## 1 Introduction  
Nowadays, mechanical systems exhibit characteristics of complicated structures and functions with the rapid development of science and technology. Therefore, machinery health management and maintenance are increasingly required in the monitoring process due to the greater demands for higher performance of mechanical systems. Modern machinery's dynamic reactions are a complex synthesis of a variety of dynamic phenomena, including mechanical vibrations, electrical oscillations, hydraulic fluctuations, and the consequences of their connection [1, 2]. As a result, there is a lot of interference with the health information present in equipment data. The crucial step in extracting health information and transforming it into a usable format is feature extraction. The solution to this problem is to find a suitable signal analysis technique to extract significant characteristics that are implicit in the gathered machinery information [3, 4]. Deep learning has also received widespread attention. By using deep network structure and deep learning methods, the representative features can be learned layer by layer, and the error of fitting complex models is very small. However, although the diagnostic performance is improved, it is easy to over-fit, resulting in poor diagnosis, and it needs powerful computing resources to realize model training, which increases the hardware cost. In current research on fault diagnosis for reciprocating compressors, researchers have employed a variety of methods and techniques. These include time-frequency adaptive variational decomposition (VTFAD) [5], empirical mode decomposition (EMD) [6], and variational mode decomposition (VMD) for signal decomposition and fault detection. Deep learning techniques such as CNN and LSTM are used for dynamic characteristic prediction [7; 8], along with digital image processing and artificial neural networks for fault detection [9; 10]. However, as a research focus, adaptive signal extraction has the advantages of data adaptation, mode independence, high accuracy and easy implementation and is the current research focus of signal fault feature extraction [11]. Time-frequency-based signal analysis techniques are frequently used as a suitable way to extract fault information for nonstationary signals because they may disclose frequency components and identify their time-variant characteristics present in the signal. Many self-adaptive time-frequency analysis techniques, such as empirical mode decomposition (EMD), have recently been developed [12]; local mean decomposition (LMD) [13] and variational mode decomposition (VMD) [14] are widely utilized to extract machinery operational status information and exhibit excellent performances in the field of fault diagnosis due to their signal-driven and non-parametric characteristics. Empirical mode decomposition (EMD), as a typical adaptive decomposition method, was proposed by the Chinese American [PERSON] in 1998. The combination of EMD and Hilbert transform constitutes the Hilbert-Huang transform, which successfully realizes the self-adapted time-frequency analysis of complex signals. In 2006, [PERSON] proposed a new adaptive signal time-frequency analysis method--intrinsic time-scale decomposition (ITD). Based on the good adaptability of ITD, it has been widely used in biomedical signal analysis and has achieved corresponding results. Compared with EMD, ITD has the advantages of fast calculation speed and online signal decomposition. LMD is a typical adaptive decomposition method used in feature extraction in the last few decades, and it can decompose a signal into a form of multiple product functions (PFs), which can be expressed as an envelope function multiplied by a pure frequency modulation function. LMD has shown excellent performance in the end effect, mode mixing, and envelope accuracy issues compared to EMD and has been widely used in the field of machinery fault diagnosis [15; 16; 17].