CN107886111A - A kind of modified part mean decomposition method - Google Patents

A kind of modified part mean decomposition method Download PDF

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CN107886111A
CN107886111A CN201711009198.XA CN201711009198A CN107886111A CN 107886111 A CN107886111 A CN 107886111A CN 201711009198 A CN201711009198 A CN 201711009198A CN 107886111 A CN107886111 A CN 107886111A
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贾林山
张庆
何小高
姚普林
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Xian Jiaotong University
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Abstract

A kind of modified part mean decomposition method, Local modulus maxima and the local minizing point of primary signal are looked for first, then Local modulus maxima and local minizing point are carried out to cubic spline interpolation respectively and obtains signal coenvelope function and signal lower envelope function, selective goal is set according to the upper and lower envelope function of signal afterwards, if selective goal meets batten envelope requirement, local mean value function and envelope estimation function are calculated using foregoing upper and lower envelope function；If selective goal is unsatisfactory for batten envelope requirement, local mean value function and envelope estimation function are calculated using moving average method；Spline method and moving average method are used in combination to build local mean value function and envelope estimation function the present invention, can be obviously improved the computational efficiency of LMD methods, strengthen the convergent stability of LMD methods.

Description

A kind of modified part mean decomposition method
Technical field
The invention belongs to part mean decomposition method technical field, and in particular to a kind of modified local mean value decomposition side Method.
Background technology
Multi -components can be decomposed into some by part mean decomposition method (Local Mean Decomposition, LMD) The AM/FM amplitude modulation/frequency modulation signal sum of individual simple component, and the instantaneous frequency of each component can be obtained, instantaneous amplitude, so as to To the complete time-frequency distributions of primary signal, it is very suitable for handling multicomponent AM/FM amplitude modulation/frequency modulation signal (Cheng Junsheng, Yang Yu, Yu De Jie a kind of new Time-Frequency Analysis Method-part mean decomposition method [J] vibrations and impact, 2008,27 (S):129-131.). However, in actual calculating process, because the requirement of LMD methods uses moving average method generation local mean value function and envelope letter Number, operand is very big, and there have that arithmetic speed is slow, iteration time is long, phase offset, Decomposition Accuracy are poor, convergence success rate is low etc. to be scarce Point, limit application of the LMD methods in Practical Project.
Empirical mode decomposition (Empirical Mode Decomposition, EMD) is similar with local mean value decomposition A kind of adaptive signal decomposition method (Huang N E, Shen Z, Long S R, et al.The Empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis.Proc.R.Soc.Lond.A,1998:903-995), this method uses cubic spline to signal extreme value Point interpolation obtains signal local mean value, and efficiency of algorithm is high but envelope be present, owes the problems such as envelope.Meanwhile existing scholar will Envelope thought during EMD is decomposed is applied among LMD improvement (Hu Jingsong, Yang Shixi, vibration letters of unconventional and unrestrained thousand, based on batten Number local average decomposition method [J] data acquisition and processions, 2009,24 (1):82-86.), but due to foregoing cubic spline sheet The defects of body, causes it to have impact on LMD convergence in some cases, and the moving average method in classical LMD methods can be preferable Solve the problems, such as cubic spline cross envelope and deficient envelope (Jonathan S.Smith, The local mean decomposition and its application to EEG perception data,J.R.Soc.Interface 2 (5),2005:443–454.)。
It yet there are no and ask for local mean value function and envelope function in combination with spline method and moving average method Relevant report.
The content of the invention
The shortcomings that in order to overcome above-mentioned prior art, it is an object of the invention to provide a kind of decomposition of modified local mean value Method, arithmetic speed, Decomposition Accuracy and the convergence success rate of LMD methods can be effectively improved.
