## 4 Discussion and Results  
Section 3 presents the numerical simulation results that validate the performance of the channel estimation and equalization methods based on the FrFT. Additionally, communication conducted at a real well site further confirms that the proposed method significantly reduces the BER in the 3000 m experiment and achieves the recovery of the original signal. A comparison with traditional DFE methods also highlights the superior performance of the fractional Fourier transform-based approach in terms of accuracy and robustness. Numerous iterative channel estimation techniques have been proposed in the wireless communication domain. However, these methods typically incur high computational costs due to their iterative nature and often exhibit degraded performance under strong interference conditions. In contrast, the proposed FrFT-based method achieves comparable estimation accuracy using a single transform operation, thereby significantly reducing computational complexity. By jointly performing channel estimation and equalization in the fractional Fourier domain, our approach delivers superior bit error rate (BER) performance, particularly in the presence of severe multipath and structured interference as observed in real-world downhole environments. Beyond comparisons with advanced iterative algorithms, we also benchmark our method against conventional equalization schemes such as time-domain RLS-DFE, MMSE, and ZF equalizers. While RLS-DFE is capable of adaptively tracking channel variations and performs adequately under moderate fading, it is susceptible to instability and requires \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Modulation Mode**} & \multirow{2}{*}{**Depths (m)**} & \multirow{2}{*}{**Data Rate (bit/s)**} & \multicolumn{2}{c}{**BER**} \\ \cline{3-4}  & & & & **DFE** & **DFE + FrFT** \\ \hline
1 & BFSK & 887 & 8 & 0.004 & 0 \\
2 & BFSK & 887 & 10 & 0.12 & 0.04 \\
3 & BFSK & 2515 & 10 & 0 & 0 \\
4 & BFSK & 2602 & 9 & 0.50 & 0.05 \\
5 & BFSK & 2632 & 6 & 0.12 & 0.01 \\
6 & BFSK & 2890 & 12 & 0.14 & 0.03 \\
7 & BFSK & 2980 & 10 & 0.23 & 0.02 \\
8 & BFSK & 3016 & 12 & 0.08 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: BER comparisons between two techniques. Figure 11: Estimated channel response in experiment. intensive computation for recursive coefficient updates and matrix operations. In contrast, our FrFT-based approach reformulates the equalization problem in the fractional Fourier domain, effectively pre-whitening the channel response and mitigating multipath distortion without the need for iterative adaptation. MMSE equalization offers a trade-off between noise suppression and ISI mitigation but suffers from residual inter-symbol interference in channels with deep fades. ZF equalizers can eliminate ISI under ideal channel conditions; however, their sensitivity to spectral nulls leads to significant noise amplification in practice. The proposed method circumvents these limitations by leveraging the energy compaction properties of linear frequency-modulated (LFM) signals in the FrFT domain. This enables precise channel impulse response estimation and efficient spectral flattening prior to time-domain DFE processing. In addition to its effectiveness in handling multipath and frequency-selective fading, the proposed FrFT-based method demonstrates intrinsic robustness against line-spectral interference--a prevalent challenge in mud pulse telemetry due to the periodic activity of mud pumps and rotary machinery. Such interference introduces narrowband components that overlap with the signal spectrum, severely impacts traditional time- and frequency-domain estimation methods. By design, the FrFT provides optimal energy localization for LFM signals when the transform order aligns with the chirp rate. Consequently, the desired signal components are sharply focused in the FrFT domain, whereas narrowband interferers remain dispersed or shifted in the time-frequency plane.