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:

| f | Z ∞ −(t − b)2 f 2 STx( f , b) = √ x(t)exp( )exp(−i2π f t)dt. (1) 2π −∞ 2

The basic wavelet function in the ST is defined as follows:

| f | −(t − b)2 f 2 w f (t) = √ 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 corresponding to the ST is as follows:

Z ∞ Z ∞ x(t) = ST( f , b)exp(i2π f t)dbd f (3) −∞ −∞

The ST is similar to the STFT but with a Gaussian window whose width scales inversely and whose height scales linearly with frequency. The ST can obtain a varying TF resolution of the signals and inherits the advantages of both the STFT and the WT.

2.2. Synchrosqueezed S Transform (SSST) The ST of x(t) is defined in Equation (1). Thus, the following is true:

1  t2  ψ(t) = √ exp exp(i2πt) (4) 2π 2

According to the Parseval theorem, Equation (1) can also be expressed as follows:

Z +∞ 1 −1 STx( f , b) = exp(−i2π f b) X(ξ)ψ( f ξ)exp(ibξ)dξ (5) 2π −∞ where ξ is the angular frequency, X(ξ) denotes the FT result of the signal x(t), and ψ(ξ) denotes the complex conjugate of the FT result of ψ(t). In particular, if ξ < 0, X(ξ) = 0. For a purely harmonic wave x(t) = A cos(2π f0t), its FT is presented as follows:

X(ξ) = Aπ[δ(ξ − 2π f0) + δ(ξ + 2π f0)] (6) where A and f0 denote, respectively, the amplitude and frequency. Appl. Sci. 2019, 9, 777 4 of 17

Then, we can obtain the instantaneous frequency of the signal x(t) as follows:

∂ ST ( f , b) = −iπA( f − f )exp(−i2π( f − f )b)ψ(2π f −1 f ) (7) ∂b x 0 0 0

− ∂ fe( f , b) = f + [i2πST ( f , b)] 1 ST ( f , b). (8) x ∂b x For the harmonic wave x(t), the instantaneous frequency can be obtained by the following:

fe( f , b) = f0 (9) that is, the instantaneous frequency of the standard sinusoidal signal is the frequency of the signal itself. Given the candidate instantaneous frequency fe( f , b) at each location ( f , b) on the spectrogram, the TF spectrogram now becomes the time–instantaneous frequency spectrogram, i.e., locations ( f , b) are mapped to ( fe( f , b),b)[21]. The next step is to reassign (squeeze) the value of the time–instantaneous frequency spectrogram. Note that the STx( f , b) is only computed at discrete values fk, and frequencies on the “squeezed” TF spectrogram are fel. Hence, we squeeze the TF coefficients STx( f , b) together via the same instantaneous frequency fe( f , b). By superimposing the time–frequency spectra values among the   ∆ fel ∆ fel frequency range fel − 2 , fel + 2 on the frequency point fel, we then obtain a sharpened TF representation [22,23], thus getting the SSST. For signal x(t), the SSST [24] is defined as follows:

−1 SSSTx( fel, b) = (∆ fel) × ∑ |STx( fk, b)| fk∆ fk (10) 1 fk:| fel ( fk,b)− fel |≤ 2 ∆ fel where fk is the discrete frequency obtained by the ST; ∆ fk = fk − fk−1; fel denotes the center frequency of the SSST; ∆ fel denotes the bandwidth; and ∆ fel = fel − fel−1. Equation (10) shows that the SSST has higher accuracy in its TF decomposition ability. Similar to the ST, the signal can be reconstructed from the SSSTx [25]. The signal can be reconstructed as follows: −1 x(b) = RE[(CψCϕ) ∑ SSSTx(efl, b)∆efl] (11) l where RE means the real part and

Z +∞ −1 2 Cψ = 0.5 −ψˆ(ξ)ξ dξ, Cϕ = exp(−i[2π f b + ϕ( f , b)]) f (12) 0 Details of the derivation can be found in the AppendixA.

2.3. Time Series Decomposition Algorithm (TD) In order to transform the multi-correlation analysis from the one-dimensional time domain to the TF domain, it is necessary to firstly transform the signal into a two-dimensional time domain. A time series decomposition algorithm (TD) can be derived from the SSST to achieve this. If the SSST is transformed by the inverse FT, a time series decomposition is obtained. For the TF spectrum attained by the SSST, if the time b in the TF domain is fixed but the frequency f is variable, then the TD is as follows: Z +∞ TDt0, b
= SSST( f , b)expi2π f t0
d f (13) −∞ where t’ denotes the time attained by the inverse FT on the frequency f. Appl. Sci. 2019, 9, 777 5 of 17

Similar to the SSST, the TD is reversible:

Z +∞ TDt0, b
dt0 = x(t). (14) −∞

As the elements of the SSST are complex valued numbers, the elements of the TD are also complex valued. The TD can transform an arbitrary time series into a set of time-limited, localized two-dimensional time series composed of each component, which can highlight the local feature of the signal. The TD decomposes the signal into N-components, and the sum amplitude of these components is equal to that of the original signal. Therefore, in the process of decomposition, the amplitude of each decomposed component will be reduced compared with the original, but each component can highlight the local characteristics.

2.4. Correlation Analysis In traditional correlation analysis, the ordinary coherence is the basis of the correlation analysis. The partial coherence is the ordinary coherence between the conditional signals, where the conditional signals mean the casual effects of the others removed in a linear least square sense and the virtual coherence is the ordinary coherence between a signal and a principle component. For x and y, the magnitude-squared coherence function (MSCF) [26] is defined as follows:

2 2 Gxy( f ) γxy ( f ) = (15) Gxx( f ) · Gyy( f ) where Gxx( f ) and Gyy( f ) denote the power spectral density and Gxy( f ) denotes the cross spectral density of x and y. The MSCF is a function of the frequency with values between 0 and 1. These values indicate how well the two signals correspond to each other at different frequencies. The greater the values, the stronger the correlation. The traditional correlation method is inaccurate for non-stationary signals. Thus, a time–frequency correlation method is proposed to process such non-stationary signals. As the TD is a two-dimensional time series matrix, if we fix time in the b-axis in the TF domain and compute time in the t’-axis using Equation (15), then we can get a local correlation between the signals at each time b and frequency f, and that is the synchrosqueezed S transform–time series decomposition time–frequency (SSST-TD TF) correlation, an extension to the traditional correlation analysis.

2.5. Steps of the SSST-TD TF Correlation Analysis To obtain the local correlation between the signals, a synchrosqueezed S transform–time series decomposition time–frequency correlation method is proposed. The main steps are as follows: (1) For the time domain signal x and y, obtain the time–frequency spectrum by the SSST. (2) Perform time series decomposition on the time–frequency spectrum. That is, the inverse Fourier transform is applied to the frequency-axis of the SSST to obtain the two-dimensional complex valued time series matrix. (3) Carry out correlation analysis on each local time b between the two-dimensional time series matrix of x and y, thus getting the local correlation between the signals. (4) Observe the time–frequency distributions of the signals in the ST/SSST time–frequency spectrum and then analyze and judge the local correlation between the signals.

3. Synthetic Examples

3.1. Simulation of Cosine Signals To verify the SSST-TD TF correlation method, a simulation was conducted. Because stationary signals, particularly the cosine ones, are a special case of the signals, analyzing stationary signals Appl. Sci. 2019, 2, x FOR PEER REVIEW 5 of 16

159 conditional signals mean the casual effects of the others removed in a linear least square sense and 160 the virtual coherence is the ordinary coherence between a signal and a principle component. For 𝑥 161 and 𝑦, the magnitude-squared coherence function (MSCF) [26] is defined as follows: 𝐺(𝑓) 𝛾 (𝑓)= (15) 𝐺(𝑓)⋅𝐺(𝑓)

162 where 𝐺(𝑓) and 𝐺(𝑓) denote the power spectral density and 𝐺(𝑓) denotes the cross spectral 163 density of 𝑥 and 𝑦. The MSCF is a function of the frequency with values between 0 and 1. These 164 values indicate how well the two signals correspond to each other at different frequencies. The 165 greater the values, the stronger the correlation. 166 The traditional correlation method is inaccurate for non-stationary signals. Thus, a time– 167 frequency correlation method is proposed to process such non-stationary signals. As the TD is a two- 168 dimensional time series matrix, if we fix time in the b-axis in the TF domain and compute time in the 169 t’-axis using Equation (15), then we can get a local correlation between the signals at each time b and 170 frequency f, and that is the synchrosqueezed S transform–time series decomposition time–frequency 171 (SSST-TD TF) correlation, an extension to the traditional correlation analysis.

172 2.5. Steps of the SSST-TD TF Correlation Analysis 173 To obtain the local correlation between the signals, a synchrosqueezed S transform–time series 174 decomposition time–frequency correlation method is proposed. The main steps are as follows: 175 (1) For the time domain signal 𝑥 and 𝑦, obtain the time–frequency spectrum by the SSST. 176 (2) Perform time series decomposition on the time–frequency spectrum. That is, the inverse 177 Fourier transform is applied to the frequency-axis of the SSST to obtain the two-dimensional 178 complex valued time series matrix. 179 (3) Carry out correlation analysis on each local time b between the two-dimensional time series 180 matrix of 𝑥 and 𝑦, thus getting the local correlation between the signals. 181 (4) Observe the time–frequency distributions of the signals in the ST/SSST time–frequency 182 spectrum and then analyze and judge the local correlation between the signals.

