CN103198053A - Instantaneous wavelet bicoherence method based on phase randomization - Google Patents

Abstract

The invention discloses an instantaneous wavelet bicoherence method based on phase randomization. The instantaneous wavelet bicoherence method based on the phase randomization comprises the following steps: firstly, selecting a mother wavelet function for continuous wavelet transform; secondly, performing continuous wavelet transform on a signal to be analyzed to obtain a time and frequency domain expression form of the signal; thirdly, computing an instantaneous wavelet bispectrum at each scale, computing the instantaneous wavelet bispectrum again, repeating computation, and summing computed results to take an expected instantaneous wavelet bispectrum based on the phase randomization; fourthly, computing the obtained instantaneous wavelet bispectrum based on the phase randomization, computing the obtained instantaneous wavelet bispectrum based on the phase randomization, and performing integration at a scale of s2 to obtain the instantaneous wavelet bispectrum based on the phase randomization; sixthly, computing to obtain wavelet bicoherence based on the phase randomization; and finally, computing to obtain instantaneous wavelet bicoherence based on the phase randomization. During the computation of the wavelet bicoherence, the amplitude and phase information are taken into account at the same time, so that defects of the conventional method such as low accuracy are avoided.

Description

A kind of based on phase place instantaneous small echo bicoherence method at random

Technical field

The invention belongs to the mechanical signal processing technology field of mechanical system fault diagnosis and control, particularly a kind of based on phase place instantaneous small echo bicoherence method at random.

Background technology

Two spectrums are proposed to use by Brillinger and Hinich the earliest.It is as the effective ways of system responses abnormal characteristic (the abnormal feature of system responses is the nonlinear sign of system) identification, and the signal that is widely used in fields such as geophysics, biological doctor's electricity, mechanical industry and radar signal is handled.In mechanical field, when there is fault in physical construction, the impact that produces owing to faults such as local damages will cause non-stationary, abnormal and the nonlinear characteristic of structural response signal.All there are the quadratic phase coupling usually in this class non-stationary, abnormal signal, and therefore, two spectrums and bicoherence method often are applied to this class Signal Processing and analysis.Traditional double spectrum and bicoherence method are based on all that Fast Fourier Transform (FFT) and Short Time Fourier Transform realize, defective such as have a little less than the signal transient characteristic recognition capability.For overcoming this defective, the two spectrums of small echo with signal transient characteristic recognition capability are suggested and widespread use with the bicoherence method.This method combines two spectrums and bicoherence method, and nonlinear characteristic detectability advantage and continuous wavelet transform thereby can obtain more efficient, objective testing result in actual applications to the advantage of signal transient feature recognition capability efficiently.

Usually, the two spectrums of small echo and the bicoherence of calculating signal all comprise following steps, at first, be the data segment of N equal time (establishing the time interval is T) with division of signal to be analyzed, then, each data segment is carried out the calculating of the two spectrums of small echo and bicoherence, last, every section result of calculation summation is got average as final result of calculation.When coherence time of signal itself with respect to the time interval T that gets more in short-term, the phase component of each data segment is separate.Yet, for most mechanical signal, all have long coherence time usually, therefore, the method for calculating the two spectrums of small echo and bicoherence by simple dividing data section can cause quadratic phase modulation result false on the spectrogram.Simultaneously, when signal existed quadratic phase coupling and non-quadratic phase coupling information simultaneously on identical bifrequency, the two spectrums of traditional small echo can not effectively be identified with the bicoherence computing method.Above problem all can significantly reduce the accuracy of result of calculation.Therefore, press for a kind of otherwise effective technique more to improve the accuracy that detects.

Summary of the invention

In order to overcome the shortcoming of above-mentioned prior art, the object of the present invention is to provide a kind ofly based on phase place instantaneous small echo bicoherence method at random, when calculating the small echo bicoherence, consider its amplitude and phase information simultaneously, avoided the classic method defects of low accuracy.

In order to achieve the above object, the technical scheme taked of the present invention is:

A kind of based on phase place instantaneous small echo bicoherence method at random, may further comprise the steps:

Step 1: select to be used for the mother wavelet function of continuous wavelet transform, when the non-stationary signal in the mechanical signal was detected, female small echo adopted the Morlet small echo shown in the formula (1),

$\mathrm{\&psi;}\left(t\right)=e^{-{\mathrm{\&sigma;}}^2{t}^2}{e}^{-\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">i</figure-callout>}2\mathrm{\&pi;ft}}$ (1)

Wherein: σ---decay factor;

The female little wave frequency of f---Morlet;

Step 2: adopt formula (2) to treat analytic signal and make continuous wavelet transform, the time-frequency domain expression-form of picked up signal:

$W_{\mathrm{\&psi;}}(s,t)=\frac{1}{\sqrt{\left|s\right|}}{\&Integral;}_{-\&infin;}^{\&infin;}x\left(t^{\&prime;}\right){\mathrm{\&psi;}}^{*}\left(\frac{t^{\&prime;}-t}{s}\right)d{t}^{\&prime;}---\left(2\right)$

Wherein: ψ (t)---selected female small echo;

X (t)---signal to be analyzed;

The s---scale factor;

The t---time factor;

*--conjugation is got in-expression;

Step 3: calculate two spectrums of instantaneous small echo under each yardstick according to formula (3):

B _W,T(s ₁,s ₂,t)=W _ψ(s ₁,t)W _ψ(s ₂,t)W _ψ ^*(s,t) (3)

Wherein: s, s ₁With s ₂Satisfy relation

Owing to calculate the two spectrums of instantaneous small echo for plural number, so formula (3) also can be expressed as the form of formula (4)

Wherein: A (s ₁, s ₂, t)---bifrequency (s ₁, s ₂) time the two spectral amplitude ratio of instantaneous small echo;

---bifrequency (s ₁, s ₂) time the two spectrum of instantaneous small echo phase places;

Step 4: will be calculated the two spectrums of instantaneous small echo by the instantaneous small echo quarter-phase substitution formula (5) that formula (3) calculate:

Wherein: the R---interval is the sequence of random variables of [π, π];

The instantaneous small echo quarter-phase that---by formula (3) calculates;

Step 5: step 4 is repeated 150 times, and the result that will calculate summation, get expectation and obtain based on the two spectrums of phase place instantaneous small echo at random, as shown in Equation (6):

Wherein: E{.}---represents to get desired operation;

Step 6: will be obtained based on the two spectrums of phase place small echo at random as shown in Equation (7) in time interval T upper integral by composing based on phase place instantaneous small echo at random is two that formula (6) calculate:

Step 7: will be by composing based on phase place instantaneous small echo at random is two that formula (6) calculate, at yardstick s ₂On carry out integration and obtain based on the two spectrums of phase place instantaneous small echo at random as shown in Equation (8):

Step 8: will be calculated by the formula of substitution as a result (9) that formula (7) calculate based on phase place small echo bicoherence at random:

$b_{W,T}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)=\frac{{\left|B_{W,T}^{\&prime;}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)\right|}^2}{{\&Integral;}_T{\left|W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2,t)\right|}^2\mathrm{dt}{\&Integral;}_T{\left|W_{\mathrm{\&psi;}}(s,t)\right|}^2\mathrm{dt}}---\left(9\right)$

Step 9: will be calculated by the formula of substitution as a result (10) that formula (8) calculate based on phase place instantaneous small echo bicoherence at random:

$b_{W,T}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)=\frac{{\left|B_{W,T}^{\&prime;}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)\right|}^2}{{\left|W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2,t)\right|}^2{\left|W_{\mathrm{\&psi;}}(s,t)\right|}^2}---\left(10\right)$

Advantage of the present invention is: traditional small echo bicoherence computing formula (11) is;

$b_{W,T}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)=\frac{{\left|B_{W,T}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)\right|}^2}{{\&Integral;}_T{\left|W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2,t)\right|}^2\mathrm{dt}{\&Integral;}_T{\left|W_{\mathrm{\&psi;}}(s,t)\right|}^2\mathrm{dt}}---\left(11\right)$

Wherein: yardstick s, s ₁With s ₂Satisfy relation

W _ψ(s, t)---yardstick is the continuous wavelet transform of s;

B _{W, T}(s ₁, s ₂)---be that traditional two spectrums of small echo are calculated, as shown in Equation (12),

Wherein: *--conjugation is got in-expression,

A (s ₁, s ₂, t)---bifrequency is (s ₁, s ₂) time the two spectral amplitude ratio of instantaneous small echo;

---bifrequency is (s ₁, s ₂) time the two spectrum of instantaneous small echo phase places.

The small echo bicoherence that is calculated by formula (11) is yardstick s ₁With s ₂Function, span is 0 to 1.Desirable small echo bicoherence computing formula need satisfy following two conditions:

If I. signal is at bifrequency (s ₁, s ₂) locate to exist the quadratic phase coupling, then at bifrequency (s ₁, s ₂) locate small echo bicoherence value b _{W, T}(s ₁, s ₂)=1 is simultaneously at bifrequency (s ₁, s ₂) quarter-phase located

If II. signal is at bifrequency (s ₁, s ₂) locate not exist the quadratic phase coupling, then at bifrequency (s ₁, s ₂) locate the value b of small echo bicoherence _{W, T}(s ₁, s ₂)=0 is simultaneously at bifrequency (s ₁, s ₂) quarter-phase located

Usually, the traditional small echo bicoherence computing method requirement of II that all can not satisfy condition.Therefore, the present invention considers its amplitude and phase information simultaneously when calculating the small echo bicoherence, avoided defectives such as the classic method accuracy is low.

