###### Abstract
Compared with a grid search, which imposes a high computational burden on a modern [33], heuristic-based intelligent optimization algorithms have remarkable advantages, including greater adaptability and higher solution efficiency [34]. Therefore, they have allowed great progress in channel estimation [35; 36; 37; 38; 39]. The differential evolution (DE) algorithm is a prevalent example of intelligent optimization algorithms. It has a simple structure with few parameters, but exhibits excellent performance [35; 37]. The DE algorithm has a feature that distinguishes it from other intelligent optimization algorithms: it is especially suitable for solving problems where there are dependencies between the adjacent variables to be optimized [40; 41]. The dependencies refer to the fact that variables are linked by a similar influencing factor, such as Rosenbrock, Schaffer F7, and Whitley's composite functions in the benchmark suites. Coincidentally, underwater acoustic channel conditions do not change suddenly under normal circumstances. Even when the CFO changes over time, the CFO value on a subsequent block changes on the basis of the previous CFO value in terms of the OFDM structure. In other words, time-varying CFO estimation for OFDM systems falls squarely into the category of problem optimization with adjacent variables depending. In summary, the specific mutation and crossover operations of the DE algorithm can be used to track the time-varying CFO values of adjacent blocks in an OFDM data frame. After a limited number of iterations, the best estimate of the time-varying subcarrier frequency offset can be obtained. From the literature, an OFDM model analogous to the MC-CDMA system can realize perfect ICI cancellation without losing any system overhead [26; 27]. Considering the significant advantages and development potential of this scheme in underwater acoustic communication, we extended the model to time-varying environments and obtained a TVCFO estimation and thus compensation in this study. Simultaneously, the crossover characteristic of the DE algorithm was used to track the TVCFO and obtain the best estimation values based on its strong optimization ability. Specifically, the CFO value of each block was regarded as a variable to be optimized in the DE. The spread code matrix in the MC-CDMA system was obtained based on the variables by analogy, and then the received signal was reconstructed using the estimated CFO values. The Euclidean distance between the reconstructed and received signals was used to form the cost function. The optimal estimation of the TVCFO was achieved by the iterative use of the genetic operators of the DE. The objective of this investigation was to study the performance of time-varying CFO estimation in OFDM systems based on the DE algorithm due to its advantages in solving problems with adjacent variables depending. The main contribution of this paper is to provide a feasible idea for time-varying CFO estimation, and the proposed method not only has the general applicability of modulation methods but also can improve the noise tolerance in demodulation. The remainder of this paper is organized as follows: Section 2 introduces the OFDM system model and DE. Section 3 details the proposed OFDM TVCFO estimation methods. The results and discussion are presented in Section 4. Finally, we conclude this paper and make suggestions for future research in Section 5.