# Testbed for Experimental Characterization of Indoor Visible Light Communication Channels
## 2 Indoor Visible Light Channel  
Figure 1 shows an illustration of the indoor VLC environment, where each lamp is composed of several LED chips whose light radiation contributes in the total received power. The radiation pattern of a LED is usually modeled as generalized Lambertian of order \(n\)[18]:  
\[R_{E}(\phi)=P_{E}\frac{n+1}{2\pi}\cos^{n}\phi,\qquad-\frac{\pi}{2}\leq\phi\leq \frac{\pi}{2}, \tag{1}\]  
where \(P_{E}\) is the total power emitted by the LED and \(n=-\ln 2/\ln(\cos\phi_{1/2})\), being \(\phi_{1/2}\) the transmitter semiange (at half power). The LOS contribution of this LED to the received power at detector is obtained as an integration over the wavelengths \(\lambda\) of the emitted visible light [13]:  
\[P_{R}=R_{E}(\phi)\int_{\lambda}p^{\prime}(\lambda)A_{\text{eff}}(\varphi)d \lambda/d^{2}=P_{E}\frac{n+1}{2\pi d^{2}}\cos^{n}\phi A_{r}\cos\varphi\int_{ \lambda}p^{\prime}(\lambda)d\lambda. \tag{2}\]  
In the above equation, \(\phi\) is the outgoing angle of light from the emitter towards the receiver, \(A_{\text{eff}}=A_{r}\cos\varphi\) the _effective area_ of the detector of physical area \(A_{r}\) with the observation angle \(\varphi\) being the incoming angle of the light with respect to the normal of the detector surface, \(d\) the distance between emitter and receiver, and \(p^{\prime}(\lambda)=p(\lambda)/P_{E}\) the normalized power spectral density (PSD) of the WLED [13]. In Equation (2), \(\int_{\lambda}p^{\prime}(\lambda)d\lambda=1\), thus the LOS contribution does not depend on the specific shape of the spectral emission characteristic of the LED. However, it is relevant when reflections on surfaces are also considered as we will see below. Moreover, this LOS contribution only occurs if the LED is in the field of view (FOV) of the receiver, that is, \(\varphi\leq\text{FOV}\leq\pi/2\). In an indoor environment, there can be contributions to the total received power from emitted radiation that is reflected on walls or other surfaces inside the room. The reflection pattern of a surface at the wavelength \(\lambda\) can be approximated by the Phong's model, which takes into account both diffuse and specular components of the light reflection [19]:  
\[R_{s}(\phi^{\prime},\theta^{\prime},\lambda)=\rho(\lambda)P_{l}R_{ s}^{\prime}(\phi^{\prime},\theta^{\prime})\] \[= \rho(\lambda)P_{l}\Big{[}r_{d}\frac{1}{\pi}\cos\phi^{\prime}+(1-r _{d})\frac{m+1}{2\pi}\cos^{m}(\phi^{\prime}-\theta^{\prime})\Big{]}.\]  
Here, \(r_{d}\) is the percentage of the received power that is reflected diffusely, \(\rho(\lambda)\) the reflection coefficient of the surface, \(m\) the mode number of the specular component which defines its directivity, \(\theta^{\prime}\) the incoming angle of the incident ray, which impinges the surface with power \(P_{l}\) and \(\phi^{\prime}\) the outgoing angle of observation (see Figure 1). To calculate the contribution after reflections, Monte Carlo based algorithms [20] randomly generate \(N\) rays according to the radiation pattern described in Equation (1) and determine the contribution of that ray after each collision and reflection off a surface. These algorithms have been demonstrated to be very accurate [21] and simultaneously with a much shorter simulation time than deterministic methods [22].