## 3 Analysis of Optical OFDM for Underwater Communications
### Dco-Ofdm
The calculation of \(\alpha_{\text{D}}(b_{\text{DC}})\) is derived in the Appendix A, the result of which is shown as:  
\[\alpha_{\text{D}}(b_{\text{DC}})=1-\frac{1}{2}\left(\text{erfc}\left(\frac{-P _{\text{m}}+b_{\text{DC}}}{\sqrt{2\sigma_{x}^{2}}}\right)+\text{erfc}\left( \frac{b_{\text{DC}}}{\sqrt{2\sigma_{x}^{2}}}\right)\right). \tag{10}\]  
In Equation (9), \(c_{\text{D}}[m]\) is clipping noise, and \(n[m]\) is the sum of shot noise and thermal noise. \(c_{\text{D}}[m]\) and \(n[m]\) are both modeled as Gaussian random variables with zero mean. The variance of the clipping noise can be calculated by:  
\[ \begin{split}\sigma_{\text{D}}^{2}&=\int_{-\infty} ^{0}x^{2}\cdot f_{\text{D}}(x)dx+\int_{P_{\text{m}}}^{\infty}(x-P_{\text{m}})^{ 2}\cdot f_{\text{D}}(x)dx\\ &=\sqrt{\frac{\sigma_{x}^{2}}{8\pi}}(b_{\text{DC}}-P_{\text{m}}) \exp\left(\frac{-(b_{\text{DC}}-P_{\text{m}})^{2}}{2\sigma_{x}^{2}}\right)\\ &-\sqrt{\frac{\sigma_{x}^{2}}{8\pi}}b_{\text{DC}}\exp\left(\frac{ -b_{\text{DC}}^{2}}{2\sigma_{x}^{2}}\right)+\frac{1}{2}\left(\sigma_{x}^{2}+( b_{\text{DC}}-P_{\text{m}})^{2}\right)\\ &+\frac{1}{4}\text{erfc}\left(\frac{b_{\text{DC}}}{\sqrt{2 \sigma_{x}^{2}}}\right)-\frac{1}{4}\text{erfc}\left(\frac{b_{\text{DC}}-P_{ \text{m}}}{\sqrt{2\sigma_{x}^{2}}}\right)\end{split}  \tag{11}\]  
Figure 5: Simulation and analytical results of the pdf of the DCO-OFDM signal, \(\beta=20\) and \(P_{\text{m}}=5\,\text{mW}\). and the variance of \(n[m]\) is:  
\[\sigma_{n}^{2}=N_{o}R_{s}+2 qr\bar{P}R_{s}, \tag{12}\]  
where \(N_{o}\) and \(R_{s}\) are thermal noise spectral density and symbol rate, respectively. \(q\) and \(\bar{P}\) are the electronic charge and average received optical power, respectively. \(\bar{P}\) is derived in the Appendix A. Therefore, the signal to noise ratio (SNR) using DCO-OFDM after FFT in the receiver, as shown in Figure 4b, can be calculated as:  
\[\gamma_{\text{D}}(d,\beta,b_{\text{DC}})=\frac{(\bar{h}h_{o}(d)\beta a_{\text{ D}}(b_{\text{DC}}))^{2}\sigma_{x}^{2}}{N((\bar{h}^{2}h_{o}^{2}(d)+\sigma_{h}^{2}) \sigma_{\text{D}}^{2}+\sigma_{n}^{2})}, \tag{13}\]  
which is a function of \(\beta\), \(d\), and the DC bias. Considering the fading underwater channel, the average SNR can be calculated as:  
\[\bar{\gamma}_{\text{D}}(d,\beta,b_{\text{DC}})=\int_{0}^{\infty}\gamma_{\text {D}}(d,\beta,b_{\text{DC}})f_{\bar{h}}(\bar{h})d\bar{h} \tag{14}\]  
Since the propagation distance is a concern in underwater optical wireless communication systems, we can obtain the optimal transmission distance, \(d^{*}\), by optimizing the coefficient, \(\beta\). The optimal transmission distance can be found by:  
\[ \begin{split}& d^{*}=\max_{\beta,b_{\text{DC}}}d\\ & s.t.\ \text{BER}(d,\beta,b_{\text{DC}})<B^{\text{max}},\end{split}  \tag{15}\]  
where \(B^{\text{max}}\) is the allowed maximum bit error rate (BER) and \(\text{BER}(d,\beta)\) represents the BER using M-QAM given by a transmission distance and \(\beta\), which approximates as:  
\[\text{BER}(d,\beta,b_{\text{DC}}){\approx}\frac{\sqrt{M}-1}{\sqrt{M}\text{log }_{2}\Big{(}\sqrt{M}\Big{)}}\text{erfc}\left(\sqrt{\frac{3\bar{\gamma}_{\text {D}}(d,\beta,b_{\text{DC}})}{2(M-1)}}\right), \tag{16}\]  
where \(M\) is the modulation constellation size for M-QAM.