# Multi-Wavelength Path Loss Model for Indoor VLC with Mobile Human Blockage
## 3 Multi-Wavelength Path Loss Analysis and Modeling under Free Space Propagation
### Multi-wavelength Path Loss Model of Free Space Propagation in Multi-LED Scenario  
According to the theoretical definition, path loss is related to the power of the transmitted signal and the power of the received signal. In the multi-LED scenario, the transmitted power is the sum of the power of the three LEDs which is a pre-known value. The received power can be regarded as the superposition of received power from every single LED. Therefore, the path loss in the multi-LED scenario can be obtained by  
\[PL[\mathrm{dB}]=10\mathrm{log}(\frac{P_{T1}+\cdots+P_{TN}}{P_{R1}+\cdots+P_{RN} }). \tag{13}\]  
According to the (9), the received power from a single LED can be expressed by  
\[P_{RN}=\frac{P_{TN}}{\frac{P_{TN}}{10\frac{P_{t1}}{10}\frac{(P_{t2}+\cdots+P_{ TN})}{10}}}, \tag{14}\]  
Figure 4: Muti-wavelength path loss model for the single-LED scenario.  
where \(PL_{single}\) is the proposed ABG model in the single-LED scenario. \(d_{N}\) indicates the distance from the receiver to LED\({}_{N}\). \(\lambda_{N}\) and \(P_{TN}\) indicates the wavelength and the power of LED\({}_{N}\), respectively. By substituting (14) to (13), the path loss in the multi-LED scenario can be derived as  
\[PL_{multi}^{{}^{\prime}}(d_{1},\ldots,d_{N};\lambda_{1},\ldots, \lambda_{N})\] \[=10\text{log}10(\frac{P_{T1}+\cdots+P_{TN}}{\frac{P_{T1}}{10} \frac{P_{T1}}{10}+\cdots+\frac{P_{TN}}{10}+\frac{P_{TN}}{10})}). \tag{15}\]  
Due to the utilization of the ABG model, the proposed path loss model in the multi-LED scenario takes into consideration the wavelength dependence and the distance dependence. Moreover, the relationship between path loss and the deployment of multiple LEDs is included through the application of all the transceiver distances. Also, the proposed model is applicable to scenarios with various transceiver deployments.  
The path loss of a 3-LED scenario with typical deployment is obtained using both the simulation method and the proposed model by fixing the wavelength at 455 nm. In the indoor scenario, the transmitters are uniformly arranged on the \(X\)-axis of the room with a spacing of 1.5 m which is illustrated in Figure 5.  
The simulated path loss and the path loss derived from the proposed model are shown in Figure 6. It is found that the path loss shows an obvious correlation with the transmitter deployment. In the scenario with the LED arranged on the \(X\)-axis of the indoor scenario, the path loss is also axisymmetric about the room. At the direction of the \(Y\)-axis, the path loss on both sides of the \(X\)-axis is 56.44 dB for receiving positions with the same distance of 3.16 m from the \(X\)-axis. Moreover, the path loss has an increasing trend with the increase of the distance between the receiving position and the \(X\)-axis. For example, at the direction of the \(Y\)-axis, the increasing distance of 0.91 m can lead to an increase of 4 dB for path loss. And, both the axisymmetric feature and the increasing trend reflect the rationality of the relationship between path loss and transmitter deployment in the multi-LED scenario.  
To verify the accuracy of the proposed model, root mean square error (RMSE) is calculated by  
\[RMSE=\sqrt{\frac{1}{NUM}\sum_{i=1}^{NUM}(PL_{sim}-PL_{mod})^{2}}, \tag{16}\]  
Figure 5: LED layout for the 3-LED scenario.  
where \(NUM\) is the number of simulated data. \(PL_{sim}\) and \(PL_{mod}\) indicate the simulated data and the data from the proposed model, respectively. The RMSE is 0.0094 indicating that the proposed model is in good agreement with the simulated data.