## 6 Conclusions  
In this work, we present a robust channel estimation method for UAC in impulsive noise environments. Our approach redefines channel estimation as an \(\ell_{1}-\ell_{1}\) optimization problem, leveraging UWA channel sparsity and addressing impulsive noise outliers. Using the ADMM framework, we decompose this convex but non-differentiable problem into two subproblems, each efficiently solved using the APG method. We also introduce a non-monotone backtracking line search strategy to adaptively adjust the step-size parameter, avoiding the need for complex evaluation of the Lipschitz constant.  
Figure 14: Constellation diagrams of raw received symbols and equalization outputs based on various channel estimates: (**a**) raw received symbols, (**b**) OMP, (**c**) A-OMP, (**d**) FISTA, (**e**) FM-SBL, (**f**) ADMM, and (**g**) nmADMM.  
Figure 15: Convergence analysis of the nmADMM algorithm: Iteration curves of (**a**) the objective function and (**b**) the primal residue (**top**) and the dual residue (**bottom**).  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Method** & **\# Iter.** & **Runtime (s)** & **SER (\%)** \\ \hline OMP [15] & 74 & 0.3152 & 10.68 \\ A-OMP [14] & 85 & 0.56 & 7.45 \\ FISTA [23] & 24 & 1.0852 & 5.87 \\ FM-SBL [48] & 57 & 9.73 & 5.83 \\ ADMM [25] & 47 & 13.1543 & 6.61 \\ nmADMM & 32 & 1.4681 & 4.48 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance comparison for channel estimation in the at-sea experiment including average number of iterations, runtime for single estimation, and SER.  
The proposed nmADMM algorithm, with a complexity of \(\mathcal{O}(MN)\) and devoid of matrix inversion or multiplication operations, is tested across various estimation tasks and communication schemes. Numerical simulations show that only a slight NMSD increase of 0.73 dB occurs in impulsive noise environments, compared to AWGN conditions, at an SNR of 10 dB and an INR of 30 dB for CIR estimation. Compared with its monotone counterpart, the mADMM algorithm, the proposed algorithm demonstrates faster convergence speed and higher estimation accuracy for both CIR and DD spread function estimations across various SNR and INR levels. At-sea experiments further validate that the nmADMM algorithm outperforms the OMP, A-OMP, FISTA, ADMM, and FM-SBL algorithms, achieving the lowest raw SER with simple MMSE equalization at 4.48%. Consistently, the nmADMM algorithm converges within tens of iterations, proving its robustness in both AWGN and impulsive noise environments.  
Conceptualization, T.T., K.Y. and F.-Y.W.; methodology, T.T., K.Y. and F.-Y.W.; software, T.T.; validation, T.T., K.Y., F.-Y.W. and Y.Z.; formal analysis, T.T. and Y.Z.; investigation, T.T. and F.-Y.W.; resources, K.Y. and F.-Y.W.; data curation, T.T.; writing--original draft preparation, T.T.; supervision, K.Y., F.-Y.W. and Y.Z.; project administration, K.Y.; funding acquisition, K.Y. and F.-Y.W. All authors have read and agreed to the published version of the manuscript.  
This work was supported in part by the National Natural Science Foundation of China under Grant 62171369 and in part by the Key Program of the National Natural Science Foundation of China under Grant 52231013.  
Dataset will be made available on request from the authors.  
The authors would like to thank [PERSON] and his team at the College of Ocean and Earth Science, Xiamen University, for their invaluable assistance in setting up the experiment at Wuyuan Bay.  
The authors declare no conflicts of interest.