# A Symmetric Matrix-Aided MIMO to Improve Reliability for Maritime Visible Light Communications
## 2 Maritime MIMO-VLC Channel Models  
In this section, we will describe three channel models. Referring to [16; 17], we assume that the deviation caused by the small displacement of the receiver or transmitter or the aperture reception is negligible. For the atmospheric and underwater channels, denote \(h_{p}\) and \(h_{t}\) as the path loss coefficient and channel fading coefficient, respectively. Then the channel coefficient \(h\) can be expressed as [18]:  
\[h=h_{p}\cdot h_{t}. \tag{1}\]  
For the underwater channel, we use the Monte Carlo (MC) method to approximate \(h_{p}\)[19]. For the atmospheric channel, the path loss coefficient \(h_{p}\) depends on the geometric loss led by the receiver aperture area, transmitter divergence angle and the atmospheric turbulence caused by the absorption and scattering [20]. The \(h_{p}\) can be received as follows:  
\[h_{p}=\frac{D}{\pi(\phi L_{au}/2)^{2}}\exp(-\beta_{v}L_{au}), \tag{2}\]  
where \(D\) is the area of the receiver's aperture, \(L_{au}\) means the propagation distance, \(\phi\) represents the divergence angle, and \(\beta_{v}\) is on behalf of the atmospheric extinction coefficient. The channel response \(h_{t}\) is related to turbulence. The double gamma model is used to describe the moderate to strong turbulence, while the turbulence intensity is weak, the log-normal model is applied [13; 20]. The log-irradiance variance \(\sigma_{I}^{2}\) is used to reshape the turbulence intensity, which can be [20; 21]:  
\[\sigma_{I}^{2}=1.23C_{n}^{2}k^{7/6}L_{au}^{11/6}, \tag{3}\]  
where \(C_{n}^{2}\) is the turbulence strength parameter, whose value ranges from \(10^{-17}\) to \(10^{-12}\). \(k=2\pi/\lambda\) is the phase constant. To describe the statistical characteristics of the weak turbulence, with reference to [22], we will use the log-normal distribution, which is the most widely used model of irradiance probability density function(PDF). Namely, the PDF of \(h_{t,w}\) can be expressed as [22]:  
\[f(h_{t,w})=\frac{1}{\sqrt{2\pi\sigma_{I}}h_{t,w}}\exp(-\frac{(\ln(h_{t,w})+ \sigma_{I}^{2}/2)^{2}}{2\sigma_{I}^{2}}). \tag{4}\]  
On the other hand, in the case of moderate to strong turbulence, the log-normal distribution data show a significant deviation from the measured data, because the log-normal distribution underestimates the role of the tail. In the recent research work, a gamma-gamma probability density function model is proposed in [23], which is a two-parameter distribution derived based on the theory of double random scintillation. Let \(h_{t,s}\) denote the channel condition corresponding to the moderate to strong turbulence, which can be modeled as a stationary stochastic process characterized by a gamma-gamma distribution [23]:  
\[f(h_{t,s})=\frac{2(\xi\gamma)^{(\xi+\gamma)/2}}{\Gamma(\xi)\Gamma(\gamma)}h_{t,s}^{(\xi+\gamma)/2-1}B_{\xi-\gamma}(2\sqrt{\xi\gamma h_{t,s}}), \tag{5}\]  
where \(\Gamma(\cdot)\) represents the gamma function, \(B_{\xi-\gamma}(\cdot)\) represents the modified Bessel function, and the order is \(\xi-\gamma\)[18]. Here, \(\xi\) and \(\gamma\) are the effective number of small-scale and large-scale eddies of the turbulent environment, where \(\xi=\{\exp[\frac{0.49\sigma_{R}^{2}}{(1+1.11\sigma_{R}^{2/5})^{7/6}}]-1\}^{-1}\), \(\gamma=\{\exp[\frac{0.51\sigma_{R}^{2}}{(1+1.69\sigma_{R}^{2/5})^{5/6}}]-1\}^ {-1}\)[18]. The air-water interface channel is more sophisticated because the medium on both sides is different and the reflection or refraction and other phenomena in the propagation process will happen, resulting in distortion [24]. Here we use the Monte Carlo method to describe the air-water interface channel regime [25]. When the light propagates the interface, the direction of light is determined by the azimuth angle \(\theta_{a}\) and the polar angle \(\theta_{p}\), if wind is present.