## 2 System Model and Channel Model
### DF Transmission Protocol  
The communication between S and D is accomplished in two time slots, \(T_{1}\) and \(T_{2}\), for the first and second hop, respectively. The relay harvests energy during \(T_{1}\) only, as simultaneous harvesting and discharging of power increases the complexity at the node. The S employs sub-carrier intensity modulation (SIM) to convert RF signal vector \(r\) withelectrical power \(\delta\) to an optical signal. A DC bias \(A\in[A_{\text{min}},A_{\text{max}}]\) (where \(A_{\text{min}}\) and \(A_{\text{max}}\) are minimum and maximum values of DC bias, respectively) is added to the RF signal to ensure a non-negative optical signal. Let \(P_{s}\) represent the electrical power of S transmitting the optical signal vector \(\mathbf{s_{I}}\); then, we can write  
\[\mathbf{s_{I}}=\sqrt{P_{s}}[\delta r+A] \tag{19}\]  
where \(\delta\) is the electrical to optical conversion coefficient. In order to prevent clipping due to non-linearity of the laser diode, such that it operates in the linear region, \(\delta\) should satisfy the following constraint [31]. \[\delta\leq\text{min}\bigg{(}\frac{A-A_{\text{min}}}{\rho},\frac{A_{\text{max}} -A}{\rho}\bigg{)} \tag{20}\]  
The electrical signal at the output of the PD can be expressed as  
\[y_{FSO}=h_{FSO}\mathbf{s_{I}}+n_{FSO} \tag{21}\]  
where \(h_{FSO}=(P_{s}^{2}\eta^{2}/\sigma_{FSO}^{2})I_{FSO}\) is the channel coefficient of S-D link, where \(\eta\) is the optical to electrical conversion coefficient. The noise \(n_{FSO}\) is due to circuit noise as well as high-intensity background illumination and is conventionally modeled as being zero mean additive white Gaussian noise. \(\sigma_{FSO}^{2}\) is the variance of additive white Gaussian noise with zero mean. In addition, since BER and outage probability are derived from atmospheric turbulence and pointing errors, we propose a method to estimate the atmospheric channel coefficient of S-D link \(h_{FSO}\) using BER and outage probability. The atmospheric channel coefficient estimation model is shown in Figure 3. We decompose the electrical signal at the output of the PD \(y_{FSO}\) using light gradient boosting machine (LightGBM) and then input the BER and outage probability together into the bi-directional long short-term memory (BiLSTM) to estimate the atmospheric channel coefficient. Figure 3: Atmospheric channel coefficient estimation model. BiLSTM is a combination of forward LSTM and backward LSTM, which uses a bidirectional input structure to capture sequence features from both positive and negative directions, allowing it to learn the effective information of features from multiple perspectives. BiLSTM has the same basic structure as LSTM, including an input gate, forgetting gate, output gate and memory unit, as shown in Figure 4.  
The function of the input gate is to control the input of new information, and the computational principle is as follows:  
\[i_{t}=\sigma(w_{i}[h_{t-1},x_{t}]+b_{i}) \tag{22}\]  
where \(h_{t-1}\) is the output vector of the LSTM at the previous moment, \(x_{t}\) is the input vector of the LSTM at the current moment, \([h_{t-1},x_{t}]\) represents the concatenation of the two vectors, \(w_{i}\), \(b_{i}\) and \(i_{t}\) are the weight vector, bias vector and calculation result of the input gate, respectively, and \(\sigma\) is the Sigmoid function. The function of the forgetting gate is to control the retention and forgetting of information and it is calculated as follows:  
\[f_{t}=\sigma(w_{f}[h_{t-1},x_{t}]+b_{f}) \tag{23}\]  
\[c_{t}=f_{t}\odot c_{t-1}+i_{t}\text{tanh}\odot(w_{c}[h_{t-1},x_{t}]+b_{c}) \tag{24}\]  
where \(w_{f}\), \(b_{f}\) and \(f_{t}\) are the weight vector, bias vector, and calculation result of the forgetting gate, respectively; \(c_{t-1}\) is the memory unit of the previous moment, \(\odot\) refers to the point-by-point multiplication of the elements in the vector, \(w_{c}\) and \(b_{c}\) are the weight vector and bias vector of the memory unit, respectively, and \(c_{t}\) is the memory unit of the current moment.