# Superimposed Perfect Binary Array-Aided Channel Estimation for OTFS Systems
## 3 Superimposed Perfect Binary Array-Aided OTFS Channel Estimator  
### Proposed Channel Estimator  
According to (9), the received symbols can be considered equivalent to weighted superpositions of the transmitted symbols with different circular shifts. Hence, in an SPBA-OTFS system, the PBA symbols also have the same shifts, which are determined by the delay and Doppler indices of the paths. As a result, the estimation of the delay and Doppler shifts can be transformed into a linear search of the indices. Therefore, a 2D linear search is performed in the DD domain at the receiver side by correlating the received symbol matrix \(\mathbf{Y}\) with the local PBA \(\mathbf{\Gamma}\) over all possible delay and Doppler indices. Furthermore, the step sizes for the linear search of the delay and Doppler indices are set as the quantization steps of the delay axis \((1/M\Delta f)\) and the Doppler axis \((1/NT)\), respectively. For each search operation, we define a search unit \(J\), which is associated with delay value \((l_{J}/M\Delta f)\) and Doppler value \((k_{J}/NT)\), where \(l_{J}\in[0,l_{m}]\) and \(k_{J}\in[-k_{m},k_{m})\) denote the delay and Doppler indices of the local PBA, respectively. Here, \(l_{m}\) and \(k_{m}\) are the maximum delay and Doppler shift indices of the channel, respectively. In this paper, we first assume an SPBA-OTFS system with ideal pulses, meaning that \(\alpha_{i}[k,l]=1\) in (7). In this case, the superimposed PBA affected by the \(q\)-th path can be expressed as  
\[\mathbf{\Lambda}_{q}=h_{q}e^{-2\pi\frac{l_{q}k_{J}}{MN}}\mathbf{P}_{k_{q},l_{ q}}, \tag{10}\]  
where \(1\leq q\leq P\). Following (10), we define the local PBA associated with search unit \(J\) as  
\[\mathbf{\Gamma}_{J}=\frac{e^{j2\pi\frac{l_{J}k_{J}}{MN}}\mathbf{P}_{k_{J},l_{ J}}^{*}}{\beta\rho}, \tag{11}\]  
where \(\mathbf{P}_{k_{J},l_{J}}^{*}\) denotes the conjugate of the matrix \(\mathbf{P}_{k_{J},l_{J}}\). For channel estimation, we correlate \(\mathbf{Y}\) with \(\mathbf{\Gamma}_{J}\) at the receiver side. The correlation value \(z\big{[}k_{J},l_{J}\big{]}\) can be expressed as  
\[z\big{[}k_{J},l_{J}\big{]}=\frac{\text{sum}(\mathbf{Y}\circ\mathbf{\Gamma}_{J} )}{MN}, \tag{12}\]  
where the operation \(\text{sum}(\mathbf{Y}\circ\mathbf{\Gamma}_{J})\) computes the sum of all elements of the matrix \(\mathbf{Y}\circ\mathbf{\Gamma}_{J}\). When the delay and Doppler indices associated with the search unit \(J\) are equal to those of the \(q\)-th path, which means that \(\big{(}k_{J},l_{J}\big{)}=\big{(}k_{q},l_{q}\big{)}\), \(\mathbf{\Gamma}_{J}\) has the maximum correlation with \(\mathbf{\Lambda}_{q}\). Consequently, a peak that reflects the channel state information (CSI) appears after the correlation calculation. In this condition, the correlation value is  
\[z\big{[}k_{J},l_{J}\big{]}=z\big{[}k_{q},l_{q}\big{]}=h_{q}+v_{q}, \tag{13}\]where \(h_{q}\) is the complex gain of the \(q\)-th path and \(v_{q}\) denotes the interference term in the correlation value, given by  
\[v_{q}=v_{q,p}+v_{q,d}+v_{q,\omega}, \tag{14}\]  
where \(v_{q,p}\), \(v_{q,d}\) and \(v_{q,\omega}\) denote the interference between the PBAs affected by different channels, between PBA and the data symbol matrix, and between PBA and the noise matrix, respectively. They can be expressed as  
\[v_{q,p}=\frac{\sum_{i=1,i\
eq q}^{P}h_{i}e^{2\pi\frac{(\{q_{i}-i\})(\{q_{i}-i\} )}{MN}}\text{sum}\Big{(}\mathbf{P}_{k_{i},l_{i}}\circ\mathbf{P}_{k_{q},l_{q}}^{* }\Big{)}}{MN(\beta\rho)}, \tag{15}\]  
\[v_{q,d}=\frac{\sum_{i=1}^{P}h_{i}e^{2\pi\frac{(\{q_{i}-i\})(\{q_{i}-i\})}{MN}} \text{sum}\Big{(}\mathbf{D}_{k_{i},l_{i}}\circ\mathbf{P}_{k_{q},l_{q}}^{*} \Big{)}}{MN(\beta\rho)}, \tag{16}\]  
\[v_{q,\omega}=\frac{e^{j2\pi\frac{l_{q,q}}{MN}}\text{sum}\Big{(}\mathbf{W} \circ\mathbf{P}_{k_{q},l_{q}}^{*}\Big{)}}{MN(\beta\rho)}. \tag{17}\]  
Furthermore, considering the autocorrelation of PBA, which will be introduced later in this section, we have  
\[\text{sum}\Big{(}\mathbf{P}_{k_{i},l_{i}}\circ\mathbf{P}_{k_{j},l_{j}}^{*} \Big{)}=\left\{