Keywords

1 Introduction

With the explosive growth of Internet of Things (IoT) devices and user services, future wireless communications network will face major challenges in high spectral efficiency, low latency and massive connectivity [1]. Against this background, a key technology for next-generation cellular communications is NOMA since it allows multiple users to allocate different power domains by sharing the same frequency/time/code resources [2, 3]. Many researchers are dedicated to studying NOMA integrated with existing advanced schemes such as cooperative relaying system (CRS) [4], device-to-device (D2D) [5] and cognitive radio (CR) [6].

Relay technology is widely used in modern wireless communication networks because of its advantages in effectively expanding network coverage and improving communication reliability. Reference [7] studies a downlink full-duplex relay system using NOMA and considers the influence of self-interference factor on the system outage probability. D2D communication has now attracted intensive attention due to it can improve the local spectral efficiency. The key idea of D2D is to allow direct communication between two adjacent devices without a base station (BS) involved [8]. The work in [9] proposes cooperative D2D communication with multiple D2D pairs, and optimal power allocation is obtained. A D2D-aided C-NOMA (DC-NOMA) was presented in [10] to enhance the outage performance of the EU link using NOMA. An appropriate power allocation factor is selected, the ergodic sum capacities of DC-NOMA becomes better than C-NOMA and C-OMA [11].

Considering the multi-user relay system, the diamond-like relay selection model is studied [12]. On the basis of the max–min scheme, performance analysis of two relay modes are carried out: amplify-and-forward (AF) and decode-and-forward (DF). The simulation results show that DF relay performs better than AF relay. In addition, the relay selection scheme can further improve the performance of DC-NOMA. In the cooperative DF relay system using NOMA, PRSS is used in literature [13] and system outage performances are improved. In DC-NOMA system, this letter proposes a TSRS and PRSS, and the outage probability of closed-form expression are derived for each user data information.

2 System Model

It was illustrated in Fig. 1, a DC-NOMA network consisting of a base station (BS), a EU, a CU, and N relays \({\text{R}}_n (n = 1,...,N)\) is considered in this paper. Due to the path loss, there is no direct link between BS and EU. Thus, BS needs send information to EU helped by N DF relays. In return, the relay can directly send itself signal to CU via D2D communication. All channels are Rayleigh fading, the channel coefficients between \({\text{BS}} \to {\text{R}}_n\), \({\text{BS}} \to {\text{CU}}\), \({\text{R}}_n \to {\text{EU}}\) and \({\text{R}}_n \to {\text{CU}}\) are represented as \(h_{sn}\), \(h_{sc}\), \(h_{ne}\) and \(h_{nc}\), variances are expressed as \(\lambda_{ij}\) and zero mean, where \(ij \in \{ sn,sc,ne,nc\}\). Moreover, noise model of receiver is complex additive white Gaussian noise (AWGN).

Fig. 1
figure 1

A DC-NOMA system model

Full size image

In the first phase, BS sends a superimposed signal \(\sqrt {P_s a_1 } x_e + \sqrt {P_s a_2 } x_c\), where \(x_e\) and \(x_c\) denote the signal required by EU and CU, respectively. \(P_s\) denotes the total transmit power of BS, and the \(a_i\) denotes the power allocation coefficient, satisfied \(a_2 < a_1\), where \(a_1 + a_2 = 1\). Therefore, the received signals at CU and the relay are respectively given by

$$y_{s \to n} = h_{sn} (\sqrt {P_s a_1 } x_e + \sqrt {P_s a_2 } x_c ) + n_n$$
(1)
$$y_{s \to c} = h_{sc} (\sqrt {P_s a_1 } x_e + \sqrt {P_s a_2 } x_c ) + n_c$$
(2)

where \(n_c\) and \(n_n\) denote the additive white Gaussian noise with \(\sigma^2\) variance at CU and the relay. According to NOMA principle, the decoding signal-to-interference-plus-noise ratio (SINR) can be expressed by

$$\gamma_{sc,x_e } = \frac{{a_1 \left| {h_{sc} } \right|^2 }}{{a_2 \left| {h_{sc} } \right|^2 + 1/\rho }}$$
(3)

where \(\rho { = }P_s /\sigma^2\) is the transmit signal-to-noise ratio (SNR). Perform SIC on the CU and SNR with signal \(x_c\) written as

$$\gamma_{sc,x_c } = \rho \left| {h_{sc} } \right|^2 a_2$$
(4)

Besides, the side information obtained by the CU through decoding is used.

