# Predicting C/N0 as a Key Parameter for Network RTK Integrity Prediction in Urban Environments
## 4 Predicting Signal Strength
### Signal Power in Non-Line-Of-Sight and Multipath  
When a signal strikes a surface, the reflection can be specular or diffuse. In the diffuse mode, which occurs at rougher surfaces, the signal is scattered into many rays with different angles. In this case, the power of the coming signal is divided into many parts, and it is quite improbable that a signal can reach the receiving antenna. In the specular reflection case, which happens at smoother surfaces, the signal is reflected as it happens in a mirror. The power of the signal is partly absorbed by the reflecting surface. If this kind of reflected signal arrives at the receiving antenna in addition to the LOS, a multipath situation is occurring and the reflected signal is the multipath component (MPC). If this reflection is the only signal arriving at the receiver without any direct signal, it is called NLOS.  
\[S_{MPC}=P_{t}\mathbf{e}_{rx}^{H}\mathbf{H}\mathbf{e}_{tx}A_{MPC}e^{-j\mathbf{ a}},P_{MPC}=|S_{MPC}|, \tag{5}\]  
where \(\mathbf{e}\) is the Jones vector (Equation (2)) with superscript H indicating the Hermitian conjugate. Matrix \(\mathbf{H}\) (Equation (6)) consists of Fresnel reflection coefficients \(\Gamma_{H}\) for horizontal and \(\Gamma_{V}\) for vertical polarization components (Equations (7) and (8)) to account for the specular reflection process, and the two rotation matrices to align the polarization ellipses of the two antennas [48]. \(A_{MPC}\) is the free space loss for the multipath reflected component. The loss due to a quite large extra path delay (EPD) of 450 m of the reflected signal is 0.999975 and is, in practice, negligible, so it can be considered equal to \(A_{LOS}\)[49].  
\[\mathbf{H}=\begin{bmatrix}cos(\psi_{rx})&sin(\psi_{rx})\\ -sin(\psi_{rx})&cos(\psi_{rx})\end{bmatrix}\begin{bmatrix}\Gamma_{H}&0\\ 0&\Gamma_{V}\end{bmatrix}\begin{bmatrix}cos(\psi_{tx})&sin(\psi_{tx})\\ -sin(\psi_{tx})&cos(\psi_{tx})\end{bmatrix}, \tag{6}\]  
\[\Gamma_{H}=\frac{sin(\theta)-\sqrt{e-cos^{2}(\theta)}}{sin(\theta)+\sqrt{e- cos^{2}(\theta)}}, \tag{7}\]  
\[\Gamma_{V}=\frac{esin(\theta)-\sqrt{e-cos^{2}(\theta)}}{esin(\theta)+\sqrt{e- cos^{2}(\theta)}}, \tag{8}\]  
where \(\epsilon=\epsilon_{r}-j60\lambda\sigma\) is the complex dielectric constant. The reflection coefficients are functions of the relative permittivity (\(e_{r}\)) and conductivity (\(\sigma\)) of the reflecting surfaces which account for the attenuation of the signal in the reflection process. They also depend on the angle of incidence (\(\theta\)). The reflecting surface in this study is assumed to be concrete, for which \(\epsilon_{r}=3\) and \(\sigma=2\times 10^{-5}\) S/m in the GNSS frequency spectrum. The rotation matrices in Equation 6 rotate the coordinates to account for the change in polarization during reflection. Angle \(\psi\) is the angle between the normal of the incident plane and the direction of the propagating signal from satellite to plane (\(tx\)) and from plane to receiver (\(rx\)) [26; 50],  
\[\psi_{tx}=tan^{-1}\left(\frac{||\hat{n}\times t\hat{x}p||}{\hat{n}\cdot t\hat{x }p}\right), \tag{9}\]  
where \(\hat{n}\) is the normal vector of the incident plane and \(t\hat{x}p\) is the unit vector between the transmitter and the reflecting point. Analogously, angle \(\psi_{rx}\) can be defined.  
The power of the compound signal which is the combination of the LOS and reflection MPC reads  
\[P_{Compound}=P_{LOS}\sqrt{1+2\left(\frac{P_{MPC}}{P_{LOS}}\right)cos(\Delta \Phi)+\left(\frac{P_{MPC}}{P_{LOS}}\right)^{2}}, \tag{10}\]  
where \(P_{MPC}/P_{LOS}\) is the relative amplitude and \(\Delta\Phi=tan^{-1}(Im(S_{MPC})/Re(S_{MPC}))\) is the relative phase of the multipath component.