Three-User Cooperative Dual-Stage Non-Orthogonal Multiple Access for Power Line Communications

We present a cooperative, dual-stage (DS) non-orthogonal multiple access (NOMA) scheme for power line communication (PLC) systems, where three PLC modems are served in two time slots. The network consists of a source (S) modem, two relay (R) modems and a destination (D) modem. In each of the two time slots, S communicates with one R using NOMA, and the other R communicates with D using the decode-and-forward technique. We consider both scenarios where a direct link may or may not exist between S and D. We derive expressions for the approximate average sum rate and overall outage probability of the network. Next, we formulate an optimization problem to find the optimal power allocation coefficients at S, such that the sum rate is maximized. We establish the accuracy of our approximations using Monte Carlo simulations. Furthermore, we show that our scheme outperforms the two-user single-stage and the DS schemes, and the three-user NOMA scheme proposed in the earlier literature, in terms of sum rate and outage probability.


I. INTRODUCTION
P OWER line communication (PLC) technology uses the existing infrastructure of power distribution networks for information transfer, which results in low deployment costs and a better performance compared to wireless technologies [1]. Some of the limitations and key issues for information transmission over a PLC channel includes the effects due to selective multipath fading, receiver noise and electromagnetic interference. Majority of PLC channel impairments due to the harsh conditions are characterized due to these impairments. Different models for signal impairments in a PLC channel have been proposed in the literature [2], [3]. Among these models, the lognormal distribution best describes the random effects due to unmatched loads observed in PLC network topologies and branches [4]. Among the noise models, a mixture of the impulsive component and the background component, for which a Bernoulli-Gaussian distribution is employed, which is amenable to analysis and captures the effect of the mixture of components [5], [6]. There has been a considerable research attention on PLC from both academia and industry recently, owing to features such as reliability and security, apart from those mentioned above [7]. Cooperative PLC has also emerged as a promising technology for applications in beyond 5G and Internet-of-Things [8]. In conventional cooperation relay based on orthogonal multiple access (OMA) schemes, such as time division multiplexing, the signal received from relay and transmitter over the two time slots is combined at the receiver [9]. This leads to an improved performance at the receiver, at the expense of spectral efficiency.
The recent surge in literature on non-orthogonal multiple access (NOMA) indicates its potential in next generation communication systems [10], [11], [12], [13]. In NOMA, multiple users are served simultaneously over same time and frequency resources, by appropriately multiplexing the transmitted signals either in code-domain or power-domain. The user symbols are assigned different transmit power levels This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ based on the user channel conditions. User with the best channel is allocated with least power, while the user with the worst channel is allocated highest power. Additionally, successive interference cancellation (SIC) is employed at the receiver, while superposition coding is used at the transmitter. Given these advantages, NOMA was proposed for cooperative PLC, which yields a higher spectral efficiency and lower latency than OMA [12], [14]. Most of the existing literature on NOMA for cooperative PLC has considered two-user networks. In [12], the authors proposed a single stage (SS) NOMA-based dual-hop decode-and-forward DF cooperative relaying scheme for narrowband PLC. The transmitter modem sends a superimposed signal to two receiver modems in first time slot. One of the receivers acts as a relay to the other, which decodes the symbols using successive interference cancellation (SIC), and assists the other node by transmitting its symbol again in the second slot. This idea was further investigated in [15], where a dual stage (DS) setup was considered, which was shown to outperform the SS case. More recently, the authors in [11] studied the outage probability performance of NOMA-PLC considering both SS and DS schemes. Furthermore, there are studies with different setups in a two-user NOMA for a hybrid wireless-PLC network [16], [17]. In particular, the authors in [16] studied the outage probability and sum rate considering DS scheme under the additive white Gaussian and Bernoulli-Gaussian noises. The source modem sends a superimposed signal to relay in the first time slot. The relay forwards the superimposed signals to both destination modems in the second slot. However, the aforementioned studies do not consider either sum rate or outage probability performance study of a three-user PLC network, under Bernoulli-Gaussian noise. The performance of three-user NOMA schemes for wireless communication were studied in [18] and [19], where the authors showed that NOMA outperforms OMA, in terms of the overall network sum rate. However, none of these works consider PLC networks. To the best of the authors' knowledge, performance of SS or DS schemes in NOMA-PLC networks with three users has not been considered in the literature.
The contributions of this paper in relation to the existing literature are as follows.
• We investigate the three-user cooperative NOMA scheme proposed for wireless networks in [19] in the context of PLC, and study its performance in terms of outage probability and network sum rate. • We derive expressions for approximate sum rate and outage probability of the considered PLC network, in two scenarios -without and with a direct link between the source and destination modems. Additionally, we establish the tightness of our approximations through Monte Carlo simulations. • Furthermore, we formulate an optimization problem and propose a solution to find the optimal power coefficients for NOMA at the source that maximizes the network sum rate.  • In terms of sum rate, we show that the considered scheme outperforms the three-user NOMA scheme considered in [18] extended to PLC, and the SS and DS two-user NOMA schemes in [15]. • Moreover, in terms of outage probability, we show that our proposed scheme outperforms the SS and DS twouser NOMA schemes in [11]. The rest of the paper is organized as follows. Section II describes the system model of the considered three-user PLC network. In Section III, analytical expressions are derived for the average sum rates without and with a direct link between the source and destination modems. The corresponding outage probabilities are derived in Section IV. Validation of our analysis is presented in Section V. Concluding remarks are presented in Section VI. The notations used in this paper are given in Table 1.

