CN114997242A - Extreme value positioning waveform continuation LMD signal decomposition method - Google Patents

Extreme value positioning waveform continuation LMD signal decomposition method Download PDF

Info

Publication number
CN114997242A
CN114997242A CN202210765149.3A CN202210765149A CN114997242A CN 114997242 A CN114997242 A CN 114997242A CN 202210765149 A CN202210765149 A CN 202210765149A CN 114997242 A CN114997242 A CN 114997242A
Authority
CN
China
Prior art keywords
point
minimum
extreme
value
signal
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Granted
Application number
CN202210765149.3A
Other languages
Chinese (zh)
Other versions
CN114997242B (en
Inventor
张英芝
宋敏侨
郑玉彬
宋杰
孙墨
张涵
程琪
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Jilin University
Original Assignee
Jilin University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Jilin University filed Critical Jilin University
Priority to CN202210765149.3A priority Critical patent/CN114997242B/en
Publication of CN114997242A publication Critical patent/CN114997242A/en
Application granted granted Critical
Publication of CN114997242B publication Critical patent/CN114997242B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
    • Y02D30/00—Reducing energy consumption in communication networks
    • Y02D30/70—Reducing energy consumption in communication networks in wireless communication networks

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Mathematical Physics (AREA)
  • Theoretical Computer Science (AREA)
  • Computational Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Optimization (AREA)
  • Pure & Applied Mathematics (AREA)
  • Data Mining & Analysis (AREA)
  • Algebra (AREA)
  • Software Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Databases & Information Systems (AREA)
  • Complex Calculations (AREA)
  • Image Analysis (AREA)

Abstract

The invention belongs to the technical field of signal processing, and relates to an extreme value positioning waveform continuation LMD signal decomposition method, which specifically comprises the following steps: 1. acquiring all extreme points of the waveform; 2. estimating an extreme value at an end point; 3. carrying out signal decomposition by using the estimated value; the invention relates to an extreme value positioning waveform continuation LMD signal decomposition method, which meets self estimation through a signal, fully considers the change rule in the signal and the information of each extreme value point, has the estimated value conforming to the fluctuation trend of the original waveform, and has higher estimation precision and stronger operability.

