## 1 Introduction
Prof. [PERSON] [14] of Australia employed a single pulse signal as the pilot, inserting a piece of the guard interval around it, and then used the threshold approach to estimate the channel response in the DD domain. Nevertheless, this channel estimation approach depends on the choice of protection interval, and the threshold is chosen based on experience, which have a significant impact on channel estimation performance. Furthermore, ref. [15] developed a set of transform domain basis functions to model the OTFS channel by constructing a low-dimensional subspace that can accurately estimate the interference caused by inter-Doppler interference (IDI), while the algorithm used in the paper is linear minimum mean square error (LMMSE). The LMMSE-based channel estimation in low-dimensional subspaces uses the prior information of the channel, which is not applicable to real-world scenarios. In addition, there are similar methods in OFDM systems to estimate the phase noise of channels in high-mobility scenarios. Ref. [16] uses the basis expansion model (BEM) to fit the time variation and phase noise of multipath channels and provides an effective approximate solution for the joint estimation problem based on maximum likelihood. Therefore, BEM can effectively and accurately fit and reduce the dimension of channel changes, which is conducive to the subsequent channel estimation. Indeed, many recent publications have presented compressed sensing-based channel estimate algorithms. Ref. [17] developed an orthogonal matching tracking algorithm for 3D structures that takes advantage of channel sparsity in three domains: the delay domain, the Doppler domain, and the angular domain. Furthermore, ref. [18] adopted three-dimensional Newtonian orthogonal matched tracking to recover the channel parameters of the time-frequency domain, incorporating channel gain, direction of arrival, delay, and Doppler shift. Reviewing the above channel estimation approaches, it is clear that most simulation environments only evaluate simple channel situations and do not conform to the channels of real scenarios. The Doppler shift in the wireless channel is not constant, but rather conforms to the variation of the U-shaped spectrum, according to the classical [PERSON]'s channel modeling method. Furthermore, unlike OFDM, the size of the grid in OTFS controls the delay-Doppler quantization resolution [19] (the larger the grid, the higher the resolution). However, the grid cannot generally take a larger value. Hence, the Doppler shift cannot be accurately quantified to the grid points, and the above channel estimation methods for this inaccurate channel model will cause certain estimation errors, degrading system performance [20]. To solve the aforementioned issues, we create a BEM-based channel model, transform the OTFS channel estimation into a basis coefficients recovery problem, and suggest generalized approximate message passing (GAMP) to lower the sparse Bayesian learning (SBL) algorithm's complexity. The primary contributions are as follows:  
1. We first give the input-output expression relationship equation in the DD domain in OTFS system, and based on this, we model the OTFS channel into a DD-domain channel response that can reflect the variation of sampling points in the Doppler domain. We then convert the channel estimation problem into a basis coefficient recovery problem using the BEM. 2. In order to estimate the DD-domain channel, including Doppler sampling points, the GAMP- SBL method is additionally proposed to estimate the basis coefficients and channel noise while simultaneously lowering the SBL's complexity. 3. The scheme is evaluated by simulation of the OTFS system. The simulation and complexity analysis show that the GAMP-SBL algorithm proposed in this paper has advantages regarding computing complexity and performance under the 3 GPP-specified V2X channel model. In summary, both simulation and experimental results demonstrate the effectiveness of the algorithm. The remainder of the paper is structured as follows.