In the second phase, when the relay forwards the decoded signal, it also sends its own signal to the D2D receiver (i.e., CU). Suppose that an optimal relay \({\text{R}}_n\) is selected from N relays, this part will be discussed in Sect. 3. The DC-NOMA allows \({\text{R}}_n\) applies the superposition coding technique to combine two independent signals \(\sqrt {P_d b_1 } x_e + \sqrt {P_d b_2 } x_d\), where \(x_e\) is intended message to EU and \(x_d\) is a D2D signal intended to CU. What’s more, \(b_1\) and \(b_2\) are the power allocation factor, assumed as \(b_1 = a_1 ,b_2 = a_2\) and \(P_d\) denotes the transmitted power of \({\text{R}}_n\), where \(P_d = \eta P_s\). Thus, the received signals at EU and CU are represented by

$$y_{n \to e} = h_{ne} (\sqrt {P_d b_1 } x_e + \sqrt {P_d b_2 } x_d ) + n_e$$
(5)
$$y_{n \to c} = h_{nc} \left( {\sqrt {P_d b_1 } x_e + \sqrt {P_d b_2 } x_d } \right) + n_c$$
(6)

where \(n_e\) and \(n_c\) denote the AWGN at CU and the relay, respectively.

After evaluating the channel information \(h_{nc}\), use the side information \(x_e\) obtained in SIC to eliminate the interference signal \(h_{nc} \sqrt {P_d b_1 } x_e\) from \(y_{n \to c}\). Therefore, the CU can decode the signal \(x_d\) without interference, and its SINR can be indicated as:

$$\gamma_{nc,x_d } = \rho_d \left| {h_{nc} } \right|^2 b_2$$
(7)

3 Relay Selection Scheme

3.1 Two-Stage Relay Selection

This relay selection scheme is usually divided into two stages. The TSRS scheme proposed in this paper selects the relay that maximizes the quality of data transmitted by the CU under the premise of ensuring the reliable reception of the signal by the EU. In the first stage, in order to ensure the reliable transmission of EU, a relay set of successfully decoded signals is established:

$$S_r = \left\{ {n:1 \le n \le N,\gamma_{sn,x_e } \ge \gamma_E^{th} ,\gamma_{ne,x_e } \ge \gamma_E^{th} } \right\}$$
(8)

where \(\gamma_E^{th} \triangleq 2^{2R_e } - 1\), and \(R_e\) is the target threshold for signal \(x_e\).

It is worth noting that, whether \(x_c\) can be successfully decoded only related to the \({\text{BS}} \to {\text{CU}}\) link. Therefore, the relay selection scheme will not enhance the outage performance of \(x_c\). In other words, it is only necessary to consider maximizing the probability of successful decoding \(x_d\) in the second stage. Select the relay \(n^\ast\) with the best \({\text{R}}_n \to {\text{CU}}\) link channel state information in the relay subset:

$$n^ \ast = \mathop {\arg }\limits_n \max \left\{ {\left| {h_{nc} } \right|^2 ,n \in S_r } \right\}$$
(9)

Next, the outage probability of the TSRS is derived to further analyze the system performance. According to formula (8), it can be seen that EU failed to decode successfully \(x_e\), only when \(\left| {S_r } \right| = 0\). Thus, the outage probability (OP) on the EU can be evinced as:

$$\begin{aligned} {\rm P}_{x_e }^{out,{\rm I}} & { = }{\rm P}\left[ {\left| {S_r } \right| = 0} \right] \\ & { = }\prod_{n = 1}^N {\left( {1 - {\rm P}\left[ {\gamma_{sn,x_e } \ge \gamma_E^{th} ,\gamma_{ne,x_e } \ge \gamma_E^{th} } \right]} \right)} \\ & { = }\prod_{n = 1}^N {\left( {1 - {\rm P}\left[ {\left| {h_{sn} } \right|^2 \ge \frac{{\Gamma_E^{th} }}{\rho }]{\rm P}[\left| {h_{ne} } \right|^2 \ge \frac{{\Gamma_E^{th} }}{\rho_d }} \right]} \right)} \\ & = \left( {1 - e^{ - \frac{\varphi_n }{\rho }} } \right)^N \\ \end{aligned}$$
(10)

