## 1 Introduction
### Received Signal Model  
Ideally, the RHCP antenna receives the RHCP component of the mixture of the LOS and MP signals, while the LHCP antenna receives the LHCP component of the MP signals. However, a practical RHCP or LHCP antenna has finite amplitude attenuation to the signals with an opposite polarization state, thus, it receives both RHCP and LHCP signals [28]. The attenuation or the polarization mismatch loss has relevance to the axial ratios (ARs) of the antennas and the signals, which defines the polarization state of an electric field as Equation (1) gives [34]:  
\[R=\frac{E_{R}+E_{L}}{E_{R}-E_{L}}\text{ or }R(\text{dB})=20\;\times\;\log 10(|R|) \tag{1}\]  
where \(E_{\text{R}}\) and \(E_{\text{L}}\) are the amplitudes of the RHCP and LHCP components, respectively. If \(R>0\), it is a right hand elliptical polarized (RHEP) wave and \(R<0\) indicates a left hand elliptical polarized (LHEP) wave. The AR of the polarization ellipse of the antenna is also defined in the same way. When the ARs of the signal and antenna are different, polarization mismatch loss occurs as given in Equation (2), which is used in [18; 34]:  
\[p(R_{\text{A}},R_{\text{S}},\Delta\tau)=\frac{1}{2}+\frac{4R_{\text{S}}R_{ \text{A}}+(R_{\text{S}}^{2}-1)(R_{\text{A}}^{2}-1)\times\cos 2\Delta\tau}{2(R_{ \text{S}}^{2}+1)(R_{\text{A}}^{2}+1)} \tag{2}\]  
where \(R_{\text{S}}\) and \(R_{\text{A}}\) are the ARs of the signal and the antenna, respectively. \(\Delta\tau\) is the relative tilt angle between the major axes of the polarization ellipses of the signal and the receiver antenna. The AR of the antenna \(R_{A}(\psi,\varphi)\) is a function of the elevation angle \(\psi\) and the azimuth angle \(\varphi\). The AR of the LOS signal depends on the AR of the satellite antenna which is no worse than 1.8 dB [35] while the AR of the MP signal is mainly determined by reflection. Equation (3) shows the complex circular reflection coefficients represented by the horizontal and vertical reflection coefficients:  
\[ \begin{array}{ll}\Gamma_{\text{o}}=|\Gamma_{\text{o}}|e^{-j\angle\Gamma_{ \text{o}}}=\frac{\Gamma_{\text{h}}+\Gamma_{\text{v}}}{2}\\ \Gamma_{\text{x}}=|\Gamma_{\text{x}}|e^{-j\angle\Gamma_{\text{x}}}=\frac{ \Gamma_{\text{h}}-\Gamma_{\text{v}}}{2}\end{array}  \tag{3}\]  
where \(|\Gamma_{\text{o}}|\) and \(\angle\Gamma_{\text{o}}\) are the relative amplitude and the phase change of the co-polar reflection coefficient, respectively, while \(|\Gamma_{\text{x}}|\) and \(\angle\Gamma_{\text{x}}\) are the corresponding cross-polar terms. The derivation of the horizontal \(\Gamma_{\text{h}}\) and vertical \(\Gamma_{\text{v}}\) reflection coefficients can be found in [18; 1]. For simplicity, we assume there is a single one-bounce reflected multipath hereinafter. The model of the complex baseband signal that a dual-polarization antenna receives is given in Equation (4):  
Figure 1: The components of the received signal by a dual-polarization antenna. \[x_{\text{RHCP}}(t) =a_{0}e^{\text{i}\beta_{0}}p(R_{\text{R}}(\psi_{0},\varphi_{0}),R_{ \text{LOS}},\Delta\tau_{0})s(t-t_{0}) \tag{4}\] \[+a_{1}e^{\text{i}\beta_{1}}|\Gamma_{0}|e^{-\text{i}\angle\Gamma_{ \text{c}}}p(R_{\text{R}}(\psi_{1},\varphi_{1}),R_{\text{MP}},\Delta\tau_{1})s(t- t_{1})+w_{\text{R}}(t)\] \[x_{\text{LHCP}}(t) =a_{0}e^{\text{i}\beta_{0}}p(R_{\text{L}}(\psi_{0},\varphi_{0}),R_ {\text{LOS}},\Delta\tau_{0})s(t-t_{0})\] \[+a_{1}e^{\text{i}\beta_{1}}|\Gamma_{\text{x}}|e^{-\text{i} \angle\Gamma_{\text{x}}}p(R_{\text{L}}(\psi_{1},\varphi_{1}),R_{\text{MP}}, \Delta\tau_{1})s(t-t_{1})+w_{\text{L}}(t)\]  
where \(s(t-t_{0})\) and \(s(t-t_{1})\) are the baseband signals of LOS and MP, respectively, \(a\) and \(\beta\) are the amplitude and the carrier phase, respectively. The subscript \(0\) represents the LOS signal and \(1\) represents the MP signal.