## 3 Optimization and Performance Evaluation  
Before introducing optimization, the assumptions made in Section 2 need to be verified. This paper compares the difference between the assumption-based correlation performance and the real situation. The simulation condition is a multi-path channel, and the channel impulse response is shown in Figure 2. Gaussian white noise is added to the signal, with \(-10\) dB SNR. The maximum Doppler factor is \(0.008\). If the sound velocity in water is \(1500\) m/s, the relative motion velocity is \(12\) m/s. The signal bandwidth is \(2000\) Hz, the carrier center frequency is \(3000\) Hz, the spreading code length is \(127\), and the symbol duration is \(63.5\) ms. Eight different codes are used, with each carrying \(3\) bits of data. The pseudo-random sequence used is a binary quantized chaotic sequence generated by the second-order Chebyshev polynomial function with different initial values [39]  
\[x_{n+1}=1-2x_{n}^{2} \tag{18}\]  
In the real case, the frequency offsets caused by the Doppler effect and the changes of data length in the time domain are both considered for the reference signal. Moreover, the received signal is intercepted with one symbol length affected by the actual present Doppler. Based on the assumptions, we only consider the influence of frequency offsets, and intercept the signal by one transmitted symbol length. The correlation results are obtained on the premise of accurate synchronization. From Figure 3, it can be seen that the position and peak amplitude of the correlation results based on the hypothesis change slightly. As described in the hypothesis above, we only consider the frequency offsets, omitting the signal length compression/expansion. The length of the one-symbol-received signal is assumed to be unchanged. According to Figure 3, this deviation can be ignored. As for the synchronization error accumulation caused by the Doppler effect, it is corrected by the previous Doppler estimation. We note that the passband spreading sequence has a large time-bandwidth product, which minimizes the error detection probability. The pseudo-random sequence also has a narrow ambiguity function on the Doppler axis [28]. Moreover, chaotic sequences with the same length have good autocorrelation and cross-correlation properties. Figure 4 shows the correlation search results of eight chaotic sequences under the condition of \(-10\) Db SNR in the Doppler variation range of \([-0.008,0.008]\). The search step is \(0.3\) Hz, and the real Doppler factor is \(0.0069\). Other simulation conditions are consistent with the above. It can be seen that the pseudo-random sequence has obvious discrimination in code and Doppler axis. This is the basis on which the characteristics of passband spreading signal are used for receiving and processing. Figure 2: Channel Impulse Response. In the process of receiving, each symbol needs to obtain the maximum correlation through a large number of correlation operations. With the symbols increasing, the large number of calculations seriously affects the data processing speed. Hence, we consider establishing a grid based on the code number and the Doppler factor in the expected range to determine the search range. Taking the two-dimensional search results as the objective function, the search and the maximum decision process are modeled as an extreme value optimization, as shown in Equation (15). Algorithm optimization is performed next. The optimization is carried out in three steps. In the first step, we reduce the number of computations by increasing the search step in the Doppler domain. Because the MSS system tolerates a small Doppler factor through despreading, we only need to make the search step within its tolerance range. The subsequent simulation results show that the search accuracy has little effect on the decoding performance within limits. Since there is an obvious difference between most search results and the optimal solution, in the second step, we set a threshold.