## 2 Methods
### Channel Equalization Based on Fractional Fourier Transform  
Channel equalization is essential for addressing ISI during demodulation. Equalizers can be broadly classified into linear and nonlinear categories. Linear equalizers, such as ZF and MMSE equalizers, aim to remove ISI under ideal conditions. In contrast, nonlinear equalizers, including the decision feedback equalizer (DFE), adaptively adjust filter parameters to accommodate time-varying or nonlinear system conditions. Furthermore, equalization techniques can be categorized based on their implementation domain: time-domain, frequency-domain, and time-frequency-domain equalization. Due to the deep fading of mud channels in the frequency domain, the signal-to-noise ratio (SNR) is very low at certain frequencies, and the signal is significantly affected by noise interference. In some cases, the exclusive use of decision feedback equalizers fails to adequately mitigate the impact of frequency-selective attenuation, which results in a high BER during data decoding. Therefore, an effective channel compensation scheme is needed. Various channel compensation strategies are available, with frequency-domain equalization constituting a common method, categorized into linear equalization and nonlinear equalization. When using a linear equalizer, the equalizer based on the ZF criterion can completely eliminate inter-symbol interference under ideal conditions. However, deep fading points in the channel frequency domain can result in increased noise and reduced SNR. The equalizer based on MMSE can find a compromise between channel noise and inter-symbol crosstalk, but it cannot completely eliminate inter-symbol crosstalk. The frequency-domain DFE based on decision feedback trades complexity for improved performance [35]. The recursive least squares decision feedback equalizer (RLS-DFE) is a type of adaptive equalizer that combines the benefits of decision feedback equalization with the RLS algorithm for efficient adaptive filtering. The RLS-DFE equalizer consists of two main components: the feedforward filter and the decision feedback filter. The RLS algorithm is applied to adaptively update the filter coefficients based on the received signal. The feedforward filter is responsible for mitigating the linear distortion caused by the channel. It processes the incoming signal by applying adaptive filter coefficients that are updated recursively using the RLS algorithm. After the feedforward filter processes the received signal, the feedback filter uses the decisions made about the transmitted symbols to remove the interference caused by past symbols. The least squares algorithm sums the squares of errors as the optimization objective, and the cost function is expressed as  
\[J_{n}=\sum_{i=0}^{n}n^{n-i}\left|e_{i}\right|^{2}=\sum_{i=0}^{n}n^{n-i}\left| I_{i}-\omega_{n}^{T}r_{i}\right|^{2} \tag{15}\]  
where \(\mu\) is the forgetting factor, with its value ranging (0,1), \(e_{i}\) is the error vector, \(\omega_{n}\) is the equalizer tap at time \(n\), \(I_{i}\) is the output vector, and \(r_{i}\) is the input vector (the received signal). Fractional Fourier equalization provides a low-complexity solution to the aforementioned issues, effectively addressing the challenge of deep fading points in the channel. It avoids the need for recursive operations, making it faster in many cases. This technique resolves the trade-off between noise power control and ISI, offering a computationally simple and easily implementable solution [25]. The channel compensation method designed in this article is to preprocess the input signal using linear equalization in the fractional Fourier domain for pre-filtering and then feed the signal into the decision feedback equalizer.