# Outdoor Visible Light Communication in Intelligent Transportation Systems: Impact of Snow and Rain
## 2 System Model
### Non-Imaging Receiver Model  
The produced light emitted by the transmitter is captured in a photodetector mounted on the vehicle as shown in Figure 4. This is an optoelectronic transducer that collects light and transforms it  
Figure 3: LED Lambertian emission pattern and coverage area represented by \(\varphi_{1/2}\) and index m.  
Figure 2: Light-emitting diode (LED)-based traffic light block diagram.  
into a small amount current proportional to the collected light. The advantages of a transimpedance amplifier (TIA) include its stable gain and low distortion. By applying a narrow band-pass filter at the input stage, it would be a good solution to remove both low and high-frequency noise from the generated voltage. At the end, additional signal conditioning is performed, setting the output signal amplitude and the direct current (DC) level in accordance with the analog to digital converter (ADC) input specifications. The sampled signal is decoded, and the digital data is extracted [14].  
In this proposed system, a single avalanche photodetector (APD) is considered with its own vertical inclination angle, \(\theta\), as shown in Figure 4. The detector is modeled as an active area collecting the radiation incident angles, \(\psi\), which are smaller than the detector FOV. The effective collection area of the detector, \(\mathrm{A_{eff}}\), is given by [13]  
\[\mathrm{A_{eff}}(\psi)=\begin{cases}\mathrm{AT_{s}}(\psi)\mathrm{g}(\psi)\cos (\psi),&0\leq\psi\leq\Psi_{\mathrm{c}}\\ 0,&\psi>\Psi_{\mathrm{c}}\end{cases} \tag{6}\]  
where A is the detector physical area, \(\mathrm{T_{s}}(\psi)\) is the optical band-pass filter of transmission, \(\mathrm{g}(\psi)\) is the non-imaging concentrator gain, and \(\Psi_{\mathrm{c}}\) is the FOV of the receiver.  
Although, ideally, a large-area detector would be suitable for an outdoor environment to collect as much power as possible, this would result in more noise entering the receiver. Therefore, the use of a non-imaging concentrator is a cost-effective solution in order to increase the overall effective collection area and to exhibit a trade-off between gain and FOV. The optical gain of an ideal non-imaging concentrator having an internal refractive index \(\mathrm{n}\)[13] is  
\[\mathrm{g}(\psi)=\begin{cases}\frac{\mathrm{n}^{2}}{\sin^{2}\,\,\Psi_{\mathrm{ c}}},&0\leq\psi\leq\Psi_{\mathrm{c}}\\ 0,&\psi>\Psi_{\mathrm{c}}\end{cases}. \tag{7}\]  
The compound parabolic concentrator (CPC) is used in the proposed system. The CPC is an angle transforming device that can absorb and focus the light from a large input area down to a smaller detector area; so, it is called non-imaging receiver, as a single photodetector is coupled with a concentrator, as shown in Figure 5. A CPC can achieve a high gain at the expense of a narrow FOV, making it suitable for direct links. CPC-based receivers have a FOV typically \(\Psi_{\mathrm{c}}\leq 90^{\circ}\)[16].  
Figure 4: Visible light communication (VLC) receiver block diagram.  
To achieve near-ideal performance, a 10 nm band-pass filter can be coupled with a CPC having a FOV = 10\({}^{\circ}\), which has been indicated in [16; 17]. In order to achieve the maximum area of reception, vertical inclination and FOV angles must fulfill the term below [18].  
\[\theta+\mathrm{FOV}\leq 90^{\circ} \tag{8}\]