CN114997242B - Extremum positioning waveform extension LMD signal decomposition method - Google Patents
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Abstract
The invention belongs to the technical field of signal processing, and relates to an extremum positioning waveform extension LMD signal decomposition method, which specifically comprises the following steps: 1. acquiring all extreme points of the waveform; 2. estimating an extremum at the endpoint; 3. performing signal decomposition by using the estimated value; the method meets self estimation by the signal itself, fully considers the change rule in the signal and the information of each extreme point, and ensures that the estimated value accords with the fluctuation trend of the original waveform, and has higher estimation precision and stronger operability.
Description
Technical Field
The invention belongs to the technical field of signal processing, and relates to an extremum positioning waveform extension LMD signal decomposition method, in particular to an extremum point information acquisition method, an extremum point estimation method and signal decomposition method.
Technical Field
The LMD (local mean decomposition) method has self-adaption in analyzing the nonstationary signals, can completely retain time-frequency information of the original signals, and is more accurate and effective in analyzing the signals of mechanical equipment compared with other similar methods. However, the end effect of LMD can negatively affect the LMD decomposition process, and to improve the end effect, the magnitude of the extreme values at both ends should be estimated. Waveform extension is a relatively efficient endpoint extremum prediction method at present. The main idea is to simulate a new small section of waveform at two ends of the signal to generate two simulated extreme points, and to use the two extreme points as extreme points to perform the next operation. The existing waveform extension method comprises image extension, neural network prediction extension, four-point waveform extension based on matching error, inner product extension and the like.
The image continuation realizes the waveform continuation by copying the original waveform to two ends through an image copying method, and the method does not consider the change rule in the signal and does not improve the end effect basically too much; the neural network prediction continuation considers the internal change rule of the signals through a neural network algorithm, but the prediction method has long calculation time, and is not suitable for practical application because parameter adjustment is performed according to different signals; compared with a neural network, the four-point waveform extension method and the inner product extension method based on the matching error are easier, the change rule of the whole signal is considered, the precision is higher, but the number of the selected nodes is more, and the matching error of the waveform is larger.
Disclosure of Invention
The end effect of the conventional LMD method can affect the stability and accuracy of signal decomposition, and further affect other analysis results based on the end effect. The invention provides an extreme value positioning waveform extension LMD method, which is characterized in that the change rule and the extreme value interval rule of extreme points are determined to position target extreme points through analyzing the whole waveform, and the extreme value extension scheme of the left and right extreme points is used as the extreme value points to restrain the end effect.
In order to solve the technical problems, the invention is realized by adopting the following technical scheme, and the technical scheme is as follows in combination with the accompanying drawings:
an extremum locating waveform continuation LMD signal decomposition method comprises the following steps:
step one, acquiring all extreme points of a waveform;
estimating extreme values at the end points;
and thirdly, performing signal decomposition 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 extremum estimating method in the second step is as follows:
given a certain discrete signal sample X (t), t=1 to N, the signal sample has a maximum pointAnd minimum point->The time nodes corresponding to the two extreme points are +.>And->By X 1 Representing the left end point, X of the sample N Representing the right end point of the sample.
