## 2 Modeling the VLC Networks: Architecture, System Model and Scenarios
### Multi-Carrier Modulations
#### 2.1.2 Receiver  
In practice, several devices can be used for receiving the transmitted optical power, such as LEDs [23], solar cells [21] or camera sensors [22]. However, for high data rate transmission schemes, the most suitable receiver is the PD because of its large bandwidth, optical responsivity and efficiency in the conversion from light to current [20]. These devices transform the received photons into an electrical current that can be measured and processed by an electronic circuit. Focusing on data transmission, there are three main parameters to consider: responsivity, bandwidth and noise [71; 72]. The responsivity is the conversion ratio between the received optical power and generated electric current, and it is expressed as A/W. In contrast to the bandwidth and noise parameters, the responsivity does not depend on the area of detection of the PD. The bandwidth is inversely proportional  
Figure 6: Time-domain DCO-OFDM signal. Figure 7: Time-domain ACO-OFDM signal. to the junction capacitance of the PD, which is greater as the area of detection increases and it is given by  
\[\Delta f_{PD}=\frac{1}{2\pi R_{\text{load}}C_{j}}, \tag{7}\]  
where \(C_{j}\) and \(R_{\text{load}}\) denote the junction capacitance and the load resistance, respectively. Moreover, the resulting noise is proportional to both the area of detection \(A_{\text{PD}}\) and the measurement bandwidth \(\Delta f_{\text{m}}\). Specifically, the noise equivalent power characterizes the sensitivity of the PD, as it is the signal power provided by the PD when signal to noise ratio (SNR) and \(\Delta f_{\text{m}}\) are unified. It is defined as  
\[\text{NEP}=\frac{E_{i}A_{\text{PD}}}{\text{SNR}\sqrt{\Delta f_{\text{m}}}} \quad\quad\left[\frac{\text{W}}{\sqrt{\text{Hz}}}\right], \tag{8}\]  
where \(E_{i}\) is the incident irradiance expressed in \(\frac{\text{W}}{\text{cm}^{2}}\). Typically, the resulting noise in VLC is modeled as the sum of both shot noise, which depends on the received optical power and the ambient light (mainly daylight), and thermal noise [18]. Optical filters and concentrators are applied to improve the incidence of the optical power into the PD. For thin-film planar filters, the optical response given by a generic angle of incidence \(\varphi\) is [73]  
\[T(\varphi;\Delta\lambda,\phi)=\frac{T_{0}}{1+\left[\frac{\lambda_{0}-\lambda( \varphi;\phi)}{\frac{\lambda}{2}}\right]^{2\
u}}, \tag{9}\]  
where \(T_{0}\) corresponds to the peak transmission, \(\lambda_{0}\) is the wavelength of the transmitted signal, \(\frac{\Delta\lambda}{2}\) is the spectral half-power bandwidth, \(\
u\) is the Lambertian radiation index and \(\lambda(\varphi;\phi)\) represents the shifting to shorter wavelengths at non-normal incidences, which is given by \(\lambda(\varphi,\phi)=\lambda_{0}\left(\frac{n_{s}^{2}-n_{s}^{2}\sin^{2} \varphi}{n_{s}^{2}-n_{s}^{2}\sin^{2}\varphi}\right)\), where \(n_{s}\) and \(n_{i}\) are the effective index of the spacer layer and input layer, respectively, and \(\phi\) is the filter orientation. The use of filters following spherical patterns allows us to create optical concentrators that generate optical gain subject to a specific field of view (FoV) [73; 74]. The most common are the hemispherical concentrators, i.e., spherical filters truncated at \(90^{\circ}\). The gain of a hemispherical concentrator is  
\[G(\Psi_{c})= \begin{cases}\frac{n_{s}^{2}}{\sin^{2}\Psi_{c}}&\text{if }0\leq \varphi\leq\Psi_{c}\\ 0&\text{if }\varphi>\Psi_{c},\end{cases}  \tag{10}\]  
where \(\Psi_{c}\) denotes the resulting FoV of the PD. Notice that the channel gain increases for narrow FoVs and vice versa. From a networking point of view, wide FoVs increase the reception coverage of each user but also the received interference. This parameter should be carefully handled in the design of the system. Typically, the works in the state of the art consider receivers equipped with a single PD. As described above, there is a trade-off between FoV and optical gain.