CN111289795A - High-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method - Google Patents

High-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method Download PDF

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CN111289795A
CN111289795A CN202010088145.7A CN202010088145A CN111289795A CN 111289795 A CN111289795 A CN 111289795A CN 202010088145 A CN202010088145 A CN 202010088145A CN 111289795 A CN111289795 A CN 111289795A
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陈辉
胡英
杨超
陈旭平
冯俊
魏友华
郭科
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Chengdu Univeristy of Technology
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Abstract

The invention discloses a high-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method, which comprises the following steps: transforming the signal X (t) into a frequency domain signal X (ω); selecting the order N and the frequency window function G (omega), and calculating the frequency window function G (omega) of X (omega) at omegakShort-time Fourier transform at G (omega) to construct a square αN(t, ω) and βN(t, ω) calculating an N-th order group delay estimate
Figure DDA0002382782130000011
The time frequency value
Figure DDA0002382782130000012
Superposition to GD estimation in the time direction
Figure DDA0002382782130000013
Computing N-order time-rearrangement synchronous extrusion transform
Figure DDA0002382782130000014
Obtaining a time frequency spectrum; and reconstructing the signal. The invention provides a new GD estimation method, which improves the estimation precision of GD which changes rapidly, and the time-frequency spectrum energy is rearranged in the time direction to gather near the real GD, thereby effectively inhibiting energy divergence and improving the time-frequency readability. And provides a reconstruction method, the reconstruction signal reduction degree is high.