(1) When tm 1 ＜tn 1 When the first extreme point of the sample is the maximum point Xm 1 . Taking Xm 1 And a first minimum point Xn 1 The time nodes corresponding to the two extreme points are tm respectively 1 And tn 1 By usingAs a reference period of time,first for the remaining maximum point +.>According to the principle of minimum differenceMatching to find Xm -1 Intermediate and Xm 1 Equal or closest maximum point Xm i ,i＝2～n Xm . These maxima points Xm are then located i Minimum value point Xn adjacent to right thereof i Time node of->Finally, according to the principle of minimum difference->Find and reference time periodMaximum point Xm corresponding to the closest time period α ,α∈2～n Xm Then take the distance maximum Xm 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 At the time, the first extreme point of the sample is the minimum point Xn 1 . Taking Xn 1 And a first maximum point Xm 1 The time nodes corresponding to the two extreme points are respectively tn as the reference points 1 And tm 1 By usingAs a reference period of time,first for the remaining minima point->According to the principle of minimum differenceMatching to find Xn -1 Intermediate and Xn 1 Equal or closest minimum point Xn i ,i＝2～n Xn . These minima points Xn are then located i Maximum point Xm adjacent to right thereof i Time node of (2), calculateFinally, according to the principle of minimum difference->Find +.>Minimum value point Xn corresponding to nearest time period α Then take the minimum value Xn α ,∈2～n Xn The nearest maximum point Xm at the left end α-1 As the left end continuation extreme point X 1 ′。
(3) When (when)At the time, the last extreme point of the sample is the minimum point +.>Get->And the last maximum point->The time nodes corresponding to the two extreme points are respectively +.>And->Use->As reference period->
First to the residual minimum pointAccording to the principle of minimum differenceMatching to find Xn -n Middle and->Equal or closest minimum point Xn i ,i＝1～n Xn -1。
Then when tm 1 ＜tn 1 Locating these minima points Xn i The maximum value point Xm adjacent to the left thereof i Time node of (2), calculateFinally, according to the principle of minimum difference->Find and reference time periodMinimum value point Xn corresponding to nearest time period α ,α∈1～n Xn -1, taking the minimum value Xn α Right nearest maximum point Xm α+1 Extension of extreme point X as right end point N '. When tm 1 ＞tn 1 Locating these minima points Xn i The maximum value point Xm adjacent to the left thereof i-1 Time node of->Finally, according to the principle of minimum difference->Find +.>Minimum value point Xn corresponding to nearest time period α ,α∈2～n Xn -1, taking the minimum value Xn α Right nearest maximum point Xm α As the right end continuation extreme point X N ′。
(4) When (when)At the time, the last extreme point of the sample is the maximum point +.>Get->And the last minimum point->The time nodes corresponding to the two extreme points are respectively +.>And->Use->As reference period->
First for the residual maximum pointAccording to the principle of minimum differenceMatching to find Xm -n Middle and->Equal or closest maximum point Xm i ,i＝1～n Xm -1。
Then when tm 1 ＜tn 1 Locating these maxima points Xm i Minimum value point Xn adjacent to left thereof i-1 Time node of (2), calculateFinally, according to the principle of minimum difference->Find +.>Maximum point Xm corresponding to the closest time period α ,α∈2～n Xm -1, then take the distance maxima Xm α Minimum value point Xn nearest to right end α Extension of extreme point X as right end point N '. When tm 1 ＞tn 1 Locating these maxima points Xm i Minimum value point Xn adjacent to left thereof i Time node of->Finally according to the principle of minimum differenceFind +.>Maximum point Xm corresponding to the closest time period α ,α∈1～n Xm -1, then take the distance maxima Xm α Minimum value point Xn nearest to right end α+1 As the 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 signal waveform extension can be achieved by extending the extreme points at two ends.
Respectively estimating and obtaining extreme points X extended by the left end and the right end according to the method 1 ' and X N ' get bagComplete extremum point information n including estimated value i (i=1, 2,) N, where N 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 ；
Average all local areas m i And local amplitude a i Respectively hooking up by line segments, and smoothing to obtain a local mean function m 11 (t) and local envelope function a 11 (t)；
m 11 (t)＝f(m i )，a 11 (t)＝f(a i )
Wherein f (·) is a smoothing function, and smoothing is performed by a sliding average method.
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) stripping local mean function m 11 A function after (t).
Using a local envelope function a 11 (t) vs. h 11 (t) performing demodulation processing:
wherein s is 11 (t) is h 11 (t) a demodulated function.
Calculating the function s according to the above principle 11 Local envelope function a of (t) 12 (t) if a 12 (t) =1, s can be described as 11 (t) is a pure frequency modulation function, otherwise the above operation steps continue to be cycled until s 1n (t) cut-off when a pure FM signal is satisfied, i.e. 1.ltoreq.s 1n (t)≤1，a 1(n+1) (t) ≡1, the specific iterative procedure is as follows:
wherein:
wherein s is 1n (t) is h 1n (t) a demodulated function.
Multiplying all obtained envelope functions to obtain an envelope signal a 1 (t)：
The obtained a 1 (t) and s 1n (t) multiplying to obtain a first PF component:
PF 1 ＝a 1 (t)s 1n (t)·············(6)
PF is set to 1 After the component (t) is separated from the original signal x (t), a further new signal c is obtained 1 (t), and c 1 (t) repeating the above steps for k times in the whole process until c k The extreme points of (t) are less than or equal to 1. Thus far, x (t) is decomposed into k PF components and c k And (t).
Wherein, c k (t) isResidual items.
Compared with the prior art, the invention has the beneficial effects that:
the extremum positioning waveform extension LMD signal decomposition method provided by the invention meets self estimation through the signal, fully considers the change rule in the signal and the information of each extremum point, and has the advantages that the estimated value accords with the fluctuation trend of the original waveform, the estimation precision is higher, and the operability is stronger.
