## 2 VLC System Model for Underground Mining
### Optical Propagation Model  
Considering that among the characteristics of the mining environments, there must be multiple \(T_{x}\)s and \(R_{x}\)s, there will also be multiple PDs distributed in \(R_{x}\). Thus, we will denote \(N_{T_{x}}\) as the total number of \(T_{x}\), whose indexes for any \(T_{x}\) is \(a=1,2\ldots,N_{T_{x}}\). We also define the number of PDs in the \(R_{x}\) as \(N_{PD}\), whose indexes for any PD is \(i=1,2\ldots,N_{PD}\). Therefore, the equivalent base-band model that we will use to describe an underground mining IM/DD VLC link is given as follows [23]\[y_{p}(t)=\tau\sigma\xi P_{a_{op}}x_{a_{op}}(t)\otimes h_{op,p}(t)+\sum_{a=1,a\
eq a _{op}}^{N_{T_{x}}}\tau\sigma\xi P_{a}x_{a}(t)\otimes h_{a,p}(t)+n_{R_{x}}(t), \tag{9}\]  
where \(y_{p}(t)\) is the electrical signal received by the \(p^{th}\) PD, \(\tau\) is the electric-optical conversion efficiency \(\sigma\) is the PD responsivity, \(\xi\) is the \(T_{x}\) modulation index, and \(\otimes\) is the convolution operator. As there are multiple \(T_{x}\) and \(R_{x}\) with several PDs, we will assume for this model that only one \(T_{x}\) should be selected as the optimum, where \(a_{op}\) is the index of this \(T_{x}\). In later sections, we will explain in more detail the selection or combination mechanisms of several signals emitted by the \(T_{x}\) of the system. Therefore, \(x_{a_{op}}(t)\) and \(x_{a}(t)\) are the electrical signals transmitted from \(T_{x_{op}}\) and \(T_{x_{a}}\), respectively, \(h_{op,p}(t)\) is the channel impulse response between the \(T_{x_{op}}\) and the \(p^{th}\) PD, \(h_{a,p}(t)\) is the channel impulse response between the \(T_{x_{a}}\) and the \(p^{th}\) PD, \(P_{a_{op}}\) and \(P_{a}\) are, respectively, the average output optical power from \(T_{x_{op}}\) and \(T_{x_{a}}\).  
An important point considered in our work for a complete propagation model is the noise. We will consider \(n_{R_{x}}\) as the additive noise in \(R_{x}\), which includes two types of noise that particularly affect mining environments, shot noise \(\sigma^{2}_{R_{x},shot}\) and thermal noise \(\sigma^{2}_{R_{x},thermal}\)[45; 46]. Both noises can be modeled as additive white Gaussian noises (AWGNs), whose variances are determined by the following expressions [11; 45; 47],  
\[\sigma^{2}_{R_{x},shot}=2q\gamma P_{R_{x}}B_{n}+2 qI_{bg}I_{2}B_{n}, \tag{10}\]  
\[\sigma^{2}_{R_{x},thermal}=\frac{8\pi\kappa T_{k}}{G}\eta A_{p}I_{2}B_{n}^{2}+ \frac{16\pi^{2}\kappa T_{k}\Gamma}{g_{m}}\eta^{2}A_{p}^{2}I_{3}B_{n}^{3}. \tag{11}\]  
\(\sigma^{2}_{R_{x},shot}\) contains two components; the former corresponds to photon fluctuation noise or quantum noise, where \(q\) is the electronic charge, \(B_{n}\) is the noise bandwidth, and \(P_{R_{x}}\) is the total power received by \(R_{x}\) that contains the power of the desired \(T_{x}\) and the power by the inter-symbol interference (ISI) caused by the others \(T_{x}\). Therefore, \(P_{R_{x}}\) can be represented as \(P_{R_{x}}\)= \(P_{a_{op}}+\sum\limits_{a=1,a\
eq a_{op}}^{N_{T_{x}}}P_{a}\). The latter corresponds to the dark current and excess noise, where \(I_{bg}\) is the background current due to the ambient light and the noise bandwidth factor is \(I_{2}\), equal to 0.562 [11; 45; 47]. In the thermal noise \(\sigma^{2}_{R_{x},thermal}\) the former and latter components represent feedback-resistor noise and FET channel noise, respectively. Here, \(\kappa\) is Boltzmann's constant, \(T_{k}\) is absolute temperature, \(G\) is the open-loop voltage gain, \(\eta\) is the fixed capacitance of PD per unit area, \(\Gamma\) is the Field-Effect Transistor (FET) channel noise factor, \(g_{m}\) is the FET trans-conductance, and the noise bandwidth factor is \(I_{3}\), equal to 0.0868 [11; 45; 47].  
It is worth noting that the shot noise is proportional to the power that \(R_{x}\) receives, and thermal noise is depends on \(A_{p}\) in a sophisticated way, so these factors are important when implementing optical components in underground mining environments.