## 4 Performance Evaluations
### Simulations
#### 4.1.1 Feasibility  
The underwater acoustic simulation channel parameters are set as follows: paths number \(N_{p}=15\), the average time interval \(\Delta\tau\) between the two paths is 2 ms, the power difference \(\Delta P_{p}\) of the closing path is 20 dB, and the average sound velocity \(v_{c}=1500\) m/s.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Structure** & **Increased Multiplications** & **Increased Additions** \\ \hline DAM-TEQ(I) & \(3\mathrm{N}(\mathrm{MN_{a}+N_{b}})^{2}+3\mathrm{N}(\mathrm{MN_{a}+N_{b}})\) & \(3\mathrm{N}(\mathrm{MN_{a}+N_{b}})^{2}+\mathrm{N}(\mathrm{MN_{a}+N_{b}})\) \\ DAM-TEQ(II) & \(3(\mathrm{M}-1)^{2}\mathrm{N_{a}^{2}}+3(\mathrm{M}-1)\mathrm{N_{a}}\) & \(3(\mathrm{M}-1)^{2}\mathrm{N_{a}^{2}}+(\mathrm{M}-1)\mathrm{N_{a}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Increased complexity.  
The relative speed of platform movement between the transmitter and receiver is \(v_{p}\) and the standard deviation \(\sigma_{v}\) of this velocity is 0.01 m/s, evenly distributed in each path. Table 2 shows the parameters of the transmitted SC signal; 1296 data bits are used as the communication sequence, which are modulated into a carrier with a frequency of 450 Hz after convolutional coding. In BPSK modulation, the bits and symbols are one-to-one, which means that the bit rate is equal to the baud rate, and this number is set to 200 in the simulation. Therefore, the channel bandwidth is 200 Hz. The SC signal sampling rate is set to 40,000 Hz, which can be applied to the transducer.  
Taking the signal-to-noise ratio (SNR) = 0 dB and \(v_{p}=1\) m/s as an example, the process of using the DAM-TEQ to process the simulation data is analyzed. In the first stage of the equalizer, set \(f_{d0}=40,000\), \(\Delta f=50\), and \(\Delta f_{\text{min}}=5\). The updated results of the equalization coefficient detection and resampling branches in the adaptive process are shown in Figure 4. The resampled branch with a relatively large amplitude of the equalization coefficient was selected as the central branch, and the adaptive sampling rate interval was narrowed if the central branch remained unchanged. The coarse Doppler estimation is realized after a few iterations, and the final value of \(f_{d}\) is 39,975. Select the resampling rate interval \(\Delta f_{d}\), and generate three branches, where the resampling rates are \(f_{d}-\Delta f_{d}\), \(f_{d}\), and \(f_{d}+\Delta f_{d}\). Then the resampled signals from the three branches are fed into the second-stage equalizer. The processing result is 0 BER, and the feasibility of the proposed equalizer is verified by simulation.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency \(f_{c}\) & 450 Hz \\ Modulation mode & BPSK \\ Sequence length & 1296 bits \\ Baud rate & 200 Baud \\ Bandwidth & 200 Hz \\ Initial sampling rate \(f_{s}\) & 40,000 Hz \\ Convolutional coding & \((2,1,3)\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: SC signal parameters.  
Figure 4: Coefficients and branch update processes.  
In order to further verify the effectiveness of the DAM-TEQ, the simulation is carried out under three different frequency-shift conditions, \(v_{p}=1\) m/s, \(v_{p}=2\) m/s, and \(v_{p}=3\) m/s, and the frequency shifts are calculated by \(\Delta f_{Doppler}=f_{c}v_{p}/v_{c}\). The simulation results are shown in Figure 5. The simulation results show that the BER with the small relative moving speed is relatively low. Still, the BER decreases with the increase in the SNR under different frequency-shift conditions, indicating that the proposed equalizer has an effective Doppler estimation and compensation ability under different frequency-shift conditions.