Drawings
FIG. 1 is a simplified flow chart of an extremum locating waveform continuation method;
FIG. 2 is an LMD signal decomposition flow chart;
FIG. 3a is a waveform diagram of a trigonometric function modulated signal;
FIG. 3b is a graph of a spectrum of a trigonometric modulated signal;
FIG. 4a is a component one after decomposition using the unmodified LMD method;
FIG. 4b is a component two after decomposition using the unmodified LMD method;
FIG. 4c is a 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 a residual component after decomposition using the unmodified LMD method;
FIG. 5a is a component one after decomposition using the mirrored LMD method;
FIG. 5b is a component two after decomposition using the mirrored LMD method;
FIG. 5c is a component three after decomposition using the mirrored LMD method;
FIG. 5d is a component four after decomposition using the mirrored LMD method;
FIG. 5e is a residual component after decomposition using the mirrored LMD method;
FIG. 6a is a component one after decomposition using the extremum located waveform extension LMD method;
FIG. 6b is a component two after decomposition using the extremum located waveform extension LMD method;
FIG. 6c is a decomposed component three using the extremum located waveform extension LMD method;
FIG. 6d is a decomposed component four using the extremum located waveform extension LMD method;
fig. 6e is a decomposed residual component using the extremum located waveform extension LMD method.
Detailed Description
The invention is further described below with reference to the accompanying drawings:
1. acquiring all extreme points of a waveform
All extreme point information of the discrete signals is obtained through a statistical means, wherein the extreme point information comprises a maximum point and a minimum point.
2. Estimating extreme values at end points
Fig. 1 is a simple flow chart of the extremum locating waveform extension method.
Given a certain discrete signal sample X (t), t=1 to N, the signal sample has a maximum pointAnd minimum point->The time nodes corresponding to the two extreme points are +.>And->By X 1 Representing the left end point, X of the sample N Representing the right end point of the sample.
(1) When tm 1 ＜tn 1 When the first extreme point of the sample is the maximum point Xm 1 . Taking Xm 1 And a first minimum point Xn 1 The time nodes corresponding to the two extreme points are tm respectively 1 And tn 1 By usingAs a reference period of time,first for the remaining maximum point +.>According to the principle of minimum differenceMatching to find Xm -1 Intermediate and Xm 1 Equal or closest maximum point Xm i ,i＝2～n Xm . These maxima points Xm are then located i Minimum value point Xn adjacent to right thereof i Time node of->Finally, according to the principle of minimum difference->Find and reference time periodMaximum point Xm corresponding to the closest time period α ,α∈2～n Xm Then take the distance maximum Xm 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 At the time, the first extreme point of the sample is the minimum point Xn 1 . Taking Xn 1 And a first maximum point Xm 1 The time nodes corresponding to the two extreme points are respectively tn as the reference points 1 And tm 1 By usingAs a reference period of time,first for the remaining minima point->According to the principle of minimum differenceMatching to find Xn -1 Intermediate and Xn 1 Equal or closest minimum point Xn i ,i＝2～n Xn . These minima points Xn are then located i Maximum point Xm adjacent to right thereof i Time node of (2), calculateFinally, according to the principle of minimum difference->Find +.>Minimum value point Xn corresponding to nearest time period α Then take the minimum value Xn α ,∈2～n Xn The nearest maximum point Xm at the left end α-1 As the left end continuation extreme point X 1 ′。
(3) When (when)At the time, the last extreme point of the sample is the minimum point +.>Get->And the last maximum point->The time nodes corresponding to the two extreme points are respectively +.>And->Use->As reference period->
First to the residual minimum pointAccording to the principle of minimum differenceMatching to find Xn -n Middle and->Equal or closest minimum point Xn i ,i＝1～n Xn -1。
Then when tm 1 ＜tn 1 Locating these minima points Xn i The maximum value point Xm adjacent to the left thereof i Time node of (2), calculateFinally, according to the principle of minimum difference->Find and reference time periodMinimum value point Xn corresponding to nearest time period α ,α∈1～n Xn -1, taking the minimum value Xn α Right nearest maximum point Xm α+1 Extension of extreme point X as right end point N '. When tm 1 ＞tn 1 Locating these minima points Xn i The maximum value point Xm adjacent to the left thereof i-1 Time node of->Finally, according to the principle of minimum difference->Find +.>Minimum value point Xn corresponding to nearest time period α ,α∈2～n Xn -1, taking the minimum value Xn α Right nearest maximum point Xm α As the right end continuation extreme point X N ′。
