## 3 The Derivation and Design of the \(k\)-\(\mu\) Estimators
\right., \tag{14}\]  
Simultaneously, we define the received power \(E(r^{2})=P_{1}+P_{2}\) in the \(k\)-\(\mu\) distribution, where \(P_{1}\) represents the energy of the line of sight (LOS) path and \(P_{2}\) corresponds to the energy of the non-line-of-sight (NLOS) paths. Parameter \(k=\sum_{1}^{n}{(p_{i}^{2}+q_{i}^{2})}/{\sum_{1}^{n}{(x_{i}^{2}+y_{i}^{2})}}=P_ {1}/P_{2}\), the second-, fourth-, and sixth-order moments can be further transformed as follows (for a detailed derivation process, please refer to Appendix B):  
\[E(r^{2})=U_{2  
The parameters in the model satisfy the following relationship:  
\[ \begin{split} a&=[{U_{2}}^{3}-3{U_{2}}({U_{2}}^{2}-{U_{4}} )-{U_{6}}],\\ b&=4[{U_{4}}^{2}-2{U_{4}}{U_{2}}^{2}+{U_{2}}^{4}],\\ c&=[{U_{2}}^{3}({U_{2}}^{2}-3{U_{4}})-3{U_{2}}(2{U_{4}}^{2 }+{U_{2}}^{4}-3{U_{4}}{U_{2}}^{2})+2{U_{6}}{U_{2}}^{2}],\\ d&=[{U_{2}}^{3}(2{U_{4}}^{2}-{U_{4}}{U_{2}}^{2})-{U_{6}}{U _{2}}^{4}],\\ A&=D^{2}-3F,\ \ B=DF-9E^{2},\\ D&=3b^{2}-8 ac,\ \ E=-b^{3}+4 abc\,\\ F&=3b^{4}+16a^{2}c^{2}-16 ab^{2}c-64a^{3}d,\end{split}  \tag{22}\]  
and we summarize the estimation process of each estimator as below:  
In Figure 1, the input to the estimator is the channel envelope data \(r\), which consists of \(N\) sample points. At a certain moment, the instantaneous envelope \(r\) is the superposition of the useful signal and noise, which are temporarily indistinguishable. Consequently, in practical applications, it is necessary to filter the received signals. However, our estimator is also applicable to the study of noisy environments. In cases where the input signal is represented by instantaneous signal-to-noise ratios (SNRs) \(\gamma\), the estimator proposed in this study remains equally valid and applicable. In signal measurement experiments conducted in practical scenarios, the transmitter sends a pseudorandom sequence, and the channel information is obtained by correlation operating the received data with the known transmitted pseudorandom sequence. The calculation process for the CFR is denoted as \(\hat{H}(f)=Y(f)\cdot{\it conj}(X(f))/\left|X(f)\right|^{2}\), where \(X(f)\) represents the transmitted frequency domain signal (PN sequence), \(Y(f)\) denotes the down-converted IQ signal of the received frequency domain signal, and \({\it conj}(\cdot)\) signifies the complex conjugate. At this point, \(\hat{H}(f)\) encompasses the transmitter system response \(H_{\rm T}(f)\), free-space channel response \(H_{\rm C}(f)\), and receiver system response \(H_{\rm R}(f)\). The noise of the transmitter system is contained within \(H_{\rm T}(f)\), the noise in the propagation medium is contained within \(H_{\rm C}(f)\), and the noise of the receiver system is contained within \(H_{\rm R}(f)\). The channel system can be represented as \(Y(f)=H_{\rm T}(f)\cdot H_{\rm C}(f)\cdot H_{R}(f)\cdot X(f)\). To  
Figure 1: Flow chart of the parameter estimation [24]. further calibrate the system, the transmitter and receiver are directly connected via a high-frequency cable to obtain the system response, where the received frequency domain signal is \(Y_{sys}(f)=H_{T}(f)\cdot H_{R}(f)\cdot X(f)\). Consequently, the calibrated transmitter and receiver channel frequency response is denoted as \(\hat{H}_{C}(f)=Y(f)/Y_{sys}(f)\). Substituting the calculation process of the CFR, the calibrated transmitter and receiver channel frequency response is represented as \(\hat{H}_{C}(f)=\hat{H}(f)\cdot X(f)/Y_{sys}(f)\). Performing the inverse Fourier transform on \(\hat{H}_{C}(f)\) results in the calibrated channel impulse response \(\hat{h}(t)\), which is a discrete variable. By taking the magnitude of \(\hat{h}(t)\), the discrete envelope signal \(r=[r_{1},r_{2},r_{3},\cdots,r_{n}]\) can be obtained. By calculating the second-, fourth-, and sixth-order moments of the envelope signal and incorporating them into the estimation process depicted in Figure 1, the \(k\) and \(\mu\) parameters can be estimated.