## 5 At-Sea Experiment  
To evaluate the real-world performance of the proposed nmADMM algorithm for UWA channel estimation, we conducted an at-sea experiment. The data were collected in the shallow waters of Wuyuan Bay, Xiamen, China, in July 2021 (GPS: 24\({}^{\circ}\)32\({}^{\prime}\)10\({}^{\prime\prime}\)N, 118\({}^{\circ}\)11\({}^{\prime}\)31\({}^{\prime\prime}\)E). Figure 11a depicts the location and deployment of the experiment. The transmitter and receiver, separated by approximately 1 km, were both anchored 2 m deep near the coastline. During the experiment, CP-OFDM signals with subcarriers modulated by Quadrature Phase Shift Keying (QPSK) symbols were transmitted. The center frequency of the system is 21 kHz, with a sampling frequency of 96 kHz. In the OFDM configuration, 256 subcarriers with 23.4 Hz spacing, an FFT size of 4096, and a CP length of 3840 were utilized. Each OFDM symbol had a length of 82.7 ms. Figure 11b presents the frame \begin{table}
\begin{tabular}{l c c c c c c} \hline  & \multicolumn{3}{c}{**INR = 15 dB**} & \multicolumn{3}{c}{**INR = 25 dB**} \\ \cline{2-7} \(\boldsymbol{q}\) & **0.01** & **0.005** & **0.001** & **0.01** & **0.005** & **0.001** \\ \hline Ground Truth & 2.10 & 0.80 & 0.30 & 18.80 & 8.85 & 2.50 \\ OMP [15] & 36.25 & 34.95 & 10.25 & 48.60 & 47.20 & 35.85 \\ FISTA [23] & 27.85 & 11.65 & 0.50 & 55.35 & 45.80 & 8.35 \\ ADMM [25] & 27.65 & 13.55 & 0.55 & 55.25 & 42.85 & 8.10 \\ mADMM [28] & 5.70 & 4.30 & 3.35 & 23.90 & 15.15 & 8.30 \\ nmADMM & 5.25 & 2.05 & 0.45 & 22.15 & 13.50 & 3.85 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of BER performance across different algorithms under INR conditions of 15 and 25 dB with impulsive noise occurrence probabilities of \(q=0.01\), 0.005, and 0.001. Figure 10: Iteration curves of the objective function for mADMM and nmADMM in DD channel estimation: (**a**) AWGN environment and (**b**) impulsive noise environment. format of the transmitted signal, where each frame of the signal consists of one Hyperbolic Frequency Modulation (HFM) preamble signal followed by OFDM symbols. A received signal sample is shown in Figure 12a. As the experiment site is near the coast, various industrial and human activities and the flow noise resulting from water currents and turbulence make the received signal contaminated by strong impulsive noise. Before channel estimation, pre-processing is conducted, including large impulsive noise reduction via peak detection and threshold clipping. Figure 12b shows the signal sample after removing large impulsive noise. Following frame synchronization through matched filtering with HFM preamble signals, large-scale Doppler spread is compensated for by resampling. The Carrier Frequency Offset (CFO) resulting from residual Doppler effects is estimated and mitigated through a one-dimensional search strategy, as detailed in [47]. The pre-processed signal is then forwarded to various channel estimators for evaluations. Consistent with the hardware configurations employed in previous numerical simulations, all algorithms are run on MATLAB 2023a, Windows 10, using a Dell-manufactured desktop PC with an Intel Core i5 2.30 GHz CPU and 16 GB RAM. We adopted block-type, pilot-based channel estimation using the first OFDM symbol of each frame. The channel order is set to the maximum value that the estimator can take, which corresponds to the length of the CP (i.e., \(N=3840\)). Pilot symbols carried by \(M=256\) subcarriers are used for channel estimation, where \(M\ll N\) indicates the problem is highly ill-defined. In addition to classic OMP, FISTA, and ADMM algorithms, we incorporate recently introduced channel estimation methods for OFDM systems, including the Adaptive OMP (A-OMP) algorithm [14] and the Fast Marginal Likelihood Maximization (FM) based Sparse Bayesian Learning (SBL) algorithm [48], for comparative analysis. Figure 13 presents the estimated channel via different algorithms.