## 3 Fast Batches Processing of Sparse Representation and Recovery in Delay-Doppler Domain  
### Fast Batches Processing of Sparse Representation  
In this section, we introduce the fast algorithm of batches processing of sparse representation and recovery in the delay-Doppler domain. The flowchart is shown in Figure 3. The steps of this algorithm are summarized at the end of this section. To construct a sparse representation model for CAF, it is assumed that there could be a surveillance signal on each unit in the delay-Doppler domain. Let \(\{\tau_{1},\tau_{2},\cdots,\tau_{L}\}\), and \(\{f_{d1},f_{d2},\cdots,f_{dM}\}\) represent the sets of delay and Doppler frequency, respectively. Where \(L\) and \(M\) are the number of delay units and Doppler frequency units, respectively. The expression for CAF in (4) can be written as  
\[\chi(\tau,f_{d})=\sum_{m=1}^{M}\sum_{l=1}^{L}\sigma_{m,l}\chi_{m,l}(\tau,f_{d} )+n_{\chi}(\tau,f_{d}) \tag{5}\]  
where \(\sigma_{m,l}\) represents the amplitude gain under the delay \(\tau_{l}\) and the Doppler frequency \(f_{dm}\) of the surveillance signal, and it can be expressed as  
Figure 3: The flowchart for fast batches processing of sparse representation and recovery in the delay-Doppler domain. \[\sigma_{m,l}=\left\{ \begin{array}{l}\dfrac{A_{p}}{A_{ref}},\ (\tau_{l},f_{dm})= \left(\tau_{p}-\tau_{0},f_{p}\right)\\ 0\,\ \text{others}\end{array} \right. \tag{6}\]  
\(\chi_{m,l}(\tau,f_{d})\) represents the CAF between the reference signal \(u_{ref}(t)\) and a copy of itself with added delay and Doppler frequency \(u_{ref}(t-\tau_{l})e^{j2\pi f_{dm}t}\), and it can be expressed as  
\[\chi_{m,l}(\tau,f_{d})=\int_{0}^{T}u_{ref}^{*}(t-\tau)u_{ref}(t-\tau_{l})e^{-j2 \pi(f_{d}-f_{dm})t}dt \tag{7}\]  
Let \(\mathbf{a}_{S}\in\mathbb{C}^{LM\times 1}\) be a column vector formed by slicing and stacking the CAF \(\chi(\tau,f_{d})\) along each Doppler frequency unit, expressed as  
\[\mathbf{a}_{S}=[\chi(\tau_{1},f_{d1})\ \chi(\tau_{2},f_{d1})\ \cdots\ \chi(\tau_{L},f_{d1})\ \chi(\tau_{1},f_{d2})\ \cdots\ \chi(\tau_{l},f_{dm})\ \cdots\ \chi(\tau_{L},f_{dM})]^{T} \tag{8}\]  
Different reference signal copies are generated by adding various delays and Doppler frequencies to the reference signal \(u_{ref}(t)\). These copies are then processed with the reference signal using CAF, and arranged in a similar manner to form a column vector. This process is repeated for each Doppler frequency unit. Consequently, this forms the measurement matrix \(\mathbf{B}_{S}\in\mathbb{C}^{LM\times LM}\), expressed as  
\[\mathbf{B}_{S}=[\chi_{1,1}\ \chi_{1,2}\ \cdots\ \chi_{1,L}\ \chi_{2,1}\ \cdots\ \chi_{m,l}\ \cdots\ \chi_{M,L}] \tag{9}\]  
where \(\chi_{m,l}\) represents a copy of the CAF obtained by adding the delay \(\tau_{l}\) and Doppler frequency \(f_{dm}\) to the reference signal. It can be expressed as follows:  
\[\chi_{m,l}= \begin{bmatrix}\chi_{m,l}(\tau_{1},f_{d1})\ \chi_{m,l}(\tau_{2},f_{d1})\ \cdots\ \chi_{m,l}(\tau_{L},f_{d1})\ \chi_{m,l}(\tau_{1},f_{d2})\\ \cdots\ \chi_{m,l}(\tau_{1},f_{dm})\ \cdots\ \chi_{m,l}(\tau_{L},f_{dM})\end{bmatrix} ^{T} \tag{10}\]  
The sparse vector \(\sigma_{m,l}\) formed by the amplitude gains \(\mathbf{\sigma}_{S}\in\mathbb{C}^{LM\times 1}\) can be expressed as  
\[\mathbf{\sigma}_{S}=[\sigma_{1,1}\ \sigma_{1,2}\ \cdots\ \sigma_{1,L}\ \sigma_{2,1}\ \cdots\ \sigma_{m,l}\ \cdots\ \sigma_{M,L}\ ]^{T} \tag{11}\]  
Therefore, \(\mathbf{a}_{S}\) can be expressed as  
\[\mathbf{a}_{S}=\mathbf{B}_{S}\mathbf{\sigma}_{S}+\mathbf{n} \tag{12}\]  
where \(\mathbf{n}\in\mathbb{C}^{LM\times 1}\) represents the impact of noise on CAF. Equation (12) represents the sparse representation model of CAF. However, when the delay-Doppler domain is large, the size of the measurement matrix \(\mathbf{B}_{S}\) becomes large, imposing high computational and memory resource requirements. Therefore, we propose improvements to the sparse representation model to reduce computational and memory resource usage.