## 5 At-Sea Experiment
The monotone strategy-based mADMM algorithm is not included here due to its slow convergence rate with high mutual coherence measurement matrices, as demonstrated in Section 4.2. Regularization factors for FISTA and ADMM were set to \(\lambda=0.01\)\(\lambda_{\infty}\), and for the nmADMM, \(\tau=1/(0.02\lambda_{\infty})\). We configured the tolerance parameters of the stopping criteria to \(\epsilon_{\text{abs}}=10^{-2}\), \(\epsilon_{\text{rel}}=10^{-1}\), and \(\epsilon_{\text{fista}}=10^{-3}\). The convergence tolerance parameter for the FM-SBL algorithm is set to \(\epsilon_{\text{fm-sbl}}=10^{-4}\). The prior sparsity information required by the OMP algorithm is derived from coarse channel estimation using matched filtering on  
Figure 11: Experiment configuration: (**a**) Experiment site map at Wuyuan Bay, Xiamen City, China, and (**b**) frame format of the transmitted signal. Figure 12: Sample of the received signal: (**a**) Raw signal and (**b**) signal after pre-processing to eliminate large impulsive noise. A-OMP algorithm does not require prior sparsity information for the channel. We set its stopping criterion, the power ratio of the residual to the received symbols, at \(5\times 10^{-2}\) through empirical evaluations. After the channel estimation, the derived results are utilized to restore the subsequent distorted data symbols using the linear MMSE equalizer. Figure 14 presents the constellation diagrams of the equalized QPSK symbols based on channel estimation from different algorithms. A total of 51 signal frames with 26,112 received QPSK data symbols were analyzed. Table 5 shows a comparison in terms of the average number of iterations, computation time, and Symbol Error Rate (SER) for the four algorithms. It can be observed that the greedy method OMP has the lowest computational complexity but with low estimation accuracy and high SER of 10.68%. The A-OMP algorithm, with an improved termination condition, lowers the SER by 3.23% compared to the OMP algorithm. When contrasted against numerical simulations of smaller scale in the previous section, the computational cost of the original ADMM algorithm increases sharply due to the evaluation of matrix inverse \((\mathbf{\Phi}^{H}\mathbf{\Phi}+\rho\mathbf{I})^{-1}\) in each estimation. Although the FM-SBL algorithm has reduced computational complexity of less than \(\mathcal{O}(M^{2}N)\) compared to the Expectation Maximization (EM)-SBL algorithm [48], it exhibits the second highest runtime at 9.73 s. The advantages of the proposed nmADMM algorithm are less pronounced compared to the simulations, as the time-to-frequency domain conversion in OFDM demodulation and the pre-processing mitigate the effects of impulsive noise. Nevertheless, it maintains high estimation accuracy, delivering the lowest SER of 4.48%. Figure 15 examines the convergence behavior of the nmADMM algorithm, showcasing iteration curves of the objective function and both primal and dual residues. One can see that the proposed algorithm can converge within tens of iterations in all estimations. Figure 13: Estimated CIR from at-sea experiment using (**a**) OMP, (**b**) A-OMP, (**c**) FISTA, (**d**) FM-SBL, (**e**) ADMM, and (**f**) nmADMM.