Description

Extreme value positioning waveform continuation LMD signal decomposition method
Technical Field
The invention belongs to the technical field of signal processing, and relates to an extreme value positioning waveform continuation LMD signal decomposition method, in particular to extreme value point information acquisition, extreme value estimation at an end point and signal decomposition.
Technical Field
The LMD (local mean decomposition) method has self-adaptability for analyzing non-stationary signals, can completely retain time-frequency information of original signals, and is more accurate and effective in analyzing signals of mechanical equipment compared with other similar methods. However, the end-point effect of the LMD negatively affects the LMD decomposition process, and in order to improve the end-point effect, the size of the extreme values at both ends needs to be estimated. Waveform continuation is a currently effective method for predicting extreme points of endpoints. The main idea is to simulate a new small segment of waveform at two ends of a signal to generate two simulated extreme points, and to use the two extreme points as end extreme values to perform the next operation. The existing waveform continuation methods include mirror continuation, neural network prediction continuation, four-point waveform continuation based on matching errors, inner product continuation and the like.
The mirror extension copies the original waveform to two ends to realize the waveform extension by a mirror copy method, and the method does not consider the change rule in the signal and does not improve the end effect too much in nature; the neural network prediction continuation considers the internal change rule of the signal through a neural network algorithm, but the prediction method has long calculation time, carries out parameter adjustment according to different signals and is not suitable for practical application; compared with a neural network, the four-point waveform continuation and inner product continuation based on the matching error are easier, the change rule of the whole signal is considered, the precision is higher, but more nodes are selected, and the matching error of the waveform is larger.
Disclosure of Invention
The endpoint effect for the traditional LMD method affects the stability and accuracy of signal decomposition, and further affects the subsequent analysis results based on the effect. The invention provides an extreme value positioning waveform continuation LMD method, which determines a change rule and an extreme value interval rule of an extreme value point to position a target extreme value point by analyzing the whole waveform, and uses the point as a left and right endpoint extreme value continuation scheme to inhibit an endpoint effect.
In order to solve the technical problems, the invention adopts the following technical scheme, which is described in the following with reference to the accompanying drawings:
an extreme value positioning waveform continuation LMD signal decomposition method comprises the following steps:
step one, acquiring all extreme points of a waveform;
estimating an extreme value at an end point;
and thirdly, decomposing the signal by using the estimated value.
In the first step, all extreme point information of the discrete signal is obtained through a statistical means, wherein the extreme point information comprises a maximum point and a minimum point.
The method for estimating the extreme value at the end point in the second step comprises the following steps:
given a certain discrete signal sample X (t), t is 1-N, the signal sample has a maximum value point
Figure BDA0003721875450000021
Sum minimum point
Figure BDA0003721875450000022
The time nodes corresponding to the two types of extreme points are respectively
Figure BDA0003721875450000023
And
Figure BDA0003721875450000024
by X 1 Denotes the left end point, X, of the sample N Representing the right end point of the sample.
(1) When tm 1 ＜tn 1 Then, the first extreme point of the sample is the maximum point Xm 1 . Take Xm 1 And a first minimum value point Xn 1 As reference points, the time nodes corresponding to the two extreme points are tm respectively 1 And tn 1 By using
Figure BDA0003721875450000025
As a reference period of time, the time period,
Figure BDA0003721875450000026
firstly, the residual maximum value point is aligned
Figure BDA0003721875450000027
According to the principle of minimum difference
Figure BDA0003721875450000028
Matching to find Xm -1 Neutralization Xm 1 Equal or closest maximum point Xm i ,i＝2～n Xm . These maximum points Xm are then located i Minimum value point Xn adjacent to the right side of the minimum value point i Time node of (2), obtaining
Figure BDA0003721875450000029
Finally, according to the minimum principle of the difference value
Figure BDA00037218754500000210
Find and reference time interval
Figure BDA00037218754500000211
Maximum value point Xm corresponding to the nearest time period α ,α∈2～n Xm Then take the maximum value Xm of distance α Minimum value point Xn nearest to left end α-1 As the left-end continuation extreme point X 1 ′。
(2) When tm 1 ＞tn 1 Then, the first extreme point of the sample is the minimum point Xn 1 . Taking Xn 1 And the first maximum point Xm 1 As a reference point, the time nodes corresponding to the two extreme points are tn respectively 1 And tm 1 By using
Figure BDA00037218754500000212
As a reference period of time, the time period,
Figure BDA00037218754500000213
firstly, the remaining minimum value point is checked
Figure BDA00037218754500000214
According to the principle of minimum difference
Figure BDA00037218754500000215
Match to find Xn -1 Neutral and Xn 1 Equal or closest minimum point Xn i ,i＝2～n Xn . These minimum points Xn are then located i Maximum value point Xm adjacent to the right side of the maximum value point i Time node of (2), obtaining
Figure BDA0003721875450000031
Finally, according to the minimum principle of the difference value
