end-to-end (e2e) SNR for s1 via the indirect link is γe2e,S1=minγSR,s1,γD,s1, and the e2e outage probability for s1is given as [26]
(12)Fe2e,s1γth=FD,s1γth+FSR,s1γth−FD,s1γthFSR,s1γth,where
(13)FSR,s1γth=PrγSR,s1<γth=1−Qμ2μκSR,2μκSR+1γthρΩSR.Finally, the total outage probability fors1 can be expressed as [20]
(14)Po,s1=FSD,s1γthFe2e,s1γth.The symbols2 will experience an outage if s2 cannot be decoded properly during the transmission via the S-D link. Furthermore, s2 will experience an outage when the indirect link experiences an outage situation. Based on the SIC concept, the symbol s1 will be detected first, and then it is removed from the received signal. After that, D detects the symbol s2. Therefore, if D detects s1 incorrectly, then s2 will not be detected. The outage probability for s2 is obtained as [27]
(15)Po,s2=Prγ<γth=1−PrγD,s1≥γth,γD,s2≥γth=1−PrγD,s1≥γthPrγD,s2≥γth=1−1−PrγD,s1<γth1−PrγD,s2<γth=1−1−FD,s1γth1−FSD,s2γth,where PrγSD,s2<γth can be obtained from (9) as
(16)FSD,s2γth=PrγSD,s2<γth=1−Qμ2μκSD,2μκSD+1γthρΩSDand FD,s1γth is the expression in (11) after replacing z with γth. Setting μ=1 in (14) and (15) yields the corresponding expressions for the Rician fading channels. When k=0 and μ=1, (14) and (15) reduce to the corresponding expressions for Rayleigh fading, which are the same expressions presented in [20].


## 3.2. Achievable Rate

In this subsection, we will study the average achievable rate for the proposed CRS-NOMA scheme overκ−μ fading channels. Expressions for the average achievable rate will be derived for both symbols s1 and s2. The total achievable rate of the symbol s1 is given by [20, 21]
(17)∁s1=12minlog21+γRD,s1,log21+γSR,s1⏟∁R,s1+12log21+γSD,s1⏟∁D,s1=12log21+minλRD,λSRρ+12log21+λSDρ,where ∁R,s1 represents the achievable rate of s1 transmitted along the dual-hop (S-R and R-D) link, ∁D,s1 represents the achievable rate of s1 transmitted directly via the S-D link, λSD≜hSD2, λSR≜hSR2, λRD≜hRD2, and ρ≜Pt/σ2 is the transmit SNR. The reason for using the parameter 1/2 in (17) is because two time slots were utilized in relaying systems. The achievable rate of the symbol s2 is given by
(18)∁s2=12log21+γSD,s2⏟∁D,s2=12log21+λSDρ=∁D,s1,which is the maximum rate at which the destination can reliably decode the source message s2 in the second time slot. The total achievable rate for the whole system is the sum of the two rates in (17) and (18), which can be presented as [15–18, 21]
(19)∁sum=∁s1+∁s2=log21+minλRD,λSRρ+log21+λSDρ.Now, letγ1≜minλRD,λSR, γ2≜λSD. Then, the average achievable rate for s1 is given as [26]
(20)∁¯s1=E12log21+minλRD,λSRρ+12log21+λSDρ=12∫0∞log21+γ1ρdFγ1+12∫0∞log21+γ2ρdFγ2.The average achievable rate fors2 is given as
(21)∁¯s2=12∫0∞log21+γ2ρdFγ2.According to [16], the expressions in (20) and (21) can be presented as
(22)∁¯s1=12∙ln2Dρ+Gρ,(23)∁¯s2=12∙ln2Gρ,where
(24)Dρ=ρ∫0∞1−Fγ11+γ1ρdγ1,(25)Gρ=ρ∫0∞1−Fγ21+γ2ρdγ2.Then, the average achievable rate for the whole system (sum rate) can be presented as(26)∁¯sum=12∙ln2Dρ+2Gρ.Letz=minx,y; then the CDF of z can be presented as [16, 28]
(27)Fz=PrZ<z=1−∫x∞fxdx∫y∞fydy.Now, in order to solve the integrals inDρ and Gρ, we need to derive Fγ1 and Fγ2. From (9), we can show that for the κ−μ distribution, ∫x∞fXxdx=Qμ2κxμ,2μκx+1x/Ωx. Then, the CDF of γ1≜minλRD,λSR, Fγ1, can be obtain as
(28)Fγ1=1−Qμ2κxμ,2μκx+1γ1ΩxQμ2κyμ,2μκy+1γ1Ωy,and the CDF of γ2≜λSD, Fγ2 can be obtained as
(29)Fγ2=1−Qμ2κzμ,2μκz+1γ2Ωz,where the subscripts x, y, and z represent the links S-R, R-D, and S-D, respectively. Inserting (28) into (24), replacing Qμa,b with its infinite series representation given in (A.4) of Appendix, and evaluating the integral with the help of [29] yields
(30)Dρ=exp−κxμexp−κyμ∑n=0∞∑r=0∞κxμnn!κyμrr!∑i=0μ+n−1∑j=0μ+r−1μ1+κx/Ωxii!μ1+κy/Ωyjj!1ρi+jexpαi+j!Γ−i−j,α,where α=μ1+κx/Ωx+μ1+κy/Ωy/ρand Γ−n,x is the upper incomplete gamma function defined in ([30], p. 900). Similarly, after inserting (29) into (25), replacing Qμa,b with its infinite series representation given in (A.4), and evaluating the integral with the help of [29], we obtain
(31)Gρ=exp−κzμ∑n=0∞κzμnn!∑i=0μ+n−1μ1+κzΩzi1ρiexpβΓ−i,β,where β=μ1+κz/Ωz/ρ. Then by substituting (30) and (31) into (22) and (26), we obtain the exact expressions of the average achievable rates for s1 and the whole system, respectively. Exact expression of the average achievable rate for s2 can be obtained by inserting (31) into (23). Inserting μ=1 in the expressions represented in (28) through (31) yields the corresponding expressions for Rician fading channels. The average achievable rates for Rician fading were considered in [21]. However, the end results were given as approximated formulas, where in this paper we present the exact expressions of the average achievable rates for s1 and s2 under Rician fading.


