## 4 SNR Estimation  
The signal-to-noise ratio (SNR) represents a very significant parameter for the communication systems. This parameter is the ratio of the signal's power to the power of the noise and can be calculated as [40; 44]  
\[SNR=\frac{P_{r}}{\sigma_{tot}^{2}} \tag{2}\]  
where \(\sigma^{2}_{tot}\) is the total noise variance and is given as [40; 44]  
\[\sigma_{tot}^{2}=\sigma_{thermal}^{2}+\sigma_{shot}^{2} \tag{3}\]  
Figure 3: Ceiling overview of the light-emitting diode (LED) pattern.  
with \(\sigma_{thermal}\) being the variance of thermal noise that is generated by the thermal agitation of charge carriers inside a conductor and \(\sigma_{shot}\) is the shot noise due to ambient light. These variances can be estimated through the following mathematical expressions [43; 47]:  
\[\sigma_{thermal}^{2}=\frac{8\pi kT_{k}nAI_{2}B^{2}}{G}+\frac{16\pi^{2}kT_{k} \Gamma n^{2}A^{2}I_{3}B^{3}}{g_{m}} \tag{4}\]  
and  
\[\sigma_{shot}^{2}=2q\gamma P_{r}B+2q\gamma P_{bg}I_{2}B \tag{5}\]  
where \(k\) stands for the Boltzmann's constant, \(G\) is the open loop voltage gain, \(A\) is the physical area of the receiver, \(T_{k}\) is the absolute temperature, \(I_{2}\) and \(I_{3}\) are noise bandwidth factors, \(n\) is the capacitance of the photo detector per unit area, \(g_{m}\) is the transconductance, \(B\) is the noise bandwidth, \(q\) is the electronic charge, and \(P_{bg}\) is the background noise power.  
Assuming that the background noise of the system depends on sunlight radiation, \(P_{bg}\), can be estimated as [14]  
\[P_{bg}=E_{\text{det}}T_{0}An_{ri}{}^{2} \tag{6}\]  
where \(T_{0}\) is the peak filter transmission coefficient, \(n_{ri}\) is the refractive index of the lens, and \(E_{det}\) is the irradiance within the spectral range of the receiver. Additionally, \(E_{det}\) is the irradiance of the receiver between the wavelengths \(\lambda_{1}\), \(\lambda_{2}\) and can be estimated as [14]  
\[E_{\text{det}}=\int\limits_{\lambda_{1}}^{\lambda_{2}}\frac{S_{\text{peak}}W( \lambda)}{\text{max}[W(\lambda)]}d\lambda \tag{7}\]  
where \(S_{\text{peak}}\) is the peak irradiance and \(W(\lambda)\) is the spectral irradiance of a black body radiation model and is given as [14]  
\[W(\lambda)=\frac{1}{\lambda^{5}}\Bigg{(}\frac{2\pi ln_{p}c^{2}}{e^{\frac{hc}{ kT_{B}}}-1}\Bigg{)} \tag{8}\]  
where \(h_{p}\) is the [PERSON]'s constant, \(c\) is the speed of light in a vacuum, and \(T_{B}\) is the average surface temperature of the Sun in Kelvin degrees. The integral of Equation (7) cannot be solved analytically, so the accurate numerical method of Monte Carlo is used in order to estimate the solar background irradiance at the receiver, using \(10^{6}\) random samples. In order to improve the SNR of the system, the VO\({}_{2}\) filter implemented can be installed at the receiver in order to reduce the sunlight radiation received. According to Figure 2, the filter is able to reduce the transmittance of the sunlight background noise power, especially for wavelengths lower that 600 nm. Furthermore, the transmittance presents its peak value at 650 nm, representing the wavelength at which the information is transmitted, so power reduction of the signal will be less than the reduction of the noise. This characteristic is very important for the selection of a suitable optical filter.