# GPS Multipath Analysis Using Fresnel Zones
## 4 Multipath Analysis from a Spatially-Limited Reflector Using Fresnel Zones
### Multipath Analysis
The pink shaded areas bounded by the solid and dashed red lines represent the transition phases, where the reflection point and only parts of the Fresnel zones are located on the roof (\(50\%\leq P_{FZ}\geq 100\%\)). In the red shaded areas (left and right of the dashed red lines), the reflection point is not located on the roof and \(P_{FZ}\leq 50\%\). Figure 17: Fresnel zones and reflection points related to antenna A (red ellipses and green dots) and B (cyan ellipses and magenta dots) for satellites G14 (**a**) and G19 (**b**) in a topocentric coordinate system. The Fresnel zones are shown for every hundreth observation epoch. Top right in left and right figures: skyplot with satellite tracks of G14 and G19, where the PRN number is located at the end of each track. It can be noticed that during the rising phase of G14, the time from which the Fresnel zones are completely located on the roof appears earlier at antenna B (\(\approx\)21:40) than at antenna A (\(\approx\)22:25). Furthermore, the typical multipath oscillations occur in these 100% sections, most visibly between 21:45 and 22:45 at antenna B. In this period, the elevation angle is below 30\({}^{\circ}\), and thus, corresponds with the considerations on the Rayleigh criterion at the beginning of this section. Contrarily, in the red shaded areas at the beginning and end of the time series, the systematic and periodic behavior disappears and the curve shape is of a more random nature. This is reasonable, since the reflection points are not located on the roof, and the overlap of the Fresnel zones and the roof is only up to 50%. Hence, if only these two regions are taken into account, the relationship between the Fresnel zone concept and multipath occurrence can be confirmed. Nevertheless, the most interesting period of time is the transition phase (50% \(\leq P_{FZ}\geq\) 100%), represented by the pink shaded region, where the reflection points are still on the roof, but the Fresnel zones are only partially overlapping the reflector. During the rising phase of G14, the \(\delta\)-SNR time series of antenna A shows more random behavior in this region. In comparison, in the setting phase of G14, the multipath oscillations are still visible but with a slightly lower amplitude. The same holds for antenna B but in reverse order. Here, the oscillations are clearly visible during the rising phase of G14. In Figure 19, the same data is shown for satellite G19. Due to the orientation of G19 with respect to the roof, the three phases of \(P_{FZ}\) are similar at both antennas, and the \(\delta\)-SNR curves show the same behavior. During the rising phase, G19 moves according to the orientation of the roof and only at the lowest elevation angles of around 5\({}^{\circ}\) are the Fresnel zones not completely located on the roof. As a consequence, the transition phase and the phase where \(P_{FZ}\leq\) 50% are very short, and the shape of the \(\delta\)-SNR time series cannot clearly be analyzed. During the complete overlapping of the Fresnel zones and the roof, the characteristic multipath oscillations appear at both antennas, whereby the amplitudes are of the same order of magnitude. From the beginning of the transition phase until the end of the time series, the systematic effects disappear, and from 23:00 onwards, the shape of both curves is of a random nature. In order to better assess the accordance between the real observations and the theory on multipath propagation, the SNR values for both antennas were simulated following Equation (4) under the assumption of a horizontal reflector. Therefore, the multipath relative phase \(\Delta\Phi_{M}\) was determined by Equation (6), and the attenuation factor \(\alpha\) was derived from the raw SNR time series of G14 and G19 in units of dB-Hz, respectively, following the procedure described in Section 2.2.