## 3 Performance Analysis for Single SWIPT Relaying System
### Average Error Rate Expressions  
By applying maximum ratio combining to \(y_{\rm sd}\) in (1) and \(y_{\rm rd}\) in (7) at the destination,  
\[\gamma_{\rm sd}\triangleq\frac{P_{\mathsf{s}}\left|h_{\rm sd}\right|^{2}}{ \sigma^{2}} \tag{16}\]  
represents the direct link SNR, whose PDF is given by \(f_{\gamma_{\rm sd}}(x)=\frac{1}{\bar{\gamma}_{\rm sd}}\exp\Bigl{(}-\frac{x}{ \bar{\gamma}_{\rm sd}}\Bigr{)}u(x)\), with \(\bar{\gamma}_{\rm sd}=\frac{P_{\mathsf{s}}\Omega_{\rm sd}}{\sigma^{2}}\) and MGF \(M_{\gamma_{\rm sd}}(\mathsf{s})=\frac{1}{\bar{\gamma}_{\rm sd}\mathsf{s}+1}\). The average BER for BPSK modulation over the SD, indirect, and combined links can be expressed as  
\[\begin{split} P_{\mathsf{B},\mathsf{sd}}&=\frac{1} {\pi}\int_{0}^{\pi/2}M_{\gamma_{\rm sd}}(\mathsf{s})d\theta\\ P_{\mathsf{B},\mathsf{id}}&=\frac{1}{\pi}\int_{0}^ {\pi/2}M_{\gamma_{\rm sd}}(\mathsf{s})d\theta\\ P_{\mathsf{B},\mathsf{cb}}&=\frac{1}{\pi}\int_{0}^ {\pi/2}M_{\gamma_{\rm sd}}(\mathsf{s})M_{\gamma_{\rm id}}(\mathsf{s})d\theta \end{split} \tag{17}\]  
where \(\mathsf{s}=g_{\rm PSK}/\sin^{2}(\theta)\) and \(g_{\rm PSK}=\sin^{2}(\pi/2)\)[43].  
By substituting \(M_{\gamma_{\rm id}}(\mathsf{s})\) in (17) with its approximation \(M_{\gamma_{\rm id}}^{\rm APP}(\mathsf{s})\) from (15), the approximated BER bounds for the indirect and combined links can be derived as  
\[\begin{split} P_{\mathsf{B},\mathsf{id}}&<P_{\mathsf{ B},\mathsf{id}}^{\rm APP}=\frac{1}{\pi}\int_{0}^{\pi/2}M_{\gamma_{\rm id}}^{ \rm APP}(\mathsf{s})d\theta\\ P_{\mathsf{B},\mathsf{cb}}&<P_{\mathsf{B},\mathsf{ cb}}^{\rm APP}=\frac{1}{\pi}\int_{0}^{\pi/2}M_{\gamma_{\rm sd}}( \mathsf{s})M_{\gamma_{\rm id}}^{\rm APP}(\mathsf{s})d\theta.\end{split} \tag{18}\]  
For the indirect link, the BER bound \(P_{\mathsf{B},\mathsf{id}}\) in (17) can be further approximated asymptotically under the high SNR assumption as  
\[\begin{split} P_{\mathsf{B},\mathsf{id}}&\approx P_{ \mathsf{B},\mathsf{id}}^{\rm Asym}\\ &=\frac{1}{\bar{\mathcal{A}}_{\rm\bar{\gamma}sr}+1}+\frac{1}{4\bar {\mathcal{V}}_{\rm\bar{\gamma}rd}\Omega_{\rm sr}/\ln(4\bar{\mathcal{V}}_{\rm rd }\Omega_{\rm sr}+1)}.\end{split} \tag{19}\]  
The detailed derivation for \(P_{\mathsf{B},\mathsf{id}}^{\rm Asym}\) is provided in Appendix A.  
The analytical BER expression in (19) introduces a new analytical regime for evaluating SWIPT-based relaying systems. It allows the SWIPT relay to be interpreted as an equivalent conventional AF relay with a dedicated power source, where the SNRs of the SR and RD links are given by \(\bar{\gamma}_{\rm sr}\) and  
\[\bar{\gamma}^{\prime}_{\rm rd}=\frac{\bar{\gamma}_{\rm rd}\Omega_{\rm sr}}{\ln(4 \bar{\gamma}_{\rm rd}\Omega_{\rm sr}+1)}, \tag{20}\]  
respectively. For example, it is observed that the BER performance of the SWIPT relay system is primarily determined by the weaker link between the SR link SNR, \(\bar{\gamma}_{\rm sr}\), and the modified RD link SNR, \(\bar{\gamma}^{\prime}_{\rm rd}\). Moreover, it is evident from (19) that the SWIPT relay does not experience any diversity loss compared to a conventional AF relay powered by its own energy source.  
By applying the result (14-4-18) in [44] with appropriate manipulations, the BER expression for the combined link, \(P_{B,\rm cb}\) in (17), can be asymptotically approximated as [24]  
\[P_{B,\rm cb}\approx P_{B,\rm cb}^{\rm Asym}=\frac{3}{4\bar{\gamma}_{\rm sd}+1 }P_{B,\rm id}^{\rm Asym}. \tag{21}\]