Graduate Institute of Electronics Engineering, NTU
Hilbert-Huang Transform(HHT)
Presenter: Yu-Hao Chen
ID:R98943021
2010/05/07
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Outline
 Author
 Motivation
 Hilbert Transform
 Instantaneous frequency(IF)
 Flow chart
 Theory
 Intrinsic Mode Function(IMF)
 Empirical Mode Decomposition(EMD)
 Time–Frequency analysis
 Application
 Problem
 Summary
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Norden E. Huang (黃鍔)
 Career and Experience
 Research Scientist, NASA (1975-2006)
 National Academy of Engineering (2000)
 Academia Sinica (2006)
 NASA Goddard Space Flight Center (2000-2006)
 Research Center for Adaptive Data Analysis (2006)
 Research topic
 Engineering Sciences
 Applied Mathematical Sciences
 Applied Physical Sciences
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Motivation
 To deal with nonlinear and non-stationary signal
 To get Instantaneous frequency(IF)
[5]
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Hilbert Transform
 The Hilbert transform can be thought of as the
convolution of s(t) with the function h(t) = 1/(πt)
1
sˆ(t )  s (t ) 
t
 Derive the analytic representation of a signal
j ( t )
ˆ
z (t )  s(t )  js(t )  m(t )  e
d
Instantane ous Frequency : f(t)   (t )
dt
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Instantaneous Frequency(IF)
 s(t) = β + cos(t)
 (1) β = 0: IF is the constant
 (2) 0 < β < 1: IF has been oscillating
 (3) β > 1: IF has been negative
[3]
[3]
[3]
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Flow Chart
[1] [4]
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Intrinsic Mode Function(IMF)
 The number of extrema and zero-crossings must either
be equal or differ at most by one.
 The mean value of the upper envelope and the lower
envelope is zero.
[5]
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Empirical Mode
Decomposition(EMD)(1/8)
[1]
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Empirical Mode
Decomposition(EMD)(2/8)
[1]
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Empirical Mode
Decomposition(EMD)(3/8)
[1]
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Empirical Mode
Decomposition(EMD)(4/8)
[1]
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Empirical Mode
Decomposition(EMD)(5/8)
[1]
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Empirical Mode
Decomposition(EMD)(6/8)

 SD < 0.1 => IMF
[4]
[1]
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Empirical Mode
Decomposition(EMD)(7/8)
[1]
Sifting Process
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Empirical Mode
Decomposition(EMD)(8/8)

[4]
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Example
[5]
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Time–Frequency Analysis
 Fast Fourier Transform (FFT)
 Wavelet Transform
 Hilbert-Huang Transform (HHT)
Basis
FFT
Wavelet
HHT
a priori
a priori
Adaptive
Nonlinear
Non-stationary
Feature Extraction
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Application
 Geoscience
 Biomedical applications
 Multimodal Pressure Flow (MMPF)
 Financial applications
 Image processing
 Audio processing
 Structural health monitoring
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Geoscience
 Length of day
1章年(19年)
[5]
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Biomedical(1/2)
 Multimodal Pressure
Flow (MMPF)
[5]
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Biomedical(2/2)
 Doppler blood flow signal analysis [14]
 Detection and estimation of Doppler shift [15]
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Image Processing
 Edge detection [10]
 Image denoise [11]
 Image fusion [12]
a
a. EMD
b. Sobel
c. Canny
b
c
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Problems of HHT
 P1: Stopping criterion
 P2: End effect problem
 Hilbert Transform
 EMD
 P3: Mode mixing problem
 Ensemble EMD (EEMD)
 Post-processing of EEMD
 P4: Speed of computing
 P5: Spline
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P1: Stopping Criterion
 Standard deviation(SD)
 SD ≤ 0.2~0.3
 S number criterion
3≤S≤5
T
SD   [
(h( k 1) (t )  hk (t ))
t 0
h
2
( k 1)
T
OrthogonalIndex(OI) 
 Three parameter method(θ1,θ2, α)
t 0
2
]
(t )
C f Cg
C C
2
f
2
g
[1]
[2]
[3]
 Mode amplitude： a(t )  (emax (t )  emin (t )) / 2
 Evaluation function： (t )  m(t ) / a(t )
 σ(t)< θ1 in (1- α)
σ(t)< θ2 in α
 α ≒ 0.05, θ1 ≒0.05,
θ2 ≒ 10θ1
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P2: End Effect Problem
 End effect of Hilbert Transform
[1]
 End effect of EMD
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P2: Solutions for End Effects
 End effect of Hilbert Transform
 Adding characteristics waves
 End effect of EMD
 Extension with linear spline fittings near the boundaries
Envelopes with end effects corrected
maxima
minima
0.6
data
upper
lower
maxima
minma
0.4
0.2
0
-0.2
-0.4
0
50
100
150
200
250
300
[6]
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P3: Mode Mixing
 Ensemble EMD (EEMD)
 Post-processing of EEMD
[1]
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P3: Ensemble EMD (EEMD)
 Noise n1-nm are identical independent distributed.
 Ensemble EMD indeed enables the signals of
similar scale collated together.
 The ensemble EMD results might not be IMFs.
X  n1
 EMD
X  n2
 EMD
IMF11
IMF21
IMF12
IMF22
…
X  nm
EEMD
 EMD
IMFm1
IMF1 
IMFm2
…
…
IMF2k
IMFmk
EEMD IMF
1 m
 IMF i1
m i 1
1 m
IMF2  i 1 IMF i 2
m
…
…
IMF1k
[7] [8]
IMFk 
1 m
IMF ik

