# GPS Multipath Analysis Using Fresnel Zones
## 2 Multipath Theory
### Fresnel Zones  
During the derivation of the formulas for the determination of the phase error \(\Phi_{C}\) and as shown in Figures 2 and 3, only a single reflection point is considered. However, GPS satellites are transmitting the signals with an aperture angle of \(\pm 13.9^{\circ}\) and thus, there are existing areas on the reflector surface that are contributing to the reflected signal. These active scattering regions are called Fresnel zones. The concept of Fresnel zones is illustrated in Figure 5.  
Between the satellite \(S\) and the mirrored antenna point \(\mathbf{A^{\prime}}\), a number (orders \(m\)) of ellipsoids of revolution can be formed, with \(S\) and \(\mathbf{A^{\prime}}\) representing the respective focal points [37]. The Fresnel ellipsoid of first order defines the region where most of the energy is transmitted. The intersection of the Fresnel ellipsoids and the reflector surface is called a Fresnel zone. In the case of a horizontal reflector, the size and shape depends on the satellite elevation angle \(el\), the carrier wavelength \(\lambda\) and the vertical distance \(h\) between the antenna and the reflector. The semi-major and semi-minor axes of the first Fresnel zone are given by  
Figure 4: (**Top**) Raw signal-to-noise ratio (SNR) time series (blue) and polynomial fit (black) representing \(A_{D}\). (**Bottom**) \(\delta\) SNR (blue) representing \(A_{M}\), after subtracting the polynomial fit from the raw SNR values, and related satellite elevation (red). Figure 5: Intersection of Fresnel ellipsoid of the first order with the horizontal reflector plane. \[b =\sqrt{\frac{\lambda h}{\sin el}+\left(\frac{\lambda}{2\sin el} \right)^{2}} \tag{11}\] \[a =\frac{b}{\sin el}\]  
and the orientation of the major axis is defined by the direction of the satellite-antenna vector [38]. In Figure 6, the size of the first Fresnel zone is shown for satellite elevation angles between 15\({}^{\circ}\) and 90\({}^{\circ}\) and antenna heights of 1 to 5 m.  
For a horizontal reflector, the size of the Fresnel zone increases as the elevation angle decreases, and it reaches its minimum at an elevation angle of 90\({}^{\circ}\). For an antenna height of \(h=1\) m, this leads to \(F\approx 0.6\) m\({}^{2}\) and for \(h=5\) m, this leads to \(F\approx 3\) m\({}^{2}\). Furthermore, the elevation angle determines the eccentricity of the ellipse (see Equation (11)). Thus, the Fresnel zone has a circular shape if the elevation angle is 90\({}^{\circ}\), and it is stretched for decreasing elevation angles. For vertical reflectors, or, in general, for reflectors with arbitrary orientation in space, as shown in Figure 7, the parameters of the Fresnel zones, \(a\) and \(b\), cannot directly be determined following Equation (11). Figure 6: Size of Fresnel zone for different satellite elevation angles and antenna heights determined for the GPS-L1 carrier wavelength. Figure 7: Intersection of Fresnel ellipsoid of the first order with an arbitrarily oriented reflector plane. Since the satellite elevation angle \(el\) no longer equals the angle between the signal and the reflector plane, it has to be substituted. For this purpose, at first, the incidence angle \(\zeta\) of the satellite signal with respect to the reflector plane is determined by  
\[\zeta=\arccos\left(\frac{\vec{n}\cdot\vec{p}}{\|\vec{n}\|\cdot\|\vec{p}\|}\right) \tag{12}\]  
where \(\vec{n}\) denotes the normal vector of the reflector plane and \(\vec{p}\) is the vector between the satellite \(S\) and the reflection point \(R\). Afterwards, the angle \(\beta\) can be expressed as  
\[\beta=90^{\circ}-\zeta. \tag{13}\]  
Moreover, the antenna height \(h\) is substituted by the orthogonal distance between the antenna and the reflector plane \(d_{o}\), leading to  
\[ \begin{split} b&=\sqrt{\frac{\lambda d_{o}}{\sin \beta}+\left(\frac{\lambda}{2\sin\beta}\right)^{2}}\\ a&=\frac{b}{\sin\beta}.\end{split}