Description of drawings

Fig. 1 is the simulate signal time domain waveform.

Fig. 2 is simulate signal continuous wavelet transform spectrum.

Fig. 3 is based on phase place small echo bicoherence spectrum at random.

Fig. 4 is traditional small echo bicoherence spectrum.

Fig. 5 is based on phase place instantaneous small echo bicoherence spectrum at random.

Fig. 6 is traditional instantaneous small echo bicoherence spectrum.

Embodiment

Below in conjunction with drawings and Examples the present invention is described in detail.

Be example with the simulate signal, the expression formula of simulate signal as shown in Equation (13)

Wherein: ε (t)---average is zero white Gaussian noise;

---interval be (π, π] even distribution random phase (j=1,2,3), and

f _i---coupling frequency (i=1,2).

The sample frequency of this simulate signal is 100Hz, wherein f ₁=9Hz, f ₂=19Hz, property is made an uproar than being 3dB.Be f at first and the 3rd ripple bag medium frequency ₁And f ₂Cosine wave (CW) satisfy the quadratic phase coupling condition, and second and the 4th ripple bag medium frequency are f ₁And f ₂Cosine wave (CW) do not satisfy the quadratic phase coupling condition.The time domain waveform of simulate signal as shown in Figure 1; For the coupling of two chi phase places in the identification signal, adopt the present invention that data are analyzed.

A kind of based on phase place instantaneous small echo bicoherence method at random, may further comprise the steps:

Step 1: select to be used for the mother wavelet function of continuous wavelet transform, when the non-stationary signal in the mechanical signal was detected, female small echo adopted the Morlet small echo shown in the formula (1),

$\mathrm{\&psi;}\left(t\right)=e^{-{\mathrm{\&sigma;}}^2{t}^2}{e}^{-\mathrm{<figure-callout\; id="2"\; label="i"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">i</figure-callout>}2\mathrm{\&pi;ft}}$ (1)

Wherein: get decay factor σ=4, the frequency f=0.95Hz of female small echo is to satisfy the admissibility condition;

Step 2: adopt formula (2) that simulate signal x (t) is made continuous wavelet transform, according to the sample frequency of simulate signal, the yardstick of getting travels through whole Fourier analysis frequency, and the time-frequency domain expression-form of picked up signal obtains the time-frequency figure of signal as shown in Figure 2;

$W_{\mathrm{\&psi;}}(s,t)=\frac{1}{\sqrt{\left|s\right|}}{\&Integral;}_{-\&infin;}^{\&infin;}x\left(t^{\&prime;}\right){\mathrm{\&psi;}}^{*}\left(\frac{t^{\&prime;}-t}{s}\right)d{t}^{\&prime;}---\left(2\right)$

Wherein: ψ (t)---selected female small echo;

X (t)---simulate signal;

The s---scale factor;

The t---time factor;

*--conjugation is got in-expression;

Step 3: calculate two spectrums of instantaneous small echo under each yardstick according to formula (3), wherein getting frequency resolution is 0.25Hz, and obtains the two spectrum of the instantaneous small echo phase places under this yardstick

B _W,T(s ₁,s ₂,t)=W _ψ(s ₁,t)W _ψ(s ₂,t)W _ψ ^*(s,t) (3)

Wherein: s, s ₁With s ₂Satisfy relation

Step 4: will be calculated the two spectrums of instantaneous small echo by the instantaneous small echo quarter-phase substitution formula (5) that formula (3) calculate;

Wherein: the R---interval is the sequence of random variables of [π, π];

The instantaneous small echo quarter-phase that---by formula (3) calculates;

Step 5: step 4 is repeated 150 times, and the result that will calculate summation, get expectation and obtain based on the two spectrums of phase place instantaneous small echo at random, as shown in Equation (6);

Wherein: E{.}---represents to get desired operation;

Step 6: will be obtained based on the two spectrums of phase place small echo at random as shown in Equation (7) in time interval T upper integral by composing based on phase place instantaneous small echo at random is two that formula (6) calculate;

Step 7: will be by composing based on phase place instantaneous small echo at random is two that formula (6) calculate, at yardstick s ₂On carry out integration and obtain based on the two spectrums of phase place instantaneous small echo at random as shown in Equation (8);

Step 8: will be calculated based on phase place small echo bicoherence at random by the formula of substitution as a result (9) that formula (7) calculate, result of calculation as shown in Figure 3;

$b_{W,T}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)=\frac{{\left|B_{W,T}^{\&prime;}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,s_2)\right|}^2}{{\&Integral;}_T{\left|W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2,t)\right|}^2\mathrm{dt}{\&Integral;}_T{\left|W_{\mathrm{\&psi;}}(s,t)\right|}^2\mathrm{dt}}---\left(9\right)$