To achieve these goals, the present invention adopts the technical scheme that：
A kind of modified part mean decomposition method, comprises the following steps：
1) all Local Extremum n of primary signal x (t) are determinedi, wherein Local modulus maxima is MaxEi, local minimum Value point is MinEi；
2) primary signal x (t) Local modulus maxima MaxEiCubic spline interpolation is carried out, forms coenvelope function envmax(t), primary signal x (t) local minizing point MinEiCubic spline interpolation is carried out, forms lower envelope function envmin(t)；Set selective goal Z, Z=envmax(t)-envmin(t)；
If meeting Z ＞ 0, local mean value function m11And envelope estimation function a (t)11(t) it is respectively：
If it is unsatisfactory for Z>0, calculate local mean value mi, i.e.,：
Then partial amplitudes a is calculatedi, i.e.,：
Using arest neighbors interpolation method by local mean value miWith partial amplitudes aiEnter row interpolation, will using moving average method Arest neighbors interpolation result is carried out smoothly, untill the sampling point value of any one sampled point both sides adjacent thereto is inconsistent, Finally give local mean value function m11And envelope estimation function a (t)11(t)；
3) by local mean value function m11(t) separate, obtain from primary signal x (t)：
h11(t)=x (t)-m11(t)
4) h is used11Divided by envelope estimation function a (t)11(t), so as to h11(t) it is demodulated, obtains：
s11(t)=h11(t)/a11(t)
By s11(t) it is used as primary signal repeat step 1) -4), until obtaining a pure FM signal s1n(t), meet 1≤ s1n(t)≤1, its envelope estimation function meets a1(n+1)(t)≈1；Therefore there is following iterative process：
In above formula,
A threshold value Δ is given, stopping criterion for iteration is set as 1- Δs≤a1n(t)≤1+Δ；Δ is set smaller, is decomposed It is more accurate；
5) step 1) -4) in caused all envelope estimation functions be multiplied to obtain envelope signal a1(t)：
a1(t)=a11(t)a12(t)···a1n(t)
6) by envelope signal a1(t) the pure FM signal s with obtaining1n(t) it is multiplied, obtains first of primary signal x (t) PF components PF1(t)：
PF1(t)=a1(t)s1n(t)
7) first PF component is separated from primary signal, obtains a new signal u1(t), by signal u1(t) As primary signal x (t) repeat steps 1) -6), circulate k times, until uk(t) untill for a monotonic function；
Most at last primary signal x (t) resolves into k PF components PFi(t), wherein i=1 ..., k, and 1 surplus uk (t), by surplus uk(t) it is denoted as R.
" selective goal Z ＞ 0 " definition is in the step 2)：Any one element z in selective goal Z sequencesiIt is full Sufficient zi＞ 0, wherein zi∈Z。
Beneficial effects of the present invention are：
(1) moving average method is used in combination in the present invention and spline method obtains local mean value function and envelope function, fills Point make use of that high frequency situations lower envelope precision is high, arithmetic speed is fast and low frequency in the case of the good advantage of moving average method convergence, It is flat to avoid operand and slip that traditional part mean decomposition method uses merely moving average method and rolled up The influence that equal method parameter selection is decomposed to whole local mean value.
(2) present invention by two methods of moving average method and spline method simultaneously applied to calculate local mean value function and Envelope function, occupation mode is using calculation used in the automatic judgements of selective goal Z, it is ensured that has in high frequency region higher Decomposition Accuracy, have higher convergence success rate in low frequency range.
Brief description of the drawings
Fig. 1 is the inventive method flow chart.
Fig. 2 is the oscillogram that embodiment emulates primary signal x (t).
Fig. 3 is embodiment primary signal x (t) Local Extremums niMark, wherein Local modulus maxima MaxEiMarked with " * " Know, local minizing point MinEiIdentified with "+".
Fig. 4 is embodiment primary signal x (t) coenvelope function envmaxAnd lower envelope function env (t)min(t)。
Fig. 5 is embodiment primary signal x (t) local mean value function m11And envelope estimation function a (t)11(t)。
Fig. 6 is that local mean value function m is subtracted from embodiment primary signal x (t)11(t) the signal h after11(t)。
Fig. 7 is from signal h11(t) envelope estimation function a is removed in11(t) the signal s after11(t)。
Fig. 8 is envelope signal a1(t)。
