# Performance Evaluation of IM/DD FSO Communication System Under Dust Storm Conditions
## 2 Dusty Channel Model
### Dusty Channel Model  
Using the collected data, we analyzed the statistical behavior of the irradiance fluctuation of the received signal under dust storm conditions. To develop a tractable and realistic statistical model, the signal irradiance fluctuations, \(I\), due to the dust-induced fading were normalized by the optical power transmitted during the measurement phase, resulting in a fading variable suitable for probabilistic analysis. Figure 2 presents a comparative analysis between the empirical histogram of normalized signal intensity measurements and the fitted probability density functions (PDFs) of four theoretical models: Beta, Weibull, Lognormal, and Power distributions. The selected distributions reflect key features of the observed histogram, including strong right skewness and bounded support. This approach enables the evaluation of models with diverse shapes and tail behaviors best to capture the underlying statistical characteristics of the data. Among all candidates, the Beta distribution exhibits the best overall fit, closely following the empirical histogram across the full range of data, particularly capturing the steep increase in density near the upper bound (\(I\approx 1\)). This alignment confirms the Beta model's flexibility in modeling bounded and right-skewed data. The Power distribution, while simpler, provides a reasonably good approximation and maintains the general shape of the tail behavior, though it slightly underestimates the peak density near the boundary. In contrast, the Weibull and Lognormal models fail to accurately represent the concentration of data at high-intensity levels, with both displaying broader shapes and heavier central masses that do not reflect the observed distribution. These discrepancies are further confirmed by two goodness-of-fit metrics: the coefficient of determination (\(R^{2}\)) and the root mean square error (RMSE). The coefficient of determination is defined as follows:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}-\bar{ y})^{2}} \tag{1}\]  
where \(y_{i}\) are the measured values, \(\hat{y}_{i}\) are the predicted values, \(\bar{y}\) is the mean of the measured values, and \(n\) is the number of observations. The RMSE is given as follows:  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}} \tag{2}\]  
A good fit is typically indicated by a high \(R^{2}\) close to 1 and a low RMSE, which reflects minimal deviation between the model and the empirical data. As shown in Table 1, the Beta distribution outperforms the other candidates, achieving the highest \(R^{2}\) value of 0.85 and the lowest RMSE of 1.30, confirming its superior alignment with the observed histogram. The estimated parameters for each distribution are listed in Table 1.  
Using the Beta PDF, the signal irradiance fluctuation under dusty conditions is modeled as follows:  
\[f(I)=\frac{1}{B(\alpha,\beta)}I^{\alpha-1}(1-I)^{\beta-1},\quad 0<I<1, \tag{3}\]  
where \(\alpha>0\) and \(\beta>0\) are the shape parameters, and \(B(\alpha,\beta)\) is the Beta function, defined as follows:  
\[B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}. \tag{4}\]  
where \(\Gamma(\cdot)\) is the Gamma function, defined as follows:  
\[\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}\,dt,\quad\text{for}\;z>0. \tag{5}\]  
The results in Figure 2 show that the fitted Beta distribution has minor deviations from the empirical irradiance histogram, particularly in the high-irradiance region, which includes a sharp peak near the upper bound. This mismatch is not primarily attributable to measurement noise or unaccounted physical effects but rather to the intrinsic shape of the observed data, which is characterized by a pronounced asymmetry with a long left-sided tail and a sharp rise near \(I\approx 1\). Such a distribution is inherently difficult to capture using a single-lobe distribution. Although composite or mixture distributions could improve fitting in this region, our objective in this study was to identify a single, analytically