CN104155134B - A kind of determination methods of the Higher Order Cumulants feature extracting method suitability - Google Patents

Abstract

The present invention relates to the determination methods of a kind of Higher Order Cumulants feature extracting method suitability, it comprises the following steps: (1) uses rotor testbed analog mechanical equipment normal operating condition, gathers rotor testbed vibration signal under normal operating conditions；(2) utilize rotor testbed analog mechanical equipment minor failure degree, moderate fault degree and three kinds of fault degrees of severe fault degree under a certain fault, and gather rotor testbed vibration signal under three kinds of fault degrees；(3) calculate in all vibration signals and often organize vibration signal 1.5 dimension spectrum；(4) judging whether 1.5 dimension spectrum signature extracting method have sensitivity and tendency for mechanical equipment fault deterioration, the feature extracting method simultaneously meeting sensitivity and tendency is applicable to the mechanical equipment fault trend prediction of this fault.The present invention can accurately judge which kind of fault type 1.5 dimension spectrum signature extracting method are applicable to, and can extensively apply in mechanical equipment fault trend prediction.

Description

A Judgment Method for Applicability of High-Order Cumulative Feature Extraction Method

technical field

The present invention relates to a method for judging the applicability of a mechanical fault extraction method in fault trend prediction, in particular to a method for applicability of a high-order cumulant feature extraction method applicable to a 1.5-dimensional spectrum feature extraction method in the field of mechanical fault diagnosis Judgment method.

Background technique

The trend prediction research on mechanical equipment failure is conducive to the active maintenance of mechanical equipment and the reduction of economic losses. The fault feature extraction method is the key to the fault trend prediction research. Therefore, people have proposed different fault feature extraction methods according to different theories in the prior art. Higher-order cumulant theory is a theoretical method applied to fault feature extraction in recent years, among which the 1.5-dimensional spectral feature extraction method is widely used. However, there are many types of mechanical faults. There is no theoretical basis for which fault type the 1.5-dimensional spectral feature extraction method is suitable for. It can only be judged based on experience. The judgment results are subjective and there are large errors.

Contents of the invention

In view of the above problems, the purpose of the present invention is to provide a method for judging the applicability of a high-order cumulant feature extraction method, which provides a theoretical basis for the selection of feature extraction methods in equipment failure trend prediction research, and can more accurately judge 1. Which fault type is suitable for the 5-dimensional spectral feature extraction method.

In order to achieve the above object, the present invention adopts the following technical solutions: a method for judging the applicability of a high-order cumulant feature extraction method, which includes the following steps: (1) using a rotor test bench to simulate the normal operating state of mechanical equipment, using existing data The acquisition equipment collects the vibration signal x _w (n)={x _1, ... x _N } of the rotor test bench under normal operating conditions, where N represents the number of data in each group, w represents the data group, and w=1 (2) Utilize the rotor test bench to simulate three fault degrees of mechanical equipment under a certain fault: mild fault degree, moderate fault degree and severe fault degree, and use the existing data acquisition equipment to collect rotor test bench in three types Vibration signal x _w (n)={x _1, ... x _N } at the fault level, where N represents the number of data in each group; w represents the data group, w=2, 3, 4, representing light (3) Calculate the 1.5-dimensional spectrum of each group of vibration signals in all vibration signals; (4) Assume that the maximum value of the 1.5-dimensional spectrum of the vibration signal under the normal operating state of the mechanical equipment is The value is S ₁ , and the maximum value of the 1.5-dimensional spectrum under the state of minor fault, moderate fault and severe fault of mechanical equipment is S ₂ , S ₃ and S ₄ respectively. When the maximum value of the 1.5-dimensional spectrum satisfies the following formula :

$\frac{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 50</span>}<span\; class="notranslate">\; \%</span><span\; class="notranslate">\; ,</span>$

Then it is judged that the _{1.5 -} dimensional spectral feature extraction method is sensitive to the failure and deterioration _of mechanical equipment _; Fault degradation has a trend; the feature extraction method that satisfies both sensitivity and trend is suitable for the failure trend prediction of mechanical equipment for this fault.

In the described step (3), the calculation steps of the 1.5-dimensional spectrum of each group of vibration signals are as follows: 1) N data in each group of data of all vibration signals are all divided into K sections, and each section has M data, and each section of data is used as One record; Ⅱ) De-average each record, and then calculate the diagonal slice of the third-order cumulant to obtain the average value of the diagonal slice of the third-order cumulant; Ⅲ) Do one-dimensional Fu on the average value of the diagonal slice of the third-order cumulant Lie transform, the 1.5-dimensional spectrum S _w,3x (ω _r ) of the vibration signal is obtained as:

In the formula, ω _r represents the frequency, r=1, 2, ... N, N is a positive integer; (τ, τ) is the average of the diagonal slice of the third-order cumulant;

τ is the time delay; 3x represents the third-order cumulant.

