# Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems
## 3 Superimposed Perfect Binary Array-Aided OTFS Channel Estimator
### Proposed Channel Estimator
\begin{array}{ll}0&\quad(i=j)\\ MN&\quad(i\
eq j)\end{array} \right., \tag{18}\]  
as a result, \(v_{q,p}=0\) in (15), indicating that there is no interference between the PBAs affected by different channels according to the perfect autocorrelation of PBA. When delay and Doppler indices associated with search unit \(J\) are not equal to those of any propagation path, \(\mathbf{\Gamma}_{J}\) has no correlation with the received symbol matrix \(\mathbf{Y}\). In this condition, only the interference terms are included in \(z\big{[}k_{J},l_{J}\big{]}\). According to (12) and (18), in this condition, \(z\big{[}k_{J},l_{J}\big{]}\) is given as  
\[z\big{[}k_{J},l_{J}\big{]}=v_{J,d}+v_{J,\omega}. \tag{19}\]  
In summary, the correlation value can be expressed as  
\[z\big{[}k_{J},l_{J}\big{]}=\left\{ \begin{array}{ll}h_{I}+v_{J,d}+v_{J,\omega }&\text{if $(k_{J},l_{J})$ is the delay-Doppler indices of the $J$-th path}\\ v_{J,d}+v_{J,\omega}&\text{else}\end{array} \right. \tag{20}\]  
The perfect autocorrelation of PBA leads to a significant amplitude difference in \(\big{|}z\big{[}k_{J},l_{J}\big{]}\big{|}\) between the two conditions. Therefore, \(\big{|}z\big{[}k_{J},l_{J}\big{]}\big{|}\) is compared to a threshold \(\gamma\); if \(\big{|}z\big{[}k_{J},l_{J}\big{]}\big{|}>\gamma\), we consider this search unit \(J\) to be associated with a valid path in the channel. Supposing that it is the \(q\)-th path, we have \(\big{(}k_{J},l_{J}\big{)}=\big{(}k_{q},l_{q}\big{)}\), and the path gain estimate is  
\[\hat{h}_{q}=z\big{[}k_{J},l_{J}\big{]}=h_{q}+v_{q,d}+v_{q,\omega}. \tag{21}\]  
The threshold \(\gamma\) corresponds to the mean squared error (MSE) of the path gain estimate. The MSE of \(\hat{h}_{q}\) is given as  
\[ \begin{split}\text{MSE}(\hat{h}_{q})&=\sigma_{v_{q,d }+v_{q,\omega}}^{2}\\ &=\text{E}\Big{\{}\Big{(}v_{q,d}+v_{q,\omega}\Big{)}\cdot\Big{(}v _{q,d}+v_{q,\omega}\Big{)}^{*}\Big{\}}\\ &=\frac{\left(\frac{\sigma_{h}^{2}(1-\beta)}{\beta}+\frac{\sigma_{h }^{2}}{\beta\rho}\right)}{MN},\end{split}  \tag{22}\]where \(\sigma_{H}^{2}=\sum_{q=1}^{P}\sigma_{h_{q}}^{2}\). According to (22), the MSE of each path gain can be rewritten as \(\text{MSE}(\hat{h})\) because it is independent of the specific path. As a result, the MSE of the threshold-based channel estimation method is given as  
\[\sigma_{\text{Thr}}^{2}=\sum_{q=1}^{P}\sigma_{v_{q}}^{2}=P\text{-MSE}(\hat{h}). \tag{23}\]  
The threshold satisfies \(\gamma^{2}\propto\text{MSE}(\hat{h})\). Algorithm 1 summarizes the proposed SPBA-OTFS channel estimation method. ```
0: Received symbol matrix \(\mathbf{Y}\), \(\text{PBA}\)\(\mathbf{\Gamma}\), maximum delay \(l_{m}\), maximum Doppler shift \(k_{m}\), threshold \(\gamma\)
0: Channel estimates \(\left\{\left(\hat{h}_{1},k_{1},l_{1}\right),\cdots,\left(\hat{h}_{i},k_{i},l_ {i}\right)\right\}\)
1: Initialize: path number \(i=1\)
2:for\(k_{j}=-k_{m}\) to \(k_{m}-1\)do
3:for\(l_{j}=0\) to \(l_{m}\)do
4: Generate the local PBA \(\mathbf{\Gamma}_{k_{j},l_{j}}\) via (11);
5: Compute the correlation value \(z\left[k_{j},l_{j}\right]\) via (12);
6:if\(\left|z\left[k_{j},l_{j}\right]\right|>\gamma\)then
7: Update \(i=i+1\);
8:\(\hat{h}_{i}\leftarrow\left|z\left[k_{j},l_{j}\right]\right|\), \(k_{i}\gets k_{j}\), \(l_{i}\gets l_{j}\);
9:endif
10:endfor
11:endfor
```  
**Algorithm 1** Superimposed Perfect Binary Array-Aided OTFS Channel Estimation Method