CN103198053B - A kind of instantaneous small echo bicoherence method random based on phase place - Google Patents

Abstract

A kind of instantaneous small echo bicoherence method random based on phase place, first select the mother wavelet function being used for continuous wavelet transform, treat analytic signal again and make continuous wavelet transform, obtain the time-frequency domain expression-form of signal, then the two spectrum of instantaneous small echo under each yardstick is calculated, calculate the two spectrum of instantaneous small echo again, repeat, and the result summation that will calculate, get and expect to obtain based on the two spectrum of the instantaneous small echo that phase place is random, then the two spectrum of the instantaneous small echo random based on phase place calculated, the two spectrum of the instantaneous small echo random based on phase place calculated, yardstick s2 carries out integration to be obtained based on the two spectrum of the instantaneous small echo that phase place is random, calculate based on the random small echo bicoherence of phase place again, finally calculate based on the random instantaneous small echo bicoherence of phase place, the present invention considers its amplitude and phase information when calculating small echo bicoherence simultaneously, avoids the defects such as classic method accuracy is low.

Description

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

Technical field

The invention belongs to Mechanical System Trouble diagnosis and the mechanical signal processing technology field controlled, a kind of particularly instantaneous small echo bicoherence method random based on phase place.

Background technology

Two spectrum proposes application by Brillinger and Hinich the earliest.The effective ways that it identifies as system responses abnormal characteristic (the abnormal feature of system responses is the mark of mission nonlinear), are widely used in the signal transacting in the fields such as geophysics, biological doctor's electricity, mechanical industry and radar signal.In mechanical field, when physical construction exists fault, the impact produced due to faults such as local damages will cause the non-stationary of structural response signal, abnormal and nonlinear characteristic.Usually all there is square phase-couple in this kind of non-stationary, abnormal signal, therefore, two spectrum and bicoherence method are often applied to the treatment and analyses of this kind of signal.Traditional double spectrum and bicoherence method, all realize based on Fast Fourier Transform (FFT) and Short Time Fourier Transform, there is the defects such as signal transient characteristic recognition capability is weak.For overcoming this defect, the two spectrum of small echo with signal transient characteristic recognition capability is suggested and widespread use with bicoherence method.The two spectrum of this methods combining and bicoherence method efficient nonlinear characteristic detectability advantage and continuous wavelet transform, to the advantage of signal transient feature recognition capability, thus can to obtain more efficient, objective testing result in actual applications.

Usually, the two spectrum of small echo and the bicoherence that calculate signal all comprise following steps, first, it is the data segment of N number of equal time (setting the time interval as T) by division of signal to be analyzed, then, each data segment is carried out to the calculating of the two spectrum of small echo and bicoherence, finally, the result of calculation of every section summation is got average as final result of calculation.When the coherence time of signal itself is relative to when the got time interval, T was shorter, the phase component of each data segment is separate.But, for most mechanical signal, usually all there is long coherence time, therefore, calculate the two spectrum of small echo by simple dividing data section and can cause quadratic phase modulation result false on spectrogram with the method for bicoherence.Meanwhile, when signal exists square phase-couple and non-square phase-couple information in identical bifrequency simultaneously, the two spectrum of traditional small echo can not effectively identify with bicoherence computing method.Above problem all significantly can reduce the accuracy of result of calculation.Therefore, in the urgent need to a kind of more effective technology to improve the accuracy of detection.

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 of instantaneous small echo bicoherence method random based on phase place, considering its amplitude and phase information when calculating small echo bicoherence simultaneously, avoiding the defect that classic method accuracy is low.

In order to achieve the above object, the technical scheme that the present invention takes is:

Based on the instantaneous small echo bicoherence method that phase place is random, comprise the following steps:

Step one: select the mother wavelet function being used for continuous wavelet transform, when detecting the non-stationary signal in mechanical signal, morther wavelet adopts the Morlet small echo shown in formula (1),

$\mathrm{\ψ}\left(t\right)=e^{-{\mathrm{\σ}}^2{t}^2}{e}^{-i2\mathrm{\πft}}$ (1)

Wherein: σ---decay factor;

The frequency of f---Morlet morther wavelet;

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

$W_{\mathrm{\ψ}}(s,t)=\frac{1}{\sqrt{\left|s\right|}}{\∫}_{-\∞}^{\∞}x\left(t^{\′}\right){\mathrm{\ψ}}^{*}\left(\frac{t^{\′}-t}{s}\right)d{t}^{\′}---\left(2\right)$

Wherein: ψ (t)---selected morther wavelet;

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

S---scale factor;

T---time factor;

*---represent and get conjugation;

Step 3: the two spectrum of the instantaneous small echo calculated 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 ₂meet relation owing to calculating the two spectrum of instantaneous small echo for plural, therefore 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 instantaneous small echo bispectrum phase;

Step 4: the instantaneous small echo quarter-phase calculated by formula (3) is substituted into formula (5) and calculate the two spectrum of instantaneous small echo:

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

---the instantaneous small echo quarter-phase calculated by formula (3);

