## 4 Simulation Results
### System Performance
is due to the fact that the norm of the matrix \(R^{-1}\) decreases when the carrier frequency \(f\) is increased, as shown in Table 3. When the carrier frequency \(f\) is increased to 5, the TDOM system with 64 carriers/symbol can achieve a communication performance similar to that of the TDOM system with 32 carriers/symbol, as shown in Figure 6b. The system communication rate at this point is twice that of Equation (18), i.e.,  
\[2(\mathrm{bit/carrier})\times 64(\mathrm{carrier/symbol})\times 4800(\mathrm{ symbol/s})=614.4(\mathrm{kbit/s}) \tag{26}\]  
When the carrier frequency is increased to 10, TDOM systems with both 96 and 128 carriers/symbol can achieve better communication performance, as shown in Figure 6c,d. The peak data transmission rate and spectral efficiency of the system at this point are shown in Equations (21) and (22), respectively:  
\[2(\mathrm{bit/carrier})\times 128(\mathrm{carrier/symbol})\times 4800(\mathrm{ symbol/s})=1.2288(\mathrm{Mbit/s}) \tag{27}\]  
\[SE_{TDOM}=1.2288(\mathrm{Mbit/s})/19.2(\mathrm{kHz})=64(\mathrm{bit/s/Hz}). \tag{28}\]  
Meanwhile, the spectral efficiency of the VDES system is shown in Equation (23). Therefore, the data transmission rate and spectral efficiency of TDOM is are 4 times and 20 times, respectively, of those of the VDES system, as shown in Equations (24) and (25):  
\[SE_{VDES}=307.2(\mathrm{kbit/s})/100(\mathrm{kHz})=3.072(\mathrm{bit/s/Hz}). \tag{29}\]  
\[\frac{Rate_{TDOM}}{Rate_{VDES}}=\frac{1.288(\mathrm{Mbps})}{307.2(\mathrm{Kbps })}=4, \tag{30}\]  
\[\frac{SE_{TDOM}}{SE_{VDES}}=\frac{64(\mathrm{bit/s/Hz})}{3.072(\mathrm{bit/s/Hz })}\approx 20. \tag{31}\]  
Figure 6: Communication performance of TDOM system under different carrier frequencies. (**a–d**) The number of carriers, i.e., \(N=32\), 64, 96, and 128. The communication performance of the TDOM system under the maritime multipath channel is shown in Figure 5:0 It can be seen that the impact of multipath interference on the communication performance of TDOM is very serious, even with the channel estimation and channel equalization method. When \(N\) is increased to more than 64, the BER is high in the high signal-to-noise (SNR) region. Figure 7b shows the communication performance of TDOM systems with \(f\) equals to 10. It can be seen that the communication performance of all TDOM systems is improved, which is consistent with the aforementioned theoretical analysis and numerical simulation. When \(f\) is increased to 20, the communication performance of the TDOM system with 64 carriers per symbol is comparable to its performance in the AWGN channel, as shown in Figure 7c. When \(f\) is raised to 30, the communication performance of the TDOM system with 96 carriers per symbol also is also comparable to its performance in the AWGN channel, as shown in Figure 7d. However, the communication performance of the TDOM system with 128 carriers per symbol is still poor. In order to further improve the performance of the TDOM system with 128 carriers per symbol, the form of the carriers is changed from sine to cosine, and the communication performance of the system is shown in Figure 5:1 It can be seen that the communication performance of the cos-TDOM system is better than that of the sin-TDOM system for all carrier frequencies. When the carrier frequency is raised to 30, the performance of the cos-TDOM with 128 carriers per symbol is comparable to the performance of the sin-TDOM system with 32 carriers per symbol in the AWGN channel. At this point, the communication rate of the TDOM system is 4 times of that of the VDES, and the spectral efficiency is 20 times higher than that of the VDES. Table 4 gives the norm of the matrix \(\mathbf{R}^{-1}\) for sin and cos-TDOM. It can be seen that the cosine form carriers can further reduce the norm of matrix \(\mathbf{R}^{-1}\) and thus improve the communication performance of the TDOM system.