CN106645947A - Time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel - Google Patents

Abstract

The invention discloses a time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel; the time-frequency analysis method combines the advantages in nonlinear mode decomposition analysis method and those in adaptive optimal kernel and comprises: decomposing a multi-component nonstable signal into a group of physically significant nonlinear mode components by using nonlinear mode composition algorithm, wherein the requirement on the anti-noise performance of subsequent analysis method is lowered since the algorithm has high noise robustness; then, enabling a kernel function to change adaptively with signal changes by using the time-frequency analysis method of adaptive optimal kernel, so that cross terms are effectively inhibited and time-frequency concentrating capacity is improved. The novel analysis method inherits the advantages in both nonlinear mode composition method and adaptive optimal kernel analysis method, and has excellent performances.

Description

Time-frequency analysis method based on nonlinear mode decomposition and self-adaptive optimal kernel

Technical Field

The invention relates to the field of time-frequency analysis of non-stationary signals, in particular to a time-frequency analysis method based on nonlinear mode decomposition and self-adaptive optimal kernels.

Background

A good time-frequency analysis method is self-evident in the evaluation of non-stationary signals. The signal itself has many properties, and for signal estimation, frequency domain characteristics are important properties.

At present, a plurality of signal processing and estimating methods exist, the most classical method is certainly Fourier transform, which can reflect the frequency characteristics of signals, the Fourier transform has good frequency domain resolution capability for stationary signals, the frequency spectrum of the signals can be obtained after the Fourier transform, but the time information is lost in the Fourier transform, namely the time of occurrence of each frequency spectrum is unknown, so the Fourier transform is not suitable for processing non-stationary signals. For non-stationary signals in an actual environment, common linear time-frequency representation methods include short-time Fourier transform, wavelet transform, S transform and the like, but the time-frequency precision of the short-time Fourier transform, the wavelet transform, the S transform and the like needs to be improved; while bilinear time-frequency distributions such as Wigner-Ville distribution, Cohen time-frequency distribution and the like have higher time-frequency precision, cross terms are inevitably introduced in the calculation process.

Empirical Mode Decomposition (EMD) can decompose a complex signal into a series of eigenmode functions (IMFs). Although EMD has a bad performance in suppressing cross terms of multi-component signals, its disadvantages are also obvious, the noise immunity is poor, and modal aliasing is caused by the presence of discontinuous signals or noise. To overcome the disadvantages of EMD, a noise-aided analysis method called Ensemble Empirical Mode Decomposition (EEMD) has emerged. While EEMD has improved noise immunity over EMD, it is still less desirable.

Time-Frequency Representation (TFR) is a very efficient instantaneous Frequency estimation method, especially for multi-component signals. The general TFR method only has a fixed window function or kernel function, so that the TFR method only has a better processing effect on a certain type of signals. The self-adaptive optimal kernel time-frequency representation (AOK TFR) is a signal-based instantaneous frequency estimation method, and adopts a radial Gaussian kernel function which can change along with the change of signals, so that the AOK has better time-frequency focusing performance and cross term inhibition capability.

EMD + AOK can reduce the effect of cross terms when analyzing multi-component signals, but is too sensitive to noise. EEMD + AOK may improve noise immunity to some extent, but is far from sufficient.

Nonlinear Mode Decomposition (NMD) can decompose a signal into a series of physically significant Nonlinear Mode components, and has strong noise robustness. The method is a novel algorithm which integrates a plurality of methods such as time-frequency analysis, data substitution inspection, harmonic identification and the like. However, the nonlinear mode component obtained by NMD decomposition is not a single component like IMF, and needs to be optimized by adapting the kernel function of the AOK method to the characteristics that change with the change of the signal.

The novel time-frequency representation method based on the NMD and the AOK not only utilizes the effective decomposition performance of the NMD on multi-component signals, but also inherits the excellent time-frequency focusing performance of the AOK and the capability of effectively inhibiting cross terms, and solves the problems that the EMD + AOK and the EEMD + AOK have poor anti-noise performance and still contain a certain amount of cross terms.

Disclosure of Invention

The purpose of the invention is as follows: in order to solve the technical problems, the invention provides a time-frequency analysis method based on Nonlinear Mode Decomposition and self-adaptive optimal kernel, which adopts Nonlinear Mode Decomposition (NMD) to decompose a signal to be processed into a series of Nonlinear Mode components to inhibit the interference of noise; then, each obtained signal component is processed by an Adaptive Optimal Kernel (AOK) analysis method to inhibit cross terms, and the time-frequency analysis performance is good because the AOK analysis method can automatically track the change of an analysis signal, the kernel function can be adaptively changed according to the change of the signal, and the kernel function generated for different signals is always optimal.

