## 4 Numerical Simulations
### Simulation II: Doubly Spread Channel Estimation  
In this section, we simulate a four-path DD spread channel with the parameters detailed in Table 2. The SNR is set at 10 dB, aligning with the moderate noise level commonly encountered in practical UWA conditions. Impulsive noise probability is configured at \(q=10^{-3}\) with an INR of 30 dB. Similar to _Simulation I_, a PRBS employing BPSK modulation serves as the probe signal, featuring a symbol rate \(R_{s}=2\) kbaud and a pulse-shaping roll-off factor of 0.4. For channel estimation, we assign an upper bound for the maximum channel delay at \(\tau_{\max}=5\) ms, where the delay domain sampling interval matches the symbol duration at \(\Delta\tau=1/R_{s}=0.5\) ms. The Doppler shift sampling interval is set at \(\Delta\
u=0.2\) Hz, with estimation bounds ranging from \(\
u_{\min}=-5\) Hz to \(\
u_{\max}=5\) Hz. This leads to a total of \(N=500\) unknown channel parameters to estimate. For sparse channel estimation, the probe symbol sequence length is \(M=\frac{2}{3}N\approx 330\), highlighting the problem is ill-posed. The second-order method, IPM-PCG, is excluded due to computational intensity and reduced accuracy in complex-valued problems. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Method** & **\# Iter.** & **Runtime (s)** & **NMSD (dB)** \\ \hline OMP [15] & 64/64 & 0.32/0.32 & \(-12.05/8.85\) \\ FISTA [23] & 43/40 & 0.11/0.17 & \(-11.89/7.85\) \\ ADMM [25] & 35/23 & 0.12/0.11 & \(-11.83/8.45\) \\ IPM-PCG [21] & 24/21 & 11.86/24.66 & \(-11.48/8.67\) \\ mADMM [28] & 54/57 & 0.23/0.25 & \(-12.20/-11.25\) \\ nmADMM & 43/42 & 0.31/0.33 & \(-12.22/-11.49\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparisons for CIR estimation in the single-carrier system with respect to the average number of iterations, runtime, and NMSD in AWGN (left side of ‘/’) and impulsive noise (right side of ‘/’) environments. Figure 4: Variation in NMSD with different numbers of measurements. (**a**) AWGN environment. (**b**) Impulsive noise environment. Figure 5: Variation in NMSD with different SNR levels. (**a**) AWGN environment. (**b**) Impulsive noise environment. Before performing sparse methods, the Cross-Ambiguity Function (CAF) serves as our benchmark for channel spread function estimation [44]. The resolution of the CAF method is highly dependent on the auto-correlation properties of the probe signal. To effectively distinguish the four paths, we extend the length of the probe signal for CAF to \(M=\frac{4}{3}N\approx 670\). The delay and Doppler resolutions approximate \(\tau_{\rm res}\approx\frac{1}{K_{s}}=0.5\) ms and \(\
u_{\rm res}\approx\frac{K_{s}}{M}=3\) Hz. Since the problem here has a similar scale to the CIR channel estimation problem in the previous section, we maintain consistent parameter settings for all algorithms except the stopping criteria. For ADMM-based methods, the absolute and relative tolerances are set at \(\epsilon_{\rm abs}=5\times 10^{-5}\) and \(\epsilon_{\rm rel}=5\times 10^{-4}\), respectively. The residue ratio tolerance for the FISTA algorithm is set at \(\epsilon_{\rm fista}=10^{-6}\). We reduced the tolerance thresholds to achieve finer estimation precision, particularly as inaccuracies in the Doppler frequency shift estimation can severely impair subsequent equalization quality. The iteration number for the OMP algorithm is set at four, aligning with the sparsity of the simulated channel. Figures 6 and 7 display the estimated DD spread functions under AWGN and impulsive noise environments. Following the probe signal, 2,000 data symbols of the same symbol rate and modulation configurations are transmitted through the channel. Based on the channel input-output relationships shown in (4)-(6), the estimated DD functions are first converted to the time-varying CIR, and then fed into the linear MMSE equalizer to retrieve the distorted data symbols. The equalization outputs using Ground Truth (GT) channel information and estimations from different methods are shown in Figures 8 and 9.