CHAPTER1
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 afirm theoretical
foundation.This chapter is an introduction to the basic method,which is fol-
adaptivelowed 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 onfinding 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 thefidelity 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(Flandrin1999;Gr¨o chenig2001)were designed for linear but non-stationary data.Additionally,various nonlinear time-series-analysis methods(see, for example,Tong1990;Kantz and Schreiber1997;Diks1999)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 a priori established basis would need to be eschewed,for the con-volution computation of the a priori basis creates more problems than solutions.
A necessary condition to represent nonlinear and nonstationary data is to have an
1
2N.E.Huang
adaptive basis.An a priori defined 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 designe
d 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 in1807but not fully proven until1933(Plancherel1933),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 Transform3 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
d2x
dt2
+x+ x3=γcos(ωt),(1.1) where is a parameter not necessarily small,andγis the amplitude of a periodic forcing function with a frequencyω.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 phe
nomena 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 as
d2x dt2+x
1+ x2
=γcos(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 f
rom 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
4N.E.Huang
x(t)of L p class can be determined(see,for example,Titchmarsh1950)by
H[x(t)]=1
πPV
∞
−∞
x(τ)
t−τ
dτ,(1.3)
in which the P V 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)=
x2+y2,andθ(t)=arctan
y
x
.(1.5)
Here,a(t)is the instantaneous amplitude,andθis the phase function,and the instantaneous frequency is simply
ω=dθ
dt
.(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 localfit 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 thefiltered 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 Transform5
Figure1.1: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 thefinal 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 m1,also shown in Fig.1.2,and the difference between the data and m1is thefirst component h1shown in Fig.1.,
h1=x(t)−m1.(1.7) The procedure is illustrated in Huang et al.(1998).

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