## 3 Processing Flow
### Phase Denoise Model  
Ideally, the phase error caused by the frequency deviation of USOs can be well eliminated by synchronization phase compensation through Equation (7). However, the synchronization signals may suffer from the thermal noise, free space attenuation, multipath effect, radio frequency interference, etc., in the link, and the synchronization phase will be disturbed by the noise. Consider a given measurement phase \(\mathbf{Y}\in\mathbb{R}^{N}\), obtained from the clean phase \(\mathbf{X}\in\mathbb{R}^{N}\) by contamination of the measurement noise \(\mathbf{V}\in\mathbb{R}^{N}\). We assume that the noise \(\mathbf{V}\) is zero-mean additive Gaussian white noise with a known STD \(\sigma\), i.e.,  
\[\mathbf{Y}=\mathbf{X}+\mathbf{V},\mathbf{V}\sim N\Big{(}\mathbf{0},\sigma^{2} \mathbf{I}\Big{)}. \tag{10}\]  
Equation (10) is the synchronization phase degradation model. The phase denoise is its inverse problem, that is, to obtain the approximation \(\mathbf{\hat{X}}\) of the unknown original phase \(\mathbf{X}\). However this inverse problem is ill-posed, which means the solution does not exist or is not unique. Fortunately, the advent of compressed sensing provides a newly developed theoretical framework for information acquisition and signal processing, which can alleviate the ill-posedness of this kind of problem. A synchronization phase segment \(\mathbf{y}_{i}=\mathbf{R}_{i}\mathbf{Y}\) with a length of \(n\) is extracted from the measured phase \(\mathbf{Y}\), where \(\mathbf{R}_{i}\in\mathbb{R}^{n\times N}\) is the matrix that extracts the \(i\)th phase segment from the measured phase. Define a known over-complete dictionary \(\mathbf{D}\in\mathbb{R}^{n\times k}\), where \(k\) denotes the number of atoms in the dictionary, and \(k>n\). According to the theory of compressed sensing, each segment of the original phase can be represented sparsely by the dictionary \(\mathbf{D}\) as  
\[\mathbf{\hat{z}}_{i}=\operatorname*{arg\,min}_{\mathbf{\hat{z}}_{i}}\left\| \mathbf{\alpha}_{i}\right\|_{0}\text{ s.t. }\left\|\mathbf{y}_{i}-\mathbf{D}\mathbf{ \alpha}_{i}\right\|_{2}^{2}\leq\varepsilon^{2} \tag{11}\]  
where \(\left\|\cdot\right\|_{0}\) represents the \(L^{0}\)-norm, which means the number of non-zero elements in the vector. \(\left\|\cdot\right\|_{2}\) represents the \(L^{2}\)-norm of the vector. \(\mathbf{\hat{z}}_{i}\in\mathbb{R}^{k}\) is the sparse representation vector for \(\mathbf{y}_{i}\) with an accuracy of \(\varepsilon\). It is a Non-deterministic Polynomial-time (NP) hard problem to obtain such a sparse representation. To solve it directly, all combinations need to be enumerated. According to Reference [38], when the solution is sparse enough, the non-convex optimization problem in Equation (11) can be transformed into a convex \(L^{1}\)-norm optimization problem as  
\[\mathbf{\hat{z}}_{i}=\operatorname*{arg\,min}_{\mathbf{\hat{z}}_{i}}\left\| \mathbf{\alpha}_{i}\right\|_{1}\text{ s.t. }\left\|\mathbf{y}_{i}-\mathbf{D}\mathbf{ \alpha}_{i}\right\|_{2}^{2}\leq\varepsilon^{2}. \tag{12}\]  
The orthogonal matching pursuit (OMP) algorithm and its improved ones are utilized to find the sparse solution of the above problem [42]. When the solution is quite sparse, the OMP algorithm can obtain quite an accurate solution efficiently. By maximum a posteriori (MAP) estimation, denoising of \(\mathbf{Y}\) is equivalent to the energy minimization problem as  
\[\left\{\mathbf{\hat{z}}_{i},\mathbf{\hat{X}}\right\}=\operatorname*{arg\,min }_{\mathbf{\hat{z}}_{i},\mathbf{X}}\lambda\left\|\mathbf{Y}-\mathbf{X}\right\| _{2}^{2}+\sum_{i}\left\|\mathbf{D}\mathbf{\alpha}_{i}-\mathbf{R}_{i}\mathbf{X }\right\|_{2}^{2}+\sum_{i}\mu_{i}\left\|\mathbf{\alpha}_{i}\right\|_{0} \tag{13}\]  
where the first term demands the proximity between the measured phase \(\mathbf{Y}\) and its denoised (and unknown) version \(\mathbf{X}\). \(\lambda\) represents the weight of the degree of proximity. The greater the \(\lambda\), the closer the estimated phase to the measured phase will be.