## 4 Performance Analysis of Closed-Form Estimator
### Impact of SNR on Estimator Performance Analysis  
Noise can cause an increase in the bias and variance of the estimators, thereby reducing the accuracy and efficiency of the estimation. In order to analyze the impact of noise on the accuracy of the estimators, we assume signal-to-noise ratios (SNRs) of 30 dB, 25 dB, 20 dB, and 15 dB and conduct simulation analysis under the assumption of channel parameters \(k=3.5\) and \(\mu=1.5\). In this case, the envelope signal satisfies the relationship \(r_{addNoise}=r_{k-\mu Signal}+r_{noise}\), \(r_{k-\mu Signal}\) is generated by the Monte Carlo method, and \(r_{noise}\) represents additive Gaussian white noise. For ease of comparison, the analysis results here are presented with a precision of four decimal places. In each experiment, the four estimators use the same sample data, and the sample size is 10,000. In multiple simulation experiments, we find that when the SNR is 30 dB, the estimation method \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) has about a 40% probability of not producing valid results; when the SNR is 25 dB, the estimation method \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) has about a 60% probability of not producing valid results; when the SNR is 20 dB, the estimation method \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) has about a 90% probability of not producing valid results; and when the SNR is 15 dB, the estimation method \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) can no longer produce valid results. In all the above experiments, the channel parameter estimator we proposed demonstrated significantly lower error probabilities under the aforementioned signal-to-noise ratio levels, indicating its robustness in various noise conditions. In the figures below, we present the valid estimation results. From Table 6, we can see that the parameters \(k\) and \(\mu\) have already become less than 0, and the \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) estimation method can no longer produce valid results. Moreover, in Figure 4d, it can be observed that the curve obtained from the \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\), estimation method has been completely deformed. This indicates that the \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) method will fail when the noise interference is significant. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**SNR (dB)**} & \multicolumn{2}{c}{**Factor**} & \(k\) & \(\mu\) & \multirow{2}{*}{**RMSE\({}_{puff}\)**} & \multirow{2}{*}{**Top**} & \multirow{2}{*}{**Figure**} \\ \cline{2-2} \cline{5-6}  & & **Truth Value** & & & & \\ \hline \multirow{3}{*}{30 dB} & \multirow{3}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) & 8.2650 & 0.7567 & 0.0109 & / & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\)/\(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 2.3838 & 1.8736 & 0.0081 & 1 & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) & 0.0777 & 3.7002 & 0.0387 & 2 & \\ \hline \multirow{3}{*}{25 dB} & \multirow{3}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) & 1.6424 & 2.2041 & 0.0205 & / & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\)/\(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 2.7958 & 1.6392 & 0.0197 & 1 & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) & 0.0809 & 3.5629 & 0.0402 & 2 & \\ \hline \multirow{3}{*}{20 dB} & \multirow{3}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) & 2.8294 & 1.4664 & 0.0554 & / & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\)/\(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 5.7361 & 0.8852 & 0.0557 & 1 & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) & 0.0903 & 3.1984 & 0.0605 & 2 & \\ \hline \multirow{3}{*}{15 dB} & \multirow{3}{*}{Estimation Method} & \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) & \(-1.4877\) & \(-20.7926\) & / & / & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\Delta}\)/\(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & 2.7161 & 1.2483 & 0.1153 & 2 & \\ \cline{3-6}  & & \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) & 0.0803 & 2.6652 & 0.1129 & 1 & \\ \hline \end{tabular}
\end{table} 
Table 6: The comparison of estimation results under different SNR conditions.