## 1 Introduction  
The dynamic response of the transmission component under time-varying conditions, which are recognized as the working environment for most mechanical equipment in the engineering field, will show strong non-stationarity [1; 2; 3]. Even in healthy conditions, the characteristic frequency and amplitude of vibration signals will vary with the operating conditions. If the transmission component fails, it will be impossible to determine whether the change of its dynamic response characteristics is caused by the failure or the varying working conditions. Therefore, one key constituent is to realize fault feature extraction and state identification under time-varying conditions [4; 5]. Recently, some researchers have put forward a machine learning and statistical framework to solve the similar problem described in this paper. However, these methods are limited to practical application because mechanical failure is typically a small sample situation [6; 7; 8]. As a very powerful signal processing method, time-frequency (TF) analysis has drawn much attention among researchers in structural health monitoring, biomedicine and other fields [9; 10]. The main reason for this is its simultaneous examination for the complicated signals on two scales of time and frequency. Built on the inner product operation between the analytical signal and basis function [11; 12; 13], traditional TF analysis methods, such as Fast Fourier transform (FFT) and Wavelet Transform (WT), can neither match the time-varying signals well, nor effectively process the non-stationary signals. Moreover, the multi-component signals always contain heavy background noise. Therefore, the characteristic components are easily masked by irrelevant signals, and thus Short-time Fourier transform (STFT) and Wigner distribution (WD) have been proposed [14; 15]. Despite an improved version of FFT, STFT remains within the limits of implementing adaptive selection of window function [16] and, inevitably, cross interference feature of WD inevitably exists, which directly leads to poor readability of the TF plane. Recently, Synchrosqueezing Transform (SST) and Linear Chirp Transform (LCT) have become research hotspots [17; 18]. As a post-processing method, SST has made it possible to synchrosqueeze and rearrange the TF coefficients along the frequency axis, which aimed to realize the concentration of TF energy [19]. Subsequently, several interesting methods have been presented, such as Synchroextracting Transform (SET) and Second order Synchrosqueezing Transform [20; 21]. These methods, however, fail to analyze the strong modulated signal with heavy noise. Since the TF coefficient rearrangement is executed from the frequency direction, it has poor processing ability for the common impulse signal of mechanical equipment. Different from SST, LCT generates a rotation angle through a fixed rotation operator for the horizontal window in STFT. When the angle between the rotated window function and the frequency modulated signal is zero, the best energy concentration is reached [22]. However, when the signal is nonlinear, the frequency modulation of the signal is time-varying. As a result, only one rotation operator cannot fully meet the requirements of energy concentration at all times. In general, when dealing with nonlinear and non-stationary signals, the energy divergence problem is unavoidable in the above methods. Reassignment method (RM) realizes the reallocation of TF transformation coefficients in two directions of time and frequency [23]. Compared with previous methods, RM not only has a higher TF resolution but it also reaches the ideal TF representations more easily. However, RM does not support signal reconstruction [24]. Moreover, the information extracted from the distribution parameters of the ridges, which are utilized for decomposing signal and reconstructing useful one, is closely related to the instantaneous frequency variation of the signal.