# A High-Resolution, Wide-Swath SAR Imaging System Based on Tandem SAR Satellites
## 3 Signal Model and Imaging Algorithm
### Signal Model of the Sat-2  
According to the SAR principle described in [20], the original SAR echo of Sat-2 can be expressed as  
\[S_{0}(\eta,\tau)=A_{0}W_{a}(\eta-\eta_{c})W_{r}\bigg{(}\tau-\frac{2R(\eta)}{c} \bigg{)}\exp\bigg{\{}-j\frac{4\pi}{\lambda}R(\eta)\bigg{\}}\cdot\exp\bigg{\{} j\pi k_{r}\bigg{(}\tau-\frac{2R(\eta)}{c}\bigg{)}^{2}\bigg{\}} \tag{3}\]  
where \(\tau\) and \(\eta\) denote the range and azimuth time, respectively, and \(\eta_{c}\) is the beam center crossing time. \(R(\eta)\) is the slant distance between the observed target and the platform at azimuth time \(\eta\), \(c\) is the speed of light, \(k_{r}\) is the pulse modulation frequency, \(A_{0}\) is a complex constant, and \(W_{a}(\cdot)\) and \(W_{r}(\cdot)\) are the azimuth and range antenna mode, respectively. \(M\) is the number of targets on the observed scene. After range compression, Equation (3) can be expressed as  
\[S_{1}(\eta,\tau)=A_{0}W_{a}(\eta-\eta_{c})\text{sin}\cbigg{[}B_{r}\bigg{(}\tau -\frac{2R(\eta)}{c}\bigg{)}\bigg{]}\exp\bigg{\{}-j\frac{4\pi}{\lambda}R(\eta) \bigg{\}} \tag{4}\]  
where \(B_{r}\) is the bandwidth of the transmitted pulse. \(S_{1}(\eta,\tau)\) can also be expressed in the matrix form as  
\[\mathbf{S}_{i}=\mathbf{A}\cdot\boldsymbol{\sigma}_{i} \tag{5}\]  
where \(\mathbf{S}_{i}\) and \(\boldsymbol{\sigma}_{i}\) are the observation vector and scene vector of the \(i\)th range gate, respectively. \(\boldsymbol{\sigma}_{i}=[\sigma_{i,1},\sigma_{i,2},\cdots,\sigma_{i,K}]^{ \text{T}}\) and \(\sigma_{i,k}\) is the equivalent backscatter coefficient of the \(k\)th pixel, \(K\) is the number of pixels in the \(i\)th range gate. \(\mathbf{A}=[\mathbf{A}_{1},\mathbf{A}_{2},\cdots,\mathbf{A}_{k},\cdots, \mathbf{A}_{K}]\) is the sensing matrix with  
\[\mathbf{A}_{k}=W_{a}(\boldsymbol{\eta}-\eta_{c,k})\cdot\exp\bigg{\{}-j\frac{4 \pi}{\lambda}R(\boldsymbol{\eta},k)\bigg{\}}\]  
where \(\boldsymbol{\eta}=[\eta_{1},\eta_{2},\cdots,\eta_{N_{a}}]^{\text{T}}\) is the azimuth time vector, \(N_{a}\) is the total number of sampling points in the azimuth direction, \(\eta_{c,k}\) is the beam center crossing time of the \(k\)th pixel, and \(R(\boldsymbol{\eta},k)\) is the slant distance vector between the \(k\)th pixel and the platform. According to Equation (5), the traditional imaging algorithms [21; 22] based on matched filtering can be expressed as  
\[\hat{\boldsymbol{\sigma}}_{i}=\mathbf{A}^{\text{H}}\cdot\mathbf{S}_{i} \tag{6}\]  
Figure 4: Traditional scanning method. Figure 5: Proposed scanning method. where \(\mathbf{A}^{\mathrm{H}}\) is the conjugate transpose matrix of \(\mathbf{A}\). Since the received echo of Sat-2 is seriously downsampled, severe ambiguity energy occurs in the imaging result of Equation (6). The CS-based SAR system introduces the sparsity prior to the scene reconstruction to compensate for the information lost by downsampling, and then selects the solution with the best sparsity as the estimation of the observed scene, as shown in (7). \[\hat{\sigma}_{i}=\underset{\hat{\sigma}_{i}}{\mathrm{argmin}}\|\hat{\sigma}_{i} \|_{0}\ \mathbf{s.t.}\ \mathbf{S}_{i}=\mathbf{A}\hat{\sigma}_{i} \tag{7}\]  
However, the estimation accuracy of (7) depends on whether the scene is sparse or whether it can be sparsely represented accurately. In order to achieve better imaging performance, this paper modifies the traditional minimum-energy criterion [23] introducing the coarse observation as prior information and constructs the optimization problem as  
\[\hat{\sigma}_{i}=\underset{\hat{\sigma}_{i}}{\mathrm{argmin}}\Big{\langle} \hat{\sigma}_{i},\mathbf{Q}^{-1}\cdot\hat{\sigma}_{i}\Big{\rangle}\ \mathbf{s.t.}\ \mathbf{S}_{i}=\mathbf{A}\hat{ \sigma}_{i} \tag{8}\]  
where \(\mathbf{Q}=\mathrm{diag}\Big{(}\sigma_{d,i}\cdot\sigma_{d,i}^{*}\Big{)}\) is a \(K\times K\) matrix and \(\sigma_{d,i}\) is the amplitude envelope of the imaging result of the \(i\)th range gate on the ground, which can be obtained by interpolating the imaging result of Sat-1.