# LED Nonlinearity Estimation and Compensation in VLC Systems Using Probabilistic Bayesian Learning
## 3 PBL-Based LED Nonlinearity Estimation and Compensation in VLC
### LED Nonlinearity Estimation and Compensation LUsing PBL Regression
To investigate the LED nonlinearity effect on the transmission performance, as shown in Figure 3, we define the MI as the ratio of the maximum current variation of the signal to the maximum current variation supported by the LED without clipping, and the DC-bias index (DI) as the ratio of the DC bias current of the signal to the maximum current variation supported by the LED without clipping. In this simulation investigation, the LED is assumed to be biased at the middle point of its dynamic range, i.e., DI = 0.5. Although a static LED nonlinearity characteristic is considered in this work, the proposed PBL-based LED nonlinearity estimation and compensation scheme can be easily generalized into an adaptive scheme by adopting the method reported in [36], which is applicable in VLC systems with dynamic LED nonlinearity characteristics. Moreover, the frequency selective fading effect of a VLC system is mainly caused by the LED, which can be easily compensated by using the frequency domain pre-equalization techniques [14; 34]. Thus, a flat system frequency response is considered here without loss of generality. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Semi-angle at half power of LED & 60\({}^{\circ}\) \\ LED output optical power & 10 W \\ Gain of optical filter & 0.9 \\ Refractive index of optical lens & 1.5 \\ Half-angle FOV of optical lens & 72\({}^{\circ}\) \\ Active area of PD & 16 mm\({}^{2}\) \\ Responsivity of PD & 0.53 A/W \\ Vertical distance between LED and PD & 2 m \\ Horizontal offset between LED and PD & 2 m \\ Modulation bandwidth & 20 MHz \\ QAM constellation order & 16 \\ Raw data rate & 80 Mbit/s \\ Size of FFT/IFFT & 512 \\ Number of data subcarriers & 128 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. The original PBL regression model proposed in [30] starts with all the \(M=N+1\) basis functions. The update rules for the hyperparameters depend on computing the posterior weight co-variance matrix which requires a Cholesky decomposition operation with a complexity of \(O(M^{3})\). As a result, the computational complexity of PBL regression could be very high for practical applications. To reduce computational complexity, a fast marginal likelihood maximization method has been proposed in [35]. Based on an accelerated training algorithm, PBL regression is initialized with a single basis function, i.e., the bias. Sequentially, the basis functions are added to increase the marginal likelihood and also modify their weightings, and meanwhile the redundant basis functions are deleted to increase the objective function. Hence, the new PBL regression model can achieve comparable performance as the original one but with greatly reduced computational complexity, which is adopted here to realize efficient LED nonlinearity estimation and compensation in OFDM-based nonlinear VLC systems. Specifically, the width parameter of the Gaussian kernel is set to \(\lambda=2\) and the maximum number of iterations is set to 10. For the purpose of comparison, the conventional TDA scheme employing multiple consecutive TSs is also considered in the simulations.