## 3 Radio Wave Propagation Analysis
### Dummy Fading Signal Generation Based on the Static Paths  
The ray-tracing method via RapLab analyzes the incoming signals at the receiving antenna for each path generated by the reflections and diffraction and then calculates the received signal by combining these paths. Since the phase fluctuations that occur over time in each path are not taken into consideration, the received signal has static characteristics, and a fading signal cannot be obtained. To address this, the dummy fading signal was generated by combining the signals taking into account the phase fluctuation generated using uniform random numbers for each path between the transmitting and receiving antennas [46; 47]. Although the speed of the UAV is slow, the fading signal including the Doppler effect was still calculated using the following equation:  
\[e(t)=\text{Re}\left[\sum_{i=1}^{N}ri(t)\exp\left(j\bigg{\{}2\pi\frac{vt}{ \lambda}\cos\alpha i+\theta i\bigg{\}}\right)\right] \tag{1}\]  
where \(r_{i}\) denotes the amplitude of each path calculated using the ray-tracing method; \(v\) indicates the movement speed of the UAV; \(\lambda\) is the wavelength; \(\alpha_{i}\) signifies the angle between the movement direction of the UAV and the arrival direction of the path; and \(\theta_{i}\) is the random initial phase of \([0,2\pi)\).  
There are many scattered paths with a power difference of 60 dB or more from the direct wave power, but it has been confirmed that the combined received power does not change significantly when the signal of these paths is added to the combined received power. Therefore, the combining received signal was calculated using \(N\) paths whose power difference from the direct wave was 60 dB or less. The dummy fading signal data were generated by performing this method at the same point and the same number of times as the sample points in the actual measurement.  
As an example of the generated dummy fading signal, Figure 9 shows the received level at \(H=10\) m in the horizontal installation in the C environment. The red-purple line shows the result of the ray tracing according to the movement of the UAV, the black line shows the result of the dummy fading signal generated with Equation (1), and the blue line shows the measurement result.  
Figure 9: Instantaneous received power for the horizontal installation at \(H=10\) m in a complex propagation environment.  
As shown in Figure 9, the measurement result shows a large drop in received power, indicating that fading is occurring. The ray-tracing result shows no large drop in received power, and the received power fluctuates slightly around \(-60\) dB, indicating that fading cannot be reproduced. In contrast, the dummy fading signal shows a large drop in received power, indicating that fading is being simulated.  
The received power of the ray-tracing result and the dummy fading signal are roughly consistent with that of the measurement results but slightly higher. The reason for this difference in received power is that the radio propagation analysis was performed using a simplified 3D model shown in Figure 7, so it is assumed that the propagation environment was not sufficiently reproduced. However, it was found that the radio wave propagation analysis using a simplified 3D model was useful. A more detailed 3D model needs to be created and analyzed as a future research task.