Top Banner

of 97

EMPIRICAL WAVELET TRANSFORM

Oct 19, 2015

Download

Documents

Christo Jacob K

a seminar on empirical wavelet transform
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
  • Empirical Wavelet Transform

    By ARJUN SS

    GECT

    Roll no 5

    March 23, 2014

    Gec Thrissur Empirical Wavelet Transform

  • Table of Contents

    Introduction

    Empirical Mode Decomposition

    Empirical Wavelet Transform Concept

    Empirical Wavelets

    Empirical Wavelet Transform

    Comparing EMD and EWT

    Conclusion

    Gec Thrissur Empirical Wavelet Transform

  • Introduction

    Fourier transform, STFT and Wavelet transforms

    Well known ability of the wavelet transforms to pack the mainsignal information into a very small number of waveletcoefficients.

    Many wavelets are present to be chosen for a specific signal.

    Basis are designed independent of signal.

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    Signals which are non stationary and non-linear requireadaptive ways.

    Adaptive methods to construct basis directly from informationin the signal.

    Empirical Mode Decomposition(EMD).

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Principle)

    Decompose a signal as a (finite) sum of N + 1 Intrinsic ModeFunctions (IMF) such that

    f (t) =N

    k=0

    fk(t)

    where IMF is an AM-FM signal

    fk(t) = Fk(t)cos(k(t))whereFk(t), k(t) > 0t

    Algorithmic method present to extract modes.

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    consider the signal, say f (t).

    Figure: f (t)

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Identify maxima and join using cubic spline interpolation.

    Figure: Computing the upper envelope

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Identify minima and join using cubic spline interpolation.

    Figure: Computing the lower envelope

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Find the mean

    Figure: Mean in bold

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Compute upper and lower envelopes,fu(t) and fl(t) ,usingcubic spline interpolation.

    Compute mean envelope, m(t) = fu(t)+fl (t)2 .

    Candidate r1(t) = f (t)m(t).

    r1(t) is the new signal and same process is continued untilfirst IMF is obtained, say f1(t) = rn(t)

    same procedure is repeated for signal f (t) f1(t).

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Compute upper and lower envelopes,fu(t) and fl(t) ,usingcubic spline interpolation.

    Compute mean envelope, m(t) = fu(t)+fl (t)2 .

    Candidate r1(t) = f (t)m(t).

    r1(t) is the new signal and same process is continued untilfirst IMF is obtained, say f1(t) = rn(t)

    same procedure is repeated for signal f (t) f1(t).

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Compute upper and lower envelopes,fu(t) and fl(t) ,usingcubic spline interpolation.

    Compute mean envelope, m(t) = fu(t)+fl (t)2 .

    Candidate r1(t) = f (t)m(t).

    r1(t) is the new signal and same process is continued untilfirst IMF is obtained, say f1(t) = rn(t)

    same procedure is repeated for signal f (t) f1(t).

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Compute upper and lower envelopes,fu(t) and fl(t) ,usingcubic spline interpolation.

    Compute mean envelope, m(t) = fu(t)+fl (t)2 .

    Candidate r1(t) = f (t)m(t).

    r1(t) is the new signal and same process is continued untilfirst IMF is obtained, say f1(t) = rn(t)

    same procedure is repeated for signal f (t) f1(t).

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Mode Decomposition (Algorithm)

    Compute upper and lower envelopes,fu(t) and fl(t) ,usingcubic spline interpolation.

    Compute mean envelope, m(t) = fu(t)+fl (t)2 .

    Candidate r1(t) = f (t)m(t).

    r1(t) is the new signal and same process is continued untilfirst IMF is obtained, say f1(t) = rn(t)

    same procedure is repeated for signal f (t) f1(t).

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: original signal

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: Finding maxima

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: join using cubic spline interpolation

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: Finding minima

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: Join using cubic spline interpolation

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: Mean

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: candidate

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure: r1(t)

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Illustration

    Figure:

    Gec Thrissur Empirical Wavelet Transform

  • Advantages

    Doesnt use any prescribed basis.

    Self adapting accordingly to the analyzed signal.

    Able to separate stationary and non-stationary componentsfrom a signal.

    Locality(between maximas and minimas) andadaptability(data driven).

    Gec Thrissur Empirical Wavelet Transform

  • Advantages

    Doesnt use any prescribed basis.

    Self adapting accordingly to the analyzed signal.

    Able to separate stationary and non-stationary componentsfrom a signal.

    Locality(between maximas and minimas) andadaptability(data driven).

    Gec Thrissur Empirical Wavelet Transform

  • Advantages

    Doesnt use any prescribed basis.

    Self adapting accordingly to the analyzed signal.

    Able to separate stationary and non-stationary componentsfrom a signal.

    Locality(between maximas and minimas) andadaptability(data driven).

    Gec Thrissur Empirical Wavelet Transform

  • Advantages

    Doesnt use any prescribed basis.

    Self adapting accordingly to the analyzed signal.

    Able to separate stationary and non-stationary componentsfrom a signal.

    Locality(between maximas and minimas) andadaptability(data driven).

    Gec Thrissur Empirical Wavelet Transform

  • Disadvantages

    Lack of mathematical theory, not well defined.

