## 2 System Model
### Propagation and Formulations
#### 2.2.1 Odd Transmission Block
\begin{subarray}{c}T_{1}^{(odd)}\\ I=\{N,\ldots,1\}\end{subarray} }^{( \begin{subarray}{c}T_{1}^{(odd)}\\ I=\{N,\ldots,1\}\end{subarray} }\right\}\right)\right). \tag{10}\]  
In the second time slot \(T_{2}^{(odd)}\) in the odd transmission block \(T^{(odd)}\), the relay \(R_{1}\) retrieves the messages \(x_{i}=\{x_{N},\ldots,x_{1}\}\) and forwards the messages to the devices in the superimposed signal while the relay \(R_{2}\) continues harvesting energy from the BS. Therefore, the received signal at devices \(D_{n}\) for \(n=\{1,\ldots,N\}\) and the EH at relay \(R_{2}\) are expressed, respectively, as follows:  
\[\max_{[A_{1}\times A_{2}]}\biggl{\{}\mathbf{Y}_{n\in N}^{\left( \begin{subarray}{c}T_{2}^{(odd)}\\ I=N\end{subarray} \right)}=\max_{[A_{1}\times A_{2}]}\{\left|\mathbf{H_{n}} \right|\}\sum\limits_{i=N}^{1}\sqrt{\alpha_{i}P_{R_{1}}}x_{i}+n_{n}, \tag{11}\]  
\[E_{R_{2}}^{\left( \begin{subarray}{c}T_{2}^{(odd)}\\ I=2\end{subarray} \right)}=\eta\lambda P_{S}\max_{[A_{0}\times A_{1}]}\Bigl{\{} \left|\mathbf{G_{0}}\right|^{2}\Bigr{\}}, \tag{12}\]  
where \(P_{R_{1}}\) is the transmission power at relay \(R_{1}\) and \(n_{n}\sim CN(0,N_{0})\) is the AWGN at device \(D_{n}\) for \(n=\{1,\ldots,N\}\) which follows zero mean and variance \(N_{0}\). Note that the pre-coding channel matrix \(\mathbf{H_{n}}\) is given by  
\[\mathbf{H}_{n=\{1,\ldots,N\}}=\left[ \begin{array}{ccc}h_{n}^{(1,1)}&\cdots&h _{n}^{(1,A_{2})}\\ \vdots&\ddots&\vdots\\ h_{n}^{(A_{1},1)}&\cdots&h_{n}^{(A_{1},A_{2})}\end{array} \right], \tag{13}\]  
where the channel \(h_{n}^{(a_{1},a_{2})}\) in the pre-coding channel matrix \(\mathbf{H}_{n}\), where \(a_{1}\in A_{1}\) and \(a_{2}\in A_{2}\), is a channel from transmitting antenna \(a_{1}\) at relay \(R_{1}\) to a receiving antenna \(a_{2}\) at a device \(D_{n}\), also applying Rayleigh distribution for propagation. Each fading channel gain is given by \(h_{n}^{(\cdot)}=d_{n}^{-\omega}\), where \(d_{n}\) is the distance from relay \(R_{1}\) to device \(D_{n}\). As a result of the combination of TAS and SIC, the SINRs are obtained at devices \(D_{n}\) for \(n=\{1,\ldots,N\}\) when the devices decode data symbols \(x_{i}\) for \(i=\{N,\ldots,n\}\) from the best received signal \(\max\{\left|\mathbf{Y_{2}}\right|\}_{A_{1}\times A_{2}}\) by treating the data symbols \(x_{j}\) for \(j=\{1,\ldots,i-1\}\) and AWGN \(n_{n}\) as interference:\[\max_{[A_{1}\times A_{2}]}\left\{ \begin{matrix}\binom{r(|dd)}{n_{2}}\\ \gamma\begin{smallmatrix}n_{m}\\ n_{m}\\ n\in N\end{smallmatrix}-\{x_{i}\}\\ i=\{N_{i},\ldots,q\}\end{matrix} \right\} =\frac{\max\limits_{[A_{1}\times A_{2}]}\Bigl{\{}|\mathbf{H}_{n} |^{2}\Bigr{\}}\alpha_{i}\rho_{R_{1}}}{\max\limits_{[A_{1}\times A_{2}]}\Bigl{\{} \bigl{|}\mathbf{H}_{n}\bigr{|}^{2}\Bigr{\}}\sum\limits_{j=1}^{i-1}\alpha_{i} \rho_{R_{1}}+1},\text{for }i>1, \tag{14}\] \[=\max\limits_{[A_{1}\times A_{2}]}\Bigl{\{}|\mathbf{H}_{n}|^{2} \Bigr{\}}\alpha_{i}\rho_{R_{1}},\hskip 42.679134 pt\text{for }i=n=1, \tag{15}\]  
where SNR \(\rho_{R_{1}}=P_{R_{1}}/N_{0}\). The achievable bit-rate reached at device \(D_{n}\) when it decodes the data symbol \(x_{i}\) for \(i=\{N,\ldots,n\}\) from the best received signal \(\max\{|\mathbf{Y_{n}}|\}\) is expressed as follows:  
\[\max_{[A_{1}\times A_{2}]}\left\{\begin{matrix}\mathbf{R}_{ \begin{subarray}{ c}n_{2}\\ n_{N}\\ n\in N\end{subarray} -i=\{N_{i},\ldots,n\}\end{subarray}}^{\left(\binom{r(|dd)}{ n_{2}}\right)}\\ =\frac{1}{2}\text{log}_{2}\left(1+\max\limits_{[A_{1}\times A_{2}]}\left\{ \gamma\begin{smallmatrix}n_{m}\\ n\in N\end{subarray}-i=\{N_{i},\ldots,n\}\end{subarray}\right\}\right). \tag{16}\]