# Design of PAM-8 VLC Transceiver System Employing Neural Network-Based FFE and Post-Equalization
## 2 Analysis of Models and Limitations in VLC Systems
### Bandwidth Limitation and Nonlinearity  
The LED model mentioned in Section 2.1 was utilized for analyzing the effect on the modulation bandwidth in the VLC system. As mentioned in Section 2.1, it could be treated as a first-order low-pass filter in the low-frequency region, as the intrinsic model was the dominant item. Considering the analysis of the impulse response, the expression of this low-pass filter in the s-domain was calculated as below:  
\[\mathrm{c(s)}=\frac{\mathrm{A}}{1+\frac{\mathrm{s}}{\omega_{0}}}, \tag{4}\]  
where A is the amplitude of the signal, and \(\omega_{0}\) is the characteristic frequency of the low-pass filter. After applying the inverse Laplace transform to Equation (5), the limited bandwidth expression in the time domain was shown as the equation below:  
\[\mathrm{c(t)}=\omega_{0}\mathrm{A}\ \times\ \mathrm{e}^{-\omega_{0}t}\times \mathrm{u(t)}, \tag{5}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**VLC Link** & **Parameters** & **Typical Value** \\ \hline \multirow{4}{*}{Transmitter (\(\mu\)-LED Model)} & Total Power & 20 W \\  & Half-power Angle & 50\({}^{\circ}\) \\  & \(-\)3 dB Bandwidth & \(\sim\)50 MHz \\ \hline \multirow{4}{*}{Receiver (PD Model S5973-01)} & PD Area & 0.12 mm\({}^{2}\) \\  & Photosensitivity & 0.51 A/W \\  & Dark Current & 100 pA \\  & Terminal Capacitance & 1.6 pF \\  & Field-of-view Angle & 30\({}^{\circ}\) \\  & \(-\)3 dB Bandwidth & 1 GHz \\ \hline \multirow{4}{*}{Free-Space Channel} & Link Range & 0.1–5 m \\  & Emission Angle \(\alpha\) & 0\({}^{\circ}\) \\  & Receiving Angle \(\beta\) & 0\({}^{\circ}\) \\ \hline \multirow{4}{*}{Underwater Channel} & Link Range & 0.1–5 m \\  & Water Type & Clear sea water \\ \cline{1-1}  & Absorption Coefficient a(\(\lambda\)) [28] & 0.114 m\({}^{-1}\) \\ \cline{1-1}  & Scattering Coefficient b(\(\lambda\)) [28] & 0.037 m\({}^{-1}\) \\ \cline{1-1}  & Tank Attenuation Coefficient \(\mathrm{a_{T}}(\lambda)\) [27] & 0.265 m\({}^{-1}\) \\ \hline \end{tabular}
\end{table}
Table 1: Simulation parameters for the VLC link model.  
Figure 4: Simulated path loss versus distance in free-space channel and water channel.  
where \(u(t)\) is the unit step function, and \(e^{-\omega_{0}t}\) is the time-dependent item complying with exponential attenuation. Therefore, \(c(t)\) caused undesired tails leading to ISI in the time domain by adding the time-dependent exponential attenuation to the input signals. This effect of bandwidth limitation on the original input signal is shown in Figure 5, which presents the generated tails spread into adjacent symbols with overlapping [29].  
To evaluate the intrinsic nonlinearity of the LED device, the I-V relationship of the LED at forward bias needed to be calculated and was [30]:  
\[I_{LED}=\left(I_{n}+I_{p}\right)\left(e^{\frac{\delta V}{kT}}-1\right), \tag{6}\]  
where \(\left(I_{n}+I_{p}\right)\) is the electrons and holes current, and \(e^{\frac{\delta V}{kT}}\) is a temperature-dependent item that has an exponential relationship with the bias voltage. It can be noticed that the driving current through the LED was nonlinear with a bias voltage of the LED that caused amplitude asymmetry with the DC bias point.