## 3 Details of the Proposed Algorithm
### Update of Primal Variable: \(\mathbf{x}\)  
Upon examining (16) and (17) against the standard form of the proximal mapping defined in (19), it is apparent that while the \(\mathbf{z}\)-subproblem aligns with the proximal mapping of function \(g_{2}(\mathbf{z})\), the \(\mathbf{x}\)-subproblem does not directly correspond due to the existence of matrix \(\boldsymbol{\Phi}\). To make the \(\mathbf{x}\)-subproblem tractable, we approximate \(f_{1}(\mathbf{x})\) using its first-order Taylor expansion at the point \(\mathbf{x}^{(k)}\)--the estimate from the preceding iteration:  
\[\tilde{f}_{1}(\mathbf{x})\triangleq f_{1}(\mathbf{x}^{(k)})+\langle\
abla f_{ 1}(\mathbf{x}^{(k)}),\mathbf{x}-\mathbf{x}^{(k)}\rangle+\frac{1}{2t_{x}}|| \mathbf{x}-\mathbf{x}^{(k)}||_{2}^{2}, \tag{21}\]  
where  
\[\
abla f_{1}(\mathbf{x}^{(k)})=\rho\boldsymbol{\Phi}^{H}(\mathbf{z}^{(k)}+ \boldsymbol{\Phi}\mathbf{x}^{(k)}-\mathbf{y}+\gamma^{(k)}/\rho). \tag{22}\]  
Replacing \(f_{1}(\mathbf{x})\) with \(\tilde{f}_{1}(\mathbf{x})\) and omitting the constant terms associated with \(\mathbf{x}^{(k)}\), (16) is reduced to an optimization problem equivalent to the proximal mapping of \(f_{2}(\mathbf{x})\) as  
\[\mathbf{x}^{(k+1)}=\text{prox}_{f_{2},t_{x}}\Big{(}\mathbf{x}^{(k)}-t_{x} \
abla f_{1}(\mathbf{x}^{(k)})\Big{)}, \tag{23}\]  
where the step-size parameter \(t_{x}\) can be set as the reciprocal of the Lipschitz constant for \(f_{1}(\mathbf{x})\), i.e., \(t_{x}=1/\lambda_{\text{max}}(\boldsymbol{\Phi}^{H}\boldsymbol{\Phi})\). \(\lambda_{\text{max}}(\cdot)\) represents the maximum eigenvalue of the given matrix. The closed-form solution of the proximal operator for \(\ell_{1}\)-norm allows (23) to be directly computed by  
\[\mathbf{x}^{(k+1)}=S_{t_{x}/\rho}\Big{(}\mathbf{x}^{(k)}-t_{x}\boldsymbol{\Phi }^{H}(\boldsymbol{\Phi}\mathbf{x}^{(k)}+\mathbf{z}^{(k)}-\mathbf{y}+\gamma^{( k)}/\rho)\Big{)} \tag{24}\]  
where \(S_{t_{x}/\rho}(\cdot)\) is the soft-thresholding function shown in (20) with the threshold parameter \(\alpha=t_{x}/\rho\). For practical applications, particularly in UWA channel estimation characterized by large delay and Doppler spread, evaluation of \(\lambda_{\max}(\mathbf{\Phi}^{H}\mathbf{\Phi})\) can be computationally intensive. In such cases, a backtracking line search strategy is a more flexible and efficient way to adjust the step-size parameters. In [28], we implemented a monotone update scheme for solving the \(\mathbf{x}\)-subproblem as detailed in Algorithm 1, where \(J(\mathbf{x})\) denotes the objective function given in (11). ```
1:Set \(\mathbf{x}^{(0)}=\mathbf{0}\), \(t_{0}>0\), and \(\eta_{x}>1\). 2:Get \(\mathbf{x}^{(k-1)}\), \(\mathbf{z}^{(k-1)}\), \(\gamma^{(k-1)}\) and \(J(\mathbf{x}^{(k-1)})\) from previous iteration. 3:Compute the gradient \(\
abla f_{1}(\mathbf{x}^{(k-1)})\) using (22). 4:Backtracking Line Search: find the minimum number of iterations \(l^{(k)}\) such that with \(t_{x}^{(k)}=t_{0}/\eta_{x}^{l^{(k)}}\) and \(\mathbf{x}_{l^{(k)}}=S_{t_{x}^{(k)}/\rho}\Big{(}\mathbf{x}^{(k-1)}-t_{x}^{(k)} /\rho\cdot\
abla f_{1}(\mathbf{x}^{(k-1)})\Big{)}\) \[J(\mathbf{x}_{l^{(k)}})\leqslant J(\mathbf{x}^{(k-1)}).\]
5:Set \(\mathbf{x}^{(k)}=\mathbf{x}_{l^{(k)}}\).