CN110347970A - Fractional order is synchronous to extract generalized S-transform Time-frequency Decomposition and reconstructing method - Google Patents
Fractional order is synchronous to extract generalized S-transform Time-frequency Decomposition and reconstructing method Download PDFInfo
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Abstract
The invention discloses a kind of synchronous extraction generalized S-transform Time-frequency Decomposition of fractional order and reconstructing methods, the following steps are included: input x (t), it chooses different rotation angle [alpha]s and Fourier Transform of Fractional Order spectrum analysis is carried out to x (t), select optimal rotation angle [alpha]optFractional order generalized S-transform is carried out to signal, obtains fractional order generalized S-transform valueModulus again, andOn the basis of seek fractional order instantaneous Frequency EstimationSynchronous extraction operator is obtained according to instantaneous Frequency EstimationOperator is extracted using synchronousFractional order generalized S-transform time-frequency spectrum is extracted, effective energy, final reconstruction signal are retained.The time-frequency of signal can be characterized and be generalized to time score order frequency characterization by the present invention, pass through the adaptive optimal rotation angle of selection, Time-Frequency Information details therein can be identified, it is operated on this basis by adjusting window function parameter and " extraction ", assemble the synchronous energy for extracting generalized S-transform time-frequency spectrum of fractional order more, to greatly improve the time frequency resolution of signal.
Description
Technical field
The present invention relates to a kind of signal decomposition and reconstructing method more particularly to a kind of synchronous extraction generalized S-transforms of fractional order
Time-frequency Decomposition and reconstructing method.
Background technique
As an important branch in nonstationary random response field, time frequency analysis is always the research of modern signal processing
One of hot spot.Common Time-Frequency Analysis Method has Short Time Fourier Transform (STFT), wavelet transformation (CWT), S-transformation (ST) and wide
Adopted S-transformation (GST) etc..Wherein, " size " and " shape " of the window function of STFT is fixed, in practical applications, window function
It is difficult to guarantee acquired results in time domain, frequency domain after selected while having sufficiently high resolution ratio；The when m- ruler that CWT passes through signal
The characteristics of degree is analyzed, can reach multiresolution, but it is substantially or a kind of based in stationary signal, adjustable Fu of window
Leaf transformation, and the selection of wavelet function directly determines the effect of wavelet transformation analysis, therefore limits to the fine of signal
Analysis.ST had both overcome the shortcomings that Short Time Fourier Transform is not adjustable time window length, also solved the phase office of wavelet transformation
Portion's problem, but because its time histories sample shape used is fixed, its shape cannot be changed according to the variation of frequency, this is certain
The application of S-transformation is limited in degree.For this purpose, generalized S-transform is improved on the basis of wavelet transformation and S-transformation, lead to
Cross parameter regulation and obtain more flexible changeable window function, in the application have higher practical and flexibility, but because by
Heisenberg-Gabor indeterminate problem influences, and spectral resolution cannot still be optimal at that time.
With the development of Time-Frequency Analysis Method, it is found that it is poly- the certain signals of Fourier transform pairs cannot reach optimal frequency
Coke, therefore Fourier Transform of Fractional Order is provided and is gone from different angles by adjusting fractional order to obtain different transformation kernels
The feature of non-stationary signal is analyzed, processing linear FM signal is especially suitable for.Fractional order generalized S-transform combines in fractional order Fu
The advantages of leaf transformation and generalized S-transform, enhances flexibility of the generalized S-transform to signal processing, improves the when frequency division of signal
Distinguish ability.
Synchronous extraction transformation (SET) is a kind of high-resolution time frequency analysis side newly proposed on the basis of traditional time frequency analysis
Method.This method establishes a kind of synchronous extraction operator, when for extracting in original time-frequency spectrum on the basis of Short Time Fourier Transform
Time-frequency coefficients at frequency crestal line substantially increase time-frequency resolving accuracy to obtain new time-frequency spectrum.And this method substantially belongs to
In time frequency analysis post-processing technology, therefore there is certain dependence to the time frequency resolution of original Time-frequency method.
Summary of the invention
It solves the above problems the object of the invention is that providing one kind, time and the frequency point of signal can be greatlyd improve
The fractional order of resolution is synchronous to extract generalized S-transform Time-frequency Decomposition and reconstructing method.
