# A Sparse Bayesian Approach for Forward-Looking Superresolution Radar Imaging
## 2 Signal Model of the Forward-Looking Scanning Radar Image
The range migration adjusting function is constructed to eliminate the effect of movement. The two-dimensional received signal is then:  
\[g(t,\tau)=\sum_{k=1}^{K}\sigma_{k}a(\theta_{k},\tau)\sin c\left[B\left(\tau- \frac{2R_{0}}{c}\right)\right]\exp\left(-j\frac{4\pi}{\lambda}R\left(t\right) \right). \tag{6}\]  
The preprocessed signal has high range resolution and low cross-range resolution. Meanwhile, the traditional matched filtering technique cannot be applied to the cross-range dimension of the forward-looking region because of the limited Doppler bandwidth. On the other hand, the cross-range received signal can be regarded as the convolution of the target scattering coefficient and antenna pattern; therefore, we can employ deconvolution approaches to improve cross-range resolution. For mathematical simplicity, the discrete received signal is rearranged and written as the matrix and vector form:  
\[\mathbf{g}=\mathbf{A}\mathbf{x}+\mathbf{n}=\left[ \begin{array}{ccc}A_{M \times K}&&\\ &\ddots&\\ &&A_{M\times K}\end{array} \right]\left[ \begin{array}{ccc}&x\left(1,\,1 \right)\\ &x\left(1,\,2\right)\\ &\vdots&\\ &x\left(1,\,K\right)\\ &\vdots&\\ &x\left(N,\,K\right)\end{array} \right]+\mathbf{n} \tag{7}\]  
where \(\mathbf{g}=\left[g\left(1,\,1\right),g\left(1,\,2\right),\cdots,g\left(1,\,M \right),\cdots,g\left(N,\,M\right)\right]^{T}\) represents the measurement range profiles, which were rearranged in the cross-range dimension with size \(NM\times 1\). \(N\) and \(M\) are the discrete sampling numbers of the received signal in range dimension and cross-range dimension, respectively. \(\mathbf{x}=\left[x\left(1,\,1\right),x\left(1,\,2\right),\cdots,x\left(1,\,K \right),\cdots,x\left(N,\,K\right)\right]^{T}\) represents the unknown scene amplitude profiles, which were rearranged in the cross-range dimension with size \(NK\times 1\). \(\mathbf{n}\) is the noise vector with dimension \(NM\times 1\), which satisfies the complex Gaussian distribution. The matrix \(\mathbf{A}\) is the measurement matrix with size \(NM\times NK\), which was composed of the convolution matrix \(A_{M\times K}\). \(A_{M\times K}=\left[\mathbf{a}_{1},\,\,\mathbf{a}_{2},\,\,\cdots,\,\,\mathbf{ a}_{K}\right]\) is the measurement matrix of the single cross-range profile, which is given by the following equation. The rows of \(A_{M\times K}\) correspond to the different discrete sampling times. The columns of \(A_{M\times K}\) express the weighted relationship of different targets and the antenna at the same time. The changes of added Doppler phase in different cross-range angles are shown in the measurement matrix. \(a_{n_{b}}\) in \(A_{M\times K}\) is the antenna pattern weighted coefficient, and \(N_{b}=\theta_{b}/\left(\omega\cdot PRI\right)\) is the sampling number of one beamwidth, where \(\theta_{b}\) is the beamwidth and \(PRI\) is the pulse recurrence interval. For the entire imaging region, the relationship of the column and row of \(A_{M\times K}\) is:  
\[M=K+N_{b}-1. \tag{8}\]  
As shown by the model can be represented by Equation (7), in addition to the Doppler history, the antenna pattern is also considered in forward-looking radar imaging. Different from the SAR imaging, the antenna pattern plays the key role to reconstruct the target distribution, and the Doppler term just makes the signal model more accurate when compared with the conventional convolution model [13; 14; 16]. Although the resolution of the superresolution method for forward-looking imaging will be lower than the SAR technique for side-looking imaging, the superresolution method is an effective way to improve the resolution of forward-looking imaging.