## 5 Numerical Results  
In this section, numerical results of the comparison of the DCO-, ACO-, and U-OFDM techniques for UOWC systems are shown. The parameters used to obtain the numerical results are shown in Table 3. This is a baseline for all the numerical results in this paper. Since the attenuation coefficient varies over temperature, concentration of salt, and spectrum of transmitted light, to simplify the problem, we assume the pure sea water condition is used [4]. The light with a wavelength from 480 nm-520 nm is used, which has the lowest absorption rate in sea water. We also assume that the thermal noise is dominated. For the photodetector (PD), a lens can be integrated to increases the effective area and collect more power. For the numerical results, the bandwidth limitation is taken into account. We mainly consider the bandwidth limitation from LEDs since the PD's bandwidth can  
Figure 10: A block diagram of the optical OFDM receiver with a single-tap equalizer. Figure 9: An illustration of the LED magnitude response. be much broader compared with the symbol rate tested in this paper [37; 38]. The numerical results illustrated in this paper are based on the theory discussed above. We have tested and verified the theoretical results by using simulation. The forward error control (FEC) codes need to be applied in a practical situation for a better BER performance, which does not change the relative results among DCO-, ACO-, and U-OFDM obtained in this paper. Therefore, there is no need to consider the advantages of FEC in this paper. In addition, we set a \(10^{-3}\) BER as the constraint when maximizing the bit rate, since, with the help of FEC, the BER can be easily and significantly improved. The numerical results of BER using Equation (16) are shown in Figure 11. In this figure, the normalized modulation index is used, which is \(\beta/N/P_{m}\). Figure 11a-c illustrates the average BER of the optical OFDM tested under different channel conditions, respectively. In general, an increasing modulation index first leads to a better BER performance for all the tested optical OFDM. Then, continued increasing of the modulation index can introduce more clipping distortion caused by the peak power constraint; therefore, the BER worsens. Since an increasing modulation index improves the SNR, yet introduces more clipping distortion, an optimal modulation index needs to be found. The ideal channel (no bandwidth constraint) case is shown in Figure 11a. Due to the less nonlinear distortion and higher power efficiency of U-OFDM compared with DCO- and ACO-OFDM, the BER performance of U-OFDM is better than DCO- and ACO-OFDM. Considering the channel bandwidth constraint, the BER performances of U- and ACO-OFDM become worse than DCO-OFDM. When the 3-dB bandwidth of the channel is 25 MHz, as shown in Figure 11b, the minimum BER of U-OFDM is similar to DCO-OFDM. For a more narrow bandwidth case (3 dB bandwidth is 10 MHz), as shown in Figure 11c, the minimum BER of U-OFDM is higher than that of DCO-OFDM since U-OFDM requires a wider bandwidth than DCO-OFDM when transmitting the same data rate. ACO-OFDM is even worse than U-OFDM due to its lower bandwidth utilization efficiency. In summary, due to different channel bandwidths, the best optical OFDM scheme can be selected to obtain a better BER performance. How to choose the optimal modulation index for DCO-, ACO-, and U-OFDM is a problem that needs to be discussed. Figure 12 shows the simulation results of the optimal normalized modulation index for different noise spectral density. In this result, the transmitted peak power and the data rate are fixed. Figure 12a,b shows the relationship between the modulation index and the noise level for two different band-limited channels, respectively. From the simulation results, we use cubic equations to fit and model the relationship between the optimal normalized modulation index and noise spectral density based on the simulation results. Comparing (a) and (b) in Figure 12, the two different bandwidths illustrate a similar phenomenon.