II. SYSTEM MODEL
Consider a three user power line communication (PLC) setup which consists of a source (S) modem and three receiver modems, as shown in Figs. 1 and 2. All modems operate in half-duplex mode. Out of the three receiver modems, two modems, namely R 1 and R 2 are closer to S with a direct link between S-R 1 and S-R 2 . The third modem D is far from S, and may not or may share a direct link with S, as shown in Fig. 1 and Fig. 2, respectively [19], [20]. The channel gains between S-D, S-R 1 , S-R 2 , R 1 -D, R 2 -D and respectively, are modeled as independent lognormal random variables [4]. The lognormal fading model is widely employed in PLC systems, which captures the statistics due to the network topology load mismatch and the signal propagation in network branches [21]. The probability density function (PDF) of these channel gains is given by Parameters μ i and σ 2 i denote the mean and the variance of 10log 10 (h i ), i ∈ {SD, SR 1 , SR 2 , R 1 D, R 2 D, R 1 R 2 }, respectively. We use the notation h i ∼ LN (μ i , σ 2 i ) to denote the above described PDF. Additionally, we consider the impairments due to distance-and frequencydependencies in the attenuation, which are denoted by where f denotes the frequency of communication and d i denotes the distances between the nodes i ∈ {SD, SR 1 , SR 2 , R 1 D, R 2 D, R 1 R 2 }. Let P S denote the total power available at S. Information transmission from S occurs over two time slots, namely the even-and odd-numbered slots. In each time slot, S uses non-orthogonal multiple access (NOMA) to transmit two symbols to either R 1 or R 2 , which will be explained in detail next.

A. ODD TIME SLOTS
During the odd time slot (2t + 1), t = 0, 1, . . . , S transmits the superimposed signal to R 1 , where s 1 (2t+1) and s 3 (2t+1) are the signals intended to R 1 and D, respectively. Parameters β 1 and β 2 are the power allocation coefficients for R 1 and D, respectively, such that β 2 > β 1 ≥ 0 and β 1 + β 2 = 1. In an odd time slot 2t + 1, R 2 also acts as a relay and forwards the decoded signal s 3 to R 1 and D. As will be shown later in (18), R 2 would receive superimposed signals from S and R 1 at the previous even slot 2t. Assuming that signals from S are successfully decoded, signal transmitted by R 2 is given by where P R 2 is the transmit power of R 2 , andŝ 3 (2t) denotes the signal destined for D, which was correctly detected by R 2 . Following this, the received signal at R 1 is given by where n k (2t + 1), k = R 1 , D represents the receiver noise random variables which are modeled by independent Bernoulli-Gaussian random variables with PDF [5], Here, f B (x) denotes the PDF for the background noise, which is the additive white Gaussian noise (AWGN) with zero mean and variance σ 2 B , and f I (x) denotes the PDF for the impulsive component. For the ease of analysis, we let f I (x) to be another independent Gaussian distribution with zero mean and variance σ 2 I . Also, 0 ≤ p < 1 is the probability of the occurrence of the impulsive component. As a consequence, the variance of n k (2t + 1) is given by where K = σ 2 I /σ 2 B is the impulsive noise index. Typically, K ∈ [10, 100] [6]. The received signal at R 1 is obtained by substituting (2) and (3) into (4) as where s 1 (2t + 1) and s 3 (2t + 1) are the desired signals, whilê s 3 (2t + 1) is the interference, which results in a degraded performance. We assume that the indirect link has worse channel conditions than the two direct links. Therefore, the modem R 1 employs successive interference cancellation (SIC) to first decode s 3 (2t + 1), followed by s 1 (2t + 1) by treatingŝ 3 (2t) as interference. Therefore the received signal-to-interference-noise ratio (SINR) at R 1 is given by where ρ S P S /σ 2 , and ρ k P k /σ 2 , k = R 1 , R 2 . Considering the transmission from S to D and R 1 , the SINR at R 1 for the detection of s 3 (2t + 1) from (6) is given by where Similarly, the SINR at D for the transmission from S-D is given by

