# A Clutter Loss Model for Satellite Communication Systems
## 5 Clutter Loss
### Analytical Model  
A simple analytical model, dependent on the frequency and elevation angle, is definitely more easily applicable than the complete simulation outlined in Section 2. For this reason, the clutter loss statistics calculated by simulating the London, Melbourne and Milan scenarios were fitted by a mathematical expression that accurately approximates the statistics for low clutter loss values and time percentages (typically lower than 60-70%), i.e., the most important ones for interference applications. The resulting fitting parameters are not associated with a single city, but they are related to the local statistics of buildings' height and the base station's height, besides the elevation angle and frequency.  
The clutter loss cumulative distributions produced at different frequencies between 1 and 30 GHz, considering the base height either uniformly distributed between 4 and 6 m or fixed at 18 m, for various elevation angles, were approximated using the following expression:  
\[L_{\text{ces}}=\frac{1}{K}\sqrt{\left(1-\frac{100-p_{min}}{100-p}\right)}\; \text{dB}, \tag{13}\]  
with  
\[K=\begin{cases}\exp(X_{1}\;\theta+X_{2})&\text{for}\;\theta>0\\ X_{3}\theta+X_{4}&\text{for}\;\theta<0\end{cases}, \tag{14}\]\[p_{min}=\left\{\begin{array}{ll}\min\bigl{(}100,\max\bigl{(}0,Y_{1}\sin\bigl{(}( \theta-5)\frac{\pi}{80}\bigr{)}+Y_{2}\theta+Y_{3}\bigr{)}\bigr{)}&\text{for $\theta>0$}\\ 0&\text{for $\theta<0$}\end{array}\right., \tag{15}\]  
\[Y_{1}=a_{1}H_{b,med}^{2}+a_{2}H_{b,med}+a_{3}, \tag{16}\]  
\[Y_{2}=a_{4}H_{b,med}+a_{5}, \tag{17}\]  
\[Y_{3}=a_{6}H_{b,med}+a_{7}. \tag{18}\]  
where \(\theta\) is the elevation angle (degrees), \(p\) is the percentage of locations (%), \(0<p<100\), and \(H_{b,med}\) is the median value of the building height (m).  
If the base station height \(H_{\text{S}}\) is randomly distributed between 4 and 6 m:  
\[X_{1}=b_{1}H_{b,med}+b_{2}, \tag{19}\]  
\[X_{2}=b_{3}H_{b,med}+b_{4}, \tag{20}\]  
\[X_{3}=b_{5}H_{b,med}+b_{6}, \tag{21}\]  
\[X_{4}=b_{7}H_{b,med}+b_{8}. \tag{22}\]  
If the base station height \(H_{\text{S}}\) is 18 m:  
\[X_{1}=c_{1}H_{b,med}+Z_{1}, \tag{23}\]  
\[X_{2}=Z_{2}H_{b,med}+c_{2}, \tag{24}\]  
\[X_{3}=c_{3}H_{b,med}+c_{4}, \tag{25}\]  
\[X_{4}=c_{5}H_{b,med}+c_{6}, \tag{26}\]  
and  
\[Z_{1}=c_{7}f^{2}+c_{8}f+c_{9}, \tag{27}\]  
\[Z_{2}=c_{10}f^{2}+c_{11}f+c_{12}. \tag{28}\]  
where \(f\) is the frequency between 2 and 10 GHz.  
The curve fitting is achieved by minimizing the overall RMS (root mean square) difference from the generated cumulative distributions by weighting the low-percentage region of the curves more than the region with the highest percentage. A similar approach was used in [11], though, in that case, the parameters were manually adjusted to achieve a good fit for low clutter loss values (no details are provided). Table 1 lists the coefficients from \(a_{1}\) to \(a_{7}\); the coefficients from \(b_{1}\) to \(b_{8}\) are reported in Table 2, and Table 3 presents the coefficients from \(c_{1}\) to \(c_{12}\).  
\begin{table}
\begin{tabular}{c c c} \hline
**Coefficient** & **Value for \(H_{\text{S}}\) = 4–6 m** & **Value for \(H_{\text{S}}\) = 18 m** \\ \hline \(a_{1}\) & \(-0.1747\) & \(0.0970\) \\ \(a_{2}\) & \(2.7079\) & \(-3.1213\) \\ \(a_{3}\) & \(18.5859\) & \(35.8211\) \\ \(a_{4}\) & \(0.0220\) & \(0.0647\) \\ \(a_{5}\) & \(0.6798\) & \(-0.4317\) \\ \(a_{6}\) & \(-2.3377\) & \(-5.7103\) \\ \(a_{7}\) & \(46.6339\) & \(137.8036\) \\ \hline \end{tabular}
\end{table}
Table 1: Coefficients of the expression approximating the clutter loss statistics for \(H_{\text{S}}\) = 4–6 m and 18 m.