## 4 Results and Discussion
### Results of Different Estimators  
Based on a model where the received OFDM signal with ICI caused by the TVCFO was analogous to the orthogonal MC- CDMA system, the estimation was conducted using DE. To verify its suitability for considering the dependencies between the CFO values in adjacent blocks of the OFDM data frame, the performance of the DE algorithm for the TVCFO estimation was compared with those of the SAMGA [36] and GA [39], which are frequently used in channel estimation. The parameters of the DE algorithm used in this study were consistent with those in [42], and the parameters of the compared algorithms were set to the values reported in the original papers. Figure 2 shows the performances of the DE algorithm, GA, and SAMGA when estimating the TVCFO in the form of a polynomial function with the same computing resources. The data in the figure are the averages obtained from 50 independent runs under different SNRs, as are the data in the following figures. The measurements show the Root-Mean-Square Error (RMSE) of the CFO values on the OFDM blocks, which can be defined as follows:  
\[\text{RMSE}(\hat{\epsilon})=\sqrt{\frac{1}{K}\sum_{i=1}^{K}(\hat{\epsilon}_{i }-\epsilon_{i})^{2}}. \tag{21}\]  
The black, blue, and red curves represent the results obtained by the SAMGA, GA, and DE estimators for the improved model, respectively. The RMSE values improved as the SNR increased. However, the SAMGA did not improve the performance of the genetic algorithm in this case. It was inferred that this lack of improvement was a result of the limited computing resources, which made the large population size unsuitable. Moreover, the SAMGA increased the diversity to prevent premature, which also wasted computing resources. Generally, the available time resources are seriously constrained in underwater acoustic communication. Therefore, the advantages of the SAMGA could not be fully utilized. The GA had a remarkable performance compared to that of the SAMGA; However, the performance of the DE algorithm was slightly better than that of the GA. The average RMSE value with the DE algorithm was approximately 9.68% smaller than that of the GA when the SNR was less than 12 dB. When the SNR exceeded 12 dB, the average RMSE of the DE estimator increased, becoming 30.02% smaller than that of the GA estimator, and finally reaching a peak that was 43.71% lower at 20 dB. This significant improvement was due, not only to the adaptive parameters of the DE algorithm used in this study [42], but also to its advantage when solving problems with dependencies between adjacent variables. The GA could not compete with the DE algorithm in terms of the dependencies between adjacent variables which is the pivotal characteristic of TVCFO estimation. The above analysis was confirmed by the gap between the two curves becoming progressively obvious as the SNR increased. Because the amplitude of the noise followed a normal distribution and had significant randomness, the dependencies between adjacent variables were diluted. However, as the SNR increased, the influence of the noise decreased, and the connections between adjacent TVCFO values could be more easily mined by the DE algorithm. The best results of the three estimators were substituted into the improved system model for demodulation. The BER values were determined based on the demodulation results, and are summarized in Figure 3. According to the curve, when the SNR did not exceed 13 dB, the performances of the three estimators were comparable, and it may be inferred that their accuracies were acceptable. Under the condition of a low SNR, the source of the error values was mainly the noise, and the CFO estimation bias was relatively negligible. The difference between the three estimators becomes clear from the results at 14 dB. When the SNR is less than 17 dB, the performance of the SAMGA estimator is comparable to those of the GA and DE estimators. After the SNR exceeds 17 dB, the SAMGA with the lowest estimation accuracy shows the worst demodulation performance.