## 4 Numerical Simulations
### Simulation II: Doubly Spread Channel Estimation
We summarize the average number of iterations, runtime for single estimation, number of error bits, and Bit Error Rate (BER) of the data sequence in Table 3. Notice that given the limitations of the MMSE equalizer in addressing impulse noise, the BER in an impulsive noise environment rises by 8.75%, even when utilizing GT channel information. \begin{table}
\begin{tabular}{l l l l l} \hline
**Path No.** & **1** & **2** & **3** & **4** \\ \hline Delays (ms) & 0.5 & 2 & 4 & 4 \\ Doppler shift (Hz) & \(-1\) & 2 & \(-4\) & 4 \\ Modulus of amplitude & 1 & 0.8 & 0.5 & 0.5 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter setting of the simulated DD channel. Figure 6: DD channel spread function estimation in AWGN environment: (**a**) CAF, (**b**) OMP, (**c**) FISTA, (**d**) ADMM, (**e**) mADMM, and (**f**) nmADMM. Figure 7: DD channel spread function estimation in impulsive noise environment: (**a**) CAF, (**b**) OMP, (**c**) FISTA, (**d**) ADMM, (**e**) mADMM, and (**f**) nmADMM. In comparison to the CIR estimation discussed in the previous section, the current problem exhibits a similar scale but features higher sparsity and a greater number of observations. However, all sparse algorithms demand more iterations, except for OMP. This is not merely due to the reduced tolerances for stopping criteria but also stems from the \"poor\" property of the measurement matrix \(\mathbf{\Phi}\). In _Simulation I_, the measurement matrix constructed directly by the PRBS had an average rank of 256 and a mutual coherence of 0.24. In contrast, for the problem here, the matrix determined by (8) has a rank of 140 and mutual coherence of 0.998. The high correlation among column vectors in \(\mathbf{\Phi}\) makes the identification of correct sparse solutions more difficult, particularly for greedy methods such as OMP. The iterative process of OMP, which selects the column most correlated with the current residue, can misidentify the correct support element due to this high coherence, further leading the entire estimate to be severely incorrect [19]. This is evident in Figure 8b, where the estimated channel components of OMP deviate from true values, leading to phase rotation during equalization even in the AWGN environment. Compared to greedy methods, convex optimization-based approaches such as FISTA, ADMM, and the proposed algorithm require less restrictive conditions on the measurement matrix for successful sparse estimation [45; 46]. One can observe from Figure 6c-f that all convex optimization methods accurately identify the DD channel path in AWGN environment. However, under the influence of impulsive noise, FISTA and the original ADMM algorithms exhibit  
Figure 8: Constellation diagrams from linear MMSE equalizer outputs using different estimation results in AWGN environment: (**a**) GT, (**b**) OMP, (**c**) FISTA, (**d**) ADMM, (**e**) ADMM, (**f**) nmADMM. Figure 9: Constellation diagrams from linear MMSE equalizer outputs using different estimation results in impulsive noise environment: (**a**) GT, (**b**) OMP, (**c**) FISTA, (**d**) ADMM, (**e**) mADMM, (**f**) nmADMM. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Method** & **\# Iter.** & **Runtime (s)** & **\# Err.** & **BER (\%)** \\ \hline Ground Truth & -/- & -/- & 0/179 & 0/8.75 \\ OMP [15] & 4/4 & 0.004/0.004 & 179/791 & 8.95/39.55 \\ FISTA [23] & 564/835 & 1.16/1.54 & 9/729 & 0.45/ 36.45 \\ ADMM [25] & 662/610 & 0.51/0.63 & 0/619 & 0/30.95 \\ mADMM [28] & 1577/2193 & 5.85/7.89 & 557/547 & 27.85/27.35 \\ nmADMM & 481/527 & 3.15/3.51 & 0/252 & 0/12.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison for DD spread function estimation in AWGN (left of ‘/’) and impulsive noise (right of ‘/’) environments with respect to average iterations, computing time, bit errors, and BER. increased estimation errors, particularly in identifying paths with smaller amplitudes. As illustrated in Figure 7c,d, the estimated locations of paths 3 and 4 by FISTA and the original ADMM algorithms deviate from their true positions.