An Introduction to HHT:
Instantaneous Frequency, Trend,
Degree of Nonlinearity and Non-stationarity
Norden E. Huang
Research Center for Adaptive Data Analysis
Center for Dynamical Biomarkers and Translational Medicine
NCU, Zhongli, Taiwan, China
Outline
Rather 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
Velocity 
distance
; mean velocity
time
Newton  v 
Frequency 
dx
dt
1
; mean frequency
period
HHT defines the phase function   
d
dt
So that both v and  can appear in differential equations.
Other Definitions of Frequency :
For any data from linear Processes
1. Fourier Analysis :
T
F (  )    x( t ) e
j
i jt
dt .
0
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 widesense stationary random process:
S(  ) 



R(  )e 2i d




S(  )d   2 ( 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 on
Instantaneous 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
According to the classic wave theory, the wave
conservation law is based on a gradually changing φ(x,t)
such that
k   ,

  
;
t
k

    0 .
t
Therefore, both wave number and frequency must have
instantaneous values and differentiable.
Hilbert Transform : Definition
For any x( t )  Lp ,
y( t ) 
1



x(  )
d ,
t 
then, x( t )and y( t ) form the analytic pairs:
z( t )  x( t )  i y( t )  a( t ) e i  ( t ) ,
where
a( t )   x 2  y 2  1 / 2 and  ( t )  tan 1
y( t )
.
x( t )
The Traditional View of the Hilbert
Transform for Data Analysis
Traditional View
a la Hahn (1995) : Data LOD
Traditional View
a la Hahn (1995) : Hilbert
Traditional Approach
a la Hahn (1995) : Phase Angle
Traditional Approach
a la Hahn (1995) : Phase Angle Details
Traditional Approach
a 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 Decomposition
Sifting : to get one IMF component
x( t )  m 1  h1 ,
h1  m 2  h2 ,
.....
.....
hk  1  m k  hk .

hk  c1
.
The Stoppage Criteria
The Cauchy type criterion: when SD is small than a preset value, where
T
SD 
h
t 0
k 1
( t )  hk ( t )
2
T
2
h
 k 1 ( t )
t 0
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 of
zero  cros sin gs and extrema ,and also having
symmetric envelopes defined by local max ima
and min ima respectively is defined as an
Intrinsic Mode Function ( IMF ).
All IMF enjoys good Hilbert Transform :
  c( t )  a( t )e i  ( t )
Empirical Mode Decomposition:
Methodology : data, r1 and m1
Empirical Mode Decomposition
Sifting : to get all the IMF components
x( t )  c1  r1 ,
r1  c2  r2 ,
. . .
rn  1  cn  rn .
 x( t ) 
n
c
j1
j
 rn .
Definition of Instantaneous Frequency
The Fourier Transform of the Instrinsic Mode
Funnction, c( t ), gives
W ( ) 
i (   t )
a(
t
)
e
dt

t
By Stationary phase approximation we have
d ( t )
 ,
dt
This is defined as the Ins tan tan eous Frequency .
An Example of Sifting
&
Time-Frequency Analysis
Length Of Day Data
LOD
:
IMF
Orthogonality Check
•
Pair-wise %
•
Overall %
•
•
•
•
•
•
•
•
•
•
•
0.0003
0.0001
0.0215
0.0117
0.0022
0.0031
0.0026
0.0083
0.0042
0.0369
0.0400
•
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 View
a la Hahn (1995) : Hilbert
Mean Annual Cycle & Envelope: 9 CEI
Cases
Mean Hilbert Spectrum : All CEs