## 2 Vehicular Visible Light Communications and Related Work
The transmitting vehicle and the receiving vehicle are simulated with two test benches, the former equipped with two off-the-shelf vehicle rear lights, and the latter with a PIN photodiode-based PDA100A optical detector, situated at the same height \(h\) as the transmitting lamps, which can be fitted with an optical collimator based on a 2-inch diameter lens, having a FOV of \(\pm 20^{\circ}\). In order to make the measurements, it is easier to consider two cases: one where the transmitting vehicle is on one of the lanes (e.g., the second one) and the receiving vehicle is on the adjacent lane (e.g., the first one) as in Figure 0(a), and the other where both cars are on the same lane (e.g., the second lane), as in Figure 0(b). It can be noted that the latter situation is a particular case of the former, where the angle of incidence \(\alpha\) has \(0^{\circ}\), and the lateral distance \(l\) is zero. Figure 1: V2V communication scenarios: (**a**) Case one, with the communication established between two vehicles on adjacent lanes; and (**b**) case two, with the communication established between two vehicles on the same lane. The simulation scenario is further divided in measurement steps where the longitudinal distance \(L\) between the vehicles is gradually increased from 1 m to 75 m, and the measurements are made alternatively for those two cases already mentioned. In order to make an analytical analysis, only the first case will be considered, as in Figure 2, the second case being easy to calculate, as already mentioned, for \(\alpha=0^{\circ}\) and \(l=0\). The receiving vehicle will occupy alternatively the positions Rx1 and Rx2, while the transmitting vehicle will occupy the position Tx, at a variable distance \(L\) from the receiving car. The position coordinates will be \(x_{Rx1}\), \(x_{Rx2}\), \(x_{Tx}\), \(y_{Rx1}\), \(y_{Rx2}\), and \(y_{Tx}\). From Figure 2, the following system of equations can be determined:  
\[x_{Rx1}=x_{Rx2}=x_{Rx}, \tag{1}\]  
\[x_{Rx}+L=x_{Tx}, \tag{2}\]  
\[y_{Rx2}=y_{Tx}, \tag{3}\]  
\[y_{Rx1}+l=y_{Rx2}, \tag{4}\]  
\[\tan\alpha=\frac{l}{L}, \tag{5}\]  
\[d=\frac{L}{\cos\alpha}, \tag{6}\]  
where \(d\) is the distance between cars. Taking into account that the measurements will be made with and without the optical collimator, when the cars are on adjacent lanes, the photodetector will have a blind zone until \(L=1/\tan\alpha_{FOV}\) for \(\alpha\geq\alpha_{FOV}\), meaning around 9 m and 3 m, respectively. An optical wireless channel with intensity modulation/direct detection (IM/DD) can be modeled as a baseband linear system that respects the following equation [62]:  
\[y(t)=R\cdot x(t)\otimes h(t)+n(t), \tag{7}\]  
where \(x(t)\) is the instantaneous optical power of the emitter, \(y(t)\) is the photocurrent at the receiver level, \(R\) is the receiver responsivity, \(h(t)\) is the impulse response, \(n(t)\) is the signal  
Figure 2: Vehicle configuration for analytical study. independent noise, and the \"\(\otimes\)\" symbol denotes convolution. Being instantaneous optical power, \(x(t)\) is non-negative, and the average transmitted power respects the equation:  
\[P_{t}=\lim_{T\rightarrow\infty}\frac{1}{2T}\int\limits_{-T}^{T}x(t)dt \tag{8}\]  
This means that the average received power can be written as:  
\[x_{Rx1}=x_{Rx2}=x_{Rx}, \tag{1}\]0  
where the channel DC-gain is:  
\[x_{Rx1}=x_{Rx2}=x_{Rx}, \tag{1}\]1  
For \(0\leq\alpha\leq\alpha_{FOV}\), in order to estimate the influence of the angle of incidence over the VLC link performance, the BER will be used as a term of comparison. First, the channel DC-gain will be determined [62]:  
\[x_{Rx1}=x_{Rx2}=x_{Rx}, \tag{1}\]2  
where \(A\) is the active area of the photodetector, \(R_{0}(\alpha)\) is the radiant intensity of the transmitter, \(g(\alpha)\) is the collimator's gain, \(T_{s}(\alpha)\) is the transmission factor of the optical filter, and \(\gamma\) is the path loss exponent.