Figure BDA0003721875450000032
Find and reference time interval
Figure BDA0003721875450000033
Minimum value point Xn corresponding to the nearest time period α Then take the minimum value Xn of distance α ,∈2～n Xn Maximum point Xm nearest to left end α-1 As the left-end continuation extreme point X 1 ′。
(3) When in use
Figure BDA0003721875450000034
When the last extreme point of the sample is the minimum point
Figure BDA0003721875450000035
Get
Figure BDA0003721875450000036
And the last maximum point
Figure BDA0003721875450000037
As a reference point, the time nodes corresponding to the two extreme points are respectively
Figure BDA0003721875450000038
And
Figure BDA0003721875450000039
by using
Figure BDA00037218754500000310
As a reference period of time, the time period,
Figure BDA00037218754500000311
firstly, the remaining minimum value point is checked
Figure BDA00037218754500000312
According to the principle of minimum difference
Figure BDA00037218754500000313
Match to find Xn -n Neutralization of
Figure BDA00037218754500000314
Equal or closest minimum value point Xn i ,i＝1～n Xn -1。
Then when tm 1 ＜tn 1 Then, these minimum value points Xn are located i Maximum point Xm adjacent to the left side thereof i Time node of (2), obtaining
Figure BDA00037218754500000315
Finally, according to the principle of minimum difference value
Figure BDA00037218754500000316
Find and reference time interval
Figure BDA00037218754500000317
Minimum value point Xn corresponding to the nearest time period α ,α∈1～n Xn 1, then take the minimum value of distance Xn α The closest maximum point Xm on the right α+1 As the continuation extreme point X of the right end point N '. When tm 1 ＞tn 1 Then, these minimum value points Xn are located i Maximum point Xm adjacent to the left side thereof i-1 Time node of (2), obtaining
Figure BDA00037218754500000318
Finally, according to the minimum principle of the difference value
Figure BDA00037218754500000319
Find and reference time interval
Figure BDA00037218754500000320
Minimum value point Xn corresponding to the nearest time period α ,α∈2～n Xn 1, then take the minimum value of distance Xn α Maximum value point Xm nearest to right end α As a right-end continuation extreme point X N ′。
(4) When the temperature is higher than the set temperature
Figure BDA00037218754500000321
Then, the last extreme point of the sample is the maximum point
Figure BDA00037218754500000322
Get
Figure BDA00037218754500000323
And the last minimum point
Figure BDA00037218754500000324
As a reference point, the time nodes corresponding to the two extreme points are respectively
Figure BDA00037218754500000325
And
Figure BDA00037218754500000326
by using
Figure BDA00037218754500000327
As a reference period of time, it is,
Figure BDA00037218754500000328
firstly, to the residual maximum value point
Figure BDA0003721875450000041
According to the principle of minimum difference
Figure BDA0003721875450000042
Matching is carried out to find Xm -n Neutralization of
Figure BDA0003721875450000043
Equal or closest maximum point Xm i ,i＝1～n Xm -1。
Then when tm 1 ＜tn 1 Then, these maximum points Xm are located i Minimum value point Xn adjacent to the left i-1 Time node of (2), obtaining
Figure BDA0003721875450000044
Finally, according to the minimum principle of the difference value
Figure BDA0003721875450000045
Find and reference time interval
Figure BDA0003721875450000046
Maximum value point Xm corresponding to the nearest time period α ,α∈2～n Xm 1, then take the maximum value Xm of distance α Minimum value point Xn nearest to right end α As extended extreme point X of right end point N '. When tm 1 ＞tn 1 Then, these maximum points Xm are located i Minimum value point Xn adjacent to the left i Time node of (2) to obtain
Figure BDA0003721875450000047
Finally, according to the minimum principle of the difference value
Figure BDA0003721875450000048
Find and reference time interval
Figure BDA0003721875450000049
Maximum value point Xm corresponding to the nearest time period α ,α∈1～n Xm -1, then take the distance maximum Xm α Minimum nearest rightPoint Xn α+1 As a right-end continuation extreme point X N ′。
According to the principle of the LMD method, only the extreme point information of the signal is used in the process of obtaining the local mean function and the envelope function, so that the purpose of extending the signal waveform can be achieved by extending the extreme points at the two ends.
Respectively estimating and obtaining the extreme points X of the continuation of the left end and the right end according to the method 1 ' and X N ' obtaining complete extreme point information n including the estimated value i (i ═ 1, 2.., N), where N is 1 ＝X 1 ′,n N ＝X N ′。
The signal decomposition process in the third step is as follows:
respectively calculating two adjacent extreme points n i And n i+1 Local mean value m between i And local amplitude a i ；
Figure BDA00037218754500000410
Figure BDA00037218754500000411
All local mean values m i And local amplitude a i Respectively hooked by line segments and smoothed to obtain a local mean function m 11 (t) and a local envelope function a 11 (t)；
m 11 (t)＝f(m i )，a 11 (t)＝f(a i )
In the formula, f (-) is a smoothing function, and a moving average method is adopted for smoothing.
Stripping the local mean function m from the original observed signal x (t) 11 (t)；
h 11 (t)＝x(t)-m 11 (t)·······················(3)
In the formula, h 11 (t) is the local mean function of the original observed signal x (t) strippingm 11 (t) after (t).
Using a local envelope function a 11 (t) to h 11 (t) performing demodulation processing:
Figure BDA0003721875450000051
in the formula, s 11 (t) is h 11 (t) the demodulated function.
Calculating function s according to the above principle 11 (t) local envelope function a 12 (t) if a 12 If (t) is 1, s can be illustrated 11 (t) is a pure frequency modulation function, otherwise the above operation steps are continued to be circulated until s 1n (t) cutoff when a pure FM signal is satisfied, i.e. 1. ltoreq. s 1n (t)≤1，a 1(n+1) (t) ≈ 1, and the specific iterative process is as follows:
