# A Clutter Loss Model for Satellite Communication Systems
## 4 Reflection and Diffraction Loss
### Reflection and Diffraction Loss Calculation  
As is clear from (1) and (2), the reflection coefficients are a function of the incidence angle, which, in turn, depends on the horizontal and vertical angles defined in Figure 6 (right side, depicting two examples of different horizontal angles). The left side of Figure 6 depicts the incidence geometry for two sample rays having the same elevation angle \(\psi_{s}\) and different horizontal angles \(\phi_{S}\). Referring to the left side of Figure 6, where the building facade is parallel to the \((x,z)\) plane, the incidence angle is given by  
\[\theta=\cos^{-1}[\cos(\psi_{s})\cos(\phi_{S})]. \tag{3}\]  
Figure 7 depicts the trend of the absolute value of the reflection coefficients at 6 GHz, for both TM and TE polarizations, as a function of the incidence angle. Results are shown  
Figure 5: Structure of the interfaces considered for building reflections: concrete/brick walls (on the left) and single glass slab (on the right).  
Figure 6: Horizontal incidence angle, \(\phi_{S}\), elevation angle, \(\psi_{s}\), and incidence angle, \(\theta\), of a single ray (left) and example of two sample rays (right) having the same elevation angle, \(\psi_{s}\), and different azimuth angles, \(\phi_{S1}\) and \(\phi_{S2}\).  
for reflection from buildings (air-concrete/brick wall and air-glass slab) and from the ground (air-dry asphalt). Overall, the TM component undergoes a lower reflection when compared to the TE polarization, especially around [PERSON]'s angle, for which the TM wave is (almost) totally transmitted [13].  
For a single discontinuity, the reflection loss (in dB) is simply calculated as  
\[L_{R}=-10\log\Bigl{(}\left|\Gamma(\theta)\right|^{2}\Bigr{)}. \tag{4}\]  
The combination of multiple reflections is explained in Section 5.2.  
Besides reflections, in an urban environment, diffraction from obstacles plays a key role in determining the overall loss of the wave on its way to the satellite. Following the method described in [11], the knife-edge model is implemented to consider the diffraction loss: an obstacle, such as a building, is approximated by a knife edge of negligible thickness.  
The calculation of the diffraction loss requires evaluating [PERSON]'s integral, which could be quite time consuming, so the proposed model uses the simple (yet effective and accurate) expression of the diffraction loss \(L_{D}\) (in dB) described in Recommendation ITU- R P.526-15 [12]:  
\[L_{D}=J(v)=\] \[=\left\{\begin{array}{cc}6.9+20\log\biggl{[}\sqrt{\left(v-0.1 \right)^{2}+1}+v-0.1\biggr{]}&\text{if }v>-0.78\\ 0&\text{otherwise}\end{array}\right., \tag{5}\]  
where  
\[v=g\sqrt{\frac{2}{\lambda d_{1}}}, \tag{6}\]  
\(\lambda\) is the wavelength, \(d_{1}\) represents the distance between the base station and the nearest vertex of the building and \(g\) is the orthogonal distance between the wave propagation direction and the vertex, as shown in Figure 8. The choice of employing the knife edge approach to calculate the diffraction loss in the proposed model is associated with the goal of preserving a good balance between modeling accuracy and simplicity, as well as of reducing the calculation time to provide reliable results in a few minutes, especially if compared to the heavy computational load associated with raytracing methods. Future enhancements of the proposed model will consider refining the approach to model the diffraction loss.  
Figure 7: Trend of the absolute value of the reflection coefficient (both TE and TM cases) at 6 GHz as a function of the incidence angle, for different discontinuities: air to ground (asphalt) and building (concrete/brick wall and glass slab).