## 2 System Model
### DF Protocol  
In case of DF relaying protocol, relay first decodes the data and then forwards it the the users. With the aid of an energy harvesting-enabled relay, the end-to- end communication occurs in two time phases. To save energy, the relay uses only single antenna to receive the signal from satellite. It then uses \(N\) antennas for maximum ratio transmission (MRT) to transmit the signal to NOMA users [32].  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(\Xi_{i}\) & The power allocation coefficient \(i\in\{1,2\}\) \\ \(P_{S}\) & The transmit power at S \\ \(P_{R}\) & The transmit power at R \\ \(n_{R}\) & The AWGN with variance \(N_{0}\) \\ \(n_{D_{i}}\) & The AWGN with variance \(N_{0}\) \\ \(\eta\) & The energy conversion efficiency and \(\eta\in(0,1]\) \\ \(\chi\) & The power splitting factor \\ \(T\) & The time duration \\ \(R_{i}\) & The target rate at \(D_{i}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key parameters of the system model.  
Figure 1: Illustration of energy harvesting HSTRN using NOMA.  
Considering the data transmission at the first sub-block in which the multiple antennas satellite S transmits an superimposed information signal \(x=\sqrt{\Xi_{1}}x_{1}+\sqrt{\Xi_{2}}x_{2}\). In the first phase  
\[y_{R}=\sqrt{P_{S}}\mathbf{h}_{SR}^{\dagger}\mathbf{w}_{SR}\left(\sqrt{\Xi_{1}}x_ {1}+\sqrt{\Xi_{2}}x_{2}\right)+n_{R}, \tag{1}\]  
where \(\mathbf{h}_{SR}=\left[h_{SR}^{1}\cdots h_{SR}^{M}\right]^{T}\) is the \(M\times 1\) channel vector from S to R. For MRT, \(\mathbf{w}_{SR}=\frac{\mathbf{h}_{SR}}{\left\|\mathbf{h}_{SR}\right\|_{F}}\), where \(\left\|.\right\|_{F}\) is Frobenius norm and \(\dagger\) denotes conjugate transpose. Following decoding of NOMA, signal decoding order is decided based on order of channel gains. We assume that \(\Xi_{1}>\Xi_{2}\). Then, the signal-to-interference-and-noise ratio (SINR) to detect \(x_{1}\) can be given as  
\[\Gamma_{R\to x_{1}}^{DF}=\frac{\tilde{\gamma}_{SR}\Xi_{1}}{\tilde{\gamma}_{SR} \Xi_{2}+1}, \tag{2}\]  
where \(\rho_{S}=\frac{P_{S}}{N_{0}}\) and \(\tilde{\gamma}_{SR}=\rho_{S}\left\|\mathbf{h}_{SR}\right\|_{F}^{2}\). After performing SIC at relay the signal-to-noise ratio (SNR) to detect \(x_{2}\) can be formulated by  
\[\Gamma_{R\to x_{2}}^{DF}=\tilde{\gamma}_{SR}\Xi_{2}. \tag{3}\]  
Next, energy harvesting is employed at the relay and energy level achieved at relay is given by [33]  
\[E_{R}=\eta P_{S}\left\|\mathbf{h}_{SR}\right\|_{F}^{2}\chi\mathcal{T}. \tag{4}\]  
From this point, it can be computed transmit power at the relay as  
\[P_{R}=\frac{E_{R}}{\left(1-\chi\right)\mathcal{T}_{/2}}=\frac{2\eta P_{S} \left\|\mathbf{h}_{SR}\right\|_{F}^{2}\chi}{\left(1-\chi\right)}. \tag{5}\]  
In the second phase, the MRT is used with beamforming vector such that we obtain \(\mathbf{w}_{i}=\frac{\mathbf{h}_{i}}{\left\|\mathbf{h}_{i}\right\|}\) in which \(\left\|.\right\|\) denotes the Euclidean norm of a matrix. Then, the received signal achieved at user \(D_{i}\) is given by  
\[y_{D_{i}}^{DF}=P_{R}\left\|\mathbf{h}_{i}\mathbf{w}_{i}\right\|\left(\sqrt{ \Xi_{1}}x_{1}+\sqrt{\Xi_{2}}x_{2}\right)+n_{D_{i}} \tag{6}\]  
The SINR at \(D_{1}\) to decode \(x_{1}\) can be expressed by  
\[\Gamma_{D_{1}\to x_{1}}^{DF}=\frac{P_{R}\|\mathbf{h}_{1}\|^{2}\Xi_{1}}{P_{R} \|\mathbf{h}_{1}\|^{2}\Xi_{2}+N_{0}}=\frac{\phi\tilde{\gamma}_{SR}\|\mathbf{h }_{1}\|^{2}\Xi_{1}}{\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{SR}\|^{2}\Xi_{2}+1}, \tag{7}\]  
where \(\phi=\frac{2\eta\chi}{\left(1-\chi\right)}\).  
In a similar way, before and after SIC performed at user \(D_{2}\), SNR computation related to detect signal \(x_{1},x_{2}\) are respectively given by  
\[\Gamma_{D_{2}\to x_{1}}^{DF}=\frac{\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{2}\|^ {2}\Xi_{1}}{\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{2}\|^{2}\Xi_{2}+1},\quad \Gamma_{D_{2}\to x_{2}}^{DF}=\phi\tilde{\gamma}_{SR}\|\mathbf{h}_{2}\|^{2}\Xi _{2}. \tag{8}\]