# Multi-Wavelength Path Loss Model for Indoor VLC with Mobile Human Blockage
## 2 Multi-Wavelength Path Loss Model with Human Blockage
### VLC Channel Model  
A VLC system consists of three parts: the transmitter, the receiver, and the visible light channel. At present, light-emitting diode (LED) and photodiode (PD) are the commonly used transmitter and receiver, respectively. As for the description of the visible light channel, the recursive model is used due to its advantages of high accuracy. Generally, LED adopts the Lambert radiation pattern [23], and the radiant intensity at each direction angle is [24]  
\[R(\varphi)=[\frac{(m+1)}{2\pi}]\cos^{m}(\varphi), \tag{1}\]  
where \(\varphi\) is the angle of irradiance which is commonly denoted as the angle of departure. \(m\) is the mode number of the radiation lobe, which expresses the directivity of the source beam and it is related to the LED semi-angle at half-power \(\varphi_{1/2}\) as  
\[m=\frac{-\text{ln}2}{\text{ln}[\cos(\varphi_{1/2})]}. \tag{2}\]  
A mode of \(m=1\) corresponds to a traditional Lambertian source. Light is emitted from the LED and transmitted through the visible light channel. Due to the multi-path effect, the light from the LED source \(S\) will reach the receiver \(R\) through multiple different paths. For the LoS component, the channel impulse response (CIR) can be calculated according to the LED radiation pattern as [25]  
\[h^{0}(t)\approx\frac{A_{det}}{D^{2}}R(\varphi)\cos(\theta)\text{rect}(\frac{ \theta}{\phi_{c}})\delta(t-\frac{D}{c}), \tag{3}\]  
where \(\theta\) is the angle of arrival. \(\phi_{c}\) is the width of the field of vision (FoV) of the PD. \(D\) is the distance between the LED and the PD. \(A_{det}\) is the area of the PD. The received power of the LoS link is [26]  
\[P_{LOS}=P_{T}\int_{0}^{\infty}h^{0}(t)dt, \tag{4}\]  
where \(P_{T}\) is the average transmitted power of LED. For the non-line-of-sight (NLoS) component, the CIR can be calculated by utilizing the recursive model. In the recursive method, the NLoS process is treated as multiple LoS process. For example, the first reflection path can be regarded as two LoS processes. Firstly, each effective scatterer is divided into small reflecting elements with a total number of \(n\). Each reflecting element is considered as both the receiver of the first LoS (\(\varepsilon_{i}^{R}\)) with an effective area \(\Delta A\) and a position of \(s_{i}\), and the transmitter of the second LoS (\(\varepsilon_{i}^{S}\)) whose radiation pattern is described by the transmitter radiation pattern. By using convolution operation to connect between these two processes, the CIR of the first reflection can be obtained. Through multiple recursive operations, k-th reflection can be obtained. Consequently, the CIR of the rays experiencing \(k\)-th reflection can be expressed as [25]  
\[ \begin{split} h^{k}(t;S,R)&\approx\sum_{i=1}^{n}\rho_{ \varepsilon_{i}^{\mathrm{F}}}l^{0}(t;S,\varepsilon_{i}^{\mathrm{R}})\otimes h^{ (k-1)}(t;\varepsilon_{i}^{S},R)\\ &=\frac{(m+1)}{2\pi}\sum_{i=1}^{n}\frac{cos^{m}(\phi)cos(\theta)} {D^{2}}\times\text{rect}(\frac{2\phi_{c}}{\pi})\\ &\times h^{(k-1)}(t-\frac{D}{c};s_{i},R)\Delta A,\end{split}  \tag{5}\]  
Here, \(\rho(\lambda)\) denotes the wavelength-dependent reflection coefficient of the reflecting element. In an indoor scenario, the typical scatterer, i.e., walls, are made of plaster materials. The wavelength-dependent reflection coefficient of plaster material is illustrated in Figure 1. It is observed that there is a significant increase in the reflection coefficient with the increasing signal wavelength which leads to the wavelength dependence of the visible light channel. Theoretically, the number of reflections is infinite for visible light signals. In previous studies, it is reported that the contribution from the second-order reflection is about 0.7 dB [27] and this is in good agreement with the result outlined in [28]. It is about a 2% decrease in the received power, which is not significant from the practical systems point of view.