The technical scheme is as follows: in order to achieve the technical effects, the technical scheme provided by the invention is as follows:

a time-frequency analysis method based on nonlinear mode decomposition and self-adaptive optimal kernel comprises the following steps:

(1) defining the signal to be processed as s (t), the sampling frequency of s (t) is f_sThe data length is N; decomposing the signal s (t) to be processed into a group of nonlinear mode components by adopting a nonlinear mode decomposition method, namely:

wherein, c_i(t) is the ith nonlinear mode component of s (t), and n (t) represents noise;

(2) analyzing each nonlinear mode component decomposed in the step (1) by adopting a self-adaptive optimal kernel analysis method, wherein the method comprises the following steps:

(2-1) construction of radial Gaussian Kernel function

In the formula,representing radial angles for controlling radial Gaussian functionsA spread function of direction;

(2-2) taking the optimal kernel which changes along with the signal self-adaption as a target problem, and constructing an optimization problem model as follows:

the constraint conditions for setting the optimization problem model are as follows:

wherein,is the ith nonlinear mode component c_i(t) ambiguity function in polar coordinates, β being the volume of the optimal kernel;the expression in the rectangular coordinate system is:

wherein A is_i(t, θ, τ) isAmount of mapping in rectangular coordinate system, s^*(t)，w^*(u) is s (t), c_i(t), complex conjugate forms of w (u); the window function w (u) is a symmetrical diamond window function with the center of T and the width of 2T, when | u | is greater than T, w (u) is 0, variables tau and theta are parameters of a general fuzzy domain { tau, u }, and | tau | is less than 2T;

(2-3) according to A_i(t, theta, tau) solving the optimization problem model to obtain a nonlinear mode component c_i(t) optimum kernel function Φ_(i)opt(t，θ，τ)；

(2-4) calculating the nonlinear mode component c_i(t) the time-frequency representation of the adaptive optimal kernel is:

in the formula,is a nonlinear mode component c_i(t) an energy value at time t;

(3) according to the time-frequency representation of the self-adaptive optimal kernels of all the nonlinear mode components obtained in the step (2), calculating the result of time-frequency analysis as follows:

further, the method for decomposing the signal s (t) to be processed into a set of nonlinear mode components by using a nonlinear mode decomposition method in the step (1) comprises the steps of:

(1-1) wavelet transform expression W of calculation signal s (t)_s(ω，t)：

Wherein,is the Fourier transform of s (t), i.e.s⁺(t) is the positive frequency portion of the s (t) signal,is a wavelet function, andare mutually Fourier transform pairs and satisfy the conditionψ^*(t)，Respectively by psi (t),the conjugate complex number of (a); omega_ψWhich represents the peak frequency of the wavelet, ω_ψ＝1，f₀the resolution parameter is used for balancing the time and frequency resolution in the transformation process;

(1-2) judging whether the wavelet transform obtained by the calculation in the step (1-1) is the optimal time-frequency representation of s (t); if the judgment result is no, calculating a windowing Fourier transform expression G of the signal s (t)_s(ω，t)：

Wherein g (t) is a windowed function of a windowed Fourier transform,is the Fourier transform of g (t), and satisfies the following conditions:

and is

(1-3) finding all ridge curves represented by the optimal time frequency of a signal s (t), and reconstructing harmonic components by using a ridge method, wherein the h-th harmonic component is as follows:

x^(h)(t)＝A^(h)(t)cosφ^(h)(t)，h∈[1，2，…，N]

in the formula, A^(h)(t)、φ^(h)(t) is the amplitude and phase of the h-th harmonic component, respectively; v. of^(h)(t) is the frequency of the h-th harmonic component, y^(h)(t)≡φ′^(h)(t)；φ′^(h)(t) is phi^(h)(t) a derivative over time t; n represents the highest order of the harmonic component, i.e., the data length;

(1-4) identifying the truth of the extracted harmonic component by using an anti-noise alternative inspection method, screening out all real harmonic components,

(1-5) adding all the true harmonic components to obtain oneNonlinear mode component c₁(t)；

(1-6) subtracting c obtained by nonlinear mode decomposition from the original signal s (t)₁(t) and repeating steps 1-1 to 1-6 on the residual component to obtain all the respective nonlinear mode components c_i(t)；

Finally, the original target signal s (t) can be expressed as:

in the formula, n (t) represents noise.

