# GPS Multipath Analysis Using Fresnel Zones
## 3 Identification of Diffracted Satellite Signals Using Fresnel Zones and OAEMs
### Identification Algorithm  
The identification algorithm uses the OAEM as an representation of possible obstructions sources in the surroundings of the antenna. For each cell \(c_{i}\) of the OAEM, the respective elevation angle \(el_{i}\) and the topocentric coordinates of the related TLS scan point, \(\mathbf{p}_{i}=\begin{bmatrix}e_{i}&n_{i}&u_{i}\end{bmatrix},\) are stored. If cells are empty because there are no obstruction sources in this azimuthal direction, \(el_{i}\) is set to a predefined standard value, e.g., \(5^{\circ}\), and the coordinates of \(\mathbf{p}_{i}\) are determined by linearly interpolating between the coordinates of the next adjacent and filled cells.  
In Figure 12, the basic features of the algorithm are illustrated. It should be noted that since the proposed algorithm is identical for every satellite \(k\) and every observation epoch \(t\), in the following description of the algorithm, the indices \(k\) and \(t\) are omitted for reasons of clarity.  
For the identification or exclusion of diffracted signals, in a first step, the grid cell \(c_{i}\) that contains the satellite azimuth angle \(az\) (\(c_{16}\) is identified, as shown in Figure 12). Afterwards, the radius \(R_{F}\) of the first Fresnel zone is determined at position \(\mathbf{p}_{i}\) following Equation (15). After \(R_{F}\) has been converted to the angular value \(\alpha_{R}\), the azimuth angles of all points of the Fresnel zone are inside the interval \(az_{F}=\{az-\alpha_{R},az+\alpha_{R}\}\). At the same time, all grid cells \(c_{j}\) that intersect with the Fresnel zone are derived from \(az_{F}\) (\(c_{13}\ldots c_{19}\)). In order to identify the obstructed areas of the Fresnel zone, it is sufficient to test if any part of the lower semicircle of the Fresnel zone boundary is lower than the elevation angles in the related grid cell. In Figure 13, an enlarged display of the relevant part of Figure 12 is shown.  
It becomes obvious that only the point with the lowest elevation angle on the Fresnel zone boundary needs to be identified in each cell. As shown in Figure 13, these points are the intersection points of the cell borders and the Fresnel zone boundary. Thus, by applying the law of Pythagoras, for each cell \(c_{j}\), one elevation angle \(el_{j}\) is determined by  
\[el_{j}=\begin{cases}el_{k}-\sqrt{R_{F}^{2}-\left[\left(az_{j-1}-az_{k}\right) \cdot d\right]^{2}},&j>i,\\ el_{k}-\alpha_{R},&j=i,\\ el_{k}-\sqrt{R_{F}^{2}-\left[\left(az_{j}-az_{k}\right)\cdot d\right]^{2}},&j<i,\end{cases} \tag{20}\]  
where \(d\) denotes the distance between the antenna and \(\mathbf{p_{i}}\), and the elevation and azimuth angles are given in radians.  
Figure 12: Diffraction identification using the Fresnel zone. The Obstruction Adaptive Elevation Mask (OAEM) is represented by the elevation angles (blue dotted line) and the respective azimuth grid cells \(c\). The half red and half black circle represent the first Fresnel zone with radius \(R_{F}\) determined for point \(D\) on the direct signal path along the LOS-vector, which is represented by the azimuth, elevation and distance to satellite \(k\). The red shaded cells indicate an overlap with the Fresnel zone, and the red and green dots denote whether the elevation angles of points on the red semicircle are lower than the OAEM or not.  
Figure 13: Enlarged display of the Fresnel zone and azimuth cell grid from Figure 12. The blue lines denote the elevation angles for the intersection points of the Fresnel zone boundary (red line) and cell borders (vertical dotted lines). The satellite azimuth angle is denoted by \(az_{k}\), and \(R_{F}\) denotes the Fresnel zone radius.  
In the last step, the elevation angles \(el_{j}\) are compared to the respective values in the OAEM. If in any of the grid cells \(el_{j}\) is smaller than the OAEM value, parts of the Fresnel zones are obstructed, and although the direct signal path is not blocked, diffraction effects can be expected (cf. Section 2.3). In Figure 12, this is the case for cells \(c_{13}\) to \(c_{15}\).