183 3. Synthetic Examples

Appl. Sci. 2019, 9, 777 6 of 17 184 3.1. Simulation of Cosine Signals 185 To verify the SSST-TD TF correlation method, a simulation was conducted. Because stationary can show the advantage and stability of the proposed method. Thus, we took two cosine signals. 186 signals, particularly the cosine ones, are a special case of the signals, analyzing stationary signals can Signal x was cosine with frequency 30 Hz, and signal y was the sum of 30 Hz and 100 Hz. The time 187 show the advantage and stability of the proposed method.1 Thus, we took two cosine signals. Signal resolution was 0.1 ms, and the sampling points were 4096. The traditional correlation coefficient 188 𝑥 was cosine with frequency 30 Hz, and signal 𝑦 was the sum of 30 Hz and 100 Hz. The time (directly computed in the time domain) between x and y is shown in Figure1. 189 resolution was 0.1 ms, and the sampling points were 14096. The traditional correlation coefficient 190 (directly computed in the time( domain) between 𝑥 and 𝑦 is shown in Figure 1. x = cos(30 × 2πt) 𝑥 = cos(30π ×2 𝑡) (16) y1 = cos(30 × 2πt) + cos(100 × 2πt) (16) 𝑦 = cos(30×2π𝑡) + cos(100π ×2 𝑡)

1.0

0.5

0.0 coeffitiont correlation 0 50 100 150 200 frequency/Hz 191 Figure 1. The traditional correlation coefficient between x and y . 192 Figure 1. The traditional correlation coefficient between 𝑥 and 1𝑦. Due to the existence of a window function and the time resolution, a correlation coefficient has a 193 Appl. Sci. Due2019, to2, x the FOR existence PEER REVIEW of a window function and the time resolution, a correlation coefficient 6 of 16 has certain bandwidth. Here, the correlation coefficient was 1 near 30 Hz, which means the two signals 194 a certain bandwidth. Here, the correlation coefficient was 1 near 30 Hz, which means the two signals 195 werewere correlated. correlated. Moreover, Moreover, it itwas was below below 0.2 0.2 near near 100 100 Hz, Hz, which which is isconsidered considered uncorrelated. uncorrelated. At At other other 196 frequencies,frequencies, the the correlation correlation coefficient coefficient fluctuated fluctuated greatly. greatly.

197 Then,Then, we we applied applied the ST ST and and the the SSST SSST to the to thesignals signalsx and 푥y 1 and. The 푦 magnitude . The magnitude of the TF of spectrum the TF is 198 spectrumshown inis Figure shown2. in Both Figure the ST 2. andBoth the the SSST ST obtainedand the signalSSST obtained distributions signal in thedistributions TF domain. in Moreover, the TF 199 domain.TF representations Moreover, TF gave representations an intuitive display gave an of intuitive the signals. display Comparing of the Figuresignals.2A–D, Comparing we can Figures conclude 200 2A–D,that thewe TFcan energy conclude bandwidth that the of TF the energy ST was bandwidth much wider of than the thatST was of the much SSST. wider The existence than that of theof the false 201 SSST.spectral The existence energies isof likely the false to cause spectral mistakes energies in interpretingis likely to cause the resultsmistakes of in the interpreting spectral decomposition, the results 202 of especiallythe spectral in thedecomposition, wide energy bandwidthespecially in around the wide 100 Hz.energy Using bandwidth the SSST, around the TF resolution100 Hz. Using was highly the 203 SSST,improved, the TF and resolution the energy was concentrated highly improved, where the and actual the frequencies energy concentrated were located. where the actual 204 frequencies were located.

205 206 (A) (B)

207 208 (C) (D)

209 FigureFigure 2. The 2. The time-frequency time-frequency (TF) (TF) representations representations of ofthe the signals: signals: (A) ( Athe) the S transform S transform (ST) (ST) of of푥, x(B,() Bthe) the

210 STST of of푦y, 1(,(C)C the) the synchrosqueezed synchrosqueezed S Stransform transform (SSST) (SSST) of of 푥x, ,and and ( (DD)) the the SSST SSST of of 푦y1.

211 Next,Next, we we decomposed decomposed the the TF TF spectrum spectrum of ofthe the ST ST and and the the SSST SSST by by the the TD, TD, that that is, is,the the ST-TD ST-TD and and 212 thethe SSST-TD, SSST-TD, and and carried carried out out a correlation a correlation analysis. analysis. The The correlation correlation coefficient coefficient is shown is shown in inFigure Figure 3. 3. 213 Obviously, near 30 Hz, both the ST-TD and the SSST-TD detected the frequency correlation between 214 the signals. However, the bandwidth of the ST-TD correlation was wider than that of the SSST-TD, 215 even worse than the traditional method. Near 100 Hz, the result of the ST was inaccurate due to the 216 low TF resolution; that is, the ST was suitable for acquiring the TF distribution of signals but not 217 suitable for calculating the correlation coefficient. Thus, the SSST-TD TF correlation analysis not only 218 obtained the same result as the traditional method, but also obtained the TF correlation distribution. 219 Figure 4 shows three SSST-TD TF correlation coefficients at different times. It is obvious that for 220 the cosine signals, the overall trend of the SSST-TD TF correlation coefficient is consistent with that 221 of the traditional method with small differences at the different frequencies. The main difference is 222 that the SSST-TD TF correlation coefficient changes with time. Appl. Sci. 2019, 9, 777 7 of 17

Obviously, near 30 Hz, both the ST-TD and the SSST-TD detected the frequency correlation between the signals. However, the bandwidth of the ST-TD correlation was wider than that of the SSST-TD, even worse than the traditional method. Near 100 Hz, the result of the ST was inaccurate due to the low TF resolution; that is, the ST was suitable for acquiring the TF distribution of signals but not suitable for calculating the correlation coefficient. Thus, the SSST-TD TF correlation analysis not only obtained the same result as the traditional method, but also obtained the TF correlation distribution. Figure4 shows three SSST-TD TF correlation coefficients at different times. It is obvious that for the cosine signals, the overall trend of the SSST-TD TF correlation coefficient is consistent with that of the traditional method with small differences at the different frequencies. The main difference is that theAppl.Appl. SSST-TD Sci.Sci. 20192019,, 22 TF,, xx FORFOR correlation PEERPEER REVIEWREVIEW coefficient changes with time. 77 ofof 1616

223223 224224 ((AA)) ((BB))

225 Figure 3. The correlation coefficients between 𝑥 and 𝑦 attained by (A) the S transform–time series 225 FigureFigure 3.3. TheThe correlation correlation coefficients coefficients between between 𝑥 andx and 𝑦 y attained1 attained by ( byA) the (A) S the transform–time S transform–time series 226226 seriesdecompositiondecomposition decomposition (ST-TD),(ST-TD), (ST-TD), andand ((BB and)) thethe ( B synchrosqueezedsynchrosqueezed) the synchrosqueezed SS transform–transform– S transform–timetimetime seriesseries decomposition seriesdecomposition decomposition (SSST-(SSST- 227227 (SSST-TD).TD).TD).

0.15s0.15s 0.1525s0.1525s 1.01.0 0.155s0.155s

0.50.5 coeffitiont correlation coeffitiont correlation 0.00.0 00 100 100 200 200 frequency/Hz 228228 frequency/Hz 229 FigureFigure 4. 4. TheThe correlation correlation coefficient coefficient between between x𝑥 andandy 𝑦1 of of different different times. times. 229 Figure 4. The correlation coefficient between 𝑥 and 𝑦 of different times. 3.2. Simulation of Different Phase Signals 230230 3.2.3.2. SimulationSimulation ofof DifferentDifferent PhasePhase SignalsSignals Considering that the SSST-TD TF correlation changes with time, the phase change must be 231 Considering that the SSST-TD TF correlation changes with time, the phase change must be 231 considered.Considering In order that to analyzethe SSST-TD the influence TF correlation of the phase, changes the with phase time, of y theis advanced phase change by π and mustπ/2: be 232 𝑦1 232 considered.considered. InIn orderorder toto analyzeanalyze thethe influenceinfluence ofof thethe phase,phase, thethe phasephase ofof 𝑦 isis advancedadvanced byby ππ andand ππ/2:/2: ( π
𝜋𝜋 y2𝑦==cos cos(30×2(30 × 2πt) +𝑡)cos + 100cos(100× 2πt + ×2 𝑡+2 ) 𝑦 = cos(30×2ππ𝑡) + cos(100ππ ×2 𝑡+ ) (17) y = cos(30 × 2πt) + cos(100 × 2πt + π) 22 (17)(17) 3𝑦 = cos(30×2 𝑡) + cos(100 ×2 𝑡+𝜋) 𝑦 = cos(30×2ππ𝑡) + cos(100ππ ×2 𝑡+𝜋) 233233 AccordingAccordingAccording to toto the thethe previous previousprevious steps, steps,steps, the thethe three threethree traditional tradittraditionalional correlation correlationcorrelation coefficient coefficientcoefficient curves curvescurves are areare shown shownshown in inin 234234 FigureFigureFigure5 .5.5. Under UnderUnder the thethe influence influenceinfluence of the ofof phase thethe phase change,phase change,change, the traditional thethe traditionaltraditional correlation correlationcorrelation coefficient coefficientcoefficient was consistent waswas 235235 nearconsistentconsistent 30 Hz nearnear and 1003030 HzHz Hz, andand with 100100 great Hz,Hz, differenceswithwith greatgreat differencesdifferences at other frequencies. atat otherother frequencies.frequencies.

xx&&yy1 xx&&yy2 xx&&yy3 1.01.0 1 2 3

0.50.5 coeffitiont correlation coeffitiont correlation 0.00.0 00 100 100 200 200 frequency/Hz 236236 frequency/Hz 237 Figure 5. The traditional correlation coefficient between 𝑥 and 𝑦, 𝑥 and 𝑦 , and 𝑥 and 𝑦 . 237 Figure 5. The traditional correlation coefficient between 𝑥 and 𝑦, 𝑥 and 𝑦, and 𝑥 and 𝑦.