Further, the method for finding all ridge curves represented by the signal s (t) in the step (1-3) is as follows:

at time t_iFinding h maximum value points, and connecting the h maximum value points to form a moment t_iThe ridge curves of (a) are:

in the above formula, i is 1, 2, …, N is data length; h_s(ω, t) is the best time-frequency representation of signal s (t), i.e. W_s(ω, t) or G_s(ω，t)；H_sAnd (omega, t) finding out ridge point connecting lines at all the time points, namely forming N ridge curves.

Further, the h-th harmonic component x is reconstructed by using a ridge method in the step (1-3)^(h)(t)＝A^(h)(t)cosφ^(h)(t) comprises the steps of:

if the best time-frequency representation of the signal s (t) to be processed is a wavelet transform, then

If the best time-frequency representation of the signal to be processed s (t) is a windowed Fourier transform, then

In the formula,andis an improved discretization impact factor obtained by parabolic interpolation.

Further, the method for screening real harmonic components in the step (1-4) comprises the steps of:

(1-4-1) construction of the recognition statistic D (α)_A，α_ν) For measuring the order of magnitude and frequency of the respective harmonic components, D (α)_A，α_v) The expression of (a) is:

in the formula,andrespectively represent A^(h)(t) and v^(h)(t) spectral entropy α_A，α_vIs a weight coefficient;

(1-4-2) creation of N for the original signal s (t)_sFourier transform the substitution data; wherein, the expression of the jth Fourier transform substitution data is as follows:

(1-4-3) calculating an optimal time-frequency expression formula of each Fourier transform alternative data, extracting each subharmonic component from the optimal time-frequency expression formula, and calculating identification statistics corresponding to each Fourier transform alternative data as follows:

defining a significance level indicatorIn the formulaTo satisfy D_s＞D₀The number of substitute data of (a); d₀The degree of order of h harmonic of the original signal;

setting the significance level index as p, and satisfying D when x substitute data exist_s＞D₀And x is more than or equal to p × N_sThen, the corresponding harmonic component is judged not to be noise; otherwise, judging that the corresponding harmonic component is noise;

(1-4-4) calculating a comprehensive metric of correlation between harmonics

Wherein,

in the formula, w_A，w_φ，w_vRepresentsWeight of rho^(h)Equal weight rho is distributed for amplitude and phase consistency^(h)≡ρ^(h)(1，1，0)；

(1-4-5) defining a threshold value for the composite metric value

(1-4-6) when a harmonic component satisfies ρ^(h)≥ρ_minAnd when the significance level index p is more than or equal to 95%, judging that the harmonic component is a real harmonic component through inspection.

Further, it is judged in the step (1-4-3) whether or not a substitute data satisfies D_s＞D₀The method comprises the following steps:

any three sets of parameters with different values are selected (α)_A，α_v) Respectively calculating the identification statistic values corresponding to the three groups of parameters, and if any one identification statistic value is larger than D₀Then the substitute data is judged to satisfy D_s＞D₀。

Has the advantages that: compared with the prior art, the invention has the following advantages:

the NMD method not only has good suppression effect on Gaussian white noise, but also can effectively suppress other types of noise signals, and has strong noise robustness and wide adaptability; however, the nonlinear mode component obtained by NMD decomposition is not a single component like IMF, and needs to be optimized by the characteristic that the kernel function of the AOK method can adaptively change with the change of the signal, and the AOK method can effectively and adaptively track the change of the non-stationary signal. The novel time-frequency representation method based on the NMD and the AOK not only utilizes the effective decomposition performance of the NMD on multi-component signals, but also inherits the excellent time-frequency focusing performance of the AOK and the capability of effectively inhibiting cross terms, and solves the problems that the EMD + AOK and the EEMD + AOK have poor anti-noise performance and still contain a certain amount of cross terms.

Drawings

FIG. 1 is a flow chart of the present invention.

Detailed Description

The following describes the embodiments and advantages of the present invention in detail by taking a multi-component non-stationary simulation signal as an example.

Let s (t) be a multi-component signal with white gaussian noise:

s(t)＝s_d(t)+n(t)

s_d(t)＝cos(20πt)+sin(200πt)+sin(400πt)+sin(100π(t-0.5)²)

wherein n (t) represents white Gaussian noise; s_d(T) is the ideal multi-component signal.set the sampling frequency to 1kHz, sample time 1s, data length to 1000.set the window length 2T to 128, and kernel volume limit to β to 5.