To achieve the goals above, the technical solution adopted by the present invention is that such: a kind of fractional order is synchronous to extract broad sense S
Convert Time-frequency Decomposition and reconstructing method, comprising the following steps:
(1) original one-dimensional signal x (t) to be analyzed is inputted；
(2) Fourier Transform of Fractional Order spectrum analysis carried out to one-dimensional signal x (t) with different rotation angle [alpha]s, and according to
The size of spectral coefficient determines the aggregation extent of signal energy, select-out signal energy accumulating corresponding rotation angle when most strong,
As optimal rotation angle [alpha]opt；
(3) optimal rotation angle [alpha] is utilizedoptFractional order generalized S-transform is carried out to signal, obtains fractional order generalized S-transform valueWherein x is one-dimensional signal x (t), and α is optimal rotation angle [alpha]opt, τ is time shaft displacement parameter, and u is point
Number order frequency；
(4) to fractional order generalized S-transform value modulus, the energy of each time frequency point is obtained, to obtain optimal rotation angle
Corresponding fractional order generalized S-transform time-frequency spectrum
(5) it is obtained according to step (3)Calculate fractional order instantaneous Frequency EstimationRoot again
According toObtain synchronous extraction operator
(6) operator is extracted using synchronousThe fractional order generalized S-transform time-frequency spectrum that step (4) obtains is extracted,
Only retain and meets set on time-frequency planeThe energy at place obtains the synchronous extraction generalized S-transform of fractional order
Value
(7) generalized S-transform value is extracted according to fractional order is synchronousReconstruct original signal.
As preferred: the step (2) specifically:
Optimal rotation angle [alpha] is chosen according to the following formulaopt
In formula, t is the time, and u is score order frequency, Kα(t, u) is the transformation kernel of Fourier Transform of Fractional Order, expression formula
Are as follows:
Whereinα=a pi/2, a are the order of fractional order, and n is integer.
As preferred: the step (3) specifically: carry out fractional order generalized S-transform to signal using following formula
In formula, τ is time shaft displacement parameter, and g (t, u) is the window function about t and u, expression formula are as follows:
Wherein λ and p is the regulatory factor of time histories sample.
As preferred: the step (5) is specially to be calculated using following formula
Wherein
As preferred: the step (6) is specially to be calculated using following formula:
As preferred: step (7) using following formula also specifically, be reconstructed:
The principle of the present invention are as follows: can adaptively choose optimal rotation angle according to signal characteristic, time-frequency is characterized and is promoted
It is characterized to time fractional order frequency, because frequency axis is rotated to suitable position, so that it may to Time-Frequency Information details therein
It is identified.The present invention passes through adjusting window function parameter on this basis, and the energy of the time-frequency spectrum made is more concentrated, same
During walking " extraction ", fractional order instantaneous Frequency Estimation value can effectively improve the computational accuracy of the estimated value, further increase
The synchronous energy accumulating for extracting generalized S-transform time-frequency spectrum of fractional order, to greatly improve time and the frequency discrimination of signal
Rate.
Wherein the purpose of step (1) is input signal.
Step (2) be it is adaptive select optimal rotation angle, time-frequency characterization is generalized to time score order frequency characterization,
Because frequency axis is rotated to position appropriate, so that it may be identified that one is available to Time-Frequency Information details therein
The time-frequency Energy distribution more assembled, secondly the computational accuracy of following fractional order instantaneous Frequency Estimation values can be improved effectively.
Step (3) utilizes optimal rotation angle [alpha]optIt is to obtain to the purpose that signal carries out fractional order generalized S-transform
Fractional order generalized S-transform value, this is one of committed step of the invention, during handling herein, passes through the tune of window function parameter
Section, the time-frequency spectrum energy on the one hand made are more assembled, and on the other hand can be further improved fractional order instantaneous Frequency Estimation
The computational accuracy of value.
The purpose of step (4) modulus is the time-frequency spectrum in order to obtain fractional order generalized S-transform.
Step (5) seeks fractional order instantaneous Frequency EstimationThen synchronous extraction is obtained according to instantaneous Frequency Estimation
OperatorThe purpose is to the computational accuracy promotions by fractional order instantaneous Frequency Estimation value, obtain more accurate synchronizing and mention
Operator is taken, is prepared and one of committed step of the invention for the synchronous generalized S-transform that extracts of fractional order below.
The purpose of step (6) is: extracting operator using synchronousFractional order generalized S-transform time-frequency spectrum is extracted,
Only retain and meets set on time-frequency planeThe energy at place, complementary energy are completely removed, so that signal
Time-frequency energy more concentrate, time frequency resolution is further promoted.