238 𝑥 𝑦 𝑥 238 UsingUsing thethe SSST-TDSSST-TD TFTF correlationcorrelation analysis,analysis, thethe correlationcorrelation coefficientscoefficients ofof 𝑥 andand 𝑦 andand 𝑥 andand 239 𝑦 239 𝑦 are are shownshown inin FigureFigure 6.6. ItIt cancan bebe seenseen thatthat thethe overalloverall correlationcorrelation coefficientcoefficient trendtrend waswas consistentconsistent 240240 withwith thethe originaloriginal phasephase withwith smallsmall differencesdifferences atat differentdifferent times.times. TakingTaking 6363 HzHz andand 100100 HzHz asas anan 241241 example,example, thethe timetime domaindomain curvescurves ofof thethe correlaticorrelationon coefficients,coefficients, afterafter removingremoving thethe twotwo ends,ends, areare 242242 shownshown inin FigureFigure 7.7. ItIt cancan bebe seenseen thatthat atat 100100 HzHz,, thethe correlationcorrelation coefficientcoefficient diddid notnot followfollow thethe phasephase 243243 change,change, remainingremaining unchanged,unchanged, whilewhile atat 63Hz,63Hz, ththee correlationcorrelation coefficiencoefficientt variedvaried withwith thethe phasephase 244244 change.change. Therefore,Therefore, comparedcompared withwith traditionaltraditional corrcorrelationelation analysis,analysis, thethe SSST-TDSSST-TD TFTF correlationcorrelation notnot Appl. Sci. 2019, 2, x FOR PEER REVIEW 7 of 16

223 224 (A) (B)

225 Figure 3. The correlation coefficients between 𝑥 and 𝑦 attained by (A) the S transform–time series 226 decomposition (ST-TD), and (B) the synchrosqueezed S transform–time series decomposition (SSST- 227 TD).

0.15s 0.1525s 1.0 0.155s

0.5 coeffitiont correlation 0.0 0 100 200 228 frequency/Hz

229 Figure 4. The correlation coefficient between 𝑥 and 𝑦 of different times.

230 3.2. Simulation of Different Phase Signals 231 Considering that the SSST-TD TF correlation changes with time, the phase change must be

232 considered. In order to analyze the influence of the phase, the phase of 𝑦 is advanced by π and π/2: 𝜋 𝑦 = cos(30×2π𝑡) + cos(100π ×2 𝑡+ ) 2 (17) 𝑦 = cos(30×2π𝑡) + cos(100π ×2 𝑡+𝜋) 233 According to the previous steps, the three traditional correlation coefficient curves are shown in Appl. Sci. 2019, 9, 777 8 of 17 234 Figure 5. Under the influence of the phase change, the traditional correlation coefficient was 235 consistent near 30 Hz and 100 Hz, with great differences at other frequencies. x&y x&y x&y 1.0 1 2 3

0.5 coeffitiont correlation 0.0 0 100 200 236 frequency/Hz

Figure 5. The traditional correlation coefficient between x and y1, x and y2, and x and y3. 237 Figure 5. The traditional correlation coefficient between 𝑥 and 𝑦, 𝑥 and 𝑦, and 𝑥 and 𝑦.

Using the SSST-TD TF correlation analysis, the correlation coefficients of x and y2 and x and y3 are 238 Using the SSST-TD TF correlation analysis, the correlation coefficients of 𝑥 and 𝑦 and 𝑥 and shown in Figure6. It can be seen that the overall correlation coefficient trend was consistent with the 239 𝑦 are shown in Figure 6. It can be seen that the overall correlation coefficient trend was consistent original phase with small differences at different times. Taking 63 Hz and 100 Hz as an example, the time 240 with the original phase with small differences at different times. Taking 63 Hz and 100 Hz as an domain curves of the correlation coefficients, after removing the two ends, are shown in Figure7. It can 241 example, the time domain curves of the correlation coefficients, after removing the two ends, are be seen that at 100 Hz, the correlation coefficient did not follow the phase change, remaining unchanged, 242 shown in Figure 7. It can be seen that at 100 Hz, the correlation coefficient did not follow the phase while at 63Hz, the correlation coefficient varied with the phase change. Therefore, compared with 243 Appl.change, Sci. 2019 remaining, 2, x FOR PEERunchanged, REVIEW while at 63Hz, the correlation coefficient varied with the phase8 of 16 traditional correlation analysis, the SSST-TD TF correlation not only reflected the correlation changing 244 change.Appl. Sci. Therefore, 2019, 2, x FOR compared PEER REVIEW with traditional correlation analysis, the SSST-TD TF correlation not8 of 16 245 withonly reflected time, but the also correlation retained the changing correlation with changing time, but with also theretained phases. the At correlation a high frequency, changing the with phase the 245 only reflected the correlation changing with time, but also retained the correlation changing with the 246 changesphases. At quickly. a high However, frequency, when the phase the frequency changes is qu low,ickly. the However, phase change when will the play frequency an important is low, rolethe 246 phases. At a high frequency, the phase changes quickly. However, when the frequency is low, the 247 andphase cannot change be will ignored. play an important role and cannot be ignored. 247 phase change will play an important role and cannot be ignored.

248 249248 (A) (B) 249 (A) (B) 250 Figure 6. The SSST-TD TF correlation coefficients of (A) 𝑥 and 𝑦 and (B) 𝑥 and 𝑦. 250 FigureFigure 6. 6.The The SSST-TD SSST-TD TF TF correlation correlation coefficients coefficients of of (A ()Ax) and𝑥 andy2 and𝑦 and (B) x (Band) 𝑥 yand3. 𝑦.

x&y x&y x&y 0.30 1 2 3 0.1 x&y1 x&y2 x&y3 x&y x&y x&y 0.30 1 2 3 0.1 x&y1 x&y2 x&y3 0.15 0.15

coeffitiont 0.0 coeffitiont

correlation 0.00 correlation

coeffitiont 0.0 coeffitiont

correlation 0.00 0.10 0.15 0.20 correlation 0.10 0.15 0.20 251 0.10time/s 0.15 0.20 0.10time/s 0.15 0.20 time/s time/s 252251 (A) (B) 252 (A) (B) 253 FigureFigure 7. 7. TheThe correlation correlation coefficients coefficients of of 𝑥x andand y𝑦3at at ( A(A)) 100 100 Hz, Hz, and and ( B(B)) 63 63 Hz. Hz. 253 Figure 7. The correlation coefficients of 𝑥 and 𝑦 at (A) 100 Hz, and (B) 63 Hz. 254 3.3. Simulation of Signals With Noise 254 3.3. Simulation of Signals With Noise Next, we added Gaussian noise to x and y1 with a mean value of 0 and a standard variance of 1/3. 255 Next, we added Gaussian noise to 𝑥 and 𝑦 with a mean value of 0 and a standard variance of 255 Next, we added Gaussian noise to 𝑥 and 𝑦 with a mean value of 0 and a standard variance of 256 The1/3. traditionalThe traditional correlation correlation coefficient coefficient is shown is shown in Figure in8 A.Figure It is clear8A. It that is underclear noisythat under conditions, noisy 256 traditional1/3. The correlation traditional can correlation recognize thecoefficient correlation is betweenshown inx andFigurey1 near 8A. 30 It Hz is andclear the that uncorrelation under noisy 257 conditions, traditional correlation can recognize the correlation between 𝑥 and 𝑦 near 30 Hz and 257 conditions, traditional correlation can recognize the correlation between 𝑥 and 𝑦 near 30 Hz and 258 nearthe uncorrelation 100 Hz. Due tonear the 100 existence Hz. Due of noise,to the theexistence correlation of noise, coefficients the correlation changed coefficients at other frequencies. changed at 258 the uncorrelation near 100 Hz. Due to the existence of noise, the correlation coefficients changed at 259 other frequencies. 259 other frequencies. 1.0 1.0

0.5 0.5 correlation coeffitiont

0.0correlation coeffitiont 0.00 50 100 150 200 0 50 100 150 200 frequency/Hz 260 frequency/Hz 261260 (A) (B) 261 (A) (B) 262 Figure 8. The correlation coefficients between 𝑥 and 𝑦 of (A) the traditional correlation and (B) the 262 Figure 8. The correlation coefficients between 𝑥 and 𝑦 of (A) the traditional correlation and (B) the 263 SSST-TD TF correlation. 263 SSST-TD TF correlation. 264 The correlation coefficient obtained by SSST-TD is shown in Figure 8B. The overall correlation 264 The correlation coefficient obtained by SSST-TD is shown in Figure 8B. The overall correlation 265 coefficient trend was consistent with the noiseless one. The correlation near 30 Hz and uncorrelation 265 coefficient trend was consistent with the noiseless one. The correlation near 30 Hz and uncorrelation 266 near 100 Hz can be accurately identified in the case of noise. Figure 9 shows the correlation 266 near 100 Hz can be accurately identified in the case of noise. Figure 9 shows the correlation 267 coefficients at different times. Except at 30 Hz and 100 Hz, the correlation coefficients at the other 267 coefficients at different times. Except at 30 Hz and 100 Hz, the correlation coefficients at the other Appl. Sci. 2019, 2, x FOR PEER REVIEW 8 of 16

245 only reflected the correlation changing with time, but also retained the correlation changing with the 246 phases. At a high frequency, the phase changes quickly. However, when the frequency is low, the 247 phase change will play an important role and cannot be ignored.

