## 2 System Model
### Single SWIPT Relaying System Model  
From this point onward, we describe the SWIPT AF relay system. Let \(h_{\text{sd}}\), \(h_{\text{sr}}\), and \(h_{\text{rd}}\) denote the complex channel gains of the source-to-destination (SD), source-to-relay (SR), and relay-to-destination (RD) links, respectively. These channels are assumed to undergo independent Rayleigh fading and are thus modeled as mutually independent complex Gaussian random variables with zero mean and variances given by \(\Omega_{\text{sd}}\triangleq E\Big{[}\left|h_{\text{sd}}\right|^{2}\Big{]}\), \(\Omega_{\text{sr}}\triangleq E\Big{[}\left|h_{\text{sr}}\right|^{2}\Big{]}\), and \(\Omega_{\text{rd}}\triangleq E\Big{[}\left|h_{\text{rd}}\right|^{2}\Big{]}\). Here, \(E[\cdot]\) denotes the stochastic expectation.  
Due to the half-duplex constraint of the relay, communication occurs over two time slots. In the first slot, the source broadcasts the signal \(x_{s}\), with \(E[x_{s}]=0\) and \(E\Big{[}\left|x_{s}\right|^{2}\Big{]}=P_{s}\), to both the relay and the destination. The received signals at the relay and the destination, denoted by \(y_{\text{sr}}\) and \(y_{\text{sd}}\), respectively, are given by  
\[y_{\text{sr}} =h_{\text{sr}}x_{s}+n_{\text{sr}} \tag{1}\] \[y_{\text{sd}} =h_{\text{sd}}x_{s}+n_{\text{sd}}\]  
where \(n_{\text{sr}}\) and \(n_{\text{sd}}\) represent the complex additive white Gaussian noise (AWGN) at the relay and destination, respectively. These are assumed to be zero-mean and mutually independent, with variances \(E\Big{[}\left|n_{\text{sr}}\right|^{2}\Big{]}=E\Big{[}\left|n_{\text{sd}} \right|^{2}\Big{]}=\sigma^{2}\).  
The relay's power splitter divides the received signal \(y_{\text{sr}}\) into two components, \(y_{\text{sr}}^{E}\) and \(y_{\text{sr}}^{I}\), for EH and IP, respectively. By introducing a PS factor \(\rho\) with \(0<\rho<1\)[28], we can express  
\[y_{\text{sr}}^{E}=\sqrt{\rho}y_{\text{sr}}=\sqrt{\rho}(h_{\text{sr}}x_{s}+n_{ \text{sr}}). \tag{2}\]  
The total energy harvested at the relay during the first time slot is then given by  
\[\text{E}_{h}=E\left[\left|y_{\text{sr}}^{E}\right|^{2}\right]=\eta\rho P_{s} |h_{\text{sr}}|^{2}T \tag{3}\]  
where \(\eta\in(0,1]\) is the energy conversion efficiency, and \(T\) denotes the time duration of the time slot.  
Since the AF relay uses an equal duration for receiving and forwarding data, the available power \(P_{r}\) at the relay during the second slot is  
\[P_{r}=\eta\rho P_{s}|h_{\text{sr}}|^{2}. \tag{4}\]  
Meanwhile, the information component \(y_{\text{sr}}^{I}\) is given by  
\[y_{\text{sr}}^{I}=\sqrt{1-\rho}y_{\text{sr}}+n_{c}=\sqrt{1-\rho}h_{\text{sr}} x_{s}+n_{r} \tag{5}\]  
where \(n_{r}=\sqrt{1-\rho}n_{\text{sr}}+n_{c}\), and \(n_{c}\) denotes the noise introduced during RF-to-baseband conversion. For simplicity, we assume \(n_{\text{sr}}\) and \(n_{c}\) are mutually independent with zero mean, and variances \(E\Big{[}\left|n_{c}\right|^{2}\Big{]}=\sigma^{2}\) and \(\sigma_{r}^{2}=E\Big{[}\left|n_{r}\right|^{2}\Big{]}=(2-\rho)\sigma^{2}\).  
During the second time slot, the relay forwards \(y_{\text{sr}}^{I}\) to the destination using the available power \(P_{r}\). The transmitted signal from the relay is  
\[x_{r}=Gy_{\text{sr}}^{I} \tag{6}\]  
where \(G=\sqrt{\frac{\eta\rho P_{s}|h_{\text{sr}}|^{2}}{(1-\rho)|h_{\text{sr}}|^{2}P _{s}+\sigma_{r}^{2}}}\) is the amplification gain. Accordingly, the signal received at the destination from the relay is\[y_{\rm rd} =h_{\rm rd}x_{r}+n_{\rm rd} \tag{7}\] \[=h_{\rm rd}G\sqrt{1-\rho}h_{\rm sr}x_{s}+h_{\rm rd}G\Big{(}\sqrt{1- \rho}n_{\rm sr}+n_{c}\Big{)}+n_{\rm rd}\]  
where \(n_{\rm rd}\) is the AWGN at the destination in the second time slot, with \(E[n_{\rm rd}]=0\) and \(E\Big{[}{|n_{\rm rd}|}^{2}\Big{]}=\sigma^{2}\). It is assumed that all noise terms \(\{n_{\rm sd},n_{\rm sr},n_{c},n_{\rm rd}\}\) are mutually independent.