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MATLAB

HHT变换

代码+例子

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CYP—test.zip （71个子文件）

CYP—test

sample_liupeng

plot_hht.m 1KB

var3.txt 10.77MB

emd.m 798B

plot_fft.m 315B

sf.jpg 14KB

s.jpg 16KB

a.jpg 42KB

af.jpg 14KB

vf.jpg 15KB

v.jpg 34KB

var2_1.txt 516KB

isimf.m 302B

var1_2.txt 7KB

原始信号与去除直流分量原始信号.fig 29KB

fs1000-fc100.h 1KB

fft.fig 19KB

ismonotonic.m 269B

integral.m 536B

var4.txt 281KB

HHT.pdf 936KB

data.dat 147KB

var1_1.txt 171KB

getspline.m 149B

var2_2.txt 7KB

Hum.wav 100KB

findpeaks.m 262B

原始信号.fig 24KB

var5.txt 293KB

var6.txt 10.57MB

license.txt 1KB

sample1

plot_hht.m 2KB

emd.m 914B

plot_hht1.m 3KB

isimf.m 292B

FFTAnalysis.m 286B

high_pass_filtertest.m 1KB

ismonotonic.m 174B

hhspectrum.m 2KB

HilbertAnalysis.m 419B

getspline.m 156B

fftfenxi.m 743B

findpeaks.m 913B

test.m 1KB

plot_imf.m 635B

sample3

8hz.mat 320KB

T9.mat 102KB

plot_hht.m 939B

emd.m 22KB

emd1.m 914B

plot_hht1.m 3KB

T4.mat 32KB

test1.m 3KB

isimf.m 292B

disp_hhs.m 2KB

FFTAnalysis.m 286B

toimage.m 3KB

ismonotonic.m 174B

T5.mat 18KB

hhspectrum.m 2KB

HilbertAnalysis.m 433B

getspline.m 156B

T6.mat 14KB

findpeaks.m 913B

test.m 937B

plot_imf.m 635B

T7.mat 6KB

T8.mat 11KB

h_plot.m 1KB

1.txt 49KB

var6.txt 10.57MB

cemd_visu.m 3KB

CHAPTER 1

INTRODUCTION TO THE HILBERT

HUANG TRANSFORM

AND ITS RELATED MATHEMATICAL PROBLEMS

Norden E. Huang

The Hilbert–Huang transform (HHT) is an empirically based data-analysis

method. Its basis of expansion is adaptive, so that it can produce physically mean-

ingful representations of data from nonlinear and non-stationary processes. The

advantage of being adaptive has a price: the diﬃculty of laying a ﬁrm theoretical

foundation. This chapter is an introduction to the basic method, which is fol-

lowed by brief descriptions of the recent developments relating to the normalized

Hilbert transform, a conﬁdence limit for the Hilbert spectrum, and a statistical

signiﬁcance test for the intrinsic mode function (IMF). The mathematical prob-

lems associated with the HHT are then discussed. These problems include (i) the

general method of adaptive data-analysis, (ii) the identiﬁcation methods of non-

linear systems, (iii) the prediction problems in nonstationary processes, which is

intimately related to the end eﬀects in the empirical mode decomposition (EMD),

(iv) the spline problems, which center on ﬁnding the best spline implementation

for the HHT, the convergence of EMD, and two-dimensional EMD, (v) the opti-

mization problem or the best IMF selection and the uniqueness of the EMD de-

composition, (vi) the approximation problems involving the ﬁdelity of the Hilbert

transform and the true quadrature of the data, and (vii) a list of miscellaneous

mathematical questions concerning the HHT.

1.1. Introduction

Traditional data-analysis methods are all based on linear and stationary assump-

tions. Only in recent years have new methods been introduced to analyze nonsta-

tionary and nonlinear data. For example, wavelet analysis and the Wagner-Ville

distribution (Flandrin 1999; Gr¨ochenig 2001) were designed for linear but non-

stationary data. Additionally, various nonlinear time-series-analysis methods (see,

for example, Tong 1990; Kantz and Schreiber 1997; Diks 1999) were designed for

nonlinear but stationary and deterministic systems. Unfortunately, in most real sys-

tems, either natural or even man-made ones, the data are most likely to be both

nonlinear and nonstationary. Analyzing the data from such a system is a daunting

problem. Even the universally accepted mathematical paradigm of data expansion

in terms of an aprioriestablished basis would need to be eschewed, for the con-

volution computation of the aprioribasis creates more problems than solutions.

