# Evaluation of Deployment Challenges of Wireless Sensor Networks at Signalized Intersections
## 2 Ray Launching Technique and Simulation Scenarios  
### Ray Launching Technique  
A deterministic method based on an in-house developed 3-D RL code has been used to analyze the radio electric behavior of the considered scenario. The 3D RL algorithm is based on Geometrical Optics (GO) and Geometrical Theory of Diffraction (GTD). It has been verified in the literature with different applications, such as the analysis of wireless propagation in closed environments [27; 28; 29; 30], interference analysis [31] or electromagnetic dosimetry evaluation in wireless systems [32]. The main principle of the RL techniques is to identify a single point on the wave front of the radiated wave with a ray that propagates in the space following a combination of optic and electromagnetic theories. This phenomenon is illustrated in Figure 1 for the considered scenario. Each ray propagates in the space as a single optic ray. The incident electric field (\(E_{i}\)) created by an antenna at a distance r in the free space can be calculated by [33]:  
\[E_{i}^{\perp\parallel}=\sqrt{\frac{P_{\mathrm{rad}}D_{t}\left(\theta_{t},\ \phi_{t}\right) \eta_{0}}{2\pi}}\frac{\mathrm{e}^{-j\beta_{0}\tau}}{r}X^{\perp\parallel}L^{ \perp\parallel} \tag{1}\]  
where \(P_{\mathrm{rad}}\) is the radiated power with a directivity \(D_{t}\left(\theta_{t},\phi_{t}\right)\), where the sub-index \(t\) refers to the transmitted angle, and polarization ratio \(\left(X^{\perp},X^{\parallel}\right)\). \(\beta_{0}=2\pi f_{c}\sqrt{\varepsilon_{0}\mu_{0}}\), \(\varepsilon_{0}=8.854\times 10^{-12}\), \(\mu_{0}=4\pi\times 10^{-7}\) and \(\eta_{0}=120\pi\). \(L^{\perp\parallel}\) are the path loss coefficients for each polarization. The parameter \(j\) in Equation (1) refers to the complex number.  
When the rays impinge with an obstacle in their path, a reflected and a transmitted ray are created with new angles provided by [PERSON]'s law [34]. It is important to take into account that the rays considered in the GO only approach are only direct, reflected and refracted rays, leading to the existence of abrupt areas, which correspond to the boundaries of the regions where these rays exist. Because of  
Figure 1: Wave front propagation with rays associated with single wave front points in the considered scenario.  
that, the diffracted rays are introduced with the GTD and its uniform extension, the Uniform GTD (UTD). The main purpose of these diffracted rays is to remove field discontinuities and to introduce proper field corrections, especially in the zero-field regions predicted by GO. The diffracted field is calculated by [35]:  
\[E_{\text{UTD}}=e_{0}\frac{e^{-j\beta_{1}}}{s_{1}}D^{\perp\parallel}\sqrt{\frac{s _{1}}{s_{2}\left(s_{1}+s_{2}\right)}}e^{-j\beta_{2}} \tag{2}\]  
where \(s_{1}\), \(s_{2}\) are the distances from the source to the edge and from the edge to the receiver point, respectively. \(D^{\perp\parallel}\) are the diffraction coefficients given by [35; 36].  
The RL algorithm is performed three-dimensionally, with angular resolution (horizontal and vertical planes) in a predefined solid angle that considers the radiation diagram of the transceivers sources. Spatial resolution is also defined by a uniform hexahedral mesh. Parameters such as frequency of operation, radiation patterns of the antennas, number of multipath reflections, separation angle between rays, and cuboid dimension can be taken into account. Besides, all the material properties for all the elements within the scenario can also be considered, given the dielectric constant and the loss tangent at the frequency range of operation of the system under analysis. When a ray impacts with an obstacle, reflection, refraction and diffraction will occur, depending on the geometry and the electric properties of the object, as is depicted in Figure 2.