## 3 Optimization and Performance Evaluation
After increasing the Doppler search step size, we change the two-dimensional search plane into \[x_{n+1}=1-2x_{n}^{2} \tag{18}\]3 The real spreading code number is 3, and the actual Doppler factor is 0.0069. The search results are consistent with \[x_{n+1}=1-2x_{n}^{2} \tag{18}\]4  
It is also found from Figure 10 that the objective function value of the optimal solution is highly differentiated from the general solution, which contains the local optimum solution, i.e., the peak with a small amplitude. Using this characteristic, this paper proposes the second optimization, the heuristic search method. Unlike the general extreme value optimization process, the optimization proposed in this paper only accepts the best solution in the limited two-dimensional search range. If the code number is incorrect, it directly affects decoding. If the Doppler factor estimation is wrong, it causes the accumulation of symbol synchronization errors. This affects the subsequent symbol correlation results and may indirectly lead to decoding errors. As is seen from Figure 10, on the Doppler axis the difference between the maximum and its two adjacent values is small. The discrimination between these three points and all the other general solutions is obvious. Using this feature, the paper sets a threshold. When optimizing, we first determine whether the maximum exceeds the threshold. If so, the search cycle is stopped. Only two more points, which are next to the maximum on the Doppler axis, are calculated. The max value between the three is taken as the final decision result. If the threshold is not exceeded, we take out a fixed number of points randomly again, and their correlations with the received signal are calculated. They are still compared with the threshold until all points in the range are taken, and the point with the max correlation result is chosen. The search process is shown in \[x_{n+1}=1-2x_{n}^{2} \tag{18}\]5  
\[x_{n+1}=1-2x_{n}^{2} \tag{18}\]6 Comparison of BER between coarse Doppler estimation and fine Doppler estimation based methods. It should be explained here that this paper uses the Fisher-Yates shuffle algorithm [32; 33; 34] to design the search process. In this way, the search is not only random, but also has a  
\[x_{n+1}=1-2x_{n}^{2} \tag{18}\]7 Optimization flow chart. \[x_{n+1}=1-2x_{n}^{2} \tag{18}\]8 Coarse Doppler estimation based two-dimensional search results. memory function. Random points are taken to find the solution exceeding the threshold with a certain probability, but while using the minimum number of searches so as to reduce the correlation computations. The memorability of the algorithm is reflected in excluding the previous points in every next cycle, so as to avoid repeated searches. It ensures that the maximum number of the random search does not exceed the exhaustive method. To avoid frequent comparison between the temporary optimal solution and the threshold due to too many cycles, the number of cycles is set to three. According to the simulation parameters, in the MSS system, there are nine Doppler factors and eight spreading code numbers to be searched for each symbol. According to the exhaustive method, there are 72 times correlation calculations needed for every single symbol. Then the maximum is obtained. It is assumed that the true value position of each symbol in the search area follows a uniform distribution. It is not considered that the search cannot exceed the threshold temporarily (when the threshold is set reasonably, the optimal solution will be found with a high probability). Since the search cycle is three times, the correlations of \(1/3\) points in the search range are calculated each time. We can stop the cycle as long one of the three points that exceed the threshold is found.