248 249 (A) (B)

250 Figure 6. The SSST-TD TF correlation coefficients of (A) 𝑥 and 𝑦 and (B) 𝑥 and 𝑦.

x&y x&y x&y 0.30 1 2 3 0.1 x&y1 x&y2 x&y3 0.15

coeffitiont 0.0 coeffitiont

correlation 0.00 correlation 0.10 0.15 0.20 0.10 0.15 0.20 251 time/s time/s 252 (A) (B)

253 Figure 7. The correlation coefficients of 𝑥 and 𝑦 at (A) 100 Hz, and (B) 63 Hz.

254 3.3. Simulation of Signals With Noise

255 Next, we added Gaussian noise to 𝑥 and 𝑦 with a mean value of 0 and a standard variance of 256 1/3. The traditional correlation coefficient is shown in Figure 8A. It is clear that under noisy 257 conditions, traditional correlation can recognize the correlation between 𝑥 and 𝑦 near 30 Hz and Appl. Sci. 2019, 9, 777 9 of 17 258 the uncorrelation near 100 Hz. Due to the existence of noise, the correlation coefficients changed at 259 other frequencies.

1.0

0.5

correlation coeffitiont 0.0 0 50 100 150 200 frequency/Hz 260 261 (A) (B)

262 FigureFigure 8. The8. The correlation correlation coefficients coefficients between betweenx and𝑥 andy1 of𝑦( Aof) ( theA) the traditional traditional correlation correlation and and (B) ( theB) the 263 SSST-TDSSST-TD TF TF correlation. correlation.

264 TheThe correlation correlation coefficient coefficient obtained obtained by by SSST-TD SSST-TD is shownis shown in Figurein Figure8B. 8B. The The overall overall correlation correlation 265 coefficientcoefficient trend trend was was consistent consistent with wi theth the noiseless noiseless one. one. The The correlation correlation near near 30 30 Hz Hz and and uncorrelation uncorrelation 266 nearAppl.near Sci. 100 2019100 Hz, 2 canHz, x FOR becan accuratelyPEER be REVIEWaccurately identified identified in the case in ofthe noise. case Figureof noise.9 shows Figure the correlation9 shows the coefficients correlation9 of 16 267 at differentcoefficients times. at different Except attimes. 30 Hz Ex andcept 100at 30 Hz, Hz the and correlation 100 Hz, the coefficients correlation at thecoefficients other frequencies at the other 268 fluctuatedfrequencies under fluctuated the influence under the of noise. influence As the of proposednoise. As methodthe proposed is a time–frequency method is a extensiontime–frequency to the 269 ordinaryextension coherence, to the ordinary the method coherence, inherits the themethod characteristics inherits the of characteristics traditional correlation; of traditional that is, correlation; under the 270 influencethat is, under of noise, the bothinfluence the traditional of noise, correlationboth the traditional analysis and correlation the proposed analysis analysis and are the influenced. proposed 271 analysis are influenced. 1.0 0.12s 0.175s 0.2s

0.5 coeffitiont correlation 0.0 0100200 272 frequency/Hz

273 FigureFigure 9. 9. TheThe correlation correlation coefficient coefficient between between 𝑥x and y𝑦1 at at different different times. times.

274 3.4. SimulationSimulation ofof Non-StationaryNon-Stationary Signals Signals 275 For stationary signals with or without noise, bothboth the traditional correlation analysis and the 276 SSST-TD TFTF correlationcorrelation analysis analysis can can identify identify the the correlation correlation between between the the signals. signals. However, However, the the SSST-TD SSST- 277 TFTD correlationTF correlation method method can bettercan better identify identify the correlation the correlation at different at different times. times. Practical Practical signals signals are often are 278 non-stationary,often non-stationary, so the so following the following signals signals were designed were designed to analyze to theanalyze advantages the advantages of the proposed of the 279 method.proposed Theymethod. consisted They consisted of cosine, of cosine cosine, frequency cosine frequency modulation modulation (FM), and (FM), shock. and Weshock. also We added also 280 Gaussianadded Gaussian noise to noise both to with both a with mean a mean value value of 0 and of 0 a and standard a standard variance variance of 1. Theof 1. timeThe time resolution resolution was 281 1was ms, 1 andms, theand sampling the sampling points points were were 4096. 4096. 282 The time domain signals are shown in Figure 1010.. The traditional correlation coefficient coefficient is shown 283 in Figure 1111.. Obviously, Obviously, it it is is impossible impossible to to distinguish distinguish the the correlation correlation between between 𝑥x and y𝑦.. For the 284 frequency 30 Hz, Hz, it it occurred occurred at at the the second second and and third third stage stage in in 𝑥x and only the third stage stage in in 𝑦y,, so the 285 correlation coefficients was expected to change and not be constant. Thus, the correlation coefficient 286 was not accurate.

𝑥 = 2𝑒𝑥𝑝(−2π𝑡)×𝑐𝑜𝑠(300×2𝜋𝑡) 𝑥 = 𝑐𝑜𝑠(30×2π𝑡 + 2 𝑐𝑜𝑠( 100 × 2𝜋𝑡 − 2 𝑐𝑜𝑠( 10×2𝜋𝑡))) 𝑥 = 𝑐𝑜𝑠(30×2π𝑡) (18) 𝑦 = 2 𝑒𝑥𝑝( − 2𝜋𝑡) × 𝑐𝑜𝑠( 300×2𝜋𝑡) 𝑦 = 2 𝑐𝑜𝑠( 100π ×2 𝑡−2𝑐𝑜𝑠(10×2𝜋𝑡)). 𝑦 = 𝑐𝑜𝑠(30×2π𝑡)

6 4 y x 0 0

-6 -4 01234 01234 287 time/s time/s

288 Figure 10. The time domain signals of 𝑥 and 𝑦.

1.0

0.5 coeffitiont correlation 0.0 0 50 100 150 200 289 frequency/Hz 290 Figure 11. The traditional correlation coefficient between 𝑥 and 𝑦.

291 The SSST of 𝑥 and 𝑦 is shown in Figure 12. It is clear that the TF distribution of 𝑥 and 𝑦 can 292 be attained by the SSST, and the energy is gathered at the actual frequency. Moreover, the SSST can Appl. Sci. 2019, 2, x FOR PEER REVIEW 9 of 16 Appl. Sci. 2019, 2, x FOR PEER REVIEW 9 of 16

268268 frequenciesfrequencies fluctuatedfluctuated underunder thethe influenceinfluence ofof noisnoise.e. AsAs thethe proposedproposed methmethodod isis aa time–frequencytime–frequency 269269 extensionextension toto thethe ordinaryordinary coherence,coherence, thethe methodmethod inheinheritsrits thethe characteristicscharacteristics ofof traditionaltraditional correlation;correlation; 270270 thatthat is,is, underunder thethe influenceinfluence ofof noise,noise, bothboth thethe traditionaltraditional correlationcorrelation analysisanalysis andand thethe proposedproposed 271271 analysisanalysis areare influenced.influenced. 0.12s 0.175s 0.2s 1.01.0 0.12s 0.175s 0.2s

0.50.5 coeffitiont correlation coeffitiont correlation 0.00.0 01002000100200 frequency/Hz 272272 frequency/Hz 273 Figure 9. The correlation coefficient between 𝑥 and 𝑦 at different times. 273 Figure 9. The correlation coefficient between 𝑥 and 𝑦 at different times.

