## 4 Simulation Results  
### System Performance  
In this subsection, we take the very-high-frequency (VHF) band as an instance with which to analyze the performance of the proposed TDOM system. The simulation is performed using Matlab R2023b. According to the M.1842-1 standard, the peak bandwidth of the maritime VHF band is 100 kHz [20]. This bandwidth is divided into 32 frequency subchannels, and the frequency spacing is 2.7 kHz. Meanwhile, 16 QAM modulation is implemented in each subchannel. Therefore, each VDES symbol carries 32 \(\times\) 2 = 64 bits of data. As the symbol rate is 2400 symbol/s, the peak data rate of VDES is 64 (bit/symbol) \(\times\) 2400 (symbol/s) = 302.7 kbps [20]. It should be noted that the reason for choosing a symbol rate of 2400 symbol/s is to obtain a long symbol duration to combat the multipath interference in maritime communication channels. The AIS system can also use a symbol rate of 4800 symbol/s to achieve the same purpose. In this paper, the TDOM symbol rate is set to be 4800 symbol/s. Meanwhile, we make \(f_{0}=f_{1}=\ldots=f_{N}\), which means that the TDOM can realize the highest spectral efficiency. The performance of the TDOM system under AWGN channel is shown in Figure 5a. It can be seen that the system has good communication performance when the number of carriers is 32. From Table 1, each carrier conveys 2 bits, so the communication rate of the TDOM system is  
\[2(\text{bit/carrier})\times 32(\text{carrier/symbol})\times 4800(\text{symbol/s})=307.2(\text{kbit/s}) \tag{24}\]  
At this point, the peak rate of the current VHF data transmission system (VDES) has been reached. It is worth mentioning that, as FFT is no longer used in the receiver, the total number of carriers within one TDOM symbol is no longer limited to be an exponent number of 2 and can be set to be any value. Here, we take 64, 96 and 128, for instance. When the total number of carriers \(N\) is increased to more than 32, the communication performance of the system starts to degrade, which is due to the fact that the norm of the matrix \(R^{-1}\) rises with the number of carriers, resulting in an increase in the BER of the system, as shown in Table 2. This coincides with the theoretical analysis results in Section 3.3. Meanwhile, the spectrum of TDOM symbols is given in Figure 5b. It can be seen that the bandwidth occupied by the whole symbol is the same as that of a single subcarrier since all carriers within the symbol have the same frequency. According to the symbol construction principle shown in Figure 2, the duration of the whole symbol is twice the duration of a single carrier, so the duration of a single subcarrier is (1/9600) s. Therefore, the bandwidth of the symbol is 19.2 kHz, and the spectrum efficiency (SE) of the system can be expressed as Equation (19). \[SE_{TDOM}=307.2(\text{kbit/s})/19.2(\text{kHz})=16(\text{bit/s/Hz}). \tag{25}\]  
Changing the form of the carrier will change the norm of the matrix \(R^{-1}\) and therefore improve the communication performance of the whole TDOM system. Results in Figure 5 default that\(f_{0}=f_{1}=\ldots=f_{\text{N}}=1\), which means that there is only one sine or cosine-wave cycle in the carrier duration \(T\). Figure 6 shows the effect of the carrier frequency \(f\) on the communication performance when \(N\) is 32, 64, 96 and 128. It can be seen that increasing the frequency \(f\) will improve the communication performance of the whole system. This \begin{table}
\begin{tabular}{c c} \hline \hline \(N\) & \(\left\lVert R^{-1}\right\rVert\) \\ \hline
32 & 0.0489 \\
64 & 0.3967 \\
96 & 1.3427 \\
128 & 3.1839 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Norm of the matrix \(R^{-1}\) for different carrier numbers. Figure 5: Communication performance and the spectrum of the TDOM system. (**a**) Communication performance under the AWGN channel. (**b**) The spectrum of a TDOM symbol.