# A Clutter Loss Model for Satellite Communication Systems
## 5 Clutter Loss
### Combination of Reflection Loss and Diffraction Loss  
Depending on the scenario, the clutter loss is obtained as a different combination of \(L_{R}\) and \(L_{D}\). Figure 9 depicts the Line-of-Sight case (LoS, no building reflections) for which the total clutter loss, \(L\), is obtained by simply summing (in dB) the diffraction losses due to the overcome buildings B1, B2, Bd3,... until building BdP, with the diffraction loss of building d(P + 1) being equal to zero:  
\[L=L_{D,B1}+L_{D,B2}+\ldots+L_{D,BdP} \tag{7}\]  
In the case of negative \(\psi_{s}\) angles, the reflection loss of the ground \(L_{R,ground}\) is added in the calculation. In the presence of reflections from buildings and/or the ground (NLoS-Non-Line-of-Sight case), the clutter loss calculation depends on the specific scenario; an example is shown in Figure 10. A key aspect is that two paths are considered whenever the ray hits either the first or second building. The first path is associated with the portion of the wave overcoming the first (or second) building (path \(S_{1}\), with total loss \(L_{1}\)), which will be affected only by further diffraction (possibly) from other buildings, until building dP (with the diffraction loss of building d(P + 1) being equal to zero). The second path is associated with the reflected wave (path \(S_{2}\), with total loss \(L_{2}\)), which will be (possibly) further reflected by other obstacles along its path to the satellite. In the latter case, \(L_{2}\) is calculated by summing  
Figure 8: Geometry of the knife-edge diffraction model. (in dB) all the reflection losses and the diffraction losses given by the further overcome buildings, after the last reflection, until building rdN (with the diffraction loss of building rd(N + 1) being equal to zero). This is clarified in Figure 10, which shows an example of how \(L_{1}\) and \(L_{2}\) are calculated. A similar calculation is performed when negative elevation angles are considered: in this case, the additional reflection from the ground will be considered for both paths \(S_{1}\) and \(S_{2}\). The ground reflection loss \(L_{R,ground}\) is added to both \(L_{1}\) and \(L_{2}\). The final step in the calculation of the total clutter loss consists of combining \(L_{1}\) and \(L_{2}\). In the following, \(l\) will indicate the unitless loss smaller than one. In principle, it should be referred to as _gain_, since the radiated power is multiplied, and not divided; nevertheless, we think that making reference to'reflection gain' and/or 'diffraction gain' could be misleading. Referring to Figure 10, if the base station radiates the power \(P\), the power associated with path \(S_{1}\) (not in dB) is \(P_{1}=P\cdot l_{D,B1}\). The remaining power, \(P_{2}=P\cdot(1-l_{D,B1})\), is associated with path \(S_{2}\). If we neglect the difference in the path lengths of \(S_{1}\) and \(S_{2}\), which is reasonable considering the geometry, the power reaching the satellite associated with paths \(S_{1}\) and \(S_{2}\) will, respectively, be  
\[P_{1}^{S}=P\cdot l_{D,B1}\cdot l_{D,B2}\cdot\ldots\cdot l_{D,BdP}=P\cdot l_{1}, \tag{8}\]  
\[P_{2}^{S} =P\cdot(1-l_{D,B1})\cdot l_{R,B1}\cdot l_{R,B73}\cdot\ldots\cdot \tag{9}\] \[\cdot l_{R,BrK}\cdot l_{D,BrM1}\cdot l_{D,BrM1}=\] \[=P(1-l_{D,B1})l_{2}\]  
The total power \(P^{S}\) reaching the satellite is  
\[P^{S}=P_{1}^{S}+P_{2}^{S}=Pl, \tag{10}\]  
Figure 10: Clutter loss calculation for a sample NLoS scenario. Figure 9: Clutter loss calculation for the LoS scenario. where \(l\) is the total clutter loss. From (8) and (9):  
\[ \begin{array}{ll}l&=&l_{D,B1}\cdot\ldots\cdot l_{D,B4P}+(1-l_{D,B1})\cdot l_{R, B1}\cdot\ldots\cdot\\ &&\cdot l_{R,Brk}\cdot l_{D,Brd1}\cdot\ldots\cdot l_{D,BrdN}\end{array} . \tag{11}\]  
Expressing \(l\) in dB yields  
\[L=-10\log\Bigl{[}10^{-L_{1}}+\Bigl{(}1-10^{-L_{D,B1}}\Bigr{)}10^{-L_{2}} \Bigr{]}, \tag{12}\]  
Following the same approach for negative angles, the impact due to the ground reflection needs to be considered in both \(L_{1}\) and \(L_{2}\).