The process of the invention is shown in figure 1 and comprises the following steps:

step A: preparing a signal s (t) to be processed with a sampling frequency f_sThe data length is N;

and B: NMD analysis of the signal s (t);

step B-1: wavelet Transform (WT) W for computing signal s (t)_s(ω，t)，

Wherein,is the Fourier transform of s (t), i.e.s⁺(t) is the positive frequency portion of the s (t) signal,ψ (t) is a wavelet function, anAre mutually Fourier transform pairs and satisfy the conditionψ^*(t)，Respectively by psi (t),the complex conjugate of (a).Representing the wavelet peak frequency.

ω_ψ＝1

Wherein f is₀Is a resolution parameter that is used to trade off the resolution of time and frequency during the transformation (usually defaults to 1). The higher the temporal resolution, the lower the frequency resolution and vice versa.

Step B-2: examining the wavelet transform W obtained in step B-1_s(ω, t) is the best time-frequency representation of the signal s (t), and if not, a Windowed Fourier Transform (WFT) G is used_s(ω, t), WFT is defined as:

where g (t) is the window function of WFT,fourier transform of g (t), satisfying the condition:selecting a Gaussian window as a window function of the WFT, wherein the expression is as follows:

step B-3 finds all ridge curves represented by the signal s (t) in the best time frequencyHere, theIs the ridge curve of the h-th harmonic. The ridge curve is a curve formed by connecting some local maximum points on a time-frequency graph. Each maximum point is called a ridge point.

At time t_iH maximum points can be found out by using the following algorithm:

in the above formula, i is 1, 2, …, and N is the data length. H_s(ω, t) is the best time-frequency representation of signal s (t), i.e. W_s(ω, t) or G_s(ω，t)。

H is to be_sAnd (omega, t) finding out ridge point connecting lines at all the moments, namely forming H ridge curves.

Step B-4: reconstructing the h-th harmonic component as x by ridge method^(h)(t)＝A^(h)(t)cosφ^(h)(t) wherein A^(h)(t)、φ^(h)(t) amplitude and phase of the h-th harmonic component, respectively, v^(h)(t) is the frequency of the h-th harmonic component, v^(h)(t)≡φ′^(h)(t)，φ′^(h)(t) is phi^(h)(t) derivative with time t.

If the best time-frequency expression of the signal s (t) to be processed is W_s(ω, t), then the h-th harmonic component x of signal s (t) can be calculated by the following equation^(h)(t)。

If the best time-frequency expression of the signal s (t) to be processed is G_s(ω, t), then the h-th harmonic-related component x of s (t) is calculated^(h)(t) is:

in the formula,andis an improved discretization impact factor obtained by parabolic interpolation.

Step B-5: and (3) identifying the authenticity of the extracted harmonic component by using an anti-noise alternative inspection method, screening out all real harmonic components, and stopping the decomposition process when three continuous harmonic components are judged to be false. The method comprises the following steps:

(1) harmonic components are extracted from the TFR and corresponding discrimination statistics D are calculated (α)_A，α_v)；

The extracted amplitude A of each harmonic component^(h)(t) and frequency v^(h)The degree of order of (t) can be determined by its spectral entropyAndto measure quantitatively, the identification statistic D and the spectral entropy Q are defined as follows:

wherein, α_A，α_vIs to calculate D (α)_A，α_v) The weight coefficient of (2).

(2) Creating N for the original signal s (t)_sFourier transform the substitution data;

order toThe amplitude of (1) is kept constant and the phase becomes N distributed uniformly over [0, 2 π)_sA random phaseThe inverse Fourier transform corresponding to each phase of the random distribution is a Fourier transform substitution data s 'of s (t)'_j(t)。

Where j is 1, 2, …, N_s。

(3) Calculating TFR related to each alternative data, and respectively extracting each subharmonic component from the TFR, thereby calculating identification statistic corresponding to each alternative data

Defining a significance level indicatorIn the formulaIs D_s＞D₀The number of substitute data of (a); d₀Is the degree of order of the h harmonic of the original signal.

Suppose to create N_sThe substitute data and significance level index are set to p, i.e. at least p × N_sOne substitute data satisfies D_s＞D₀The component can be considered not to be noise and the decomposition process can continue.

Determining whether a substitute data satisfies D_s＞D₀The method includes selecting any three sets of parameters (α) with different values_A，α_v) Respectively calculating the identification statistic values corresponding to the three groups of parameters, and if any one identification statistic value is larger than D₀Then the substitute data is judged to satisfy D_s＞D₀。

(4) Calculating a composite metric of correlation between harmonics

Wherein,

in the formula, w_A，w_φ，w_vRepresentsWhere p is used by default^(h)≡ρ^(h)(1, 1, 0) equal weight is distributed for amplitude and phase consistency, and no weight is distributed for frequency consistency.