Fig. 9 is embodiment primary signal x (t) the 1st PF components PF1(t)。
Figure 10 is the coenvelope function env that embodiment spline method calculatesmaxAnd lower envelope function env (t)min(t)。
Figure 11 is the local mean value function m that embodiment moving average method is calculated11And envelope estimation function a (t)11 (t)。
Figure 12 is embodiment primary signal x (t) the 1st PF components PF1(t)。
Figure 13 is embodiment primary signal x (t) the 2nd PF components PF2(t)。
Figure 14 is embodiment primary signal x (t) the 3rd PF components PF3(t)。
Figure 15 is embodiment primary signal x (t) the 4th PF components PF4(t)。
Figure 16 is embodiment primary signal x (t) the 5th PF components PF5(t)。
Figure 17 is embodiment primary signal x (t) surplus R.
Embodiment
Being opposed below in conjunction with drawings and examples, the present invention is further to be described in detail.
Shown in reference picture 1, a kind of modified part mean decomposition method, comprise the following steps：
1) produce an emulation signal is as primary signal x (t), formula：
X (t)=(1+0.5cos (8 π t)) cos (200 π t+2cos (10 π t))+0.8sin (π t) sin (30 π t) settings are adopted Sample frequency is 1002Hz, t ∈ [0,1], and its waveform is as shown in Figure 2；Find primary signal x (t) Local modulus maxima MaxEiWith Local minizing point MinEi, use " * " to represent Local modulus maxima, local minizing point represented using "+" number, such as Fig. 3 institutes Show；
2) primary signal x (t) Local modulus maxima MaxEiCubic spline interpolation is carried out, forms coenvelope function envmax(t), primary signal x (t) local minizing point MinEiCubic spline interpolation is carried out, forms lower envelope function envmin(t), as shown in Figure 4；Set selective goal Z, Z=envmax(t)-envmin(t), the Z ＞ 0 in this circulation, then it is local Mean value function m11And envelope estimation function a (t)11(t) it is respectively：
As shown in Figure 5；
3) by local mean value function m11(t) separated from primary signal x (t), obtain h11(t)=x (t)-m11(t), As shown in Figure 6；
4) h is used11Divided by envelope estimation function a (t)11(t), so as to h11(t) it is demodulated, obtains s11(t)=h11 (t)/a11(t), as shown in Figure 7；Ideally, s11(t) it is a pure FM signal, its envelope function a12(t) should meet a12(t)=1, if s11(t) condition is not satisfied, then by s11(t) as initial data repeat step 1) -4) iterative process, Until obtaining a pure FM signal s1n(t) 1≤s, is met1n(t)≤1, its envelope estimation function meets a1(n+1)(t)≈1； A threshold value Δ is given, stopping criterion for iteration is set as 1- Δs≤a1n(t)≤1+Δ；Δ is set smaller, and decomposition is more accurate, this Given threshold Δ=0.01 in embodiment；The present embodiment by iteration twice obtain one meet stopping criterion for iteration 1-0.01≤ a1n(t)≤1+0.01 pure FM signal s12(t)；
5) step 1) -4) in caused all envelope estimation functions be multiplied to obtain envelope signal a1(t)=a11(t)a12 (t), as shown in Figure 8；
6) by envelope signal a1(t) the pure FM signal s with obtaining12(t) it is multiplied, obtains first of primary signal x (t) PF components PF1(t)=a1(t)s1n(t), as shown in Figure 9；
7) first PF component is separated from primary signal, obtains a new signal u1(t), by signal u1(t) As primary signal x (t) repeat steps 1) -6), circulate k times, until uk(t) untill for a monotonic function；In the present embodiment In, when circulation is carried out to the 3rd time, selective goal Z no longer meets Z ＞ 0 in step 2), now it can be seen that spline method calculates Coenvelope function envmaxAnd lower envelope function env (t)min(t) nearby intersected at 0.3 second, as shown in Figure 10, therefore not The coenvelope function and lower envelope function that spline method can be used to calculate, due to being unsatisfactory for Z>0, calculate local mean value mi, i.e.,：
Then partial amplitudes a is calculatedi, i.e.,：
Using arest neighbors interpolation method by local mean value miWith partial amplitudes aiEnter row interpolation, will using moving average method Arest neighbors interpolation result is carried out smoothly, untill the sampling point value of any one sampled point both sides adjacent thereto is inconsistent, Finally give local mean value function m11And envelope estimation function a (t)11(t), as shown in figure 11；In the present embodiment, it is final to obtain To primary signal x (t) 5 PF components and a surplus R, respectively such as Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 institute Show.