(4) When (when)At the time, the last extreme point of the sample is the maximum point +.>Get->And the last minimum point->The time nodes corresponding to the two extreme points are respectively +.>And->Use->As reference period->
First for the residual maximum pointAccording to the principle of minimum differenceMatching to find Xm -n Middle and->Equal or closest maximum point Xm i ,i＝1～n Xm -1。
Then when tm 1 ＜tn 1 Locating these maxima points Xm i Minimum value point Xn adjacent to left thereof i-1 Time node of (2), calculateFinally, according to the principle of minimum difference->Find +.>Maximum point Xm corresponding to the closest time period α ,α∈2～n Xm -1, then take the distance maxima Xm α Minimum value point Xn nearest to right end α Extension of extreme point X as right end point N '. When tm 1 ＞tn 1 Locating these maxima points Xm i Minimum value point Xn adjacent to left thereof i Time node of->Finally according to the principle of minimum differenceFind +.>Maximum point Xm corresponding to the closest time period α ,α∈1～n Xm -1, then take the distance maxima Xm α Minimum value point Xn nearest to right end α+1 As the 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 signal waveform extension can be achieved by extending the extreme points at two ends.
Respectively estimating and obtaining extreme points X extended by 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 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 signal waveform extension can be achieved by extending the extreme points at two ends.
Respectively estimating and obtaining extreme points X extended by 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 1 ＝X 1 ′,n N ＝X N ′。
3. Signal decomposition using estimated values
As shown in fig. 2, after obtaining the complete extreme points, two adjacent extreme points n are calculated respectively i And n i+1 Local mean value betweenAnd local amplitude +.>Average all local areas m i And local amplitude a i Respectively hooking up by line segments, and smoothing by using a moving average method to obtain a local mean function m 11 (t) and 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). Using a local envelope function a 11 (t) vs. h 11 (t) performing demodulation processing to obtainTo s 11 (t). Calculating the function s according to the above principle 11 Local envelope function a of (t) 12 (t) if a 12 (t) =1, s can be described as 11 (t) is a pure frequency modulation function, otherwise the above operation steps continue to be cycled until s 1n (t) cut-off when a pure FM signal is satisfied, i.e. 1.ltoreq.s 1n (t)≤1，a 1(n+1) (t) ≡1, the specific iterative procedure is as follows:
multiplying all obtained envelope functions to obtain an envelope signalThe obtained a 1 (t) and s 1n (t) multiplying to obtain a first PF component PF 1 (t)。
PF is set to 1 After the component (t) is separated from the original signal x (t), a further new signal c is obtained 1 (t), and c 1 (t) repeating the above steps for k times in the whole process until c k The extreme points of (t) are less than or equal to 1. Thus far, x (t) is decomposed into k PF components and c k Sum of (t)c k And (t) is a residual term.
Examples
And selecting a certain trigonometric function modulation signal as an analysis signal to verify the effectiveness of the extreme value positioning waveform extension LMD decomposition method. The signal expression is as follows:
as can be seen from the expression, the signal is a multi-component signal synthesized from a plurality of sinusoidal signals, the signal comprising two frequencies of 3Hz, 20 Hz. The signal was discretized into 2048 points at a discretized frequency of 200Hz. The waveform diagram and the spectrogram are shown in fig. 3a and 3 b.
The signal is first decomposed using the unmodified LMD method, the decomposition results being shown in fig. 4a, 4b, 4c, 4d, 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, and two signal frequencies of 3Hz and 20Hz are successfully decomposed, which correspond to fig. 4b and 4a respectively, so that the LMD method can decompose the signal better. However, distortion phenomena are evident at both ends of fig. 4b, 4c, and 4d, and particularly in the PF2 component of fig. 4b, not only the end points fluctuate, but also the amplitude gradually decreases toward both ends. The end effect of the unmodified LMD decomposition results is seen to be more pronounced.
In order to verify the effectiveness of the LMD decomposition method, and to embody the advantages of the method, the signals are decomposed and compared by using the image continuation and the extremum positioning waveform continuation, and the decomposition results are shown in fig. 5a, 5b, 5c, 5d, 5e, 6a, 6b, 6c, 6d and 6e.
