## 2 System Model
### Propagation and Formulations
#### 2.2.2 Even Transmission Block  
As with the first time slot in the odd transmission block, by applying the PS protocol, the first time slot in the even transmission block \(T_{1}^{(even)}\) is used by the BS to transmit wireless energy and superimposed information simultaneously to coupled relays. The EH from the best channel in the pre-coding channel matrix at coupled relays \(R_{1}\) and \(R_{2}\) is expressed as follows:  
\[E_{R_{1}}^{T_{1}^{(even)}}=\eta\lambda P_{S}\max_{[A_{0}\times A_{1}]}\Bigl{\{} |\mathbf{H_{0}}|^{2}\Bigr{\}}, \tag{17}\]  
\[E_{R_{2}}^{T_{1}^{(even)}}=\eta\lambda P_{S}\max_{[A_{0}\times A_{1}]}\Bigl{\{} |\mathbf{G_{0}}|^{2}\Bigr{\}}, \tag{18}\]  
where \(T_{1}^{(odd)}=T^{(odd)}\Bigl{/}2\). The BS broadcasts a superimposed signal by combining the independent messages \(x_{i}\) of devices \(D_{i}\) for \(i=\{N,\ldots,1\}\). Therefore, the received signal at coupled relays is expressed as follows:  
\[\max_{[A_{0}\times A_{1}]}\Biggl{\{}\mathbf{Y}_{R_{1}}^{\left(T_{1}^{(even)} \right)}\Biggr{\}}=\sqrt{1-\lambda}\max_{[A_{0}\times A_{1}]}\{|\mathbf{H_{0} }|\}\sum\limits_{i=N}^{1}\sqrt{\alpha_{i}P_{S}}x_{i}+n_{R_{1}}, \tag{19}\]  
\[\max_{[A_{0}\times A_{1}]}\Biggl{\{}\mathbf{Y}_{R_{2}}^{\left(T_{1}^{(even)} \right)}\Biggr{\}}=\sqrt{1-\lambda}\max_{[A_{0}\times A_{1}]}\{|\mathbf{G_{0} }|\}\sum\limits_{i=N}^{1}\sqrt{\alpha_{i}P_{S}}x_{i}+n_{R_{2}}. \tag{20}\]  
In the second transmission block, SIC is implemented at the relay \(R_{2}\). As with the SIC in the odd transmission block, the relay \(R_{2}\) has to repeat SIC until it detect all data symbols \(x_{i}\) for \(i=\{N,\ldots,1\}\) in the best received signal as (20). The SINRs are obtained when relay \(R_{2}\) decodes the data symbols \(x_{i}=\{x_{N},\ldots,x_{1}\}\) as follows:  
\[\max_{[A_{0}\times A_{1}]}\left\{ \begin{matrix}\gamma\begin{smallmatrix}\binom{r( |con)}{n_{1}}\\ \gamma\begin{smallmatrix}R_{2}-x_{i}\\ i=\{N_{i},\ldots,1\}\end{smallmatrix}\end{matrix} \right\} =\frac{(1-\lambda)\max\limits_{[A_{0}\times A_{1}]}\Bigl{\{} \bigl{|}\mathbf{G_{0}}|^{2}\Bigr{\}}\alpha_{i}\rho_{S}}{(1-\lambda)\max\limits_ {[A_{0}\times A_{1}]}\Bigl{\{}\bigl{|}\mathbf{G_{0}}|^{2}\Bigr{\}}\rho_{S}\sum \limits_{j=1}^{i-1}\alpha_{j}+1},\text{ for }i>1, \tag{21}\] \[=(1-\lambda)\max\limits_{[A_{0}\times A_{1}]}\Bigl{\{}\bigl{|} \mathbf{G_{0}}|^{2}\Bigr{\}}\alpha_{i}\rho_{S},\hskip 42.679134 pt\text{for }i=n=1. \tag{22}\]Similar to (8) and (8), maximization of the instantaneous bit-rate threshold achieved at the relay \(R_{2}\) when the relay \(R_{2}\) decodes data symbols \(x_{i}\), where \(i=\{N,\ldots,1\}\) is expressed as follows:  
\[\max_{[A_{0}\times A_{1}]}\left\{ \begin{matrix}\mathbf{R}_{R_{2}-x_{i}}^{\left(T _{2}^{\left(even\right)}\right)}\\ i=\{N,\ldots,1\}\end{matrix} \right\}=\frac{1}{2}\text{log}_{2}\left(1+\max_{[A _{0}\times A_{1}]}\left\{ \begin{matrix}\gamma_{R_{2}-x_{i}}^{\left(T_{2}^{ \left(even\right)}\right)}\\ i=\{N,\ldots,1\}\end{matrix} \right\}\right). \tag{23}\]  
By applying a DF protocol, relay \(R_{2}\) recovers the decoded data symbols \(x_{i}\) for \(i=\{N,\ldots,1\}\) and forwards a beamforming superimposed signal to devices \(D_{n}\) for \(n=\{1,\ldots,N\}\). The received signals in the second time slot \(T_{2}^{\left(even\right)}\) in even transmission block \(T^{\left(even\right)}\) at devices \(D_{n}\) while EH at relay \(R_{1}\) are expressed, respectively, as follows:  
\[\max_{[A_{1}\times A_{2}]}\left\{\mathbf{Y}_{n\in N}^{\left(T_{2}^{\left(even \right)}\right)}\right\}=\max_{[A_{1}\times A_{2}]}\left\{\left|\mathbf{G}_{n }\right|\right\}\sum_{i=N}^{1}\sqrt{\alpha_{i}P_{R_{2}}}x_{i}+n_{n}, \tag{24}\]  
\[E_{R_{1}}^{\left(T_{2}^{\left(even\right)}\right)}=\eta\lambda P_{S}\max_{[A _{0}\times A_{1}]}\left\{\left|\mathbf{H}_{0}\right|^{2}\right\}, \tag{25}\]  
where \(P_{R_{2}}\) is the transmission power at relay \(R_{2}\) and the pre-coding channel matrix \(\mathbf{G}_{n}\) is given by  
\[\mathbf{G}_{n}=\{1,\ldots,N\}=\left[