# A Sparse Bayesian Approach for Forward-Looking Superresolution Radar Imaging
## 2 Signal Model of the Forward-Looking Scanning Radar Image  
Figure 1 shows the geometric model of the forward-looking scanning radar. The antenna sweeps across the forward-looking scene with counterclockwise mechanical movement. The airborne platform moves along the radial direction with a fixed velocity \(v\), and the platform height is \(H\). Suppose some  
targets are distributed in the forward-looking imaging region; target \(P_{i}\) is taken as an example to analyze the range history between the point target and sensor. The original distance between the sensor and target \(P_{i}\) is \(R_{0}\). The angle of the radial direction of \(P_{i}\) is \(\theta_{i}\), and \(\varphi\) is the incident angle of the antenna. The range history between \(P_{i}\) and the antenna is:  
\[R\left(t\right)=\sqrt{R_{0}^{2}+\left(vt\right)^{2}-2R_{0}\cos\theta_{i}\cos \varphi vt}. \tag{1}\]  
which can be expanded into a Taylor series as:  
\[R\left(t\right)=R_{0}-v\cos\theta_{i}\cos\varphi t+\frac{v^{2}\left(1-\cos \theta_{i}^{2}\cos\varphi^{2}\right)}{2R_{0}}t^{2}+O\left(t^{2}\right). \tag{2}\]  
In side-looking mode, \(\theta\) is within a small range around \(90^{\circ}\), and the quadratic terms in (2) play the dominant role. Then, the resolution can be improved significantly via the SAR technique [33; 34]. However, in forward-looking mode, \(\theta\) is within a small range around \(0^{\circ}\), and the linear term in (2) plays the dominant role. In this case, the Doppler bandwidth becomes much smaller, and SAR cannot work for forward-looking imaging. Since the second-order term above is very small and the product of time and velocity is also much smaller than \(R_{0}\), Equation (2) can be approximated as:  
\[R\left(t\right)\cong R_{0}-\cos\theta_{i}\cos\varphi vt. \tag{3}\]  
Equation (3) shows that targets in different cross-range angles of the forward-looking region have different range histories, which result in a different added Doppler phase of different targets. This is why we cannot employ real-valued signal recovery approaches to deal with the forward-looking imaging problem. In forward-looking imaging, we also transmit linear frequency modulation (LFM) signals in order to easily realize high range resolution. For far distance imaging regions, the signal received after the antenna scans in the forward-looking region is:  
Figure 1: Geometric model of forward-looking scanning radar. \[g(t,\tau)=\sum_{k=1}^{K}\sigma_{k}a(\theta_{k},\tau)rect\left(\frac{\tau-\frac{2R \left(t\right)}{c}}{T}\right)\exp\left(-j\frac{4\pi}{\lambda}R\left(t\right) \right)\exp\left(j\pi K_{r}[\tau-\frac{2R\left(t\right)}{c}]^{2}\right) \tag{4}\]  
where \(K\) is the number of samples in the cross-range dimension that can be calculated by \(K=\phi\cdot PRF/\omega\), wherein \(\phi\) is the scanning region, \(\omega\) is the sweep speed, and \(PRF\) is the pulse repetition frequency. \(rect\left(\cdot\right)\) denotes the unit rectangular function, and \(T\) denotes the signal duration. \(rect\left(\frac{t}{T}\right)=1\) when \(\left|t\right|\leq T\). \(t\) and \(\tau\) represent the fast and slow time in the range dimension and cross-range dimension, respectively. In radar signal processing, operations that are applied to data from a single pulse occur on the shortest time scale (often referred to as fast time), and the sampling interval between pulses is referred to as slow time. \(f_{c}\) denotes carrier frequency, and \(K_{r}\) denotes the chirp rate. \(a(\cdot)\) is the antenna pattern modulation function, and \(\theta_{k}\) denotes the antenna pointing direction of the k-th signal. Because the transmitting signals are LFM signals, high range resolution can be realized by matched filtering. After this operation, the received signal becomes:  
\[g(t,\tau)=\sum_{k=1}^{K}\sigma_{k}a(\theta_{k},\tau)\sin c\left[B\left(\tau- \frac{2R\left(t\right)}{c}\right)\right]\exp\left(-j\frac{4\pi}{\lambda}R \left(t\right)\right). \tag{5}\]  
where \(B\) is the bandwidth of transmitted signals.