# Multi-Wavelength Path Loss Model for Indoor VLC with Mobile Human Blockage
## 4 Blockage Probability Analysis and Modeling
### Human Body Shadowing Calculation Method  
There are three factors related to the human body shadowing area: the shape of the human body, the transmitter position, and the human body position. Once these three factors are determined, according to a relative geometrical relationship, the extent and position of the human body shadowing can be derived. Next in this section, the human body is modeled first, and then the shadowing area is calculated based on the geometry-based method. In this paper, the human body is modeled as a cylinder with height and radius denoted by \(H\) and \(R\), respectively [35]. The position of the human body is \(P_{H}(x_{H},y_{H},0)\). The position of the transmitter and the receiver is \(P_{L}=(x_{L},y_{L},z_{L})\) and \(P_{R}=(x_{R},y_{R},0)\), respectively. These position coordinates are their values in the global coordinate system (GCS). There are two key parameters that determine the extent of the shadowing area: angle range and radius distance. When the horizontal distance between the human body and the transmitter is less than \(R\), the angle range of the shadow is 360 degrees in the LCS of the transmitter, and the shape of the shadow is circular or oval, as shown in Figure 7.  
For a specific shadow angle in the LCS of \(P_{L}\) within the angle range, its corresponding radial length is \((0,d_{up})\). \(d_{up}\) is the distance between \(P_{L}\) and point E which is the maximum radial length of the shadow. If the distance between a receiving position and the transmitter is less than \(d_{up}\), the receiving position is in the shadowing area which is in a blockage state. \(d_{up}\) can be expressed as  
\[d_{up}=(1+\frac{H}{z_{L}-H})\sqrt{(x_{Q})^{2}+(y_{Q})^{2}}, \tag{20}\]  
where \(z_{L}\) is the height of the indoor scenario. Point \(Q\) represents the intersection point of the radial distance and the human cylinder on the two-dimensional plane. \(x_{Q}\) and \(y_{Q}\) are the abscissa and ordinate of point \(Q\). The calculation methods of the position coordinates of point \(Q\) will be introduced later. When the horizontal distance between the human body and the transmitter is greater than \(R\), the angle range is \((\theta_{low},\theta_{up})\). The shape of the shadow is a part of a fan, as shown in Figure 8.  
\(\theta_{low}\) and \(\theta_{up}\) are the minimum and maximum coverage angle of the shadow which can be calculated according to the quadrant where \(P_{H}\) is located in the LCS of \(P_{L}\), as shown in Table 3. \(\theta_{0}\) can be expressed as  
\[\theta_{0}=\arcsin(\frac{R}{d_{l}}), \tag{21}\]  
where \(d_{l}\) is the horizontal distance between \(P_{L}\) and \(P_{H}\). \(\theta_{1}\) is the angle of the line from \(P_{L}\) to \(P_{H}\) in the LCS of \(P_{L}\), which can be expressed as  
\[\theta_{1}=\arctan(\frac{x_{H}-x_{L}}{y_{H}-y_{L}}). \tag{22}\]  
Figure 8: 3-D model of the fan human body shadowing. Figure 7: 3-D model of the oval human body shadowing. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Quadrant** & \(\theta_{low}\) & \(\theta_{up}\) \\ \hline
1 & \(90-\theta_{1}-\theta_{0}\) & \(90-\theta_{1}+\theta_{0}\) \\
2 & \(90+\theta_{1}-\theta_{0}\) & \(90+\theta_{1}-\theta_{0}\) \\
3 & \(270-\theta_{1}-\theta_{0}\) & \(270-\theta_{1}+\theta_{0}\) \\
4 & \(270+\theta_{1}-\theta_{0}\) & \(270+\theta_{1}+\theta_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calculation formula of \(\theta_{low}\) and \(\theta_{up}\). Under this circumstance, the range of radial distance is \((d_{low},d_{up})\). \(d_{low}\) is the lower limit of the shadow range on every shadow angle. \(d_{low}\) and \(d_{up}\) can be calculated by [34]  
\[d_{up}=(1+\frac{H}{z_{L}-H})\sqrt{(x_{QF})^{2}+(y_{QF})^{2}}, \tag{23}\]  
\[d_{low}=\sqrt{(x_{QN})^{2}+(y_{QN})^{2}}. \tag{24}\]  
\(x_{QF}\) and \(y_{QF}\) are the abscissa and ordinate of point \(Q_{F}\), \(x_{QN}\) and \(y_{QN}\) are the abscissa and ordinate of point \(Q_{N}\).