Claims (2)

1. a kind of modified part mean decomposition method, it is characterised in that comprise the following steps：
1) all Local Extremum n of primary signal x (t) are determinedi, wherein Local modulus maxima is MaxEi, local minizing point For MinEi；
2) primary signal x (t) Local modulus maxima MaxEiCubic spline interpolation is carried out, forms coenvelope function envmax (t), primary signal x (t) local minizing point MinEiCubic spline interpolation is carried out, forms lower envelope function envmin(t)； Set selective goal Z, Z=envmax(t)-envmin(t)；
If meeting Z ＞ 0, local mean value function m11And envelope estimation function a (t)11(t) it is respectively：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>m</mi> <mn>11</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mfrac> <mrow> <msub> <mi>env</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>env</mi> <mi>min</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mn>11</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mfrac> <mrow> <msub> <mi>env</mi> <mi>max</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>env</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> </mtd> </mtr> </mtable> </mfenced>
If it is unsatisfactory for Z>0, calculate local mean value mi, i.e.,：
<mrow> <msub> <mi>m</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>n</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow>
Then partial amplitudes a is calculatedi, i.e.,：
<mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <msub> <mi>n</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mn>2</mn> </mfrac> </mrow>
Using arest neighbors interpolation method by local mean value miWith partial amplitudes aiEnter row interpolation, will be nearest using moving average method Adjacent interpolation result is carried out smoothly, untill the sampling point value of any one sampled point both sides adjacent thereto is inconsistent, finally Obtain local mean value function m11And envelope estimation function a (t)11(t)；
3) by local mean value function m11(t) separate, obtain from primary signal x (t)：
h11(t)=x (t)-m11(t)
4) h is used11Divided by envelope estimation function a (t)11(t), so as to h11(t) it is demodulated, obtains：
s11(t)=h11(t)/a11(t)
By s11(t) it is used as primary signal repeat step 1) -4), until obtaining a pure FM signal s1n(t) 1≤s, is met1n (t)≤1, its envelope estimation function meets a1(n+1)(t)≈1；Therefore there is following iterative process：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>h</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>m</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>h</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>m</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>......</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mrow> <mn>1</mn> <mrow> <mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>m</mi> <mrow> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
In above formula,
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mn>12</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>......</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>s</mi> <mrow> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
A threshold value Δ is given, stopping criterion for iteration is set as 1- Δs≤a1n(t)≤1+Δ；Δ is set smaller, is decomposed more smart Really；
5) step 1) -4) in caused all envelope estimation functions be multiplied to obtain envelope signal a1(t)：
a1(t)=a11(t)a12(t)···a1n(t)
6) by envelope signal a1(t) the pure FM signal s with obtaining1n(t) it is multiplied, obtains first PF points of primary signal x (t) Measure PF1(t)：
PF1(t)=a1(t)s1n(t)
7) first PF component is separated from primary signal, obtains a new signal u1(t), by signal u1(t) conduct Primary signal x (t) repeat steps 1) -6), circulate k times, until uk(t) untill for a monotonic function；
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mi>P</mi> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mi>P</mi> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <mo>...</mo> <mo>...</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>=</mo> <msub> <mi>u</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>-</mo> <mi>P</mi> <msub> <mi>F</mi> <mi>k</mi> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mtd> </mtr> </mtable> </mfenced>
Most at last primary signal x (t) resolves into k PF components PFi(t), wherein i=1 ..., k, and 1 surplus uk(t), will Surplus uk(t) it is denoted as R.
A kind of 2. modified part mean decomposition method according to claim 1, it is characterised in that：In the step 2) " selective goal Z ＞ 0 " definition is：Any one element z in selective goal Z sequencesiIt is satisfied by zi＞ 0, wherein zi∈Z。
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CN109342018A (en) * 2018-12-14 2019-02-15 国家电网有限公司 A kind of Turbine Cavitation Testing state monitoring method
CN110053043A (en) * 2019-03-05 2019-07-26 华南理工大学 A kind of industrial robot collision checking method, device, electronic equipment and medium
CN110289007A (en) * 2019-06-25 2019-09-27 西安交通大学 A kind of improvement part mean decomposition method extracted for voice fundamental frequency
CN114997242A (en) * 2022-06-30 2022-09-02 吉林大学 Extreme value positioning waveform continuation LMD signal decomposition method
CN114997242B (en) * 2022-06-30 2023-08-29 吉林大学 Extremum positioning waveform extension LMD signal decomposition method
CN117169591A (en) * 2023-10-25 2023-12-05 南方电网科学研究院有限责任公司 Broadband measurement method and device for power system and computer equipment
CN117169591B (en) * 2023-10-25 2024-03-12 南方电网科学研究院有限责任公司 Broadband measurement method and device for power system and computer equipment

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