# Building Penetration Losses at 3.5 GHz: Dependence on Polarization and Incidence Angle
## 4 Results and Discussion  
We computed attenuations for the different locations, angles of incidence and polarizations by comparing data gathered across the different facade sections with a free-space reference measurement using Equation (9):  
\[\text{Attenuation }(\text{dB})=\text{Main\_ray\_amplitude}_{\text{free\_space}} -\text{Main\_ray\_amplitude}_{\text{ffacade}} \tag{9}\]  
Averaging those attenuation values obtained at the different measurement points, for each type of facade and angle, yields the results shown in Figure 13 for vertical polarization. The averaging process reduces the measurement variability that may arise due to the variation of the facade composition from one measurement spot to another. Attenuation values are larger for the facade without windows than for that with windows. The attenuation values obtained for the normal incidence (0\({}^{\circ}\)) and vertical polarization for the 30 cm facades are between 8 dB and 12 dB, depending on the presence of windows. This agrees with results in the literature. In [23], a value of 8.4 \(\pm\) 1.4 dB is reported at 3.5 GHz for the external wall of a building made of light construction materials and clear glass windows, which is very similar to the no-windows facade used to obtain the results in this paper. Figure 12: Glass facade. Regarding the angular variation, the attenuation values increase with the incidence angle by about 8 dB from 0\({}^{\circ}\) to 60\({}^{\circ}\) for both types of facade. This result is in agreement with those in the literature at other frequencies. In [15], at 2.5 GHz, the normal incidence attenuation will range from 4 to 20 dB, depending on the construction material, and it will increase by 5 dB at a 60\({}^{\circ}\) incidence angle, while in [24], a 17 dB attenuation increase from a normal to 60\({}^{\circ}\) incidence is reported at 5.2 GHz. According to [6; 7; 10], the normal incidence attenuation will be 14 dB and will increase by 4 dB at 60\({}^{\circ}\) for the 2-6 GHz frequency band. However, the increase of the losses with the incidence angle is not monotonous, as the attenuation at 45\({}^{\circ}\) is slightly lower than at 30\({}^{\circ}\). This does not agree with other empirical models in the literature as most of them expect a monotonous increase of the transmission losses with the incidence angle. However, as we showed in Section 2.2. and in Figure 4a, non-monotonous variations like these may happen in complex facades, such as the ones we considered, made by inhomogeneous walls of rough hollow clay bricks joined by concrete and with rough gravel covering the exterior wall. The 15 cm facade results follow a similar trend to those of the 30 cm facade without windows, as the values increase non-monotonically with the incidence angle; however, the attenuation is lower due to the reduced width. The glass facade does follow a monotonous growing curve with the incidence angle. The attenuation for these two last facades increases up to 6 dB from a normal incidence to 60\({}^{\circ}\) incidence, which is also in line with values reported in the literature mentioned above. The glass facade attenuates more than the brick one, which agrees with the results reported in [25]. For the horizontal polarization, results are given in Figure 14. For almost every angle, the attenuation values are lower than those found for vertical polarization. Such a difference is not strange, as even for the ideal situation of an infinite slab in air, the Fresnel equations predict different transmission coefficients for parallel and perpendicular polarizations. Figure 13: Attenuation as a function of incident angle \(\theta\) for vertical polarization for 30 cm facades without windows (red solid line); and with windows (blue dashed line), 15 cm brick facade (green dash-dot line) and glass facade (purple dotted line). The values for normal incidence on the 30 cm facades are also between 8 and 12 dB; however, there is not a clear trend to increase with the incidence angle. This agrees with the theoretical results in Section 2.