# LED Nonlinearity Estimation and Compensation in VLC Systems Using Probabilistic Bayesian Learning
## 5 Results and Discussion  
In this section, simulation results based on the setup described above are presented and the corresponding discussions are also provided. Figure 4 depicts the normalized signal amplitude vs. the normalized input current for different MI values, where DI = 0.5 and the length of the sawtooth-based TS is \(N=64\). As we can observe, for both MI = 0.6 and 0.8, the amplitude of the received TS has a significant variation due to the additive noise in the VLC system. However, after performing PBL regression, the predicted power-current curve of the LED becomes much smoother which matches the actual power-current curve of the LED very well, indicating an accurate estimation of the LED nonlinearity by using only a single sawtooth-based TS. We further analyze the impact of the number of TSs on the estimation accuracy of both the conventional TDA scheme and the proposed PBL-based scheme. Figure 5 shows the average estimation error vs. the number of TSs with MI = 0.8, DI = 0.5 and \(N=64\). For the conventional TDA scheme, it can be seen that the average estimation error is gradually reduced with the increase of the number of TSs and it becomes stable at about \(1.7\times 10^{-3}\) when the number of TSs reaches 20. In contrast, for the proposed PBL regression scheme, a comparable average estimation error can be achieved by using only a single TS. Figure 3: Measured nonlinear power-current curve of white LED Cree PLCC4. The impact of the length of the TS on the BER performance of the OFDM-based nonlinear VLC system is also analyzed. Figure 6 shows the BER vs. the length of the TS for the conventional  
Figure 4: Normalized signal amplitude vs. normalized input current for (**a**) MI = 0.6 and (**b**) MI = 0.8. Figure 5: Average estimation error vs. number of training symbols. TDA scheme with different numbers of TSs and the proposed PBL-based scheme with a single TS, with MI = 0.8 and DI = 0.5. Evidently, for both conventional TDA and the proposed PBL-based scheme, the BER performance is substantially improved with the increase of the length of the TS and stable BERs can be guaranteed when the length of the TS is about 64. Based on the analysis above, we evaluate and compare the BER performance vs. MI for the OFDM-based nonlinear VLC system without and with LED nonlinearity compensation, where DI = 0.5 and the length of the TS is fixed at \(N\) = 64. As shown in Figure 7, when no LED nonlinearity compensation is considered, the BER reduces with the increase of the MI and an MI of 0.87 is required to reach the BER threshold of \(10^{-3}\). Moreover, the BER improvement becomes insignificant by further increasing MI when MI is larger than 0.9, suggesting the adverse effect of LED nonlinearity on the BER performance of the OFDM-based nonlinear VLC system. However, when the conventional TDA scheme is applied, the BER performance can be substantially improved. For TDA with 5 TSs, the required MI to reach BER = \(10^{-3}\) is reduced to 0.80. By further increasing the number of TSs, the BER performance can be further enhanced. In contrast, when the proposed PBL-based scheme with a single TS is adopted, the required MI to reach BER = \(10^{-3}\) is only about 0.77. Furthermore, nearly the same BER performance can be achieved by the conventional TDA scheme with totally 20 TSs and the proposed PBL-based scheme with a single TS. Therefore, a substantial reduction of the training overhead for LED nonlinearity mitigation can be achieved by using the proposed PBL-based scheme in comparison to the conventional TDA scheme, which indicates a greatly improved overall spectral efficiency of the OFDM-based nonlinear VLC system. The corresponding 16 QAM constellation diagrams are also shown as insets in Figure 7.  
Figure 6: BER vs. length of training symbol. Finally, we analyze the computational complexity of the proposed PBL-based LED nonlinearity estimation and compensation scheme.