where \(\Gamma_E^{th} \triangleq \gamma_E^{th} /(a_1 - \gamma_E^{th} a_2 )\) satisfied as \(a_1 - \gamma_E^{th} a_2 > 0\), and \(\varphi_n \triangleq \frac{{\Gamma_E^{th} }}{{\lambda_{sn} }} + \frac{{\gamma_E^{th} }}{{\lambda_{ne} \eta }}\) (\(\forall n \in N\)).

Since an outage event occurs when CU cannot decode successfully \(x_e\) or \(x_c\), the OP can be derived as

$$\begin{aligned} {\rm P}_{x_c }^{out,{\rm I}} & = {\rm P}\left[ {\gamma_{sc,x_e } < \gamma_E^{th} \cup \gamma_{sc,x_c } < \gamma_C^{th} } \right] \\ & = 1 - {\rm P}\left[ {\gamma_{sc,x_e } \ge \gamma_E^{th} \cap \gamma_{sc,x_c } \ge \gamma_C^{th} } \right] \\ & = 1 - {\rm P}\left[ {\left| {h_{sc} } \right|^2 \ge \frac{{\Gamma_E^{th} }}{\rho } \cap \left| {h_{sc} } \right|^2 \ge \frac{{\gamma_C^{th} }}{\rho a_2 }} \right] \\ & = 1 - e^{ - \frac{1}{{\rho \lambda_{sc} }}\max \left\{ {\Gamma_E^{th} ,\frac{{\gamma_C^{th} }}{a_2 }} \right\}} \\ \end{aligned}$$
(11)

An outage event occurs at CU when: (i) \(\left| {S_r } \right| = 0\), no relay is selected to send D2D signal \(x_d\); and (ii) \(\left| {S_r } \right| > 0\), CU cannot successfully decode \(x_e\) or \(x_d\). Hence, the outage probability of \(x_d\) at CU can be formulated as

$$\begin{aligned} {\rm P}_{x_d }^{out} & { = }{\rm P}\left[ {\left| {S_r } \right| = 0} \right] + {\rm P}\left[ {\left| {S_r } \right| > 0,\gamma_{sc,x_e } < \gamma_E^{th} \cup \gamma_{n^\ast c,x_d } < \gamma_D^{th} } \right] \\ & = {\rm P}\left[ {\left| {S_r } \right| = 0} \right] + \sum_{l = 1}^N {{\rm P}\left[ {\left| {S_r } \right| = l} \right]} \\ & {\rm P}\left[ {\gamma_{sc,x_e } < \gamma_E^{th} \cup \gamma_{n^\ast c,x_d } < \gamma_D^{th} |\left| {S_r } \right| = l} \right] \\ \end{aligned}$$
(12)

Since the channel gain satisfied the exponential distribution, according to the knowledge of probability theory, formula (12) can be divided into the following parts:

$$\begin{aligned} {\rm P}[\left| {S_r } \right| = l] & = \left( {\begin{array}{*{20}c} N \\ l \\ \end{array} } \right)\prod_{n = 1}^{N - l} {(1 - {\rm P}[\gamma_{sn,x_e } \ge \gamma_E^{th} ,\gamma_{ne,x_e } \ge \gamma_E^{th} ])} \\ & \quad \times \prod_{n = N - l + 1}^N {{\rm P}[\gamma_{sn,x_e } \ge \gamma_E^{th} ,\gamma_{ne,x_e } \ge \gamma_E^{th} ]} \\ & = \left( {\begin{array}{*{20}c} N \\ l \\ \end{array} } \right)\left[ {1 - e^{ - \frac{\varphi_n }{\rho }} } \right]^{N - l} e^{ - \frac{\varphi_n }{\rho }l} \\ \end{aligned}$$
(13)
$${\rm P}[(\gamma_{sc,x_e } < \gamma_E^{th} \cup \gamma_{n^\ast c,x_d } < \gamma_D^{th} )|\left| {S_r } \right| = l] = \left[ {1 - e^{ - \frac{{\Gamma_E^{th} }}{{\rho \lambda_{sc} }}} e^{ - \frac{{\gamma_D^{th} }}{{\rho_d a_2 \lambda_{nc} }}} } \right]^l$$
(14)

