# LED Nonlinearity Estimation and Compensation in VLC Systems Using Probabilistic Bayesian Learning
## 2 System Model  
In this section, we introduce the model of an OFDM-based nonlinear VLC system and the block diagram of the system model is illustrated in Figure 1. As we can see, the input bits are first modulated into real-valued OFDM symbols and then TSs are added for efficient LED nonlinearity estimation and compensation at the receiver side. The obtained digital signal is subsequently transformed into an analog signal through digital-to-analog conversion (DAC), and then a direct current (DC) bias is further added to convert the bipolar signal into a unipolar signal in order to generate a real-valued nonnegative driving signal for the LED transmitter. After that, the generated signal is fed into a white LED which suffers from nonlinearity. In order to support a long transmission distance and a large communication coverage area in typical indoor environments, a relatively high modulation index (MI) is generally required when modulating the signal to the LEDs in practical VLC systems. However, the input signal could be significantly distorted when using a high MI due to LED nonlinearity [34].  
The visible light radiated from the white LED propagates through the free-space VLC channel for simultaneous illumination and communication. For simplicity and without loss of generality, it is reasonable to only consider the line-of-sight (LOS) component in the system model [12]. Assuming that the LED follows a generalized Lambertian pattern, the LOS optical channel gain can be calculated by [1]  
\[h=\frac{(m+1)\rho A}{2\pi d^{2}}\text{cos}^{m}(\varphi)G_{f}G_{l}\text{cos}( \theta), \tag{1}\]  
where \(m\) is the order of Lambertian emission which is given by \(m=-\text{ln}2/\text{ln}(\text{cos}(\Psi))\) with \(\Psi\) being the semi-angle at half power of the LED transmitter; \(\rho\) and \(A\) are the responsivity and the active area of the photodetector (PD), respectively; \(d\) is the distance between the LED and the PD; \(\varphi\) and \(\theta\) are the corresponding emission angle and incident angle, respectively; \(G_{f}\) and \(G_{l}\) are the gains of the optical filter and the optical lens, respectively. The gain of the optical lens is given by \(G_{l}=\frac{n^{2}}{\text{sin}^{2}\Phi}\), where \(n\) and \(\Phi\) are the refractive index and the half-angle field-of-view (FOV) of the optical lens, respectively. Please note that the LOS channel gain becomes zero if the incident light is outside the FOV of the receiver.  
At the receiver side, the light is detected by a PD and the obtained electrical OFDM signal can be expressed by [31]  
\[y(t)=P_{0}h\xi f_{x}(t)+n(t), \tag{2}\]  
where \(P_{0}\) is the average output optical power of the LED, \(h\) is the optical channel gain defined in Equation (1), \(\xi\) is the MI of the LED, \(f_{x}(t)\) is the distorted version of the transmitted OFDM signal \(x(t)\) due to LED nonlinearity, and \(n(t)\) is the additive white Gaussian noise (AWGN) including both shot and thermal noises. The detailed expressions of the noise variances can be found in [31]. The obtained analog OFDM signal is then converted to a digital signal through analog-to-digital conversion (ADC). In order to mitigate the adverse effect of LED nonlinearity in VLC systems, PBL-based LED nonlinearity estimation and compensation are subsequently executed. The detailed procedures of LED nonlinearity estimation and compensation using PBL are described in the next section. After that, the compensated OFDM signal is achieved which is further demodulated to generate the output bits. The principle of real-valued OFDM modulation/demodulation can be found in [31], which is omitted here for brevity.  
Figure 1: Block diagram of an OFDM-based nonlinear VLC system using PBL-based LED nonlinearity estimation (est.) and compensation (comp.).