# The Deep Deterministic Policy Gradient Algorithm Based on RIS Technology in a Coal Mine Tunnel
## 2 System Model
### Coal Mine Channel Modeling and Reflection Characterization Analysis
\(f(\theta_{n})\) defined as \(f_{\text{sin}}(\theta_{n})=\left(\frac{\text{sin}(\theta_{n}-\mu)+1}{2}\right)^ {\alpha}\); therefore,  
\[\beta(\theta_{n})=(1-\beta_{\text{min}})\left(\frac{\text{sin}(\theta_{n}-\mu)+ 1}{2}\right)^{\alpha}+\beta_{\text{min}} \tag{4}\]  
where \(\mu\) represents the horizontal offset of the phase change, which determines the phase difference between the function and the angle \(\theta_{n}\), and \(\alpha\) controls the steepness of the function curve; the larger \(\alpha\) is, the steeper the curve is near \(\mu\), and vice versa, the flatter it is. \(\beta_{\text{min}},\mu\geq 0,\alpha\geq 0\) depends on the hardware implementation constants of the RIS. However, in actual situations, due to hardware limitations, the horizontal offset \(\mu\) of the phase change will produce errors, resulting in phase errors in the reflected signal, and perfect phase control cannot be achieved. In addition, the limited number of RIS reflective elements limits the accuracy of phase adjustment, making it impossible to achieve optimal beamforming, ultimately reducing system gain. If the signal is ideally reflected on the RIS, then \(\left|\phi_{n}\right|^{2}=1\), i.e., \(\beta_{\text{min}}=1\) or \(\alpha=0\). Assuming the transmitted signal is \(S(t)\), with zero mean and unit variance, \(\mathbf{S}=\left[s_{1},s_{2},\cdots,s_{K}\right]^{T}\in\mathbb{C}^{K\times 1}\), then the received signal after RIS is expressed as follows [5]:  
\[y=\sqrt{\mathbf{P}}\mathbf{H}\mathbf{G}\mathbf{S}+W=\sqrt{\mathbf{P}}\mathbf{H }_{mn}^{H}\mathbf{\Phi}\mathbf{H}_{mi}\mathbf{G}\mathbf{S}+W \tag{5}\]  
where \(\mathbf{P}\) is the transmit power, \(\mathbf{P}=diag[p_{1},p_{2},\cdots,p_{K}]\in\mathbb{R}^{K\times K}\), and \(\mathbf{G}\) is the beamforming matrix applied at BS, \(\mathbf{G}\in\mathbb{C}^{M\times K}\). \(W\) represents the total noise in the coal mine tunnel, which includes the background noise \(w_{\text{bg}}\), i.e., continuous noise generated by mechanical operation and the impulse interference noise \(w_{\text{imp}}\), i.e., some sudden noise (such as equipment failure). The total noise can be expressed as follows:  
\[W=w_{\text{bg}}+w_{\text{imp}} \tag{6}\]  
where the background noise is additive Gaussian white noise, expressed as \(w_{\text{bg}}\sim\mathcal{N}\left(0,\sigma^{2}\right)\). The impulse interference noise is \(w_{\text{imp}}=B\cdot G_{a}\), where \(B\) is a Bernoulli random process with mean 0 and variance 1, taking the value of 0 or 1, indicating whether there is impulse noise generated at a certain moment; \(G_{a}\) is a Gaussian random process related to \(B\), which represents the amplitude of the impulse and obeys \(\mathcal{N}\left(0,\sigma_{\text{imp}}^{2}\right)\). Therefore, the coal mine noise can be represented by an independent and identically distributed Bernoulli-Gaussian process [5], that is:  
\[W=w_{\text{bg}}+B\cdot G_{a} \tag{7}\]  
The received signal of the \(k\)th user can be expressed as follows:  
\[y_{k}=\sqrt{\mathbf{P}}\mathbf{H}_{nm,k}^{H}\mathbf{\Phi}\mathbf{H}_{nu} \mathbf{G}\mathbf{S}+W_{k} \tag{8}\]  
Further, it can be written as follows:  
\[y_{k}=\sqrt{P_{k}}\mathbf{H}_{nm,k}^{H}\mathbf{\Phi}\mathbf{H}_{nu}g_{k} \mathbf{s}_{k}+\sqrt{P_{n}}\sum_{n,n\
eq k}^{K}\mathbf{h}_{nm,k}^{H}\mathbf{ \Phi}\mathbf{H}_{nu}g_{n}\mathbf{s}_{n}+W_{k} \tag{9}\]  
where the first term is the expected signal of the \(k\)th user, the second term is the interference caused by the signals of all other users (\(n\
eq k\)) to the \(k\)th user, i.e., Co-Channel Interference (CCI), and \(\mathbf{g}_{k}\) is the \(k\)th column vector of the matrix \(G\).