Step (7) is signal reconstruction, by treated for step (1)-(6) signal, has signal time-frequency energy height poly-
The advantages of collection.
Step (7) is signal reconstruction, by treated for step (1)-(6) signal, has signal time-frequency energy height poly-
The advantages of collection.
Compared with the prior art, the advantages of the present invention are as follows:
1, fractional order generalized S-transform is deduced a kind of new Time-Frequency Analysis Method-point with based on synchronous extraction transformation
Number rank is synchronous to extract generalized S-transform and its reconstruct expression formula.
2, the shortcomings that present invention uses fractional order generalized S-transform, overcomes Short Time Fourier Transform, can be by difference
The Fourier Transform of Fractional Order frequency spectrum that rotation angle is calculated is analyzed, and is selected according to the aggregation extent of energy is adaptive
Time-frequency characterization is generalized to time score order frequency characterization by optimal rotation angle, because frequency axis is rotated to suitable position,
Time-Frequency Information details therein can be identified.
3, according to window parameter when signal characteristic flexible modulation, when obtaining the higher fractional order generalized S-transform of time frequency resolution
Frequency spectrum." extraction " operation is carried out to fractional order generalized S-transform time-frequency spectrum, only retains the energy in time-frequency plane near time-frequency crestal line
Amount, remaining diverging energy are completely removed, so that the time-frequency energy of signal is more concentrated, time frequency resolution is obtained further
It is promoted.
Detailed description of the invention
Fig. 1 is flow chart of the present invention；
Fig. 2 is the one-dimensional signal x (t) of two linear signals synthesis in embodiment 2；
Fig. 3 is Fig. 2 through the method for the present invention treated time-frequency spectrum；
Fig. 4 is the one-dimensional signal x (t) of two nonlinear properties synthesis in embodiment 3；
Fig. 5 is Fig. 4 through the method for the present invention treated time-frequency spectrum.
Specific embodiment
The present invention will be further described with reference to the accompanying drawings.
Embodiment 1: referring to Fig. 1, a kind of fractional order is synchronous to extract generalized S-transform Time-frequency Decomposition and reconstructing method, including with
Lower step:
(1) original one-dimensional signal x (t) to be analyzed is inputted；
(2) Fourier Transform of Fractional Order spectrum analysis carried out to one-dimensional signal x (t) with different rotation angle [alpha]s, and according to
The size of spectral coefficient determines the aggregation extent of signal energy, select-out signal energy accumulating corresponding rotation angle when most strong,
As optimal rotation angle [alpha] opt；
The step (2) specifically:
Optimal rotation angle [alpha] is chosen according to the following formulaopt
In formula, t is the time, and u is score order frequency, Kα(t, u) is the transformation kernel of Fourier Transform of Fractional Order, expression formula
Are as follows:
Whereinα=a pi/2, a are the order of fractional order, and n is integer；
(3) optimal rotation angle [alpha] is utilizedoptFractional order generalized S-transform is carried out to signal, obtains fractional order generalized S-transform valueWherein x is one-dimensional signal x (t), and α is optimal rotation angle [alpha] opt, τ is time shaft displacement parameter, and u is point
Number order frequency；
The step (3) specifically: fractional order generalized S-transform is carried out to signal using following formula
In formula, τ is time shaft displacement parameter, and g (t, u) is the window function about t and u, expression formula are as follows:
Wherein λ and p is the regulatory factor of time histories sample；
(4) to fractional order generalized S-transform value modulus, the energy of each time frequency point is obtained, to obtain optimal rotation angle
Corresponding fractional order generalized S-transform time-frequency spectrum
(5) it is obtained according to step (3)Calculate fractional order instantaneous Frequency EstimationRoot again
According toObtain synchronous extraction operator
The step (5) is specially to be calculated using following formula
Wherein,
(6) operator is extracted using synchronousThe fractional order generalized S-transform time-frequency spectrum that step (4) obtains is extracted,
Only retain and meets set on time-frequency planeThe energy at place obtains the synchronous extraction generalized S-transform of fractional order
Value
The step (6) is specially to be calculated using following formula:
(7) generalized S-transform value is extracted according to fractional order is synchronousReconstruct original signal.
The step (6) using following formula specifically, be reconstructed:
Embodiment 2:
(1) original one-dimensional signal x (t) to be analyzed is inputted, in the present embodiment, one-dimensional signal x (t), which is shown in Fig. 2, is
The signal sig of two linear signals sig1 and sig2 synthesis.
