## 4 Performance Analysis of Closed-Form Estimator
### Impact of Pilot Signal on Estimator Performance Analysis  
In general, the amount of information required by the pilot signal depends on the specific estimation methods and the channel environment. The impact of the pilot signal on the estimation performance also varies with different estimation algorithms and channel conditions. Basically, a higher pilot signal power or density can lead to better channel estimation accuracy, but also at the cost of higher overhead and decreased spectral efficiency. On the other hand, a lower pilot signal power or density can result in lower accuracy and higher estimation error, especially in fast-varying channels.  
Consequently, it is indeed true that with a longer pilot signal, more samples can be obtained, and as the sample size increases, the estimations of \(k\) and \(\mu\) parameters will approach unbiasedness and become more accurate. In practical applications, extending the pilot signal duration and increasing the sampling rate can enhance estimation accuracy. However, this comes at the cost of consuming more communication resources, which in turn results in a reduced data rate for wireless communication. It is essential to weigh these trade-offs carefully.  
To address this question, we conducted related parameter measurements and fitting simulation experiments in an indoor environment. The wireless communication scenario parameters are set as follows: The operating frequency band is the C-band (5 GHz); assuming a Rician channel with a channel length of 20, the transmitter employs a pilot PN sequence of 2048 symbols (25% of the total amount of transmitted data), BPSK modulation, and a code rate of 20 Mbps transmission scheme. On the receiver. At the receiving end, we used a receiving demodulation and processing device equipped with high-speed sampling, fiber-optic broadband transmission, and high-speed storage, achieving a transmission rate of 5 Gbps. The sampling frequency can reach up to 500 Msps.  
The left column in Figure 5 shows the time on the x-axis and signal power on the y-axis (within a one-second acquisition period, we detected 10 instances of robust signal waves; however, only a single representative example is displayed here). On the right column, we estimate the received signal, where the gray area represents the frequency histogram of envelope \(r\), and the two curves are estimation curves, with the \(x\)-axis representing the normalized envelope and the \(y\)-axis representing the probability. In the right column, using the estimator proposed in this paper and defining \(\mu=1\), we can obtain the estimated curve--\"Rice distribution\" (by taking advantage of the \(k\)-\(\mu\) distribution transforming into the Rice distribution when \(\mu=1\)). Using the estimator without restricting the values of \(k\) and \(\mu\), we can obtain the curve--\(k\)-\(\mu\) distribution\". As can be seen from the graphs, when the sample size is sufficient, the \(k\)-\(\mu\) distribution approximates the Rice distribution. When the sample size is not enough, the \(k\)-\(\mu\) distribution estimator exhibits excellent adaptability.  
Figure 5: _Cont_.  
As Figure 5 indicates, as the number of sampling points increases, the \(k\)-\(\mu\) distribution becomes more closely aligned with the actual probability distribution, and the greater the number of samples, the more evident the alignment. In this experimental scenario, when the number of samples \(N\) obtained from pilot signal sampling is greater than 8000, the PDF estimation curve generated by the sample points essentially matches the histogram and theoretical curve. Consequently, a consistent correlation between the theoretical analysis and experimental measurement results concerning the impact of the pilot signal on estimator performance is observed.