## 3 Fast Batches Processing of Sparse Representation and Recovery in Delay-Doppler Domain
### Fast Batches Processing of Sparse Representation
However, as constructing \(\mathbf{B}_{S,1,1}\) requires calculating \(L^{\prime}M^{\prime}\) CAFs, even constructing only \(\mathbf{B}_{S,1,1}\) involves a huge computational load. For any \(m^{\prime}\) and \(l^{\prime}\), the difference between different CAF copies \(\mathbf{\chi}^{r,c}_{m^{\prime},l^{\prime}}\) lies only in the different peak positions. Therefore, we propose a method of self ambiguity function cropping to construct the measurement matrix \(\mathbf{B}_{S,1,1}\) and reduce the computational load. Let the self ambiguity function of the reference signal \(u_{ref}(t)\) be \(\chi_{ref}(\tau,f_{d})\), i.e.,  
\[\chi_{ref}(\tau,f_{d})=\int_{0}^{T}u_{ref}^{*}(t-\tau)u_{ref}(t)e^{-2\pi f_{d} t}dt \tag{19}\]Suppose the matrix \(A_{ref}\in\mathbb{C}^{(2L^{\prime}M^{\prime}-1)\times(2L^{\prime}M^{\prime}-1)}\) formed by \(\chi_{ref}\) corresponds to the delay and Doppler frequency \(\left\{\tau_{-L^{\prime}+1}^{ref},\tau_{-L^{\prime}+2}^{ref},\cdots,\tau_{L^{ \prime}-1}^{ref}\right\}\), and \(\left\{f_{-M^{\prime}+1}^{ref},f_{-M^{\prime}+2}^{ref},\cdots,f_{M^{\prime}-1 }^{ref}\right\}\), where \(\left(\tau_{0}^{ref},f_{0}^{ref}\right)=(0,0)\). Then, \(A_{ref}\) can be expressed as  
\[\mathbf{A}_{ref}= \begin{bmatrix}\chi_{ref}\left(\tau_{-L^{\prime}+1}^{ref},\tau_{- M^{\prime}+1}^{ref}\right)&\chi_{ref}\left(\tau_{-L^{\prime}+1}^{ref},f_{-M^{ \prime}+2}^{ref}\right)&\cdots&\chi_{ref}\left(\tau_{-L^{\prime}+1}^{ref},f_{ M^{\prime}-1}^{ref}\right)\\ \chi_{ref}\left(\tau_{-L^{\prime}+2}^{ref},f_{-M^{\prime}+1}^{ref}\right)&\chi _{ref}\left(\tau_{-L^{\prime}+2}^{ref},f_{-M^{\prime}+2}^{ref}\right)&\cdots& \chi_{ref}\left(\tau_{-L^{\prime}+2}^{ref},f_{M^{\prime}-1}^{ref}\right)\\ \vdots&\vdots&\ddots&\vdots\\ \chi_{ref}\left(\tau_{L^{\prime}-1}^{ref},f_{-M^{\prime}+1}^{ref}\right)&\chi _{ref}\left(\tau_{L^{\prime}-1}^{ref},f_{-M^{\prime}+2}^{ref}\right)&\cdots& \chi_{ref}\left(\tau_{L^{\prime}-1}^{ref},f_{M^{\prime}-1}^{ref}\right)\\ \end{bmatrix}  \tag{20}\]  
Let the matrix form of the CAF copy \(\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1}\) be \(\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1}\in\mathbb{C}^{L^{\prime}\times M^{\prime}}\), then \(\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1}\) can be obtained by cropping from \(A_{ref}\), i.e.,  
\[\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1}= \begin{bmatrix}\chi_{ref}\left(\tau_{1- l^{\prime}}^{ref},f_{1-m^{\prime}}^{ref}\right)&\chi_{ref}\left(\tau_{1- l^{\prime}}^{ref},f_{2-m^{\prime}}^{ref}\right)&\cdots&\chi_{ref}\left(\tau_{1- l^{\prime}}^{ref},f_{M^{\prime}-m^{\prime}}^{ref}\right)\\ \chi_{ref}\left(\tau_{2-l^{\prime}}^{ref},f_{1-m^{\prime}}^{ref}\right)&\chi _{ref}\left(\tau_{2-l^{\prime}}^{ref},f_{2-m^{\prime}}^{ref}\right)&\cdots& \chi_{ref}\left(\tau_{2-l^{\prime}}^{ref},f_{M^{\prime}-m^{\prime}}^{ref}\right) \\ \vdots&\vdots&\ddots&\vdots\\ \chi_{ref}\left(\tau_{L^{\prime}-l^{\prime}}^{ref},f_{1-m^{\prime}}^{ref}\right)& \chi_{ref}\left(\tau_{L^{\prime}-l^{\prime}}^{ref},f_{2-m^{\prime}}^{ref}\right) &\cdots&\chi_{ref}\left(\tau_{L^{\prime}-l^{\prime}}^{ref},f_{M^{\prime}-m^{ \prime}}^{ref}\right)\\ \end{bmatrix}  \tag{21}\]  
Therefore, \(\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1}\) can be expressed as  
\[\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1}= \begin{bmatrix}\chi_{ref}\left(\tau_{1- l^{\prime}}^{ref},f_{1-m^{\prime}}^{ref}\right)\chi_{ref}\left(\tau_{2- l^{\prime}}^{ref},f_{1-m^{\prime}}^{ref}\right)\\ \cdots&\chi_{ref}\left(\tau_{L^{\prime}-l^{\prime}}^{ref},f_{1-m^{\prime}}^{ref} \right)\chi_{ref}\left(\tau_{1-l^{\prime}}^{ref},f_{2-m^{\prime}}^{ref}\right) \\ \cdots&\chi_{ref}\left(\tau_{L^{\prime}-l^{\prime}}^{ref},f_{M^{\prime}-m^{ \prime}}^{ref}\right)\\ \end{bmatrix}  \tag{22}\]  
\(\mathbf{B}_{S,1,1}\) can be expressed as  
\[\mathbf{B}_{S,1,1}= \begin{bmatrix}\mathbf{X}_{1,1}^{1,1}&\mathbf{X}_{1,2}^{1,1}&\cdots&\mathbf{ X}_{1,1}^{1,1}&\mathbf{X}_{2,1}^{1,1}&\cdots&\mathbf{X}_{m^{\prime},l^{\prime}}^{1,1} \end{bmatrix}  \tag{23}\]  
So far, we have completed the rapid construction of the measurement matrix.