High-Precision and High-Resolution Synchrosqueezing Transform via Time-Frequency Instantaneous Phases

: Synchrosqueezing transform (SST) can effectively improve time-frequency precision and resolution by squeezing time-frequency spectra via instantaneous frequencies, and it has been applied in many diverse disciplines; however, the precision of estimated instantaneous frequencies during SST is usually affected by the time-sample interval of the inputted signal; this usually leads to low-precision or inaccurate SST results and limits its further application. To obtain high-precision and high-resolution SST results with high efﬁciency, we propose a high-precision and high-resolution SST via time-frequency instantaneous phases (HSST); in HSST, time-frequency instantaneous phases with period-jumps removal are used for high-precision instantaneous frequencies estimation and SST. Two synthetic signal examples show that HSST can minimize the impact of the time-sample interval to achieve high-precision and high-resolution SST results with high efﬁciency. A real 3D seismic data application demonstrates that HSST has fantastic performance in time-frequency precision and resolution enhancement, and it can be widely used in digital signals processing and interpretation ﬁelds.

Compared with these classical time-frequency methods [1][2][3][4][5][6][7][8][9], SST can effectively improve time-frequency precision and resolution by time-frequency spectra reassignment via instantaneous frequencies, and this makes SST more suitable for non-linear and nonstationary signals. However, precisions of estimated instantaneous frequencies during SST are decreased with the increasing time-sample interval of the inputted signal; this usually leads to low-precision or inaccurate SST results [15,18,19,21] and limits its further application.
To obtain high-precision and high-resolution SST results, Oberlin et al. proposed a second-order SST via STFT [15], Huang et al. proposed a second-order SST via S transform [18], Pham and Meignen proposed a high-order SST via STFT [19]. In second-order SST, instantaneous phases of the inputted signal are considered second-order functions of time and expressed by second-order Taylor expansion, and second-order partial derivatives of time-frequency spectra with respect to time and frequency are used for instantaneous frequency estimation; whereas in high-order SST, instantaneous phases are considered high-order functions of time and expressed by high-order Taylor expansion, and high-order partial derivatives of time-frequency spectra with respect to time and frequency are used for instantaneous frequency estimation. The second or high-order functions cannot well match complex instantaneous phases or complex signals, and second or high-order SST has low processing efficiency due to multiple times partial derivative operations. In order to minimize the impact of the time-sample interval to obtain desirable high-precision and high-resolution SST results, Zhang et al. proposed an adaptive resampled high-resolution SST [21], which is referred to as ASST. In ASST, the resampled time-sample interval is inversely proportional to the highest frequency in the effective frequency band [9,21] of the inputted signal, it can minimize the impact of time-sample interval to obtain high-precision and high-resolution SST results but at the expense of processing efficiency.
In this paper, we propose a high-efficiency, high-precision, and high-resolution SST via time-frequency instantaneous phases (instantaneous phases [22,23] of time-frequency spectra) with period-jumps removal, which is referred to as HSST; in HSST, time-frequency instantaneous phases with period-jumps removal are used for high-precision instantaneous frequencies estimation and SST. Synthetic signals and real 3D seismic data examples demonstrate that HSST can be widely used for seismic signal [8,9,13,16], microseismic signal [24,25], gear vibration signal [6], gravitational-wave signal [19], and other nonstationary digital signals processing and interpretation.

Conventional SST (CSST)
If complex numbers G(t, f ) denote the time-frequency spectra of the inputted signal g(t), then SST results can be expressed as and where v and f are both frequencies; ∆v is the frequency increment; Re[ ] takes the real part of a complex number; V 1 (t, f ) denote the instantaneous frequencies of the timefrequency spectra G(t, f ), which are referred to as time-frequency instantaneous frequen- where Im[ ] takes an imaginary part of a complex number, respectively, then For actual discrete signals, Equation (4) should be modified as (see Appendix A) where ∆t is the time-sample interval of the inputted signal g(t); and V 2 (t, f ) can be considered the estimated time-frequency instantaneous frequencies of G(t, f ); correspondingly, Equation (1) can be rewritten as which is considered conventional SST (CSST).
Precisions of V 2 (t, f ) are inversely proportional to the time-sample interval, this usually leads to inaccurate SST results [21]. To minimize the impact of the time-sample interval, Zhang et al. proposed ASST [21], which can minimize the impact of the timesample interval to obtain high-precision and high-resolution SST results, but at the expense of processing efficiency.  Table 1 shows the time consuming of MSTFT, CSST, and ASST for the 50 Hz harmonic signal with 4 s length. The peak frequencies in Figure 1a-f are 50, 48, 50, 50, 37, and 50 Hz, respectively; we can see that CSST obtains incorrect SST results, and ASST obtains higher precision and resolution results than MSTFT but with lower processing efficiency.

