CN104992063A - Noise reduction method for vibration signal of mechanical equipment - Google Patents

Abstract

The invention relates to a noise reduction method for a vibration signal of mechanical equipment. The method comprises the steps of: performing local mean decomposition on a non-stable vibration signal; according to PF components obtained by local mean decomposition, calculating a cross correlation coefficient of each PF component and the non-stable vibration signal, comparing the cross correlation coefficient with a preset value, and performing superposition reconstruction on each PF component when the cross correlation coefficient is smaller than the preset value to obtain a virtual noise channel signal, wherein the virtual noise channel signal serves as an input signal of an FastICA algorithm; and performing blind source separation on the vibration signal and the virtual noise channel signal according to the FastICA algorithm to obtain a source signal and a noise signal of the vibration signal, so that the noise reduction of the vibration signal is realized. The method can effectively reduce noise interference in the vibration signal, enable fault character frequency to be more obvious and then facilitate extraction of fault characteristics, and can be widely applied to the field of fault diagnosis of mechanical equipment.

Description

A kind of noise-reduction method of mechanical equipment vibration signal

Technical field

The present invention relates to a kind of noise-reduction method, particularly about a kind of noise-reduction method of mechanical equipment vibration signal.

Background technology

Plant equipment is a complicated system, relative motion is there is between each parts, and the operating condition of plant equipment often changes, these factors cause in the vibration signal collected except reflecting the useful signal of equipment running status, noises a large amount of in addition.The existence of noise all brings very large interference to the monitoring of equipment running status, fault diagnosis, therefore needs to carry out noise reduction process to vibration signal.

Traditional vibration signal noise-reduction method uses wave filter to arrange different passband usually, reaches the suppression to certain certain wave band signal or filtering.But this kind of noise-reduction method is only applicable to the situation that signal and noise are in different frequency bands.For complication system vibration signal, the phenomenon that mutual aliasing occurs on frequency band for system signal and noise signal is unavoidable often, and this makes traditional noise-reduction method there is very large limitation when processing complication system signal, and effect is often not good.

Summary of the invention

For the problems referred to above, the object of this invention is to provide a kind of noise-reduction method of mechanical equipment vibration signal, can effectively reduce noise in vibration signal, make fault characteristic frequency more obvious, and then be conducive to the extraction of fault signature.

For achieving the above object, the present invention takes following technical scheme: a kind of noise-reduction method of mechanical equipment vibration signal, it is characterized in that, said method comprising the steps of: (1) is carried out local mean value to Non-stationary vibration signal x (t) and decomposed as follows: (1.1) determine all Local Extremum n on vibration signal x (t) _i, calculate the mean value m of all adjacent two extreme points _i:

$m_i=\frac{n_i+n_{i+1}}{2},$

In formula, n _i+1be and Local Extremum n _iadjacent Local Extremum; (1.2) by all mean value m _iconnect with straight line, then adopt running mean method to do smoothing processing, obtain local mean value function m ₁₁(t); (1.3) by Local Extremum n _iobtain corresponding envelope estimated value a _i, and by all envelope estimated value a _iconnect with straight line, then adopt running mean method to do smoothing processing, obtain corresponding envelope estimation function a ₁₁(t); (1.4) from original Non-stationary vibration signal x (t) by local mean value function m ₁₁t () separates, obtain residual signal h ₁₁t (), uses residual signal h ₁₁t () is divided by envelope estimation function a ₁₁t (), to residual signal h ₁₁t () carries out demodulation:

h ₁₁(t)＝x(t)-m ₁₁(t)， (1)

S ₁₁(t)=h ₁₁(t)/a ₁₁(t), in (2) formula, s ₁₁t () is process signal, demodulation is until this process signal s ₁₁t () is for till pure FM signal; (1.5) envelope estimation function iteration obtained all is multiplied, and obtains an instantaneous amplitude function a ₁(t):

$a_1\left(t\right)=a_{11}\left(t\right)a_{12}\left(t\right)...a_{1n}\left(t\right)=\underset{j=1}{\overset{n}{\Π}}{a}_{1j}\left(t\right);$

(1.6) by instantaneous amplitude function a ₁(t) and pure FM signal s _1nt () is multiplied, obtain first PF of original Non-stationary vibration signal x (t) ₁component: PF ₁(t)=a ₁(t) s _1n(t); (1.7) from original Non-stationary vibration signal x (t) by first PF ₁component obtains new signal u after being separated ₁t (), by signal u ₁t (), as new original signal, is repeated above-mentioned steps, is recycled to signal u _kt () is a monotonic quantity:

$\left\{\begin{array}{c}{u}_1\left(t\right)=x\left(t\right)-{\mathrm{PF}}_1\left(t\right)\\ u_2\left(t\right)=u_1\left(t\right)-{\mathrm{PF}}_2\left(t\right)\\ ...\\ u_k\left(t\right)=u_{k-1}\left(t\right)-{\mathrm{PF}}_k\left(t\right)\end{array}\right.,$

In formula, PF _kt () represents a kth PF component, u _kt () represents signal margin; (2) the PF component obtained after decomposing according to local mean value in step (1), calculate the cross-correlation coefficient of each PF component and Non-stationary vibration signal x (t), by cross-correlation coefficient with preset numerical value and compare, and cross-correlation coefficient is less than each PF component presetting numerical value carry out superposition reconstruct, obtain virtual noise channel signal v (t); This virtual noise channel signal v (t) is as the input signal of FastICA algorithm; (3) according to FastICA algorithm, vibration signal x (t) and virtual noise channel signal v (t) are carried out blind source separating, obtain vibration signal source signal and noise signal, realize the noise reduction process to vibration signal

In described step (1.3), described envelope estimated value a _ifor:

In described step (1.4), judge described process signal s ₁₁t whether () is the method for pure FM signal: if process signal s ₁₁the envelope estimation function a of (t) ₁₂t ()=1, then judge this process signal s ₁₁t () is a pure FM signal, otherwise repeat to carry out n iteration according to formula (1)-(2), until process signal s _1nt () is a pure FM signal till, that is:

$\left\{\begin{array}{c}{h}_{11}\left(t\right)=x\left(t\right)-m_{11}\left(t\right)\\ h_{12}\left(t\right)=s_{11}\left(t\right)-m_{12}\left(t\right)\\ ...\\ h_{1n}\left(t\right)=s_{1(n-1)}\left(t\right)-m_{1n}\left(t\right)\end{array}\right.,$

Wherein, $\left\{\begin{array}{c}{s}_{11}\left(t\right)=h_{11}\left(t\right)/a_{11}\left(t\right)\\ s_{12}\left(t\right)=h_{12}\left(t\right)/a_{12}\left(t\right)\\ ...\\ s_{ln}\left(t\right)=h_{1n}\left(t\right)/a_{1n}\left(t\right)\end{array}\right.;$ The end condition of iteration is set to:

Described stopping criterion for iteration is preferably: 1-Δ≤a _1nt ()≤1+ Δ, wherein Δ is the deviate arranged according to actual needs.

In described step (3), the blind source separation method based on described FastICA algorithm is as follows: (3.1) suppose to there is the individual separate vibration source s of m _j(t), wherein j=1,2 ..., m, obtains m vibration signal x by signals collecting _j(t), and m vibration signal x _jt the vector form X ' of () represents; (3.2) to m vibration signal x _jt the vectorial X ' of () carries out centralization process, even X ' is-E [X ']=X ", make its average be 0, wherein, E is mathematical expectation; (3.3) according to the albefaction formula in FastICA algorithm, vectorial X " being carried out whitening processing, being obtained for solving approximate signal source y _jthe vectorial X of (t); Whitening processing is as follows: a) first solve vectorial X " covariance matrix C _x, C _x=E [X " (X ") ^t]; B) according to covariance matrix C _x, obtain with covariance matrix C _xunit norm proper vector be row matrix F=(e ₁e _n), wherein, e _i(i=1,2 ..., n) be covariance matrix C _xunit norm proper vector; C) according to covariance matrix C _x, obtain with covariance matrix C _xeigenwert be the diagonal matrix D=diag (d of diagonal element ₁d _n), wherein, d _i(i=1,2 ..., n) be covariance matrix C _xeigenwert; D) by step b) and step c) in matrix F and diagonal matrix D substitute into albefaction formula in FastICA algorithm obtain vectorial X; (3.4) approximate signal source y is constructed according to the separation matrix W in FastICA algorithm and vectorial X _jt the vectorial Y=WX of (), makes Y and original independent signal source s _it () is similar to.