(5) To reduce false positives for true harmonic components, thresholds for the composite metric are defined

(6) When a harmonic component satisfies rho^(h)≥ρ_minAnd when the significance level index p is more than or equal to 95%, the harmonic component is considered to be a real harmonic component through inspection.

Step B-6: adding all the real harmonic components to obtain a nonlinear mode component c₁(t)。

Step B-7: subtracting c from the original signal s (t) by nonlinear mode decomposition₁(t) and repeating steps B-1 through B-6 on the residual component to obtain all of the respective nonlinear mode components c_i(t)。

Finally, the original target signal s (t) can be expressed as:

in the formula, n (t) represents noise.

And C: each nonlinear mode component c decomposed by NMD method_i(t) performing AOK analysis by the following procedure:

step C-1: first, a radial gaussian kernel function is selected, which is defined as follows:

in the formula,representing radial angles for controlling radial Gaussian functionsA spreading function of the direction.

Step C-2: in order to obtain an optimal kernel that varies adaptively with the signal, the following constrained optimization problem should be solved.

The constraint conditions are as follows:

wherein,is a function of the ambiguity in polar coordinates,for each non-linear mode componentβ is the volume of the optimal nucleus.

Definition A in rectangular coordinate System_i(t; θ, τ) is:

wherein s is^*(t)，w^*(u) is s (t), c_i(t), complex conjugate forms of w (u); the window function w (u) is a symmetrical diamond window function with the center of T and the width of 2T, and when | u | is greater than T, w (u) is 0; the variables τ and θ are parameters of the general fuzzy domain { τ, u }, and | τ | < 2T.

Step C-3: handleSubstituting the equations (4) and (5) in the step C-2 can obtain an optimal kernel function by solving the constrained optimization problemIt is associated with short term blurringThe function will also vary with time.

Step C-4: calculating a certain nonlinear mode component c at the current time point (t moment)_i(t) time-frequency representation of adaptive optimal Kernel AOK TFR_i：

In the formula,is a certain component c of the signal_i(t) the energy value at a certain time t.

Step C-5: and (4) repeating the step C-1 to the step C-4 to obtain the self-adaptive optimal kernel time-frequency representation of all decomposed nonlinear mode components.

Step D: AOK TFR of all the obtained signal components_i(i.e. the) Summing to obtain final time-frequency analysis result P_NMD-AOK(t，ω)：

The result of NMD + AOK time-frequency analysis can be finally obtained by the formula.

Compared with the prior art, the technical scheme provided by the invention can obtain the following analysis results:

only the AOK method can generate a certain amount of cross terms, and the noise resistance performance is poor; the EMD + AOK and EEMD + AOK methods can inhibit cross terms to a certain extent, but are still too sensitive to noise; the NMD + AOK method adopted by the invention is obviously superior to other methods in terms of noise removal and cross term inhibition.

The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (6)

1. A time-frequency analysis method based on nonlinear mode decomposition and self-adaptive optimal kernel is characterized by comprising the following steps:

(1) defining the signal to be processed as s (t), the sampling frequency of s (t) is f_sThe data length is N; decomposing the signal s (t) to be processed into a group of nonlinear mode components by adopting a nonlinear mode decomposition method, namely:

$s\left(t\right)=\underset{i}{\&Sigma;}{c}_i\left(t\right)+n\left(t\right)$

wherein, c_i(t) is the ith nonlinear mode component of s (t), and n (t) represents noise;

(2) analyzing each nonlinear mode component decomposed in the step (1) by adopting a self-adaptive optimal kernel analysis method, wherein the method comprises the following steps:

(2-1) construction of radial Gaussian Kernel function

In the formula,representing radial angles for controlling radial Gaussian functionsA spread function of direction;

(2-2) taking the optimal kernel which changes along with the signal self-adaption as a target problem, and constructing an optimization problem model as follows:

the constraint conditions for setting the optimization problem model are as follows:

wherein,is the ith nonlinear mode component c_i(t) ambiguity function in polar coordinates, β being the volume of the optimal kernel;the expression in the rectangular coordinate system is:

$A_i(t,\mathrm{\&theta;},\mathrm{\&tau;})={\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}{c}_i^{*}(u-\frac{\mathrm{\&tau;}}{2})w^{*}(u-t-\frac{\mathrm{\&tau;}}{2})\&CenterDot;c_i(u-\frac{\mathrm{\&tau;}}{2})w(u-t-\frac{\mathrm{\&tau;}}{2})e^{j\mathrm{\&theta;}u}du$

wherein A is_i(t, θ, τ) isAmount of mapping in rectangular coordinate system, s^*(t)，w^*(u) is s (t), c_i(t), complex conjugate forms of w (u); the window function w (u) is a symmetrical diamond window function with the center of T and the width of 2T, when | u | is greater than T, w (u) is 0, variables tau and theta are parameters of a general fuzzy domain { tau, u }, and | tau | is less than 2T;

(2-3) according to A_i(t, theta, tau) solving the optimization problem model to obtain a nonlinear mode component c_i(t) optimum kernel function Φ_(i)opt(t，θ，τ)；

(2-4) calculating the nonlinear mode component c_i(t) the time-frequency representation of the adaptive optimal kernel is:

$P_{\left(i\right)AOK}(t,\mathrm{\&omega;})=\frac{1}{4\mathrm{\&pi;}}{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}{A}_i(t,\mathrm{\&theta;},\mathrm{\&tau;}){\mathrm{\&Phi;}}_{\left(i\right)opt}(t,\mathrm{\&theta;},\mathrm{\&tau;})e^{-j\mathrm{\&theta;}t-j\mathrm{\&tau;}\mathrm{\&omega;}}d\mathrm{\&theta;}d\mathrm{\&tau;}$

in the formula, P_(i)AOK(t, ω) is a nonlinear mode component c_i(t) an energy value at time t;

(3) according to the time-frequency representation of the self-adaptive optimal kernels of all the nonlinear mode components obtained in the step (2), calculating the result of time-frequency analysis as follows:

$P_{NMD-AOK}(t,\mathrm{\&omega;})=\underset{i}{\&Sigma;}{P}_{{\left(i\right)}_{AOK}}(t,\mathrm{\&omega;}).$

2. the time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel as claimed in claim 1, wherein the method for decomposing the signal to be processed s (t) into a set of nonlinear mode components by using nonlinear mode decomposition in step (1) comprises the steps of:

(1-1) wavelet transform expression W of calculation signal s (t)_s(ω，t)：

$W_s(\mathrm{\&omega;},t)\&equiv;{\&Integral;}_{-\mathrm{\&infin;}}^{+\mathrm{\&infin;}}{s}^{+}\left(u\right){\mathrm{\&psi;}}^{*}\&lsqb;\frac{\mathrm{\&omega;}(u-t)}{{\mathrm{\&omega;}}_{\mathrm{\&psi;}}}\&rsqb;\frac{\mathrm{\&omega;}du}{{\mathrm{\&omega;}}_{\mathrm{\&psi;}}}=\frac{1}{2\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&infin;}}{e}^{i\mathrm{\&xi;}t}\hat{s}\left(\mathrm{\&xi;}\right){\widehat{\mathrm{\&psi;}}}^{*}\left(\frac{{\mathrm{\&omega;}}_{\mathrm{\&psi;}}\mathrm{\&xi;}}{\mathrm{\&omega;}}\right)d\mathrm{\&xi;}$

Wherein,is the Fourier transform of s (t), i.e.s⁺(t) is the positive frequency portion of the s (t) signal,ψ (t) is a wavelet function, anAre mutually Fourier transform pairs and satisfy the condition Respectively by psi (t),the conjugate complex number of (a); omega_ψWhich represents the peak frequency of the wavelet, ω_ψ＝1，f₀the resolution parameter is used for balancing the time and frequency resolution in the transformation process;

(1-2) judging whether the wavelet transform obtained by the calculation in the step (1-1) is the optimal time-frequency representation of s (t); if the judgment result is no, calculating a windowing Fourier transform expression G of the signal s (t)_s(ω，t)：

$G_s(\mathrm{\&omega;},t)\&equiv;\underset{-\mathrm{\&infin;}}{\overset{\mathrm{\&infin;}}{\&Integral;}}{s}^{+}\left(u\right)g(u-t)e^{-i\mathrm{\&omega;}(u-t)}dt=\frac{1}{2\mathrm{\&pi;}}\underset{0}{\overset{\mathrm{\&infin;}}{\&Integral;}}{e}^{i\mathrm{\&xi;}t}\hat{s}\left(\mathrm{\&xi;}\right)\hat{g}(\mathrm{\&omega;}-\mathrm{\&xi;})d\mathrm{\&xi;}$