It can be seen from fig. 5a, 5b, 5c, 5d, 5e and 6a, 6b, 6c, 6d, and 6e that both the image extension LMD method and the extremum located waveform extension LMD method can quickly separate two frequency signals in the target signal, indicating that both methods are effective. In contrast to fig. 5b and fig. 4b, the distortion phenomena across the PF2 component in fig. 5b have been alleviated, and the image continuation has improved the end point effect to some extent, but the control capability is limited. Comparing fig. 6b with fig. 4b, in fig. 6b, almost no distortion and jitter phenomenon occur at both ends of the PF2 component, and the amplitude and frequency are stable, and the variation trend extends toward both ends according to the original waveform, so that it can be seen that fig. 6a, fig. 6b, fig. 6c, fig. 6d, fig. 6e are greatly improved compared with fig. 4a, fig. 4b, fig. 4c, fig. 4d, and stable decomposition components are obtained.
Through the verification and comparison, compared with the traditional LMD method and the image continuation LMD method, the extremum positioning waveform continuation LMD method not only improves distortion and jitter phenomena, but also maintains the original trend of signals, so that the extremum positioning waveform continuation LMD method has certain advantages in the aspect of improving end point effects.
Assessment of end point effects
The extent of influence of the end point effect on the decomposition can be assessed by comparing the change in signal energy before and after the LMD decomposition.
The decomposition result of LMD is:
the effective value of the signal is expressed as:
wherein: s (i) is a specific signal, where n is the number of sampling points.
θ was used as an evaluation index of the end effect:
wherein R is 0 Is the effective value of the original signal; r is R j Is the effective value of the jth PF component; r is R c Is the effective value of the residual component.
When θ=0, there is no end effect influence; there is an end effect when θ > 0, and the larger θ, the more serious the influence of the end effect.
According to the actual analysis situation, the position with the most obvious influence of the end effect is positioned at 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 comprehensive index result is obtained as shown in the following table 1:
table 1 results of the respective indicators
As can be seen from Table 1, the extremum locating waveform extension LMD method has advantages in terms of three indexes, and the comprehensive index is optimal, so that the improvement of the end effect is most obvious.
Through the verification and comparison, compared with the traditional LMD method and the mirror image extension LMD method, the extreme value positioning waveform extension LMD method can effectively improve the end effect.
The foregoing is merely illustrative of specific embodiments of the present invention, and the scope of the invention is not limited thereto, but any modifications, equivalents, improvements and alternatives falling within the spirit and principles of the present invention will be apparent to those skilled in the art within the scope of the present invention. And all that is not described in detail in this specification is well known to those skilled in the art.
Claims (2)
1. An extremum locating waveform continuation LMD signal decomposition method comprises the following steps:
step one, acquiring all extreme points of a waveform;
estimating extreme values at the end points;
step three, performing signal decomposition by using the estimated value;
the extremum estimating method in the second step is as follows:
given a certain discrete signal sample X (t), t=1 to N, the signal sample has a maximum pointAnd minimum point->The time nodes corresponding to the two extreme points are +.>And->By X 1 Representing the left of the sampleEndpoint, X N Representing the right end point of the sample;
(1) When tm 1 ＜tn 1 When the first extreme point of the sample is the maximum point Xm 1 The method comprises the steps of carrying out a first treatment on the surface of the Taking Xm 1 And a first minimum point Xn 1 The time nodes corresponding to the two extreme points are tm respectively 1 And tn 1 By usingAs a reference period of time,first for the remaining maximum point +.>According to the principle of minimum differenceMatching to find Xm -1 Intermediate and Xm 1 Equal or closest maximum point Xm i ,i＝2～n Xm The method comprises the steps of carrying out a first treatment on the surface of the These maxima points Xm are then located i Minimum value point Xn adjacent to right thereof i Time node of->Finally, according to the principle of minimum difference->Find and reference time periodMaximum point Xm corresponding to the closest time period α ,α∈2～n Xm Then take the distance maximum Xm 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 At the time, the first sampleThe extreme point is the minimum point Xn 1 The method comprises the steps of carrying out a first treatment on the surface of the Taking Xn 1 And a first maximum point Xm 1 The time nodes corresponding to the two extreme points are respectively tn as the reference points 1 And tm 1 By usingAs a reference period of time,first for the remaining minima point->According to the principle of minimum differenceMatching to find Xn -1 Intermediate and Xn 1 Equal or closest minimum point Xn i ,i＝2～n Xn The method comprises the steps of carrying out a first treatment on the surface of the These minima points Xn are then located i Maximum point Xm adjacent to right thereof i Time node of->Finally, according to the principle of minimum difference->Find +.>Minimum value point Xn corresponding to nearest time period α Then take the minimum value Xn α ,∈2～n Xn The nearest maximum point Xm at the left end α-1 As the left end continuation extreme point X 1 ′；
