# 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  
The principle of the proposed PBL-based LED nonlinearity estimation and compensation scheme is depicted in Figure 2. For simple and efficient LED nonlinearity (i.e., the nonlinear power-current curve of the LED) estimation, a sawtooth-based vector \(\mathbf{z}\) with \(N\) samples, i.e., \(\mathbf{z}=[z_{1},z_{2},\cdots,z_{N}]^{T}\), is adopted as the TS for LED nonlinearity estimation using PBL regression in this work. The TS is known and shared by all the receivers within the coverage of the VLC system. After transmitting through the nonlinear VLC system, a corresponding vector \(\mathbf{r}=[r_{1},r_{2},\cdots,r_{N}]^{T}\) can be detected at the receiver side, which can give the raw estimation of the LED nonlinearity. In order to obtain an accurate estimation of the LED nonlinearity, PBL regression is performed which takes the raw estimation of the LED nonlinearity \(\mathbf{r}\) as the input and the actual LED nonlinearity \(\mathbf{\tau}\) as the target. Although the design matrix is generated by using the input in [30; 35], it has been found in our study that the sawtooth-based training vector \(\mathbf{z}\) can be directly used to generate the design matrix \(\mathbf{\Phi}(\mathbf{z})\), which can achieve comparable performance as that using the input \(\mathbf{r}\). Taking \(\mathbf{r}\) as the input and using the design matrix \(\mathbf{\Phi}(\mathbf{z})\), PBL regression can be successfully performed to obtain an accurate estimation of \(\mathbf{\tau}\), i.e., \(\hat{\mathbf{\tau}}\). As shown in Figure 2, after obtaining \(\mathbf{z}\) and \(\hat{\mathbf{\tau}}\), a corresponding look-up table (LUT), i.e., \(\mathcal{L}=[\hat{\mathbf{\tau}}\mathbf{z}]\), can be generated. By using the generated LUT, LED nonlinearity compensation can be executed regarding the received OFDM signal. First, the amplitude of the received OFDM signal is scaled by a factor \(P_{0}h\xi\) so as to match the amplitude of the transmitted OFDM signal, and hence the scaled OFDM signal is expressed by  
\[\hat{y}(t)=\frac{y(t)}{P_{0}h\xi}=f_{x}(t)+\frac{n(t)}{P_{0}h\xi}. \tag{10}\]  
Then, using the obtained \(\hat{\mathbf{\tau}}=[\hat{\tau}_{1},\hat{\tau}_{2},\cdots,\hat{\tau}_{N}]^{T}\), the index \(i(t)\) of the element in \(\hat{\mathbf{\tau}}\) which is closest to \(\tilde{y}(t)\) can be identified as  
\[i(t)=\arg\min_{k}\{\hat{y}(t)-The PD has an active area of 16 mm\({}^{2}\) and a responsivity of 0.53 A/W. The vertical distance between LED and PD is set to 2 m and the horizontal offset between LED and PD is also assumed to be 2 m. A modulation bandwidth of 20 MHz is considered and 16 QAM constellation is adopted in real-valued OFDM modulation/demodulation. Hence, the raw data rate of the OFDM-based nonlinear VLC system is 80 Mbit/s. In OFDM modulation/demodulation, the size of fast Fourier transform (FFT)/inverse fast Fourier transform (IFFT) is set to 512. Due to the Hermitian symmetry constraint, only 128 subcarriers are used to carry valid data. A total of 1000 OFDM symbols are transmitted for bit-error rate (BER) calculation. In the simulation setup, a commercially available white LED (Cree PLCC4) is used and the measured nonlinear power-current curve is illustrated in Figure 3. It can be observed that the normalized output optical power of the LED exhibits strong nonlinearity with the normalized input current. Since only unipolar signal can be modulated onto the luminous intensity, a DC bias current is generally applied to convert the bipolar OFDM signal to a unipolar one. Hence, the nonlinear distortion is mainly caused by two factors: the DC bias current and the peak-to-peak current of the signal.