Wherein g (t) is a windowed function of a windowed Fourier transform,is the Fourier transform of g (t), and satisfies the following conditions:

and is

(1-3) finding all ridge curves represented by the optimal time frequency of a signal s (t), and reconstructing harmonic components by using a ridge method, wherein the h-th harmonic component is as follows:

x^(h)(t)＝A^(h)(t)cosφ^(h)(t)，h∈[1，2，…，N]

in the formula, A^(h)(t)、φ^(h)(t) is the amplitude and phase of the h-th harmonic component, respectively; v is^(h)(t) is the frequency of the h-th harmonic component, v^(h)(t)≡φ^′(h)(t)；φ^′(h)(t) is phi^(h)(t) a derivative over time t; n represents the highest order of the harmonic component, i.e., the data length;

(1-4) identifying the truth of the extracted harmonic component by using an anti-noise alternative inspection method, screening out all real harmonic components,

(1-5) adding all the real harmonic components to obtain a nonlinear mode component c₁(t)；

(1-6) subtracting c obtained by nonlinear mode decomposition from the original signal s (t)₁(t) and repeating steps 1-1 to 1-6 on the residual component to obtain all the respective nonlinear mode components c_i(t)；

Finally, the original target signal s (t) can be expressed as:

$s\left(t\right)=\underset{i}{\&Sigma;}{c}_i\left(t\right)+n\left(t\right)$

in the formula, n (t) represents noise.

3. The time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel as claimed in claim 2, wherein all ridge curves of optimal time-frequency representation of signal s (t) are found in the steps (1-3)The method comprises the following steps:

at time t_iFinding h maximum value points, and connecting the h maximum value points to form a moment t_iThe ridge curves of (a) are:

${\mathrm{\&omega;}}_p\left(t\right)=\underset{\mathrm{\&omega;}\&Element;\&lsqb;{\mathrm{\&omega;}}_{-}\left(t_i\right),{\mathrm{\&omega;}}_{+}\left(t_i\right)\&rsqb;}{\arg max}\left|H_s(\mathrm{\&omega;},t)\right|$

in the above formula, i is 1, 2, …, N is data length; h_s(ω, t) is the best time-frequency representation of signal s (t), i.e. W_s(ω, t) or G_s(ω，t)；H_sAnd (omega, t) finding out ridge point connecting lines at all the time points, namely forming N ridge curves.

4. The time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel as claimed in claim 3, wherein the h-th harmonic component x is reconstructed by ridge method in the steps (1-3)^(h)(t)＝A^(h)(t)cosφ^(h)(t) comprises the steps of:

if the best time-frequency representation of the signal s (t) to be processed is a wavelet transform, then

If the best time-frequency representation of the signal to be processed s (t) is a windowed Fourier transform, then

$\left\{\begin{array}{c}{v}^{\left(h\right)}\left(t\right)={\mathrm{\&omega;}}_p^{\left(h\right)}\left(t\right)+\mathrm{\&delta;}{v}_d^{\left(h\right)}\left(t\right)\\ A^{\left(h\right)}{e}^{{\mathrm{i\&phi;}}^{\left(h\right)}\left(t\right)}=\frac{2G_s\left({\mathrm{\&omega;}}_p^{\left(h\right)}\right(t),t)}{\hat{g}\&lsqb;{\mathrm{\&omega;}}_p^{\left(h\right)}\left(t\right)-v^{\left(h\right)}\left(t\right)\&rsqb;}\end{array}\right.$

In the formula,seed of a plantIs an improved discretization impact factor obtained by parabolic interpolation.

5. The time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel as claimed in claim 4, wherein the method for screening true harmonic components in the steps (1-4) comprises the steps of:

(1-4-1) construction of the recognition statistic D (α)_A，α_ν) For measuring the order of magnitude and frequency of the respective harmonic components, D (α)_A，α_v) The expression of (a) is:

$D({\mathrm{\&alpha;}}_A,{\mathrm{\&alpha;}}_v)\&equiv;{\mathrm{\&alpha;}}_{A}Q\&lsqb;{\hat{A}}^{\left(h\right)}\left(\mathrm{\&xi;}\right)\&rsqb;+{\mathrm{\&alpha;}}_{v}Q\&lsqb;{\hat{v}}^{\left(h\right)}\left(\mathrm{\&xi;}\right)\&rsqb;,$

