## 1 Introduction  
Over the past decades, high-speed rail has undergone rapid development, and it has become a popular mode of transportation for intercity and cross-regional travel. With its speed, safety, and convenience, high-speed railways have significantly reduced the travel time between cities and fostered economic and cultural exchanges. A number of countries have constructed high-speed rail infrastructures to improve their technical level and service quality. However, during daily train operations, various damage, such as cracks, spalling, shelling, and fractures, often occur on the rails, severely affecting operational safety [1]. Therefore, it is necessary to strengthen the overhaul and maintenance of steel rails to ensure the safe operation of high-speed railways. At present, traditional defect detection methods for critical structures in high-speed railways struggle to identify internal or early-stage defects due to blind spots, low sensitivity, and other issues. Therefore, acoustic emission (AE) technology has drawn increasing attention; it collects and analyzes acoustic signals passively to achieve fast and accurate detection of rail cracks, with its natural advantages of high sensitivity and real-time detectability [2]. However, there normally exist complex noise components in the actual railway environment. The acquired AE signals need to be denoised or demixed before the defect source can be identified, and blind signal separation (BSS) technology is thus introduced in this study to effectively extract the crack signal from the received acoustic mixtures. Blind signal separation (BSS) is a powerful multi-channel signal processing technique that offers non-parametric optimization and requires no prior knowledge of the mixing process or source signals. Considering their strong capability and flexibility in separating the acoustic mixtures, BSS algorithms have been studied in depth by the research community and broadly applied in industrial or medical fields, such as in fault diagnosis in bearings/gearboxes [3], EEG/ECG event extraction [4], wireless communication [5], and speech separation, enhancement, and dereverberation [6]. Independent component analysis (ICA) is one of the most cited BSS methods that originated from multi-sensor processing, and it retrieves the demixing weights through maximizing the independence or the non-Gaussianity of the demixed signals. For example, [PERSON] presented a singular spectrum analysis integrated with ICA technique for single-channel EEG signal separation [7]. However, the traditional ICA method has a low separating accuracy in the existence of non-stationary noise or high amplitude Gaussian noise. To solve this problem, [PERSON] et al. modified the adaptive ICA into a noise cancellation method for a speech signal in communication devices. The simulation results showed that the interference of the noise is significantly reduced in the proposed method [8]. Aside from ICA, there is also another representative branch of BSS methods based on the statistic metrics, i.e., joint approximate diagonalization of eigenmatrices (JADE) and second order blind identification (SOBI). [PERSON] proposed an improved SOBI method involving a Hilbert transform for blind identification of bandlimited signals [9]. Theoretically, JADE makes use of the fourth order cumulants to automatically suppress Gaussian background noises and enhance non-Gaussian sources. To cope with the problem when two subjects of Doppler radar cross the angular resolution limit, [PERSON] et al. introduced JADE into the array signal processing of radar, and experimental results demonstrated JADE superseded the performance of the other extraction methods [10]. However, conventional BSS methods rely highly on the optimization process with a recursive fixed-step quasi-Newton method, which is inefficient and suffers from slow computing, oscillation, and instability of the convergence.