Described separation matrix W is utilized to solve described approximate signal source y _jduring the vectorial Y of (t), the iterative formula adopting Finland scientist to be permitted Wei Lining to separation matrix W carry out restraining calculate after the separation matrix W that obtains.

The convergence method of described separation matrix W is: a) first give random starting values to separation matrix W; B) iterative formula W (k+1)=E{Xg [W (k) of Hyvarinen is utilized ^tx] }-E{g [W (k) ^tx] } W (k) ^tcalculate W (k+1), wherein E represents and asks mathematical expectation, and g represents the derivative of non-quadratic function, and k is iterations; C) utilize W (k+1) ← W (k+1)/|| W (k+1) || ₂standardization W (k+1); D) iterate until W convergence, obtain the separation matrix W after restraining.

The present invention is owing to taking above technical scheme, it has the following advantages: 1, the present invention is based on local mean value decomposition and combine noise reduction with independent component analysis, realize carrying out noise reduction to vibration signal noise, system signal and noise signal are had on frequency band to the vibration signal of aliasing situation, effectively can reduce the interference of noise in vibration signal.2, to combine the signal fault characteristic frequency after noise-reduction method process more obvious in the present invention, is conducive to the extraction of fault signature.The present invention can extensively apply in mechanical fault diagnosis field.

Accompanying drawing explanation

Fig. 1 is overall flow schematic diagram of the present invention.

Embodiment

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

As shown in Figure 1, the invention provides a kind of noise-reduction method of mechanical equipment vibration signal, the method decomposes based on local mean value the method with independent component analysis, vibration signal being realized to associating noise reduction, and it comprises the following steps:

(1) carry out local mean value decomposition to Non-stationary vibration signal x (t), its step is as follows:

(1.1) all Local Extremum (comprising maximum value and the minimal value) n on vibration signal x (t) is determined _i, calculate the mean value m of all adjacent two extreme points _i:

$m_i=\frac{n_i+n_{i+1}}{2},---\left(1\right)$

In formula, n _i+1be and Local Extremum n _iadjacent Local Extremum.

(1.2) by all mean value m _iconnect with straight line, then adopt running mean method to do smoothing processing, obtain local mean value function m ₁₁(t).

(1.3) by Local Extremum n _iobtain corresponding envelope estimated value a _i, and by all envelope estimated value a _iconnect with straight line, then adopt running mean method to do smoothing processing, obtain corresponding envelope estimation function a ₁₁(t).

Envelope estimated value a _ifor:

$a_i=\frac{\left|n_i-n_{i+1}\right|}{2}.---\left(2\right)$

(1.4) from original Non-stationary vibration signal x (t) by local mean value function m ₁₁t () separates, obtain residual signal h ₁₁t (), uses residual signal h ₁₁t () is divided by envelope estimation function a ₁₁t (), to residual signal h ₁₁t () carries out demodulation, that is:

h ₁₁(t)＝x(t)-m ₁₁(t)， (3)

s ₁₁(t)＝h ₁₁(t)/a ₁₁(t)， (4)

In formula, s ₁₁t () is process signal, need to judge this process signal s ₁₁t whether () be pure FM signal, and decision method is as follows:

If process signal s ₁₁the envelope estimation function a of (t) ₁₂t ()=1, then can judge this process signal s ₁₁t () is a pure FM signal, otherwise repeat to carry out n iteration according to formula (3)-(4), until process signal s _1nt () is a pure FM signal till, that is:

$\left\{\begin{array}{c}{h}_{11}\left(t\right)=x\left(t\right)-m_{11}\left(t\right)\\ h_{12}\left(t\right)=s_{11}\left(t\right)-m_{12}\left(t\right)\\ ...\\ h_{1n}\left(t\right)=s_{1(n-1)}\left(t\right)-m_{1n}\left(t\right)\end{array}\right.,---\left(5\right)$

Wherein, $\left\{\begin{array}{c}{s}_{11}\left(t\right)=h_{11}\left(t\right)/a_{11}\left(t\right)\\ s_{12}\left(t\right)=h_{12}\left(t\right)/a_{12}\left(t\right)\\ ...\\ s_{1n}\left(t\right)=h_{1n}\left(t\right)/a_{1n}\left(t\right)\end{array}\right.;$

The end condition of iteration is set to:

In the present embodiment, owing to considering computational load actually becomes, under the prerequisite not affecting discomposing effect, therefore stopping criterion for iteration is preferably: 1-Δ≤a _1nt ()≤1+ Δ, wherein Δ is the deviate arranged according to actual needs.

