###### Abstract
proposed that OFDM with ICI could be regarded as a multi-carrier-code division multiple access (MC-CDMA) system on the receiver side [26] because MC-CDMA is a technology that combines OFDM and code division multiple access (CMDA). Moreover, the ICI matrix in OFDM is an orthogonal matrix. An \(N\times N\) ICI matrix acts as a spreading code matrix; thus \(N\) data symbols in the OFDM are spread over all \(N\) subcarriers in the MC-CDMA system [26]. The data symbols quantize the normalized CFO, obtaining M spreading code matrices as candidates. The received signal is then decoded, decided, and reconstructed using candidates. The Euclidean distance is utilized to measure the difference between the reconstructed and received signals. The result with the smallest difference value corresponds to the best-normalized CFO estimation. The scheme should be able to perfectly cancel the ICI caused by the CFO. In 2013, a joint estimation method for the CFO and channel information that embedded the channel estimation block was proposed on the basis of previous work [27]. Concurrently, the computational complexity was also improved by adopting the \"golden section search\" [28]. Simulation results showed that the proposed scheme achieved a promising performance, which was comparable to that of OFDM in the absence of ICI. Although much research is available in the literature related to CFO estimation and compensation, few schemes have been presented for time-varying CFO (TVCFO) estimation in underwater acoustic communication. Based on the above analysis, relative motion causes a Doppler effect or CFO, and a constant CFO can only be caused by a relatively uniform motion. However, in a complex and changeable underwater environment, it is difficult to guarantee a relatively uniform motion. Instead, an unpredictable acceleration often exists, which leads to a TVCFO in OFDM. This TVCFO is very difficult to deal with because its tracking and equalization usually require high computational complexity [29], which tends to increase linearly with the number of symbols in the OFDM frame. An introduction to the existing research is given below. [PERSON] et al. proposed a method for estimating the constant Doppler shift or slowly changing the time-varying Doppler shift within the packet by transmitting two identical OFDM symbols and employing a bank of parallel autocorrelators at the receiver [30]. This work was extended by [PERSON] et al. [31], who suggested tracking the Doppler variation with the help of marginal maximum likelihood estimation between symbols. [PERSON] et al. [32] proposed a TVCFO estimation method that predicted and tracked the nonuniform Doppler shift between adjacent data blocks based on the frequency and time correlation. [PERSON] [29] assumed that the Doppler shift varied linearly among OFDM symbols, based on the principle that a Doppler shift changes the signal length in the time domain. This requires frequent sampling and interpolation to estimate and compensate for the Doppler shift within the duration of a symbol. Simultaneously, the first-order statistics of the cyclic prefix are also estimated to reduce the ICI. In 2016, [PERSON] studied the effect of sudden changes in the acceleration and velocity directions on the cyclic prefix correlation and proposed an adaptive time-varying Doppler estimation scheme. This adaptive mechanism was embedded in the automatic selection of the best estimation method based on CP window center positioning, first-order expectation, and autocorrelation of the CP [33]. [PERSON] first studied the closed-form expression of the received signal that passed through a double-spread channel based on the least squares criterion. The value of the constant CFO was obtained using a root-based and eigenvalue decomposition-based CFO estimator [25]. In 2022, [PERSON] extended the constant CFO to a polynomial and piecewise model variation over time [9], and implemented a method for estimating the TVCFO based on previous research.