## 2 System Model and Transmission Process
### Transmission Process
#### 2.2.1 Direct Transmission Stage of the First Time Slot  
In the first time slot, BS broadcasts the superimposed signals of U\({}_{1}\) and U\({}_{2}\) via the superimposed coding (SC). After receiving the signal, decoding result from U\({}_{1}\) will be sent to all relays, and the decodable relays will be divided into two candidate sets. The details are as follows. BS broadcasts a superimposed signal, \(x_{S}=\sqrt{\alpha_{1}}x_{1}+\sqrt{\alpha_{2}}x_{2}\), which can be received by U\({}_{1}\) and the potential relays with the transmit power, \(P_{S}\). Here, \(x_{1}\) and \(x_{2}\) are the signals for U\({}_{1}\) and U\({}_{2}\) with \(E\Big{\{}\left|x_{1}|^{2}\right\}\!\!=E\Big{\{}\left|x_{2}\right|^{2}\!\!\Big{\}} \!\!=1\), and \(\alpha_{1}\) and \(\alpha_{2}\) denote the power allocation coefficients at BS for messages \(x_{1}\) and \(x_{2}\) with \(\alpha_{1}+\alpha_{2}=1\) and \(\alpha_{2}\geq\alpha_{1}\geq 0\), respectively. Therefore, the received signal at the \(n\)-th relay can be expressed as  
\[y_{SR_{n}}=\sqrt{P_{T}}h_{TR_{n}}x_{p}+\sqrt{P_{S}}h_{SR_{n}}x_{S}+\eta_{R_{n}} \tag{1}\]  
Figure 1: System model. where \(\eta_{R_{n}}\) is AWGN at the \(n\)-th relay \(R_{n}\). Then, SIC technique is used to try to decode the received superimposed signal. The signal with high power (or poor channel quality) is first decoded, i.e., at the \(n\)-th relay, \(x_{2}\) is first decoded and \(x_{1}\) is treated as interference, and then \(x_{1}\) can be decoded only after \(x_{2}\) successfully decodes and cancels. Therefore, the achievable rates of \(x_{2}\) and \(x_{1}\) at the relay \(R_{n}\) are given by  
\[R_{SR_{n}}^{x_{2}}=\frac{1}{2}log_{2}\Bigg{(}1+\frac{\alpha_{2} \rho_{S}|h_{SR_{n}}|^{2}}{\rho_{T}|h_{TR_{n}}|^{2}+\alpha_{1}\rho_{S}|h_{SR_{n} }|^{2}+1}\Bigg{)}\text{ and }R_{SR_{n}}^{x_{1}}=\frac{1}{2}log_{2}\Bigg{(}1+\frac{ \alpha_{1}\rho_{S}|h_{SR_{n}}|^{2}}{\rho_{T}|h_{TR_{n}}|^{2}+1}\Bigg{)} \tag{2}\]  
respectively, where \(\rho_{T}=P_{T}/\sigma^{2}\) and \(\rho_{S}=P_{S}/\sigma^{2}\) are the transmit signal-to-noise ratio (SNR). Relays that can successfully decode \(x_{1}\) and \(x_{2}\) are divided into two candidate sets, \(S_{1}=\left\{n\big{|}n\in N,R_{SR_{n}}^{x_{2}}\geq R_{2}\right\}\) and \(S_{2}=\left\{n\big{|}n\in N,R_{SR_{n}}^{x_{2}}\geq R_{2},R_{SR_{n}}^{x_{1}}\geq R _{1}\right\}\), respectively, where \(R_{1}\) and \(R_{2}\) denote the target data rates of \(x_{1}\) and \(x_{2}\) at the receiving nodes, respectively. The received signal at U\({}_{1}\) can be expressed as  
\[y_{SU_{1}}=\sqrt{P_{T}}h_{TU_{1}}x_{p}+\sqrt{P_{S}}h_{SU_{1}}x_{S}+\eta_{U_{1}} \tag{3}\]  
where \(\eta_{U_{1}}\) is AWGN at U\({}_{1}\). After using SIC, the achievable rates of \(x_{2}\) and \(x_{1}\) at U\({}_{1}\) are given, respectively, as  
\[R_{SU_{1}}^{x_{2}}=\frac{1}{2}log_{2}\Bigg{(}1+\frac{\alpha_{2} \rho_{S}|h_{SU_{1}}|^{2}}{\rho_{T}|h_{TU_{1}}|^{2}+\alpha_{1}\rho_{S}|h_{SU_{1} }|^{2}+1}\Bigg{)}\text{ and }R_{SU_{1}}^{x_{1}}=\frac{1}{2}log_{2}\Bigg{(}1+\frac{ \alpha_{1}\rho_{S}|h_{SU_{1}}|^{2}}{\rho_{T}|h_{TU_{1}}|^{2}+1}\Bigg{)} \tag{4}\]  
Note that the prerequisite for a receiving node to decode \(x_{1}\) is that \(x_{2}\) has been successfully detected and cancelled, i.e., \(R_{SR_{n}}^{x_{1}}\) and \(R_{SU_{1}}^{x_{1}}\) are achievable under the conditions that \(R_{SR_{n}}^{x_{2}}\geq R_{2}\) and \(R_{SU_{1}}^{x_{2}}\geq R_{2}\) in Equation (2) and Equation (4), respectively. At the end of the first time slot, the decoding result is sent to relays as a signaling. There are two possible cases: (1) U\({}_{1}\) fails to get the desired message in the first time slot. Then, it will send a signaling to relays and an optimal relay from the candidate set will be selected to forward the superimposed signal to U\({}_{1}\) and U\({}_{2}\) in the next time slot; (2) U\({}_{1}\) successfully decodes the desired message. Then, a signaling will be sent by U\({}_{1}\) and an optimal relay from the candidate set or U\({}_{1}\) will be selected to forward the signal that U\({}_{2}\) expects.