(1.5) the envelope estimation function that previous step iteration obtains all is multiplied, obtains an instantaneous amplitude function a ₁(t):

$a_1\left(t\right)=a_{11}\left(t\right)a_{12}\left(t\right)...a_{1n}\left(t\right)=\underset{j=1}{\overset{n}{\Π}}{a}_{1j}\left(t\right).\text{---}\left(6\right)$

(1.6) by instantaneous amplitude function a ₁(t) and the pure FM signal s in step (1.4) _1nt () is multiplied, obtain first PF of original Non-stationary vibration signal x (t) ₁component:

PF ₁(t)＝a ₁(t)s _1n(t)。(7)

(1.7) from original Non-stationary vibration signal x (t) by first PF ₁component can obtain new signal u after being separated ₁t (), by signal u ₁t (), as new original signal, is repeated above-mentioned steps, is recycled to signal u _kt () is a monotonic quantity.

$\left\{\begin{array}{c}{u}_1\left(t\right)=x\left(t\right)-{\mathrm{PF}}_1\left(t\right)\\ u_2\left(t\right)=u_1\left(t\right)-{\mathrm{PF}}_2\left(t\right)\\ ...\\ u_k\left(t\right)=u_{k-1}\left(t\right)-{\mathrm{PF}}_k\left(t\right)\end{array}\right.,\text{---}\left(8\right)$

In formula, PF _kt () represents a kth PF component, u _kt () represents signal margin.

(2) the PF component obtained after decomposing according to local mean value in step (1), calculate the cross-correlation coefficient of each PF component and Non-stationary vibration signal x (t), by cross-correlation coefficient with preset numerical value and compare, and cross-correlation coefficient is less than each PF component presetting numerical value carry out superposition reconstruct, obtain new signal and be virtual noise channel signal v (t); This virtual noise channel signal v (t) is as the input signal of FastICA algorithm (Fast ICA algorithm, ICA is independent component analysis).

(3) according to FastICA algorithm, vibration signal x (t) and virtual noise channel signal v (t) are carried out blind source separating, obtain vibration signal source signal and noise signal, realize the noise reduction process to vibration signal.

Wherein, the blind source separation method based on FastICA algorithm is as follows:

(3.1) suppose to there is the individual separate vibration source s of m _j(t), wherein j=1,2 ..., m, obtains m vibration signal x by signals collecting _j(t), and m vibration signal x _jt the vector form X ' of () represents;

(3.2) to m vibration signal x _jt the vectorial X ' of () carries out centralization process, even X ' is-E [X ']=X ", make its average be 0, wherein, E is mathematical expectation;

(3.3) according to the albefaction formula in FastICA algorithm, vectorial X " being carried out whitening processing, being obtained for solving approximate signal source y _jthe vectorial X of (t); Whitening processing is as follows:

A) first solve vectorial X " covariance matrix C _x, C _x=E [X " (X ") ^t];

B) according to covariance matrix C _x, obtain with covariance matrix C _xunit norm proper vector be row matrix F=(e ₁e _n), wherein, e _i(i=1,2 ..., n) be covariance matrix C _xunit norm proper vector;

C) according to covariance matrix C _x, obtain with covariance matrix C _xeigenwert be the diagonal matrix D=diag (d of diagonal element ₁d _n), wherein, d _i(i=1,2 ..., n) be covariance matrix C _xeigenwert;

D) by step b) and step c) in matrix F and diagonal matrix D substitute into albefaction formula in FastICA algorithm obtain vectorial X.

(3.4) approximate signal source y is constructed according to the separation matrix W in FastICA algorithm and vectorial X _jt the vectorial Y=WX of (), makes Y and original independent signal source s _it () is similar to;

Because the separation matrix W initial value in FastICA algorithm is random imparting, therefore solves vectorial Y with initial separation matrix W, approximate signal source y can be affected _j(t) and Independent Vibration source s _it the approximation quality of () is lower, therefore, the iterative formula adopting Finland scientist to be permitted Wei Lining (Hyvarinen) is carried out convergence to separation matrix W and calculated, and to obtain true and reliable separation matrix W, and then improves approximate signal source y _jthe approximation quality of (t).Then the convergence method of separation matrix W is:

A) first random starting values is given to separation matrix W;

B) iterative formula W (k+1)=E{Xg [W (k) of Hyvarinen is utilized ^tx] }-E{g [W (k) ^tx] } W (k) ^tcalculate W (k+1), wherein E represents and asks mathematical expectation, and g represents the derivative of non-quadratic function, and k is iterations;

C) utilize W (k+1) ← W (k+1)/|| W (k+1) || ₂standardization W (k+1);

D) iterate until W convergence, obtain the separation matrix W after restraining.