High-Precision and High-Resolution SST via Time-Frequency Instantaneous Phases (HSST)
According to Appendix B, we can use instantaneous phases of G(t, f ) for highprecision time-frequency instantaneous frequency estimation and finally obtain highprecision and high-resolution SST results with high efficiency.
If θ(t, f ) and θ(t, f ) ∈ (−∞, +∞) denote the argument angles of G(t, f ), θ 1 (t, f ) and θ 1 (t, f ) ∈ [0, 2π] denote the principal argument angles of G(t, f ), then we have where k denote integer numbers; θ(t, f ) can be considered time-frequency instantaneous phases of G(t, f ); θ 1 (t, f ) can be considered time-frequency principal instantaneous phases of G(t, f ) given by According to Equations (A6) and (A16), the time-frequency instantaneous frequencies of G(t, f ) can be given by [22] but parameter k in Equation (7) cannot be determined; thus, θ(t, f ) cannot be used for time-frequency instantaneous frequencies estimation. If θ 0 (t, f ) are used for time-frequency instantaneous frequencies estimation, according to Equations (A16)-(A19), V 1 (t, f ) in Equation (10) can be obtained by computing the derivative of the arctangent function itself to avoid period-jumps of θ 0 (t, f ) as and Then, the estimated time-frequency instantaneous frequencies can be given by Thus, according to [21] and Figure 1, it avoids period-jumps of θ 0 (t, f ), but it cannot avoid the impact of the time-sample interval.
If θ 1 (t, f ) is used for instantaneous frequencies estimation, then according to Equation (A21), the estimated time-frequency instantaneous frequencies can be given by Appl. Sci. 2021, 11, 11760 5 of 16 and the corresponding SST results can be expressed as which is considered SST via the time-frequency principal instantaneous phases (PSST); PSST can avoid the impact of the time-sample interval to obtain high-resolution SST results, except for time-points corresponding to period-jumps of θ 1 (t, f ). According to Equations (A21)-(A27), if we considered (16) then, estimated high-precision time-frequency instantaneous frequencies can be given by and corresponding SST results can be expressed as which is considered HSST; HSST can avoid the impacts of the time-sample interval and period-jumps of θ 1 (t, f ) to obtain high-precision and high-resolution SST results with high-efficiency. Figure 2a shows the time-frequency principal instantaneous phases of the 50 Hz harmonic signal; Figure 2b,c show the time-frequency instantaneous amplitudes obtained by PSST with ∆t = 2 and 4 ms, respectively; Figure 2d,e show the time-frequency instantaneous amplitudes obtained by HSST with ∆t = 2 and 4 ms, respectively. Table 2 shows the time consuming of PSST and HSST for the 50 Hz harmonic signal with 4 s length.   Comparison of Figures 1 and 2 shows that PSST has higher precision and resolution than MSTFT and CSST, except for time-points corresponding to period-jumps of timefrequency principal instantaneous phases; HSST avoids period-jumps during PSST, and it has higher precision and resolution than MSTFT, CSST, and PSST.

Examples
In this section, a synthetic signal and 3D real seismic data are used to demonstrate the high-precision and high-resolution abilities of HSST with comparisons of CSST, ASST, and PSST; and the time-frequency spectra for CSST, ASST, PSST, and HSST are obtained by MSTFT [7] given by (19) and where z(t) is the analytic signal [22] of the inputted signal g(t); a > 0 and b ≥ 0.

Synthetic Signal Example
We consider a synthetic signal as and its instantaneous frequencies can be expressed as Figure 3a,b show the synthetic signal and its instantaneous frequencies, respectively. Figure 4a-e show time-frequency instantaneous amplitudes obtained by MSTFT, CSST, ASST, PSST, and HSST with ∆t = 1 ms, respectively; Figure 4f shows the differences between Figure 4d,e. We can see that the peak frequencies in Figure 4a,c,e are almost equal to the instantaneous frequencies shown in Figure 3b, whereas the peak frequencies in Figure 4b show some deviations from the instantaneous frequencies shown in Figure 3b, and these deviations become more obvious with the increasing instantaneous frequencies.
A comparison of Figure 4a Figure 4d. Above all, HSST has higher precision and resolution than MSTFT, CSST, and PSST, and almost the same results as ASST.

3D Field Seismic Data Example
We applied MSTF, CSST, PSST, and HSST to a 3D field seismic data with ∆t = 2 ms. Figure 5a-c show amplitudes, instantaneous amplitudes, and principal instantaneous phases of crossline 1 extracted from the field 3D seismic data, respectively.     Figure 8a shows a time slice extracted from the field 3D seismic data; Figure 8b,c show the same time slices of instantaneous amplitudes and principal instantaneous phases, respectively, obtained by Hilbert transform [22]. As shown in Figure 8b,c, the main channels are represented but with low resolution.   Figure 8a shows a time slice extracted from the field 3D seismic data; Figure 8b, c show the same time slices of instantaneous amplitudes and principal instantaneous phases, respectively, obtained by Hilbert transform [22]. As shown in Figure 8b, c, the main channels are represented but with low resolution.  Thus, HSST has higher precisions and resolutions than MSTFT, CSST and PSST, and it achieves desirable high-precision and high-resolution SST results. Thus, HSST has higher precisions and resolutions than MSTFT, CSST and P it achieves desirable high-precision and high-resolution SST results. Thus, HSST has higher precisions and resolutions than MSTFT, CSST and PSST, and it achieves desirable high-precision and high-resolution SST results.

