An Introduction to HHT: Instantaneous Frequency, Trend,

Degree of Nonlinearity and Non-stationarity

Norden E. Huang

Research Center for Adaptive Data AnalysisCenter for Dynamical Biomarkers and Translational Medicine

NCU, Zhongli, Taiwan, China

OutlineRather than the implementation details,

I will talk about the physics of the method.

• What is frequency?

• How to quantify the degree of nonlinearity?

• How to define and determine trend?

What is frequency?

It seems to be trivial.

But frequency is an important parameter for us to understand many physical phenomena.

Definition of Frequency

Given the period of a wave as T ; the frequency is defined as

1.

T

Instantaneous Frequency

distanceVelocity ; mean velocity

time

dxNewton v

dt

1Frequency ; mean frequency

period

dHH

So that both v and

T defines the p

can appear in differential equations.

hase functiondt

Other Definitions of Frequency : For any data from linear Processes

j

Ti t

j 0

1. Fourier Analysis :

F( ) x( t ) e dt .

2. Wavelet Analysis

3. Wigner Ville Analysis

Definition of Power Spectral Density

Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case.

Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a wide-sense stationary random process:

2i 2S( ) R( )e d S( )d ( t )

Fourier Spectrum

Problem with Fourier Frequency• Limited to linear stationary cases: same

spectrum for white noise and delta function.

• Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost.

• Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum.

Surrogate Signal:Non-uniqueness signal vs. Power Spectrum

I. Hello

The original data : Hello

The surrogate data : Hello

The Fourier Spectra : Hello

The Importance of Phase

To utilize the phase to define

Instantaneous Frequency

Prevailing Views onInstantaneous Frequency

The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer.

J. Shekel, 1953

The uncertainty principle makes the concept of an Instantaneous Frequency impossible.

K. Gröchennig, 2001

The Idea and the need of Instantaneous Frequency

k , ;

t

tk

0 .

According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that

Therefore, both wave number and frequency must have instantaneous values and differentiable.

p

2 2 1 / 2 1

i ( t )

For any x( t ) L ,

1 x( )y( t ) d ,

t

then, x( t )and y( t ) form the analytic pairs:

z( t ) x( t ) i y( t ) ,

wherey( t )

a( t ) x y and ( t ) tan .x( t )

a( t ) e

Hilbert Transform : Definition

The Traditional View of the Hilbert Transform for Data Analysis

Traditional Viewa la Hahn (1995) : Data LOD

Traditional Viewa la Hahn (1995) : Hilbert

Traditional Approacha la Hahn (1995) : Phase Angle

Traditional Approacha la Hahn (1995) : Phase Angle Details

Traditional Approacha la Hahn (1995) : Frequency

The Real World

Mathematics are well and good but nature keeps dragging us around by the nose.

Albert Einstein

Why the traditional approach does not work?

Hilbert Transform a cos + b : Data

Hilbert Transform a cos + b : Phase Diagram

Hilbert Transform a cos + b : Phase Angle Details

Hilbert Transform a cos + b : Frequency

The Empirical Mode Decomposition Method and Hilbert Spectral Analysis

Sifting

(Other alternatives, e.g., Nonlinear Matching Pursuit)

Empirical Mode Decomposition: Methodology : Test Data

Empirical Mode Decomposition: Methodology : data and m1

Empirical Mode Decomposition: Methodology : data & h1

Empirical Mode Decomposition: Methodology : h1 & m2

Empirical Mode Decomposition: Methodology : h3 & m4

Empirical Mode Decomposition: Methodology : h4 & m5

Empirical Mode DecompositionSifting : to get one IMF component

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.

The Stoppage Criteria

The Cauchy type criterion: when SD is small than a pre-set value, where

T2

k 1 kt 0

T2

k 1t 0

h ( t ) h ( t )SD

h ( t )

Or, simply pre-determine the number of iterations.

Empirical Mode Decomposition: Methodology : IMF c1

Definition of the Intrinsic Mode Function (IMF): a necessary condition only!

Any function having the same numbers ofzero cros sin gs and extrema,and also havingsymmetric envelopes defined by local max imaand min ima respectively is defined as anIntrinsic Mode Function( IMF ).

All IMF enjoys good Hilbert Transfo

i ( t )

rm :

c( t ) a( t )e

Empirical Mode Decomposition: Methodology : data, r1 and m1

Empirical Mode DecompositionSifting : to get all the IMF components

1 1

1 2 2

n 1 n n

n

j nj 1

x( t ) c r ,r c r ,

x( t ) c r

. . .r c r .

.

Definition of Instantaneous Frequency

i ( t )

t

The Fourier Transform of the Instrinsic ModeFunnction, c( t ), gives

W ( ) a( t ) e dt

By Stationary phase approximation we have

d ( t ) ,dt

This is defined as the Ins tan taneous Frequency .

An Example of Sifting &

Time-Frequency Analysis

Length Of Day Data

LOD : IMF

Orthogonality Check

• Pair-wise % • 0.0003• 0.0001• 0.0215• 0.0117• 0.0022• 0.0031• 0.0026• 0.0083• 0.0042• 0.0369• 0.0400

• Overall %

• 0.0452

LOD : Data & c12

LOD : Data & Sum c11-12

LOD : Data & sum c10-12

LOD : Data & c9 - 12

LOD : Data & c8 - 12

LOD : Detailed Data and Sum c8-c12

LOD : Data & c7 - 12

LOD : Detail Data and Sum IMF c7-c12

LOD : Difference Data – sum all IMFs

Traditional Viewa la Hahn (1995) : Hilbert

Mean Annual Cycle & Envelope: 9 CEI Cases

Mean Hilbert Spectrum : All CEs