## 2 Modeling the VLC Networks: Architecture, System Model and Scenarios
### System Model  
Modeling indoor VLC systems considers \(M\), \(m=\{1,\ldots,M\}\), optical APs providing data service to \(K\), \(k=\{1,\ldots,K\}\), users, while ensuring constant illumination within the considered indoor mean, as shown in Figure 10. The signal associated with the \(M\) optical APs at time \(t\) can be written as \(\mathbf{x}(t)= \begin{bmatrix}x_{1}&\ldots&x_{M}\end{bmatrix} ^{T}\in\mathbb{R }_{+}^{M\times 1}\), where \(x_{m}\) denotes the signal of optical transmitter \(m\). Thus, assuming a flat channel response, the signal received by user \(k\) at time \(t\) is given by  
\[y^{[k]}(t)=\mathbf{h}^{[k]}(t)^{T}\mathbf{x}(t)+z^{[k]}(t), \tag{11}\]  
where \(\mathbf{h}^{[k]}(t)= \begin{bmatrix}h_{1}^{[k]}&\ldots&h_{M}^{[k]}\end{bmatrix} ^ {T}\in\mathbb{R}_{+}^{M\times 1}\) is the channel vector between the \(M\) optical APs and user \(k\) at time \(t\), \(h_{m}^{[k]}\) corresponds to the channel response between AP \(m\) and user \(k\), and \(z^{[k]}(t)\) is real-valued additive white Gaussian noise for user \(k\) with zero mean and variance \(\sigma_{z}^{2}\)[18; 93]. Figure 10: VLC scenario. There are two main components that contribute to the channel impulse response in the time domain of a VLC channel: the front-end and the free-space components. Thus, the time-domain channel impulse response is given by  
\[h(t)=h_{\text{fe}}(t)\otimes h_{\text{fs}}(t), \tag{12}\]  
where \(h_{\text{fe}}(t)\) represents the channel impulse response of the front-end devices that performs a low-pass filtering with a typical 3 dB bandwidth in the range between 3 and 60 MHz [94; 95; 96]. The frequency-domain channel transfer function of \(h_{\text{fe}}(t)\) can be written as  
\[\left|H_{\text{fe}}(f)\right|=\sqrt{\exp\left(-\frac{f}{F_{\text{fe}}}\right)}, \tag{13}\]  
where \(F_{\text{fe}}\) is the so-called bandwidth factor that controls the frequency-domain characteristics of the front-end devices. On the other hand, \(h_{\text{fs}}(t)\) denotes the free-space channel impulse response that comprises a flat channel component plus a diffuse channel contribution depending on the propagation conditions. The flat and diffuse components correspond to the LoS and non-LoS (NLoS) propagation, respectively. Moreover, the LoS propagation may be subject to blocking effects [97]. According to [98; 99], the channel gain between the optical transmitter \(m\) and user \(k\) at time \(t\) is given by  
\[h_{\text{fs},m}^{[k]}(t)=\Big{(}h_{\text{LoS},m}^{[k]}\delta(t)\mathbf{1}_{ \text{block},m}^{[k]}+h_{\text{diff}}(t-\Delta T)\Big{)}, \tag{14}\]  
where \(h_{\text{LoS},m}^{[k]}\) denotes the LoS component from optical AP \(m\) to user \(k\), \(h_{\text{diff}}\) is the contribution of the NLoS component given by diffuse reflections, \(\Delta T\) is the delay between the LoS and NLoS components and  
\[\mathbf{1}_{\text{block},m}^{[k]}=\left\{ \begin{array}{ll}1&\text{if no obstacle between PD}\,k\,\text{and AP}\,m\\ 0&\text{if obstacle between PD}\,k\,\text{and AP}\,m\end{array} \right.. \tag{15}\]  
Notice that the LoS component theoretically provides a flat channel response in the frequency domain. However, this flat channel response decays in practice due to the front-end device frequency response (\(\left|H_{\text{fe}}(f)\right|\)). The LoS component is determined by the geometry between the optical AP and the PD, as shown in Figure 11. The distance between optical AP \(m\) and user \(k\) is denoted as \(d_{km}\) and the irradiance and incidence angles (For ADRs, assuming that the size of the PDs is much smaller than the distance to any optical AP, the irradiance angles between optical AP \(m\) and all the PDs that compose the ADR of user \(k\) are the same. On the other hand, distinct incidence angles are obtained in each of the PDs of the ADR of user \(k\).) are denoted as \(\phi_{m}^{[k]}\) and \(\phi_{m}^{[k]}\), respectively. Thus, the LoS contribution to the optical channel between transmitter \(m\) and user \(k\) is [61],  
\[h_{\text{LoS},m}^{[k]}=