## 3 Proposed Iterative Fractional Doppler Shift and Channel Estimation Algorithm  
Many existing joint estimation algorithms [20,21] for frequency offset and channel information can exhibit an excellent performance in the OFDM system. However, in the OTFS system with fractional Dopplers, their performance can hardly meet the requirements. To obtain the full channel information with a high accuracy for the channel with fractional Doppler shift paths, we propose an iterative fraction Doppler offset and channel coefficient joint estimation scheme, which includes the search for the strongest path, the fraction Doppler shift and channel coefficient joint estimation, reconstructed interference cancellation, and channel information reconstructed. In the proposed method, the strongest path is searched in the guard interval in the DD domain. If the searched position of the strong path is \((\hat{l}_{i},\hat{k}_{i})\), and the actual Doppler offset of the path is \(k_{i}+\kappa_{i}\). \(\hat{k}_{i}\) is the closest integer to it. The fractional Doppler of this path is estimated through search-based compensation. The received pilot signal \(Y_{dd}^{pilot}(l,k)\) is represented as a vector \(\mathbf{Y}_{D}^{pilot}(\hat{l}_{i})\) at the delay \(\hat{l}_{i}\), as follows:  
\[\mathbf{Y}_{D}^{pilot}(\hat{l}_{i})=[\mathbf{Y}_{dd}^{pilot}(\hat{l}_{i},0),...,\mathbf{ Y}_{dd}^{pilot}(\hat{l}_{i},N-1)] \tag{12}\]  
and its corresponding vector in the time domain is \(\mathbf{r_{t}}(\hat{l}_{i})=F^{-1}\mathbf{Y}_{D}^{pilot}(\hat{l}_{i})\), where \(F^{-1}\) is the inverse matrix of the Fourier transform. By compensating a fractional frequency offset \(v_{f}\in[-1,1]\) to \(\mathbf{r_{t}}(\hat{l}_{i})\), the compensated DD domain signal at \((\hat{l}_{i},\hat{k}_{i})\) can be obtained as follows:  
\[\hat{\chi}_{dd}^{pilot}(\hat{l}_{i},\hat{k}_{i})=\sum_{j=0}^{N-1}\mathbf{r_{t}}( \hat{l}_{i})e^{j2\pi\frac{i|\hat{(k_{i}-v_{f})}}{N}},\;v_{f}\in[-1,1] \tag{13}\]  
When the fractional Doppler of this path is compensated to an integer, it satisfies: \((k_{i}+\kappa_{i})-\hat{\kappa}_{i}=\hat{k}_{i}\). The signal energy at point \((\hat{l}_{i},\hat{k}_{i})\) reaches its maximum. Therefore, we can estimate the fractional Doppler of this path as follows:  
\[\hat{\kappa}_{i}=\operatorname*{arg\,max}_{v_{f}}E(v_{f}) \tag{14}\] \[s.t.v_{f}\in[-1,1]\]  
where \(E(v_{f})=|\hat{\gamma}_{dd}^{pilot}(\hat{l}_{i},\hat{k}_{i})|\) represents the signal amplitude at \((\hat{l}_{i},\hat{k}_{i})\). Based on the compensated signal \(\hat{\gamma}_{dd}^{pilot}(\hat{l}_{i},\hat{k}_{i})\) by \(\hat{\kappa}_{i}\), the coefficient of the path \(\hat{h}_{i}\) can be estimated as follows:  
\[\hat{h}_{i}=\frac{\hat{\chi}_{dd}^{pilot}(\hat{l}_{i},\hat{k}_{i})}{X_{dd}^{ pilot}(m_{p},n_{p})e^{j2\pi\frac{m_{p}\hat{\kappa}_{i}}{MN}}} \tag{15}\]  
Based on (14) and (15), the signal on \((\hat{l}_{i},\hat{k}_{i})\) and IDI of the path on other Doppler bins can be reconstructed, and subtracted from \(\mathbf{Y_{D}}(\hat{l}_{i})\):\[\mathbf{Y}_{\text{D}}^{\textit{pilot(\textbf{1})}}(\hat{l}_{i})=\mathbf{Y}_{\text{D}}^{ \textit{pilot}}(\hat{l}_{i})-\mathbf{A}\left(\hat{h}_{i}e^{i2\pi\frac{m_{p}(\hat{l}_ {i}+\xi_{i})}{MN}}X_{dd}^{pilot}(m_{p},n_{p})\right) \tag{16}\]  
where \(\mathbf{A}=[\alpha_{1},\alpha_{1},...,\alpha_{N}]\) represents the IDI caused by \(\hat{\kappa}_{i}\), and \(\mathbf{Y}_{\text{D}}^{\textit{pilot(\textbf{1})}}(\hat{l}_{i})\) is the renewed signal. The renewed signal in the guard interval is employed to search for the strongest path once more. The searched strong path has the same operation as the above. After \(a\) iterations, the operation will stop when the following condition is satisfied:  
\[\underset{(\hat{l}_{i},\hat{k}_{i})}{\arg\max}|\mathbf{Y}_{\text{D}}^{\textit{ pilot(\textbf{a})}}(l,k)|<\beta\left(\frac{1}{MN}\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}|Y_{dd} (l,k)|\right) \tag{17}\]  
where \(\beta\) is a preset threshold that is often employed in the channel estimation of OTFS systems with the embedded pilot.