## 2 Theoretical Background  
This section presents the required formulas to model the V2V VLC system scenario in MATLAB. The system consists of two LED transmitters, Tx1 and Tx2, and a photodiode receiver, Rx. The transmitted light signal is OOK-modulated. The signal travels through the channel and reaches the receiver. For simplicity, we assume the LEDs to be Lambertian emitters with a certain FOV. In addition, we assume that only additive white Gaussian noise (AWGN) is present in the system, and ignore effects such as weather conditions, air turbulence, ambient noise from sunlight, and multipath fading. Instantaneous transmitted power from a Lambertian emitter can be written as [41]  
\[X(\phi,t)=X(\phi)e^{i\omega t} \tag{1}\]  
where \(\phi\) is the angle of irradiance, \(\omega\) is the angular frequency, and  
\[X(\phi)=X_{0}\cos^{m}(\phi) \tag{2}\]  
where m is the order of the Lambertian emission:  
\[m=-\frac{\ln(2)}{\ln(\cos(\phi_{\frac{1}{2}}))} \tag{3}\]  
and \(\phi_{\frac{1}{2}}\) is the transmitter's half FOV. The output signal \(Y(t)\) of an optical wireless channel can be expressed as [41]  
\[Y(t)=\gamma X(t)\otimes h(t)+N(t) \tag{4}\]  
where \(\gamma\) represents the photodetector responsivity, \(h(t)\) is the impulse response, \(\otimes\) denotes convolution, and \(N(t)\) is the AWGN. The average transmitter power is calculated as follows [5]:  
\[P_{t}=\frac{1}{T}\int_{0}^{T}X(t)\,dt \tag{5}\]  
where \(T=\frac{2\pi}{\omega}\) is the period. Once \(P_{t}\) is known, the received power can be calculated by [5]  
\[P_{r}=H(0)P_{t} \tag{6}\]  
where \(H(0)=\int_{-\infty}^{\infty}h(t)\,dt\) is the DC channel gain, which is given by [5]  
\[H(0)= \begin{cases}\frac{(m+1)A}{2\pi D^{2}}\cos^{m}(\phi)T_{s}(\psi)g(\psi) \cos(\psi),&0\leq\psi\leq\psi_{c}\\ 0,&\text{otherwise}\end{cases}  \tag{7}\]  
where \(A\) is the area of the photodetector, \(D\) is the distance between the transmitter and receiver, \(\phi\) is the angle of irradiance, \(\psi\) is the angle of incidence, \(T_{s}(\psi)\) is the signal transmission coefficient of an optical filter, \(g(\psi)\) is the gain of the optical concentrator, and \(\psi_{c}\) is the receiver's half FOV. The gain achieved by an idealized concentrator with an internal refractive index \(n\) is given by [5]  
\[g(\psi)= \begin{cases}\frac{n^{2}}{sin^{2}(\psi_{c})},&0\leq\psi\leq\psi_{c}\\ 0,&\text{otherwise}\end{cases}  \tag{8}\]  
The SNR represents the performance quality of a given channel. The signal component is given by [41]  
\[S=\gamma^{2}P_{rSignal}^{2} \tag{9}\]  
where  
\[P_{rSignal}=\frac{1}{T}\int_{0}^{T}\Biggl{(}\sum_{i=1}^{L\text{EDs}}h_{i}(t) \otimes X(t)\Biggr{)}dt \tag{10}\]  
The Gaussian noise is the sum of the contributions from shot noise, thermal noise, and intersymbol interference due to an optical path difference. The total variance \(N\) is given as [41]  
\[N=\sigma_{\text{shot}}^{2}+\sigma_{thermal}^{2}+\gamma^{2}P_{rISI}^{2} \tag{11}\]  
where  
\[P_{rISI}=\lim_{T^{\prime}\rightarrow\infty}\frac{1}{T^{\prime}-T}\int_{T}^{T^ {\prime}}\bigl{(}\sum_{i=1}^{LEDs}h_{i}(t)\otimes X(t)\bigr{)}\,dt \tag{12}\]  
The shot noise and thermal noise contributions are given as [41]  
\[\sigma_{\text{shot}}^{2}=2 qR\Bigl{(}P_{\text{Signal}}+P_{r,\text{ISI}}\Bigr{)} B_{\text{en}}+2 qI_{\text{bg}}I_{2}B_{\text{en}} \tag{13}\]  
\[\sigma_{\text{thermal}}^{2}=\frac{8\pi kT_{k}}{G}\delta AI_{2}B_{\text{en}}^{2 }+\frac{16\pi^{2}kT_{k}E}{g_{m}}\delta^{2}A^{2}I_{3}B_{\text{en}}^{3} \tag{14}\]  
where \(q\) is the electron charge, \(B_{\text{en}}\) is the equivalent noise bandwidth, \(I_{\text{bg}}\) is the background current, \(I_{2}\) is the noise bandwidth factor, \(k\) is the Boltzmann's constant, \(T_{k}\) is the absolute temperature, \(G\) is the open-loop voltage gain, \(\delta\) is the fixed capacitance of the photodetector per unit area, \(E\) is the field-effect transistor (FET) channel noise factor, \(g_{m}\) is the FET transconductance, \(I_{3}=0.0868\), and \(P_{r,\text{ISI}}\) is the power received through intersymbol interference.