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A Time–Frequency Correlation Analysis Method of Time Series Decomposition Derived from Synchrosqueezed S Transform

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A Time–Frequency Correlation Analysis Method of Time Series Decomposition Derived from Synchrosqueezed S Transform applied sciences Article A Time–Frequency Correlation Analysis Method of Time Series Decomposition Derived from Synchrosqueezed S Transform Gaoyuan Pan , Shunming Li * and Yanqi Zhu College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210000, China; [email protected] (G.P.); [email protected] (Y.Z.) * Correspondence: [email protected]; Tel.: +86-136-0519-9671 Received: 23 January 2019; Accepted: 6 February 2019; Published: 22 February 2019 Abstract: Traditional correlation analysis is analyzed separately in the time domain or the frequency domain, which cannot reflect the time-varying and frequency-varying characteristics of non-stationary signals. Therefore, a time–frequency (TF) correlation analysis method of time series decomposition (TD) derived from synchrosqueezed S transform (SSST) is proposed in this paper. First, the two-dimensional time–frequency matrices of the signals is obtained by synchrosqueezed S transform. Second, time series decomposition is used to transform the matrices into the two-dimensional time–time matrices. Third, a correlation analysis of the local time characteristics is carried out, thus attaining the time–frequency correlation between the signals. Finally, the proposed method is validated by stationary and non-stationary signals simulation and is compared with the traditional correlation analysis method. The simulation results show that the traditional method can obtain the overall correlation between the signals but cannot reflect the local time and frequency correlations. In particular, the correlations of non-stationary signals cannot be accurately identified. The proposed method not only obtains the overall correlations between the signals, but can also accurately identifies the correlations between non-stationary signals, thus showing the time-varying and frequency-varying correlation characteristics. The proposed method is applied to the acoustic signal processing of an engine–gearbox test bench. The results show that the proposed method can effectively identify the time–frequency correlation between the signals. Keywords: time–frequency correlation; synchrosqueezed S transform; time series decomposition; coherence 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 Appl. Sci. 2019, 9, 777; doi:10.3390/app9040777 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 777 2 of 17 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 Heisenberg, 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. The existence of the false spectral energies may very likely cause mistakes in interpreting the results of the spectral decomposition [12]. Moreover, the “false bandwidth” of the frequency axis has a great influence on the correlation analysis. One possible way to minimize such errors is to improve the TF resolution. The generalized S transform [13] and the modified S transform have been proposed to deal with the TF resolution, but they have not solved the problem perfectly. Synchrosqueezed transform (SST) [14,15] has recently been identified by Daubechies et al. to serve as an empirical mode decomposition-like tool. The SST, a special case of reassignment methods, aims to ‘sharpen’ TF representation but has the advantage of allowing the reconstruction of the signals. The SST not only has a similar TF resolution to the EMD, but also has a rigorous mathematical foundation, which provides us with the possibility of making modifications on the formula of the SST. An ST-based SST, which we refer to as the synchrosqueezed S transform (SSST) [16,17], has been proposed in recent years to improve the TF resolution. The SSST retains the characteristics of the ST and the SST, so it is considered to be useful. The advantages of the SSST can be summarized as follows: (a) the results are directly in the TF domain and not the time–scale domain; (b) the frequency axis presents a linear and uniform distribution, which meets the basic requirements of the inverse Fourier transform; (c) the algorithm is reversible and reconfigurable; (d) the TF resolution is improved; and (e) the SSST can be used to compute complex valued time signals. The frequency axis of the SSST is directly in the frequency domain. The frequency domain signals cannot be computed by correlation analysis directly, so they need to be converted into the time domain. Appl. Sci. 2019, 9, 777 3 of 17 By using the inverse Fourier transform, the SSST can be transformed from the TF domain into the time–time domain, and a time series decomposition (TD) [18,19] has been derived to achieve this. In this paper, a time series decomposition algorithm derived from the SSST is proposed, and TF correlation is analyzed in the TF plane, thus obtaining the correlations varying with time and frequency. In Section2, we derive formulas for the SSST and the TD, respectively. Then, the main steps of the time–frequency correlation analysis are presented. In Sections3 and4, we apply traditional and the SSST-TD correlation analysis to both synthetic and experimental data and compare the corresponding results. We present our conclusions in Section5. 2. Principles 2.1. S Transform (ST) Before presenting with the details of the synchrosqueezed S transform, the S transform is first introduced. The ST is a signal processing method proposed by Stockwell [20]. For x(t), the ST is defined as follows: j f j Z ¥ −(t − b)2 f 2 STx( f , b) = p x(t)exp( )exp(−i2π f t)dt. (1) 2π −¥ 2 The basic wavelet function in the ST is defined as follows: j f j −(t − b)2 f 2 w f (t) = p exp( ) (2) 2π 2 where x(t) represents a signal, STx represents the ST of x(t), f denotes frequency, t denotes time, b denotes the time shift, and i is the imaginary unit. The ST is invertible. The inverse transform
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