1) WITHOUT DIRECT LINK TO D
Considering that there is no direct link between S and D (as shown in Fig. 1), the received signal at D is given by For the relayed transmission from R 2 to D, the received SINR at D is given by 2) WITH DIRECT LINK TO D As shown in Fig. 2, let us consider a direct link between S and D. During the time slot (2t+1), t = 0, 1, . . . , S transmits the superimposed signal (2) to both R 1 and D. The received signal at D is given by Substituting (2) and (3) into (12), we get For the relayed transmission from R 2 to D, the received SINR at D is given by Next, we consider the communication in even time slots.

B. EVEN TIME SLOTS
During the even time slot (2t), t = 0, 1, . . . , S transmits the superimposed signal to R 2 , where s 2 (2t) and s 3 (2t) are the signals intended to R 2 and D, respectively. As mentioned earlier, The signal detected by R 2 at this time slot is forwarded to D in the next time slot. That is, R 1 forwards the decoded signal received from the previous odd time slot to R 2 and D, which is The received signals at R 2 is given by The received signal at R 2 is obtained by substituting (16) and (17) into (18) as Similar to the previous case, R 2 employs SIC to first decode s 3 (2t), and then s 2 (2t) by treatings 3 (2t − 1) as interference.
The detected and re-encoded signalŝ 3 (2t) will be forwarded to D at the next odd time slot as in (3). Therefore, the SINR at R 2 is The SINR at R 2 for the transmission from S to D and R 2 , and the detection of s 3 (2t) from (19) is given by where and the SINR at D is equivalent to (9).

1) WITHOUT DIRECT LINK TO D
The received signal at D is given by For the relayed transmission from R 1 to D, the received SINR at D is given by

2) WITH DIRECT LINK TO D
The received signal at D is given by (24) where n k (2t), k = R 2 , D denotes the Bernoulli-Gaussian noise. Similar to the previous case, the received signal at D is obtained by substituting (16) and (17) into (24) as For the relayed transmission from R 1 to D, the received SINR at D is given by

III. SUM RATE ANALYSIS
In this section, we provide an analysis on the sum rate and the overall outage probability for the considered three user PLC network, both without and with direct link to D. We first consider the sum rate in the following subsection.

A. ACHIEVABLE RATES AND SUM RATE
Note that the achievable rate at D is given by where C B,D and C I,D represent the capacities of the corresponding PLC channels in the presence of AWGN only, and a mixture of Gaussian and impulsive noises, respectively. As shown in (6) and (19), S transmits superimposed NOMA signals on the direct links SR 1 , SR 2 and SD, which has an interference term due to the signals decoded at R 2 and R 1 , respectively. We assume that the signal intended to D is perfectly decoded and canceled at R 1 and R 2 . The achievable rate for the direct-link receiver R k is given by where l = R 2 , Obtaining an expression for the average sum rate is difficult. In the following, we consider the scenarios without and with direct links to D separately, and in each case describe a lemma which gives an expression for the corresponding approximate average sum rate. Later in Section V, we show that these approximations are tight.

B. WITHOUT DIRECT LINK TO D
Recall that in this scenario, D receives signals forwarded from R 1 and R 2 at both odd-and even-numbered time slots. Since R 1 and R 2 operate in the decode and forward mode to transmit the signal s 3 in odd and even time slot, the achievable rate from S to D through R 1 is given by [22] Likewise, the achievable data rate for the relayed transmission from the S to D through R 2 is given by Proposition 1: The approximate expressions for C R 1 D , C R 2 D and the sum rate of the three user cooperative NOMA network without a direct link to D are given by (2) .
Proof: See Appendix A.

C. WITH DIRECT LINK TO D
Recall that in this case, D receives the signals from S as well as signals which are forwarded from R 1 and R 2 , at both odd-and even-numbered time slots. Since R 1 operates in the decode and forward mode to transmit the signal s 3 (2t + 1), the achievable rate from S to D through R 1 is given by [22] C SD,R 1 = 1 2 log 2 1 + min γ D,S→R 1 , γ D,S→D s 3 .
Likewise, the achievable data rate for the relayed transmission from the S to D through R 2 is given by Proposition 2: The approximate expressions for C SD,R 1 , C SD,R 2 and the sum rate of the three user cooperative NOMA network with a direct link to D are given by Proof: The proof is similar to the proof of Proposition 1 given in Appendix A, and is skipped for brevity.

D. OPTIMAL POWER ALLOCATION
Now, we discuss an optimization problem to find the optimal power allocation coefficient β 1 ∈ (0, 1) that maximizes the sum rate. The problem is formulated as We subsequently show through simulations in Section V that the problem in (38) is concave in β 1 . Note that finding an expression for the optimal β 1 in the above optimization problem is hard, for both with and without direct link cases. Therefore, we propose the following constrained optimization problem for a simpler case when the distances between the source to both relays are equal, and a minimum rate requirement on each communication link is imposed [23]. The modified optimization problem for the case of without direct link to D is formulated as: Similarly, a corresponding optimization problem can be formulated for the case with direct link to D as well. In both cases, it is interesting to note that the optimal β 1 values are equal. The following proposition discusses the solution to the optimal β 1 for the case of no direct link. The details for the other case are similar and are omitted for brevity. Proposition 3: When d SR 1 = d SR 2 , the optimal value of β 1 which is the solution to the optimization problem in (39) for the case of without direct link to D is given by Proof: See Appendix B.
Note that the solution in (40) corresponds to the scenario where the constraint C SR 1 ≥ r 1 is satisfied with equality, and C R 1 D is maximized. Such a scheme is desirable in practice.

IV. OUTAGE PROBABILITY ANALYSIS
In this subsection, we derive the expressions for outage probabilities at R 1 , R 2 and D. Let the thresholds on SINRs for a given requirements on the data rate at R 1 , R 2 and D for symbols s 1 , s 2 and s 3 be denoted by α 1 , α 2 and α 3 , respectively. First, the outage at R 1 occurs if R 1 fails to decode either of the symbol s 1 or s 3 , which was sent from SR 1 in the odd time slot. Therefore, the outage probability at R 1 is expressed as To derive an expression for (41), we employ the technique discussed in [11], which can be summarized as next. We skip the mathematical details for brevity. Following the properties 1)-3) of a lognormal distribution discussed in Appendix A, it can be shown that the outage at R 1 is the approximate cumulative distribution function (CDF) of another lognormal distribution, which is given by where μ R 1 and σ R 1 are obtained using (53) and (54). Further, outage at R 2 occurs if R 2 fails to decode either of the symbol s 2 or s 3 , following which we get using the same set of arguments as before. In the following, we describe two propositions, which characterize the outage probability at the destination for without and with direct link to D.

Proposition 4:
The outage probability at D for without the direct link is given by (46) in the bottom of the page.
Proof: See Appendix C. Consequently, the overall outage probability for the scenario without a direct link to D is given by As mentioned earlier, we show that the derived expressions are tight through Monte Carlo simulations in Section V.

Proposition 5:
The outage probability at D with a direct link is given by (49) given in the bottom of next page.
Proof: The outage at D occurs when D fails to decode the symbol s 3 from both the links S and R 1 in odd time slot, and S and R 2 in even time slot. The proof is similar to the proof of Proposition 4 given in Appendix C, and is skipped for brevity. In this case, the overall outage probability is given as out P (SD) out . (50)

C. OPTIMAL POWER ALLOCATION
Now, we discuss an optimization problem to find the optimal power allocation coefficient β 1 ∈ (0, 1) that minimizes the overall outage probability. This is an alternative way to calculate β 1 . The problem is formulated as Even in this case, obtaining an expression for the optimal value of β 1 is hard, for both with and without direct link cases. However, we discuss the convexity of the problem (51) in Section V, following which the optimal β 1 can be obtained using numerical evaluation.

V. SIMULATION RESULTS
In this section, we present a study through Monte Carlo simulations to corroborate our analysis on the sum rate and outage probability. The PLC cable attenuation model is considered to be where η = a 0 + a 1 f m is the attenuation factor, a 0 and a 1 are constants determined from measurements as a 0 = 9.4×10 −3 , a 1 = 4.2×10 −7 and f = 30 MHz is the center frequency, m = 0.7 is the exponent of the attenuation factor [12]. Unless specified otherwise, ) m, α 1 = α 2 = α 3 = 2 2r − 1 and r = 2 bps/Hz. Fig. 3 and Fig. 4 show the effect of the Bernoulli-Gaussian noise impulsive component parameter p on the sum rate and overall outage probability, for without and with direct link to D. It is observed that as the value of p increases, the performance of the system (in terms of both sum rate and outage probability) deteriorates, which translates to the inherent impulsive nature present in the noise. Moreover, as expected, the presence of a direct link from S to D improves the sum rate and outage probability. For the rest of the results, we set p = 0, without loss of generality.   Fig. 6 show the sum rate performances obtained due to Monte Carlo simulations and our approximate analysis given in Section III-A, in terms of the SNR for different values of distance d SR 1 and d SD for without direct link to D and with direct link to D, respectively. The near-perfect match between the two in both cases establishes the accuracy of our analysis. Further, as expected, the sum rate increases with SNR and decreases with an increase in d SR 1 and d SD . Moreover, it is shown that the sum rate performance of the considered three-user network without a direct link to D is better than the three-user NOMA benchmark, which we have extended from [18] to a PLC scenario. In the NOMA benchmark scheme, S sends superimposed signal to R 1 in the first time slot, which forwards the signal to D at the second slot. In the third slot, S communicates with R 2 directly. The transmit power and the power allocation coefficients of the S in the NOMA benchmark scheme are optimized and set up similar to the employed scheme. Fig. 7 and Fig. 8 highlights the outage probabilities obtained through Monte Carlo simulations and the approximate analysis discussed in Section IV for the system models considered on Fig. 1 and Fig. 2, respectively. Even in this case, the near-perfect match between the two highlights the accuracy of our analysis. Fig. 9 shows the overall outage probability of the cooperative NOMA scheme without direct link, where as the inter-user channel distance between R 1 and R 2 increases the performance decreases and remains constant after a particular distance. The concave behavior of the sum rate with respect to the power allocation coefficient β 1 is observed in Fig. 10, as discussed in Section III. The optimal power allocation coefficients are chosen using the steepest ascent algorithm.    shows the variation of the optimal β * 1 with respect to parameter μ. Since C sum is directly dependent on the parameter μ, small values of β * 1 maximize the sum rate for larger values of μ. Next, Fig. 13 shows the variation of sum rate for different transmit SNR. It can be observed that there is a slight mismatch between theoretical and simulated sum rates for low values of SNR and low values of μ. This trend is expected, since the Maclaurin series expansion used in the derivation of the sum rate becomes tighter for higher values of μ. Fig. 14 highlights the convexity of the outage probability with respect to β 1 , as discussed in Section IV. Fig. 15 and Fig. 16 compares the sum rate and the overall outage probability of the three-user network for both models, i.e., without direct link to D and with direct link to D. Recall that p = 0. It is observed that the performance of the system with a direct link to D is higher as compared to the system for without direct link to D for the given SNR and d SR 1 , as expected.