## 4. Numerical Results

In this section, we present some numerical examples to illustrate the main results obtained in this paper. The results presented in the examples here are validated via simulations. Figures reveal that simulation and analytical results are in full agreement.Figures2 and 3 plot the total outage probability of s1 (Po,s1) and s2 (Po,s2) over κ−μ fading as a function of ρ for μ=2, ΩSD=1, ΩSR=10, ΩRD=10, and the threshold SNR γth=0 dB. The values of k selected for Figure 2 are kSD=0.3, kSR=0.4, and kRD=0.5, whereas kSD=1, kSR=1, and kRD=2 are the values used in Figure 3. It can be seen in both figures that as ρ increases Po,s1 and Po,s2 decrease (i.e., Po,s1 and Po,s2 improve with increasing ρ). It can also be seen that the decrease in Po,s1 is more pronounced than in the case of Po,s2. Furthermore, it can be observed from the figures that increasing k will improve the outage probability.Figure 2
Outage probability overκ−μ fading for γth=0 dB, μ=2, ΩSD=1, ΩSR=10, ΩRD=10, κSD=0.3, κSR=0.4, and κRD=0.5.Figure 3
Outage probability overκ−μ fading for γth=0 dB, μ=2,ΩSD=1, ΩSR=10, ΩRD=10, κSD=1, κSR=1, and κRD=2.Figure4 shows Po,s1 and Po,s2 versus ρ for the same parameters values considered in Figure 3 except for μ=3. The comparison of Figure 3 with Figure 4 reveals that the outage performance is improved with increasing μ.Figure 4
Outage probability overκ−μ fading for γth=0 dB, μ=3,ΩSD=1, ΩSR=10, ΩRD=10, κSD=1, κSR=1, and κRD=2.Figure5 shows the variation of the outage probability for different values of ΩRD. It can be noted that as ΩRD increases the outage probability improves for both s1 and s2. However, the improvement is more obvious in the case of s1 than that of s2. Furthermore, it can be noticed that the improvement is more pronounced in the case of increasing ΩRD from 2 to 5 than that from 5 to 10, which implies that the benefit of increasing ΩRD on the performance of the outage probability diminishes as ΩRD increases.Figure 5
Outage probability overκ−μ fading for γth=0 dB, μ=2, ΩSD=1, ΩSR=10, ΩRD=2, 5, 10, κSD=1, κSR=1, and κRD=2.Figure6 presents a comparison of the outage probability between Rayleigh (kSD=kSR=kRD=0 and μ=1), Rician (μ=1), and κ−μ (μ=2) fading channels. It can be seen from the figure that κ−μ fading channel exhibits better outage results compared with the other two fading channels as expected, especially when ρ=10 dB and beyond.Figure 6
Outage probability overκ−μ, Rician, and Rayleigh fading channels for γth=0 dB, ΩSD=1, ΩSR=10, ΩRD=10, κSD=1, κSR=1, and κRD=2.In summary, it can be noticed from all figures mentioned above that bothPo,s1 and Po,s2 improve as the values of ρ or k or μ or Ω increase. However, the improvement is more noticeable in the case of Po,s1 than that of Po,s2. Furthermore, it can be easily seen in all figures that the improvement of Po,s2 will begin to diminish when ρ=10 dB and even the improvement stops as ρ reaches 15 dB and beyond, which means that increasing ρ beyond 15 dB will not help improving Po,s2. However, improving the channel conditions and/or average power may help in improving Po,s2.Figures7 and 8 plot the average achievable rates of the proposed CRS-NOMA scheme utilized in the study for s1,s2 and for the whole system combine (sum rate) over κ−μ fading as a function of ρ, where μ=1, ΩSD=1, ΩSR=10, and ΩRD=10. The values of k used to plot Figure 7 are kSD=kSR=kRD=0, whereas in Figure 8kSD=0.7, kSR=0.8, and kRD=1.5 are used. It is obvious from these two figures that Figure 7 represents the average achievable rate over Rayleigh fading, whereas Figure 8 represents the average achievable rate over Rician fading channels. As expected, there is a small increase in the average achievable rate in Figure 8 over that of Figure 7 as a result of a slight increase in the values of k.Figure 7
Average achievable rate overκ−μ fading forμ=1, kSD=0, kSR=0, kRD=0 (Rayleigh fading), ΩSD=1, ΩSR=10, and ΩRD=10.Figure 8
Average achievable rate overκ−μ fading for μ=1, κSD=0.7, κSR=0.8, κRD=1.5 (Rician fading), ΩSD=1, ΩSR=10, and ΩRD= 10.Figure9 shows the average achievable rate over κ−μ fading for μ=2, kSD=2,kSR=5,kRD=5,ΩSD=3, ΩSR=10, and ΩRD=10. Comparing Figure 8 with Figure 9, it can be easily seen that increasing μ and the values of k will result in improving the achievable rate of the system.Figure 9
