## 2 System Model
### Propagation and Formulations
#### 2.2.1 Odd Transmission Block  
As shown in Figure 1, it is important to note that both the BS and coupled relays (\(R_{1}\) and \(R_{2}\)) are equipped with multiple antennas, where \(A_{0}>1\) and \(A_{1}>1\). \(A_{0}\) and \(A_{1}\) are the number of antennas at the BS and coupled relays, respectively. [PERSON] et al. [23] designed a pre-coding channel matrix size of [number of transmitting antennas \(\times\) number of receiving antennas]. We therefore designed the pre-coding channel matrices from the \(A_{0}\) transmitting antennas at the BS to the \(A_{1}\) receiving antennas on the coupled relays \(R_{1}\) and \(R_{2}\), respectively, as follows:  
\[\mathbf{H}_{0}=\left[ \begin{array}{ccc}h_{0}^{(1,1)}&\cdots&h_{0}^{(1,A_{1}) }\\ \vdots&\ddots&\vdots\\ h_{0}^{(A_{0},1)}&\cdots&h_{0}^{(A_{0},A_{1})}\end{array} \right], \tag{1}\]  
\[\mathbf{G}_{0}=\left[ \begin{array}{ccc}g_{0}^{(1,1)}&\cdots&g_{0}^{(1,A_{1}) }\\ \vdots&\ddots&\vdots\\ g_{0}^{(A_{0},1)}&\cdots&g_{0}^{(A_{0},A_{1})}\end{array} \right], \tag{2}\]  
where \(h_{0}^{(a_{0},a_{1})}\in\mathbf{H}_{0}\) and \(g_{0}^{(a_{0},a_{1})}\in\mathbf{G}_{0}\) for \(a_{0}\in A_{0}\) and \(a_{1}\in A_{1}\) are channels from a transmitting antenna \(a_{0}\) at the BS to a receiving antenna \(a_{1}\) at the coupled relays \(R_{1}\) and \(R_{2}\). In addition, the fading channels are modeled over Rayleigh distributions with \(h_{0}^{(-)}\) and \(g_{0}^{(-)}\) following \(h_{0}^{(-)}=d_{R_{1}}^{-\omega}\) and \(g_{0}^{(-)}=d_{R_{2}}^{-\omega}\), where \(d_{R_{1}}\) and \(d_{R_{2}}\) are the distances from the BS to coupled relays \(R_{1}\) and \(R_{2}\), respectively, and the coefficient \(\omega\) is the path-loss exponent factor. By applying the PS protocol, the first time slot in the odd transmission block \(T_{1}^{(odd)}\) is used for the BS to transmit wireless energy and superimposed information simultaneously. To illustrate, two phases take place. In the first phase, the BS sends wireless energy to the coupled relay with the PS factor \(\lambda\). Therefore, the EH from the best channel in the pre-coding channel matrices given by (1) and (2) in the first time slot \(T_{1}^{(odd)}\) in odd transmission block \(T^{(odd)}\) at coupled relays \(R_{1}\) and \(R_{2}\) are expressed as follows:  
\[E_{R_{1}}^{\left(T_{1}^{(odd)}\right)}=\eta\lambda P_{S}\max_{|A_{0}\times A_ {1}|}\Bigl{\{}|\mathbf{H}_{0}|^{2}\Bigr{\}}, \tag{3}\]  
Figure 2: Diagram of two transmission blocks. \[E_{R_{2}}^{\left(T_{1}^{(odd)}\right)}=\eta\lambda P_{S}\max_{\left[A_{0}\times A _{1}\right]}\Bigl{\{}\left|\mathbf{G}_{0}\right|^{2}\Bigr{\}}, \tag{4}\]  
where \(\eta\) is the collection factor, \(\lambda\) is the PS factor, and \(P_{S}\) is the transmission power at the BS. In the second phase of the first time slot \(T_{1}^{(odd)}\) in odd transmission block \(T^{(odd)}\), in a major advantage of NOMA theories, the BS broadcasts a superimposed signal by superimposing the messages \(x_{i}\) of devices \(D_{i}\) for \(i=\{1,\ldots,N\}\) to the coupled relays \(R_{1}\) and \(R_{2}\). We assume that no direct down-link exists from the BS to the devices. The received signal at the coupled relays is expressed as follows:  
\[\max_{\left[A_{0}\times A_{1}\right]}\Biggl{\{}\mathbf{Y}_{R_{1}}^{\left(T_{1 }^{(odd)}\right)}\Biggr{\}}=\sqrt{1-\lambda}\max_{\left[A_{0}\times A_{1} \right]}\{\left|\mathbf{H}_{0}\right|\}\sum_{i=N}^{1}\sqrt{\alpha_{i}P_{S}}x_ {i}+n_{R_{1}}, \tag{5}\]  
\[\max_{\left[A_{0}\times A_{1}\right]}\Biggl{\{}\mathbf{Y}_{R_{2}}^{\left(T_{1 }^{(odd)}\right)}\Biggr{\}}=\sqrt{1-\lambda}\max_{\left[A_{0}\times A_{1} \right]}\{\left|\mathbf{G}_{0}\right|\}\sum_{i=N}^{1}\sqrt{\alpha_{i}P_{S}}x_ {i}+n_{R_{2}}, \tag{6}\]  
where \(n_{\mathcal{R}}\sim\mathrm{CN}(0,N_{0})\) is the Additive White Gaussian Noise (AWGN) at the coupled relays \(\mathcal{R}=\{R_{1},R_{2}\}\) with zero mean and variance \(N_{0}\).