## 4 Performance Analysis of Closed-Form Estimator  
### Computational Complexity Analysis of Closed-Form Estimator  
Algorithmic complexity is a criterion for evaluating the execution efficiency of an effective algorithm. When the sample size is \(N\), the total number of basic operations performed by different algorithms is denoted as time complexity, while the required memory space is denoted as space complexity. Necessary procedures that both types of complexities need to address are not considered in the comparison of complexities. We present the complexity analysis of four closed-form solutions in Table 2:  
In Table 2, \(O_{1}(\cdot)\) represents the number of basic operations required for computing other variables (excluding \(k\) and \(\mu\)) before the closed-form calculation. For example, the \(O_{1}(\cdot)\) value for Estimation Method \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) indicates the complexity of solving for variable \(Var\), \(\eta\), \(\Delta\); similarly, the \(O_{1}(\cdot)\) value for Estimation Method \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) also represents the complexity of solving for variable \(a\), \(b\), \(c\), \(D\), \(P_{1}\). \(O_{2}(\cdot)\) denotes the complexity of the \(k\) and \(\upmu\) estimation functions, such as the closed-form expressions for Method \(\hat{k}\)-\(\hat{\mu}_{vacuum}\) given by Equations (6) and (7). \(O_{3}(\cdot)\) refers to the memory space required to store temporary variables during the estimation process. For instance, Estimation Method \(\hat{k}\)-\(\hat{\mu}_{vacuum}\) only needs to store the variance operator \(Var[\cdot]\), thus \(O_{3}(\cdot)=1\). Since each moment estimation requires the calculation of the second-order, fourth-order, and sixth-order moments of the envelope, the computational cost of \(E(r^{2})\), \(E(r^{4})\), \(E(r^{6})\) is not included in the complexity analysis.  
To further analyze time complexity, we employ the Monte Carlo method to generate channel data following the kappa-mu distribution and compare estimation efficiency based on the time taken for the estimation. As shown in Figure 1, we use the channel impulse response \(h(t)\) as the data sample. The channel data generated using the Monte Carlo method in this section does not include noise.  
We set the sample size to 10,000, with each round of estimation experiments using the same data. The efficiency of the estimator is evaluated based on the total runtime of multiple estimations, and the runtime is recorded in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Estimator Name** & **Time Complexity \(T(n)\)** & **Space Complexity \(S(n)\)** & **Estimation Efficiency** \\ \hline \(\hat{k}\)-\(\hat{\mu}_{vacuum}\)[24] & \(O_{1}(2)+O_{2}(16)=O(n^{0})\) & \(O_{3}(1)=O(n^{0})\) & 1 \\ \hline \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & \(O_{1}(15)+O_{2}(10)=O(n^{0})\) & \(O_{3}(3)=O(n^{0})\) & 2 \\ \hline \(\hat{k}\)-\(\hat{\mu}_{\Delta}\) & \(O_{1}(101)+O_{2}(8)=O(n^{0})\) & \(O_{3}(10)=O(n^{0})\) & 4 \\ \hline \(\hat{k}\)-\(\hat{\mu}_{\text{B}}\) & \(O_{1}(50)+O_{2}(8)=O(n^{0})\) & \(O_{3}(5)=O(n^{0})\) & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Algorithm complexity characteristics table.  
As can be seen from Tables 2 and 3, Estimation Method \(\hat{k}\)-\(\hat{\mu}_{\lambda}\), while Method \(\hat{k}\)-\(\hat{\mu}_{\lambda}\) is the least efficient. However, in the authors' numerous simulation studies, it was found that when the estimation error of Method \(E(r^{2}),E(r^{4}),E(r^{6})\) is relatively large, the estimator \(\hat{k}\)-\(\hat{\mu}_{Yacoub}\) may produce \(k\) and \(\mu\) values that are less than zero or complex numbers, which can be considered erroneous estimates. In practical applications, repeated estimations may generally consume more time.