The vibration fault signal is composed of multiple instantaneous non-stationary vibration components, and how to accurately describe the local information of the signal and effectively extract the signal features are the keys to fault identification [
3]. With regard to the diagnosis and identification analysis of vibration fault signals, two kinds of analysis methods are usually adopted. One is time-frequency analysis [
4,
5,
6], such as when the short-time Fourier transform (
STFT), fast Fourier transform (
FFT), wavelet transform (
WT), Hilbert–Huang transform (
HHT) and Wigner–Ville distribution (
WVD) take form in the original or various improved transforms. When the window width of the
STFT is fixed, the time resolution and frequency resolution are mutually restricted and cannot simultaneously reach the optimum, and the time-frequency concentration is not dense. However, there is no cross-term interference [
7]. Reference [
8] proposed an analytical method for aircraft vibration fault signals based on the
STFT. By analyzing various influencing factors, including engine operation, external aerodynamic excitation, equipment operation and electrical interference, the time-frequency characteristics of flight vibration signals can be effectively obtained. In terms of algorithm improvement, reference [
9] combined empirical mode decomposition (
EMD) with the
STFT to address selecting a multi-component signal window function and window width.
EMD is used to carry out the adaptive decomposition of cylinder head vibration signals, and then the
STFT is performed by selectively targeting the window function. The optimal
STFT time-frequency distribution of the original signal is obtained by linear superposition, on the basis of which the time-frequency resolution of the
STFT is effectively advanced. The spectrum leakage and fence effect of
FFT damping characteristics and local characteristics cannot be analyzed. To facilitate algorithm improvement [
10,
11], reference [
12] proposed a characteristic extraction method for rotor vibration signal characteristics based on the
FFT and empirical wavelet transform (
EWT), combining the rotor characteristic frequency with
EWT modal components to form multi-dimensional characteristic vectors, and the rotor states were identified by the K-means clustering method. The
WT has multi-resolution characteristics, can characterize the local details of the signal, and is sensitive to singularity. It also has the ability to quickly capture fault mutations and has no cross-term interference, but it has difficulty extracting the attenuation characteristics of the signal.
WVD can extract the edge characteristics and instantaneous frequency of vibration signals well, but it cannot accurately demonstrate the multi-component signals due to the existence of cross-term interference. The
HHT decomposes the vibration signal by
EMD. Then, each decomposed intrinsic mode function (
IMF) component is transformed by the Hilbert transform, and the instantaneous frequency and amplitude of the signal are obtained, thus fully characterizing the time-frequency distribution of the signal. However, the endpoint effect and mode aliasing effect of the
HHT are highlighted. The original S transform (
ST) is based on the
STFT and
WT, and exhibits the excellent characteristics of both; that is, it has variable multi-resolution and keeps the absolute phase of each component unchanged. It has strong sensitivity to the non-stationary characteristics of the signal transients, but the time-frequency concentration is not high. Since the window function of the
ST is constant, it lacks adaptability to time-frequency characteristics. For this reason, the amplitude stretch factor and frequency-scale stretch factor of the window function are introduced to obtain the generalized S transform (
GST). The
GST can adaptively adjust window width with frequency change, but the non-optimal value of adjusting parameters has a great impact on the window function [
13,
14]. These time-frequency analysis methods complete fault diagnosis by extracting the characteristic information of the vibration fault signal. The application of this method does not need prior knowledge, but it cannot accurately represent the more complex time-frequency characteristics of non-stationary signals [
15]. Additionally, the vibration fault signal is usually weak, and is easily concealed by noise or other feature information, which affects the identification of rotor faults [
16]. G.Yu’s two references have clearly shown that the time-frequency effect of the synchroextracting transform (
SET) is better than that of the synchrosqueezing transform [
17,
18]. For
ST, there is no need to compare and analyze the synchrosqueezing S transform and the synchroextracting S transform. Based on the
WT, the synchrosqueezing wavelet transform (
SWT) uses the synchrosqueezing method to redistribute the energy of the time-scale plane and “squeeze” the energy of the original time-frequency spectrum to the vicinity of the time-frequency ridge, which is not suitable for processing high-frequency signals [
19,
20].
As the non-parametric spectral estimation method, the most classic is the periodogram method. In 1967, Welch improved the periodogram method and proposed the Welch method, which uses smoothing and windowing measures to reduce the variance of frequency spectrum estimation, and has been effectively applied in practice [
21]. With the development of modern spectrum estimation methods, Schmidt proposed multiple signal classification (
Music) in 1986 [
22,
23,
24] and Roy et al. proposed estimation of signal parameters by rotational invariance techniques (
ESPRIT) in 1987 [
25]. The minimum variance spectrum estimation method, which was proposed by Capon in 1969, can adaptively construct filter banks through the observed signals to make the frequency signals of interest pass through without distortion, and can suppress the interference and noise of other frequency components to the greatest extent at the same time [
26]. In 2009, Stoica et al. proposed an iterative adaptive approach (
IAA) based on weighted least squares. Compared with Capon,
IAA obtains higher resolution through cyclic iteration, and can process spectrum estimation in the case of uniform sampling and non-uniform sampling at the same time [
27,
28,
29].
The other analysis method is intelligent identification; for example, back propagation neural network, deep belief network, convolutional neural network and other algorithms. Since the vibration fault of the motor rotor is unpredictable compared with intelligent identification methods, time-frequency analysis methods are more universal in the identification of vibration fault signals, and time-frequency concentration is often used as an evaluation index. Through the above analysis, in order to promote the signal extraction of tiny vibration faults in a motor rotor and meet the needs of multi-component processing without cross-term interference, energy attenuation or false information interference while achieving high time-frequency concentration, in this paper, based on the original
ST, the
SET is introduced to extract the time-frequency coefficients at the time-frequency ridge line of the original rotor vibration fault signal to obtain a new time-frequency spectrum with high concentration [
30]. The combination of the original
ST and
SET is called the synchroextracting S transform (
SEST). The time-frequency distribution of the
SEST is presented as a clear straight line. Compared with the
STFT,
GST,
HHT,
WT and
ST, the time-frequency concentration of the
SEST is noticeably increased, the anti-noise effect is significant, and the rotor vibration fault signal can be clearly identified.