In summary, the closed-form expression for the outage probability of CU can be obtained as

$$\begin{aligned} {\rm P}_{x_d }^{out,{\rm I}} & {\text{ = }}{\rm P}[\left| {S_r } \right| = 0] + {\rm P}[\left| {S_r } \right| > 0,\gamma _{sc,x_e } < \gamma _E^{th} \cup \gamma _{n^ \ast c,x_d } < \gamma _D^{th} ] \\ & = {\rm P}[\left| {S_r } \right| = 0] + \sum_{l = 1}^N {{\rm P}[\left| {S_r } \right| = l]{\rm P}[(\gamma _{sc,x_e } < \gamma _E^{th} \cup \gamma _{n^ \ast c,x_d } < \gamma _D^{th} )|\left| {S_r } \right| = l]} \\ & = (1 - e^{ - \frac{{\varphi _n }}{\rho }} )^N {\text{ + }}\sum_{l = 1}^N {\left( {\begin{array}{*{20}c} N \\ l \\ \end{array} } \right)\left[ {1 - e^{ - \frac{{\varphi _n }}{\rho }} } \right]^{N - l} e^{ - \frac{{\varphi _n }}{\rho }l} \left[ {1 - e^{ - \frac{{\Gamma _E^{th} }}{{\rho \lambda _{sc} }}} e^{ - \frac{{\gamma _D^{th} }}{{\rho _d a_2 \lambda _{nc} }}} } \right]^l } \\ \end{aligned}$$
(15)

3.2 Partial Relay Selection

The PRSS selects the best relay based on the channel quality of BS to the relay link, which can be expressed as

$$n^* = \mathop {\arg }\limits_n \max \{ \left| {h_{sn} } \right|^2 ,n \in N\}$$
(16)

According to the principle of relay selection, the probability density function of \(\left| {h_{sn} } \right|^2\) can be defined as

$$f_{\left| {h_{sn} } \right|^2 } (x) = \sum_{n = 1}^N {\left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)} \left( { - 1} \right)^{n - 1} \frac{n}{{\lambda_{sn} }}e^{ - \frac{n}{{\lambda_{sn} }}x}$$
(17)

Hence, the outage probability of \(x_e\) at EU can be formulated as

$$\begin{aligned} {\rm P}_{x_e }^{out,{\rm I}{\rm I}} & = 1 - {\rm P}\left[ {\gamma _{sn,x_e } \ge \gamma _E^{th} ,\gamma _{ne,x_e } \ge \gamma _E^{th} } \right] \\ & = 1 - {\rm P}\left[ {\left| {h_{sn} } \right|^2 \ge \frac{{\Gamma _E^{th} }}{\rho }} \right]{\rm P}\left[ {\left| {h_{ne} } \right|^2 \ge \frac{{\Gamma _E^{th} }}{{\rho _d }}} \right] \\ & = 1 - \sum_{n = 1}^N {\left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)\left( { - 1} \right)^{n + 1} e^{ - \frac{{n\Gamma _E^{th} }}{{\rho \lambda _{sn} }}} e^{ - \frac{{\Gamma _E^{th} }}{{\rho _d \lambda _{ne} }}} } \\ \end{aligned}$$
(18)

