## 4 Simulation Results and Discussions  
The performance of the proposed channel estimation method is investigated with that of traditional embedded pilot algorithm [8] and two-stage basis function correlation algorithm [13], two-step modified OMPFR algorithm [16], and tensor decomposition-based channel estimation [18]. Our work focuses on the LEO satellite communication network scenarios specified in 3 GPP standards, utilizing the OTFS simulation parameters listed in Table 1. The message passing (MP) detector in [22] is used in the data detection. The normalized mean square error (NMSE) is employed to evaluate the performance of the channel estimation. The NTN-TDL-D channel model is an extension of the non-terrestrial network (NTN) model based on the 3 GPP TR 38.901 standard [23], which is specifically designed for LEO satellite high-mobility scenarios. Based on this, we adjusted the multipath delay to better suit our simulation system, and others all refer to the 3 GPP protocol. The power-delay profile (PDP) of this channel is characterized in Table 2.  
Figure 2 compares the Doppler estimation metric functions \(E(v_{f})\) of the proposed algorithm and the basis function correlation algorithm [13] when there are two paths on the same delay tap at \(-6\) dB. The normalized Doppler shifts of the two paths are [6.6, 9.5]. Both algorithms estimate the Doppler shifts of multiple paths by iterative cancellation processing. Figure 2a,b present the metric functions when estimating two paths, respectively. In Figure 2a, the initial metric functions obtained by the two algorithms are shown. The proposed algorithm exhibits two distinct and sharp peaks, whereas the basis function correlation algorithm suffers from significant sidelobes and blurred peaks due to IDI and noise. In Figure 2b, the metric functions after one iteration are given, where the strongest path has been canceled and the remaining paths have a Doppler shift of 6.6. The proposed algorithm still maintains a higher and sharper peak value compared to the basis function correlation algorithm in this scenario. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of sub-carriers & \(M=64\) \\ Number of time-domain symbols & \(N=32\) \\ subcarrier spacing & \(\Delta f=62.5\) kHz \\ Carrier frequency & \(f_{c}=10\) GHz \\ Max Doppler shift & \(k_{max}=50\) kHz \\ Coding scheme & LDPC \\ Coding rate & \(0.667\) \\ Pilot information & \(X_{dd}(6,8)=100\) \\ Doppler compensation range and step & \(v_{f}=[-1,1],step=0.01\) \\ channel model & \(NTN-TDL-D\) \\ stopping threshold coefficient & \(\beta=3\) \\ guard symbols overhead & \(0.031\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation Parameter. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Delay (ns) & 0 & 430 & 470 & 890 & 1110 & 1190 \\ Power (dB) & 0 & \(-1.5\) & \(-4.5\) & \(-7.4\) & \(-10.4\) & \(-15.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Power-delay profile of the channel. Figure 3 shows the MSE performances of the proposed algorithm, the two-stage basis function correlation algorithm [13], two-step modified OMPFR algorithm [16], and tensor decomposition-based channel estimation [18]. It can be seen that as the SNR increases, the MSE of the proposed algorithm approaches \(10^{-4}\), which outperforms the other three algorithms. Among the available algorithms, when SNR = 20 dB, the MSE of the tensor decomposition-based algorithm approaches \(10^{-3}\), but at low SNR levels, its MSE is higher than those of the two-stage basis function correlation algorithm and the two-step modified OMPFR algorithm. Figure 4 shows the NMSE performances of the proposed algorithm, the embedded pilot algorithm [8], the two-stage basis function correlation algorithm [13], two-step modified OMPFR algorithm [16], and tensor decomposition-based channel estimation [18].