A necessary condition to represent nonlinear and nonstationary data is to have an

2 N. E. Huang

adaptive basis. An apriorideﬁned function cannot be relied on as a basis, no matter

how sophisticated the basis function might be. A few adaptive methods are available

for signal analysis, as summarized by Windrow and Stearns (1985). However, the

methods given in their book are all designed for stationary processes. For nonsta-

tionary and nonlinear data, where adaptation is absolutely necessary, no available

methods can be found. How can such a basis be deﬁned? What are the mathemat-

ical properties and problems of the basis functions? How should the general topic

of an adaptive method for data analysis be approached? Being adaptive means

that the deﬁnition of the basis has to be data-dependent, an a posteriori-deﬁned

basis, an approach totally diﬀerent from the established mathematical paradigm

for data analysis. Therefore, the required deﬁnition presents a great challenge to

the mathematical community. Even though challenging, new methods to examine

data from the real world are certainly needed. A recently developed method, the

Hilbert–Huang transform (HHT), by Huang et al. (1996, 1998, 1999) seems to be

able to meet some of the challenges.

The HHT consists of two parts: empirical mode decomposition (EMD) and

Hilbert spectral analysis (HSA). This method is potentially viable for nonlinear

and nonstationary data analysis, especially for time-frequency-energy representa-

tions. It has been tested and validated exhaustively, but only empirically. In all the

cases studied, the HHT gave results much sharper than those from any of the tradi-

tional analysis methods in time-frequency-energy representations. Additionally, the

HHT revealed true physical meanings in many of the data examined. Powerful as it

is, the method is entirely empirical. In order to make the method more robust and

rigorous, many outstanding mathematical problems related to the HHT method

need to be resolved. In this section, some of the problems yet to be faced will be

listed, in the hope of attracting the attention of the mathematical community to

this interesting, challenging and critical research area. Some of the problems are

easy and might be resolved in the next few years; others are more diﬃcult and will

probably require much more eﬀort. In view of the history of Fourier analysis, which

was invented in 1807 but not fully proven until 1933 (Plancherel 1933), it should be

anticipated that signiﬁcant time and eﬀort will be required. Before discussing the

mathematical problem, a brief introduction to the methodology of the HHT will

ﬁrst be given. Readers interested in the complete details should consult Huang et

al. (1998, 1999).

1.2. The Hilbert

Huang transform

The development of the HHT was motivated by the need to describe nonlinear

distorted waves in detail, along with the variations of these signals that naturally

occur in nonstationary processes. As is well known, the natural physical processes

are mostly nonlinear and nonstationary, yet the data analysis methods provide very

limited options for examining data from such processes. The available methods are

either for linear but nonstationary, or nonlinear but stationary and statistically de-

Introduction to the Hilbert–Huang Transform 3

terministic processes, as stated above. To examine data from real-world nonlinear,

nonstationary and stochastic processes, new approaches are urgently needed, for

nonlinear processes need special treatment. The past approach of imposing a linear

structure on a nonlinear system is just not adequate. Other then periodicity, the

detailed dynamics in the processes from the data need to be determined because

one of the typical characteristics of nonlinear processes is their intra-wave frequency

modulation, which indicates the instantaneous frequency changes within one oscil-

lation cycle. As an example, a very simple nonlinear system will be examined, given

by the non-dissipative Duﬃng equation as

+ x + x

= γ cos(ωt) , (1.1)

where  is a parameter not necessarily small, and γ is the amplitude of a periodic

forcing function with a frequency ω. In (1.1), if the parameter  were zero, the system

would be linear, and the solution would be easily found. However, if  were non-

zero, the system would be nonlinear. In the past, any system with such a parameter

could be solved by using perturbation methods, provided that   1. However,

if  is not small compared to unity, then the system becomes highly nonlinear,

and new phenomena such as bifurcations and chaos will result. Then perturbation

methods are no longer an option; numerical solutions must be attempted. Either

way, (1.1) represents one of the simplest nonlinear systems; it also contains all the

complications of nonlinearity. By rewriting the equation in a slightly diﬀerent form

+ x



1+x



= γ cos(ωt) , (1.2)

its features can be better examined. Then the quantity within the parenthesis can

be regarded as a variable spring constant, or a variable pendulum length. As the

frequency (or period) of the simple pendulum depends on the length, it is obvious

that the system given in (1.2) should change in frequency from location to location,

and time to time, even within one oscillation cycle. As Huang et al. (1998) pointed

out, this intra-frequency frequency variation is the hallmark of nonlinear systems. In

the past, when the analysis was based on the linear Fourier analysis, this intra-wave

frequency variation could not be depicted, except by resorting to harmonics. Thus,

any nonlinear distorted waveform has been referred to as “harmonic distortions.”

