## 4 Simulation Results  
In this section, we assess the performance of the proposed GWOEIS solution and other key benchmarks for comparisons in terms of the root mean square error (RMSE) or the normalized MSE (NMSE) over a total number of Monte-Carlo runs \(M_{c}=1000\). Following the IEEE 802.11 ac standard (see [7] and first reference therein), we consider a bandwidth \(B=80\) MHz with a sub-carrier spacing \(\Delta_{f}=312.5\) KHz giving a total of 256 sub-carriers, among which 11 are exploited for network signaling purposes and the remainder are payload carriers (i.e., \(K=122\)). We also consider \(P=6\) antennas and \(Q=2\) equi-powered paths with AoAs \(20^{\circ}\) and \(45^{\circ}\) and TDs \(25\) ns and \(62.5\) ns, respectively. Moreover, we set \(\rho_{1}=4\), \(\delta_{\tau}=1.25\) ns, \(\delta_{\theta}=0.1^{\circ}\), \(\Delta_{\tau}=18.75\) ns, and \(\Delta_{\theta}=10^{\circ}\). In Figures 3 and 4, we evaluate the RMSE/NMSE performance versus the SNR to compare the new GWOEIS solution against the Cramer-Rao lower bound (CRLB) of [6], the UMP algorithm in [7], the classic GWO in [28], the classic DE [13], the IS technique in [11], IS-DE that simply combines DE with IS in [13], and another benchmark version developed here by simply combining this time IS with GWO, referred to as IS-GWO. Figure 2: Flow chart of GWOEIS algorithm. As shown in Figures 3 and 4, our approach outperforms all other estimation techniques, both in terms of TD and AoA estimations. Additionally, it reaches the CRLB, even at low SNR levels and even with a very low value either parameter \(R=30\) and \(T_{H}=30\). We also observe a severe performance degradation of the original GWO and DE techniques due to the high dimension (i.e., \(2Q\)) of the optimization problem. When combined with IS, IS-DE improves a little but is outperformed by IS-GWO, and more so by GWOEIS. In Figures 3c and 4c, we compare the channel NMSE using the estimates of the TDs and AoAs assuming a perfect knowledge of the channel gains. Our approach remarkably outperforms all other estimation techniques and reaches the lower bound, even at very low SNR levels and even with a very low value either parameter \(R=30\) and \(T_{H}=30\). In Figures 5 and 6, we assess the impact of the samples size \(R\) and the hunting duration \(T_{H}\) on channel estimation performance versus the SNR. Here, the sample size \(R\) refers to the number of realizations with IS, to the wolves pack size with GWO, IS-GWO, and GWOEIS, or the number of individuals with DE and IS-DE. The number of iterations (e.g., \(T_{H}\)) is defined only for iterative approaches, i.e., GWO, IS-GWO, GOWEIS, DE, and IS-DE. Figure 4: MSE vs. the SNR in dB for \(Q=2\) (\(\mathbf{\theta}=[20^{\circ},\ 45^{\circ}]\); \(\mathbf{\tau}=[25\ \mathrm{ns},\ 62.5\ \mathrm{ns}]\)), \(R=100\), and \(T_{H}=30\) of: (**a**) the \(Q\) TDs, (**b**) the \(Q\) AoAs, and (**c**) the \(P\times K\) channel coefficients (on average, per element, for all three parameter types). Figure 3: MSE vs. the SNR in dB for \(Q=2\) (\(\mathbf{\theta}=[20^{\circ},\ 45^{\circ}]\); \(\mathbf{\tau}=[25\ \mathrm{ns},\ 62.5\ \mathrm{ns}]\)), \(R=30\), and \(T_{H}=100\) of: (**a**) the \(Q\) TDs, (**b**) the \(Q\) AoAs, and (**c**) the \(P\times K\) channel coefficients (on average, per element, for all three parameter types). solutions. We notice that both parameters can have a detrimental effect on the exploration abilities of these techniques. Thus, they need to be set as large as possible to cover the widest area of the multidimensional space. In Figure 5(a,i,b,i,c,i), we assess the estimation performance of the TD, the AoA, and the channel, respectively. GWOEIS, IS-GWO, and IS performs better than DE, GWO, and IS-DE. Figure 5(a,i,b,ii,c,ii) show the minimum RMSE over all techniques for TD, AoA, and channel estimation, respectively. The RMSE decreases when \(R\) and SNR increase. In Figure 5(a,iii,b,iii,c,iii), we can see that GWOEIS achieves the best performance in terms of TD, AoA, and channel estimation over \(R\geq 20\) and all SNR values.