## 1 Introduction
The block estimation method using LFM signals can only calculate the average Doppler within a single frame and is not suitable for long frame sequences [14]. Furthermore, coding iterative techniques require a high signal-to-noise ratio, rendering them unsuitable for underwater acoustic communication under low SNR ratio conditions [15; 16]. [PERSON] et al. proposed a receiver based on the Minimum Mean Square Error (MMSE) criterion for the acquisition of Direct Sequence Spread Spectrum (DSSS) signals. The core method involves using an adaptive Decision Feedback Equalizer (DFE) in combination with spreading codes for synchronization and tracking. This approach is particularly suitable for complex multipath propagation environments in multi-user underwater acoustic communications [17]. [PERSON] et al. analyzed the sea trial results of the Thorcys V0 underwater acoustic communication waveform, validating its performance across multiple frequency bands and complex channel environments, and evaluated the performance of different receiver structures [18]. [PERSON] et al. proposed a method based on the measurement and updating of passive matched filters to enhance spreading gain by mitigating variable multipath effects and performing Doppler estimation within symbols; however, the calculation speed and efficiency of Doppler estimation significantly decreased under low SNR ratios [5]. [PERSON] et al. introduced a novel linear prediction equation suitable for sparse underwater acoustic channels, calculating linear prediction coefficients by estimating the Doppler frequency shift for each symbol. However, the computational load and prediction accuracy are significantly influenced by the signal-to-noise ratio, and the algorithm exhibits high complexity [19]. Due to the relatively low speed of sound in the ocean, even minor velocity changes in underwater platforms can induce substantial Doppler frequency shifts. Methods for Doppler estimation in underwater acoustic communication primarily involve estimating and predicting the velocity of movement, which entails significant computational load and complexity. When there are deviations in velocity estimation, the Doppler estimation is also significantly affected and constrained by the accuracy of the estimation. During the process of Doppler compensation, it is common to perform a search within a certain range of the Doppler estimation results before updating them into the matched filter bank, a process referred to as Doppler precise estimation. Thus,even if there is a certain deviation in Doppler estimation, it will not significantly impact the spreading gain; the permissible range of deviation in Doppler estimation is referred to as Doppler-tolerant [16]. Influenced by the studies in [9; 10], this research introduces the CIM-SS technology from the field of radio communication into the realm of underwater acoustic communication, innovatively designing an M-SS communication transmission module based on polarity and spreading code index modulation, which partitions the transmitted information into polarity data and index data for separate modulation, achieving a transmission rate that is \(log_{2}(M+n)\) times higher than that of a DSSS communication system transmission module with the same signal length, where M denotes the size of the spreading code index library and n represents the number of transmitted symbols, thereby further enhancing spectral efficiency. In response to the issue of carrier frequency offset (CFO) caused by time-varying Doppler frequency shifts in complex channel environments, this paper focuses on enhancing Doppler-tolerant, traditional methods typically involve conducting a search within a certain range of the Doppler estimation results, followed by performing correlation-matched filtering based on the various search values and selecting the search position corresponding to the highest output peak as the final Doppler estimation result.