The various embodiments described above are only for illustration of the present invention; the structure of each parts, size, setting position and shape all can change to some extent; on the basis of technical solution of the present invention; all improvement of carrying out individual part according to the principle of the invention and equivalents, all should not get rid of outside protection scope of the present invention.

Claims (7)

1. a noise-reduction method for mechanical equipment vibration signal, is characterized in that, said method comprising the steps of:

(1) carrying out local mean value to Non-stationary vibration signal x (t) decomposes as follows:

(1.1) all Local Extremum n on vibration signal x (t) are determined _i, calculate the mean value m of all adjacent two extreme points _i:

$m_i=\frac{n_i+n_{i+1}}{2},$

In formula, n _i+1be and Local Extremum n _iadjacent Local Extremum;

(1.2) by all mean value m _iconnect with straight line, then adopt running mean method to do smoothing processing, obtain local mean value function m ₁₁(t);

(1.3) by Local Extremum n _iobtain corresponding envelope estimated value a _i, and by all envelope estimated value a _iconnect with straight line, then adopt running mean method to do smoothing processing, obtain corresponding envelope estimation function a ₁₁(t);

(1.4) from original Non-stationary vibration signal x (t) by local mean value function m ₁₁t () separates, obtain residual signal h ₁₁t (), uses residual signal h ₁₁t () is divided by envelope estimation function a ₁₁t (), to residual signal h ₁₁t () carries out demodulation:

h ₁₁(t)＝x(t)-m ₁₁(t)， (1)

s ₁₁(t)＝h ₁₁(t)/a ₁₁(t)， (2)

In formula, s ₁₁t () is process signal, demodulation is until this process signal s ₁₁t () is for till pure FM signal;

(1.5) envelope estimation function iteration obtained all is multiplied, and obtains an instantaneous amplitude function a ₁(t):

$a_1\left(t\right)=a_{11}\left(t\right)a_{12}\left(t\right)...a_{1n}\left(t\right)=\underset{j=1}{\overset{n}{\Π}}{a}_{1j}\left(t\right);$

(1.6) by instantaneous amplitude function a ₁(t) and pure FM signal s _1nt () is multiplied, obtain first PF of original Non-stationary vibration signal x (t) ₁component:

PF ₁(t)＝a ₁(t)s _1n(t)；

(1.7) from original Non-stationary vibration signal x (t) by first PF ₁component obtains new signal u after being separated ₁t (), by signal u ₁t (), as new original signal, is repeated above-mentioned steps, is recycled to signal u _kt () is a monotonic quantity:

$\left\{\begin{array}{c}{u}_1\left(t\right)=x\left(t\right)-{\mathrm{PF}}_1\left(t\right)\\ u_2\left(t\right)=u_1\left(t\right)-{\mathrm{PF}}_{\text{2}}\left(t\right)\\ ...\\ u_k\left(t\right)=u_{k-1}\left(t\right)-{\mathrm{PF}}_k\left(t\right)\end{array}\right.,$

In formula, PF _kt () represents a kth PF component, u _kt () represents signal margin;

(2) the PF component obtained after decomposing according to local mean value in step (1), calculate the cross-correlation coefficient of each PF component and Non-stationary vibration signal x (t), by cross-correlation coefficient with preset numerical value and compare, and cross-correlation coefficient is less than each PF component presetting numerical value carry out superposition reconstruct, obtain virtual noise channel signal v (t); This virtual noise channel signal v (t) is as the input signal of FastICA algorithm;

(3) according to FastICA algorithm, vibration signal x (t) and virtual noise channel signal v (t) are carried out blind source separating, obtain vibration signal source signal and noise signal, realize the noise reduction process to vibration signal.

