###### Abstract
In [22], code modulation is used in the spread spectrum system so that the mapping bit does not reduce the Euclidean distance between transmission bits as it did in the traditional modulation, resulting in improved BER performance. The scheme proposed in this paper is based on \(M\)-ary spread spectrum (MSS) code modulation and combines symbol-by-symbol Doppler estimation and synchronization to meet the needs of mobile UWA communication. In addition, the MSS has a higher data rate and is more secure than the conventional direct sequence spread spectrum (DSSS), making it more practical [3; 23]. To deal with the time-varying characteristics of the mobile UWA channel, the receiver needs to estimate the channel state information. The commonly employed Doppler estimation method is to use the thumb-tack ambiguity function of Doppler sensitive waveforms to calculate the cross-ambiguity function (CAF) on the two-dimensional grid of channel delay and Doppler compensation factor [24; 25]. The maximum amplitude position of CAF correlation on Doppler grid is the estimated Doppler factor. However, this method has two obvious defects. One is a large amount of calculation. The other is the estimation accuracy limited by the time-bandwidth product of Doppler sensitive waveforms [26]. Inspired by [27], on the premise synchronization, we set up a group of discrete correlators. With different Doppler factors in the transmitted symbol as reference, we use these correlators to search in the expected Doppler range, obtaining the Doppler estimation. Simultaneously, it is assumed that the multi-path delay in the UWA channel is less than the duration of one symbol spreading code, ignoring the impact of inter-symbol interference on the propagated signal, and focusing on the signal distortion caused by Doppler [15; 26]. In the spread spectrum system, the spreading sequences carrying information are Doppler sensitive waveforms [28]. Because the MSS method uses M distinctive sequences with the same length, it also needs to be set as a bank of discrete correlators. Therefore, the method presented needs to implement the correlation maximum search in the two-dimensional grid composed of spreading code numbers and different Doppler factors for Doppler factor estimation. The estimated Doppler is used to adjust the data length of the current symbol and synchronize the next. When an idea similar to CAF is used, its disadvantages also become that of the proposed method, namely, a large amount of calculation and insufficient estimation accuracy. For estimation accuracy, the MSS technique is different from OFDM and SC-MPSK. It tolerates a certain degree of Doppler through the gain brought by spreading and despreading. Therefore, it does not need a high estimation accuracy if the residual Doppler is within its tolerance. The verification will be given in Section 3. This paper proposes a three-step optimization scheme to solve the problem of high computational complexity, which is also the innovation of this paper. The first step is to adjust the frequency-domain search step size. Usually, a small search step size is required in the two-dimensional grid to get the maximum of all the correlation results in the traversal range. In this paper, we find that when searching in the frequency domain, within a certain range, this method is more tolerant to relatively large step sizes. Moreover, through this approach, we reduce the number of calculations to nearly 1/20 of the original. The second step of the optimization scheme is to use a heuristic search method [29]. Observing the correlation results, we find that the maximum and the two adjacent values on the Doppler axis have obvious discrimination with other correlation results. This paper takes this as heuristic information and sets a threshold during the search. To reduce the amount of calculations, we present a random search method [30], which is different from the exhaustive method [31]. Some points are taken randomly from the search range in each iteration.