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

Wherein: E{.}---represents and gets desired operation;

Step 6: the two spectrum of the instantaneous small echo random based on phase place that will be calculated by formula (6), integration obtains composing as shown in formula (7) based on the small echo that phase place is random is two over time intervalt:

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

Step 8: the result calculated by formula (7) is substituted into formula (9) and calculates based on the random small echo bicoherence of phase place:

$b_{W,T}(s_1,s_2)=\frac{{\left|B_{W,T}^{\′}(s_1,s_2)\right|}^2}{{\∫}_T{\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t)\right|}^2\mathrm{dt}{\∫}_T{\left|W_{\mathrm{\ψ}}(s,t)\right|}^2\mathrm{dt}}---\left(9\right)$

Step 9: the result calculated by formula (8) is substituted into formula (10) and calculates based on the random instantaneous small echo bicoherence of phase place:

$b_{W,T}(s_1,t)=\frac{{\left|B_{W,T}^{\′}(s_1,t)\right|}^2}{{\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t)\right|}^2{\left|W_{\mathrm{\ψ}}(s,t)\right|}^2}---\left(10\right)$

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

$b_{W,T}(s_1,s_2)=\frac{{\left|B_{W,T}(s_1,s_2)\right|}^2}{{\∫}_T{\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t)\right|}^2\mathrm{dt}{\∫}_T{\left|W_{\mathrm{\ψ}}(s,t)\right|}^2\mathrm{dt}}---\left(11\right)$

Wherein: yardstick s, s ₁with s ₂meet relation

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

B _w,T(s ₁, s ₂)---for the two spectrum of traditional small echo calculates, as shown in formula (12),

Wherein: *---represent and get conjugation,

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 instantaneous small echo bispectrum phase.

The small echo bicoherence calculated by formula (11) is yardstick s ₁with s ₂function, span is 0 to 1.Desirable small echo bicoherence computing formula need meet following two conditions:

If I. signal is at bifrequency (s ₁, s ₂) place exists square phase-couple, then at bifrequency (s ₁, s ₂) place small echo two-phase dry values b _w,T(s ₁, s ₂)=1, simultaneously at bifrequency (s ₁, s ₂) quarter-phase at place

If II. signal is at bifrequency (s ₁, s ₂) place do not exist square phase-couple, then at bifrequency (s ₁, s ₂) the value b of place's small echo bicoherence _w,T(s ₁, s ₂)=0, simultaneously at bifrequency (s ₁, s ₂) quarter-phase at place

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

Accompanying drawing explanation

Fig. 1 is simulate signal time domain waveform.

Fig. 2 is simulate signal continuous wavelet transform spectrum.

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

Fig. 4 is Traditional Wavelet bicoherence spectrum.

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

Fig. 6 is traditional instantaneous small echo bicoherence spectrum.

Embodiment

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

For simulate signal, the expression formula of simulate signal is as shown in formula (13)

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

---interval be (-π, π] be uniformly distributed 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, to-noise ratio is 3dB.Be f first and the 3rd ripple bag medium frequency ₁and f ₂cosine wave (CW) meet square phase-couple condition, and second and the 4th ripple bag medium frequency are f ₁and f ₂cosine wave (CW) do not meet square phase-couple condition.The time domain waveform of simulate signal as shown in Figure 1; In order to two chi phase coupling estimations in identification signal, adopt the present invention to data analysis.

Based on the instantaneous small echo bicoherence method that phase place is random, comprise the following steps:

Step one: select the mother wavelet function being used for continuous wavelet transform, when detecting the non-stationary signal in mechanical signal, morther wavelet adopts the Morlet small echo shown in formula (1),

$\mathrm{\ψ}\left(t\right)=e^{-{\mathrm{\σ}}^2{t}^2}{e}^{-i2\mathrm{\πft}}$ (1)

Wherein: get decay factor σ=4, the frequency f=0.95Hz of morther wavelet is to meet admissibility condition;

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

$W_{\mathrm{\ψ}}(s,t)=\frac{1}{\sqrt{\left|s\right|}}{\∫}_{-\∞}^{\∞}x\left(t^{\′}\right){\mathrm{\ψ}}^{*}\left(\frac{t^{\′}-t}{s}\right)d{t}^{\′}---\left(2\right)$

Wherein: ψ (t)---selected morther wavelet;

X (t)---simulate signal;

S---scale factor;

T---time factor;

*---represent and get conjugation;

Step 3: the two spectrum of the instantaneous small echo calculated under each yardstick according to formula (3), wherein getting frequency resolution is 0.25Hz, and the instantaneous small echo bispectrum phase under obtaining 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 ₂meet relation

Step 4: the instantaneous small echo quarter-phase calculated by formula (3) is substituted into formula (5) and calculate the two spectrum of instantaneous small echo;

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

---the instantaneous small echo quarter-phase calculated by formula (3);

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

Wherein: E{.}---represents and gets desired operation;

Step 6: the two spectrum of the instantaneous small echo random based on phase place that will be calculated by formula (6), integration obtains based on the two spectrum of the small echo that phase place is random as shown in formula (7) over time intervalt;

