## 3 Optimization and Performance Evaluation
The result is shown as the black solid line at the lower left corner in Figure 6.  
Due to the narrow available bandwidth of underwater acoustic communication, the signal duration is generally in the order of a few seconds to tens of seconds, and the motion speed is difficult to maintain unchanged within the signal duration. This situation is an ideal state, in reality. We only regard it as a reference result. Through this comparison, we find the limitations of the conventional MSS method. Because the method proposed in this paper performs signal processing within the duration of symbol level, the method we use has a wider applicability. The simulation parameters are consistent with the above. Two conclusions are drawn from the simulation results. 1. If the relative motion speed changes slowly within the signal duration, with only spreading/despreading and sliding correlation symbol synchronization, the conventional MSS method tolerates the Doppler of 0.6-2.4 m/s (Doppler factor range, 1e\(-\)4-1.6e\(-\)3). If Doppler estimation and average Doppler compensation are added to the receiver, the MSS method can cope with a slightly larger motion velocity (2.4 m/s). 2. When the relative velocity changes irregularly in the whole signal duration, even if the average speed is only 1.5 m/s, the MSS method using sliding correlation for symbol synchronization cannot effectively deal with it. The first conclusion is the reason why we can appropriately increase the frequency-domain search step. In Section 4, the second conclusion is confirmed. The distinction between the coarse and fine Doppler estimation of the MSS method is shown in Figures 7 and 8. In the process of a two-dimensional correlation search, if the  
Figure 6: BER curves of the conventional MSS method under different motion scenarios. frequency shift of the reference signal is set to a small step, the correlation computations greatly increase. As shown in Figures 7 and 8, we note that even if the search step size is adjusted larger, it is enough to describe the signal variation in the frequency-domain. If setting the Doppler estimation step to 0.002, the difference between the real Doppler factor and the estimated value shall not exceed 0.001. The system should still be able to decode correctly according to the results in \[x_{n+1}=1-2x_{n}^{2} \tag{18}\]0  
Next, Monte Carlo simulations are conducted to verify the rationality of the hypothesis. As shown in Figure 9, the Doppler factor is roughly estimated (Doppler step size is 0.002) and accurately estimated (Doppler step size is 0.0001) in a fixed frequency domain. The SNR ranges from \(-15\) dB to \(-8\) dB. Each method is calculated 3000 times, and other simulation parameters remain unchanged. The BER performance of coarse Doppler estimation is the same as that of fine Doppler estimation. For each code number of one symbol, \([-0.008,0.008]\) is taken as the Doppler factor range for the search. A total of 161 correlation operations are required for fine Doppler estimation and only nine for coarse  
\[x_{n+1}=1-2x_{n}^{2} \tag{18}\]1 Doppler estimation symbol-by-symbol. \[x_{n+1}=1-2x_{n}^{2} \tag{18}\]2 Single symbol Doppler estimated curves. Doppler estimation. In other words, simply changing the Doppler search step size reduces the correlation computations to 9/161. Using the above conclusions, we redescribe the search problem as follows. The sequence numbers and the Doppler factors are discrete. The search range is described as a grid of \(M\frac{2f_{d}}{T}\). \(M\) is the code numbers used in the MSS method. \(2\frac{f_{d}}{T}\) is the number of Doppler factors. The Doppler search range is \([-f_{d},f_{d}]\), and the search step in the Doppler domain is \(U\). A group of reference signals are generated with different frequency offsets and spread spectrum codes. The objective function is the peak value of the correlation results between the reference signals and the received signal.