Conclusions
To obtain high-precision and high-resolution SST results with high-efficiency, we propose a high-precision and high-resolution SST via time-frequency instantaneous phases, which is referred to as HSST. Compared with CSST, ASST, and PSST, HSST avoids the impact of the time-sample interval during CSST and period-jumps of time-frequency principal instantaneous phases during PSST, and finally, it obtains desirable high-precision and high-resolution SST results with high efficiency. Synthetic signal and field 3D seismic data examples indicate that HSST can be extensively applied in non-linear and non-stationary digital signals analysis, processing, and interpretation fields.

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Conflicts of Interest:
The authors declare no conflict of interest.
As principal periods (or period-jumps) of θ 0 (t) and θ 1 (t) are π and 2π, respectively; thus, θ 0 (t) and θ 1 (t) are both discontinuous functions, and their derivatives do not exist. Therefore, instantaneous frequencies of c(t) can be given by [22,23] v(t) = 1 2π If taking natural logarithm to Equation (A2), we have Then, Equation (A6) can be expressed as Usually, x(t) and y(t) are both implicit, and we cannot directly obtain their derivatives; but, if Fourier transform [22] results of c(t), x(t), and y(t) all exist, we can use them to obtain the corresponding derivatives; it means that if and where Re[ ] takes the real part of a complex number; c 0 (t),c 1 (t), and c 2 (t) are derivatives of c(t),x(t), and y(t), respectively; η is the frequency; C(η), X(η), and Y(η) denote the Fourier transform results of c(t), x(t), and y(t), respectively; thus, Equation (A8) can be rewritten as v(t) = Re 1 2πi Due to impacts of dynamic range [9,21] and amplitude amplified scale factor 2πη, background noises beyond effective bandwidths [9,21] of C(η), X(η), and Y(η) will seriously affect the stabilities of c 0 (t), c 1 (t), and c 2 (t), respectively, and finally reduce the stabilities and precisions of v(t). Usually, we can replace derivatives c 0 (t), c 1 (t), and c 2 (t) in Equation (A12) with their corresponding differences; thus, estimated instantaneous frequencies can be expressed as and where ∆t denotes the time-sample interval.
If x(t) and y(t) do not meet the requirements expressed in Equation (A9), such as x(t) = t 2 and so on, then x(t) and y(t) do not have Fourier transform results; therefore, we cannot obtain derivatives c 0 (t), c 1 (t), and c 2 (t), but we can replace them in Equation (A12) with their corresponding differences. Correspondingly, the estimated instantaneous frequencies can be expressed as Equations (A13) and (A14).
Above all, the estimated instantaneous frequencies can be expressed as v 1 (t) = Re 1 2π∆t which is considered the conventional instantaneous frequencies estimation method (CIF). Equation (A15) shows that the estimated instantaneous frequencies are affected by the timesample interval, and precisions of estimated instantaneous frequencies are decreased with the increasing time-sample interval [21]; thus, CIF may lead to inaccurate instantaneous frequency estimation results.

Appendix B. High-Precision Instantaneous Frequencies Estimation Methods via Principal Instantaneous Phases
Equation (A6) can be rewritten as v(t) = 1 2π as parameter k in Equation (A3) cannot be determined, thus, θ(t) cannot be directly used for instantaneous frequencies estimation; but if we can avoid period-jumps of θ 0 (t) and θ 1 (t), then they can be both directly used for instantaneous frequencies estimation. If θ 0 (t) is used for instantaneous frequencies estimation, then according to Equation (A16), the estimated instantaneous frequencies can be obtained by computing the derivative of arctangent function itself as [23] p 1 (t) = 1 2π ∂θ 0 (t) ∂t = 1 2π ∂arctan x −1 (t) · y(t) ∂t (A17) and p 1 (t) = 1 2π x(t) ∂y(t) ∂t − y(t) ∂x(t) ∂t x 2 (t) + y 2 (t) = v(t).
Therefore, according to Equations (A9)-(A14), the estimated instantaneous frequencies can be expressed as p 2 (t) = 1 2π∆t and it means that this method can avoid phase period-jumps, but it cannot avoid the impact of the time-sample interval and may lead to inaccurate instantaneous frequencies estimation results. If θ 1 (t) is used for instantaneous frequencies estimation, then according to Equation (A16), the estimated instantaneous frequencies can be given by and it avoids the impact of the time-sample interval to obtain high-precision instantaneous frequencies, except for time-points corresponding to period-jumps of θ 1 (t).
Thus, we can modify ∆θ 1 (t) into ∆θ 2 (t) to meet the requirement expressed in Equation (A24) as Then − π ≤ ∆θ 2 (t) ≤ π (A26) and estimated high-precision instantaneous frequencies can be given by and it avoids impacts of time-sample interval and period-jumps of θ 1 (t).