## 2 System Model
### Node Deployment  
At time instant \(n(n>1)\), \(M_{T}\) data streams \(\mathbf{s}_{i}[n]\in\mathbb{C}^{M_{T}\times 1}\) with a normalized power are transmitted through the beamformer \(\mathbf{F}_{i}\in\mathbb{C}^{M_{T}\times M_{T}}(i\in\{1,2\})\) from \(S_{i}\) simultaneously and the relay \(R\) forwards its received signal \(\mathbf{y}_{R}[n]\in\mathbb{C}^{N_{R}\times 1}\) after multiplying it by a beamforming matrix \(\mathbf{F}_{r}\in\mathbb{C}^{N_{T}\times N_{R}}\). In practice, a \(\tau(\tau\geqslant 1)\)-symbol processing delay is unavoidable at \(R\) when it processes the received signals. Accordingly, in time slot \(n\), the received signal \(\mathbf{y}_{R}\left[n\right]\) at \(R\) can be expressed as  
\[\mathbf{y}_{R}\left[n\right]=\sum_{i=1}^{2}\mathbf{H}_{S_{i}R}\mathbf{F}_{i} \mathbf{s}_{i}[n]+\mathbf{H}_{RR}\mathbf{x}_{R}[n]+\mathbf{n}_{r}[n], \tag{1}\]  
where \(\mathbf{n}_{r}[n]\sim CN\left(0,\sigma_{r}^{2}\mathbf{I}_{N_{r}}\right)\) represents the additive white Gaussian noise (AWGN) at \(R\). Assuming that \(\tau\) is kept small and that the SI can be cancelled perfectly or almost perfectly with the knowledge of the signal transmitted by the relay itself [37], the transmitted signal \(\mathbf{x}_{R}[n]\in\mathbb{C}^{N_{T}\times 1}\) at \(R\) can be written as  
\[\mathbf{x}_{R}[n]=\mathbf{F}_{r}\left(\mathbf{y}_{R}[n-\tau]-\mathbf{H}_{RR} \mathbf{x}_{R}[n-\tau]\right). \tag{2}\]  
Substituting Equation (1) into Equation (2), the overall relay output can be given by  
\[\mathbf{x}_{R}\left[n\right]=\mathbf{F}_{r}\left(\sum_{i=1}^{2}\mathbf{H}_{S_{ i}R}\mathbf{F}_{i}\mathbf{s}_{i}[n-\tau]+\mathbf{n}_{r}[n-\tau]\right). \tag{3}\]  
The signal received by \(S_{i}\), \(i\in\{1,2\}\) can be written as  
\[\widehat{\mathbf{y}}_{S_{i}}[n]= \mathbf{H}_{RS_{i}}\left[\mathbf{F}_{r}\left(\mathbf{H}_{S_{i}R} \mathbf{F}_{\widehat{i}}[n-\tau]\mathbf{s}_{\widehat{i}}[n-\tau]+\mathbf{H}_{ S_{i}R}\mathbf{F}_{i}\mathbf{s}_{i}[n-\tau]\right)\right]+\mathbf{H}_{S_{i}S_{i}} \mathbf{x}_{S_{i}}[n]+\widehat{\mathbf{n}}_{i}[n], \tag{4}\]  
where, the same as below, \(\widehat{i}=2\) if \(i=1\) and vice versa. \(\widehat{\mathbf{n}}_{i}[n]\) is the equivalent noise vector representing \(\widehat{\mathbf{n}}_{i}[n]=\mathbf{H}_{RS_{i}}\mathbf{F}_{r}\mathbf{n}_{r}[n -\tau]+\mathbf{n}_{i}[n]\), where \(\mathbf{n}_{i}[n]\) denotes an AWGN at source nodes \(S_{i}\) with \(\mathbf{n}_{i}[n]\sim CN\left(0,\sigma_{S_{i}}^{2}\mathbf{I}_{M_{R}}\right)\). For simplicity, we assume that the full channel state information (CSI) is known and that both \(S_{1}\) and \(S_{2}\) know their own transmitted signals; thus, the SI at \(S_{i}\) herein can be cancelled. After subtracting the back-propagated self-interference term \(\mathbf{H}_{RS_{i}}\mathbf{F}_{r}\mathbf{H}_{S_{i}R}\mathbf{F}_{i}\mathbf{s} _{i}[n-\tau]\) from Equation (4), the received signal at \(S_{i}\) becomes  
\[\widehat{\mathbf{y}}_{S_{i}}[n]=\mathbf{H}_{RS_{i}}\mathbf{F}_{r}(\mathbf{H}_{ S_{i}R}\mathbf{F}_{\widehat{i}}\mathbf{s}_{\widehat{i}}[n-\tau]+\mathbf{n}_{r}[n- \tau])+\mathbf{n}_{i}[n]. \tag{5}\]To implement SWIPT, a portion \(\beta_{i}\in(0,1)\) of the signal power is applied to Equation (5), which splits \(\mathbf{\hat{y}}_{S_{i}}[n]\) into two parts, \(\beta_{i}\) portion for ID and the remaining \((1-\beta_{i})\) portion for EH. Then, the signals for ID at each source node can be represented as  
\[\mathbf{y}_{S_{1}}[n]=\sqrt{\beta_{i}}\mathbf{H}_{RS_{i}}\mathbf{F}_{r} \mathbf{H}_{S_{i}R}\mathbf{F}_{\mathbf{\hat{y}}}\mathbf{\hat{s}}_{i}[n-\tau]+ \mathbf{n}_{i}^{{}^{\prime}}[n,\tau], \tag{6}\]  
where \(\mathbf{n}_{i}^{{}^{\prime}}[n,\tau]=\sqrt{\beta_{i}}\left(\mathbf{H}_{RS_{i} }\mathbf{F}_{r}\mathbf{n}_{r}[n-\tau]+\mathbf{n}_{i}[n]\right)+\mathbf{n}_{p} [n]\) denotes the equivalent noise vector and \(\mathbf{n}_{p}[n]\sim\text{CN}\left(0,\sigma_{p}^{2}\mathbf{I}_{M_{R}}\right)\) is the AWGN caused by the power splitter.