# Two-Way Multi-Antenna Relaying with Simultaneous Wireless Information and Power Transfer
## 3 PSR Protocol
### PS-DF Scheme  
In the MA phase, the received RF signal at the relay node can be modeled as:  
\[\begin{array}{rcl}\mathbf{y_{r}}&=&\left[ \begin{array}{c}y_{r,1}\\ y_{r,2}\\ \vdots\\ y_{r,M}\end{array} \right]\\ &=&\left[ \begin{array}{c}\sqrt{P_{1}}h_{1,1}x_{1}+\sqrt{P_{2}}h_{2,1}x_{1}+n _{r,1}^{[\mathrm{a}]}\\ \sqrt{P_{1}}h_{1,2}x_{1}+\sqrt{P_{2}}h_{2,2}x_{2}+n_{r,2}^{[\mathrm{a}]}\\ \vdots\\ \sqrt{P_{1}}h_{1,M}x_{1}+\sqrt{P_{2}}h_{2,M}x_{2}+n_{r,M}^{[\mathrm{a}]}\end{array}  \right]\\ &=&\left[ \begin{array}{c}\sqrt{P_{1}}h_{1,1}\sqrt{P_{2}}h_{2,1}\\ \sqrt{P_{1}}h_{1,2}\sqrt{P_{2}}h_{2,2}\\ \vdots\vdots\\ \sqrt{P_{1}}h_{1,M}\sqrt{P_{2}}h_{2,M}\end{array} \right]\left[ \begin{array}[] {c}x_{1}\\ x_{2}\end{array} \right]+\left[ \begin{array}{c}n_{r,1}^{[\mathrm{a}]}\\ n_{r,2}^{[\mathrm{a}]}\\ \vdots\\ n_{r,M}^{[\mathrm{a}]}\end{array} \right]\\ &=&\left[ \begin{array}{c}\sqrt{P_{1}}h_{1}\sqrt{P_{2}}h_{2}\end{array}  \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+\mathbf{n_{r}^{[\mathrm{a}]}} \tag{2}\]  
where \(P_{1}=\zeta_{1}P\) and \(P_{2}=\zeta_{2}P\) are the transmitted power from T\({}_{1}\) and T\({}_{2}\), respectively; \(0<\zeta_{1},\zeta_{2}<1\) are the power ratios of T\({}_{1}\) and T\({}_{2}\), respectively (i.e., \(\zeta_{1}+\zeta_{2}=1\)); \(x_{1}\) and \(x_{2}\) are the normalized information signals from T\({}_{1}\) and T\({}_{2}\), respectively (i.e., \(\mathrm{E}\left[\left|x_{1}\right|^{2}\right]=\mathrm{E}\left[\left|x_{2} \right|^{2}\right]=1\)); and \(n_{r,m}^{[\mathrm{a}]}\sim CN\big{(}0,\sigma_{a}^{2}\big{)}\) is the additive white Gaussian noise (AWGN) at the \(m\)-th antenna of R. The energy harvesting receiver in Figure 2b rectifies the RF signal \(\sqrt{\rho}y_{r,m}\) directly and gets the direct current to charge up the battery. Therefore, the harvested energy at the \(m\)-th antenna of the relay node during the MA phase is given by:  
\[E_{m}=\eta\rho N\Big{(}P\sigma^{2}+\sigma_{a}^{2}\Big{)},\ m=1,2,...,M \tag{3}\]  
where \(0<\eta\leq 1\) is the energy conversion efficiency (which depends on the rectification process and the energy harvesting circuitry [9]). Meanwhile, the information receiver in Figure 2b down-converts the RF signal \(\sqrt{1-\rho}y_{r,m}\) to baseband and processes the baseband signal, where \(n_{r,m}^{[\mathrm{c}]}\sim CN\big{(}0,\sigma_{\mathrm{c}}^{2}\big{)}\) is the AWGN due to RF-band-to-baseband signal conversion. After down conversion, the sampled baseband signal vector at the relay node is given by:  
\[ \begin{array}{rcl}\widetilde{\mathbf{y}}_{r}&=&\sqrt{1-\rho}\mathbf{y_{r}}+ \mathbf{n_{r}^{[\mathrm{c}]}}\\ &=&\sqrt{1-\rho}\Bigg{(}\left[\begin{array}{cc}\sqrt{P_{1}}\mathbf{h_{1}}& \sqrt{P_{2}}\mathbf{h_{2}}\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]+\mathbf{n_{r}^{[\mathrm{a}]}}\Bigg{)}+\mathbf{n_{r}^{[ \mathrm{c}]}}\\ &=&\Big{[}\begin{array}{cc}\sqrt{(1-\rho)P_{1}}\mathbf{h_{1}}&\sqrt{(1-\rho )P_{2}}\mathbf{h_{2}}\end{array}\Big{]}\left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]+\Big{(}\sqrt{(1-\rho)}\mathbf{n_{r}^{[\mathrm{a}]}}+ \mathbf{n_{r}^{[\mathrm{c}]}}\Big{)}\\ &=&\boldsymbol{\
abla}\Bigg{[}\begin{array}{c}x_{1}\\ x_{2}\end{array}\Bigg{]}+\mathbf{n_{r}}.\end{array}  \tag{4}\]  
Assuming that \(\rho\), \(P_{1}\), \(P_{2}\), \(\{h_{1,m}\}_{m=1}^{M}\), and \(\{h_{2,m}\}_{m=1}^{M}\) are known at the relay node and applying zero-forcing (ZF) detection, estimates of \(x_{1}\) and \(x_{2}\), denoted by \(\hat{x}_{1}\) and \(\hat{x}_{2}\), respectively, are obtained as:  
\[\left[ \begin{array}{c}\hat{x}_{1}\\ \hat{x}_{2}\end{array} \right]=\left(\boldsymbol{\Psi}^{\mathrm{H}}\boldsymbol{ \Psi}\right)^{-1}\boldsymbol{\Psi}\widetilde{\mathbf{y}}_{r} \tag{5}\](For discussions of how the relay node can obtain \(\{h_{1,m}\}_{m=1}^{M}\) and \(\{h_{2,m}\}_{m=1}^{M}\), interested readers are referred to [24; 25]). Let \(\gamma_{ir}\) denote the instantaneous SNR of \(\hat{x}_{i}\), where \(i=1,2\). It is straightforward to show that:  
\[