Figure BDA0003721875450000052
wherein:
Figure BDA0003721875450000053
in the formula, s 1n (t) is h 1n (t) the demodulated function.
Multiplying all the obtained envelope functions to obtain an envelope signal a 1 (t)：
Figure BDA0003721875450000054
A to be obtained 1 (t) and s 1n (t) multiplying to obtain a first PF component:
PF 1 ＝a 1 (t)s 1n (t)·························(6)
mixing PF 1 (t) the component is obtained after separation from the original signal x (t)Another new signal c 1 (t) and c 1 (t) repeating the above steps, and circulating the whole process for k times until c k (t) is not more than 1 extreme point. To this end, x (t) is decomposed into k PF components and c k (t) sum of.
Figure BDA0003721875450000061
In the formula, c k (t) is a residual term.
Compared with the prior art, the invention has the beneficial effects that:
the extreme value positioning waveform continuation LMD signal decomposition method meets self estimation through the signal, fully considers the change rule in the signal and the information of each extreme value point, ensures that the estimated value conforms to the fluctuation trend of the original waveform, and has high estimation precision and strong operability.
Drawings
FIG. 1 is a simplified flow chart of an extremum locating waveform continuation method;
FIG. 2 is a flow chart of LMD signal decomposition;
FIG. 3a is a waveform diagram of a trigonometric modulation signal;
FIG. 3b is a graph of a trigonometric modulation signal spectrum;
FIG. 4a is a graph of the decomposed component one using the unmodified LMD method;
FIG. 4b is a graph of component two after decomposition using the unmodified LMD method;
FIG. 4c shows component three after decomposition using the unmodified LMD method;
FIG. 4d is a component four after decomposition using the unmodified LMD method;
FIG. 4e is the residual component after decomposition using the unmodified LMD method;
FIG. 5a is a component one after decomposition using the mirror LMD method;
FIG. 5b is a component two after decomposition using the mirror LMD method;
FIG. 5c shows component three after decomposition using the mirror LMD method;
FIG. 5d is a component four after decomposition using the mirror LMD method;
FIG. 5e is the residual component after decomposition using the mirror LMD method;
FIG. 6a is a component one after decomposition using an extremum-located waveform continuation LMD method;
FIG. 6b shows the second component after decomposition by the extremum-localized waveform continuation LMD method;
FIG. 6c is a component three after decomposition using an extremum-located waveform continuation LMD method;
FIG. 6d is a component four after decomposition using the extremum-located waveform continuation LMD method;
fig. 6e shows the residual component after decomposition using the extremum-localized waveform continuation LMD method.
Detailed Description
The invention is further described below with reference to the accompanying drawings:
first, all extreme points of the waveform are obtained
And obtaining all extreme point information of the discrete signal by a statistical means, wherein the extreme point information comprises a maximum point and a minimum point.
Second, estimate extreme value at endpoint
Fig. 1 is a simple flow chart of the extremum-locating waveform continuation method.
Given a certain discrete signal sample X (t), t is 1-N, the signal sample has a maximum value point
Figure BDA0003721875450000071
Sum minimum point
Figure BDA0003721875450000072
The time nodes corresponding to the two types of extremum points are respectively
Figure BDA0003721875450000073
And
Figure BDA0003721875450000074
by X 1 Denotes the left end point, X, of the sample N Representing the right end point of the sample.
(1) When tm 1 ＜tn 1 Then, the first extreme point of the sample is the maximum pointXm 1 . Take Xm 1 And a first minimum value point Xn 1 As reference points, the time nodes corresponding to the two extreme points are tm respectively 1 And tn 1 By using
Figure BDA0003721875450000075
As a reference period of time, the time period,
Figure BDA0003721875450000076
firstly, the residual maximum value point is aligned
Figure BDA0003721875450000077
According to the principle of minimum difference
Figure BDA0003721875450000078
Matching is carried out to find Xm -1 Neutralization Xm 1 Equal or closest maximum point Xm i ,i＝2～n Xm . These maximum points Xm are then located i Minimum value point Xn adjacent to the right side of the minimum value point i Time node of (2), obtaining
Figure BDA0003721875450000079
Finally, according to the minimum principle of the difference value
Figure BDA00037218754500000710
Find and reference time interval
Figure BDA00037218754500000711
Maximum value point Xm corresponding to the nearest time period α ,α∈2～n Xm Then take the maximum value Xm of distance α Minimum value point Xn nearest to left end α-1 As the left-end continuation extreme point X 1 ′。
(2) When tm 1 ＞tn 1 Then, the first extreme point of the sample is the minimum point Xn 1 . Taking Xn 1 And the first maximum value point Xm 1 As reference points, the time nodes corresponding to the two extreme points are tn respectively 1 And tm 1 By using
Figure BDA0003721875450000081
As a reference period of time, it is,
Figure BDA0003721875450000082
firstly, the remaining minimum value point is checked
Figure BDA0003721875450000083
According to the principle of minimum difference
Figure BDA0003721875450000084
Match to find Xn -1 Neutral and Xn 1 Equal or closest minimum value point Xn i ,i＝2～n Xn . These minimum points Xn are then located i Maximum value point Xm adjacent to the right side of the maximum value point i Time node of (2), obtaining
Figure BDA0003721875450000085
Finally, according to the principle of minimum difference value
Figure BDA0003721875450000086
Find and reference time interval
Figure BDA0003721875450000087