$Q\&lsqb;f\left(x\right)\&rsqb;\&equiv;-\&Integral;\frac{|f\left(x\right){|}^2}{\&Integral;|f\left(x\right){|}^{2}dx}\ln \frac{|f\left(x\right){|}^2}{\&Integral;|f\left(x\right){|}^{2}dx}dx$

in the formula,andrespectively represent A^(h)(t) and v^(h)(t) spectral entropy α_A，α_vIs a weight coefficient;

(1-4-2) creation of N for the original signal s (t)_sFourier transform the substitution data; wherein, the expression of the jth Fourier transform substitution data is as follows:

$s_j^{\&prime;}\left(t\right)=\frac{1}{2\mathrm{\&pi;}}\&Integral;e^{i\mathrm{\&xi;}t}\left|\hat{s}\left(\mathrm{\&xi;}\right)\right|e^{{\mathrm{i\&phi;}}_{{\mathrm{\&xi;}}_j}}d\mathrm{\&xi;},j=1,2,...,N_s;$

(1-4-3) calculating an optimal time-frequency expression formula of each Fourier transform alternative data, extracting each subharmonic component from the optimal time-frequency expression formula, and calculating identification statistics corresponding to each Fourier transform alternative data as follows:

$D_j({\mathrm{\&alpha;}}_A,{\mathrm{\&alpha;}}_v)\&equiv;{\mathrm{\&alpha;}}_{A}Q\&lsqb;{\hat{A}}^{\left(j\right)}\left(\mathrm{\&xi;}\right)\&rsqb;+{\mathrm{\&alpha;}}_{v}Q\&lsqb;{\hat{v}}^{\left(j\right)}\left(\mathrm{\&xi;}\right)\&rsqb;,j=1,2,...,N_s$

defining a significance level indicatorIn the formulaTo satisfy D_s＞D₀The number of substitute data of (a); d₀The degree of order of h harmonic of the original signal;

setting the significance level index as p, when presentx number of alternative data satisfy D_s＞D₀And x is more than or equal to p × N_sThen, the corresponding harmonic component is judged not to be noise; otherwise, judging that the corresponding harmonic component is noise;

(1-4-4) calculating a comprehensive metric of correlation between harmonics

${\mathrm{\&rho;}}^{\left(h\right)}(w_A,w_{\mathrm{\&phi;}},w_v)={\left(q_A^{\left(h\right)}\right)}^{w_A}{\left(q_{\mathrm{\&phi;}}^{\left(h\right)}\right)}^{w_{\mathrm{\&phi;}}}{\left(q_v^{\left(h\right)}\right)}^{w_v}$

Wherein,

$q_A^{\left(h\right)}\&equiv;\exp \{-\frac{\sqrt{<{\&lsqb;A^{\left(h\right)}\left(t\right)<A^{\left(1\right)}\left(t\right)>-A^{\left(1\right)}\left(t\right)<A^{\left(h\right)}\left(t\right)>\&rsqb;}^2>}}{<A^{\left(1\right)}\left(t\right)A^{\left(h\right)}\left(t\right)>}\},$

$q_{\mathrm{\&phi;}}^{\left(h\right)}\&equiv;a|<\exp \{i\&lsqb;{\mathrm{\&phi;}}^{\left(h\right)}\left(t\right)-{\mathrm{h\&phi;}}^{\left(1\right)}\left(t\right)\&rsqb;\}>|,$

$q_v^{\left(h\right)}\&equiv;\exp \{-\frac{\sqrt{<{\&lsqb;v^{\left(h\right)}\left(t\right)-{\mathrm{hv}}^{\left(1\right)}\left(t\right)\&rsqb;}^2>}}{<v^{\left(h\right)}\left(t\right)>}\}.$

in the formula, w_A，w_φ，w_vRepresentsWeight of rho^(h)Equal weight rho is distributed for amplitude and phase consistency^(h)≡ρ^(h)(1，1，0)；

(1-4-5) defining a threshold value for the composite metric value

(1-4-6) when a harmonic component satisfies ρ^(h)≥ρ_minAnd when the significance level index p is 95%, the harmonic component is determined to be the true harmonic component through the inspection.

6. The time-frequency analysis method based on nonlinear mode decomposition and adaptive optimal kernel as claimed in claim 5, characterized in thatJudging whether a substitute data satisfies D in the above step (1-4-3)_s＞D₀The method comprises the following steps:

any three sets of parameters with different values are selected (α)_A，α_v) Respectively calculating the identification statistic values corresponding to the three groups of parameters, and if any one identification statistic value is larger than D₀Then the substitute data is judged to satisfy D_s＞D₀。