Description

High-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method
Technical Field
The invention relates to a signal processing method, in particular to a high-precision high-order time rearrangement synchronous extrusion transform time-frequency analysis method.
Background
Time-frequency analysis is always an important research method for modern non-stationary signal analysis and processing. The current common time-frequency analysis methods mainly comprise short-time Fourier transform (STFT), Continuous Wavelet Transform (CWT), S Transform (ST) and the like, and the methods based on window functions have certain defects and are limited by the time-frequency resolution of the window functions, the energy aggregation of time frequency spectrums is limited, and the characteristics of signals are difficult to accurately extract.
In order to improve the effect of time-frequency analysis, Kodera and the like propose a time-frequency rearrangement method, and improve the time-frequency aggregation property by redistributing the time-frequency spectrum energy, but the method does not have the reconfigurability and is an irreversible time-frequency analysis method. In subsequent researches, Daubechies et al propose Synchronous Squeeze Transform (SST), which not only effectively improves time-frequency focusing, but also realizes accurate reconstruction of signals by redistributing time-frequency spectrum energy in the frequency direction to the vicinity of the true frequency of the signals. SST is only suitable for processing weak time-varying signals and therefore many scholars have improved and perfected this approach. To address the problem that SST cannot handle impulse-like signals well, Dong He et al propose a time-rescheduling synchronous crush transform (TSST) that "crushes" the time-spectrum energy in the time direction, redistributing it around the true GD. However, TSST is a method based on a time domain signal model, and Zhoujie He and the like consider that a frequency domain signal model can better reflect the characteristics of an impulse-like signal, so that a second-order time rearrangement synchronous extrusion transformation is provided on the basis. However, this method has a limited accuracy for GD estimation, which affects the final signal processing effect.
Group delay, which is called GroupDelay in english, abbreviated as GD.
Disclosure of Invention
The invention aims to solve the problems, and provides a high-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method which can obtain a GD estimation value with higher precision, improve the time-frequency energy aggregation and realize high-precision signal reconstruction.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows: a high-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method comprises the following steps:
(1) acquiring a time domain signal X (t), and performing Fourier transform to obtain a frequency domain signal X (omega), wherein t is time and omega is frequency;
(2) selecting the order N and the frequency window function G (omega), and calculating the frequency window function G (omega) of X (omega) at omegakG (omega) short-time Fourier transform to obtain different time frequency values corresponding to time frequency points (t, omega)
Figure BDA0002382782110000021
Where N is a positive integer, k is 0,1,2, …, max {1,2N-2}, and when k is 0, G (ω) is ω0G(ω)，
Figure BDA0002382782110000022
Figure BDA0002382782110000023
By using
Figure BDA0002382782110000024
Constructing an N-th order square αN(t, ω) and βN(t,ω)；
Figure BDA0002382782110000025
Figure BDA0002382782110000031
In the formula (I), the compound is shown in the specification,
Figure BDA0002382782110000032
s is an adjustment factor in the window function G (ω);
(3) using an N-th order square αN(t, ω) and βN(t, ω) computing an N-th order group delay estimate
Figure BDA0002382782110000033
When βNWhen the determinant value of (t, ω) is not zero, the matrix α is dividedN(t, ω) and βNDividing the determinant of (t, omega), taking the imaginary part of the determinant, adding the imaginary part of the determinant to the time t corresponding to the time frequency point (t, omega), and calculating to obtain the product
Figure BDA0002382782110000034
A value of (d);
when matrix βNWhen the determinant value of (t, ω) is zero,
Figure BDA0002382782110000035
the value of (d) is infinite;
(4) the time frequency value
Figure BDA0002382782110000036
Superposition to GD estimation in the time direction
Figure BDA0002382782110000037
Computing N-order time-rearrangement synchronous extrusion transform
Figure BDA0002382782110000038
(5) To pair
Figure BDA0002382782110000039
Modulus is taken to obtain N-order time rearrangement synchronous extrusionTime frequency spectrum corresponding to pressure transformation
Figure BDA00023827821100000310
Preferably, the method comprises the following steps: and (6) obtaining a reconstructed signal x' (t) by N-order time rearrangement synchronous extrusion inverse transformation.
Preferably, the method comprises the following steps: in the step (2),
Figure BDA00023827821100000311
is obtained by the following formula;
Figure BDA00023827821100000312
Figure BDA00023827821100000313
wherein v is a frequency variable independent of the value of ω, i is an imaginary unit, and G*(ω) represents the complex conjugate of G (ω).
Preferably, the method comprises the following steps: in step (3), the N-order group delay estimation is calculated by the following formula
Figure BDA00023827821100000314
Figure BDA00023827821100000315
In the formula, Im represents the imaginary part of the complex number.
Preferably, the method comprises the following steps: in the step (4), N-order time rearrangement synchronous extrusion transformation is obtained by adopting the following formula
Figure BDA0002382782110000041
Figure BDA0002382782110000042
In the formula, u is a time variable and is a different time variable independent of the value of t, and the function σ (x) is a Dirac function.
Preferably, the method comprises the following steps: in the step (6), a reconstructed signal x' (t) is obtained specifically by adopting the following formula;
Figure BDA0002382782110000043
in the formula, τ is also a time variable and is a different time variable independent of the value of t.
In step (2), G (omega) is a frequency window function, and since k is a variable, omega iskG (omega) represents different frequency window functions, the frequency domain signal X (omega) is multiplied by the moving frequency window function, then inverse Fourier transform is carried out on the multiplied signals, the corresponding value of the short-time Fourier transform result is called as a short-time Fourier transform time-frequency value, different time-frequency values, two N-order square matrixes α can be constructedN(t, ω) and βN(t,ω)。
And the step (4) is used for calculating real time-frequency information corresponding to the time-frequency points (t, omega), so that the time-frequency spectrum energy is superposed near the real GD, and the time-frequency focusing performance is improved.
Compared with the prior art, the invention has the advantages that:
(1) according to the invention, on the basis of short-time Fourier transform in a frequency domain form, a high-order time rearrangement synchronous extrusion operator is introduced, and time-frequency spectrum energy is rearranged in the time direction to be gathered near the real GD, so that energy divergence is inhibited, and time-frequency readability is improved.
(2) Compared with other time rearrangement synchronous extrusion methods, the method disclosed by the invention utilizes the Taylor formula to expand the phase of the signal to the N order on the basis of using the frequency domain signal model, so that the GD estimation of the N order can be calculated, and the precision of the GD estimation is improved.
(3) The data after the development and the high-order time rearrangement can also realize the high-precision reconstruction of the signal, and the reconstructed signal and the original signal have high reduction degree and good consistency.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a time domain diagram of an original signal according to embodiment 4;
FIG. 3 is a time-frequency spectrum of FIG. 2 after short-time Fourier transform;
FIG. 4 is a time-frequency spectrum of the simultaneous extrusion transform of FIG. 2 after 1-order time rearrangement;
FIG. 5 is a time-frequency spectrum of the 2-step time rearrangement synchronous extrusion transform of FIG. 2;
FIG. 6 is a time-frequency spectrum of the transformation of FIG. 2 by 3-order time rearrangement and synchronous extrusion;
FIG. 7 is a time-frequency spectrum of the 4-order time rearrangement synchronous extrusion transform of FIG. 2;
FIG. 8 is a graph of the reconstructed signal of FIG. 4 after the 1 st order method of the present invention;
FIG. 9 is a graph of the reconstructed signal of FIG. 5 after the 2-stage method of the present invention;
FIG. 10 is a graph of the reconstructed signal of FIG. 6 after the 3-stage method of the present invention;
FIG. 11 is a diagram of the reconstructed signal of FIG. 7 after the 4-stage method of the present invention.
Detailed Description
The invention will be further explained with reference to the drawings.
Example 1: referring to fig. 1, a high-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method includes the following steps:
(1) acquiring a time domain signal X (t), and performing Fourier transform to obtain a frequency domain signal X (omega), wherein t is time and omega is frequency;
(2) selecting the order N and the frequency window function G (omega), and calculating the frequency window function G (omega) of X (omega) at omegakG (omega) short-time Fourier transform to obtain different time frequency values corresponding to time frequency points (t, omega)
Figure BDA0002382782110000061
Where N is a positive integer, k is 0,1,2, …, max {1,2N-2}, and when k is 0, G (ω) is ω0G(ω)，
Figure BDA0002382782110000062
Figure BDA0002382782110000063
By using
Figure BDA0002382782110000064
Constructing an N-th order square αN(t, ω) and βN(t,ω)；
Figure BDA0002382782110000065
Figure BDA0002382782110000066
In the formula (I), the compound is shown in the specification,
Figure BDA0002382782110000067
s is an adjustment factor in the window function G (ω);
(3) using an N-th order square αN(t, ω) and βN(t, ω) computing an N-th order group delay estimate
Figure BDA0002382782110000068
When βNWhen the determinant value of (t, ω) is not zero, the matrix α is dividedN(t, ω) and βNDividing the determinant of (t, omega), taking the imaginary part of the determinant, adding the imaginary part of the determinant to the time t corresponding to the time frequency point (t, omega), and calculating to obtain the product
Figure BDA0002382782110000069
A value of (d);
when matrix βNWhen the determinant value of (t, ω) is zero,
Figure BDA00023827821100000610
the value of (d) is infinite;
(4) the time frequency value
Figure BDA00023827821100000611
Superposition to GD estimation in the time direction
Figure BDA00023827821100000612
Computing N-order time-rearrangement synchronous extrusion transform
Figure BDA00023827821100000613
(5) To pair
Figure BDA00023827821100000614
Modulus is taken to obtain a time frequency spectrum corresponding to N-order time rearrangement synchronous extrusion transformation
Figure BDA00023827821100000615
Example 2: and (6) obtaining a reconstructed signal x' (t) by N-order time rearrangement synchronous extrusion inverse transformation. The rest is the same as in example 1.
Example 3: this example is the same as example 2, except that, in step (2),
Figure BDA0002382782110000071
is obtained by the following formula;
Figure BDA0002382782110000072
Figure BDA0002382782110000073
wherein v is a frequency variable independent of the value of ω, i is an imaginary unit, and G*(ω) represents the complex conjugate of G (ω).
In step (3), the N-order group delay estimation is calculated by the following formula
Figure BDA0002382782110000074
Figure BDA0002382782110000075
In the formula, Im represents the imaginary part of the complex number.
In the step (4), the concrete application isObtaining N-order time rearrangement synchronous extrusion transformation by the following formula
Figure BDA0002382782110000076
Figure BDA0002382782110000077
In the formula, u is a time variable and is a different time variable independent of the value of t, and the function σ (x) is a Dirac function.
In the step (6), a reconstructed signal x' (t) is obtained specifically by adopting the following formula;
Figure BDA0002382782110000078
in the formula, τ is also a time variable and is a different time variable independent of the value of t.
Example 4: referring to fig. 2 to 7, the present embodiment is the same as embodiment 2, wherein specifically,
in the step (1), the time domain signal X (t) is fourier-transformed to obtain the frequency domain signal X (ω), the time domain diagram of the original time domain signal X (t) is shown in fig. 2, and the short-time fourier-transform time-frequency spectrogram under the window function G (ω) is shown in fig. 3.
In step (4), N-order time rearrangement synchronous extrusion transformation
Figure BDA0002382782110000081
The time-frequency spectrograms are shown in fig. 4-7, wherein fig. 4-7 are time-frequency spectrograms corresponding to 1 order, 2 order, 3 order and 4 order respectively.
It can be seen from the figures that the time-frequency spectrogram obtained from fig. 4-7 is obviously better than the time-frequency spectrogram of fig. 3 after the processing by the method of the present invention, that is, the time-frequency spectrogram obtained after the processing by the method of the present invention is obviously better than the original time-frequency spectrogram, and the higher the order, the clearer and more concentrated the distribution of time-frequency energy.
Example 5:
referring to fig. 8 to 11, fig. 8 to 11 are diagrams of signals reconstructed by the 1-4 order method of the present invention, respectively, and comparing fig. 8 to 11 with fig. 2, it can be seen that the signals expanded and reconstructed by the present invention have a higher restoration degree than the original signals, which illustrates that the method of the present invention can realize the precision reconstruction of the signals.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (6)