Minimum value point Xn corresponding to the nearest time period α Then take the minimum value Xn of distance α ,∈2～n Xn Maximum point Xm nearest to left end α-1 As the left-end continuation extreme point X 1 ′。
(3) When in use
Figure BDA0003721875450000088
Then, the last extreme point of the sample is the minimum point
Figure BDA0003721875450000089
Get
Figure BDA00037218754500000810
And the last maximum point
Figure BDA00037218754500000811
As a reference point, the time nodes corresponding to the two extreme points are respectively
Figure BDA00037218754500000812
And
Figure BDA00037218754500000813
by using
Figure BDA00037218754500000814
As a reference period of time, the time period,
Figure BDA00037218754500000815
firstly, the remaining minimum value point is checked
Figure BDA00037218754500000816
According to the principle of minimum difference
Figure BDA00037218754500000817
Match to find Xn -n Neutralization of
Figure BDA00037218754500000818
Equal or closest minimum point Xn i ,i＝1～n Xn -1。
Then when tm 1 ＜tn 1 Then, these minimum value points Xn are located i Maximum point Xm adjacent to the left side thereof i Time node of (2) to obtain
Figure BDA00037218754500000819
Finally, according to the principle of minimum difference value
Figure BDA00037218754500000820
Find and reference time interval
Figure BDA00037218754500000821
Minimum value point Xn corresponding to the nearest time period α ,α∈1～n Xn 1, then take the minimum value of distance Xn α Pole with nearest right endMaximum value point Xm α+1 As extended extreme point X of right end point N '. When tm 1 ＞tn 1 Then, these minimum value points Xn are located i Maximum point Xm adjacent to the left side thereof i-1 Time node of (2), obtaining
Figure BDA00037218754500000822
Finally, according to the minimum principle of the difference value
Figure BDA00037218754500000823
Find and reference time interval
Figure BDA00037218754500000824
Minimum value point Xn corresponding to the nearest time period α ,α∈2～n Xn 1, then take the minimum value of distance Xn α Maximum value point Xm nearest to right end α As the extended extreme point X on the right end N ′。
(4) When in use
Figure BDA0003721875450000091
Then, the last extreme point of the sample is the maximum point
Figure BDA0003721875450000092
Get
Figure BDA0003721875450000093
And the last minimum point
Figure BDA0003721875450000094
As a reference point, the time nodes corresponding to the two extreme points are respectively
Figure BDA0003721875450000095
And
Figure BDA0003721875450000096
by using
Figure BDA0003721875450000097
As a reference period of time, the time period,
Figure BDA0003721875450000098
firstly, the residual maximum value point is aligned
Figure BDA0003721875450000099
According to the principle of minimum difference
Figure BDA00037218754500000910
Matching is carried out to find Xm -n Neutralization of
Figure BDA00037218754500000911
Equal or closest maximum point Xm i ,i＝1～n Xm -1。
Then when tm 1 ＜tn 1 Then, these maximum points Xm are located i Minimum value point Xn adjacent to the left i-1 Time node of (2), obtaining
Figure BDA00037218754500000912
Finally, according to the minimum principle of the difference value
Figure BDA00037218754500000913
Find and reference time interval
Figure BDA00037218754500000914
Maximum value point Xm corresponding to the nearest time period α ,α∈2～n Xm 1, then take the maximum value Xm of distance α Minimum value point Xn nearest to right end α As extended extreme point X of right end point N '. When tm 1 ＞tn 1 Then, these maximum points Xm are located i Minimum value point Xn adjacent to the left i Time node of (2), obtaining
Figure BDA00037218754500000915
Finally, according to the minimum principle of the difference value
Figure BDA00037218754500000916
Find and reference time interval
Figure BDA00037218754500000917
Maximum value point Xm corresponding to the nearest time period α ,α∈1～n Xm -1, then take the distance maximum Xm α Minimum value point Xn nearest to right end α+1 As a right-end continuation extreme point X N ′。
According to the principle of the LMD method, only extreme point information of the signal is used in the process of obtaining the local mean function and the envelope function, so that the purpose of extending the signal waveform can be achieved by extending the extreme points at the two ends.
Respectively estimating and obtaining the extreme points X of the continuation of the left end and the right end according to the method 1 ' and X N ', obtaining complete extreme point information n including the estimated value i (i ═ 1, 2.., N), where N is 1 ＝X 1 ′,n N ＝X N ′。
According to the principle of the LMD method, only the extreme point information of the signal is used in the process of obtaining the local mean function and the envelope function, so that the purpose of extending the signal waveform can be achieved by extending the extreme points at the two ends.
Respectively estimating and obtaining the extreme points X of the continuation of the left end and the right end according to the method 1 ' and X N ', obtaining complete extreme point information n including the estimated value i (i ═ 1, 2.., N), where N is 1 ＝X 1 ′,n N ＝X N ′。
Thirdly, signal decomposition is carried out by utilizing the estimated value
After obtaining the complete extreme point, two adjacent extreme points n are calculated respectively as shown in FIG. 2 i And n i+1 Local mean value between
Figure BDA0003721875450000101
And local amplitude
Figure BDA0003721875450000102
All local mean values m i And local amplitude a i Are respectively hooked by line segments and use sliding platesSmoothing by mean square method to obtain local mean function m 11 (t) and a local envelope function a 11 (t)；
Stripping the local mean function m from the original observed signal x (t) 11 (t) obtaining h 11 (t) of (d). Using a local envelope function a 11 (t) to h 11 (t) demodulating to obtain s 11 (t) of (d). Calculating function s according to the above principle 11 (t) local envelope function a 12 (t) if a 12 If (t) is 1, s can be illustrated 11 (t) is a pure FM function, otherwise the above steps are continued to be cycled through s 1n (t) cutoff when a pure FM signal is satisfied, i.e. 1. ltoreq. s 1n (t)≤1，a 1(n+1) (t) ≈ 1, the specific iterative process is as follows:
