# Two-Way Multi-Antenna Relaying with Simultaneous Wireless Information and Power Transfer
## 3 PSR Protocol
### PS-DF Scheme
\begin{array}{ll}\gamma_{ir}&=\frac{1}{\left(\left(1-\rho\right)\sigma_{a}^{ 2}+\sigma_{\rm c}^{2}\right)\left|\left(\left(\mathbf{r}^{\rm H}\mathbf{r} \right)^{-1}\right|_{ii}\right.}\\ &=\frac{\left(1-\rho\right)\xi_{i}\mathbb{P}_{k=1}^{M-1}\sum\limits_{i=1}^{M} \left|h_{1,h}h_{2,l}-h_{1,l}h_{2,k}\right|^{2}}{\left(\left(1-\rho\right)\sigma _{\rm c}^{2}+\sigma_{\rm c}^{2}\right)\sum\limits_{m=1}^{M}\left|h_{j,m}\right| ^{2}}.\end{array}  \tag{6}\]  
Defining \(\varphi:=\frac{\sum_{k=1}^{M-1}\sum_{i=1}^{M}\left|h_{1,k}h_{2,l}-h_{1,l}h_{2, k}\right|^{2}}{\sum\limits_{m=1}^{M}\left|h_{j,m}\right|^{2}}\) and using (Equation (16), [26]), the PDF of \(\varphi\) can be expressed as:  
\[f_{\varphi}(x)=\frac{x^{M-2}}{\Gamma(M-1)\sigma^{2M-2}}\exp\left(-\frac{x}{ \sigma^{2}}\right);\ x\geq 0. \tag{7}\]  
From Equations (6) and (7), we have:  
\[f_{\gamma_{ir}}(x)=\frac{\left(\left(1-\rho\right)\sigma_{\rm a}^{2}+\sigma_{ \rm c}^{2}\right)^{M-1}}{\Gamma(M-1)\sigma^{2M-2}(\left(1-\rho\right)\zeta_{i }P)^{M-1}}x^{M-2}\exp\left(-\frac{\left(1-\rho\right)\sigma_{\rm a}^{2}+\sigma _{\rm c}^{2}}{\left(1-\rho\right)\zeta_{i}P\sigma^{2}}\,x\right);\ x\geq 0. \tag{8}\]  
Hence, the corresponding average BER can be derived as follows:  
\[ \begin{array}{ll}\overline{P}_{\rm e,ir}&=\int\limits_{0}^{\infty}\,Q\Big{(} \sqrt{2x}\Big{)}f_{\gamma_{ir}}(x)\ dx\\ &=\frac{1}{2\sqrt{\pi}}\int\limits_{0}^{\infty}\,\Gamma\Big{(}\frac{1}{2},x \Big{)}f_{\gamma_{ir}}(x)\ dx\\ &=\frac{\left(\left(1-\rho\right)\sigma_{\rm a}^{2}+\sigma_{\rm c}^{2}\right) ^{M-1}}{2\sqrt{\pi}\Gamma(M-1)\sigma^{2M-2}(\left(1-\rho\right)\zeta_{i}P)^{M -1}}\\ &\times\int\limits_{0}^{\infty}\,\Gamma\Big{(}\frac{1}{2},x\Big{)}\,x^{M-2} \exp\left(-\frac{\left(1-\rho\right)\sigma_{\rm a}^{2}+\sigma_{\rm c}^{2}}{ \left(1-\rho\right)\zeta_{i}P\sigma^{2}}\,x\right)\ dx\\ &=\frac{\sigma\,\Gamma(M-1/2)(\left(1-\rho\right)\zeta_{i}P)^{1/2}(\left(1- \rho\right)\zeta_{i}^{2}+\sigma_{\rm c}^{2})^{M-1}}{2\sqrt{\pi}\,\Gamma(M) \left(\left(1-\rho\right)\left(\zeta_{i}P\sigma^{2}+\sigma_{\rm a}^{2}\right) +\sigma_{\rm c}^{2}\right)^{M-1/2}}\\ &\times{}_{2}\mathbb{F}_{1}\left(1,M-\frac{1}{2};M;\frac{1-\rho}{\left(1- \rho\right)\left(\zeta_{i}P\sigma^{2}+\sigma_{\rm a}^{2}\right)+\sigma_{\rm c }^{2}}{\left(1-\rho\right)\left(\zeta_{i}P\sigma^{2}+\sigma_{\rm a}^{2}\right) +\sigma_{\rm c}^{2}}\right)\end{array}  \tag{9}\]  
where the second equality is obtained by using the fact that \(Q(x)=\operatorname{erfc}\left(x\,/\sqrt{2}\right)/2\) and (Equation (8.359.3), [23]), and the fourth equality is obtained by using (Equation (6.455.1), [23]). The relay node then performs NC of \(\hat{x}_{1}\) and \(\hat{x}_{2}\) at bit level to obtain the composite signal. Specifically, let \(\hat{b}_{i}=\operatorname{demod}(\hat{x}_{i})\) be the estimated information bit sequence corresponding to \(\hat{x}_{i}\), where \(i=1,2\). The composite signal is given by \(x_{\rm r}=\operatorname{mod}(\hat{b}_{1}\oplus\hat{b}_{2})\). As in [9], we assume that the processing power required by the transmit/receive circuitry at the relay node is negligible as compared to the power used for transmitting the composite signal in the broadcast (BC) phase. From Equation (3), the latter power is given by:  
\[ \begin{array}{ll}P_{\rm r}&=\frac{\sum_{m=1}^{M}\frac{F_{m}}{E_{m}}}{N}\\ &=\eta pM\big{(}P\sigma^{2}+\sigma_{\rm a}^{2}\big{)}\end{array}  \tag{10}\]  
and the sampled received (baseband) signal at the end node T\({}_{i}\) (\(i=1,2\)) in the BC phase can be expressed as:  
\[y_{i}=\sqrt{P_{\rm r}}\sum_{m=1}^{M}h_{i,m}x_{\rm r}+n_{i}^{[\rm a]}\!+\!n_{i}^ {[\rm c]} \tag{11}\]where \(n_{i}^{[a]}\sim CN\left(0,\sigma_{a}^{2}\right)\) and \(n_{i}^{[c]}\sim CN\left(0,\sigma_{c}^{2}\right)\) are the AWGN due to the antenna and that due to RF-band-to-baseband signal conversion, respectively. Assuming that \(\left\{h_{i,m}\right\}_{m=1}^{M}\) is known at T\({}_{i}\), an estimate of \(x_{r}\) is obtained as:  
\[