# Sparsity-Inducing Super-Resolution Passive Radar Imaging with Illuminators of Opportunity
## 2 Parametric Passive Radar Imaging Model  
Consider a two-dimensional imaging model as shown in Figure 1. The reference point \(O\) is taken at the target center,\(T\left(r_{i},\theta_{i}\right)\) and \(R\left(r_{0},\theta_{0}\right)\) denote the \(i\)-th (\(i=1,2\)) illuminator and receiver coordinates, respectively; \(P_{k}\left(r_{k},\varphi_{k}\right)\) denotes the \(k\)-th (\(k=1,2,...,K\)) scatterer's coordinates on the target, and the distances from the \(k\)-th scatterer to the \(i\)-th illuminator and receiver are denoted by \(R_{i}^{k}\) and \(R_{0}^{k}\), respectively. Suppose the \(i\)-th illuminator transmits a monochromatic continuous signal [19; 20; 21]  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]  
where \(t\) is the fast time and \(f_{i}\) is the carrier frequency. The received signal reflected from the \(k\)-th scatterer is  
\[s_{i}\left(t,\tau\right)=A_{k}\exp\left[j2\pi f_{i}\left(t-\tau\right)\right], \tag{2}\]  
where \(A_{k}\) is the reflectivity of the \(k\)-th scatterer, and \(\tau\) is the signal propagation time determined by  
\[\tau=\frac{R_{i}^{k}+R_{0}^{k}}{c}, \tag{3}\]  
with \(c\) being the speed of light. In the far-field condition, since \(R_{i}^{k}\gg r_{k}\) and \(R_{0}^{k}\gg r_{k}\), we can use the following approximations  
\[R_{i}^{k} \approx r_{i}-r_{k}\cos(\varphi_{k}-\theta_{i}), \tag{4}\] \[R_{0}^{k} \approx r_{0}-r_{k}\cos(\varphi_{k}-\theta_{0}), \tag{5}\]  
Then, Equation (2) can be rewritten as  
\[s\left(t\right)=A_{k}\exp\left[j2\pi\left(f_{i}t-\frac{r_{i}+r_{0}}{\lambda_{ i}}\right)\right]\times\exp\left\{j\frac{2\pi}{\lambda_{i}}\left[r_{k}\cos( \varphi_{k}-\theta_{i})+r_{k}\cos\left(\varphi_{k}-\theta_{0}\right)\right] \right\}, \tag{6}\]  
with \(\lambda_{i}=c/f_{i}\) being the wavelength. Assuming that the referenced signal is  
\[s_{ref}\left(t\right)=\exp\left[j2\pi\left(f_{i}t-\frac{r_{i}+r_{0}}{\lambda_ {i}}\right)\right], \tag{7}\]  
the instantaneous complex envelope of the returned signal after demodulation and translational motion compensation can be represented by  
\[s=A_{k}\exp\left\{j\frac{2\pi}{\lambda_{i}}\left[r_{k}\cos(\varphi_{k}-\theta _{i})+r_{k}\cos\left(\varphi_{k}-\theta_{0}\right)\right]\right\}. \tag{8}\]  
Note that stationary illuminators and receiver and moving target are assumed. In this case, the rotation velocity can be estimated by the method presented in [19], wherein both a short observation  
Figure 1: Passive radar imaging model. time and corresponding small rotation angle that can be regarded as uniform rotation are assumed. In this paper, we suppose the target moves with a rotation angle \(\Theta\) during the observation time and define the instantaneous rotation angle as \(\varphi_{\Theta}\in\left[\varphi_{k}-\Theta,\varphi_{k}\right]\). Thus, Equation (8) can be rewritten as  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]0  
Let \(\left(\theta_{0}-\theta_{i}\right)/2=\beta\), \(\left(\theta_{0}+\theta_{i}\right)/2=\alpha\), \(\cos\left(\beta\right)/\lambda=\zeta\) and \(\varphi_{\Theta}=\varphi_{k}-\Theta\), (9) be reformulated through the trigonometrical transform into  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]1  
For such a small rotation angle, we have  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]2 \[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]3  
Substituting Equation (11) into Equation (10) yields  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]4  
Let \(h\left(k\right)=A_{k}\exp\left[j4\pi\zeta r_{k}\cos\left(\varphi_{k}-\alpha \right)\right]\) and \(\omega_{k}=-4\pi\zeta r_{k}\sin\left(\varphi_{k}-\alpha\right)\), (12) can be rewritten as  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]5  
Suppose the target consists of \(K\) scatterers, the received signal can then be represented by  
\[s_{i}\left(t\right)=\exp\left(j2\pi f_{i}t\right), \tag{1}\]6  
where \(h\left(k\right)\) and \(\omega_{k}\) are the reflected amplitude and spatial frequency of the \(k\)-th scatterer, respectively.