Figure BDA0003721875450000103
multiplying all the obtained envelope functions to obtain an envelope signal
Figure BDA0003721875450000104
A to be obtained 1 (t) and s 1n (t) multiplying to obtain a first PF component PF 1 (t)。
Mixing PF 1 (t) after separation of the component from the original signal x (t), a new signal c is obtained 1 (t) and c 1 (t) repeating the above steps, and circulating the whole process for k times until c k (t) is less than or equal to 1 extreme point. To this end, x (t) is decomposed into k PF components and c k (t) sum of
Figure BDA0003721875450000105
c k (t) is a residual term.
Examples
And selecting a certain trigonometric function modulation signal as an analysis signal for verifying the effectiveness of the extremum positioning waveform continuation LMD decomposition method. The signal expression is as follows:
Figure BDA0003721875450000111
as can be seen from the expression, the signal is a multi-component signal synthesized from a plurality of sinusoidal signals, the signal including two frequencies of 3Hz, 20 Hz. The signal was discretized into 2048 points at a discretization frequency of 200 Hz. The oscillogram and the spectrogram are shown in fig. 3a and 3 b.
The signal is first decomposed using the unmodified LMD method, and the decomposition results are shown in fig. 4a, 4b, 4c, 4d, and 4 e.
As can be seen from fig. 4a, 4b, 4c, and 4d, the LMD method decomposes the target signal into 4 effective components and a decomposition margin, successfully decomposes two signal frequencies of 3Hz and 20Hz, and corresponds to fig. 4b and 4a, respectively, which illustrates that the LMD method can decompose the signal well. However, distortion phenomena are clearly seen at both ends of fig. 4b, 4c and 4d, and particularly in the PF2 component of fig. 4b, not only do the end points fluctuate, but the amplitude gradually decreases as the end points are closer to both ends. The endpoint effect of the unmodified LMD decomposition result is obvious.
In order to verify the effectiveness of the extremum locating waveform continuation LMD decomposition method and to embody the advantages of the method, the signal is decomposed and compared by using mirror image continuation and extremum locating waveform continuation, and the decomposition results are shown in fig. 5a, 5b, 5c, 5d, 5e, 6a, 6b, 6c, 6d, and 6 e.
As can be seen from fig. 5a, 5b, 5c, 5d, 5e, 6a, 6b, 6c, 6d, and 6e, both the mirror extension LMD method and the extremum-locating waveform extension LMD method can rapidly separate two frequency signals in the target signal, which indicates that both methods are effective. Comparing fig. 5b with fig. 4b, the distortion phenomena across the PF2 component in fig. 5b has been mitigated and the mirror extension has improved the endpoint effect to some extent, but the control capability is limited. Comparing fig. 6b and fig. 4b, the PF2 component in fig. 6b has almost no distortion and jitter at both ends, and has stable amplitude and frequency, and the variation trend extends to both ends according to the original waveform, so that fig. 6a, fig. 6b, fig. 6c, fig. 6d, and fig. 6e are greatly improved compared with fig. 4a, fig. 4b, fig. 4c, and fig. 4d, and a stable decomposition component is obtained.
Through the verification and comparison, compared with the traditional LMD method and the mirror image continuation LMD method, the extreme value positioning waveform continuation LMD method not only improves the distortion and the jitter phenomena, but also keeps the original trend of the signal, so the extreme value positioning waveform continuation LMD method has certain advantages in the aspect of improving the end effect.
Evaluating end-point effects
The degree of influence of the endpoint effect on the decomposition can be evaluated by comparing the change in signal energy before and after the LMD decomposition.
The decomposition result of the LMD is as follows:
Figure BDA0003721875450000121
the effective value of the signal is represented as:
Figure BDA0003721875450000122
in the formula: s (i) is the specific signal, where n is the number of sample points.
Using θ as an evaluation index of the end-point effect:
Figure BDA0003721875450000123
in the formula, R 0 Is the effective value of the original signal; r j Is the effective value of the jth PF component; r c Is the effective value of the residual component.
When θ is 0, there is no endpoint effect; the endpoint effect is present when θ > 0, and the larger θ, the more severe the effect of the endpoint effect.
The actual analysis shows that the most significant end effect is at the two ends of the PF component, so that three points at the head and the tail of the waveform are additionally taken for index analysis, and the following table 1 is obtained as a comprehensive index result:
table 1 results of the respective indices
Figure BDA0003721875450000124
As can be seen from Table 1, the extreme value positioning waveform continuation LMD method has advantages in three indexes, and the comprehensive index is optimal, so that the improvement on the end effect is most obvious.
Through the verification and comparison, compared with the traditional LMD method and the mirror image continuation LMD method, the extreme value positioning waveform continuation LMD method can effectively improve the end effect.
The above description is only for the purpose of illustrating the present invention and the appended claims are not to be construed as limiting the scope of the invention, which is intended to cover all modifications, equivalents and improvements that are within the spirit and scope of the invention as defined by the appended claims. And those not described in detail in this specification are well within the skill of the art.

