## 1 Introduction  
Petroleum is a crucial resource that plays an essential role in industrial production and the functioning of modern society [1]. However, decades of intensive extraction have progressively depleted conventional oil reserves, which were both high-quality and easily accessible. This depletion has brought considerable challenges to the oil industry while simultaneously catalyzing rapid advancements in drilling technology [2]. Among these developments, measurement while drilling (MWD) systems provide real-time data while drilling, cut down the time needed for logging operations after drilling, and ensure the collection of dependable data in high-difficulty wells. The transmission of data in these systems is broadly classified into wired and wireless telemetry. Wired telemetry primarily includes optical fibers, cables, and so on, which offer high data rates but are costly [3]. Conversely, wireless telemetry mainly includes mud pulse telemetry (MPT), electromagnetic telemetry, and acoustic telemetry, among which mud pulse telemetry is the most widely used [4]. Notably, electromagnetic telemetry suffers from significant signal attenuation and poor noise immunity, rendering it suitable only for short-distance transmission under specific formation resistivity conditions [5]. In addition, acoustic telemetry encounters challenges due to the substantial attenuation of mechanical waves, compounded by the complexities of the drilling environment and the interference from background noise, which hinder the detection and extraction of information from acoustic signals. In contrast, mud pulse telemetry offers longer transmission distances and reliable performance, although its data transmission speed is relatively low compared to other methods [3]. MPT uses the circulating mud within the drill string for communication between the underground environment and the surface system. Activated by the underground control circuit, the generator is modified with the transmitted data, thereby inducing periodic pressure variations and converting electrical signals into pressure signals effectively. After a series of processing steps by the demodulation system, the surface system completes the recovery of underground measured data. With its reliable performance over long communication distances, mud pulse data transmission technology stands as the major wireless transmission manner deployed in real well applications [6]. Additionally, MPT systems do not necessitate specific alterations to the drill string, resulting in relatively low costs. In addition, the technology exhibits robust adaptability to the bottom environment because of the location of the mud channel inside the drill string, which mitigates the influence of geological factors [7]. Consequently, MPT has become one of the most-used wireless drilling transmission techniques [8]. During the transmission process from the bottom of the well to the ground through mud pipelines, significant interference and noise are generated in the transmission of mud pulses. These interferences primarily originate from the periodic motion of piston pumps and various mechanical vibrations underground, causing the useful mud pulse signals to become obscured. Furthermore, over long distances, the mud signal undergoes varying degrees of amplitude attenuation, resulting in significantly weakened signals. In this context, accurately identifying and decoding weak signals from high-energy and irregular noise has become a critical issue in MPT technology. Moreover, in the field environment, pressure signals are subject to frequency-selective attenuation, leading to severe inter-symbol interference (ISI) [9]. Consequently, channel equalization is essential to mitigating the effects of frequency-selective fading and reducing ISI. Research on the characteristics of mud channels primarily focuses on two aspects. Firstly, the Lamb acoustic attenuation model is commonly employed to describe amplitude attenuation as a function of distance, characterizing propagation loss.