# Sparsity-Inducing Super-Resolution Passive Radar Imaging with Illuminators of Opportunity
## 3 Super-Resolution Imaging for Passive Radar
### CS-Based Passive Radar Super-Resolution Imaging  
It is widely recognized that in the CS framework, the optimal reconstruction of an unknown sparse signal can be achieved from limited noisy measurements by solving a sparsity-constrained optimization problem [23]. The CS framework has been successfully applied in conventional ISAR and bistatic ISAR imaging [24; 25; 26; 27]. In this paper, we use such a CS technique to accomplish super-resolution passive radar imaging with fewer television illuminators and a smaller rotation angle. Taking additive noise into account, Equation (16) can then be rewritten as  
\[\mathbf{s}=\mathbf{\Phi}\mathbf{h}+\mathbf{\sigma}, \tag{22}\]  
where \(\mathbf{h}{=}[h\left(0\right),h\left(1\right),\cdots,h\left(N-1\right)]^{T}\), \(\mathbf{\sigma}\) is the measured noise, \(\mathbf{\Phi}\) is the dictionary matrix, and the \(\left(k,m\right)\) element of \(\mathbf{\Phi}\) is defined as:  
\[\phi_{k}\left(m\right)=\exp\left(-j\omega_{k}\cdot m\Delta\Theta\right). \tag{23}\]  
Note that the dictionary \(\mathbf{\Phi}\) is characterized by unknown parameters \(\left\{\omega_{k}\right\}\), which should be estimated together with the unknown complex amplitude \(\left\{h_{k}\right\}\). Estimating \(\left\{\omega_{k}\right\}\) and \(\left\{h_{k}\right\}\) can then be formulated as a sparse recovery problem:\[ \begin{array}{ll}\min\limits_{\mathbf{h}}&\left\|\mathbf{h}\right\|,\\ \textit{s.t.}&\left\|\mathbf{s-\Phi h}\right\|_{2}<\delta,\end{array}  \tag{24}\]  
where \(\left\|\cdot\right\|_{1}\) denotes the \(\ell_{1}\) norm, \(\delta\) is the noise level, and \(\mathbf{\Phi}\in C^{M\times N}\) (\(M\ll N\), \(N\) is the length of the recovered signal) is an over-complete dictionary. Accurate reconstruction of the sparse signal can be achieved through existing recovery algorithms, such as orthogonal matching pursuit (OMP) [28] and basis pursuit (BP) [29]. In this paper, we use a log-sum sparsity-encouraging function for the sparse reconstruction. The log-sum penalty function has been extensively applied for sparse signal recovery [30; 31; 32], and its superiority over the conventional \(\ell_{1}\)-type algorithm has been widely verified. In doing so, Equation (24) can be recast as  
\[ \begin{array}{ll}\min\limits_{\mathbf{h}}&L\left(\mathbf{h}\right)=\sum\limits_{i=1 }^{N}\log\left(\left|h_{i}\right|^{2}+\varepsilon^{\prime}\right),\\ \textit{s.t.}&\left\|\mathbf{s-\Phi h}\right\|_{2}<\delta,\end{array}  \tag{25}\]  
where \(h_{i}\) denotes the \(i\)-th component of the vector \(\mathbf{h}\), and \(\varepsilon^{\prime}>0\) is a positive parameter to ensure that the function is well-defined. Thus, the \(\{\omega_{k}\}\) and \(\{h_{k}\}\) can be estimated by the algorithm proposed in [33]. Once the amplitude and spatial frequency of each scatterer are obtained, the target position can be determined by searching the vector \(\mathbf{\omega}^{\prime}=\{\omega_{k}\}_{k=1}^{K}\). Here, it should be pointed out that the passive radar image achieved in this paper is a two-dimensional image. Thus, at least two illuminators are needed. Thus, the scatterers' position estimation problem can be further formulated as  
\[\{\mathbf{r}^{\prime},\mathbf{\varphi}^{\prime}\}=\operatorname*{arg\,min}\limits_{ \mathbf{r},\mathbf{\varphi}}\left\|\mathbf{\omega}_{\textit{ref}}-\mathbf{\omega}^{\prime} \right\|_{2}, \tag{26}\]  
where \(\{\mathbf{r}^{\prime},\mathbf{\varphi}^{\prime}\}\) are the estimates of \(\{r_{i},\varphi_{i}\}_{i=1}^{i=K}\), and \(\mathbf{\omega}_{\textit{ref}}\) is the referenced spatial frequency. The position information of different scatterers can then be obtained by solving Equation (26). In summary, the flowchart of the proposed CS-based method for passive radar imaging is shown in Figure 2. After searching for positions using two sets of \(\mathbf{\omega}^{\prime}\), and adding the scattering amplitude information in the reconstructed vector \(\mathbf{h}^{\prime}\), we can obtain the passive radar image of the target.