Step (2)-(7) are the same as embodiment 1.
The present embodiment further includes between step (6) (7), further includes a step are as follows: carries out modulus to result in step (6)
Operation obtains the synchronous extraction generalized S-transform time-frequency spectrum of final fractional orderThe step obtains that treated at this time
Time-frequency spectrum such as Fig. 3.From figure 3, it can be seen that the time-frequency spectrum of two linear subsignals is able to correctly after the method for the present invention is handled
It presents, frequency linearly increases with the time, and frequency resolution is very high, and energy is concentrated.
Embodiment 3:
(1) original one-dimensional signal x (t) to be analyzed is inputted, in the present embodiment, one-dimensional signal x (t), which is shown in Fig. 4, is
The signal sig of two nonlinear properties sig1 and sig2 synthesis.
Step (2)-(7) are the same as embodiment 1.
The present embodiment further includes between step (6) (7), further includes a step are as follows: carries out modulus to result in step (6)
Operation obtains the synchronous extraction generalized S-transform time-frequency spectrum of final fractional orderThe step obtains that treated at this time
Time-frequency spectrum such as Fig. 5.From fig. 5, it can be seen that treated, time-frequency spectrum has two curves, respectively corresponds two nonlinear properties
Time-frequency spectrum, for frequency with time curved variation, frequency resolution is very high, energy concentrate.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all in essence of the invention
Made any modifications, equivalent replacements, and improvements etc., should all be included in the protection scope of the present invention within mind and principle.
Claims (6)
1. a kind of fractional order is synchronous to extract generalized S-transform Time-frequency Decomposition and reconstructing method, it is characterised in that: the following steps are included:
(1) original one-dimensional signal x (t) to be analyzed is inputted；
(2) Fourier Transform of Fractional Order spectrum analysis is carried out to one-dimensional signal x (t) with different rotation angle [alpha]s, and according to frequency spectrum
The size of coefficient determines the aggregation extent of signal energy, select-out signal energy accumulating corresponding rotation angle when most strong, by it
As optimal rotation angle [alpha]opt；
(3) optimal rotation angle [alpha] is utilizedoptFractional order generalized S-transform is carried out to signal, obtains fractional order generalized S-transform valueWherein x is one-dimensional signal x (t), and α is optimal rotation angle [alpha]opt, τ is time shaft displacement parameter, and u is point
Number order frequency；
(4) to fractional order generalized S-transform value modulus, the energy of each time frequency point is obtained, so that it is corresponding to obtain optimal rotation angle
Fractional order generalized S-transform time-frequency spectrum
(5) it is obtained according to step (3)Calculate fractional order instantaneous Frequency EstimationFurther according toObtain synchronous extraction operator
(6) operator is extracted using synchronousThe fractional order generalized S-transform time-frequency spectrum that step (4) obtains is extracted, is only protected
It stays and meets set on time-frequency planeThe energy at place obtains the synchronous extraction generalized S-transform value of fractional order
(7) generalized S-transform value is extracted according to fractional order is synchronousReconstruct original signal.
2. the synchronous extraction generalized S-transform Time-frequency Decomposition of fractional order according to claim 1 and reconstructing method, feature exist
In: the step (2) specifically:
Optimal rotation angle [alpha] is chosen according to the following formulaopt
In formula, t is the time, and u is score order frequency, Kα(t, u) is the transformation kernel of Fourier Transform of Fractional Order, expression formula are as follows:
Whereinα=a pi/2, a are the order of fractional order, and n is integer.
3. the synchronous extraction generalized S-transform Time-frequency Decomposition of fractional order according to claim 2 and reconstructing method, feature exist
In: the step (3) specifically: fractional order generalized S-transform is carried out to signal using following formula
In formula, τ is time shaft displacement parameter, and g (t, u) is the window function about t and u, expression formula are as follows:
Wherein λ and p is the regulatory factor of time histories sample.
4. the synchronous extraction generalized S-transform Time-frequency Decomposition of fractional order according to claim 3 and reconstructing method, feature exist
In: the step (5) is specially to be calculated using following formula
Wherein
5. the synchronous extraction generalized S-transform Time-frequency Decomposition of fractional order according to claim 4 and reconstructing method, feature exist
In: the step (6) is specially to be calculated using following formula:
6. the synchronous extraction generalized S-transform Time-frequency Decomposition of fractional order according to claim 5 and reconstructing method, feature exist
In: step (7) is specially to be reconstructed using following formula:
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