## 3 OTFS Channel Estimation
### Proposed P-IPIC Algorithm  
The proposed variant of the DDIPIC algorithm aims to reduce the complexity of DDIPIC and is based on a RCF maximization and performs an iterative search as in the original algorithm until the maximum number of iterations is reached or the stopping criterion is met. The pseudocode of the proposed P-IPIC algorithm is provided in Algorithm 2. The main parameters are as for the DDIPIC algorithm. ```
Input:\(P_{\max}\), \(\epsilon\), \(s_{\max}\), \(\epsilon_{\tau}\), \(e_{\
u}\), \(m_{\tau}\), \(n_{\
u}\), \(N_{0}\), \(\mathbf{y}\)
1\(\mathcal{E}^{(0)}=\mathbf{y}\); for\(i=1\)to\(P_{\max}\)do
2\(\tau_{\text{coarse}}^{(i)}\), \(\
u_{\text{coarse}}^{(i)}=\arg\max_{(\tau,\
u)\in I_{DD}}\Phi\big{(}\mathbf{r} (\tau,\
u),\mathcal{E}^{(i-1)}\big{)}\);
3\(\hat{\tau}_{\text{fine}}^{(0)}=\tau_{\text{coarse}}^{(i)}\);
4\(\hat{\
u}_{\text{coarse}}^{(0)}=\
u_{\text{coarse}}^{(i)}\); for\(s=1\)to\(s_{\max}\)do
5\(W_{\tau}^{(s)}=\frac{\tau_{\text{fine}}}{m_{\tau}^{s}}\): delay search width;
6\(W_{\
u}^{(s)}=\frac{\tau_{\text{fine}}}{m_{\tau}^{s}}\): Doppler search width;
7\(F_{DD}^{(s)}=\left\{W_{\tau}^{(s)}T+\hat{\tau}_{\text{fine}}^{(s-1)}\right\} \times\left\{W_{\
u}^{(s)}\Lambda+\hat{\
u}_{\text{fine}}^{(s-1)}\right\}\);
8\(\hat{\tau}_{\text{fine}}^{(s)}\), \(\hat{\
u}_{\text{fine}}^{(s)}=\arg\max_{(\tau,\
u)\in F_{DD}^{(s)}}\Phi\big{(} \mathbf{r}(\tau,\
u),\mathcal{E}^{(i-1)}\big{)}\);
9 if\(s=s_{\max}\)or\(\left(\left|\hat{\tau}_{\text{fine}}^{(s)}-\hat{\tau}_{\text{fine}}^{(s)} \right|<\epsilon_{\tau}\)and\(\left|\hat{\
u}_{\text{fine}}^{(s)}-\hat{\
u}_{\text{fine}}^{(s-1)} \right|<\epsilon_{\
u}\)\)then
10 break;
11
12\(\hat{\tau}[i]=\hat{\tau}_{\text{fine}}^{(s)}\);
13\(\hat{\
u}[i]=\hat{\
u}_{\text{fine}}^{(s)}\);
14\(\hat{\mathbf{g}}=\left(\mathbf{R}^{H}(\hat{\tau},\hat{\
u})\mathbf{R}(\hat{ \tau},\hat{\
u})+N_{0}\mathbf{I}\right)^{-1}\mathbf{R}^{H}(\hat{\tau},\hat{ \
u})\mathbf{y}\);
15\(\mathcal{E}^{(i)}=\mathbf{y}-\mathbf{R}(\hat{\tau},\hat{\
u})\mathbf{g}\);
16 if\(i>1\)then
17 if\(\|\mathcal{E}^{(i)}\|<\epsilon\)then
18\(\hat{P}=i\);
19 break;
20
21 Refinement procedure (\(i_{\text{ref}}=1,\dots,\hat{P}\))\(\rightarrow\)\(\hat{P}\);
22 Output:\(\hat{\mathbf{H}}=\sum_{i=1}^{\hat{P}}\delta_{i}\mathbf{G}_{i}(\hat{\tau}_{i}, \hat{\
u}_{i})\);
```  
**Algorithm 2**Proposed P-IPIC  
The proposed algorithm features several key differences compared to Algorithm 1:  
* _Coarse and fine estimation based on the residuals (lines 3 and 10):_ The key idea is that once a path is detected and the residue vector is obtained, during the next search, a new path is discovered looking at the residual pilot signal of the previous iteration. Thus, at the \(i\)-th iteration, the algorithm looks for a new path maximizing an RCF given by \[\Phi\big{(}\mathbf{r}(\tau,\
u),\mathcal{E}^{(i-1)}\big{)} = \left(\mathcal{E}^{(i-1)}\right)^{H}\mathbf{r}(\tau,\
u)\left( \mathbf{r}^{H}(\tau,\
u)(\tau,\
u)\right)\right)^{-1}\mathbf{r}^{H}(\tau,\
u) \mathcal{E}^{(i-1)}\] (20) \[= \frac{\big{(}\mathbf{r}^{H}(\tau,\
u)\mathcal{E}^{(i-1)}\big{)}^ {*}\mathbf{r}^{H}(\tau,\
u)\mathcal{E}^{(i-1)}}{\|\mathbf{r}(\tau,\
u)\|^{2}} = \frac{|\mathbf{r}^{H}(\tau,\
u)\mathcal{E}^{(i-1)}|^{2}}{\|\mathbf{r}(\tau,\
u) \|^{2}}.\] (21) In this way, at each iteration, the IPI between the paths is directly canceled, facilitating the new search. * _Global refinement (line 21):_ It is important to notice that this solution is prone to error propagation across all estimates. This may lead to residue vectors with residual peaks that are interpreted as small-amplitude paths by the algorithm. In order to avoid false alarms or multidetections due to residual uncancelled IPI, the P-IPIC algorithm needs a final refinement step in which all the estimates are refined while the residue in monitored. If the residue satisfies the stopping criterion, all remaining unrefined estimates are declared false alarms and, therefore, discarded. Specifically, during the _search phase_, a number of paths equal to \(\hat{P}\) is detected.