The OP of \(x_d\) at CU can be derived by

$$\begin{aligned} {\rm P}_{x_d }^{out,{\rm I}{\rm I}} & = 1 - {\rm P}\left[ {\gamma _{sn,x_e } \ge \gamma _E^{th} \cap \gamma _{nc,x_d } \ge \gamma _D^{th} \cap \gamma _{sc,x_e } \ge \gamma _E^{th} } \right] \\ & = 1 - {\rm P}\left[ {\left| {h_{sn} } \right|^2 \ge \frac{{\Gamma _E^{th} }}{\rho } \cap \left| {h_{nc} } \right|^2 \ge \frac{{\gamma _D^{th} }}{{\rho _d b_2 }} \cap \left| {h_{sc} } \right|^2 \ge \frac{{\Gamma _E^{th} }}{\rho }} \right] \\ & = 1 - \left[ {\sum_{n = 1}^N {\left( {\begin{array}{*{20}c} N \\ n \\ \end{array} } \right)\left( { - 1} \right)^{n + 1} e^{ - \frac{{n\Gamma _E^{th} }}{{\rho \lambda _{sn} }}} \times e^{ - \frac{{\gamma _D^{th} }}{{\rho _d b_2 \lambda _{nc} }}} \times e^{ - \frac{{\Gamma _E^{th} }}{{\rho \lambda _{sc} }}} } } \right] \\ \end{aligned}$$
(19)

Furthermore, the outage probability of \(x_c\) has nothing to do with the relay selection, so \({\rm P}_{x_c }^{out,{\rm I}{\rm I}} = {\rm P}_{x_c }^{out,{\rm I}}\).

4 Performance Analysis

To confirm and compare the outage performance analysis, numerical results are provided in this section. Aiming at the proposed TSRS DC-NOMA network, Monte-Carlo computer simulation is carried out in a Rayleigh fading channel. Unless otherwise specified, the proposed system simulation parameters are set by default as: normalized distance \(d_{sc} = d_{sn} = 0.6\), \(d_{ne} = 0.4\), \(d_{nc} = 0.25\).We consider the fixed power allocation factors, \(a_1 = b_1 = 0.65\) and \(a_2 = b_2 = 0.35\). Besides, the pass loss exponent are set as \(a_{sc} = a_{sn} = a_{ne} = a_{nc} = 3\), and \(\eta = 0.5\). Without loss of generality, \(R_e = 0.7{\text{bit/s}}\), \(R_c = 0.4{\text{bit/s}}\), and \(R_d = 0.4{\text{bit/s}}\) denote the target rate.

Figures 2 and 3 show the outage probability of \(x_e\) and \(x_d\) under the PRSS and TSRS relay selection schemes versus SNR, respectively. It can be clearly observed that analytical curves match with simulation results. It is worth noting that compared to the PRSS scheme, the proposed TSRS scheme can significantly improve the outage performance of \(x_e\) and \(x_d\). For the \(x_e\), the PRSS scheme only selects the relay with the best BS to the relay link quality, and only guarantees the outage probability of successful decoding \(x_e\) at the relay. However, the proposed TSRS in this paper is based on the premise of ensuring successful decoding on the relay and EU. For the signal \(x_d\), the reason is that the TSRS scheme selects the relay with the best \({\text{R}}_n \to {\text{CU}}\) link quality under guarantee the reliable reception of the relay, which significantly reduces the OP of \(x_d\).

Fig. 2
figure 2

Outage probability \(x_e\) versus of \(\rho\)

Full size image
Fig. 3
figure 3

Outage probability \(x_d\) versus of \(\rho\)

Full size image

Figure 4 illustrates how the number of relays affect outage probability of TSRS and PRSS scheme. Observe that the OP of PRSS scheme remains unchanged when the relays reaches a certain value. Under the TSRS strategy, as the number of relays is increasing, the OP declines linearly. Noted that the superiority of TSRS scheme is apparent and the outage probability can be effectively reduced with the number of relays increasing. From a practical point of view, it is important to consider multiple relays scenarios. Furthermore, the TSRS scheme is especially significant for improving the outage performance of EU with poor channel quality.

Fig. 4
figure 4

Outage probability \(x_e\) and \(x_d\) versus of \(N\)

Full size image

5 Conclusion

In this paper, D2D communication is applied in the cooperative relaying NOMA network. In the future, the DC-NOMA can be exploited in the nearly local communication of the wireless communication network. Based on the DC-NOMA system, we have proposed a TSRS multiple relays selection scheme. The outage probabilities have been derived and the derivation results are fully consistent with simulation results. It was also shown that, as compared to PRSS scheme, the proposed TSRS scheme can effectively improve the outage performance of the DC-NOMA system. Noted that the increase in the number of relays greatly reduced the signal outage probability, so it can meet the application of large-scale relay scenarios in massive connections in the future.