Harmonics distortions are a mathematical artifact resulting from imposing a linear

structure on a nonlinear system. They may have mathematical meaning, but not

a physical meaning (Huang et al. 1999). For example, in the case of water waves,

such harmonic components do not have any of the real physical characteristics of a

real wave. The physically meaningful way to describe the system is in terms of the

instantaneous frequency, which will reveal the intra-wave frequency modulations.

The easiest way to compute the instantaneous frequency is by using the Hilbert

transform, through which the complex conjugate y(t) of any real valued function

4 N. E. Huang

x(t)ofL

class can be determined (see, for example, Titchmarsh 1950) by

H[x(t)] =



∞

−∞

x(τ)

t − τ

dτ , (1.3)

in which the PV indicates the principal value of the singular integral. With the

Hilbert transform, the analytic signal is deﬁned as

z(t)=x(t)+iy(t)=a(t)e

iθ(t)

, (1.4)

where

a(t)=



+ y

, and θ(t) = arctan





. (1.5)

Here, a(t) is the instantaneous amplitude, and θ is the phase function, and the

instantaneous frequency is simply

ω =

dθ

. (1.6)

A description of the Hilbert transform with the emphasis on its many mathemat-

ical formalities can be found in Hahn (1996). Essentially, (1.3) deﬁnes the Hilbert

transform as the convolution of x(t)with1/t; therefore, (1.3) emphasizes the lo-

cal properties of x(t). In (1.4), the polar coordinate expression further clariﬁes the

local nature of this representation: it is the best local ﬁt of an amplitude and phase-

varying trigonometric function to x(t). Even with the Hilbert transform, deﬁning

the instantaneous frequency still involves considerable controversy. In fact, a sen-

sible instantaneous frequency cannot be found through this method for obtaining

an arbitrary function. A straightforward application, as advocated by Hahn (1996),

will only lead to the problem of having frequency values being equally likely to be

positive and negative for any given dataset. As a result, the past applications of the

Hilbert transform are all limited to the narrow band-passed signal, which is narrow-

banded with the same number of extrema and zero-crossings. However, ﬁltering in

frequency space is a linear operation, and the ﬁltered data will be stripped of their

harmonics, and the result will be a distortion of the waveforms. The real advantage

of the Hilbert transform became obvious only after Huang et al. (1998) introduced

the empirical mode decomposition method.

1.2.1. The empirical mode decomposition method (the sifting

process)

As discussed by Huang et al. (1996, 1998, 1999), the empirical mode decomposition

method is necessary to deal with data from nonstationary and nonlinear processes.

In contrast to almost all of the previous methods, this new method is intuitive,

direct, and adaptive, with an a posteriori-deﬁned basis, from the decomposition

method, based on and derived from the data. The decomposition is based on the

Introduction to the Hilbert–Huang Transform 5

200 250 300 350 400 450 500 550 600

−10

−8

−6

−4

−2

Test Data

Time : second

Amplitude

Figure 1.1: The test data.

simple assumption that any data consists of diﬀerent simple intrinsic modes of os-

cillations. Each intrinsic mode, linear or nonlinear, represents a simple oscillation,

which will have the same number of extrema and zero-crossings. Furthermore, the

oscillation will also be symmetric with respect to the “local mean.” At any given

time, the data may have many diﬀerent coexisting modes of oscillation, one su-

perimposing on the others. The result is the ﬁnal complicated data. Each of these

oscillatory modes is represented by an intrinsic mode function (IMF) with the fol-

lowing deﬁnition:

(1) in the whole dataset, the number of extrema and the number of zero-crossings

must either equal or diﬀer at most by one, and

(2) at any point, the mean value of the envelope deﬁned by the local maxima and

the envelope deﬁned by the local minima is zero.

An IMF represents a simple oscillatory mode as a counterpart to the simple

harmonic function, but it is much more general: instead of constant amplitude

and frequency, as in a simple harmonic component, the IMF can have a variable

amplitude and frequency as functions of time. With the above deﬁnition for the

IMF, one can then decompose any function as follows: take the test data as given

in Fig. 1.1; identify all the local extrema, then connect all the local maxima by

a cubic spline line as shown in the upper envelope. Repeat the procedure for the

local minima to produce the lower envelope. The upper and lower envelopes should

cover all the data between them, as shown in Fig. 1.2. Their mean is designated as

, also shown in Fig. 1.2, and the diﬀerence between the data and m

is the ﬁrst

component h

shown in Fig. 1.3; i. e.,

= x(t) − m

. (1.7)

The procedure is illustrated in Huang et al. (1998).

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asdxiaer

2021-05-24

挺好用的，对每个IMF分量有相位变化调制信号瞬时频率图，就是调制信号是什么我不太懂

皮皮CHEN

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