## 4 Performance Analysis of Closed-Form Estimator
### The Simulation for Estimator Accuracy Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number** & **Factor** & \(k\) & \(\mu\) & **RMSE\({}_{pdf}\)** & **Top** & **Figure** \\ \hline \multirow{4}{*}{1} & \multirow{4}{*}{Estimation Method} & Truth value & 0 & 1 & / & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[24] & 2.697529972710 & 0.491148658790 & 0.094653847993 & 4 th \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 1.658816467801 & 0.650138179259 & 0.065828663572 & 2 nd \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 1.658816467801 & 0.650138179259 & 0.065828663572 & 2 nd \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.069035813789 & 1.060038794578 & 0.019338029254 & 1 st \\ \hline \multirow{4}{*}{2} & Truth value & 0 & 1.5 & / & / \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[24] & 1.171262682065 & 1.062944179515 & 0.043970366233 & 4 th \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.606908589722 & 1.302013307991 & 0.021477554324 & 2 nd \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.606908589723 & 1.302013307991 & 0.021477554324 & 2 nd \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.037025064065 & 1.516708286752 & 0.003692816335 & 1 st \\ \hline \multirow{4}{*}{3} & Truth value & 0 & 2 & / & / \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\)[24] & 2.442853784456 & 0.971494141133 & 0.056600362496 & 4 th \\ \cline{3-6}  & Estimation Method & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.395558380575 & 1.842234181266 & 0.012151341100 & 2 nd \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.395558380576 & 1.842234181266 & 0.012151341100 & 2 nd \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 0.024242478356 & 2.002043886281 & 0.000434639810 & 1 st \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance comparison table of estimators for \(k\)-\(\mu\) fading channels when \(k=0\). At the same time, Estimator \(\hat{k}\)-\(\hat{\mu}_{\gamma acoub}\) continues to exhibit incorrect estimates with negative or complex values for \(k\) and \(\mu\). In cases of correct estimation, as shown in Table 5, the RMSE\({}_{pdf}\) of the estimator is sufficiently small. From the last column in Table 5, it can be observed that under the condition of \(k=0\), Estimator \(\hat{k}\)-\(\hat{\mu}_{B}\) has the best estimation accuracy, while Estimator \(\hat{k}\)-\(\hat{\mu}_{\gamma acoub}\) has the worst. Therefore, Estimator \(\hat{k}\)-\(\hat{\mu}_{\gamma acoub}\) is more suitable for situations with extremely poor channel quality, where there are numerous obstacles between the transmitting and receiving antennas and no line-of-sight component (\(k\) approaching 0). Figure 3: Estimation accuracy analysis of \(k\), \(\mu\) parameter estimation algorithm based on Monte Carlo method generated \(k\)-\(\mu\) channel data; (**a**) Comparison the error of theoretical PDF and estimated PDF curves at \(k=0\), \(\mu=1\); (**b**) Comparison the error of theoretical PDF and estimated PDF curves at \(k=0\), \(\mu=1.5\); (**c**) Comparison the error of theoretical PDF and estimated PDF curves at \(k=0\), \(\mu=2\). In summary, through complexity analysis and simulation, we find that Estimator \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) has the lowest estimation efficiency. When \(k\) approaches 0, using Estimator \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) is the best choice; in other cases, Estimator \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) is the most optimal option.