## 3 Processing Flow
### Phase Denoise Model
The second term demands that each phase segment can be represented up to a bounded error by coefficients\(\mathbf{\alpha}_{i}\), with respect to the dictionary \(\mathbf{D}\). The last term demands that the coefficients of each phase should be as sparse as possible. \(\mu_{i}\) is the weight of corresponding synchronization phase segment. The OMP algorithm gather one atom at a time, and stop when the error \(\|\mathbf{y}_{i}-\mathbf{D}\mathbf{\alpha}_{i}\|_{2}^{2}\) goes below \(\varepsilon^{2}\) or the number of iterations reaches the set sparsity threshold \(m\). Therefore, after obtaining all sparse representations in Equation (12), Equation (13) is transformed into the following optimization problem as  
\[\mathbf{\hat{X}}=\operatorname*{arg\,min}_{\mathbf{X}}\lambda\|\mathbf{Y}- \mathbf{X}\|_{2}^{2}+\sum_{i}\|\mathbf{D}\mathbf{\alpha}_{i}-\mathbf{R}_{i}\mathbf{ X}\|_{2}^{2}. \tag{14}\]  
The objective function above is a typical quadratic term, and its derivative to \(\mathbf{X}\) can be written as  
\[\frac{\partial f(\mathbf{X})}{\partial\mathbf{X}}= 2\lambda(\mathbf{X}-\mathbf{Y})+2\sum_{i}\left(\mathbf{R}_{i}^{ \top}\mathbf{R}_{i}\mathbf{X}-\mathbf{R}_{i}^{\top}\mathbf{D}\mathbf{\alpha}_{i}\right)\] \[= 2\Bigg{(}\lambda\mathbf{I}+\sum_{i}\mathbf{R}_{i}^{\top} \mathbf{R}_{i}\Bigg{)}\mathbf{X}-2\Bigg{(}\lambda\mathbf{Y}+\sum_{i}\mathbf{ R}_{i}^{\top}\mathbf{D}\mathbf{\alpha}_{i}\Bigg{)}. \tag{15}\]  
So the analytical solution of Equation (14) can be obtained by making Equation (15) equal to \(\mathbf{0}\), i.e.,  
\[\mathbf{\hat{X}}=\Bigg{(}\lambda\mathbf{I}+\sum_{i}\mathbf{R}_{i}^{\top} \mathbf{R}_{i}\Bigg{)}^{-1}\Bigg{(}\lambda\mathbf{Y}+\sum_{i}\mathbf{R}_{i}^{ \top}\mathbf{D}\mathbf{\alpha}_{i}\Bigg{)}. \tag{16}\]  
As an important parameter of the algorithm, the weight of proximity \(\lambda\) is negatively correlated with the standard deviation of the noise \(\mathbf{V}\). The higher the noise level, the lower the \(\lambda\) to reduce the similarity between \(\mathbf{\hat{X}}\) and \(\mathbf{Y}\) for a better denoise effect. According to experience, \(\lambda=0.01/\sigma\). In fact, the standard deviation \(\sigma\) of the noise is affected by the SNR error, the interpolation error, and the aliasing error. The LT-1 system has a high synchronization frequency, so the interpolation and aliasing errors can be ignored. \(\sigma\) is mainly determined by the SNR error which is given by [21]  
\[\frac{1}{2}\sigma^{2}=\frac{1}{4f_{\text{syn}}\text{SNR}}\int\limits_{-f_{ \text{syn}}/2}^{f_{\text{syn}}/2}\big{|}H_{\text{syn}}(f)H_{\text{az}}(f) \big{|}^{2}df \tag{17}\]  
where \(f_{\text{syn}}\) denotes the synchronization frequency. \(H_{\text{syn}}(f)\) is the noise spectrum. For additive Gaussian white noise, \(H_{\text{syn}}(f)=1\). \(H_{\text{az}}(f)\) is the azimuth transfer function depending on the SAR imaging, which can be expressed as [21]  
\[H_{az}(f)=\frac{\sin(\pi T_{a}f)}{\pi T_{a}f} \tag{18}\]  
where \(T_{a}\) is synthetic aperture time. SNR can be estimated from the synchronization signal after pulse compression. Therefore, the value of the parameter \(\lambda\) can refer to Equation (17). After applying CS and MAP estimation, a high-precision synchronization phase can be obtained. The above analysis assumes that the over-complete dictionary \(\mathbf{D}\) for sparse coding is already known. In the next subsection, the dictionary \(\mathbf{D}\) will be trained by the K-SVD algorithm, and the training strategy is described in detail.