The collection of illustrative plates that Fig. 4 obtains for traditional small echo bicoherence computing method, as can be seen from Figure 4, spectrogram exists at a plurality of frequencies place and false peak value occurs, is difficult to judge real quadratic phase coupling frequency composition.Yet, based on phase place instantaneous small echo bicoherence method at random only at the frequency content place (9Hz and 19Hz) that has the quadratic phase coupling peak value appears, as shown in Figure 3, therefore, can overcome classic method effectively under certain conditions based on phase place instantaneous small echo bicoherence method at random, the defective of the false quadratic phase coupling testing result that exists has improved the accuracy that detects;

Step 9: will be calculated by the formula of substitution as a result (10) that formula (8) calculate based on phase place instantaneous small echo bicoherence at random, result of calculation as shown in Figure 5;

$b_{W,T}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)=\frac{{\left|B_{W,T}^{\&prime;}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)\right|}^2}{{\left|W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900031.png"\; state="\{\{state\}\}">s</figure-callout>}}_1,t)W_{\mathrm{\&psi;}}({\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA00002944299900011.png,HDA00002944299900012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2,t)\right|}^2{\left|W_{\mathrm{\&psi;}}(s,t)\right|}^2}---\left(10\right)$

The spectrogram that is calculated by the instantaneous bicoherence method of traditional small echo as shown in Figure 6, as seen from Figure 6, no matter ripple is surrounded by does not exist the quadratic phase coupling, testing result all demonstrates bigger small echo bicoherence coefficient.And as shown in Figure 5, only there are first and the 3rd the bigger small echo two-phase responsibility of ripple bag place appearance of quadratic phase coupling based on phase place instantaneous small echo bicoherence spectrum at random, and at second and the 4th ripple bag place not having the quadratic phase coupling, its small echo bicoherence coefficient is then very little.What this patent proposed can overcome classic method effectively based on phase place instantaneous small echo bicoherence computing method at random, when quadratic phase coupling exists simultaneously with non-quadratic phase coupling under identical bifrequency and the defective that can't differentiate, improve the accuracy that detects, be applicable to each associated field.

Claims (1)

1. one kind based on phase place instantaneous small echo bicoherence method at random, it is characterized in that, may further comprise the steps:

Step 1: select to be used for the mother wavelet function of continuous wavelet transform, when the non-stationary signal in the mechanical signal was detected, female small echo adopted the Morlet small echo shown in the formula (1),

(1)

Wherein: σ---decay factor;

The female little wave frequency of f---Morlet;

Step 2: adopt formula (2) to treat analytic signal and make continuous wavelet transform, the time-frequency domain expression-form of picked up signal:

Wherein: ψ (t)---selected female small echo;

X (t)---signal to be analyzed;

The s---scale factor;

The t---time factor;

*--conjugation is got in-expression;

Step 3: calculate two spectrums of instantaneous small echo under each yardstick according to formula (3):

B _W,T(s ₁,s ₂,t)=W _ψ(s ₁,t)W _ψ(s ₂,t)W _ψ ^*(s,t) (3)

Wherein: s, s ₁With s ₂Satisfy relation

Owing to calculate the two spectrums of instantaneous small echo for plural number, so formula (3) also can be expressed as the form of formula (4)

Wherein: A (s ₁, s ₂, t)---bifrequency (s ₁, s ₂) time the two spectral amplitude ratio of instantaneous small echo;

---bifrequency (s ₁, s ₂) time the two spectrum of instantaneous small echo phase places;

Step 4: will be calculated the two spectrums of instantaneous small echo by the instantaneous small echo quarter-phase substitution formula (5) that formula (3) calculate:

Wherein: the R---interval is the sequence of random variables of [π, π];

The instantaneous small echo quarter-phase that---by formula (3) calculates;

Step 5: step 4 is repeated 150 times, and the result that will calculate summation, get expectation and obtain based on the two spectrums of phase place instantaneous small echo at random, as shown in Equation (6):

Wherein: E{.}---represents to get desired operation;

Step 6: will be obtained based on the two spectrums of phase place small echo at random as shown in Equation (7) in time interval T upper integral by composing based on phase place instantaneous small echo at random is two that formula (6) calculate:

Step 7: will be by composing based on phase place instantaneous small echo at random is two that formula (6) calculate, at yardstick s ₂On carry out integration and obtain based on the two spectrums of phase place instantaneous small echo at random as shown in Equation (8):

Step 8: will be calculated by the formula of substitution as a result (9) that formula (7) calculate based on phase place small echo bicoherence at random:

Step 9: will be calculated by the formula of substitution as a result (10) that formula (8) calculate based on phase place instantaneous small echo bicoherence at random:

。

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