# Optical OFDM for SiPM-Based Underwater Optical Wireless Communication Links
## 5 Description of the Considered Signaling Schemes
### Adapting the Signal Amplitude to the LED DR  
In practice, before intensity modulation of the LED, the signal needs to be fit to the LED I-V characteristics, or in other words to its DR. Figure 4 shows the I-V characteristics of the LED that we consider in this work [43]. For the sake of simplicity, in our analysis, we ignore the I-V non-linearity within the LED DR and consider the approximate linearized characteristics (the red plot in Figure 4) in the sequel. Note that a digital pre-distortion device can be used to mitigate the non-linear distortion. The study of the effect of non-linear LED characteristics [44; 45] is out of the scope of this work. Firstly, power-normalized QAM constellations are considered, see Figures 1 and 2. It can be shown that the signal \(x_{n}\) after IFFT is also power normalized, i.e., \(\mathbb{E}\left\{x_{n}^{2}\right\}=1\), where \(\mathbb{E}\{.\}\) denotes the expected value. After adding the CP, the scale of the signal is changed by multiplying by factor \(\alpha\) to obtain \(\hat{x}_{n}\). Afterwards, the DC bias, denoted by \(B_{\text{DC}}\), is added to it. Denote the lower and upper limits for the LED bias voltage by \(V_{\text{min}}\) and \(V_{\text{max}}\), respectively. It is obvious that a DC bias of \(V_{\text{min}}\) is needed for the cases of ACO- and LACO-OFDM. Different criteria can be considered for fixing the scaling factor \(\alpha\)[46; 47]. In particular, for LACO, a different \(\alpha\) can be used for each layer. Unless otherwise specified, here it is fixed so as to obtain the same transmit electrical power, \(P_{\text{Tx,e}}\), for the different considered OFDM schemes in order to make a fair comparison between them. This choice of fixing \(P_{\text{Tx,e}}\) is justified by the fact that it determines the power consumption of the Tx, which is an essential factor for mobile units (we will further relax this condition and discuss the obtained results in Section 6). Note that for LACO, we use the same \(\alpha\) in both layers, although in general it can be adjusted within each layer to further optimize the performance. Figure 4: I-V characteristics of the LED [43]. Blue curve: real characteristic; red line: approximate linearized characteristic with \(V_{\text{min}}=2.75\,\text{V}\), \(V_{\text{max}}=4.03\,\text{V}\), \(I_{\text{min}}=1\,\text{mA}\), \(I_{\text{max}}=110\,\text{mA}\). \(P_{\rm T,x,e}\) primarily depends on the scaling factor \(\alpha\) and the DC bias \(B_{\rm DC}\). Having fixed these two parameters, we calculate \(P_{\rm T,x,e}\) by averaging the power corresponding to each symbol, which is obtained from its product by the bijective current according to the linearized LED I-V characteristics. As explained, for the cases of ACO- and LACO-OFDM, \(B_{\rm DC}\) is simply set to \(V_{\rm min}\). For DCO-OFDM, based on a given \(\alpha\), we consider two ways to set the bias, as explained in the following. Firstly, as a simple method, we use the classical approach of considering the so-called _clipping factor_\(\mathcal{K}\)[40], which is defined in the following equation, where the effect of the scaling factor is taken into account:  
\[B_{\rm DC}=\mathcal{K}\sqrt{\mathbb{E}\{\tilde{x}_{n}^{2}\}}+V_{\rm min}\approx \mathcal{K}\,\alpha+V_{\rm min}. \tag{11}\]  
In the sequel, the clipping factor in decibel is used, i.e., \(\mathcal{K}_{\rm dB}=10\,\log_{10}\mathcal{K}\). As a second approach, we consider the idea of [48] to calculate the the optimum value of \(B_{\rm DC}\) so as to minimize the mean square error \(\mathcal{E}\) between \(\tilde{x}_{n}\) and double-side clipped signal \(\tilde{x}_{n}\), while taking \(V_{\rm min}\) and \(V_{\rm max}\) of the LED into consideration:  
\[\mathcal{E}=\mathbb{E}\left\{\sum_{n=0}^{N-1}(\tilde{x}_{n}-\tilde{x}_{n})^{2 }\right\}. \tag{12}\]  
For given \(\alpha\), \(V_{\rm min}\), and \(V_{\rm max}\), the optimum bias is calculated numerically by setting \(\partial\mathcal{E}/\partial B_{\rm DC}\) to zero. More details are provided in Appendix A.