## 3 Analysis of Optical OFDM for Underwater Communications
### Dco-Ofdm  
DCO-OFDM is a basic method of optical OFDM and is widely used due to its low complexity [31; 32]. Since only the non-negative signals can be transmitted, DCO-OFDM signals are derived by adding a DC bias to the bipolar OFDM signals. Figure 4 shows the block diagram of DCO-OFDM. In this figure, Parts (a) and (b) illustrate the transmitter and receiver structures, respectively. Hermitian symmetry is used to generate real signals for optical wireless communication systems at the transmitter. However, considering the peak power limit, the added DC bias cannot totally avoid zero clipping. In the meantime, adding the DC can introduce peak power clipping distortion. At the receiver, the DC bias needs to be removed first, then the M-QAM data can be recovered and demodulated. Figure 3: Illustration of zero and peak power limits for bipolar signals. Figure 2: Pointing error caused by the fluctuation of transmitters and receivers. In Figure 4a, after the inverse fast Fourier transform (IFFT), the discrete data for the \(k\)th subcarrier component, \(x[k]\), can be represented as:  
\[x[k]=\sum_{i=0}^{N-1}X_{i}\exp\left(\frac{j2\pi ki}{N}\right),\;\forall\;k=0,1, \cdots,N-1, \tag{5}\]  
where \(N\) represents the number of subcarriers used for modulation. \(X_{i}\) is the data after M-QAM mapping for the \(i\)th subcarrier. Due to the Hermitian symmetry, we set \(X_{N-1-i}=X_{i}^{*}\), \(\forall\;i=0,1,\cdots,N-1\). After the parallel to serial (P/S) converter, the \(m\)th sample of one DCO-OFDM can be represented as:  
\[x_{\text{D}}[m]=\beta\sum_{k=0}^{N-1}x[k]\delta[m-k]+b_{\text{DC}}, \tag{6}\]  
where \(b_{\text{DC}}\) is the additive DC bias and \(\beta\) is a coefficient that controls the scale of the DCO-OFDM signals. Since we assume that the number of subcarriers, \(N\), is usually very large (at least 64), \(x_{\text{D}}[m]\) can be modeled as a Gaussian random variable with \(b_{\text{DC}}\) mean and \(\sigma_{x}^{2}\) variance. Considering the zero and peak power clipping, the probability density function (pdf) of the DCO-OFDM signal can be represented as:  
\[f_{\text{D}}(x,b_{\text{DC}})= \frac{1}{\sqrt{2\pi\sigma_{x}^{2}}}\exp\left(\frac{-(x-b_{\text{ DC}})^{2}}{2\sigma_{x}^{2}}\right)\Pi(x,P_{\text{m}}) \tag{7}\] \[+\frac{1}{2}\text{erfc}\left(\frac{b_{\text{DC}}}{\sqrt{2\sigma_ {x}^{2}}}\right)\delta(x)\] \[+\frac{1}{2}\text{erfc}\left(\frac{P_{\text{m}}-b_{\text{DC}}}{ \sqrt{2\sigma_{x}^{2}}}\right)\delta(x-P_{\text{m}}),\]  
where \(\Pi(x,P_{\text{m}})=u(x)-u(x-P_{\text{m}})\) is defined as a rectangular function and \(u(x)\) is the unit step function. \(\text{erfc}(x)\) is the complementary error function, which is defined as:  
\[\text{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}\exp(-y^{2})dy. \tag{8}\]  
Figure 5 shows the pdf of the DCO-OFDM signal numerically. In this paper, simulation and analytical results are perfectly fit. From the results, zero clipping distortion is smaller than the peak  
Figure 4: Block diagram of DCO-OFDM. (**a**) Structure of the transmitter; (**b**) structure of receiver [33]. P, parallel; S, serial; M-QAM, \(M\)-ary quadrature amplitude modulation. clipping distortion when the DC bias is greater than half of the peak transmitted power. Otherwise, zero clipping distortion is greater than the peak clipping distortion. Since the DCO-OFDM signal is modeled as a Gaussian random variable and the LED works as a nonlinear device, the sampled received signal then can be modeled based on the Bussgang theorem as [26; 34]:  
\[ \begin{split} y_{\text{DCO}}[m]&=rh\left(\alpha_{ \text{D}}(b_{\text{DC}})\cdot x_{\text{D}}[m]+c_{\text{D}}[m]\right)+n[m],\\ &=r(\hbar h_{o}+\Delta h)\left(\alpha_{\text{D}}(b_{\text{DC}}) \cdot x_{\text{D}}[m]+c_{\text{D}}[m]\right)+n[m]\end{split}  \tag{9}\]  
where \(r\) is the optical power to current converting rate. \(\alpha_{\text{D}}(b_{\text{DC}})\) represents the clipping coefficient, which is a function of the DC bias, \(b_{\text{DC}}\).