## 3 Fast Batches Processing of Sparse Representation and Recovery in Delay-Doppler Domain
### Fast Batches Processing of Sparse Representation
Let the CAF matrix \(\mathbf{A}_{S}\in\mathbb{C}^{L\times M}\) be the rearranged form of CAF \(\chi(\tau,f_{d})\) before being organized into the column vector \(\mathbf{a}_{S}\), i.e.,  
\[\mathbf{A}_{S}= \begin{bmatrix}\chi(\tau_{1},f_{d1})&\chi(\tau_{1},f_{d2})&\cdots &\chi(\tau_{1},f_{dM})\\ \chi(\tau_{2},f_{d1})&\chi(\tau_{2},f_{d2})&\cdots&\chi(\tau_{2},f_{dM})\\ \vdots&\vdots&\ddots&\vdots\\ \chi(\tau_{L},f_{d1})&\chi(\tau_{L},f_{d2})&\cdots&\chi(\tau_{L},f_{dM})\end{bmatrix}  \tag{13}\]\(\mathbf{A}_{S}\) is divided into \(R\times C\) equally sized sub-blocks, each with dimensions \(L^{\prime}\times M^{\prime}\), where \(L=RL^{\prime}\) and \(M=CM^{\prime}\). Then, \(\mathbf{A}_{S}\) can be expressed as  
\[\mathbf{A}_{S}= \begin{bmatrix}A_{S,1,1}&A_{S,1,2}&\cdots&\mathbf{A}_{S,1,C}\\ A_{S,2,1}&A_{S,2,2}&\cdots&A_{S,2,C}\\ \vdots&\vdots&\ddots&\vdots\\ A_{S,R,1}&A_{S,R,2}&\cdots&A_{S,R,C}\end{bmatrix}  \tag{14}\]  
Without loss of generality, consider the case where, for any \(r\in\{1,2,\cdots,R\}\), and \(c\in\{1,2,\cdots,C\}\), there are targets present in the \((r,c)\)-th sub-block \(\mathbf{A}_{S,r,c}\in\mathbb{C}^{L^{\prime}\times M^{\prime}}\). Let \(\{\tau_{r,c,1},\tau_{r,c,2},\cdots,\tau_{r,c,L^{\prime}}\}\) and \(\{f_{r,c,1},f_{r,c,2},\cdots,f_{r,c,M^{\prime}}\}\) represent the delay and Doppler frequency corresponding to \(\mathbf{A}_{S,r,c}\). Then, \(\mathbf{A}_{S,r,c}\) can be expressed as  
\[\mathbf{A}_{S,r,c}= \begin{bmatrix}\chi(\tau_{r,c,1},f_{r,c,1})&\chi(\tau_{r,c,1},f _{r,c,2})&\cdots&\chi(\tau_{r,c,1},f_{r,c,M^{\prime}})\\ \chi(\tau_{r,c,2},f_{r,c,1})&\chi(\tau_{r,c,2},f_{r,c,2})&\cdots&\chi(\tau_{r,c,2},f_{r,c,M^{\prime}})\\ \vdots&\vdots&\ddots&\vdots\\ \chi(\tau_{r,c,L^{\prime}},f_{r,c,1})&\chi(\tau_{r,c,L^{\prime}},f_{r,c,2})& \cdots&\chi(\tau_{r,c,L^{\prime}},f_{r,c,M^{\prime}})\end{bmatrix}  \tag{15}\]  
Let \(\mathbf{a}_{S,r,c}\in\mathbb{C}^{L^{\prime}M^{\prime}\times 1}\) be the column vector form of \(\mathbf{A}_{S,r,c}\), i.e.,  
\[\mathbf{a}_{S,r,c}= \begin{bmatrix}\chi(\tau_{r,c,1},f_{r,c,1})&\chi(\tau_{r,c,2}, f_{r,c,1})&\cdots&\chi(\tau_{r,c,L^{\prime}},f_{r,c,1})&\chi(\tau_{r,c,1},f_{r,c,2})\\ \cdots&\chi(\tau_{r,c,l^{\prime}},f_{r,c,m^{\prime}})&\cdots&\chi(\tau_{r,c,L ^{\prime}},f_{r,c,M^{\prime}})\end{bmatrix} ^{T} \tag{16}\]  
The measurement matrix \(\mathbf{B}_{S,r,c}\in\mathbb{C}^{L^{\prime}M^{\prime}\times L^{\prime}M^{\prime}}\) corresponding to \(\mathbf{a}_{S,r,c}\) is expressed as  
\[\mathbf{B}_{S,r,c}= \begin{bmatrix}\chi^{r,c}_{1,1}\,\mathbf{x}^{r,c}_{1,2}&\cdots&\chi ^{r,c}_{1,L^{\prime}},\,\chi^{r,c}_{2,1}&\cdots&\chi^{r,c}_{m^{\prime},p^{ \prime}}&\cdots&\chi^{r,c}_{M^{\prime},L^{\prime}}\end{bmatrix}  \tag{17}\]  
where \(\chi^{r,c}_{m^{\prime},l^{\prime}}\in\mathbb{C}^{L^{\prime}M^{\prime}\times 1}\) represents a copy of the CAF obtained by adding the delay \(\tau_{r,c,l^{\prime}}\) and Doppler frequency \(f_{r,c,m^{\prime}}\) to the reference signal. It can be expressed as  
\[\chi^{r,c}_{m^{\prime},l^{\prime}}= \begin{bmatrix}\chi_{m^{\prime},l^{ \prime}}(\tau_{r,c,1},f_{r,c,1})&\chi_{m^{\prime},l^{\prime}}(\tau_{r,c,2},f_ {r,c,1})\\ \cdots&\chi_{m^{\prime},l^{\prime}}(\tau_{r,c,l^{\prime}},f_{r,c,1})&\chi_{m^ {\prime},l^{\prime}}(\tau_{r,c,1},f_{r,c,2})\\ \cdots&\chi_{m^{\prime},l^{\prime}}(\tau_{r,c,l^{\prime}},f_{r,c,m^{\prime}} )&\cdots&\chi_{m^{\prime},l^{\prime}}(\tau_{r,c,L^{\prime}},f_{r,c,M^{\prime }})\end{bmatrix} ^{T} \tag{18}\]  
From (7) and (18), it can be observed that the peak position of the CAF copy \(\chi^{r,c}_{m^{\prime},l^{\prime}}\) is only related to \(m^{\prime}\) and \(l^{\prime}\), and is independent of \(r\) and \(c\). Therefore, when performing sparse recovery for each \(\mathbf{a}_{S,r,c}\), the measurement matrix \(\mathbf{B}_{S,r,c}\) can be reused. In other words, after obtaining \(\mathbf{B}_{S,1,1}\), for any \(r\) and \(c\), use \(\mathbf{B}_{S,1,1}\) instead of \(\mathbf{B}_{S,r,c}\).