# Outdoor Visible Light Communication in Intelligent Transportation Systems: Impact of Snow and Rain
## 2 System Model
### Channel Modeling in Outdoor VLC Environment and Received Power  
Most wireless communication systems can be modeled as having an output signal Y(t) and an input signal X(t). The output signal can be calculated as  
\[\mathrm{Y(t)}=\mathrm{X(t)}\otimes\mathrm{h(t)}+\mathrm{N(t)} \tag{9}\]  
where \(\bigotimes\) denotes convolution, h(t) is the channel impulse response, and N(t) is additive noise. In optical wireless communication channels, the direct current (DC) gain, H\({}_{\mathrm{LoS}}\), [13] is given by:  
\[\mathrm{H_{LoS}}(0)= \begin{cases}\frac{(\mathrm{m}+1)\mathrm{A}}{2\pi\mathrm{ d}^{2}}\cos^{\mathrm{m}}(\varphi)\mathrm{T_{s}}(\psi)g(\psi)\cos(\psi),&0\leq\psi\leq \Psi_{\mathrm{c}}\\ 0,&\psi>\Psi_{\mathrm{c}}\end{cases} . \tag{10}\]  
The outdoor VLC channel is strongly affected by atmospheric turbulence due to the presence of molecular and aerosols in the air, as shown in Figure 6. These molecules, such as rain and snow, attenuate the light wave as the same way as electromagnetic radiation or RF signals, causing light scattering. Moreover, they attenuate the transmitted light signal, which reduces the received power in the photodiode. Figure 5: The compound parabolic concentrator (CPC) non-imaging concentrator model. Figure 6: Simplified outdoor VLC model in an intelligent transport system (ITS). The absorption and scattering of light through the atmosphere are figured by the exponential Beers-Lambert Law [19]. \[\tau(d)=e^{-\gamma(\lambda)d} \tag{11}\]  
where \(\tau(d)\) is the transmittance at the distance d from the transmitter, and \(\gamma(\lambda)\) is the extinction coefficient per unit of length. Therefore, Equation (19) can be written as  
\[H_{\text{los}}(0)= \begin{cases}\frac{(m+1)A}{2\pi d^{2}}\cos^{m}(\varphi)T_{s} (\psi)g(\psi)\tau(d)\cos(\psi),&0\leq\psi\leq\Psi_{c}\\ 0,&\psi>\Psi_{c}\end{cases} . \tag{12}\]  
Thus, the desired power in the APD receiver can be obtained as [13]  
\[P_{\text{rx-los}}(t)=P_{\text{tr}}(\varphi)\cdot H_{\text{los}}(0). \tag{13}\]  
The received signal is given by [13]  
\[Rx_{s}=P_{\text{rx-los}}(t)+Noise(t). \tag{14}\]  
Outdoor VLC applications are subject to multiple external noise sources that influence the communication channel. The major noise source in VLC is represented by the background light. The background light can be either from artificial sources or from natural sources. The bright skylight and the direct sunlight are the natural light sources that can saturate the photo-element and make it blind. Additionally, there are other background noise sources at the receiver side, such as thermal noise (generated in the receiver electronic circuit components such as resistors and capacitors). Following the analysis of [13] in estimating the receiver noise, the total noise variance is approximately given by  
\[\sigma_{\text{tot}}^{2}\approx 2 qr(P_{\text{rx-los}}+P_{a})B+\frac{8\pi k_{B}T}{ G}\eta AI_{2}B^{2}+\frac{16\pi^{2}k_{B}T\,\Gamma}{g_{m}}\eta^{2}A^{2}I_{3}B^{3}. \tag{15}\]  
The first term is the shot noise, which is considered to be the predominant noise source in wireless optical communications [13], \(P_{a}\) is the ambient light power, \(r\) is the detector responsivity, \(q\) is the electronic charge, and \(I_{2}\) and \(I_{3}\) are constants referring to the noise bandwidth factor, while \(B\) is the desired bandwidth, relying on the modulation technique used and the bit rate. The thermal noise of the feedback resistor is represented in the second term, \(k_{B}\) the is Boltzmann constant, \(\eta\) is the capacitance per unit area, \(T\) is the absolute temperature, and \(G\) is the open-loop voltage gain. Both the shot and the thermal noise are modeled as additive white Gaussian noise (AWGN). The third term describes the thermal noise from the field-effect transistor (FET) channel resistance, \(\Gamma\) is the FET channel noise factor, and \(g_{m}\) is the FET conductance.