# Multi-Wavelength Path Loss Model for Indoor VLC with Mobile Human Blockage
## 2 Multi-Wavelength Path Loss Model with Human Blockage
### Multi-Wavelength Path Loss with Human Blockage in Multi-LED Scenario  
To meet the lighting requirements, multiple LEDs are usually used for lighting simultaneously, which enables multi-LED communication in VLC systems. In this subsection, we discuss path loss with the presence of human blockages in the multi-LED scenario. The use of multiple LEDs also increases the complexity of the blockage conditions. In the single-LED scenario, the path loss with human blockage shows the two-states properties, while in the multi-LED scenario, signals from different transmitters form different shadowing areas on the receiving plane and thus, result in two-state for each transmitter. For a receiving position, it may receive optical signals from some transmitters, and optical signals from other transmitters will be blocked. The number of blockage conditions depends on the number of transmitters. According to the principle of permutation and combination, when the number of LEDs is \(N\), the number of blockage conditions is \(C_{N}^{0}+C_{N}^{1}+\cdots+C_{N}^{N}\). Therefore, at each receiving position, the possible case for path loss is also \(C_{N}^{1}+C_{N}^{1}+\cdots+C_{N}^{N}\). Taking \(N=3\) as an example, for the receiving position in the 3-LED scenario, there are 8 cases of blockage conditions, as illustrated in Figure 3. Different from the situation in the single-LED scenario that, once the blockage happens the receiving positions can not receive any optical signal from the transmitter, in the multi-LED scenario, only under the blockage condition where all the signals from the three LEDs are cut off, the receiving position will be in a complete blockage state. While under the other seven cases of blockage conditions, which can be classified into three categories, i.e., no blockage, blockage for one LED, and blockage for two LEDs, the signal can be partially or totally received. Figure 3: Human blockage in the 3-LED scenario (This figure represents typical human blockage conditions in the 3-LED scenario. If there is a change in the human body position, the blockage conditions will change accordingly). Therefore, the multi-wavelength path loss model with human blockage in the 3-LED scenario is proposed in the form of a piecewise function with 8 cases, which is given by  
\[PL_{multi}(d_{1},d_{2},d_{3};\lambda_{1},\lambda_{2},\lambda_{3})=\left\{ \begin{array} []{ll}PL^{{}^{\prime}}_{multi}(d_{1},d_{2},d_{3};\lambda_{1},\lambda_{2}, \lambda_{3}),&p_{0}\\ PL^{{}^{\prime}}_{multi}(d_{2},d_{3};\lambda_{2},\lambda_{3}),&p_{11}\\ PL^{{}^{\prime}}_{multi}(d_{1},d_{3};\lambda_{1},\lambda_{3}),&p_{12}\\ PL^{{}^{\prime}}_{multi}(d_{1},d_{2};\lambda_{1},\lambda_{2}),&p_{13}\\ PL^{{}^{\prime}}_{multi}(d_{3};\lambda_{3}),&p_{21}\\ PL^{{}^{\prime}}_{multi}(d_{2};\lambda_{2}),&p_{22}\\ PL^{{}^{\prime}}_{multi}(d_{3};\lambda_{3}),&p_{23}\\ PL^{{}^{\prime}}_{multi}(d_{3};\lambda_{3}),&p_{23}\\ \end{array} \right. \tag{12}\]  
Here, the path loss for the 8 cases of blockage conditions can be derived by utilizing the multi-wavelength path loss model of free space propagation in the multi-LED scenario. For instance, the cases in which only the signal from LED\({}_{1}\) is blocked can be regarded as the free space propagation of the signal from LED\({}_{2}\) and LED\({}_{3}\). The path loss of other cases can be derived in a similar way. Moreover, the probability that the 8 cases of blockage conditions occur can be described by the BP. \(p_{0}\) is the non-blockage probability that all the signals from the three transmitters are received. \(p_{1N}\) is the BP that signals from LED\({}_{N}\) are blocked and the signals from the other LEDs are received. \(p_{2N}\) is the BP that signals from LED\({}_{N}\) are received and the signals from the other LEDs are blocked. \(p_{3}\) is the complete blockage probability that the signals from three LEDs are all blocked.