# GPS Multipath Analysis Using Fresnel Zones
## 2 Multipath Theory
### Multipath Propagation  
The superimposed multipath signal is composed by components of direct and reflected GNSS signals. This phenomena can be illustrated by means of a vector diagram, where the superimposed signal is the sum of two or more vectors representing the different signal components. Moreover, the length of the vectors denotes the amplitude of the respective signal components. In Figure 1, the respective vector diagrams for the superimposition of the direct signal and one reflected signal are shown. The amplitude of the compound signal \(A_{C}\) is the sum of the amplitudes of the direct signal \((A_{D})\) and the reflected signal \((A_{M})\), where \(A_{M}=\alpha A_{D}\), with \(\alpha=A_{M}/A_{D}\) representing the attenuation factor of the reflection process and the antenna gain pattern. \(\Delta\Phi_{M}\) denotes the phase difference between the direct signal and the reflected signal due to the additional signal path and is called the multipath relative phase. The phase error resulting from the superimposition is denoted by \(\Phi_{C}\). The phase error between the direct and the compound signal \(\Phi_{C}\) can be written as  
\[\tan\Phi_{C}=\frac{b}{A+a} \tag{1}\]  
where the auxiliary values \(a\) and \(b\) can be derived from the multipath relative phase \(\Delta\Phi_{M}\) and the amplitude \(A_{M}\) by  
\[ \begin{split} a&=A_{M}\cos\Delta\Phi_{M}=\alpha A _{D}\cos\Delta\Phi_{M}\\ b&=A_{M}\sin\Delta\Phi_{M}=\alpha A_{D}\sin\Delta \Phi_{M}.\end{split}  \tag{2}\]  
After inserting Equation (2) into (1), the phase error of the compound signal \(\Phi_{C}\) is given by  
\[\Phi_{C}=\arctan\left(\frac{\alpha\sin\Delta\Phi_{M}}{1+\alpha\cos\Delta\Phi_{ M}}\right). \tag{3}\]  
Using the amplitude \(A_{D}\) and the expressions for \(a\) and \(b\), the amplitude of the compound signal \(A_{C}\) can be written as  
\[A_{C}^{2}=(A_{D}+\alpha A_{D}\cos\Delta\Phi_{M})^{2}+(\alpha A_{D}\sin\Delta \Phi_{M})^{2} \tag{4}\]  
and after rearranging Equation (4), \(A_{C}\) can be determined by  
\[A_{C}=A_{D}\sqrt{1+2\alpha\cos\Delta\Phi_{M}+\alpha^{2}}. \tag{5}\]  
Using Equation (3), the maximal and minimal phase errors can be estimated. Under the assumption that the reflected signal is not attenuated (\(\alpha=1\)), the amplitude of the direct signal \(A_{D}\) equals the amplitude of the reflected signal \(A_{M}\). Thus, for \(\Delta\Phi_{M}=180^{\circ}\), the phase error \(\Phi_{C}\) reaches  
Figure 1: Resulting amplitude and phase error after superimposition of a direct signal and one reflected signal. its maximum of 90\({}^{\circ}\), which corresponds to one quarter of the respective signal wavelength (\(\approx\)4.8 cm for GPS-L1). On the other hand, for \(\Delta\Phi_{M}=0^{\circ}\), the phase error becomes zero. All of the parameters introduced above change over time and strongly depend on the geometrical configuration between the satellite, reflector and user antenna. In Figure 2, the geometrical configuration in a local horizontal coordinate frame is shown for a single reflector with arbitrary orientation in space. Here, \(az^{R}\) and \(el^{R}\) denote the azimuth and elevation angles of the reflection point, and \(az^{S}\) and \(el^{S}\) denote the azimuth and elevation angles of the satellite, respectively. The distance between the antenna along the signal path of the reflected signal is denoted by \(d\), and \(d_{h}\) denotes the respective horizontal component. Using the angle \(\gamma\), the path delay \(\delta\) (red part of the signal path) and the multipath relative phase \(\Delta\Phi_{M}\) can be determined by  
\[\delta\left(t\right) =d\left(1+\cos\gamma\left(t\right)\right) \tag{6}\] \[\Delta\Phi_{M}\left(t\right) =\frac{2\pi}{\lambda}\delta\left(t\right)=\frac{2\pi d}{\lambda} \left(1+\cos\gamma\left(t\right)\right)\]  
where \(\lambda\) denotes the respective signal wavelength.