## 4 Discussion and Results
This intrinsic separation facilitates enhanced discrimination and extraction of useful signals without requiring interference modeling or adaptive filtering. Field data further verify the method's robustness under periodic pump induced interference, underscoring its practical applicability in telemetry systems operating under harsh and noise-prone downhole conditions. To illustrate the superiority of our work, we make quantitative comparisons between the proposed equalization schemes and some of the mentioned equalization works in Section 1 to create a scientific discussion. We use the same data in Table 1, for which the traditional DFE method fails to achieve correct demodulation. Table 2 displays the BER results after demodulation with three different equalizers (FrFT, ZF, and MMSE) at various depths. From the table, it can be seen that all equalizers can successfully demodulate in case 1. However, for cases 2-6, the ZF equalizer cannot demodulate properly due to the increase in well depth, while the MMSE equalizer suffers from residual ISI in channels with deep fades and fails to achieve correct demodulation at data rates of 10 and 12 bit/s. The results indicate that our proposed equalizer has better performance in improving the BER in a real well test demodulation. In general, the main innovation points of this article are as follows:  
1. A channel estimation method tailored to the characteristics of mud channels is proposed using FrFT, along with an optimal order search strategy that significantly reduces computational complexity. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Depths (m)**} & \multirow{2}{*}{**Data Rate (bit/s)**} & \multicolumn{3}{c}{**BER**} \\ \cline{3-5}  & & & **ZF** & **MMSE** & **FrFT** \\ \hline
1 & 887 & 10 & 0.02 & 0.04 & 0.04 \\
2 & 2602 & 9 & 0.13 & 0.07 & 0.05 \\
3 & 2632 & 6 & 0.08 & 0.01 & 0.01 \\
4 & 2890 & 12 & 0.15 & 0.14 & 0.03 \\
5 & 2980 & 10 & 0.12 & 0.17 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BER comparisons between three equalizers. 2. A compensation scheme for channel frequency-selective fading is developed using FrFT domain equalization, which enhances the overall communication performance. The BER of experimental results after demodulation has decreased by 90% compared to the traditional DFE method. Moreover, compared with ZF and MMSE equalization, our method obtains a lower BER in real well tests. In general, the BER after demodulation is reduced to below 0.5%, which meets the engineering requirements. Although the current study has achieved some good results, the overall work is still not perfect and exists limitations which needs to be further researched. Specifically, when using FrFT for LFM signal detection, currently there is no adaptive threshold method due to the impact of noise. In addition, the high computational complexity for selecting the optimal fractional order in FrFT should be reduced. In addition, we need to test and compare the performance of this study with more advanced published works. Finally, further experiments are needed to verify the proposed method in different drilling environments.