(3) When (when)The last extreme point of the sample is the poleSmall value dot +.>Get->And the last maximum point->The time nodes corresponding to the two extreme points are respectively +.>And->Use->As reference period->
First to the residual minimum pointAccording to the principle of minimum differenceMatching to find Xn n- Middle and->Equal or closest minimum point Xn i ,i＝1～n Xn -1；
Then when tm 1 ＜tn 1 Locating these minima points Xn i The maximum value point Xm adjacent to the left thereof i Time node of (2), calculateFinally, according to the principle of minimum difference->Find +.>Minimum value point Xn corresponding to nearest time period α ,α∈1～n Xn -1, taking the minimum value Xn α Right nearest maximum point Xm α+1 Extension of extreme point X as right end point N 'A'; when tm 1 ＞tn 1 Locating these minima points Xn i The maximum value point Xm adjacent to the left thereof i-1 Time node of->Finally according to the principle of minimum differenceFind +.>Minimum value point Xn corresponding to nearest time period α ,α∈1～n Xn -1, taking the minimum value Xn α Right nearest maximum point Xm α As the right end continuation extreme point X N ′；
(4) When (when)At the time, the last extreme point of the sample is the maximum point +.>Get->And the last minimum point->The time nodes corresponding to the two extreme points are respectively +.>And->Use->As reference period->
First for the residual maximum pointAccording to the principle of minimum differenceMatching to find Xm -n Middle and->Equal or closest maximum point Xm i ,i＝1～n Xm -1；
Then when tm 1 ＜tn 1 Locating these maxima points Xm i Minimum value point Xn adjacent to left thereof i-1 Time node of (2), calculateFinally, according to the principle of minimum difference->Find +.>Maximum point Xm corresponding to the closest time period α ,α∈1～n Xm -1, then take the distance maxima Xm α Minimum value point Xn nearest to right end α Extension of extreme point X as right end point N 'A'; when tm 1 ＞tn 1 Locating these maxima points Xm i Minimum value point Xn adjacent to left thereof i Time node of->Finally according to the principle of minimum differenceFind +.>Maximum point Xm corresponding to the closest time period α ,α∈1～n Xm -1, then take the distance maxima Xm α Minimum value point Xn nearest to right end α+1 As the 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 value function and the envelope function, so that the purpose of signal waveform extension can be achieved by extending the extreme points at two ends;
respectively estimating and obtaining extreme points X extended by 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 1 ＝X 1 ′,n N ＝X N ′。
2. The extremum locating waveform extension 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 ；
Average all local areas m i And local amplitude a i Respectively hooking up by line segments, and smoothing to obtain a local mean function m 11 (t) and local envelope function a 11 (t)；
m 11 (t)＝f(m i )，a 11 (t)＝f(a i )
Wherein, f (·) is a smoothing function, and smoothing is performed by a sliding average method;
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) stripping local mean function m 11 A function after (t);
using a local envelope function a 11 (t) vs. h 11 (t) performing demodulation processing:
wherein s is 11 (t) is h 11 (t) a demodulated function;
calculating the function s according to the above principle 11 Local envelope function a of (t) 12 (t) if a 12 (t) =1, s can be described as 11 (t) is a pure frequency modulation function, noThe above operation steps are continued to be circulated until s 1n (t) cut-off when a pure FM signal is satisfied, i.e. 1.ltoreq.s 1n (t)≤1，a 1(n+1) (t) ≡1, the specific iterative procedure is as follows:
wherein:
wherein s is 1n (t) is h 1n (t) a demodulated function;
multiplying all obtained envelope functions to obtain an envelope signal a 1 (t)：
The obtained a 1 (t) and s 1n (t) multiplying to obtain a first PF component:
PF 1 ＝a 1 (t)s 1n (t)··············(6)
PF is set to 1 After the component (t) is separated from the original signal x (t), a further new signal c is obtained 1 (t), and c 1 (t) repeating the above steps for k times in the whole process until c k The extreme points of (t) are less than or equal to 1; thus far, x (t) is decomposed into k PF components and c k The sum of (t);
wherein, c k And (t) is a residual term.
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