A Spectral Efficient NOMA-based Two-Way Relaying Scheme for Wireless Networks with Two Relays

Abstract

This paper proposes a novel two-way relaying (TWR) approach for a two-relay wireless network based on non-orthogonal multiple access (NOMA), where two terminals exchange messages with a cellular base station (BS) via two intermediate relay stations (RSs). We propose a NOMA-based TWR approach with two relaying schemes, i.e., amplify-and-forward (AF) and decode-and-forward (DF), referred to as NOMA-AF and NOMA-DF. The sum-rate performance of our proposed NOMA-AF and NOMA-DF is analyzed. A closed-form sum-rate upper bound for the NOMA-AF is obtained, and the exact ergodic sum-rate of NOMA-DF is also derived. The asymptotic sum-rate of NOMA-AF and NOMA-DF is also analyzed. Simulation results show that the proposed scheme outperforms conventional orthogonal multiple access based transmission schemes. It is also shown that increasing the transmit power budget of the relays only cannot always improve the sum-rates.

1. Introduction

The non-orthogonal multiple access (NOMA) technique has attracted a lot of research interests recently as a promising solution to meet the increasing demand for low latency and massive connectivity [1-3]. Compared with orthogonal multiple access (OMA), NOMA improves transmission efficiency by allowing multiple wireless users to share the same radio resources via superposition signaling.

Recently, the NOMA technique has been successfully applied in relay-assisted wireless communication systems [4-7]. The authors in [4] proposed a cooperative NOMA approach, where a user with better link quality was selected as a relay to forward the signals. In [5], the authors considered a two-user system that the full-duplex (FD) near user helps the far user, and the outage performance for two NOMA users was analyzed. For networks with dedicated relays, the authors in [6,7] proposed relaying schemes that the relay station and the direct link user forms a NOMA pair, and the relay forwards signal to the indirect link user with decode-and-forward (DF) [6], or amplify-and-forward (AF) [7] relaying. For NOMA-based transmission with multiple relays, a relay selection approach can be employed to obtain spatial diversity gain [8-10]. For instance, based on the global channel information, a two-stage relay selection scheme was proposed in [10], and the outage probability as well as the diversity gain was analyzed.

All these aforementioned NOMA-based relaying schemes considered one-way relaying only. It is well known that two-way relaying (TWR) [11-13] is more efficient than one-way relaying. Hence, it is natural to combine NOMA with TWR technique [14-17]. In [16], the authors considered a NOMA two-way relay network (TWRN) and analyzed the outage performance. In [17], a hybrid two-way relaying scheme was proposed, which combined the NOMA and network coding techniques. The authors in [18] studied the spectral efficiency of NOMA-based TWRN with opportunistic relay selection. It was shown in [18] that the energy efficiency of NOMA-based TWR transmission can be greatly improved. In [19], a full-duplex DF relaying based cooperative NOMA TWR transmission was proposed to improve the system throughput.

For more sophisticated two-way relaying systems, such as multi pair TWRNs [20-22] and multi-way relay networks (MWRNs) [23,24], the system throughput can also be improved by applying NOMA. In [20], the authors considered a NOMA multipair TWRN, in which multiple pairs of users communicate with each other with the help of several relay nodes. A rate splitting scheme with successive decoding was proposed to combat the interference. A cognitive radio inspired NOMA-based network coding scheme was proposed in [21] to improve the spectral efficiency of multipair TWRNs. In [22], the authors designed beamforming matrices for performance optimization in a NOMA-based multi-pair TWRN. In [24], the authors investigated an unmanned aerial vehicle (UAV)-aided NOMA MWRN. Numerical results showed that these proposed approaches [22-24] can significantly improve the sum-rates by reducing the required time slots for information exchange.

Most of these existing works [20-25] considered NOMA transmission with a single relay node only. For systems with multiple relays, a relay selection approach was commonly used such that only one relay was active for each round of transmission [8-10]. In this work, we consider a two-user TWRN with two relay nodes (RNs), as shown in Fig. 1. Unlike the existing works that the users in a NOMA pair are connected to the same relay, the two terminals associated with the two different relays form a NOMA pair. Compared with one-way relaying with one active relay node, the terminals, relays and the BS will suffer from serve inter-stream interference in the considered TWRN, due to the increased number of data streams.

To this end, we design an efficient NOMA-based relaying scheme that only three time slots are required for each round of transmission, as shown in Fig. 1. The proposed scheme can simultaneously provide wireless connectivity to isolated users with no connections to the BS, thereby improving spectral utilization. The achievable sum rates for the proposed scheme with both AF and DF relaying are analyzed. For AF-based NOMA TWR scheme, referred to as NOMA-AF, a closed-form expression for the ergodic sum-rate upper bound is derived. For DF-based NOMA TWR scheme, referred to as NOMA-DF, we derived the exact ergodic sum- rate. We further study the performance of the proposed NOMA-AF and NOMA-DF schemes in the high signal-to-noise ratio (SNR) region. Numerical results demonstrated the accuracy of the analytical results. It is shown that the sum-rate of the schemes is independent of the transmit power of the relays in the high SNR regions, given the transmit powers of the BS and users.

Fig. 1. Two-way relaying based on NOMA

2. System Model

Fig. 1 shows a TWRN consist of one BS, two terminals and two relays. It is assumed that both two terminals are far from the BS and the direct links are unavailable. Letℎ_{𝐵𝑅, 𝑘} be the channel between the BS and relay 𝑅_𝑘, 𝑘=1, 2 , and ℎ_{𝑈𝑅, 𝑘}be the channel between terminal𝑈_𝑘 and relay 𝑅_𝑘. In this work, we assume time division duplex (TDD) that the channels between any two nodes are reciprocal. The channels remain the same during a transmission block, but may vary from block to block independently [4-7]. We also assume that terminal𝑈₁ (or 𝑈₂) is far away from the relay 𝑅₂ (or 𝑅₁), and no direct link exists between them. This is often the case, as the two relays are usually deployed with a large distance apart and buildings between terminal𝑈₁ (or 𝑈₂) and the relay 𝑅₂ (or 𝑅₁) can block their radio propagation to each other.

For conventional TDMA-based transmissions, four time slots are usually needed for the considered TWRN, as two slots are used for each terminal to exchange messages with the BS. In this work, we design a NOMA-based TWR scheme that can reduce the number of required time slots into three. Specifically, in the first time slot, the BS and 𝑈₁send signals to relay 𝑅₁. In the second time slot, the BS and 𝑈₂send signals to relay 𝑅₂ simultaneously, while the two relays broadcast the signals to the BS and the terminalsat the third slot.

2.1 The First Time Slot

The transmit signal of the BS is given by𝑥_{𝐵, 1}=√𝑃_𝐵𝑠_{𝐵, 1} , where 𝑃_𝐵 is the power budget of the BS, and 𝑠_{𝐵, 1} is the symbol to be send to terminal𝑈₁. The transmit signal of 𝑈₁ is given by 𝑥_{𝑈, 1}=√𝑃₁𝑠_{𝑈, 1}, where 𝑃₁ is the transmit power. As mentioned earlier, the BS and terminal𝑈₁ send signals to 𝑅₁ in the first slot. The received signals at 𝑅₁ is given by

𝑦_{𝑅, 1}=ℎ_{𝐵𝑅, 1}𝑥_{𝐵, 1}+ℎ_{𝑈𝑅, 1}𝑥_{𝑈, 1}+𝑛_{𝑅, 1},       (1)

where 𝑛_{𝑅, 1}∼𝐶𝑁(0, 𝜎²)denotes the additive white Gaussian noise (AWGN) at 𝑅₁.

2.2 The Second Time Slot

Both the BS and terminalU₂send signals to 𝑅₂ in the second slot, while terminalU₁ and relay R₂ keep idle. The received signals at relay R₂ is given by

𝑦_{𝑅, 2}=ℎ_{𝐵𝑅, 2}𝑥_{𝐵, 2}+ℎ_{𝑈𝑅, 2}𝑥_{𝑈, 2}+𝑛_{𝑅, 2},       (2)

where x_{B, 2}=√P_Bs_{B, 2} is transmit signal of the BS, x_{U, 2}=√P₂s_{U, 2} is the transmit signal of terminalU₂, and n_{R, 2}∼𝐶𝑁(0, 𝜎²) is the AWGN at relay R₂.

2.3 The Third Time Slot

In the third time slot, let 𝑥_{𝑅, 𝑘} be the transmit signal of relay 𝑅_𝑘 with a power of 𝛼_𝑘𝑃_𝑅, 𝑘= 1, 2, where α₁ and α₂ are power allocation factors with α₁+α₂=1 and 𝑃_𝑅 denotes the total power budget of the two relays. As mentioned before, the two relays broadcast the signals to the BS and the two terminals. The received signals at the BS and the two terminals are respectively given by

\(y_{B}=\sum_{k=1}^{2} h_{B R, k} x_{R, k}+n_{B}\)      (3)

and

𝑦_{𝑈, 𝑘}=ℎ_{𝑈𝑅, 𝑘}𝑥_{𝑅, 𝑘}+𝑛_{𝑈, 𝑘}, 𝑘=1, 2,       (4)

where 𝑛_𝐵∼𝐶𝑁(0, 𝜎²) and 𝑛_{𝑈, 𝑘}∼𝐶𝑁(0, 𝜎²) are AWGNs.

3. AF-based NOMA Two-Way Relaying

3.1 The NOMA-AF Scheme

For NOMA-AF, the transmit signal at𝑅_𝑘 in the third slot is generated by

𝑥_{𝑅, 𝑘}=𝜃_𝑘𝑦_{𝑅, 𝑘}, 𝑘=1, 2,       (5)

where 𝜃_𝑘 is the amplification coefficient. 𝜃_𝑘 can be calculated as

\(\theta_{k}=\sqrt{\frac{\alpha_{k} P_{R}}{\left|h_{B R, k}\right|^{2} P_{B}+\left|h_{U R, k}\right|^{2} P_{k}+\sigma^{2}}}\),       (6)

3.1.1 Uplink of NOMA-AF

Substituting (1), (2), (5) and (6) into (3), 𝑦_𝐵at the BS is given by

𝑦_𝐵=∑² _𝑘=1𝜃_𝑘ℎ_{𝐵𝑅, 𝑘}(ℎ_{𝐵𝑅, 𝑘}𝑥_{𝐵, 𝑘}+ℎ_{𝑈𝑅, 𝑘}𝑥_{𝑈, 𝑘}+𝑛_{𝑅, 𝑘})+𝑛_𝐵.       (7)

Note that the signal related to 𝑥_{𝐵, 𝑘} in the right hand side (RHS) of (7) is the self-interference known by the BS, and can be removed completely. After removing the self-interference, the signals at the BS can be expressed by

\(\begin{aligned} \tilde{y}_{B} &=y_{B}-\sum_{k=1}^{2} \theta_{k} h_{B R, k}^{2} x_{B, k} \\ &=\theta_{1} h_{B R, 1} h_{U R, 1} x_{U, 1}+\theta_{2} h_{B R, 2} h_{U R, 2} x_{U, 2}+\tilde{n}_{B} \end{aligned}\),       (8)

where 𝑛̃_𝐵 = ∑²_𝑘=1 𝜃_𝑘ℎ_𝐵𝑅,_𝑘𝑛_𝑅,𝑘 + 𝑛_𝐵 is the total noise with a variance of 𝜎̃_𝐵² = ∑²_𝑘=1 |𝜃_𝑘ℎ_{𝐵𝑅,𝑘}|²𝜎² + 𝜎² . Assuming that 𝛼₁>𝛼₂, i.e., the transmit power of 𝑅₁ is higher than 𝑅₂, the BS detects the signal 𝑠_{𝑈, 1} first by treating 𝑠_{𝑈, 2} as noise. Then, 𝑠_{𝑈, 1}will be subtracted from 𝑦̃_𝐵to detect 𝑠_{𝑈, 2}. The achievable uplink rates of the two terminals are respectively given by

\(R_{U L, 1}^{A F}=\frac{1}{3} \log _{2}\left(1+\frac{\left|\theta_{1} h_{U R, 1} h_{B R, 1}\right|^{2} P_{1}}{\left|\theta_{2} h_{U R, 2} h_{B R, 2}\right|^{2} P_{2}+\tilde{\sigma}_{B}^{2}}\right),\)       (9)

and

\(R_{U L, 2}^{A F}=\frac{1}{3} \log _{2}\left(1+\frac{\left|\theta_{2} h_{U R, 2} h_{B R, 2}\right|^{2} P_{2}}{\widetilde{\sigma}_{B}^{2}}\right)\).       (10)

3.1.2 Downlink of NOMA-AF

From (1), (2), (5) and (6), the received signal at terminal𝑈_𝑘is

𝑦_{𝑈, 𝑘}=𝜃_𝑘ℎ_{𝑈𝑅, 𝑘}(ℎ_{𝐵𝑅, 𝑘}𝑥_{𝐵, 𝑘}+ℎ_{𝑈𝑅, 𝑘}𝑥_{𝑈, 𝑘}+𝑛_{𝑅, 𝑘})+𝑛_{𝑈, 𝑘},       (11)

Note that the signal 𝑥_{𝑈, 𝑘} in the RHS of (11) is the self-interference, which can be removed completely. After removing the self-interference, 𝑈_𝑘 is able to detect the desired signal directly. The achievable downlink rate of 𝑈_𝑘 is

\(R_{D L, k}^{A F}=\frac{1}{3} \log _{2}\left(1+\frac{\left|\theta_{k} h_{B R, k} h_{U R, k}\right|^{2} P_{B}}{\left|\theta_{k} h_{U R, k}\right|^{2} \sigma^{2}+\sigma^{2}}\right)\).       (12)

Remark 1: In the NOMA-AF scheme, SIC is employed at the BS only. The signal detection complexity at the terminals and relays are the same as in conventional TDMA-based transmission scheme.

3.2 Sum-Rate of NOMA-AF

3.2.1 Sum-Rate Upperbound of NOMA-AF

Define 𝜌_𝑘=𝑃_𝑘/𝜎², 𝑘=1, 2, 𝜌_𝑅=𝑃_𝑅/𝜎², and 𝜌_𝐵=𝑃_𝐵/𝜎². Let 𝐺_{𝐵𝑅, 𝑘}=𝐸(|ℎ_{𝐵𝑅, 𝑘}|²) denote the average gain of the channel between the BS and 𝑅_𝑘, 𝑘=1, 2, and 𝐺_{𝑈𝑅, 𝑘}= 𝐸(|ℎ_{𝑈𝑅, 𝑘}|²)be the average channel gain between relay 𝑅_𝑘 and terminal𝑈_𝑘, 𝑘=1, 2. By averaging over all possible channel realizations, we have the following result on the ergodic sum-rate of the proposed NOMA-AF scheme.

Proposition 1: The ergodic sum-rate of NOMA-AF is upper bounded by:

\(\begin{array}{l} \tilde{R}_{\text {Sum }}^{A F, U B}=\sum_{k=1}^{2} \frac{\log _{2}(e)}{3} \Phi\left(\frac{1}{a_{k} \rho_{R} G_{U R, k}}+\frac{1}{\mu_{k} \rho_{B} G_{B R, k}}\right) \\ +\frac{\log _{2}(e)}{3\left(1-c_{1}\right)}\left(\Phi\left(\frac{1}{\alpha_{1} \mu_{3} \rho_{1} G_{U R, 1}}\right)-\Phi\left(\frac{1}{\alpha_{1} \mu_{3} \rho_{1} G_{U R, 1} c_{1}}\right)\right) \\ +\frac{\log _{2}(e)}{3} \Phi\left(\frac{1}{\mu_{4} \rho_{2} G_{U R, 2}}+\frac{1}{\mu_{4} \rho_{B} G_{B R, 2}}\right) \end{array}\) ,      (13)

where Φ(𝑥)≜𝑒^x𝐸₁(𝑥), 𝐸₁(𝑥) is the exponential integral function, 𝜇_𝑘=𝛼_𝑘𝜌_𝑅/(𝛼_𝑘𝜌_𝑅+ 𝜌_𝑘), 𝑘=1, 2, 𝜇₃=𝜌_𝑅/(𝜌_𝐵+𝜌_𝑅), 𝜇₄=𝛼₂𝜌_𝑅/(𝜌_𝐵+𝜌_𝑅), and 𝑐₁=𝛼₂𝜌₂𝐺_{𝑈𝑅, 2}/𝛼₁𝜌₁𝐺_{𝑈𝑅, 1}.

Proof: See Appendix A.

3.2.2 Asymptotic Sum-Rate Upper bound of NOMA-AF

As will be shown later, the derived upper bound in (13) is tight. In the following, we further investigate the asymptotic sum-rate performance of NOMA-AF in the high power (SNR) range. Note that 𝐸₁(𝑥)≃−𝐶₀−ln(𝑥)and 𝑒^𝑥≃1as𝑥→0, where 𝐶₀=0.5772... is the Euler constant [26] and (1, 5, 11). We have

Φ(𝑥)=𝑒^𝑥𝐸₁(𝑥)≃−𝐶₀−ln(𝑥), 𝑥→0.       (14)

As a result, the asymptotic sum-rate upper bound (13) for NOMA-AF can be expressed as

\(\begin{array}{l} \tilde{R}_{\text {sum }}^{A F, U B} \simeq \frac{1}{3\left(1-c_{1}\right)} \log _{2}\left(\frac{1}{c_{1}}\right)-\log _{2}(e) C_{0} \\ -\sum_{k=1}^{2} \frac{1}{3} \log _{2}\left(\frac{1}{\alpha_{k} \rho_{R} G_{U R, k}}+\frac{1}{\mu_{k} \rho_{B} G_{B R, k}}\right) \\ -\frac{1}{3} \log _{2}\left(\frac{1}{\mu_{4} \rho_{2} G_{U R, 2}}+\frac{1}{\mu_{4} \rho_{B} G_{B R, 2}}\right) . \end{array}\)       (15)

Remark 2: From (15), the sum-rate upper bound of NOMA-AF depends on the power allocation factors 𝛼₁ and 𝛼₂.

Remark 3: When the powers of the terminals and the BS are fixed, i.e., 𝑃_𝐵, 𝑃₁, and 𝑃₂ are fixed, as 𝑃_𝑅→+∞, we have 𝜇_𝑘=𝛼_𝑘𝜌_𝑅/(𝛼_𝑘𝜌_𝑅+𝜌_𝑘)≃1, 𝑘=1, 2, 𝜇₃=𝜌_𝑅/(𝜌_𝐵+ 𝜌_𝑅)≃1, and 𝜇₄≃𝛼₂. Then, the asymptotic sum-rate upper bound in (15) can be written as which is a constant independent of 𝑃_𝑅. This suggests that increasing the power budget of the relays only can not always improve the sum-rate of NOMA-AF.

4. DF-based NOMA Two-Way Relaying

4.1 The NOMA-DF Scheme

In NOMA-DF, the two relays first decode the received signal with SIC. Take the decoding operation at relay 𝑅₁ as an example. Since the BS’s transmit power is usually higher than the terminal𝑈₁, 𝑅₁ detects 𝑠_{𝐵, 1} first by treating 𝑠_{𝑈, 1} as noise. Then, 𝑠_{𝐵, 1} is removed from 𝑦_{𝑅, 1} to detect 𝑠_{𝑈, 1}. The detection process of relay 𝑅₂ is the same as that of 𝑅₁.

From (1) and (2), the achievable uplink rate from terminal𝑈_𝑘 to the relay 𝑅_𝑘 is

\(R_{U L, k}^{U_{k}, R_{k}}=\frac{1}{3} \log _{2}\left(1+\frac{\left|h_{U R, k}\right|^{2} P_{k}}{\sigma^{2}}\right)\),       (17)

provided that 𝑠_{𝐵, 𝑘}was successfully decoded, i.e.

\(R_{D L, k}^{B, R_{k}} \leq \frac{1}{3} \log _{2}\left(1+\frac{\left|h_{B R, k}\right|^{2} P_{B}}{\left|h_{U R, k}\right|^{2} P_{k}+\sigma^{2}}\right)\),       (18)

After decoding, the relay 𝑅_𝑘 broadcasts signal 𝑥_{𝑅, 𝑘}=√𝛼_𝑘𝑃_𝑅𝑠_{𝑅, 𝑘} to the BS and 𝑈_𝑘 in the third slot, where 𝑠_{𝑅, 𝑘}=𝑠_{𝐵, 𝑘}⊕ 𝑠_{𝑈, 𝑘} is the XOR encoded signal.

4.1.1 Uplink of NOMA-DF

In the third slot, the signal 𝑦_𝐵 at the BS for NOMA-DF is

𝑦_𝐵=∑² _𝑘=1ℎ_{𝐵𝑅, 𝑘}√𝛼_𝑘𝑃_𝑅𝑠_{𝑅, 𝑘}+𝑛_𝐵.       (19)

As in the NOMA-AF scheme, the BS detects 𝑠_{𝑅, 1} first, then 𝑠_{𝑅, 1} is removed to detect 𝑠_{𝑅, 2}. The data rate for the link from 𝑅₂ to the BS given by

\(R_{U L, 2}^{R_{2}, B}=\frac{1}{3} \log _{2}\left(1+\frac{\alpha_{2} P_{R}\left|h_{B R, 2}\right|^{2}}{\sigma^{2}}\right)\),       (20)

under the constraint that the rate from 𝑅₁ to the BS satisfies

\(R_{U L, 1}^{R_{1}, B} \leq \frac{1}{3} \log _{2}\left(1+\frac{\alpha_{1} P_{R}\left|h_{B R, 1}\right|^{2}}{\alpha_{2} P_{R}\left|h_{B R, 2}\right|^{2}+\sigma^{2}}\right)\).       (21)

4.1.2 Downlink of NOMA-DF

For the down link transmission from 𝑅_𝑘 to terminal 𝑈_𝑘, from (4), we have

\(R_{D L, k}^{R_{k}, U_{k}}=\frac{1}{3} \log _{2}\left(1+\alpha_{k} P_{R}\left|h_{U R, k}\right|^{2} / \sigma^{2}\right), \quad k=1,2\).       (22)

Now consider the achievable rates for the proposed NOMA-DF scheme. From (17) and (21), the uplink rate of terminal 𝑈₁ is given by

\(R_{U L, 1}^{D F}=\frac{1}{3} \log _{2}\left(1+\gamma_{U L, 1}^{D F}\right)\)\(R_{U L, 1}^{D F}=\frac{1}{3} \log _{2}\left(1+\gamma_{U L, 1}^{D F}\right)\).       (23)

where

\(\gamma_{U L, 1}^{D F}=\min \left(P_{1}\left|h_{U R, 1}\right|^{2} / \sigma^{2}, \frac{\alpha_{1} P_{R}\left|h_{B R, 1}\right|^{2}}{\alpha_{2} P_{R}\left|h_{B R, 2}\right|^{2}+\sigma^{2}}\right)\),       (24)

Similarly, from (17) and (20), the uplink rate of 𝑈₂ is given by

\(R_{U L, 2}^{D F}=\frac{1}{3} \log _{2}\left(1+\gamma_{U L, 2}^{D F}\right)\),       (25)

where

\(\gamma_{U L, 2}^{D F}=\min \left(P_{2}\left|h_{U R, 2}\right|^{2} / \sigma^{2}, \alpha_{2} P_{R}\left|h_{B R, 2}\right|^{2} / \sigma^{2}\right) .\)       (26)

For downlink transmission, from (18) and (22), we have

\(R_{D L, k}^{D F}=\frac{1}{3} \log _{2}\left(1+\gamma_{D L, k}^{D F}\right)\),       (27)

where

\(\gamma_{D L, k}^{D F}=\min \left(\frac{\left|h_{B R, k}\right|^{2} P_{B}}{\left|h_{U R, k}\right|^{2} P_{k}+\sigma^{2}}, \frac{\alpha_{k} P_{R}\left|h_{U R, k}\right|^{2}}{\sigma^{2}}\right)\).       (28)

Remark 4: In NOMA-DF, SIC is employed at the BS and the relays. The signal detection at the two terminals is as low as in TDMA-based schemes.

4.2 Sum-Rate Performance of NOMA-DF

4.2.1 The Exact Sum-Rate of NOMA-DF

Proposition 2: By averaging over all possible channel realizations, the ergodic sum-rate of the proposed NOMA-DF scheme for the TWRN is given by:

\(\begin{array}{l} \tilde{R}_{\text {Sum }}^{D F}=\frac{\log _{2}(e)}{3\left(1-c_{2}\right)}\left(\Phi\left(\psi_{1}\right)-\Phi\left(\frac{\psi_{1}}{c_{2}}\right)\right)+\frac{\log _{2}(e)}{3} \Phi\left(\psi_{2}\right) \\ +\sum_{k=1}^{2} \frac{\log _{2}(e)}{3\left(1-d_{k}\right)}\left(\Phi\left(\eta_{k}\right)-\Phi\left(\frac{\eta_{k}}{d_{k}}\right)\right) \end{array}\),       (29)

\(\begin{array}{l} \text { where } c_{2}=\alpha_{2} G_{B R, 2} / \alpha_{1} G_{B R, 1}, \psi_{k}=\frac{1}{\alpha_{k} \rho_{R} G_{B R, k}}+\frac{1}{\rho_{k} G_{U R, k}}, \eta_{k}=\frac{1}{\rho_{B} G_{B R, k}}+\frac{1}{\alpha_{k} \rho_{R} G_{U R, k}}, \text { and } d_{k}= \\ \rho_{k} G_{U R, k} / \rho_{B} G_{B R, k}, k=1,2 \end{array}\)

Proof: See Appendix B.

4.2.2 The Asymptotic Sum-Rate of NOMA-DF

Consider the asymptotic performance in the high SNR region. Using the result in (14), the asymptotic sum-rate of NOMA-DF is

\(\begin{aligned} \tilde{R}_{\text {Sum, asym }}^{D F} &=\frac{\log _{2}(e)}{3\left(1-c_{2}\right)} \ln \left(\frac{1}{c_{2}}\right)-\frac{\log _{2}(e)}{3}\left(C_{0}+\ln \left(\psi_{2}\right)\right) \\ &-\sum_{k=1}^{2} \frac{\log _{2}(e)}{3\left(1-d_{k}\right)}\left(C_{0}+\ln \left(\eta_{k}\right)+\Phi\left(\frac{\eta_{k}}{d_{k}}\right)\right) . \end{aligned}\)      (30)

Remark 5: From (30), the sum-rate of NOMA-DF also depends on the power allocation factors 𝛼₁ and 𝛼₂, as in NOMA-AF.

Remark 6: When the powers of the terminals and the BS are fixed, i.e., 𝑃_𝐵, 𝑃₁, and 𝑃₂ are fixed, as 𝑃_𝑅→+∞, we have 𝜓_𝑘≃ \(\frac{1}{\rho_{k} G_{U R, k}}\), and 𝜂_𝑘≃\(\frac{1}{\rho_{B} G_{B R, k}}\), 𝑘 = 1,2. Then, the asymptotic sum-rate of NOMA-DF is given by

\(\begin{array}{l} \tilde{R}_{\text {Sum,asym }}^{D F}=\frac{\log _{2}(e)}{3\left(1-c_{2}\right)} \ln \left(\frac{1}{c_{2}}\right)-\frac{\log _{2}(e)}{3} C_{0} \\ -\frac{\log _{2}(e)}{3} \ln \left(\frac{1}{\rho_{2} G_{U R, 2}}\right)-\sum_{k=1}^{2} \frac{\log _{2}(e)}{3\left(1-d_{k}\right)} C_{0} \\ -\sum_{k=1}^{2} \frac{\log _{2}(e)}{3\left(1-d_{k}\right)}\left(\ln \left(\frac{1}{\rho_{B} G_{B R, k}}\right)+\Phi\left(\frac{1}{d_{k} \rho_{B} G_{B R, k}}\right)\right) \end{array}\).       (31)

From (31), we conclude that the asymptotic sum-rate of NOMA-DF approaches to a constant as in NOMA-AF scheme in this case.

5. Simulation Results

During the simulations, the channels are independent Rayleigh distributed, and the average channel gains are 𝐺_{𝐵𝑅, 1}=𝐺_{𝐵𝑅, 2}=0 dB, 𝐺_{𝑈𝑅, 1}=−5 dB, and 𝐺_{𝑈𝑅, 2}=−10 dB. Unless otherwise specified, the power allocation factors are chosen as 𝛼₁=0.9, and 𝛼₂=0.1. The powers of the two terminals are the same, i.e., 𝑃₁=𝑃₂=𝑃. The noise variance is 𝜎²=0 dBm. The transmit powers for the BS and the two relays are given in the following.

In Fig. 2, we plot the sum-rates of the proposed NOMA-AF and NOMA-DF schemes. The SNR in the figure is defined as 𝑆𝑁𝑅=𝑃/𝜎². The powerof the BS is 30 dB higher than that of the terminals, while the total power of the relays is 20 dB higher. For NOMA-AF, the sum- rate upper bound (13) and the asymptotic upper bound (15) are also shown in the figure. While for NOMA-DF, we plot the analytical ergodic sum-rate (29) and the asymptotic sum-rate (30). From the figure, we see that the derived upper bound is tight across the whole SNR region for NOMA-AF, and the asymptotic upper bound for NOMA-AF is tight. For NOMA-DF, the analyzed exact and asymptotic ergodic sum-rates are also accurate. It can also be seen that a much higher sum-rate can be achieved for NOMA-DF as compared with NOMA-AF in the whole SNR range.

Fig. 2. Sum-rate performance of the NOMA-AF and NOMA-DF schemes

Fig. 3 compares the performance of the proposed schemes with conventional TDMA-based TWR transmissions. While Fig. 4 shows the individual terminal rates of these schemes. Note that in TDMA-based AF and DF relaying schemes (labeled as “TDMA-AF" and “TDMA-DF" in the figure), four time slots are required for information exchange between the two terminals and the BS. From Fig. 3, we see that much higher sum-rates can be achieved by the NOMA- AF and NOMA-DF schemes as compared with TDMA-AF and TDMA-DF. For instance, a sum-rate gain of 30% is observed for NOMA-AF as compared with TDMA-AF scheme when the SNR is above 10 dB. For the individual rates shown in Fig. 4, we see that terminal𝑈₁ achieves a higher data rate for the proposed schemes in the whole SNR region. While for terminal𝑈₂, the proposed schemes outperform TDMA-based schemes when the SNR is above 15 dB.

Fig. 3. Sum-rates performance comparison between NOMA and TDMA-based schemes

Fig. 4. Individual rates comparison between NOMA and TDMA-based schemes

Next, Fig. 5 investigates the impact of the relay’s power budget on NOMA-AF and NOMA- DF. Here, the transmit power of the BS is fixed to be 𝑃_𝐵=40 dBm, and the powers of the two terminals are 𝑃₁=𝑃₂=10 dBm, respectively. From the figure, we see that the derived upper bound for NOMA-AF is tight, and the analyzed sum-rate for NOMA-DF is accurate. When the power budget of the relays is smaller than 50 dBm, the achievable sum-rates of both schemes increase with the relay’s transmit power. However, the sum-rates approach to certain constants in the high power region for both NOMA-AF and NOMA-DF, which agrees with the analytical results in Remarks 3 and 6.

Fig. 5. Sum-rates under different relay power budgets

Similar results can be observed in Fig. 6, which shows the performance of NOMA-AF and NOMA-DF with different BS power budgets. The powers of the relay stations and the two terminals are fixed to be 𝑃_𝑅=30 dBm and 𝑃₁=𝑃₂=10 dBm, respectively. The NOMA-DF scheme outperforms NOMA-AF when 𝑃_𝐵≥25 dBm. It can be seen that there is an optimal BS power value to maximize the sum-rates of NOMA-AF. This is due to the fact that the downlink rate of NOMA-AF increases with the transmit power of the BS, while the uplink rate decreases with 𝑃_𝐵 for lowtomoderate BS power budgets, see (9, 10, 12). While for high BS power budgets, both the uplink and downlink rates approach to certain constants. For NOMA-DF, the achievable downlink rate increases while the uplink data rates remain constant for low to moderate BS power budgets. Hence, the sum-rate of NOMA-DF increases with the transmit power of the BS when 𝑃_𝐵<40 dBm.The sum-rates of both schemes approach to certain constants in the high BS power region.

Fig. 6. Sum-rates under different BS power budgets

Finally, Fig. 7 investigates the sum-rates under different power allocation factors 𝛼₁. In the simulations, we set𝑃_𝐵=40 dBmand 𝑃₁=𝑃₂=10 dBm. While the total power of the two relays is set to be 𝑃_𝑅=15 or 30 dBm. From the figure, we see that when 𝑃_𝑅=15 dBm, the achievable sum-rates of both NOMA-AF and NOMA-DF remain almost unchanged. However, when 𝑃_𝑅=30 dBm, we see that the sum-rates of both schemes depend on 𝛼₁. The individual rates of the proposed schemes with different 𝛼₁ are shown in Fig. 8, where 𝑃_𝐵=40 dBm, 𝑃_𝑅=30 dBm, and 𝑃₁=𝑃₂=10 dBm. From the figure, it is clear that as 𝛼₁ becomes large, more power is allocated to relay 𝑅₁, which results in a higher rate for terminal 𝑈₁, while the data rate of terminal𝑈₂ decreases.

Fig. 7. Sum-rates under different power allocation factors (𝛼₁), 𝑃_𝐵=40 dBm and 𝑃₁=𝑃₂=10 dBm

Fig. 8. Individual rates under different power allocation factors (𝛼₁), 𝑃_𝐵=40dBm, 𝑃₁=𝑃₂=10 dBm, and 𝑃_𝑅=30 dBm

6. Conclusion

This paper designed efficient NOMA-based TWR schemes for a two-terminal two-relay system. With the proposed NOMA-AF and NOMA-DF schemes, the BS and the terminals are able to exchange messages within three time slots. We analyzed the sum-rates of NOMA-AF and NOMA-DF, and derived a closed-form upper bound for NOMA-AF, and the exact ergodic sum-rate of NOMA-DF. We further investigated the asymptotic sum-rates of NOMA-AF and NOMA-DF in the high SNR region. We reveal that the sum-rates of the proposed NOMA- based TWR schemes are independent of the transmit power of the relays in the high SNR region. Simulation results showed that NOMA-AF and NOMA-DF outperform the existing TDMA-based schemes significantly.

Appendix A: Proof of Proposition 1

For the downlink transmission of 𝑈₁, the received SINR at terminal 𝑈_𝑘, 𝑘=1, 2, can be expressed as

\(\begin{aligned} \gamma_{D L, k}^{A F} &=\frac{\alpha_{k}\left|\theta_{k} h_{B R, k} h_{U R, k}\right|^{2} \rho_{B}}{\left|\theta_{k} h_{U R, k}\right|^{2}+1} \\ &=\frac{\alpha_{k} \rho_{R} \rho_{B}\left|h_{B R, k}\right|^{2}\left|h_{U R, k}\right|^{2}}{\left(\rho_{k}+\alpha_{k} \rho_{R}\right)\left|h_{U R, k}\right|^{2}+\rho_{B}\left|h_{B R, k}\right|^{2}+1} \end{aligned}\),       (32)

which can be upper bounded by

\(\gamma_{D L, k}^{A F} \leq \gamma_{D L, k}^{A F, U B}=\mu_{k} \min \left(\left(\alpha_{k} \rho_{R}+\rho_{k}\right)\left|h_{U R, k}\right|^{2}, \rho_{B}\left|h_{B R, k}\right|^{2}\right)\)\(\gamma_{D L, k}^{A F} \leq \gamma_{D L, k}^{A F, U B}=\mu_{k} \min \left(\left(\alpha_{k} \rho_{R}+\rho_{k}\right)\left|h_{U R, k}\right|^{2}, \rho_{B}\left|h_{B R, k}\right|^{2}\right)\),       (33)

where 𝜇_𝑘=𝛼_𝑘𝜌_𝑅/(𝛼_𝑘𝜌_𝑅+𝜌_𝑘), 𝑘=1, 2.

The CDF of the SINR upper bound \(\gamma_{D L, k}^{A F, U B}\) is

\(F_{\gamma_{D L, k}^{A F, U B}}(x)=1-e^{-\frac{x}{a_{k} \rho_{R} G_{U R, k}}} e^{\frac{x}{\mu_{k} \rho_{B} G_{B R, k}}}\).       (34)

As a result, the downlink rate upper bound of terminal 𝑈_𝑘 can be calculated as

\(\begin{aligned} \tilde{R}_{D L, k}^{A F, U B} &=\int_{0}^{+\infty} \frac{1}{3} \log _{2}(1+x) f_{\gamma_{D L, k}^{A F, U B}}(x) d x \\ &=\frac{\log _{2}(e)}{3} \int_{0}^{+\infty} \frac{{ }^{1-F}{ }_{\nu L, k}^{A F, U B}(x)}{1+x} d x \\ &=\frac{\log _{2}(e)}{3} \Phi\left(\frac{1}{\alpha_{k} \rho_{R} G_{U R, k}}+\frac{1}{\mu_{k} \rho_{B} G_{B R, k}}\right), \end{aligned}\)      (35)

where Φ(𝑥)=𝑒^𝑥𝐸₁(𝑥), 𝐸₁(𝑥) is the exponential integral function, and the last step is due to ∫ 𝑒^−𝑎𝑥/(𝑏+𝑥)𝑑𝑥=𝑒^𝑎𝑏𝐸₁(𝑎𝑏) as in [26] and (1, 5, 28).

Now consider the uplink transmission rate of terminal𝑈₁. Note that in the second step we have used the approximation that \(\left|\theta_{k} h_{B R, k}\right|^{2} \simeq \frac{\alpha_{k} P_{R}}{P_{B}} \text { as } P_{B} \gg P_{k}, k=1,2\) holds for practical systems. Then, the received SINR for 𝑈₁ at the BS can be expressed as

\(\begin{aligned} \gamma_{U L, 1}^{A F} &=\frac{\left|\theta_{1} h_{U R, 1} h_{B R, 1}\right|^{2} P_{1}}{\left|\theta_{2} h_{U R, 2} h_{B R, 2}\right|^{2} P_{2}+\tilde{\sigma}_{B}^{2}} \\ &=\frac{\alpha_{1} \mu_{3} \rho_{1}\left|h_{U R, 1}\right|^{2}}{\alpha_{2} \mu_{3} \rho_{2}\left|h_{U R, 2}\right|^{2}+1} \end{aligned}\)       (36)

where 𝜇₃=𝜌_𝑅/(𝜌_𝐵+𝜌_𝑅). The CDF of \(\gamma_{U L, 1}^{A F}\) is

\(F_{\gamma_{U L, 1}^{A F}}(x)=1-\frac{1}{1+c_{1} x} e^{-\frac{x}{\alpha_{1} \mu_{3} \rho_{1} G_{U R, 1}}}\)       (37)

where 𝑐1=𝛼2𝜌₂𝐺𝑈𝑅, 2/𝛼1𝜌₁𝐺𝑈𝑅, 1. Hence, the uplink rate of terminal 𝑈1 is given by

\(\begin{array}{l} \tilde{R}_{U L, 1}^{A F}=\int_{0}^{+\infty} \frac{1}{3} \log _{2}(1+x) f_{\gamma_{U L, 1}^{A F}(x) d x} \\ =\frac{\log _{2}(e)}{3} \int_{0}^{+\infty} \frac{1}{(1+x)\left(1+c_{1} x\right)} e^{-\frac{x}{\alpha_{1} \mu_{3} \rho_{1} G_{U R, 1}}} d x \\ =\frac{\log _{2}(e)}{3\left(1-c_{1}\right)}\left(\Phi\left(\frac{1}{\alpha_{1} \mu_{3} \rho_{1} G_{U R, 1}}\right)-\Phi\left(\frac{1}{\alpha_{1} \mu_{3} \rho_{1} G_{U R, 1} c_{1}}\right)\right) \end{array}\).       (38)

For the uplink transmission of 𝑈₂, the received SINR can be expressed as

\(\begin{aligned} \gamma_{U L, 2}^{A F} & \simeq \frac{\left|\theta_{2} h_{U R, 2} h_{B R, 2}\right|^{2} \rho_{2}}{1+\frac{P_{R}}{P_{B}}} \\ &=\mu_{4} \frac{\rho_{2} \rho_{B}\left|h_{U R, 2}\right|^{2}\left|h_{B R, 2}\right|^{2}}{\rho_{2}\left|h_{U R, 2}\right|^{2}+\rho_{B}\left|h_{B R, 2}\right|^{2}+1} \end{aligned}\),       (39)

where 𝜇₄=𝛼2𝜌_𝑅/(𝜌_𝐵+𝜌_𝑅) .

Using the inequality that \(\frac{X Y}{X+Y+1}\)≤min(𝑋, 𝑌) for any non-negative variables 𝑋 and 𝑌, the uplink received SINR of terminal𝑈₂ is upper bound by

\(\gamma_{U L, 2}^{A F} \leq \gamma_{U L, 2}^{A F, U B}=\mu_{4} \min \left(\rho_{2}\left|h_{U R, 2}\right|^{2}, \rho_{B}\left|h_{B R, 2}\right|^{2}\right)\),       (40)

whose CDF is given by

\(F_{\gamma_{U L, 2}^{A F, U B}}(x)=1-e^{-\frac{x}{\mu_{4} \rho_{2} G_{U R, 2}}} e^{-\frac{x}{\mu_{4} \rho_{B} G_{B R, 2}}}\).       (41)

Then, the upper bound for the data rate of 𝑈₂ in the uplink is

\(\begin{array}{l} \tilde{R}_{U L, 2}^{A F, U B}=\int_{0}^{+\infty} \frac{1}{3} \log _{2}(1+x) f_{\gamma_{U L, 2}^{A F, U B}}(x) d x \\ =\frac{\log _{2}(e)}{3} \int_{0}^{+\infty} \frac{1}{1+x} e^{-\frac{x}{\mu_{4} \rho_{2} G_{U R, 2}}} e^{-\frac{x}{\mu_{4} \rho_{B} G_{B R, 2}}} d x \\ =\frac{\log _{2}(e)}{3} \Phi\left(\frac{1}{\mu_{4} \rho_{2} G_{U R, 2}}+\frac{1}{\mu_{4} \rho_{B} G_{B R, 2}}\right) \end{array}\).       (42)

From (35), (38), and (42), we obtain the sum-rate upper bound in Proposition 1.

Appendix B: Proof of Proposition 2

For terminal𝑈₁, the CDF of 𝛾₁=\(\frac{\left|h_{B R, 1}\right|^{2} \alpha_{1} \rho_{R}}{\left|h_{B R, 2}\right|^{2} \alpha_{2} \rho_{R}+1}\) can be calculated as

\(F_{\gamma_{1}}(x)=1-\frac{1}{1+c_{2} x} e^{-\frac{x}{\alpha_{1} \rho_{R} G_{B R, 1}}}\),       (43)

where 𝑐₂=𝛼₂𝐺_{𝐵𝑅, 2}/𝛼₁𝐺_{𝐵𝑅, 1}. Then, the CDF of 𝛾_𝑈^𝐷_𝐿^𝐹 _{, 1} =min(𝛾₂, 𝛾₁) is

\(F_{\gamma_{U L, 1}^{D F}}(x)=1-\frac{1}{1+c_{2} x} e^{-\frac{x}{\alpha_{1} \rho_{R} G_{B R, 1}}} e^{-\frac{x}{\rho_{1} G_{U R, 1}}}\)       (44)

Based on (44), the uplink data rate of 𝑈₁ is given by

\(\begin{array}{l} \tilde{R}_{U L, 1}^{D F}=\int_{0}^{+\infty} \frac{1}{3} \log _{2}(1+x) f_{\gamma_{U L, 1}^{D F}}(x) d x \\ =\frac{\log _{2}(e)}{3} \int_{0}^{+\infty} \frac{1}{(1+x)\left(1+c_{2} x\right)} e^{-\psi_{1} x} d x \\ =\frac{\log _{2}(e)}{3\left(1-c_{2}\right)}\left(\Phi\left(\psi_{1}\right)-\Phi\left(\frac{\psi_{1}}{c_{2}}\right)\right) \end{array}\)       (45)

where \(\psi_{1}=\frac{1}{\alpha_{1} \rho_{R} G_{B R, 1}}+\frac{1}{\rho_{1} G_{U R, 1}}\)

For terminal 𝑈₂ in the uplink, the CDF of 𝛾_{𝑈𝐿, 2} =min(𝜌₂|ℎ_{𝑈𝑅, 2}|², 𝛼₂𝜌_𝑅|ℎ_{𝐵𝑅, 2}|²) is given by

\(F_{\gamma_{U L, 2}^{D F}}(x)=1-e^{-\frac{x}{\rho_{2} G_{U R, 2}}} e^{-\frac{x}{\alpha_{2} \rho_{R} G_{B R, 2}}}\).       (46)

The uplink data rate of 𝑈₂ is given by

\(\begin{aligned} \tilde{R}_{U L, 2}^{D F} &=\int_{0}^{+\infty} \frac{1}{3} \log _{2}(1+x) f_{\gamma_{U L, 2}^{D F}}(x) d x \\ &=\frac{\log _{2}(e)}{3} \Phi\left(\psi_{2}\right) \end{aligned}\)       (47)

where \(\psi_{2}=\frac{1}{\rho_{2} G_{U R, 2}}+\frac{1}{\alpha_{2} \rho_{R} G_{B R, 2}}\)

For 𝑈_𝑘, 𝑘=1, 2 in the downlink, note that the CDF of \(\frac{\left|h_{B R, k}\right|^{2} \rho_{B}}{\left|h_{U R, k}\right|^{2} \rho_{k}+1}\) is given by

\(F_{k}(x)=1-\frac{1}{1+d_{k} x} e^{-\frac{x}{\rho_{B} G_{B R, k}}}\),       (48)

where =𝜌_𝑘𝐺_{𝑈𝑅, 𝑘}/𝜌_𝐵𝐺_{𝐵𝑅, 𝑘}, 𝑘=1, 2. Hence, the CDF of 𝛾_{𝐷𝐿, 𝑘} is then given by

\(F_{\gamma_{D L, k}^{D F}}(x)=1-\frac{1}{1+d_{k} x} e^{-\eta_{k} x}\),       (49)

where \(\eta_{k}=\frac{1}{\rho_{B} G_{B R, k}}+\frac{1}{\alpha_{k} \rho_{R} G_{U R, k}}\)

The achievable downlink rate of 𝑈_𝑘 is now upper bounded by

\(\begin{array}{l} \tilde{R}_{D L, k}^{D F}=\int_{0}^{+\infty} \frac{1}{3} \log _{2}(1+x) f_{\gamma_{D L, K}^{D F}}(x) d x \\ =\frac{\log _{2}(e)}{3} \int_{0}^{+\infty} \frac{1}{(1+x)\left(1+d_{k} x\right)} e^{-\eta_{k} x} d x \\ =\frac{\log _{2}(e)}{3\left(1-d_{k}\right)}\left(\Phi\left(\eta_{k}\right)-\Phi\left(\frac{\eta_{k}}{d_{k}}\right)\right) \end{array}\).       (50)

Based on (45), (47) and (50), we can obtain the result in Proposition 2.