Fig. 5 and
Finally, we present a performance comparison of the three-user cooperative NOMA network without a direct link to D with the two-user SS-, DS-NOMA and OMA networks  considered in [11] and [12], [15]. Fig. 17 and Fig. 18 compares the sum rate and the overall outage probability of the considered three-user network without a direct link to D, respectively. It is observed that the considered three-user network outperforms all others schemes in terms of sum rate and overall outage probability. It should be further noted that the performance comparison with a direct link to D is not included, since the performance of that case is better than that of the scenario without the direct link.

VI. CONCLUSIONS
We studied the performance of three-user cooperative NOMA for PLC, with a single source modem, two relays and a single destination modem. We derived closed form expressions for the approximate sum rates and the overall outage probability of the network, in the presence and absence of a direct link between the source and destination. We presented an analysis to find the optimal power allocation co-efficient, such that the sum rate is maximized. Further, we provided insights of the impact of the parameters such as the distance, fading and impulsive noise component on the overall on the  PLC system performance, and established the accuracy of our analysis through Monte Carlo simulations. Our scheme was shown to outperform existing schemes in the literature, in terms of sum rate and outage probability.

APPENDIX A PROOF OF PROPOSITION 1
We start by considering the rate of R 1 . The cumulative distribution function (CDF) of the received SINR at R 1 is given by where, following the properties of a lognormal distribution, it is easy to see that h 2 ); μ l = μ i − μ j , and σ l = σ i + σ j , and 3) H l = H i + H j can be well-approximated by another lognormal distribution, with parameters μ l and σ 2 l , which can be calculated as [24], where the moments E[H −1 l ] and E[H −2 l ] can be calculated either through simulations, or numerically as where f H 1 (h 1 ) and f H 2 (h 2 ) are the PDFs of H i and H j . Using the aforementioned properties, we obtain the CDF of γ R 1 using (52) as where μ R 1 and σ R 1 are evaluated using (53) and (54), respectively. Thus, the average achievable rate at the link SR 1 is given by Changing the variable ln x = t in (56), we get Further, we employ the Maclaurin series expansion of the exponential function inside the integral in (57), and evaluate the integral by truncating the series to the first term. We obtain the expression for the average rate as: Similarly, the average rate of the link S-R 2 is given by where μ R 2 is obtained from (53). Next, we consider the links connecting the D modem. Similar to what is considered in [19], we assume that {min(γ D,S→R 1 , γ D,R 1 →D )} ≈ γ D,S→R 1 to evaluate C R 1 D , and {min(γ D,S→R 2 , γ D,R 2 →D )} ≈ γ D,S→R 2 to evaluate C R 2 D . As a result, expressions of C R 1 D and C R 2 D given in (30) and (31) can be approximated as C SD,R 1 ≈ 1 2 log 2 (1 + γ D,S→R 1 ). (60) 6) 0 < β 1 < 1 2 , 7) λ 1 , λ 2 ≥ 0, where g(β 1 ) = −f (β 1 ) ≤ 0 and λ 1 , λ 2 are the Lagrange multipliers for the constraints C SR 1 ≥ r 1 and C R 1 D ≥ r 3 respectively. For λ 1 > 0 and λ 2 > 0, both KKT conditions 2) and 3) must be satisfied. Setting λ 1 > 0 and λ 2 = 0, the optimal β 1 can be obtained from conditions 1) and 2) as This solution corresponds to maximizing the rate at D, by equating the rate constraint on R 1 . For completeness, setting λ 2 > 0 and λ 1 = 0 gives the optimal value of β 1 following condition 3), which corresponds to maximizing the rate at R 1 , by equating the rate constraint at D.

APPENDIX C PROOF OF PROPOSITION 4
The outage at D occurs when D fails to decode the symbol s 3 from either of the links from R 1 or R 2 . That is, where F k (α) denotes the CDF of the random variable γ k evaluated at a point α ∈ R. Next, to derive the outage probability of D, we need to obtain the CDFs of γ D,S→R 1 and γ D,S→R 2 . To this end, both random variables γ D,S→R 1 and γ D,S→R 2 are approximated as lognormal random variables with parameters obtained by following (53) and (54), given in Appendix A. These parameters are calculated using the properties 1)-3) described in Appendix A. Substituting the corresponding parameters in the CDFs and simplifying gives the required result.