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

Step 8: the result calculated by formula (7) is substituted into formula (9) and calculates based on the random small echo bicoherence of phase place, result of calculation as shown in Figure 3;

$b_{W,T}(s_1,s_2)=\frac{{\left|B_{W,T}^{\′}(s_1,s_2)\right|}^2}{{\∫}_T{\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t)\right|}^2\mathrm{dt}{\∫}_T{\left|W_{\mathrm{\ψ}}(s,t)\right|}^2\mathrm{dt}}---\left(9\right)$

Fig. 4 is the collection of illustrative plates that Traditional Wavelet bicoherence computing method obtain, and as can be seen from Figure 4, spectrogram exists the peak value occurring falseness at multiple frequency place, is difficult to judge real square phase-couple frequency content.But, only there is peak value at the frequency content place (9Hz and 19Hz) that there is square phase-couple in the instantaneous small echo bicoherence method random based on phase place, as shown in Figure 3, therefore, the instantaneous small echo bicoherence method random based on phase place can effectively overcome classic method under certain conditions, the defect of the false square phase-couple testing result existed, improves the accuracy of detection;

Step 9: the result calculated by formula (8) is substituted into formula (10) and calculates based on the random instantaneous small echo bicoherence of phase place, result of calculation as shown in Figure 5;

$b_{W,T}(s_1,t)=\frac{{\left|B_{W,T}^{\′}(s_1,t)\right|}^2}{{\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t)\right|}^2{\left|W_{\mathrm{\ψ}}(s,t)\right|}^2}---\left(10\right)$

As shown in Figure 6, as seen from Figure 6, no matter ripple is surrounded by not square phase-couple to the spectrogram calculated by the instantaneous bicoherence method of Traditional Wavelet, and testing result all demonstrates larger small echo bicoherence coefficient.And as shown in Figure 5, only there is larger small echo two-phase responsibility first and the 3rd ripple bag place there is square phase-couple in the instantaneous small echo bicoherence spectrum random based on phase place, and at second and the 4th ripple bag place there is not square phase-couple, its small echo bicoherence coefficient is then very little.The instantaneous small echo bicoherence computing method random based on phase place that this patent proposes can overcome classic method effectively, the defect cannot differentiated when square phase-couple and non-square phase-couple exist simultaneously under identical bifrequency, improve the accuracy of detection, be applicable to associated each field.

Claims (1)

1., based on the instantaneous small echo bicoherence method that phase place is random, it is characterized in that, comprise the following steps:

Step one: select the mother wavelet function being used for continuous wavelet transform, when detecting the non-stationary signal in mechanical signal, morther wavelet adopts the Morlet small echo shown in formula (1),

$\mathrm{\ψ}\left(t\right)=e^{-{\mathrm{\σ}}^2{t}^2}{e}^{-i2\mathrm{\π}ft}---\left(1\right)$

Wherein: σ---decay factor;

The frequency of f---Morlet morther wavelet;

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

$W_{\mathrm{\ψ}}(s,t)=\frac{1}{\sqrt{\left|s\right|}}{\∫}_{-\mathrm{\∞}}^{\mathrm{\∞}}x\left(t^{\′}\right){\mathrm{\ψ}}^{*}\left(\frac{t^{\′}-t}{s}\right){\mathrm{dt}}^{\′}---\left(2\right)$

Wherein: ψ (t)---selected morther wavelet;

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

S---scale factor;

T---time factor;

*---represent and get conjugation;

Step 3: the two spectrum of the instantaneous small echo calculated 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 ₂meet relation owing to calculating the two spectrum of instantaneous small echo for plural, therefore 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 instantaneous small echo quarter-phase;

Step 4: the instantaneous small echo quarter-phase calculated by formula (4) is substituted into formula (5) and calculate the two spectrum of instantaneous small echo:

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

---the instantaneous small echo quarter-phase calculated by formula (4);

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

Wherein: E{}---represents and gets desired operation;

Step 6: the two spectrum of the instantaneous small echo random based on phase place that will be calculated by formula (6), integration obtains composing as shown in formula (7) based on the small echo that phase place is random is two over time intervalt:

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

Step 8: the result calculated by formula (7) is substituted into formula (9) and calculates based on the random small echo bicoherence of phase place:

$b_{W,T}(s_1,s_2)\frac{|B_{W,T}^{\′}(s_1,s_2){|}^2}{{\∫}_T\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t){|}^{2}dt{\∫}_T\right|W_{\mathrm{\ψ}}(s,t){|}^{2}dt}---\left(9\right)$

Step 9: the result calculated by formula (8) is substituted into formula (10) and calculates based on the random instantaneous small echo bicoherence of phase place:

$b_{W,T}(s_1,t)=\frac{|B_{W,T}^{\′}(s_1,t){|}^2}{\left|W_{\mathrm{\ψ}}(s_1,t)W_{\mathrm{\ψ}}(s_2,t){|}^2\right|W_{\mathrm{\ψ}}(s,t){|}^2}.---\left(10\right)$ 。