Average achievable rate overκ−μ fading for μ=2,κSD=2, κSR=5, κRD=5, ΩSD=3, ΩSR=10, and ΩRD=10.Figures10 and 11 show the variations in the average achievable sum rate affected by ΩRD and ΩSD, respectively. It can be noted that as the value of ΩRD increases in Figure 10 the sum rate improves. Similar effect was observed on the sum rate in Figure 11 with increasing ΩSD. It can be noticed from the figures that the sum rate at ρ=30 dB improves by about 1 bit/sec/Hz when increasing ΩRD from 2 to 10, whereas increasing ΩSD from 2 to 3 improves the sum rate by almost 0.7 bit/sec/Hz. Even though there is about 0.3 bit/sec/Hz more improvement in the case of increasing ΩRD over that of ΩSD, but that small improvement came at the expense of increasing ΩRD alot more than ΩSD. It can be concluded that increasing ΩSD will improve the sum rate more than that of ΩRD. That is because, increasing ΩSD will impact the rates of s1 and s2 in first time slot and the second time slot, respectively, whereas increasing ΩRD will only impact the rate of s1 in the second time, which can be confirmed from equations (22) and (23).Figure 10
Average achievable rate overκ−μ fading for μ=2,κSD=2, κSR=2, κRD=5, ΩSD= 3, ΩSR=10, and ΩRD=2,10.Figure 11
Average achievable rate overκ−μ fading for μ=2,κSD=2, κSR=2, κRD=5, ΩSD=2,3, ΩSR=10, and ΩRD=10.Figure12 presents the average achievable sum rate over κ−μ fading for three different values of μ, namely, μ=1,2,and3. It is obvious from the figure that the sum rate increases as μ increases. However, it can be seen that the sum rate improvement is more pronounced when μ increases from 1 to 2 as ρ reaches beyond 15 dB, which implies that the improvement of the rate diminishes as μ increases.Figure 12
Average achievable rate overκ−μ fading for μ=1,2,3,κSD=2, κSR=5, κRD=5, ΩSD=3, ΩSR=8, and ΩRD=8.Figure13 plots the average achievable rates of the conventional CRS-NOMA scheme presented in Figure 1(a) over Rician fading. The expressions utilized to plot this figure are presented in [16], and the parameter values selected for this plot (i.e.,κSD=2, κSR=5, κRD=5, ΩSD=9, ΩSR=144, and ΩRD=36) are the same ones used to plot Figure 3 in [16]. The power allocation coefficients are set as a1=0.6 and a2=0.4. For the purpose of comparison, the average achievable rates of the proposed CRS-NOMA used in this study are plotted in Figure 14 over κ−μ fading when μ=1Ricianfading for the same parameters values used in Figure 13, which is a regeneration of Figure 3 of [16]. When comparing Figure 13 with Figure 14, we can easily observe that the proposed scheme used here outperforms the conventional one significantly in terms of average achievable sum rate. That is because, two copies of s1 and one copy of s2 were acquired by the destination over two time slots under the proposed scheme, where s1 was transmitted in the first time slot from the source to the destination with full power and s1 and s2 were transmitted in the second time slot from the relay and the source, respectively, to the destination node each with full power as well. On the other hand, in the first time slot of the conventional scheme only s1 was decoded by the destination, where s2 was treated as noise [15, 16]. Since SIC was not utilized at the destination in the first time slot for s1 detection process, s1 is subject to interference [19], and therefore, s1 rate performance is very limited and almost constant in comparison with s2 performance as shown in Figure 13. Furthermore, we should keep in mind that s1 was transmitted from the source to the destination with partial transmission power (i.e., power allocation coefficient a1was assigned to it), which will limit its rate performance further.Figure 13
Average achievable rate of the conventional CRS-NOMA over Rician fading forκSD=2, κSR=5, κRD=5, ΩSD=9, ΩSR=144, and ΩRD=36.Figure 14
Average achievable rate of the proposed CRS-NOMA overκ−μ fading for μ=1Ricianfading,κSD=2, κSR=5, κRD=5,ΩSD=9, ΩSR=144, and ΩRD=36.