1. A high-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method is characterized by comprising the following steps: the method comprises the following steps:
(1) acquiring a time domain signal X (t), and performing Fourier transform to obtain a frequency domain signal X (omega), wherein t is time and omega is frequency;
(2) selecting the order N and the frequency window function G (omega), and calculating the frequency window function G (omega) of X (omega) at omegakG (omega) short-time Fourier transform to obtain different time frequency values corresponding to time frequency points (t, omega)
Figure FDA0002382782100000011
Where N is a positive integer, k is 0,1,2, …, max {1,2N-2}, and when k is 0, G (ω) is ω0G(ω)，
Figure FDA0002382782100000012
Figure FDA0002382782100000013
By using
Figure FDA0002382782100000014
Constructing an N-th order square αN(t, ω) and βN(t，ω)；
Figure FDA0002382782100000015
Figure FDA0002382782100000016
In the formula (I), the compound is shown in the specification,
Figure FDA0002382782100000017
s is an adjustment factor in the window function G (ω);
(3) using an N-th order square αN(t, ω) and βN(t, ω) computing an N-th order group delay estimate
Figure FDA0002382782100000018
When βNWhen the determinant value of (t, ω) is not zero, the matrix α is dividedN(t, ω) and βNDividing the determinant of (t, omega), taking the imaginary part of the determinant, adding the imaginary part of the determinant to the time t corresponding to the time frequency point (t, omega), and calculating to obtain the product
Figure FDA0002382782100000019
A value of (d);
when matrix βNWhen the determinant value of (t, ω) is zero,
Figure FDA00023827821000000110
the value of (d) is infinite;
(4) the time frequency value
Figure FDA00023827821000000111
Superposition to GD estimation in the time direction
Figure FDA00023827821000000112
Computing N-order time-rearrangement synchronous extrusion transform
Figure FDA00023827821000000113
(5) To pair
Figure FDA0002382782100000021
Modulus is taken to obtain a time frequency spectrum corresponding to N-order time rearrangement synchronous extrusion transformation
Figure FDA0002382782100000022
2. The high-precision high-order time rearrangement synchronous extrusion transform time-frequency analysis method according to claim 1, characterized in that: and (6) obtaining a reconstructed signal x' (t) by N-order time rearrangement synchronous extrusion inverse transformation.
3. The high-precision high-order time rearrangement synchronous extrusion transform time-frequency analysis method according to claim 1, characterized in that: in the step (2),
Figure FDA0002382782100000023
is obtained by the following formula;
Figure FDA0002382782100000024
Figure FDA0002382782100000025
wherein v is a frequency variable independent of the value of ω, i is an imaginary unit, and G*(ω) represents the complex conjugate of G (ω).
4. The high-precision high-order time rearrangement synchronous extrusion transform time-frequency analysis method according to claim 1, characterized in that: in step (3), the N-order group delay estimation is calculated by the following formula
Figure FDA0002382782100000026
Figure FDA0002382782100000027
In the formula, Im represents the imaginary part of the complex number.
5. According to the rightThe high-precision high-order time rearrangement synchronous extrusion transformation time-frequency analysis method according to claim 1, characterized in that: in the step (4), N-order time rearrangement synchronous extrusion transformation is obtained by adopting the following formula
Figure FDA0002382782100000029
Figure FDA0002382782100000028
In the formula, u is a time variable and is a different time variable independent of the value of t, and the function σ (x) is a Dirac function.
6. The high-precision high-order time rearrangement synchronous extrusion transform time-frequency analysis method according to claim 2, characterized in that: in the step (6), a reconstructed signal x' (t) is obtained specifically by adopting the following formula;
Figure FDA0002382782100000031
in the formula, τ is also a time variable and is a different time variable independent of the value of t.
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CN114563824A (en) * 2022-02-25 2022-05-31 成都理工大学 Identification method for second-order multiple synchronous extrusion polynomial chirp transform thin reservoir
CN114563824B (en) * 2022-02-25 2024-01-30 成都理工大学 Second-order multiple synchronous extrusion polynomial chirp let transformation thin reservoir identification method
CN115356108A (en) * 2022-10-10 2022-11-18 成都阿普奇科技股份有限公司 Method and device for diagnosing mechanical fault of modulation high-order horizontal extrusion transformation
CN115356108B (en) * 2022-10-10 2023-02-10 成都阿普奇科技股份有限公司 Method and device for diagnosing mechanical fault of modulation high-order horizontal extrusion transformation

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