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Graduate Institute of Electronics Engineering, NTU

Hilbert-Huang Transform(HHT)Hilbert-Huang Transform(HHT)

Presenter: Yu-Hao ChenID:R98943021

2010/05/07

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OutlineOutline 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 (Norden E. Huang ( 黃鍔黃鍔 ))

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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MotivationMotivation To deal with nonlinear and non-stationary signal To get Instantaneous frequency(IF)

[5]

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Hilbert TransformHilbert Transform The Hilbert transform can be thought of as the

convolution of s(t) with the function h(t) = 1/(πt)

Derive the analytic representation of a signal

)()()(ˆ)()( tjetmtsjtstz

ttsts

1)()(ˆ

)(f(t) :Frequency ousInstantane tdtd

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Instantaneous Frequency(IF)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 ChartFlow Chart

[4][1]

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Intrinsic Mode Function(IMF)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.

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(1/8)(1/8)

[1]

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(2/8)(2/8)

[1]

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(3/8)(3/8)

[1]

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(4/8)(4/8)

[1]

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(5/8)(5/8)

[1]

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(6/8)(6/8)

SD < 0.1 => IMF

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(7/8)(7/8)

[1]Sifting Process

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Empirical Mode Decomposition(EMD)Empirical Mode Decomposition(EMD)(8/8)(8/8)

[4]

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ExampleExample

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Time–Frequency AnalysisTime–Frequency Analysis

Fast Fourier Transform (FFT) Wavelet Transform Hilbert-Huang Transform (HHT)

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ApplicationApplication Geoscience Biomedical applications

Multimodal Pressure Flow (MMPF) Financial applications Image processing Audio processing Structural health monitoring

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GeoscienceGeoscience Length of day

[5]

1 章年 (19 年 )

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Biomedical(1/2)Biomedical(1/2) Multimodal Pressure Flow (MMPF)

[5]

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Biomedical(2/2)Biomedical(2/2) Doppler blood flow signal analysis [14] Detection and estimation of Doppler shift [15]

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Image ProcessingImage Processing Edge detection [10] Image denoise [11] Image fusion [12]

P. 23

a. EMDb. Sobelc. Canny

a b

c

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Problems of HHTProblems 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 CriterionP1: Stopping Criterion Standard deviation(SD)

SD ≤ 0.2~0.3 S number criterion

3 ≤ S ≤ 5

Three parameter method(θ1,θ2, α) Mode amplitude ： Evaluation function ： σ(t)< θ1 in (1- α)

σ(t)< θ2 in α α ≒ 0.05, θ1 ≒0.05,

θ2 ≒ 10θ1

T

t k

kk

th

ththSD

0 )1(2

2

)1( ])(

))()(([

2/))()(()( minmax teteta )(/)()( tatmt

[1]

[2]

[3]

T

t gf

gf

CCCC

022Index(OI) Orthogonal

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P2: End Effect ProblemP2: End Effect Problem End effect of Hilbert Transform

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End effect of EMD

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End effect of Hilbert Transform Adding characteristics waves End effect of EMD Extension with linear spline fittings near the boundaries

0 50 100 150 200 250 300-0.4

-0.2

0

0.2

0.4

0.6Envelopes with end effects corrected

dataupperlowermaximaminma

P2: Solutions for End EffectsP2: Solutions for End Effects

maxima minima

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Ensemble EMD (EEMD) Post-processing of EEMD

P3: Mode MixingP3: Mode Mixing

[1]

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P3: Ensemble EMD (EEMD)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.

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EEMD IMF

…… …

… …

ik

m

i

i

m

i

i

m

i

IMFm

IMFm

IMFm

1k

212

111

1IMF

1IMF

1IMF

EEMD

1k

12

11

1

IMF

IMFIMF

EMD

X

n

2k

22

21

2

IMF

IMFIMF

EMD

X

n

mk

m2

m1

IMF

IMFIMF

EMD

X

mn

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Post-processing EEMD can get real IMFs.

P3: Post-Processing of EEMDP3: Post-Processing of EEMD

…

…

trend

residualresidual

residualresidualresidual

kk

k

1jk

k1k

2212

111

pIMFX(t)

pIMFIMF

pIMFIMFpIMFIMF

EEMD of processing-Post

ik

m

i

i

m

i

i

m

i

IMFm

IMFm

IMFm

1k

212

111

1IMF

1IMF

1IMF

EEMD

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P4: Speed of ComputingP4: 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: SplineP5: Spline Cubic B-Spline

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ConclusionConclusion 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)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)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-Processing- Technique 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)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