# A High-Resolution, Wide-Swath SAR Imaging System Based on Tandem SAR Satellites
## 3 Signal Model and Imaging Algorithm
### Minimum-Energy-Based Imaging Algorithm
#### 3.2.5 Minimum-Energy-Based Estimation  
To obtain the minimum weighted energy solution required in Equation (8), we define a new norm in the space consisted of band-limited sequences as  
\[\left\|\hat{\sigma}_{i}\right\|_{\text{new}}=\left\langle\hat{\sigma}_{i},\mathbf{ Q}^{-1}\cdot\hat{\sigma}_{i}\right\rangle \tag{13}\]  
The validation of the new norm can be found in Appendix A. Then, the solution with minimum weighted energy is the solution with the smallest new norm. According to the projection theorem in [25], \(\hat{\sigma}_{i}\) must be orthogonal in \(\left\langle\cdot,\cdot\cdot\right\rangle_{\text{new}}\) to \(\aleph(\mathbf{A})\), where \(\aleph(\mathbf{A})\) is the null space of \(\mathbf{A}\) and \(\left\langle\cdot\cdot\cdot\right\rangle_{\text{new}}\) is the new inner product corresponding to \(\left\|\cdot\right\|_{\text{new}}\). So, we obtain  
\[\hat{\sigma}_{i}\in\Re(\mathbf{A}^{*}) \tag{14}\]  
where \(\Re(\mathbf{A}^{*})\) is the range space of \(\mathbf{A}^{*}\), and \(\mathbf{A}^{*}\) is the adjoint operator of \(\mathbf{A}\), which can be calculated by  
\[\mathbf{A}^{*}=\mathbf{Q}\mathbf{A}^{\text{H}} \tag{15}\]  
The specific derivation of Equation (15) can be found in the appendix of [26]. According to Equations (14) and (15), \(\hat{\sigma}_{i}\) can be expressed as  
\[\hat{\sigma}_{i}=\mathbf{Q}\mathbf{A}^{\text{H}}\cdot\mathbf{v} \tag{16}\]  
where \(\mathbf{v}\) is a \(N_{a}\times 1\) vector. According to \(\mathbf{S}_{i}=\mathbf{A}\hat{\sigma}_{i}\) in Equation (8), \(\mathbf{v}\) can be calculated by  
\[\mathbf{v}=\left(\mathbf{A}\mathbf{Q}\mathbf{A}^{\text{H}}\right)^{-1}\mathbf{ S}_{i} \tag{17}\]  
Then, the \(\hat{\sigma}_{i}\) can be obtained by  
\[\hat{\sigma}_{i}=\mathbf{Q}\mathbf{A}^{\text{H}}\Big{(}\mathbf{A}\mathbf{Q} \mathbf{A}^{\text{H}}\Big{)}^{-1}\mathbf{S}_{i} \tag{18}\]  
It can be divided into the following two steps:  
\[\hat{\sigma}_{\text{tem}}=\Big{(}\mathbf{A}\mathbf{Q}\mathbf{A}^{\text{H}} \Big{)}^{-1}\mathbf{S}_{i} \tag{19}\]  
\[\hat{\sigma}_{i}=\mathbf{Q}\cdot\mathbf{A}^{\text{H}}\hat{\sigma}_{\text{tem}} \tag{20}\]  
Step one normalizes the amplitude of the targets in \(\mathbf{S}_{i}\) as shown in Equation (19), and step two is to image the normalized result of step one as shown in Equation (20). Due to the  
Figure 8: Illustration of the nearest neighbor interpolation algorithm, where \((x,y)\) is the 2D coordinate of pixels in the original image.  
inevitable model error and noise in the real data, we modify Equations (19)-(21) to achieve more robust imaging performance.  
\[\hat{\sigma}_{\text{tem}}=\mathbf{Q}_{s}{}^{-1}\mathbf{S}_{i} \tag{21}\]  
where \(\mathbf{Q}_{s}\) is a diagonal matrix, its diagonal elements are \(\left\{|S_{i,1}|^{\rho},|S_{i,2}|^{\rho},\cdots,|S_{i,N_{k}}|^{\rho}\right\}\), and \(|S_{i,n}|\) is the absolute value of the \(n\)th element in \(\mathbf{S}_{i}\), \(\rho\) is the normalization factor and this paper sets \(\rho=0.5\). At the same time, Equation (20) can be modified to an iterative form, as shown in Equation (22).  
\[\hat{\sigma}_{i,j}=\mathbf{Q}_{j-1}\cdot\mathbf{A}^{\text{H}}\hat{\sigma}_{ \text{tem}} \tag{22}\]  
where \(\hat{\sigma}_{i,j}\) is the \(j\)th reconstruction result and \(\hat{\sigma}_{i,0}=\sigma_{d,i}\), \(\sigma_{d,i}\) is the \(i\)th column of the coarse observation of Sat-1, \(\mathbf{Q}_{j-1}=\text{diag}\Big{(}\hat{\sigma}_{i,j-1}\cdot\hat{\sigma}_{i,j -1}^{*}\Big{)}\). This paper sets the iteration order to 2, and the detailed steps are listed in Table 1.