2. the noise-reduction method of a kind of mechanical equipment vibration signal as claimed in claim 1, is characterized in that: in described step (1.3), described envelope estimated value a _ifor:

$a_i=\frac{|n_i-n_{i-1}|}{2}.$

3. the noise-reduction method of a kind of mechanical equipment vibration signal as claimed in claim 1, is characterized in that: in described step (1.4), judges described process signal s ₁₁t whether () is the method for pure FM signal: if process signal s ₁₁the envelope estimation function a of (t) ₁₂t ()=1, then judge this process signal s ₁₁t () is a pure FM signal, otherwise repeat to carry out n iteration according to formula (1)-(2), until process signal s _1nt () is a pure FM signal till, that is:

$\left\{\begin{array}{c}{h}_{11}\left(t\right)=x\left(t\right)-m_{11}\left(t\right)\\ h_{12}\left(t\right)=s_{11}\left(t\right)-m_{\text{12}}\left(t\right)\\ ...\\ h_{1n}\left(t\right)=s_{1(n-1)}\left(t\right)-m_{\text{1n}}\left(t\right)\end{array}\right.,$

Wherein, $\left\{\begin{array}{c}{s}_{11}\left(t\right)=h_{11}\left(t\right)/a_{11}\left(t\right)\\ s_{12}\left(t\right)=h_{12}\left(t\right)/a_{12}\left(t\right)\\ ...\\ s_{1n}\left(t\right)=h_{1n}\left(t\right)/a_{1n}\left(t\right)\end{array}\right.;$ The end condition of iteration is set to: $\underset{n\→\mathrm{\∞}}{lim}{a}_{1n}\left(t\right)=1.$

4. the noise-reduction method of a kind of mechanical equipment vibration signal as claimed in claim 3, is characterized in that: be preferably by described stopping criterion for iteration: 1-Δ≤a _1nt ()≤1+ Δ, wherein Δ is the deviate arranged according to actual needs.

5. the noise-reduction method of a kind of mechanical equipment vibration signal as described in any one of Claims 1 to 4, is characterized in that: in described step (3), and the blind source separation method based on described FastICA algorithm is as follows:

(3.1) suppose to there is the individual separate vibration source s of m _j(t), wherein j=1,2 ..., m, obtains m vibration signal x by signals collecting _j(t), and m vibration signal x _jt the vector form X ' of () represents;

(3.2) to m vibration signal x _jt the vectorial X ' of () carries out centralization process, even X ' is-E [X ']=X ", make its average be 0, wherein, E is mathematical expectation;

(3.3) according to the albefaction formula in FastICA algorithm, vectorial X " being carried out whitening processing, being obtained for solving approximate signal source y _jthe vectorial X of (t); Whitening processing is as follows:

A) first solve vectorial X " covariance matrix C _x, C _x=E [X " (X ") ^t];

B) according to covariance matrix C _x, obtain with covariance matrix C _xunit norm proper vector be row matrix F=(e ₁e _n), wherein, e _i(i=1,2 ..., n) be covariance matrix C _xunit norm proper vector;

C) according to covariance matrix C _x, obtain with covariance matrix C _xeigenwert be the diagonal matrix D=diag (d of diagonal element ₁d _n), wherein, d _i(i=1,2 ..., n) be covariance matrix C _xeigenwert;

D) by step b) and step c) in matrix F and diagonal matrix D substitute into albefaction formula in FastICA algorithm obtain vectorial X;

(3.4) approximate signal source y is constructed according to the separation matrix W in FastICA algorithm and vectorial X _jt the vectorial Y=WX of (), makes Y and original independent signal source s _it () is similar to.

6. the noise-reduction method of a kind of mechanical equipment vibration signal as claimed in claim 5, is characterized in that: utilize described separation matrix W to solve described approximate signal source y _jduring the vectorial Y of (t), the iterative formula adopting Finland scientist to be permitted Wei Lining to separation matrix W carry out restraining calculate after the separation matrix W that obtains.

7. the noise-reduction method of a kind of mechanical equipment vibration signal as claimed in claim 6, is characterized in that: the convergence method of described separation matrix W is:

A) first random starting values is given to separation matrix W;

B) iterative formula of Hyvarinen is utilized calculate W (k+1), wherein E represents and asks mathematical expectation, and g represents the derivative of non-quadratic function, and k is iterations;

C) utilize W (k+1) ← W (k+1)/|| W (k+1) || ₂standardization W (k+1);

D) iterate until W convergence, obtain the separation matrix W after restraining.