## 2 The 3D A2A MIMO Channel Model
### Specular and Diffuse Components
#### 2.2.2 Powers of the Rays  
The complex reflection coefficient of the \(n\)th ray is calculated according to the Fresnel law as follows [37]:  
\[\Gamma_{n}=\frac{\cos\theta_{i,n}-Z}{\cos\theta_{i,n}+Z}. \tag{47}\]  
Here, \(Z\) is expressed as \(Z_{v}\) and \(Z_{h}\) for the vertical and horizontal polarizations, respectively, i.e.,  
\[Z_{v}=\frac{1}{\eta}\sqrt{\eta-(\sin\theta_{i,n})^{2}} \tag{48}\] \[Z_{h}=\sqrt{\eta-(\sin\theta_{i,n})^{2}} \tag{49}\]  
where \(\eta\) is the relative permittivity of the reflection material and \(\theta_{i,n}\) accounts for the incident angle of the \(n\)th ray and is obtained as \(\theta_{i,n}=\pi-\theta_{n}\). It is important to note that (47) holds true only for smooth surfaces. In the case of a rough surface, the incident wave splits into a specular reflected ray and multiple diffuse rays scattered in various directions, as depicted in Figure 4. Here, a scattering loss factor is introduced to account for the power attenuation via specular reflections due to the scattering effect, as follows [32]:  
\[\rho_{s,n}=\exp\left(-8\pi^{2}\left(\frac{\sigma_{h}}{\lambda}\right)^{2}\cos ^{2}(\theta_{i,n})\right) \tag{50}\]  
where \(\sigma_{h}\) is the standard deviation of the Gaussian-distributed surface heights. Thus, the modified reflection coefficient is obtained as  
\[\Gamma_{\text{mod},n}=\rho_{s,n}\Gamma_{n}. \tag{51}\]  
Therefore, the power of the specular ray can be calculated as  
\[P_{0}=PL_{0}^{-1}G_{t}G_{r}|\Gamma_{\text{mod},0}|^{2} \tag{52}\]  
Figure 4: Geometry of specular reflection and diffuse scattering. where \(PL_{0}\) is the path loss applied to the specular ray, i.e.,  
\[PL_{0}^{-1}=\bigg{(}\frac{\lambda}{4\pi}\bigg{)}^{2}d_{0}^{-\gamma}. \tag{53}\]  
It can be observed that by reducing the wavelength or the incident angle of the wave or by increasing the roughness of the surface, more energy will be scattered rather than reflected. The power of the \(n\)th (\(n=1,...,N\)) diffuse ray is calculated as  
\[P_{n}=S_{n}^{2}PL_{n}^{-1}G_{t}G_{r}|\Gamma_{\text{mod},n}|^{2}[S_{0}f(\psi_{n} )]^{2} \tag{54}\]  
where \(PL_{n}\) stands for the path for the \(n\)th diffuse ray, which is determined as  
\[Z_{v}=\frac{1}{\eta}\sqrt{\eta-(\sin\theta_{i,n})^{2}} \tag{48}\]0  
Furthermore, \(S_{n}^{2}\) is the proportion of the power scattered in all directions with respect to the reflected power. Note that both \(\rho_{s,n}\) and \(S_{n}\) fall within the range of \([0,1]\). Due to energy conservation, the total energy of the incident wave must be conserved after scattering, which means that the sum of the energies of the specular reflected and diffuse scattered waves is equal to the energy of the incident wave. Thus, \(\rho_{s,n}\) and \(S_{n}\) must satisfy the following constraint:  
\[Z_{v}=\frac{1}{\eta}\sqrt{\eta-(\sin\theta_{i,n})^{2}} \tag{48}\]1  
In (54), \(f(\psi_{n})\) describes the relative power of the diffuse rays with respect to the specular direction and can be modeled by a single-lobe directive model as follows [22]:  
\[Z_{v}=\frac{1}{\eta}\sqrt{\eta-(\sin\theta_{i,n})^{2}} \tag{48}\]2  
with  
\[Z_{v}=\frac{1}{\eta}\sqrt{\eta-(\sin\theta_{i,n})^{2}} \tag{48}\]3  
As shown in Figure 4, \(\psi_{n}\) is the angle between the \(n\)th diffuse ray and the specular reflected ray originating from the same reflection point, and \(\theta_{s}\) and \(\varphi_{s}\) are the zenith and azimuth angles of the diffuse rays, respectively. The parameter \(\alpha_{R}\) governs the width of the scattering lobe. Based on measurements, a typical value for \(\alpha_{R}\) ranges from 1 to 5 [22; 44; 45]. To further illustrate this, Figure 5 shows the polar representation of the normalized scattering radiation pattern for an incident angle of \(\pi/4\). It is evident that the scattering radiation pattern achieves its maximum value in the specular direction. A larger value of \(\alpha_{R}\) leads to a narrower scattering lobe, which means more scattered energy is concentrated around the specular reflection direction.