###### Abstract  
Orthogonal frequency division multiplexing (OFDM) is the preferred scheme for high-speed communication in the field of underwater acoustic communication. However, it is very sensitive to the carrier frequency offset (CFO). This study used a time-varying CFO estimation method aided by the differential evolution (DE) algorithm to accurately estimate the CFO of an OFDM system. This method was based on the principle that the received OFDM signal with inter-carrier interference could be considered by a Multi Carrier-code division multiple access (MC-CDMA) system on the receiver side because MC-CDMA is a technology that combines OFDM and code division multiple access (CMDA). Because it is suitable for solving problems where there are dependencies between adjacent variables, the DE algorithm was used to capture the varying CFO values on the adjacent blocks. The spreading code of the MC-CDMA was obtained based on the estimated CFO values, which were elements in the DE solutions. Then the received signal was reconstructed. The Root-Mean-Square Error between the reconstructed and actual received signals was used as the cost function, and the CFO was estimated using the DE algorithm because of its powerful parallel search capability. The simulation results showed that the proposed method had a high estimation accuracy. Compared with other intelligent optimization algorithms such as the genetic algorithm and simulated annealing mutated-genetic algorithm, the time-varying CFO estimation performance of the DE algorithm was better because of its unique ability to solve problems with dependencies between adjacent variables. Specifically, under the condition of a high signal-to-noise ratio, the improvement of estimation accuracy reaches 36.13%, and the Bit Error Rate of demodulation is thus reduced by 75%, compared with the reference algorithms. In addition, the proposed method also has good applicability to modulation methods. For phase-shift keying and quadrature amplitude modulation, in particular, the proposed method not only achieved high-precision time-varying CFO estimation values, but also reduced the demodulation deterioration caused by noise. Orthogonal frequency division multiplexing: differential evolution algorithm; time-varying carrier frequency offset estimation; code division multiple access 20221018263090  
estimating and compensating for the CFO before demodulating and decoding the received signal is an important research objective for OFDM systems. Because of factors such as the sea surface fluctuation effect, turbulence, and internal wave motion, the transmitter and receiver terminals used in underwater communication usually have relative velocities, which leads to the Doppler effect. An approximate Doppler value is commonly estimated for each OFDM frame, and a resampling method is used for compensation after receiving the signal [6]. Because the transmitted OFDM symbol is a wideband signal, there is a residual Doppler frequency shift in the system after resampling, which can be regarded as a result of the CFO [7]. Presently, the methods used for CFO estimation can be mainly divided into two categories [8; 9]: data-aided estimation [10; 11; 12] and non-data-aided estimation [13]. When the data-aided estimation is performed in the time domain, it utilizes a training sequence [10], whereas if it is performed in the frequency domain, it uses pilot symbols [11; 12; 14]. The non-data-aided estimation methods are also commonly referred to as blind estimation. Some auxiliary information about the received signal is used for blind channel estimation, but this does not increase the overhead of the channel resources. This information usually includes pre-set structures and characteristics such as the cyclic prefix (CP) in OFDM [15; 16; 17; 18], responses on the null subcarrier [19; 20; 21], constant modulus [22; 23; 24], and non-Gaussianity of modulated signals [25]. In addition to the blind estimation method, [PERSON] et al.