EMD empirical mode decomposition

The original text comes from here

https://zhuanlan.zhihu.com/p/40005057
https://zhuanlan.zhihu.com/p/121803211

1. EMD Cause of

In terms of signal processing, we know the time-domain processing method （ Such as effective value 、 Kurtosis ）、 Frequency domain processing method （ Such as spectrum 、 Power spectrum ） And some time-frequency domain processing methods （ Such as wavelet analysis ）（ See the column about signal processing for details ）. Time domain and frequency domain have various advantages and scope of application , Not much ,EMD（Empirical Mode Decomposition） As a time-frequency domain processing method , What are the advantages of wavelet analysis compared with the same time-frequency domain method ？

EMD The most striking feature , It overcomes the problem that the basis function has no self adaptability . What do you mean ？ Recall the content of wavelet analysis , We will know that wavelet analysis needs to select a wavelet base , The selection of wavelet base has a great influence on the results of wavelet analysis , Once the wavelet base is determined , Cannot be replaced throughout the analysis , Even if the wavelet base may be optimal in the whole world , But in some parts, it may not be , Therefore, the basis function of wavelet analysis is lack of adaptability .

Popular said , use EMD What are the benefits ？ For an unknown signal , There is no need to do pre analysis and Research , You can just start decomposing . This method will automatically divide them into layers according to some solid patterns , Without human setting and intervention .

A little more general ,EMD Like a machine , Throw in a pile of coins mixed together , He will automatically follow 1 element 、5 hair 、1 hair 、5 branch 、1 Divide into several parts .

2. Components of each eigenfunction

Culvert modal component （Intrinsic Mode Functions, IMF） The original signal is EMD The signal components of each layer obtained after decomposition .EMD Huang e, the originator of , Any signal can be divided into the sum of several connotative modal components . The connotative modal component has two constraints ：

1） Within the entire data segment , The number of extreme points and the number of zero crossing points must be equal or differ by no more than one .

2） At any moment , The average value of the upper envelope formed by the local maximum point and the lower envelope formed by the local minimum point is zero , That is to say 、 The lower envelope is locally symmetric with respect to the time axis .

What do you mean by that? ？

Use loose language and soul painter to explain ：

1. The lines should be crossed repeatedly x Axis , like this ：

Within the entire data segment , The number of extreme points and the number of zero crossing points must be equal or differ by no more than one

Instead, multiple poles cannot appear after crossing the zero point ：

2. The envelope should be symmetrical , like this ：
   The envelope is symmetrical

Not like this ：

Take a normal example ：

Above by 7 It's made up of pictures , Among them the first 1 Zhang is the original signal , The following are EMD After decomposition 6 Weight , named IMF1~IMF5, The last picture shows the residuals , every last IMF Component represents a kind of connotative modal component in the original signal . It can be seen that , Every IMF The components satisfy these two constraints .

3. EMD Decomposition steps

EMD The decomposition process of is simple and intuitive ：

1） According to the upper and lower extreme points of the original signal , Draw the upper 、 The lower envelope .
On 、 The lower envelope

2） Please 、 The mean value of the lower envelope , Draw the mean envelope .
Mean envelope

3） Original signal minus mean envelope , Get an intermediate signal .

Original signal minus mean envelope

4） Judge whether the intermediate signal meets IMF Two conditions of , If meet , The signal is a IMF component ; If not , Based on this signal , To do 1）~4） Analysis of .IMF The acquisition of components usually requires several iterations .

Constraint not satisfied 2, Need to continue iterating

Use the above method to get the first IMF after , Subtract from the original signal IMF1, As a new raw signal , Re pass 1）~4） Analysis of , You can get IMF2, And so on , complete EMD decompose .

The figure in the above example is from http://perso.ens-lyon.fr/patrick.flandrin/emd.ppt

This ppt It uses dozens of pages of pictures to show EMD The process of decomposition , Look at this document and I believe you will have a clearer understanding of the process .