## 2 Hybrid PLC and VLC Channel Based on Distinct Optical Beam Configurations
### Hybrid Channel Based on Baseline Lambertian Optical Beam Configuration  
In the first case, the discussed channel model of the baseline hybrid system of concern is the cascade of the PLC and baseline Lambertian VLC portions. Accordingly, the channel model of the PLC portion can be expressed as follows [25]:  
\[H_{\text{PLC}}(f)=\sum_{l=1}^{L}g_{l}e^{-(\alpha_{0}+\alpha_{1}f^{k})\eta_{l}v_{ p}}e^{-j2\pi f/\eta_{l}}, \tag{1}\]  
where \(g_{l}\) denotes the weighting factor of the \(l\)-th path, which consists the reflection and transmission factor along this path, \(\alpha_{0}\) and \(\alpha_{1}\) are the attenuation parameters, \(k\) is the exponent of the attenuation factor (with usual values between 0.2 and 1), \(v_{p}\) denotes the group velocity, \(\tau_{l}\) is the delay of \(l\)-th path, and \(L\) represents the number of the paths. For this baseline case, since the conventional Lambertian optical spatial beam is adopted to construct the VLC part, the respective channel model of this Lambertian VLC part consists of one line-of-sight (LOS) component and one diffuse or non-line-of-sight (NLOS) component, which can be written as follows [26]:  
\[H_{\text{VLC}}^{\text{Lam}}(f)=\eta_{\text{LOS}}^{\text{Lam}}e^{-j2\pi f\Delta t _{\text{LOS}}}+\eta_{\text{DIFF}}\frac{e^{-j2\pi f\Delta t_{\text{DIFF}}}}{1+ jf/f_{\text{C}}}, \tag{2}\]  
where \(\eta_{\text{LOS}}^{\text{Lam}}\), \(\eta_{\text{DIFF}}\), \(\Delta t_{\text{LOS}}\) and \(\Delta t_{\text{DIFF}}\) denote the gains and delays of the Lambertian LOS and diffuse signals, respectively. \(f_{\text{C}}\) denotes the 3-dB cutoff frequency of the purely diffuse optical channel. Actually, the mentioned \(f_{\text{C}}\) can be identified by \(f_{\text{C}}=1/(2\pi\tau)\), where the exponential decay time is given by the following:  
\[\tau=-\frac{\langle t\rangle}{\text{In}\langle\rho\rangle}. \tag{3}\]  
The figure \(\langle t\rangle\) can be viewed as the average time between two reflections. In a typical rectangular room, \(\langle t\rangle\) is given as follows [26]:  
\[\langle t\rangle=\frac{4\text{V}_{\text{ROOM}}}{c\text{A}_{\text{ROOM}}}=\frac {2}{c}\frac{l\cdot w\cdot h}{l\cdot w+l\cdot h+w\cdot h}, \tag{4}\]  
where \(l\), \(w\), and \(h\) are the length, the width, and the height, respectively. Under this baseline Lambertian beam configuration, the LOS optical channel gain can be specifically represented by the following [13; 22]:  
\[\eta_{\text{LOS}}^{\text{Lam}}=\left\{ \begin{array}{ll}\frac{A_{R}}{d^{2}} I_{\text{Lam}}(\phi)\cos(\theta),&0\leq\theta\leq\theta_{\text{FOV}}\\ 0,&\theta>\theta_{\text{FOV}}\end{array} \right., \tag{5}\]  
where \(A_{R}\) denotes the detection area of receiver; \(d\) denotes the distance between the optical source LED and the receiver; \(\phi\) denotes the irradiance angle from the optical source; \(\theta\) denotes the angle of incidence to the receiver, and \(\theta_{\text{FOV}}\) is the field of view (FOV) of the receiver. As presented in (3), the Lambertian emission intensity \(R_{\text{Lam}}(\phi)\) is the key metric to characterize the Lambertian radiation performance of the LED optical spatial beams in LOS channel gain, as given by the following:  
\[I_{\text{Lam}}(\phi)=\frac{m_{\text{Lam}}+1}{2\pi}\cos^{m_{\text{Lam}}}(\phi), \tag{6}\]  
where \(m_{\text{Lam}}\) denotes the Lambertian beam index. And, it can be written as follows:  
\[m_{\text{Lam}}=-\frac{\text{In}2}{\text{In}(\cos\phi_{1/2})}, \tag{7}\]where \(\phi_{1/2}\) denotes the half-intensity beam angle of the LED. When the Lambertian index is set as 1, then the respective 3D radiation pattern is as is shown in Figure 1. In this Figure, one indoor hybrid PLC and VLC scenario with a single Lambertian beam is illustrated as well. By substituting (4) into (3), the baseline Lambertian LOS optical channel gain for the hybrid PLC and VLC can be renewed as follows:  
\[\eta_{\text{LOS}}^{\text{Lam}}\ =\ \left\{