Among the steps II), the specific steps are as follows: (a) Assumption is the i-th record, where i=1,...K, h=0,1,...M-1; find the diagonal slice of the third-order cumulant for the i-th record for:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}}\underset{\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\overset{\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

In the formula, M ₁ ＝max(0,-τ); M ₂ ＝min(M-1,M-1-τ), τ is time delay; (b) Diagonally slice all third-order cumulants take the average, get the average for:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; K</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; K</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$

The present invention has the following advantages due to the adoption of the above technical scheme: the present invention judges whether it is applicable to the fault trend prediction of a certain fault type according to the maximum value of the 1.5-dimensional spectrum, and its judgment result is the same as the experience adopted in the prior art The judgment is relatively accurate, which can provide a theoretical basis for the selection of feature extraction methods in the research of equipment failure trend prediction, and solve the problem of lack of theoretical basis for the selection of feature extraction methods in the research of fault trend prediction. The invention can be widely applied in the prediction of failure trend of mechanical equipment.

Description of drawings

Fig. 1 is a schematic diagram of the overall process of the present invention.

detailed description

The present invention will be described in detail below in conjunction with the accompanying drawings and embodiments.

As shown in Figure 1, the present invention provides a method for judging the applicability of the high-order cumulant feature extraction method, which is mainly used for judging the applicability of the 1.5-dimensional spectral feature extraction method in fault trend prediction, which includes the following step:

(1) The rotor test bench is used to simulate the normal operation state of mechanical equipment, and the existing data acquisition equipment is used to collect the vibration signal x _w (n)={x ₁ ,...x _N } of the rotor test bench under normal operation state, where , N represents the number of data in each group, w represents the data group, w=1, which means normal operation.

(2) Use the rotor test bench to simulate three failure degrees of mechanical equipment under a certain kind of failure: mild failure degree, moderate failure degree and severe failure degree, and use the existing data acquisition equipment to collect the three failure degrees of the rotor test bench. Vibration signal x _w (n)={x ₁ ,...x _N } at the fault level, where N represents the number of data in each group; w represents the data group, w=2, 3, 4, representing light The state of severe fault degree, the state of moderate fault degree and the state of severe fault degree.

(3) Calculate the 1.5-dimensional spectrum of each group of vibration signals in all vibration signals, the specific steps are as follows:

Ⅰ) The N data in each group of data of all vibration signals are divided into K segments, each segment has M data, and each segment of data is regarded as a record.

Ⅱ) For each record, remove the mean value, and then calculate the diagonal slice of the third-order cumulant to obtain the average value of the diagonal slice of the third-order cumulant. The steps are as follows:

(a) assumption is the i(i=1,...K) record, where h=0,1,...M-1; find the diagonal slice of the third-order cumulant for the i record for:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}}\underset{\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\overset{\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; h</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$

In the formula, M ₁ =max(0,-τ); M ₂ =min(M-1,M-1-τ), where τ is the time delay; 3x represents the third-order cumulant.

(b) Diagonally slice all third-order cumulants take the average, get the average for:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; K</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; K</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$

Ⅲ) Diagonal slice average of third-order cumulants Do one-dimensional Fourier transform to get the 1.5-dimensional spectrum S _w,3x (ω _r ) of the vibration signal as:

${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; \&infin;</span>}{\overset{<span\; class="notranslate">\; \&infin;</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j\&omega;\&tau;</span>}}<span\; class="notranslate">\; ,</span>$

In the formula, ω _r represents frequency, r=1, 2,...N, N is a positive integer.

(4) Assume that the maximum value of the 1.5-dimensional spectrum S _w,3x (ω _r ) of the vibration signal under the normal operating state of the mechanical equipment is S ₁ , and the maximum value of the mechanical equipment under the state of minor failure, moderate failure and severe failure is 1.5 The maximum values of the dimensional spectra S _w,3x (ω _r ) are S ₂ , S ₃ and S ₄ respectively.

If the maximum value of the 1.5-dimensional spectrum S _w,3x (ω _r ) satisfies the following formula:

$\frac{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 50</span>}<span\; class="notranslate">\; \%</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

It is judged that the 1.5-dimensional spectral feature extraction method is sensitive to the failure and degradation of mechanical equipment;

If the maximum value of the 1.5-dimensional spectrum S _w,3x (ω _r ) satisfies the following formula:

S ₁ < S ₂ < S ₃ < S ₄ , (2)

It is judged that the 1.5-dimensional spectral feature extraction method has a tendency for mechanical equipment failure degradation. The feature extraction method that satisfies both sensitivity and trend is suitable for the fault trend prediction of mechanical equipment, and the judgment result is relatively accurate.

Above-mentioned each embodiment is only for illustrating the present invention, wherein the structure of each component, connection mode etc. all can be changed to some extent, all equivalent transformations and improvements carried out on the basis of the technical solution of the present invention, all should not be excluded from the present invention. outside the scope of protection of the invention.

Claims (3)

1. a determination methods for the Higher Order Cumulants feature extracting method suitability, the method is mainly used in 1.5 dimension spectrum signatures Extracting method is the determination methods of the suitability in failure trend prediction, and it comprises the following steps:

(1) use rotor testbed analog mechanical equipment normal operating condition, utilize available data collecting device to gather rotor real Test platform vibration signal x under normal operating conditions_w(n)={ x₁,...x_N, wherein, N represents and often organizes data amount check, and w represents number According to group, w=1；

(2) utilize rotor testbed analog mechanical equipment minor failure degree under a certain fault, moderate fault degree and Three kinds of fault degrees of severe fault degree, and utilize available data collecting device to gather rotor testbed under three kinds of fault degrees Vibration signal x_w(n)={ x₁,...x_N, wherein, N represents and often organizes data amount check；W represents data group, w=2,3,4, difference Represent minor failure level state, moderate fault degree state and severe fault degree state；

(3) calculate in all vibration signals and often organize vibration signal 1.5 dimension spectrum；

(4) assume that the maximum that 1.5 dimensions of vibration signal under plant equipment normal operating condition are composed is S₁, plant equipment slightly event The maximum of 1.5 dimension spectrums under barrier degree, moderate fault degree, severe fault degree state is respectively S₂、S₃And S₄, when 1.5 dimensions The maximum of spectrum meets following formula:

$\frac{S_2-S_1}{S_1}\&GreaterEqual;50\%,$

Then it is judged as that 1.5 dimension spectrum signature extracting method have sensitivity for mechanical equipment fault deterioration；If the maximum of 1.5 dimension spectrums Value meets S₁＜ S₂＜ S₃＜ S₄, then it is judged as that 1.5 dimension spectrum signature extracting method have trend for mechanical equipment fault deterioration Property；The feature extracting method simultaneously meeting sensitivity and tendency is applicable to the mechanical equipment fault trend prediction of this fault.

The determination methods of a kind of Higher Order Cumulants feature extracting method suitability the most as claimed in claim 1, it is characterised in that: In described step (3), often the calculation procedure of group vibration signal 1.5 dimension spectrum is as follows:

I) N number of data in the often group data of all vibration signals being all divided into K section, every section of M data, every segment data is as one Record；

II) each record is gone average, then calculate three-order cumulant, obtain three-order cumulant Meansigma methods；

III) three-order cumulant meansigma methods is done one-dimensional Fourier transform, obtain 1.5 dimension spectrum S of vibration signal_w,3x (ω_r) it is:

$S_{w,3x}\left({\mathrm{\&omega;}}_r\right)=\underset{\mathrm{\&tau;}=-\mathrm{\&infin;}}{\overset{\mathrm{\&infin;}}{\&Sigma;}}{\hat{c}}_{w,3x}(\mathrm{\&tau;},\mathrm{\&tau;})e^{-j\mathrm{\&omega;}\mathrm{\&tau;}},$

In formula, ω_rRepresent frequency, r=1,2 ... N, N are positive integer；For three-order cumulant meansigma methods； τ is time delay；3x represents Third-order cumulants.

The determination methods of a kind of Higher Order Cumulants feature extracting method suitability the most as claimed in claim 2, it is characterised in that: In described step II) in, it specifically comprises the following steps that

A () is assumedIt is i-th record, wherein, i=1 ... K, h=0,1 ... M-1；I-th record is asked its three rank Cumulant diagonal slicesFor:

$x_{w,3x}^i(\mathrm{\&tau;},\mathrm{\&tau;})=\frac{1}{M}\underset{h=M_1}{\overset{h=M_2}{\&Sigma;}}{x}_w^i\left(h\right)x_w^i(h+\mathrm{\&tau;})x_w^i(h+\mathrm{\&tau;}),$

In formula, M₁=max (0 ,-τ)；M₂=min (M-1, M-1-τ), τ are time delay, and 3x represents Third-order cumulants；

B () is to all three-order cumulantAverage, obtain meansigma methodsFor:

${\hat{c}}_{w,3x}(\mathrm{\&tau;},\mathrm{\&tau;})=\frac{1}{K}\underset{i=1}{\overset{K}{\&Sigma;}}{c}_{w,3x}^i(\mathrm{\&tau;},\mathrm{\&tau;}).$