## 5 Conclusions  
This paper proposed the channel estimation and equalization method based on fractional Fourier transform. The architecture of mud pulse telemetry is introduced, along with a brief overview of the processes involved in signal generation, transmission, and acquisition. The continuous pressure wave signal used in this project is generated by a swinging mud pulse generator and is transmitted to the ground through mud fluid. During the transmission process, the pressure wave decays exponentially with distance and exhibits significant frequency-selective attenuation. A systematic analysis of the noise and channels within the mud of the actual well was conducted using field data. We developed a channel estimation method based on FrFT to estimate path delays and attenuation coefficients. Compared to conventional LFM-based time-domain estimation techniques, the proposed method offers lower computational complexity and improved robustness, especially in the presence of unknown frequency offsets, and achieves significantly higher estimation accuracy. Moreover, an FrFT domain equalizer is designed to mitigate the effects of frequency-selective fading. The equalizer effectively compensates for channel distortion without amplifying background noise. The overall effectiveness of the proposed technique is validated through data collected from a real well site, demonstrating enhanced decoding performance. The BER of experimental results after demodulation is decreased by 90% compared to the traditional DFE method and is reduced to below 0.5%, which meets the engineering requirements, ensuring that our approach is not only theoretically but also practically feasible. In addition, our proposed equalizer achieves better performance in improving the BER compared to the ZF and MMSE equalizers. Finally, we achieve advanced data rate communication (over 12 bit/s) in a 3000 m drilling system. Future research will focus on expanding and refining our current methodology. Specifically, we should explore the potential of FrFT-based equalization algorithms to reduce training sequence length and eliminate redundant information, thereby lowering computational complexity. In addition, practical deployment remains an important direction for continued investigation. Key aspects include analyzing the sensitivity of fractional order selection under dynamic channel conditions, developing lightweight and adaptive order estimation strategies, and implementing FrFT-based signal processing on embedded or low-power DSP platforms. Finally, more experiments will be conducted to verify the proposed method in the changing behavior of the mud and drilling environment and test the channel characteristics with varying temperature, pressure, and depth. These efforts are expected to bridge the gap between theoretical algorithm design and real-timefield deployment, ultimately enabling robust and efficient telemetry systems for harsh downhole environments. Conceptualization, [PERSON] ([PERSON]) and F.Q. ; methodology, [PERSON] ([PERSON]), Z.S., and [PERSON] ([PERSON]); software, [PERSON] ([PERSON]), [PERSON] ([PERSON]), and Z.S. ; validation, [PERSON] ([PERSON]), L.W., and Y.S. ; formal analysis, [PERSON] ([PERSON]), Z.S., [PERSON] ([PERSON]), and F.Q. ; investigation, [PERSON] ([PERSON]), L.W., and Y.S. ; resources, [PERSON] ([PERSON]), L.W., and Y.S. ; data curation, L.W., Y.S., and F.Q. ; writing--original draft preparation, [PERSON] ([PERSON]), [PERSON] ([PERSON]), and Z.S. ; writing--review and editing, [PERSON] ([PERSON]), Z.S., F.Q., L.W., and Y.S. ; visualization, [PERSON] ([PERSON]), L.W., and Y.S. ; supervision, [PERSON] ([PERSON]), Z.S., and F.Q. ; project administration, L.W., Y.S., and F.Q. ; funding acquisition, F.Q. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Fund for Distinguished Young Scholars, grant number 62225114. Data Availability Statement: Data are contained within the article.