# Fuzzy Logic-Based Performance Enhancement of FSO Systems Under Adverse Weather Conditions
## 2 Modeling of Rain and Fog Impairments in FSO System
### Fog  
A mathematical expression for the fog and haze attenuation in the FSO system is given in Equation (2) [6]. In the case of different fog and haze conditions, attenuation is more compared to the rain attenuation as the dust and smoke particles remain longer in the atmosphere. \[\alpha=\frac{13}{V}\bigg{(}\frac{\lambda}{550\text{ nm}}\bigg{)}^{-x} \tag{2}\]  
In Equation (1), \(\alpha\) shows the fog or haze attenuation in (dB/km), V is the visibility in (km), \(\lambda\) is the wavelength in nm, and \(x\) is the size distribution of the scattering particle. According to [PERSON] model, \(x\) can be determined as follows [2; 3; 4; 5; 6]:  
\[x=\left\{ \begin{array}{l}1.6V>50\\ 1.36<V<50\\ 0.16V+0.341<V<6\\ V-0.5<V<1\\ 0V<0.5\end{array} \right. \tag{3}\]  
Table 2 shows haze and fog conditions with their attenuation [8]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Rain Intensity** & **Attenuation (dB/km)** \\ \hline Storm & 59.7 \\ Cloud burst & 20.99 \\ Heavy rain (HR) & 19.28 \\ Medium rain (MR) & 9.64 \\ Light rain (LR) & 6.27 \\ Drizzle & 0.44 \\ Very clear & 0.22 \\ Clear & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relation between rain intensity and attenuation. Figure 1: FSO under different rain scenarios. As the received power decreases due to these weather conditions, the relation between the received and transmitted power can be expressed as follows [9]:  
\[P_{RX}=P_{TX}\bigg{(}\frac{d_{RX}}{d_{TX}+\theta L}\bigg{)}^{2}10^{-\frac{\theta L }{10}} \tag{4}\]  
where \(P_{RX}\), \(P_{TX}\), \(d_{RX}\), \(d_{TX}\), \(\theta\), and \(L\) respectively indicate received power, transmitted power, received aperture diameter, transmitted aperture diameter, beam divergence, and link range, while \(\beta\) refers to the atmospheric attenuation which is \(\gamma_{train}\) in case of rain condition and \(\alpha\) in case of fog or haze conditions. In our model, geometric losses are accounted for using fixed system parameters, consistent with prior literature. We assume a transmitted power of 10 dBm, a receiver aperture of 20 cm, a transmitter aperture of 8 cm, and a beam divergence angle of 2 mrad, as reported in [2; 6; 8]. These parameters define the baseline configuration of the PSO system and are used consistently in analytical and fuzzy control evaluations. The received signal, after propagating through the FSO channel, is directed to the photodetector (PD) to be converted back into an electrical signal. Then, it enters the low-pass filter (LPF), which has a cutoff frequency of 0.75\(\times\) data rate Hz to block the unwanted signal. Furthermore, it passes through a bit error rate (BER) analyzer to compute the performance of the received data. As the BER depends on the signal-to-noise ratio (SNR), it is necessary to initially examine the SNR in detail. The SNR can be characterized in terms of shot noise variance, \(\sigma_{s}^{2}\), and thermal noise, \(\sigma_{t}^{2}\), as [2; 13]  
\[\text{SNR}=\frac{I_{\text{PD}}^{2}}{\sigma_{s}^{2}+\sigma_{\text{th}}^{2}} \tag{5}\]  
where \(I_{\text{PD}}\) represents the resultant current from the photodetector [2]. \[I_{P}=RP_{\text{rx}} \tag{6}\]  
where \(R\) represents the responsivity of the photodetector and has a value of 0.8 A/W. \(\sigma_{s}^{2}\) and \(\sigma_{\text{th}}^{2}\) can be calculated as [8; 13]  
\[\sigma_{s}^{2}=2q_{e}(I_{P}+I_{D})B_{e} \tag{7}\]  
\[\sigma_{\text{th}}^{2}=\frac{4k_{B}TB_{e}}{R_{L}} \tag{8}\]  
Here, \(q_{e}\) represents the electron charge in C, \(B_{e}\) denotes the electrical bandwidth (taken as 0.75\(\times\) data rate in Hz), \(I_{\text{D}}\) is the dark current and has a value of 10 nA. \(k_{B}\) is the Boltzmann constant, the parameter \(T\) stands for the receiver noise temperature and is assumed to be 298 K, while \(R_{L}\) refers to the receiver load resistance, set at 1030 \(\Omega\).