## 5 Numerical Results
The fitting equations for DCO-, U-, and ACO-OFDM are represented as:  
\[ \begin{split}\text{DCO-OFDM}:&\frac{\beta}{NP_{ \text{max}}}=6.7\times 10^{21}\times N_{0}^{3}-3.6\times 10^{14}\times N_{0}^{2}+7.8 \times 10^{6}\times N_{0}+0.061\\ \text{U-OFDM}:&\frac{\beta}{NP_{\text{max}}}=1.3 \times 10^{22}\times N_{0}^{3}-6.9\times 10^{14}\times N_{0}^{2}+1.2\times 10^{7} \times N_{0}+0.12\\ \text{ACO-OFDM}:&\frac{\beta}{NP_{\text{max}}}=2.5 \times 10^{22}\times N_{0}^{3}-1.1\times 10^{15}\times N_{0}^{2}+1.8\times 10^{7} \times N_{0}+0.17,\end{split}  \tag{27}\] \begin{table}
\begin{tabular}{l l} \hline \hline Responsivity & 0.45 A/W \\ Equivalent area of the PD using a lens & \(10^{-4}\) m\({}^{2}\) \\ Thermal noise density, \(N_{0}\) & \(1.3\times 10^{-13}\) mW/Hz \\ Radiation optical power of each lamp, \(P_{m}\) & 10 W \\ LED semi-angle & \(30^{o}\) \\ \(a(\lambda)\) & 0.053 (m\({}^{-1}\)) \\ \(b(\lambda)\) & 0.003 (m\({}^{-1}\)) \\ \(c(\lambda)\) & 0.056 (m\({}^{-1}\)) \\ Number of subcarriers, \(N\) & 128 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters used in this paper. where these equations can be used to predict the optimal modulation index for DCO-, ACO-, and U-OFDM in noisy environments with different noise levels. Figure 13 shows analytical results of the required power using DCO-, ACO-, and U-OFDM for different propagation distances with a 10-MHz, 3-dB bandwidth. In this figure, the cases with and without the bitloading algorithm are compared. Since the bandwidth tested is narrow, DCO-OFDM has a better performance than ACO- and U-OFDM, which is consistent, as shown in Figure 11. Without using the bitloading algorithm, the required power of the three optical OFDM schemes tested is about 30% higher than the case with bitloading. Since a single LED can only provide about tens of Watts [39; 40], a LED-array with multiple LEDs needs to be used when a high power is required. Figure 14 shows the maximum propagation distance achieved using different transmission bit rates through a band-limited channel. In this result, a bitloading algorithm is used [41]. When the symbol rate is low compared to the channel bandwidth, U-OFDM can support a longer propagation distance than ACO- and DCO-OFDM due to its higher power efficiency and less nonlinear distortion than ACO- and DCO-OFDM. For the higher bit rates case, DCO-OFDM outperforms ACO- and U-OFDM due to its higher spectral efficiency. Depending on different required bit rates and channel bandwidth, we can choose the proper optical OFDM technique to obtain a good propagation and BER performance. Figure 11: Average BER comparison of DCO-, ACO-, and U-OFDM with \(R_{s}=30\) Msps and 64-QAM. Propagation distance is 1 m. (**a**) Ideal channel with no bandwidth limit; (**b**) the 3-dB bandwidth of the channel is 25 MHz; (**c**) the 3-dB bandwidth of the channel is 10 MHz. A bit rate comparison of DCO-, ACO-, and U-OFDM in underwater optical wireless communication systems is shown in Figure 15. In this figure, the channel estimation error caused by pointing error is considered, and the bitloading algorithm is used to improve the throughput. In general, the maximum data rate for all the techniques tested drops as the channel uncertainty variance, \(\sigma_{h}^{2}\), increases. Based on the theoretical analysis, the SNR dramatically decreases for an increasing channel uncertainty variance. For this result, since a severely-limited LED bandwidth is used, DCO-OFDM outperforms ACO- and U-OFDM, which is consistent with the previous analysis and results. Figure 12: Normalized modulation index of the minimum BER for different noise levels. (**a**) The 3-dB bandwidth of the channel is 25 MHz; (**b**) the 3-dB bandwidth of the channel is 10 MHz. Figure 13: Required transmitted power for different propagation distances for a \(10^{-3}\) BER. The 3-dB bandwidth is 10 MHz. The bit rate is 100 Mbps.