## 5. Conclusions

The performance of the CRS-NOMA system was investigated in this paper in terms of outage probability and achievable rate overκ−μ fading channels. This system was analyzed without the usage of the complicated power allocation method used in the other CRS-NOMA schemes. Furthermore, the data received by the system destination was transmitted over two time slots, where the second time slot was utilized by the relay node and by the source node directly to transmit data. This is different from the other CRS-NOMA schemes, where the second time slot was used by the relay only. Analytical expressions for the outage probability and the average achievable rates were derived for two symbols s1 and s2. The analytical results obtained here were all validated through simulations. The expressions for the outage probability and the average achievable rates include infinite sum series, but those series will converge rapidly. It was noticed that the outage probability and the achievable rates of s1 and s2 improve with increasing the values of ρ or k or μ or Ω. However, the improvement for s1 was more pronounced than that of s2. It was also noticed that increasing ρ beyond 15 dB will not improve the outage probability of s2, but increasing k and/or μ and/or Ω will. Therefore, improving the outage probability of s2 can be achieved by adjusting the average power of the links and/or the channel condition. Furthermore, it was easily observed through the numerical results that the proposed scheme analyzed in this paper outperforms the conventional one significantly in terms of average achievable sum rate.



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*Source: 1319687-2022-04-23.xml*