i 1
m
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P3: Post-Processing of EEMD
 Post-processing EEMD can get real IMFs.
EEMD
Post - processing of EEMD
…
1 m
IMFk  i 1 IMF ik
m
IMF1  pIMF1  residual1
IMF2  residual1  pIMF2  residual2
…
1 m
 IMF i1
m i 1
1 m
IMF2  i 1 IMF i 2
m
IMF1 
IMFk  residualk 1  pIMFk  residualk
k
 X(t)   pIMFk  trend
j1
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P4: Speed of Computing
 The processing time of HHT is dependent on
complexity of the data and criterions of the algorithm
 HHT data processing system(HHT-DPS)
 Implementation of HHT based on DSP [13]
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P5: Spline
 Cubic B-Spline
[5]
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Conclusion
 The definition of an IMF guarantees a well-behaved
Hilbert transform of the IMF
 IMF represents intrinsic signature of physics behind the
data
 Although there are still many problems in HHT,HHT
has lots of applications in all aspects
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Reference(1/3)
[1] N. E. Huang, Z. Shen, etc. “The empirical mode deomposition and the Hilbert
spectrum for nonlinear and non-stationary time series analysis,” Proceedings
of the Royal Society, vol. 454, no. 1971, pp. 903–995, March 8 1998.
[2 ] N. E. Huang, M. C. Wu, S. R. Long, S. S. P. Shen, W. Qu, P. Gloersen and K. L.
Fan, “A Confidence Limit for the Empirical Mode Decomposition and Hilbert
Spectrum Analysis”, Proc. R. Soc. Lond. A, vol. 459, 2003, pp. 2317- 2345.
[3] G. Rilling, P. Flandrin and P. Gonçalvés, “On Empirical Mode Decomposition
and Its Algorithms”, IEEE-EURASIP Work- shop on Nonlinear Signal and Image
Processing NSIP-03, Grado, Italy, 8-11 Jun. 2003.
[4] J. Cheng, D. Yu and Y. Yang, “Research on the Intrinsic Mode Function (IMF)
Criterion in EMD Method”, Mechanical Systems and Signal Processing, vol. 20,
2006, pp. 817-824.
[5] Z. Xu, B. Huang and S. Xu, “Exact Location of Extrema for Empirical Mode
Decomposition”, Electronics Letters, vol. 44, no. 8, 10 Apr. 2008, pp. 551-552.
[6] 國立中央大學 數據分析研究中心 (RCADA)
Available: http://rcada.ncu.edu.tw/intro.html
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Reference(2/3)
[7] Z. WU and N. E. HUANG , “ENSEMBLE EMPIRICAL MODE
DECOMPOSITION:A NOISE-ASSISTED DATA ANALYSIS METHOD”,
Advances in Adaptive Data Analysis, Vol. 1, No. 1 pp 1–41,2009
[8] Master thesis: Applications of Ensemble Empirical Mode Decomposition (EEMD)
and Auto-Regressive (AR) Model for Diagnosing Looseness Faults of Rotating
Machinery
[9] Y. Deng, W. Wang, C. Qian, Z. Wang and D. Dai, ”Boundary-ProcessingTechnique in EMD Method and Hilbert Transform”, Chinese Science Bulletin,
vol. 46, no. 1, Jan. 2001, pp. 954-960.
[10] J. Zhao and D. Huang, “Mirror Extending and Circular Spline Function for
Empirical Mode Decomposition Method”, Journal of Zhejiang University,
Science, vol. 2, no.3, July-Sep. 2001, pp. 247-252.
[11] K. Zeng and M. He, “A simple Boundary Process Technique for Empirical
Mode Decomposition”, IEEE International Geoscience and Remote Sensing
Symposium IGARSS '04, vol. 6, 2004, pp. 4258-4261.
[12] Z. Zhao and Y. Wang, “A New Method for Processing End Effect in
Empirical Mode Decomposition”, IEEE International Conference on Circuits
and Systems for Communications ICCSC 2007, 2007, pp. 841-845.
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Reference(3/3)
 [13] H. Li and Z. Li, etc. ,” Implementation of Hilbert-Huang Transform (HHT)
Based on DSP”, International Conference on Signal Processing, vol.1, 2004
 [14] Z. Zhidong and W. Yang ,”A New Method for Processing End Effect In
Empirical Mode Decomposition”, International Conference on
Communications, Circuits and Systems, ICCCAS , pp 841-845, July 2007
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Thank you
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