## 3 Optimization and Performance Evaluation
Therefore, from the perspective of probability calculation, let the number of all combinations of \(m\) (\(m\leq n\)) elements be taken out from \(n\) different elements, expressed by \(C_{n}^{n}\), and the number of all possible permutations of \(m\) elements be taken out from \(n\) elements, represented by \(A_{n}^{m}\), and the equation is  
\[x_{n+1}=1-2x_{n}^{2} \tag{18}\]9  
where \(x!\) denotes factorial of \(x\). Taking the above parameters as an example, there are 72 elements, and 24 elements are taken out each cycle. Since the points are taken randomly, there are \(C_{72}^{24}\) possibilities in total. As long as one, two, or three elements are taken out from the three points with high correlation, and the remaining 23, 22, and 21 elements are selected from other general solutions, the optimal solution can be found in the first cycle. The probability of finding the temporary optimal solution exceeding the threshold in the first cycle is calculated as  
Figure 2:0  
In the second search, because we use the Fisher-Yates shuffle method to exclude the points searched in the first iteration, we only need to extract 24 points from the remaining 48 points to calculate the correlations. Then the probability that the optimal solution is not found in the primary cycle and instead found in the second cycle is as follows  
Figure 2:1  
The probability of finding the optimal solution in the first cycle is 71%, and the calculations are reduced to about \(1/3\) of the original. The probability of finding the peak in two cycles is about 96%. That means that through this optimization step, the number of calculations is probably reduced to at least 2/3 of the original. If the threshold decision fails, the computation will be higher than the above results, but it is still similar to that of the exhaustive method. The calculation results are only an example of specific parameter conditions. In fact, as long as the distribution of search results conforms to the heuristic information, this optimization method can be used for different search ranges to reduce the calculations. Based on the motion feature of the UUVs, this paper puts forward a further optimization scheme. Due to the considerable resistance of water, the speed and acceleration of UUVs are limited to a certain extent, since these factors do not change as fast as for surface ships or aircrafts. In this paper, the motion state of an underwater moving target is modeled as a Markov process [35]. It is assumed that the Doppler factor of each symbol is affected by the previous Doppler estimation. Coupled with the premise that the velocity of the underwater moving target does not change much in a short time, we only need to search the Doppler region neighboring the prior estimated Doppler. For example, if the estimated Doppler factor of one symbol is 0.004, the search range of the next is all the code numbers with the Doppler factors of 0.002, 0.004, and 0.006. Hence, compared with the whole search area, \([-0.008,0.008]\), the search area is reduced to the original 1/3. If the Doppler factor estimation of one symbol is 0.008, the next search area is only 0.006 and 0.008. In this way, the number of calculations is reduced to about 1/3 of the original. Using the three-step optimization method, according to the search range parameters utilized in the simulation, the correlation times of the method with no optimization are reduced to  
Figure 2:2  
The original two-dimensional search for each symbol required 1288 correlation operations; now about 16 correlation operations are needed. The MSS method using sliding correlation only needs \(M\) correlation operations for each symbol. It takes the synchronization signal estimation as the average Doppler of the whole frame signal. Therefore, the conventional method does not search on the Doppler axis and cannot adapt to the Doppler change within the signal duration, as shown in Figure 2:3 Compared with this method, the optimized method needs \(2M\) correlation operations for each symbol.