Claims (3)

1. An extreme value positioning waveform continuation LMD signal decomposition method comprises the following steps:
step one, acquiring all extreme points of a waveform;
estimating an extreme value at an end point;
and thirdly, decomposing the signal by using the estimated value.
2. The extremum-locating waveform continuation LMD signal decomposition method of claim 1, wherein:
the method for estimating the extreme value at the end point in the second step comprises the following steps:
given a certain discrete signal sample X (t), t is 1-N, the signal sample has a maximum value point
Figure FDA0003721875440000011
And minimum point
Figure FDA0003721875440000012
The time nodes corresponding to the two types of extreme points are respectively
Figure FDA0003721875440000013
And
Figure FDA0003721875440000014
by X 1 Denotes the left end point, X, of the sample N Represents the right end of the sample;
(1) when tm 1 ＜tn 1 Then, the first extreme point of the sample is the maximum point Xm 1 (ii) a Take Xm 1 And a first minimum value point Xn 1 As reference points, the time nodes corresponding to the two extreme points are tm respectively 1 And tn 1 By using
Figure FDA0003721875440000015
As a reference period of time, the time period,
Figure FDA0003721875440000016
firstly, the residual maximum value point is aligned
Figure FDA0003721875440000017
According to the principle of minimum difference
Figure FDA0003721875440000018
Matching is carried out to find Xm -1 Neutralization Xm 1 Equal or closest maximum point Xm i ,i＝2～n Xm (ii) a These maximum points Xm are then located i Minimum value point Xn adjacent to the right side of the minimum value point i Time node of (2) to obtain
Figure FDA0003721875440000019
Finally, according to the minimum principle of the difference value
Figure FDA00037218754400000110
Find and reference time interval
Figure FDA00037218754400000111
Maximum value point Xm corresponding to the nearest time period α ,α∈2～n Xm Then take the maximum value Xm of distance α Minimum value point Xn nearest to left end α-1 As the left-end continuation extreme point X 1 ′；
(2) When tm 1 ＞tn 1 Then, the first extreme point of the sample is the minimum point Xn 1 (ii) a Taking Xn 1 And the first maximum value point Xm 1 As a reference point, the time nodes corresponding to the two extreme points are tn respectively 1 And tm 1 By using
Figure FDA00037218754400000112
As a reference period of time, it is,
Figure FDA00037218754400000113
firstly, to the remaining minimum value point
Figure FDA00037218754400000114
According to the principle of minimum difference
Figure FDA00037218754400000115
Match to find Xn -1 Neutral and Xn 1 Equal or closest minimum point Xn i ,i＝2～n Xn (ii) a These minimum points Xn are then located i Maximum value point Xm adjacent to the right side of the maximum value point i Time node of (2), obtaining
Figure FDA0003721875440000021
Finally, according to the principle of minimum difference value
Figure FDA0003721875440000022
Find and reference time interval
Figure FDA0003721875440000023
Minimum value point Xn corresponding to the nearest time period α Then take the minimum value Xn of distance α ,∈2～n Xn Maximum point Xm nearest to left end α-1 As the left-end continuation extreme point X 1 ′；
(3) When the temperature is higher than the set temperature
Figure FDA0003721875440000024
Then, the last extreme point of the sample is the minimum point
Figure FDA0003721875440000025
Get the
Figure FDA0003721875440000026
And the last maximum point
Figure FDA0003721875440000027
As a reference point, the time nodes corresponding to the two extreme points are respectively
Figure FDA0003721875440000028
And
Figure FDA0003721875440000029
by using
Figure FDA00037218754400000210
As a reference period of time, the time period,
Figure FDA00037218754400000211
firstly, the remaining minimum value point is checked
Figure FDA00037218754400000212
According to the principle of minimum difference
Figure FDA00037218754400000213
Match to find Xn -n Neutralization of
Figure FDA00037218754400000214
Equal or closest minimum point Xn i ,i＝1～n Xn -1；
Then when tm 1 ＜tn 1 Then, these minimum value points Xn are located i Maximum value point Xm adjacent to the left side of the maximum value point i Time node of (2) to obtain
Figure FDA00037218754400000215
Finally, according to the minimum principle of the difference value
Figure FDA00037218754400000216
Find and reference time interval
Figure FDA00037218754400000217
Minimum value point Xn corresponding to the nearest time period α ,α∈1～n Xn 1, then take the minimum value of distance Xn α The closest maximum point Xm on the right α+1 As the continuation extreme point X of the right end point N '; when tm 1 ＞tn 1 Then, these minimum value points Xn are located i Maximum value point Xm adjacent to the left side of the maximum value point i-1 Time node of (2), obtaining
Figure FDA00037218754400000218
Finally, according to the minimum principle of the difference value
Figure FDA00037218754400000219
Find and reference time interval
Figure FDA00037218754400000220
Minimum value point Xn corresponding to the nearest time period α ,α∈1～n Xn -1, then taking the minimum value Xn of distance α Maximum value point Xm nearest to right end α As a right-end continuation extreme point X N ′；
(4) When in use
Figure FDA00037218754400000221
Then, the last extreme point of the sample is the maximum point
Figure FDA00037218754400000222
Get
Figure FDA00037218754400000223
And the last minimum point
Figure FDA00037218754400000224
As a reference point, the time nodes corresponding to the two extreme points are respectively
Figure FDA00037218754400000225
And
Figure FDA0003721875440000031
by using
Figure FDA0003721875440000032
As a reference period of time, it is,
Figure FDA0003721875440000033
firstly, the residual maximum value point is aligned
Figure FDA0003721875440000034
According to the principle of minimum difference
Figure FDA0003721875440000035
Matching is carried out to find Xm -n Neutralization of
Figure FDA0003721875440000036
Equal or closest maximum point Xm i ,i＝1～n Xm -1；
Then when tm 1 ＜tn 1 Then, these maximum points Xm are located i Minimum value point Xn adjacent to the left i-1 Time node of (2), obtaining
Figure FDA0003721875440000037
Finally, according to the minimum principle of the difference value
Figure FDA0003721875440000038
Find and reference time interval
Figure FDA0003721875440000039
Maximum value point Xm corresponding to the nearest time period α ,α∈1～n Xm 1, then take the maximum value Xm of distance α Minimum value point Xn nearest to right end α As extended extreme point X of right end point N '; when tm 1 ＞tn 1 Then, these maximum points Xm are located i Minimum value point Xn adjacent to the left i Time node of (2), obtaining
Figure FDA00037218754400000310
Finally, according to the principle of minimum difference value
Figure FDA00037218754400000311
Find and reference time interval
Figure FDA00037218754400000312
Maximum value point Xm corresponding to the nearest time period α ,α∈1～n Xm 1, then take the maximum value Xm of distance α Minimum value point Xn nearest to right end α+1 As a right-end continuation extreme point X N ′；
According to the principle of the LMD method, only the extreme point information of the signal is used in the process of obtaining the local mean function and the envelope function, so that the purpose of extending the signal waveform can be achieved by extending the extreme points at the two ends;
respectively estimating and obtaining the extreme points X of the continuation of the left end and the right end according to the method 1 ' and X N ', obtaining complete poles including the estimated valuesValue point information n i (i ═ 1, 2., N) where N 1 ＝X 1 ′,n N ＝X N ′。
3. The extremum-locating waveform continuation LMD signal decomposition method of claim 1, wherein:
the signal decomposition process in the third step is as follows:
respectively calculating two adjacent extreme points n i And n i+1 Local mean value m between i And local amplitude a i ；
Figure FDA00037218754400000313
Figure FDA00037218754400000314
All local mean values m i And local amplitude a i Respectively hooked by line segments and smoothed to obtain a local mean function m 11 (t) and a local envelope function a 11 (t)；
m 11 (t)＝f(m i )，a 11 (t)＝f(a i )
In the formula, f (-) is a smoothing function, and a moving average method is adopted for smoothing;
stripping the local mean function m from the original observed signal x (t) 11 (t)；
h 11 (t)＝x(t)-m 11 (t)……………………………(3)
In the formula, h 11 (t) is the original observed signal x (t) the local mean function of the strip m 11 A function after (t);
using a local envelope function a 11 (t) to h 11 (t) performing demodulation processing:
Figure FDA0003721875440000041
in the formula, s 11 (t) is h 11 (t) a demodulated function;
calculating function s according to the above principle 11 (t) local envelope function a 12 (t) if a 12 If (t) is 1, s can be illustrated 11 (t) is a pure FM function, otherwise the above steps are continued to be cycled through s 1n (t) satisfies a pure FM signal cut-off, i.e. 1. ltoreq. s 1n (t)≤1，a 1(n+1) (t) ≈ 1, and the specific iterative process is as follows:
Figure FDA0003721875440000042
wherein:
Figure FDA0003721875440000043
in the formula, s 1n (t) is h 1n (t) a demodulated function;
multiplying all the obtained envelope functions to obtain an envelope signal a 1 (t)：
Figure FDA0003721875440000051
A to be obtained 1 (t) and s 1n (t) multiplying to obtain a first PF component:
PF 1 ＝a 1 (t)s 1n (t)………………………………(6)
mixing PF 1 (t) after separation of the component from the original signal x (t), a new signal c is obtained 1 (t) and c 1 (t) repeating the above steps, and circulating the whole process for k times until c k (t) until the number of extreme points is less than or equal to 1; to this end, x (t) is decomposed into k PF components and c k (t) sum;
Figure FDA0003721875440000052
in the formula, c k (t) is a residual term.
CN202210765149.3A 2022-06-30 2022-06-30 Extremum positioning waveform extension LMD signal decomposition method Active CN114997242B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202210765149.3A CN114997242B (en) 2022-06-30 2022-06-30 Extremum positioning waveform extension LMD signal decomposition method

