## 1 Introduction  
When analyzing multi-input and multi-output vibration systems, most of the inputs are thought to be independent and to have no influence on each other. However, due to the physical and other connections in the system, mutual influence is inevitable, and a correlation inevitably exists [1]. In near-field measurements, the main signals measured can usually ignore the influence of others. However, when there are different sources with the same frequencies, such as the multiples of the fundamental frequency, correlation analysis becomes necessary. Moreover, blind source separation (BSS) [2] methods, such as independent component analysis (ICA) [3] and fast independent component analysis, have attracted more and more attention in the field of correlation analysis, especially in the cases of medium and strong correlations [4]. Thus, in order to separate vibration and noise sources more accurately, it is necessary to perform correlation analysis. In the time domain, correlation analysis mainly uses an auto-correlation function and a cross-correlation function to denoise and filter signals and to describe the correlation delay of random signal processes [5]. When it comes to the frequency domain, the correlation function is transformed into a coherency function to indicate how well the signals correspond to each other at different frequencies. However, traditional correlation analysis (including correlation and coherence) generally analyzes signals in the time and frequency domain separately and cannot obtain local correlation for non-stationary signals, thus limiting their potential. In practice, signals are usually time-varying and non-stationary. Analyzing signals in only the time or frequency domain will not disclose essential time-varying and frequency-varying characteristics, so the correlation analysis is not optimal. Compared with the Fourier transform (FT), which can only obtain the frequency components in general, the time-frequency (TF) analysis can better uncover the local characteristics and simultaneously provide insight on the time and frequency information of the signal [6]. TF representations provide a powerful tool for analyzing non-stationary time signals and can be used in correlation analysis. The common TF transforms used in spectral decomposition include the short-time Fourier transform (STFT), the Gabor transform, the wavelet transform (WT), the S transform (ST), and the empirical mode decomposition (EMD) [7]. The window function of the STFT is fixed, and the STFT cannot yield multi-resolution information. As for the Wigner distribution and the Hilbert transformation, cross-term and mode mixing will cause some problems [8]. The wavelet transform correlation (WTC) [9] gives a time-frequency correlation. However, the TF representations attained by the WT are in the time-scale(period) domain and not directly in the TF domain. In addition, the scale of the WT is generally not linear, and it is difficult to choose a suitable mother wavelet. In contrast, the ST is able to provide a better TF spectrum than the WT. The ST [10] is superior due to its higher time resolution in the TF spectrogram, ability to realize lossless inverse transformation, and property of retaining the original signal phase. More importantly, the results of the ST are in the time-frequency spectra, which are more intuitive than those in the time-scale spectra of the WT. Moreover, the ST can also be used to analyze complex, valued time series, and it satisfies the principle of linear superposition. Moreover, high-frequency low-amplitude signal is better illustrated in the ST spectrum. However, according to [PERSON], the uncertainty principle, the spectral values transformed from an instantaneous frequency component by these TF transforms and the WTC, would spread over a ribbon, which means the TF resolution is limited [11]. Consequently, false spectral energies would be observed on the TF plane at locations where no spectral energy should exist.