274274 3.4.3.4. SimulationSimulation ofof Non-StationaryNon-Stationary SignalsSignals 275275 ForFor stationarystationary signalssignals withwith oror withoutwithout noise,noise, bobothth thethe traditionaltraditional correlationcorrelation analysisanalysis andand thethe 276276 SSST-TDSSST-TD TFTF correlationcorrelation analysisanalysis cancan identifyidentify thethe cocorrelationrrelation betweenbetween thethe signals.signals. However,However, thethe SSST-SSST- 277277 TDTD TFTF correlationcorrelation methodmethod cancan betterbetter identifyidentify thethe cocorrelationrrelation atat differentdifferent times.times. PracticalPractical signalssignals areare 278278 oftenoften non-stationary,non-stationary, soso thethe followingfollowing signalssignals wewerere designeddesigned toto analyzeanalyze thethe advantagesadvantages ofof thethe 279279 proposedproposed method.method. TheyThey consistedconsisted ofof cosine,cosine, cosinecosine frequencyfrequency modulationmodulation (FM),(FM), andand shock.shock. WeWe alsoalso 280280 addedadded GaussianGaussian noisenoise toto bothboth withwith aa meanmean valuevalue ofof 00 andand aa standardstandard variancevariance ofof 1.1. TheThe timetime resolutionresolution 281281 Appl.waswas Sci.11 ms,ms,2019 andand, 9, 777 thethe samplingsampling pointspoints werewere 4096.4096. 10 of 17 282282 TheThe timetime domaindomain signalssignals areare shownshown inin FigureFigure 10.10. TheThe traditionaltraditional correlationcorrelation coefficientcoefficient isis shownshown 283283 inin FigureFigure 11.11. Obviously,Obviously, itit isis impossibleimpossible toto distinguishdistinguish thethe correlationcorrelation betweenbetween 𝑥𝑥 andand 𝑦𝑦.. ForFor thethe correlation coefficients was expected to change and not be constant. Thus, the correlation coefficient 284284 frequencyfrequency 3030 Hz,Hz, itit occurredoccurred atat thethe secondsecond andand thirdthird stagestage inin 𝑥𝑥 and and onlyonly thethe thirdthird stagestage inin 𝑦𝑦,, soso thethe was not accurate. 285285 correlationcorrelation coefficientscoefficients waswas expectedexpected toto changechange anandd notnot bebe constant.constant. Thus,Thus, thethe correlationcorrelation coefficientcoefficient 286 was not accurate.  286 was not accurate. x = 2exp(−2πt) × cos(300 × 2πt)  1 𝑥x𝑥2 ===cos 2𝑒𝑥𝑝(−22𝑒𝑥𝑝(−2(30 × 2ππtπ+𝑡)×𝑐𝑜𝑠(300×2𝜋𝑡)𝑡)×𝑐𝑜𝑠(300×2𝜋𝑡)2 cos(100 × 2πt − 2 cos(10 × 2πt)))  𝑥x𝑥3 === cos 𝑐𝑜𝑠(30×2𝑐𝑜𝑠(30×2(30 × 2πtπ) 𝑡𝑡 + + 2 2 𝑐𝑜𝑠( 𝑐𝑜𝑠( 100 100 × × 2𝜋𝑡 2𝜋𝑡 − − 22 𝑐𝑜𝑠( 𝑐𝑜𝑠( 10×2𝜋𝑡))) 10×2𝜋𝑡)))  π (18) 𝑥𝑥 == y𝑐𝑜𝑠(30×2𝑐𝑜𝑠(30×2= 2exp(ππ−𝑡)𝑡)2πt) × cos(300 × 2πt)  1 (18) 𝑦 = 2 𝑒𝑥𝑝( − 2𝜋𝑡) × 𝑐𝑜𝑠( 300×2𝜋𝑡) (18) y𝑦2 = =2 cos 2 𝑒𝑥𝑝((100 × 2π t−− 2𝜋𝑡)2 cos(10 ×× 𝑐𝑜𝑠(2πt)) 300×2𝜋𝑡) . 𝑦 = 2 𝑐𝑜𝑠( 100π ×2 𝑡−2𝑐𝑜𝑠(10×2𝜋𝑡)).  y𝑦3 = =cos 2( 𝑐𝑜𝑠(30 × 2πt) 100π ×2 𝑡−2𝑐𝑜𝑠(10×2𝜋𝑡)). 𝑦 = 𝑐𝑜𝑠(30×2π𝑡) 𝑦 = 𝑐𝑜𝑠(30×2π𝑡)

66 44 y x 0

y 0 x 0 0 -6 -6 -4-4 012340123401234 time/s 01234 287287 time/s timetime//ss 𝑥 𝑦 288288 FigureFigure 10.10. TheThe timetime domaindomain signalssignals ofof x𝑥 andandandy .𝑦..

1.01.0

0.50.5 coeffitiont correlation 0.0 coeffitiont correlation 0.0 00 50 50 100 100 150 150 200 200 frequency/Hz 289289 frequency/Hz 𝑥 𝑦 290290 FigureFigure 11.11. TheTheThe traditionaltraditional traditional correlationcorrelation coefficientcoefficient coefficient betweenbetween x𝑥 andandandy .𝑦..

The SSST of x and y is shown in Figure 12. It is clear that the TF distribution of x and y can be 291291 TheThe SSSTSSST ofof 𝑥𝑥 andand 𝑦𝑦 is is shownshown inin FigureFigure 12.12. ItIt isis clearclear thatthat thethe TFTF distributiondistribution ofof 𝑥𝑥 andand 𝑦𝑦 cancan attained by the SSST, and the energy is gathered at the actual frequency. Moreover, the SSST can obtain 292292 Appl.bebe Sci. attainedattained 2019, 2, xbyby FOR thethe PEER SSST,SSST, REVIEW andand thethe energyenergy isis gatheredgathered atat thethe actualactual frequency.frequency. Moreover,Moreover, thethe SSSTSSST10 of 16 cancan the time-varying distribution more accurately. The correlation coefficient obtained by the SSST-TD TF 293 obtaincorrelation the time-varying analysis is showndistribution in Figure more 13A. accurately. The correlation coefficient obtained by the 294 SSST-TDComparing TF correlation the SSST analysis spectrum is shown with in the Figure correlation 13A. coefficient, the signals were correlated in the 295 firstComparing and third stages.the SSST The spectrum 30 Hz uncorrelation with the correlation was also coefficient, obtained inthe the signals second were stage. correlated The correlation in the 296 firstcoefficient and third of stages. the frequency The 30 modulationHz uncorrelation was obtained was also as obtained well, as canin the be second seen in thestage. local, The enlarged correlation detail 297 coefficientin Figure of 13 theB. Thefrequency outline modulati was clear,on and was this obtained cannot as be well, attained as can by be traditional seen in the correlation local, enlarged analysis. 298 detailFor thein Figure non-stationary 13B. The signals, outline traditional was clear, correlation and this cannot analysis be cannot attained obtain by traditional the accurate correlation correlation. 299 analysis.Similarly, For the the correlation non-stationary coefficients signals, were traditional affected bycorrelation noise. The analysis SSST-TD cannot TF correlation obtain the analysis accurate not 300 correlation.only obtained Similarly, the correlation the correlation between coefficients the signals, were but affected also the by correlation noise. The as SSST-TD it changed TF with correlation time and 301 analysisfrequency. not only Thus, obtained by analyzing the correlation the correlation between in th thee signals, TF plane, but the also correlations the correlation between as it signalschanged are 302 withmore time accurate and frequency. and specific. Thus, by analyzing the correlation in the TF plane, the correlations between 303 signals are more accurate and specific.

304 305 (A) (B)

306 FigureFigure 12. 12.TheThe SSST SSST of signals of signals (A) (A𝑥 )andx and (B ()B 𝑦).y .

307

308 (A) (B)

309 Figure 13. (A) The SSST-TD TF correlation coefficient of 𝑥 and 𝑦 and (B) the local, enlarged detail of 310 the SSST-TD TF correlation coefficient.

311 Through a time series decomposition algorithm derived from the SSST, the SSST-TD TF 312 correlation was carried out in the TF plane, and the local correlation between the signals was attained. 313 Compared with the traditional correlation analysis method using sinusoidal signals, sinusoidal 314 signals with noise, and non-stationary signals, the proposed method was validated by stationary and 315 non-stationary signals simulation. The simulation results show that the traditional method can obtain 316 the overall correlation between the signals but cannot reflect the local time and frequency 317 correlations, especially the correlations between the non-stationary signals. The proposed method 318 not only obtained the overall correlations between the signals, but also accurately identified the 319 correlations between the non-stationary signals, revealing more specific and essential correlations 320 between the signals. Moreover, by comparing the ST and the SSST, it can be concluded that a more 321 energy-concentrated TF result denotes a better TF location and a better characterization of the time- 322 varying feature.

323 4. Gearbox Noise Signal Analysis Appl. Sci. 2019, 2, x FOR PEER REVIEW 10 of 16

293 obtain the time-varying distribution more accurately. The correlation coefficient obtained by the 294 SSST-TD TF correlation analysis is shown in Figure 13A. 295 Comparing the SSST spectrum with the correlation coefficient, the signals were correlated in the 296 first and third stages. The 30 Hz uncorrelation was also obtained in the second stage. The correlation 297 coefficient of the frequency modulation was obtained as well, as can be seen in the local, enlarged 298 detail in Figure 13B. The outline was clear, and this cannot be attained by traditional correlation 299 analysis. For the non-stationary signals, traditional correlation analysis cannot obtain the accurate 300 correlation. Similarly, the correlation coefficients were affected by noise. The SSST-TD TF correlation 301 analysis not only obtained the correlation between the signals, but also the correlation as it changed 302 with time and frequency. Thus, by analyzing the correlation in the TF plane, the correlations between 303 signals are more accurate and specific.

304 305Appl. Sci. 2019, 9, 777 (A) (B) 11 of 17 306 Figure 12. The SSST of signals (A) 𝑥 and (B) 𝑦.

307

308 (A) (B)

309 FigureFigure 13. (A 13.) The (A) SSST-TDThe SSST-TD TF correlationTF correlation coefficient coefficient of ofx and𝑥 andy and𝑦 and (B ()B the) the local, local, enlarged enlarged detaildetail of of the 310 SSST-TDthe TFSSST-TD correlation TF correlation coefficient. coefficient.

311 ThroughThrough a time a seriestime decompositionseries decomposition algorithm algori derivedthm derived from thefrom SSST, the theSSST, SSST-TD the SSST-TD TF correlation TF 312 was carriedcorrelation out inwas the carried TF plane, out in and the TF the plane, local and correlation the local betweencorrelation the between signals the was signals attained. was attained. Compared 313 Compared with the traditional correlation analysis method using sinusoidal signals, sinusoidal with the traditional correlation analysis method using sinusoidal signals, sinusoidal signals with noise, 314 signals with noise, and non-stationary signals, the proposed method was validated by stationary and 315and non-stationarynon-stationary signals signals, simulation. the proposed The simulation method re wassults validated show that the by traditional stationary method and non-stationary can obtain 316signals the simulation. overall correlation The simulation between resultsthe signals show but that cannot the traditional reflect the method local time can obtainand frequency the overall 317correlation correlations, between especially the signals the correlations but cannot between reflect the the local non-stationary time and frequencysignals. The correlations, proposed method especially 318the correlationsnot only obtained between the the overall non-stationary correlations signals.between the The signals, proposed but also method accurately not only identified obtained the the 319overall correlations correlations between between the non-stationary the signals, but signals, also accuratelyrevealing more identified specific the and correlations essential correlations between the 320non-stationary between the signals, signals. revealing Moreover, more by specificcomparing and the essential ST and correlationsthe SSST, it can between be concluded the signals. that a Moreover, more 321by comparingenergy-concentrated the ST and TF the result SSST, denotes it can a bebetter concluded TF location that and a a more better energy-concentrated characterization of the TFtime- result 322 denotesvarying a better feature. TF location and a better characterization of the time-varying feature. 323 4. Gearbox4. Gearbox Noise Noise Signal Signal Analysis Analysis Appl. Sci. 2019, 2, x FOR PEER REVIEW 11 of 16 4.1. Experimental Equipment 324 4.1. Experimental Equipment In order to verify the SSST-TD TF correlation analysis, experiments were carried out on an 325 engine–gearboxIn order totest verify bench. the TheSSST-TD test bench TF correlation was composed analysis, of a four-cylinder experiments engine, were carried a gearbox, out a on brake, an 326 aengine–gearbox base, etc., as shown test bench. in Figure The 14 test. This bench test benchwas co wasmposed designed of a forfour-cylinder identifying engine, vibration a gearbox, and noise a 327 sourcesbrake, a andbase, the etc., fault as diagnosisshown in Figure of gearboxes, 14. This bearings, test bench and was shafts designed driven for by identifying engines or motors.vibration Thus, and 328 thenoise noise sources signals and were the fault computed. diagnosis of gearboxes, bearings, and shafts driven by engines or motors. 329 Thus, the noise signals were computed.

330 331 Figure 14. The test bench and its components: ( 1)) the the engine, engine, ( (2–3) the the sound sound pressure sensors, (4) the 332 gearbox, and (5) the base.

333 The gearbox was a primary planetary gearbox, and the details are shown in Table1 1.. TheThe sunsun 334 gear was connected to the engine with the ring gear fixed.fixed. The operating condition condition was was 2600 2600 rpm. rpm. 335 The sensors were two sound pressure sensors. The sampling frequency was 5.12 kHz, and the 336 sampling points were 4096.

337 Table 1. Gear parameters of the test gearbox.

Gear Name Number Teeth Number Gear Pressure Angle (°) Gear Modulus (mm) ring 1 72 20 2 pinion 3 18 20 2 sun 1 36 20 2

338 4.2. Experimental Signal Analysis 339 To reduce the influence between the sources, the signals were measured near-field. The 340 measured near-field sound pressures of the gearbox 𝑥 and the engine 𝑦 are shown in Figure 15. The 341 FT of 𝑥 and 𝑦 are shown in Figure 16, and the traditional correlation coefficient was calculated as 342 shown in Figure 17. The most common signals of the rotating machinery were the fundamental 343 frequency and the multiples of the fundamental frequency. As can be seen in Figure 16, the amplitude 344 peaks of gearbox occurred mainly at 86.25 Hz, 173.75 Hz, and 606 Hz, which were multiples of the 345 rotating frequency. The amplitude peaks of the engine occurred mainly at 86.25 Hz, 173.75 Hz, 260 346 Hz, 372 Hz, and 693 Hz, which were multiples of the rotating frequency (2600/60 = 43.33 Hz) and 347 other frequencies. Taking 86.25 Hz and 173.75 Hz as an example, the traditional correlation 348 coefficients at 86.25 Hz and 173.75 Hz were over 0.8, showing good compliance with the FT values. 8 15

0 0

-8 -15 0.0 0.4 0.8 0.0 0.4 0.8 sound pressure/pa sound sound pressure/pa 349 time/s time/s 350 (A) (B)

351 Figure 15. The time domain sound pressure signals of (A) 𝑥 and (B) 𝑦. Appl. Sci. 2019, 2, x FOR PEER REVIEW 11 of 16

324 4.1. Experimental Equipment 325 In order to verify the SSST-TD TF correlation analysis, experiments were carried out on an 326 engine–gearbox test bench. The test bench was composed of a four-cylinder engine, a gearbox, a 327 brake, a base, etc., as shown in Figure 14. This test bench was designed for identifying vibration and 328 noise sources and the fault diagnosis of gearboxes, bearings, and shafts driven by engines or motors. 329 Thus, the noise signals were computed.

330 331 Figure 14. The test bench and its components: (1) the engine, (2–3) the sound pressure sensors, (4) the 332 Appl.gearbox, Sci. 2019 and, 9, 777(5) the base. 12 of 17

333 The gearbox was a primary planetary gearbox, and the details are shown in Table 1. The sun 334 gearThe was sensors connected were twoto the sound engine pressure with sensors.the ring Thegear sampling fixed. The frequency operating was condition 5.12 kHz, was and 2600 the sampling rpm. 335 Thepoints sensors were were 4096. two sound pressure sensors. The sampling frequency was 5.12 kHz, and the 336 sampling points were 4096. Table 1. Gear parameters of the test gearbox.

337 Gear Name NumberTable Teeth1. Gear Number parameters Gearof the Pressure test gearbox. Angle (◦) Gear Modulus (mm) Gearring Name Number 1 Teeth Number 72 Gear Pressure Angle20 (°) Gear Modulus (mm) 2 pinionring 1 3 72 18 20 20 2 2 pinionsun 3 1 18 36 20 20 2 2 sun 1 36 20 2 4.2. Experimental Signal Analysis 338 4.2. Experimental Signal Analysis To reduce the influence between the sources, the signals were measured near-field. The measured 339 near-fieldTo reduce sound the pressuresinfluence ofbetween the gearbox the sources,x and the the engine signalsy are were shown measured in Figure near-field. 15. The FTThe of x 340 measuredand y are near-field shown insound Figure pressures 16, and of the the traditional gearbox 𝑥 correlation and the engine coefficient 𝑦 are was shown calculated in Figure as 15. shown The in 341 FT Figureof 𝑥 and 17. The𝑦 are most shown common in Figure signals 16, ofand the the rotating traditional machinery correlation were thecoefficient fundamental was calculated frequency as and 342 shownthe multiplesin Figure of17. the The fundamental most common frequency. signals Asof the can rotating be seen inmachinery Figure 16 were, the amplitudethe fundamental peaks of 343 frequencygearbox and occurred the multiples mainly of at the 86.25 fundamental Hz, 173.75 freque Hz, andncy. 606 As Hz,can whichbe seen were in Figure multiples 16, the of amplitude the rotating 344 peaksfrequency. of gearbox The occurred amplitude mainly peaks ofat the86.25 engine Hz, 173.75 occurred Hz, mainly and 606 at Hz, 86.25 which Hz, 173.75 were Hz,multiples 260 Hz, of 372 the Hz, 345 rotatingand 693 frequency. Hz, which The were amplitude multiples peaks of the of rotating the en frequencygine occurred (2600/60 mainly = 43.33 at 86.25 Hz) Hz, and 173.75 other frequencies.Hz, 260 346 Hz,Taking 372 Hz, 86.25 and Hz 693 and Hz, 173.75 which Hz were as an multiples example, of the the traditional rotating frequency correlation (2600/60 coefficients = 43.33 at 86.25 Hz) Hzand and 347 other173.75 frequencies. Hz were overTaking 0.8, showing86.25 Hz good and compliance 173.75 Hz withas an the example, FT values. the traditional correlation 348 coefficients at 86.25 Hz and 173.75 Hz were over 0.8, showing good compliance with the FT values. 8 15

0 0

-8 -15 0.0 0.4 0.8 0.0 0.4 0.8 sound pressure/pa sound sound pressure/pa 349 time/s time/s 350 (A) (B) Appl. Sci. 2019, 2, x FOR PEER REVIEW 12 of 16 351 Appl. Sci. 2019, 2, x FigureFORFigure PEER 15. REVIEWThe 15. Thetime time domain domain sound sound pressure pressure signals signals of (A of) (𝑥A )andx and (B)( B𝑦). y. 12 of 16

1.8 4 1.8 4 0.9 2 0.9 2 0.0 0 0.0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 sound pressure/pa 0 500 1000 1500 2000 2500 sound pressure/pa 0 500 1000 1500 2000 2500 sound pressure/pa 352 frequency/Hz sound pressure/pa frequency/Hz 352 frequency/Hz frequency/Hz 353 (A) (B) 353 (A) (B) 354 Figure 16. The frequency spectrums of (A) 𝑥 and (B) 𝑦. 354 FigureFigure 16. 16. TheThe frequency frequency spectrums spectrums of of (A (A) )𝑥x and (B) y𝑦..

1.0 1.0

0.5 0.5

0.0 0.0 0 500 1000 1500 2000 2500 correlation coefficient 0 500 1000 1500 2000 2500 355 correlation coefficient frequency/Hz 355 frequency/Hz Figure 17. The traditional correlation coefficient between x and y. 356 Figure 17. The traditional correlation coefficient between 𝑥 and 𝑦. 356 Figure 17. The traditional correlation coefficient between 𝑥 and 𝑦.

357 357 358 (A) (B) 358 (A) (B)

359 359 360 (C) (D) 360 (C) (D) 361 Figure 18. The TF representations of signals: (A) the ST of 𝑥, (B) the ST of 𝑦, (C) the SSST of 𝑥, and 361 Figure 18. The TF representations of signals: (A) the ST of 𝑥, (B) the ST of 𝑦, (C) the SSST of 𝑥, and 362 (D) the SSST of 𝑦. 362 (D) the SSST of 𝑦. 363 Then, we applied the ST and the SSST to 𝑥 and 𝑦, and the TF spectra obtained are shown in 363 Then, we applied the ST and the SSST to 𝑥 and 𝑦, and the TF spectra obtained are shown in 364 Figure 18. As can be seen, the ST energy below 500 Hz was mainly gathered at the frequencies around 364 Figure 18. As can be seen, the ST energy below 500 Hz was mainly gathered at the frequencies around 365 86 Hz and 173.65 Hz, which corresponds well with multiple rotating frequencies. For the SSST, the 365 86 Hz and 173.65 Hz, which corresponds well with multiple rotating frequencies. For the SSST, the Appl. Sci. 2019, 2, x FOR PEER REVIEW 12 of 16

1.8 4

0.9 2

0.0 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 2500 sound pressure/pa sound pressure/pa 352 frequency/Hz frequency/Hz

353 (A) (B)

354 Figure 16. The frequency spectrums of (A) 𝑥 and (B) 𝑦.

Appl. Sci. 2019, 9, 777 13 of 17 1.0

Then, we applied the ST0.5 and the SSST to x and y, and the TF spectra obtained are shown in Figure 18. As can be seen, the ST energy below 500 Hz was mainly gathered at the frequencies around 86 Hz and 173.65 Hz, which corresponds well with multiple rotating frequencies. For the 0.0 SSST, the bandwidth around 86 Hz and 173.65 Hz was much smaller, showing the advantages of the 0 500 1000 1500 2000 2500 SSST. The energy distributionscorrelation coefficient on the TF spectrum were concentrated around the real instantaneous 355 frequency/Hz frequency of the signal. Then, we carried out the SSST-TD TF correlation analysis. The correlation coefficient is shown in Figure 19. 356 Figure 17. The traditional correlation coefficient between 𝑥 and 𝑦.

357

358 (A) (B)

359 Appl. Sci. 2019, 2, x FOR PEER REVIEW 13 of 16

366360 bandwidth around 86 Hz and(C 173.65) Hz was much smaller, showing (D) the advantages of the SSST. The 367 energy distributions on the TF spectrum were concentrated around the real instantaneous frequency 361 FigureFigure 18. 18.The The TF TF representations representations of of signals: signals: (A ()A the) the ST ST of xof,( B𝑥,) ( theB) the ST ofSTy ,(ofC 𝑦), the(C) SSSTthe SSST of x, of and 𝑥, ( Dand) 368 of the signal. Then, we carried out the SSST-TD TF correlation analysis. The correlation coefficient is 362 the(D SSST) the ofSSSTy. of 𝑦. 369 shown in Figure 19. 363 Then, we applied the ST and the SSST to 𝑥 and 𝑦, and the TF spectra obtained are shown in 364 Figure 18. As can be seen, the ST energy below 500 Hz was mainly gathered at the frequencies around 365 86 Hz and 173.65 Hz, which corresponds well with multiple rotating frequencies. For the SSST, the

370 371 FigureFigure 19. 19. TheThe SSST-TD SSST-TD TF TF correlation correlation coefficient coefficient between between 𝑥x andand y𝑦. .

372 Taking 86.25 Hz and 173.75 Hz as an example. From the ST and the SSST, it can be seen that the 373 signals were basically continuous at 86.25 Hz, so the correlation coefficient was also expected to be 374 continuous. The result shows consistency with that of the traditional method, as shown in Figure 375 20A. However, there were many discontinuities and fluctuations near 173.65 Hz, which means the 376 correlation coefficient was not as continuous as that at 86.25 Hz, as shown in Figure 20B. The other 377 main frequencies can also be analyzed similarly. 1.0 1.0

0.5 0.5

0.0 0.0 correlation coefficient correlation coefficient 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 378 time/s time/s 379 (A) (B)

380 Figure 20. Correlation coefficient between 𝑥 and 𝑦 at frequencies of (A) 86.25Hz and (B) 173.75Hz.

381 As previously analyzed, the correlation coefficient varied with the phase. The ST/SSST time 382 frequency spectrum should be considered to judge the correlation between the signals. Therefore, the 383 traditional method is the overall correlation coefficient, which cannot reflect the time-varying 384 characteristics, while the SSST-TD TF correlation method is able to obtain the time-varying 385 correlation and obtain a better local correlation between the signals.

386 5. Conclusion 387 A more energy-concentrated TF result denotes a better TF location and better characterization 388 of time-varying features. In this paper, a time series decomposition algorithm derived from the SSST 389 was proposed, and TF correlation was analyzed in the TF plane. The local correlation between the 390 signals in the TF domain were attained, and the correlations varying with time and frequency were 391 able to reveal more specific and essential correlations between the signals. By comparing the 392 traditional correlation analysis with the proposed method using sinusoidal signals, sinusoidal signals 393 with noise, and non-stationary signals, the superiority of the proposed method is that the correlation 394 can be obtained more intuitively and accurately. Similar to the traditional method, the SSST-TD TF 395 correlation method is also affected by noise. The influence of noise must be further studied. As an 396 extension to the traditional correlation analysis, the proposed method not only obtains the correlation Appl. Sci. 2019, 2, x FOR PEER REVIEW 13 of 16

366 bandwidth around 86 Hz and 173.65 Hz was much smaller, showing the advantages of the SSST. The 367 energy distributions on the TF spectrum were concentrated around the real instantaneous frequency 368 of the signal. Then, we carried out the SSST-TD TF correlation analysis. The correlation coefficient is 369 shown in Figure 19.

370 Appl. Sci. 2019, 9, 777 14 of 17 371 Figure 19. The SSST-TD TF correlation coefficient between 𝑥 and 𝑦. Taking 86.25 Hz and 173.75 Hz as an example. From the ST and the SSST, it can be seen that 372 the signalsTaking were 86.25 basically Hz and continuous 173.75 Hz as at an 86.25 example. Hz, so Fr theom correlation the ST and coefficientthe SSST, it was can alsobe seen expected that the 373 tosignals be continuous. were basically The result continuous shows at consistency 86.25 Hz, so with the thatcorrelation of the traditionalcoefficient was method, also expected as shown to in be 374 Figurecontinuous. 20A. However, The result there shows were consistency many discontinuities with that andof the fluctuations traditional near method, 173.65 as Hz, shown which in means Figure 375 the20A. correlation However, coefficient there were was many not as discontinuities continuous as that and at fluctuations 86.25 Hz, as near shown 173.65 in Figure Hz, which 20B. The means other the 376 maincorrelation frequencies coefficient can also was be analyzednot as continuous similarly. as that at 86.25 Hz, as shown in Figure 20B. The other 377 main frequencies can also be analyzed similarly. 1.0 1.0

0.5 0.5

0.0 0.0 correlation coefficient correlation coefficient 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 378 time/s time/s 379 (A) (B)

380 FigureFigure 20. 20.Correlation Correlation coefficient coefficient between betweenx and𝑥 andy at 𝑦 frequencies at frequencies of (A of) 86.25(A) 86.25Hz Hz and and (B) ( 173.75B) 173.75Hz. Hz.

381 AsAs previously previously analyzed, analyzed, the the correlation correlation coefficient coefficient varied varied with with the the phase. phase. The The ST/SSST ST/SSST time time 382 frequencyfrequency spectrum spectrum should should be be considered considered toto judgejudge the correlation correlation between between the the signals. signals. Therefore, Therefore, the 383 thetraditional traditional method method is is the the overall overall correlation coefficient,coefficient, whichwhich cannotcannot reflect reflect the the time-varying time-varying 384 characteristics,characteristics, while while the SSST-TDthe SSST-TD TF correlation TF correlation method method is able to is obtain able theto time-varyingobtain the time-varying correlation 385 andcorrelation obtain a betterand obtain local correlationa better loca betweenl correlation the signals. between the signals.

386 5.5. Conclusions Conclusion 387 AA more more energy-concentrated energy-concentrated TF TF result result denotes denotes a bettera better TF TF location location and and better better characterization characterization 388 ofof time-varying time-varying features. features. InIn this paper, paper, a a time time se seriesries decomposition decomposition algorithm algorithm derived derived from from the theSSST 389 SSSTwas was proposed, proposed, and and TF TFcorrelation correlation was was analyzed analyzed in inthe the TF TF plane. plane. The The local local correlation correlation between between the 390 thesignals signals in inthe the TF TF domain domain were were attained, attained, and and the the correlations correlations varying varying with with time time and and frequency frequency were 391 wereable able to toreveal reveal more more specific specific and and essential essential corre correlationslations between thethe signals.signals. ByBy comparing comparing the the 392 traditionaltraditional correlation correlation analysis analysis with with the the proposed proposed method method using using sinusoidal sinusoidal signals, signals, sinusoidal sinusoidal signals signals 393 withwith noise, noise, and and non-stationary non-stationary signals, signals, the the superiority superiority of of the the proposed proposed method method is is that that the the correlation correlation 394 cancan be be obtained obtained more more intuitively intuitively and and accurately. accurately. Similar Similar to to the the traditional traditional method, method, the the SSST-TD SSST-TD TF TF 395 correlationcorrelation method method is is also also affected affected by by noise. noise. The The influence influence of of noise noise must must be be further further studied. studied. As As an an 396 extensionextension to to the the traditional traditional correlation correlation analysis, analysis, the th proposede proposed method method not not only only obtains obtains the the correlation correlation between the signals in accordance with the traditional correlation analysis method, but also obtains the time–frequency correlation distribution and can be used for non-stationary signals. Thus, the following major conclusions can be drawn.

1 For stationary signals, both the traditional correlation and the SSST-TD TF correlation methods can identify the correlations between signals, and the SSST-TD TF correlation method can also obtain the correlation as it changes with time and frequency. 2 Noise and the phase will affect the correlation coefficient. The traditional correlation method cannot recognize the influence of the phase, but the SSST-TD TF correlation method can get the effect of the phase of the cosine signals. 3 For time-varying signals, the correlation between the signals cannot be recognized by the traditional correlation method. The SSST-TD TF correlation method can obtain the correlation of non-stationary signals and can also reflect the time-varying and frequency-varying characteristics between the signals. Appl. Sci. 2019, 9, 777 15 of 17

Author Contributions: Conceptualization, G.P. and S.L.; Data curation, Y.Z.; Formal analysis, G.P.; Project administration, S.L.; and Software, G.P. and Y.Z. Funding: The research was funded by the National Natural Science Foundation of China (51675262), the Advance Research Field Fund Project of China (6140210020102), the Fundamental Research Funds for the Central Universities (NP2018304), and the Major National Science and Technology Projects (2017-IV-0008-0045). Acknowledgments: The authors wish to thank E. Brevdo for his MATLAB Synchrosqueezing Toolbox, and three anonymous reviewers for their constructive comments and suggestions. Conflicts of Interest: No conflicts of interest exist in the submission of this manuscript, and the manuscript has been approved by all the authors for publication. The work described herein is original research that has not been published previously and is not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A

Derivation of the Formula of Inverse SSST The definition of the SSST is as follows:

−1 SSSTx( fel, b) = (∆ fel) × ∑ |STx( fk, b)| fk∆ fk (A1) 1 fk:| fel ( fk,b)− fel |≤ 2 ∆ fel

According to Parseval theorem, the ST of the signal can be expressed as follows:

| f | R ∞ −(t−b)2 f 2 STx( f , b) = √ x(t) exp( ) exp(−i2π f t)dt 2π −∞ 2 (A2) = |STx( f , b)| exp(iϕ( f , b))

The following is true: −i[2π f b+ϕ( f ,b)] 2 Cϕ = e f (A3) We found the following:   SSSTx fel, b −1 = ( ) × (− π ) −1 ∆ fe ∑ |STx( fk, b)|exp i2 fkb fk ∆ fk (A4) Cϕ 1 fk:| fe( fk,b)− fel |≤ 2 ∆ fe

According to Equation (5) and using the continuous form of the right side of Equation (A4), we found the following:

R ∞ π −1 0 STx( f , b)|exp(i2 f b) f d f 1 R ∞ R ∞ ˆ −1 −1 = 2π −∞ 0 X(ξ)ψ( f ξ) f exp(ibξ)d f dξ (A5) R +∞ ˆ −1 1 R +∞ = 0 −ψ(ξ)ξ dξ · 2π 0 X(ξ)exp(ibξ)dξ

R +∞ ˆ −1 Let Cψ = 0.5 0 −ψ(ξ)ξ dξ and assuming that the signal is real-valued, the signal can then be reconstructed by the following:

1 R ∞  x(b) = π RE xˆ(ξ) exp(ibξ)dξ h 0 i (A6) −1R ∞ π −1 = RE Cψ 0 STx( f , b) exp(i2 f b) f d f

In the piecewise constant approximation corresponding to the binning in f, the following is true:

−1 x(b) = RE[(CψCϕ) ∑ SSSTx(efl, b)∆efl] (A7) l Appl. Sci. 2019, 9, 777 16 of 17

References

1. Darsena, D.; Gelli, G.; Paura, L.; Verde, F. Blind Channel Shortening for Space-Time-Frequency Block Coded MIMO-OFDM Systems. IEEE Trans. Wireless Commun. 2012, 11, 1022–1033. [CrossRef] 2. Yi, C.; Lv, Y.; Xiao, H.; You, G.; Dang, Z. Research on the Blind Source Separation Method Based on Regenerated Phase-Shifted Sinusoid-Assisted EMD and Its Application in Diagnosing Rolling-Bearing Faults. Appl. Sci. 2017, 7, 414. [CrossRef] 3. Delorme, A.; Makeig, S. EEGLAB: An open source toolbox for analysis of single-trial EEG dynamics including independent component analysis. J. Neurosci. Methods 2004, 134, 9–21. [CrossRef][PubMed] 4. Wang, F.; Wang, Z.; Li, R.; Gao, X.; Jin, J. Survey on statistical dependent source separation model and algorithms. Control Decision 2015, 30, 1537–1545. [CrossRef] 5. Liang, M.; Xi-Hai, L.; Wan-Gang, Z.; Dai-Zhi, L. The generalized cross-correlation Method for time delay estimation of infrasound signal. In Proceedings of the Fifth International Conference on Instrumentation & Measurement, Computer, Communication, and Control (IMCCC), Qinhuangdao, China, 18–20 September 2015; pp. 1320–1323. [CrossRef] 6. Mallat, S. A Wavelet Tour of Signal Processing, The Sparse Way, 3rd ed.; Academic Press: Cambridge, MA, USA, 2008. 7. Zhang, X.; Zhao, J. Application of the DC Offset Cancellation Method and S Transform to Gearbox Fault Diagnosis. Appl. Sci. 2017, 7, 207. [CrossRef] 8. Suzuki, T.; Matsui, H.; Choi, S.; Sasaki, O. Low-coherence interferometry based on continuous wavelet transform. Opt Commun 2013, 311, 172–176. [CrossRef] 9. Zhang, Z.; Leng, Y.; Yang, Y.; Xiao, X.; Ge, S. Comparative study on wavelet coherence and HHT coherence. In Proceedings of the IEEE International Congress on Image & Signal Processing, Dalian, China, 14–16 October 2014; pp. 867–872. [CrossRef] 10. Pinnegar, C.R.; Mansinha, L. The S-transform with windows of arbitrary and varying shape. Geophys. 2003, 68, 381–385. [CrossRef] 11. Daubechies, I.; Lu, J.; Wu, H.T. Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Appl. Comput. Harmon. Anal. 2011, 30, 243–261. [CrossRef] 12. Chen, H.; Chen, Y.; Sun, S.; Hu, Y.; Feng, J. A High-Precision Time-Frequency Entropy Based on Synchrosqueezing Generalized S-Transform Applied in Reservoir Detection. Entropy 2018, 20, 428. [CrossRef] 13. Liu, Y.; Zhou, W.; Li, P.; Yang, S.; Tian, Y. An Ultrahigh Frequency Partial Discharge Signal De-Noising Method Based on a Generalized S-Transform and Module Time-Frequency Matrix. Sensors 2016, 16, 941. [CrossRef] 14. Daubechies, I.; Wang, Y.; Wu, H.T. ConceFT: Concentration of Frequency and Time via a multitapered synchrosqueezed transform. Philos. Trans. Math. Phys. Eng. Sci. 2015, 374, 2065. [CrossRef][PubMed] 15. Wang, P.; Gao, J.; Wang, Z. Time-Frequency Analysis of Seismic Data Using Synchrosqueezing Transform. IEEE Geosci. Remote Sens. Lett. 2014, 11, 2042–2044. [CrossRef] 16. Wang, Q.; Gao, J.; Liu, N.; Jiang, X. High-Resolution Seismic Time-Frequency Analysis Using the Synchrosqueezing Generalized S Transform. IEEE Geosci. Remote Sens. Lett. 2018, 15, 374–378. [CrossRef] 17. Huang, Z.; Zhang, J.; Zhao, T. Synchrosqueezing based-transform and Its Application in Seismic Spectral Decomposition. IEEE Trans. Geosci. Remote Sens. 2015, 54, 817–825. [CrossRef] 18. Pinnegar, C. Time-frequency and time-time filtering with the S-transform and TT-transform. Digital Signal Process. 2005, 15, 604–620. [CrossRef] 19. Jashfar, S.; Esmaeili, S.; Zareian-Jahromi, M.; Rahmanian, M. Classification of power quality disturbances using S-transform and TT-transform based on the artificial neural network. Turk. J. Electr. Eng. Computer Sci. 2013, 21, 1528–1538. [CrossRef] 20. Stockwell, R.G.; Mansinha, L.; Lowe, R.P. Localization of the complex spectrum: The S transform. IEEE Trans. Signal. Process. 2002, 44, 998–1001. [CrossRef] 21. Thakur, G.; Brevdo, E.; Fuˇckar, N.S.; Wu, H.T. The Synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications. Signal. Process. 2013, 93, 1079–1094. [CrossRef] 22. Wang, Z.; Ren, W.; Liu, J. A synchrosqueezed wavelet transform enhanced by extended analytical mode decomposition method for dynamic signal reconstruction. J. Sound Vib. 2013, 332, 6016–6028. [CrossRef] Appl. Sci. 2019, 9, 777 17 of 17

23. Auger, F.; Flandrin, P.; Lin, Y.T.; McLaughlin, S.; Meignen, S.; Oberlin, T.; Wu, H.T. Time-frequency reassignment and synchrosqueezing: An overview. IEEE Signal Process. Mag. 2013, 30, 32–41. [CrossRef] 24. Huang, Z.; Zhang, J. Synchrosqueezing S-Transform and Its Application in Seismic Spectral Decomposition. IEEE Trans. Geosci. Remote Sens. 2016, 54, 817–825. [CrossRef] 25. Iatsenko, D.; McClintock, P.V. Linear and synchrosqueezed time–frequency representations revisited: Overview, standards of use, resolution, reconstruction, concentration, and algorithms. Digital Signal Process. 2015, 42, 1–26. [CrossRef] 26. Nadarajah, S. Comments on “Approximation of statistical distribution of magnitude squared coherence estimated with segment overlapping”. Signal Process. 2007, 87, 2834–2836. [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).