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202210765149.3A CN114997242B (en) 2022-06-30 2022-06-30 Extremum positioning waveform extension LMD signal decomposition method

Publications (2)

Publication Number Publication Date
CN114997242A true CN114997242A (en) 2022-09-02
CN114997242B CN114997242B (en) 2023-08-29

Family

ID=83020020

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202210765149.3A Active CN114997242B (en) 2022-06-30 2022-06-30 Extremum positioning waveform extension LMD signal decomposition method

Country Status (1)

Country Link
CN (1) CN114997242B (en)

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115204243A (en) * 2022-09-15 2022-10-18 西南交通大学 LMD endpoint effect improvement method based on similar triangular waveform matching continuation

Citations (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6862558B2 (en) * 2001-02-14 2005-03-01 The United States Of America As Represented By The Administrator Of The National Aeronautics And Space Administration Empirical mode decomposition for analyzing acoustical signals
CN103994062A (en) * 2014-05-13 2014-08-20 山东理工大学 Hydraulic-pump fault feature signal extraction method
CN104236905A (en) * 2014-08-26 2014-12-24 中国直升机设计研究所 Bearing fault diagnosis method
CN105424057A (en) * 2015-10-30 2016-03-23 东南大学 Fiber-optic gyroscope vibration signal analysis method based on improved LMD
CN106328120A (en) * 2016-08-17 2017-01-11 重庆大学 Public place abnormal sound characteristic extraction method
CN107783200A (en) * 2017-11-21 2018-03-09 吉林大学 Joint EMD and TFPF algorithms a kind of all-wave magnetic resonance signal random noise method for reducing
CN107886111A (en) * 2017-10-25 2018-04-06 西安交通大学 A kind of modified part mean decomposition method
CN111947040A (en) * 2020-08-24 2020-11-17 重庆邮电大学 Pipeline leakage signal time delay estimation method based on mean decomposition and signal definition

Patent Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6862558B2 (en) * 2001-02-14 2005-03-01 The United States Of America As Represented By The Administrator Of The National Aeronautics And Space Administration Empirical mode decomposition for analyzing acoustical signals
CN103994062A (en) * 2014-05-13 2014-08-20 山东理工大学 Hydraulic-pump fault feature signal extraction method
CN104236905A (en) * 2014-08-26 2014-12-24 中国直升机设计研究所 Bearing fault diagnosis method
CN105424057A (en) * 2015-10-30 2016-03-23 东南大学 Fiber-optic gyroscope vibration signal analysis method based on improved LMD
CN106328120A (en) * 2016-08-17 2017-01-11 重庆大学 Public place abnormal sound characteristic extraction method
US20190120995A1 (en) * 2017-10-20 2019-04-25 Jilin University Method for random noise reduction from mrs oscillating signal using joint algorithms of emd and tfpf
CN107886111A (en) * 2017-10-25 2018-04-06 西安交通大学 A kind of modified part mean decomposition method
CN107783200A (en) * 2017-11-21 2018-03-09 吉林大学 Joint EMD and TFPF algorithms a kind of all-wave magnetic resonance signal random noise method for reducing
CN111947040A (en) * 2020-08-24 2020-11-17 重庆邮电大学 Pipeline leakage signal time delay estimation method based on mean decomposition and signal definition

Non-Patent Citations (2)

* Cited by examiner, † Cited by third party
Title
JONATHAN S. SMITH: "The local mean decomposition and its application to EEG perception data", 《INTERFACE》, pages 443 - 454 *
魏永合;牛保国;刘雪丽;赵旭宁;: "波形曲率延拓在局域均值分解中的应用", 中国机械工程, vol. 27, no. 03, pages 343 - 354 *

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115204243A (en) * 2022-09-15 2022-10-18 西南交通大学 LMD endpoint effect improvement method based on similar triangular waveform matching continuation
CN115204243B (en) * 2022-09-15 2023-02-07 西南交通大学 LMD endpoint effect improvement method based on similar triangular waveform matching continuation

Also Published As

Publication number Publication date
CN114997242B (en) 2023-08-29

Similar Documents

Publication Publication Date Title
Moussaoui et al. Separation of non-negative mixture of non-negative sources using a Bayesian approach and MCMC sampling
CN108804731B (en) Time series trend feature extraction method based on important point dual evaluation factors
CN111445108B (en) Data-driven power distribution network line variation relation diagnosis method, device and system
CN109270345A (en) A kind of detection method of mains by harmonics signal
CN111934711B (en) Parameter estimation method of time-frequency aliasing frequency hopping signal
CN111125885A (en) ASF correction table construction method based on improved kriging interpolation algorithm
CN114997242B (en) Extremum positioning waveform extension LMD signal decomposition method
CN112232011B (en) Wide-frequency-band electromagnetic response self-adaptive determination method and system of integrated circuit
CN109444844A (en) A kind of method and apparatus for extracting target scattering center feature
Aguirre et al. Nonlinear identification and cluster analysis of chaotic attractors from a real implementation of Chua's circuit
Djurovic Estimation of the sinusoidal signal frequency based on the marginal median DFT
Luo et al. Entropy-based Training Methods for Scalable Neural Implicit Samplers
CN112200000A (en) Welding stability recognition model training method and welding stability recognition method
CN113095353B (en) Underdetermined blind source separation method based on AP clustering
CN105978733A (en) Network flow modelling method and system based on Weibull distribution
DE112014006345T5 (en) System identification device
CN111611686A (en) Detection method for communication signal time-frequency domain
Blumensath et al. Monte Carlo methods for adaptive sparse approximations of time-series
CN115310478A (en) Method for extracting effective Raman spectrum of natural gas based on wavelet optimal decomposition layer number
Hanke An ϵ-free a posteriori stopping rule for certain iterative regularization methods
Kronvall et al. Multi-pitch estimation via fast group sparse learning
Wang et al. Blind source separation based on variational Bayesian independent component analysis
Baka et al. Confidence of model based shape reconstruction from sparse data
Barber On solving ordinary differential equations using Gaussian processes
Troudi et al. The generalised Plug-in algorithm for the diffeomorphism kernel estimate

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant