## 1 Introduction
Secondly, the frequency-selective attenuation caused by factors such as reflection on the transmission path is analyzed to help understand the channel model. Channel estimation is an important method to acquire the channel structure, with current methods performing cross-correlation on linear sweep signals to estimate the delay and attenuation coefficient of the multipath effect [10]. To address the inter-symbol interference caused by multipath effects, researchers have proposed a time-domain equalization method [11]. The commonly used types of equalizers include linear equalizers and decision feedback equalizers (DFEs). At present, time-domain equalization, such as decision feedback equalization based on recursive least squares (RLSs), is extensively used in mud communication systems [12]. In the early 1990s, single-carrier frequency-domain equalization (SC-FDE) technology was proposed, which lowered the computation burden of traditional time-domain equalizers [13]. Although the complexity of SC-FDE is relatively low, it requires the insertion of redundant information such as cyclic prefixes in data frames, leading to a reduction in the communication rate [14]. In addition, due to the frequency-selective attenuation of the mud channel, the attenuation is relatively severe at certain frequency points, and the frequency-domain equalization is greatly affected by noise at deep fading points [15]. With advancements in efficient numerical methods for fractional Fourier transform, fractional Fourier domain equalization has emerged as a solution to address the channel frequency-selective attenuation [16]. It has been shown that fractional Fourier domain equalization is less susceptible to noise at deep fading points and has a computational complexity comparable to Fourier transform. In recent years, many experts have come up with their own solutions to channel estimation and equalization. [PERSON], [PERSON] et al. [17] proposed a channel parameter estimation scheme leveraging fractional Fourier transform (FrFT) and compressed sensing (CS) technology to achieve the non-cooperative perception of the FrFT optimal transform order. [PERSON], [PERSON] et al. [18] utilized a multi-armed bandit problem to deal with the uncertainty of channel state information (CSI) and proposed a channel estimation algorithm to explore the global changing channel information. [PERSON], [PERSON] et al. [19] adopted a sparse pilot structure and proposed a multi-blocks sparse Bayesian learning channel estimation algorithm to reduce the use of pilots and improve the transmission efficiency. [PERSON], [PERSON] et al. [20] applied the golden section method, parabolic interpolation, and Brent method to search for the optimal fractional-order domain to accurately estimate the parameters of the linear frequency modulation (LFM) signal. [PERSON], [PERSON] et al. [21] studied the FRFT normalized domain of polyphase-coded signals and derived a mathematical model to prove the impulse characteristic of the polyphase-coded signals under optimal order. [PERSON], [PERSON] et al. [22] studied the energy aggregation characteristics of LFM signal in different FRFT domains, combining the FRFT with fractional power spectrum accumulation to accurately determine the optimal order and capture the signal with the first order Doppler rate of change. [PERSON], [PERSON] et al. [23] created a hybrid carrier communication system based on weighted-type fractional Fourier transform to make the integration of Orthogonal Frequency Division Multiplexing (OFDM) and SC-FDE possible. [PERSON], [PERSON] et al. [24] realized feedforward adaptive equalization based on superimposed FrFT training sequences. [PERSON], [PERSON] et al. [25] applied FrFT to channel equalization of the Sweep-Spread carrier (SSC) system and improved its ability to resist the Doppler effect and ISI. [PERSON], [PERSON] et al. [26] proposed a Radio Frequency Fingerprint Identification (RFFI) method integrating Zero-forcing (ZF) equalization to effectively reduce the interference of time-varying channels on RFFI.