## 3 Fast Batches Processing of Sparse Representation and Recovery in Delay-Doppler Domain
### Target Detection Based on Sparse Recovery  
The solution to sparse vectors requires sparse recovery. In recent years, many sparse recovery algorithms [36; 37; 38; 39; 40; 41; 42; 43; 44] have been applied to address this issue. However, these algorithms need the measurement matrix to satisfy the restricted isometry property (RIP) condition, and the verification of RIP condition is an NP-hard problem. Therefore, we employed a sparse recovery algorithm based on the \(l_{0}\) pseudonorm iterative solution [45] to reduce the theoretical limitations. This algorithm uses the following pseudonorm as a replacement for the \(l_{0}\) norm:  
\[\left\|\mathbf{x}\right\|_{g_{\alpha}}=\sum_{d=1}^{D}g_{\alpha}(x_{d}),\,\mathbf{x} \in\mathbb{R}^{D\times 1} \tag{28}\]  
In (28), \(x_{d}\) represents the \(d\)-th element of \(\mathbf{x}\). Where \(g_{\alpha}(x)\) is  
\[g_{\alpha}(x)=\frac{\left|x\right|}{\sqrt{x^{2}+\alpha}} \tag{29}\]  
The relationship between \(g_{\alpha}(x)\), \(x\), and \(\alpha\) is shown in Figure 4. Obviously, as \(\alpha\) approaches \(0\), the properties of \(g_{\alpha}(x)\) tend to \(0\) norm.  
Because the domain of definition of \(g_{\alpha}(x)\) is a set of real numbers, Equation (25) needs to be converted into real form.  
\[\mathbf{a}_{r,c}=\begin{bmatrix}\text{Re}(\mathbf{a}_{S,r,c})\\ \text{Im}(\mathbf{a}_{S,r,c})\end{bmatrix} \tag{30}\]  
\[\mathbf{B}_{1,1}=\begin{bmatrix}\text{Re}(\mathbf{B}_{S,1,1})&-\text{Im}(\mathbf{B}_{S,1, 1})\\ \text{Im}(\mathbf{B}_{S,1,1})&\text{Re}(\mathbf{B}_{S,1,1})\end{bmatrix} \tag{31}\]  
\[\mathbf{\sigma}_{r,c}=\begin{bmatrix}\text{Re}(\mathbf{\sigma}_{S,r,c})\\ \text{Im}(\mathbf{\sigma}_{S,r,c})\end{bmatrix} \tag{32}\]  
In summary, based on (25) and (30) (32), when the influence of noise is neglected, we have  
\[\mathbf{a}_{r,c}=\mathbf{B}_{1,1}\mathbf{\sigma}_{r,c} \tag{33}\]Therefore, as long as we obtain the sparse vector \(\sigma_{r,c}\in\mathbb{R}^{2L^{\prime}M^{\prime}\times 1}\) and recover \(\sigma_{S,r,c}\) from \(\sigma_{r,c}\), the positions of the non-zero values in \(\sigma_{S,r,c}\) can determine the cells in the \((r,c)\)-th sub-block where the surveillance signal exists.  
To estimate the sparse vector \(\sigma_{r,c}\) consider the following problem  
\[\begin{split}\min_{\sigma_{r,c}\in\mathbb{R}^{2L^{\prime}M^{ \prime}\times 1}}\left\|\sigma_{r,c}\right\|_{g_{\alpha}}\\ \text{s.t.}&\mathbf{a}_{r,c}=\mathbf{B}_{1,1}\sigma_{r,c} \end{split} \tag{34}\]  
According to the derivation in reference [45], the expression for iteratively solving \(\sigma_{r,c}\) can be obtained as  
\[\sigma_{r,c}=\Gamma(\sigma_{r,c})\mathbf{B}_{1,1}^{T}\left(\mathbf{B}_{1,1}\Gamma( \sigma_{r,c})\mathbf{B}_{1,1}^{T}\right)^{\dagger}\mathbf{a}_{r,c} \tag{35}\]  
where \(\Gamma(\sigma_{r,c})\in\mathbb{R}^{2L^{\prime}M^{\prime}\times 2L^{\prime}M^{ \prime}}\) is a diagonal matrix, and its element at the \(i\)-th row and \(i\)-th column is given by  
\[\Gamma(\sigma_{r,c})_{i,i}=\frac{\left|\sigma_{r,c,i}\right|}{g_{\alpha^{ \prime}}(\sigma_{r,c,i})}=\frac{\left|\sigma_{r,c,i}\right|\left(\sigma_{r,c,i }^{2}+\alpha\right)^{1.5}}{\alpha} \tag{36}\]  
where \(\sigma_{r,c,i}\) is the \(i\)-th element of \(\sigma_{r,c}\).  
By analyzing (35), it can be observed that this equation is formulated as a fixed-point iteration. Performing a finite number of iterations on (35) yields the solution \(\sigma_{r,c}^{*}\), indicating that the iteration is convergent.