    Algorithmic approach, difficult to model.

    Gec Thrissur Empirical Wavelet Transform

  • Disadvantages

    Lack of mathematical theory, not well defined.

    Algorithmic approach, difficult to model.

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Wavelet Transform Concept

    EMD

    Signal=Fast oscillation + Slow oscillation

    Iteration

    separation of fast and slow oscillations.

    data driven.

    local analysis based on extremas.

    adaptability but no mathematical theory.

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Wavelet Transform Concept

    WAVELETS

    Signal=Approximation + Details

    Iteration

    separation of approximation and details.

    based on apriori filtering.

    global analysis.

    strong mathematical background.

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Wavelet Transform Concept

    EWT

    EWT = EMD + WAVELETS

    Idea is to combine the strength of wavelets formalism and EMDsadaptability.

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    wavelets are equivalent to filter banks

    Figure: dyadic decomposition of Fourier spectrum

    EWT gives adaptive decomposition of Fourier spectrum

    Figure: adaptive decomposition of Fourier spectrum

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Wavelets

    Instead of dyadic decomposition of Fourier spectrum, why notdecompose it adaptively.

    Method to build a family of wavelets adapted to a signal isequivalent to building a set of bandpass filters in the Fourierdomain, such that their support depends on informationpresent in the signal being analyzed .

    Lets see how spectrum can be segmented depending oninformation present in signal.

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    Figure: Fourier spectrum of a signal

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    Figure: Fourier spectrum segmented

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    Figure: Fourier spectrum segmented(another signal)

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    Aim is to separate different portion of the spectrum whichcorresponds to modes.

    Assume N no of segments which means N 1 extraboundaries excluding for 0 and pi .

    Algorithm

    detect M local maximas, sort in decreasing order.

    M N : the algorithm found enough maxima to define the wantednumber of segments, keep only the first N 1 maxima.M < N : the signal has less modes than expected, keep all thedetected maxima and reset N to the appropriate value.

    Gec Thrissur Empirical Wavelet Transform

  • Filter Bank Construction

    assume the Fourier support [0, pi] is segmented into Ncontiguous segments.

    denote n to be the limits between each segments (where0 = 0 and N = pi ) and each segment is denoted asn = [n1, n].

    Figure: Filter bank construction

    n=half length of transition phase, in practice, n = n

    Gec Thrissur Empirical Wavelet Transform

  • Filter Bank Construction

    Scaling function spectrum

    n() =

    1 if || (1 )ncos

    [pi

    2

    (1

    2n(|| (1 )n)

    )]if(1 )n || (1 + )n

    0 otherwise

    wavelet spectrum

    n() =

    1 if (1 + )n || (1 )n+1

    cos

    [pi

    2

    (1

    2n+1(|| (1 )n+1)

    )]if(1 )n+1 || (1 + )n+1

    sin

    [pi

    2

    (1

    2n(|| (1 )n)

    )]if(1 )n || (1 + )n

    0 otherwise

    Gec Thrissur Empirical Wavelet Transform

  • Filter Bank Reconstruction

    Scaling function spectrum, n = 1, = 0.5

    Wavelet spectrum, n = 1, n+1 = 2.5and = 0.2

    Gec Thrissur Empirical Wavelet Transform

  • Tight frame and an Example

    proposition for a tight frame is < minn

    (n+1 nn+1 + n

    )Examples

    n {0, 1.5, 2, 2.8, pi} with = 0.05

    Figure: Fourier partitioning of an empirical filter bank

    Gec Thrissur Empirical Wavelet Transform

  • Empirical Wavelet Transform

    Detail coefficient

    W f (n, t) = f , n =(f ()n()

    )Approximation coefficient

    W f (0, t) = f , 1 =(f ()n()

    )Reconstruction

    f (t) = W f (0, t) 1(t) +Nn=1

    W f (n, t) n(t)

    =

    (W f (0, )1() +

    Nn=1

    W f (n, )n()

    )

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    We have already seen

    f (t) =N

    k=0

    fk(t)

    Empirical mode from above reconstructed form fk

    f0(t) = Wf (0, t) 1(t)

    fk(t) = Wf (k, t) k(t)

    Gec Thrissur Empirical Wavelet Transform

  • Cont..

    Algorithm

    Finding Fourier transform of signal, f f .Compute the local maxima of f on [0, pi] and find the set n.

    Choose < minn

    (n+1 nn+1 + n

    ).

    Build filter bank.

    Filter the signal to get each component.

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: signal 1

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: signal 2

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: signal 3

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: real ECG signal

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: real ECG signal

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: signal 1 modes by EWT and EMD

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: signal 2 modes by EWT and EMD

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: signal 3 modes by EWT and EMD

    Gec Thrissur Empirical Wavelet Transform

  • Examples

    Figure: real ECG signal modes by EWT and EMD

    Gec Thrissur Empirical Wavelet Transform

  • Comparing EMD and EWT

    EMD automatically estimates number of modes whereas inEWT, we set number of modes.

    EMD overestimates number of modes but EWT givesdifferent components which are closer to original signal .

    Gec Thrissur Empirical Wavelet Transform

  • Comparing EMD and EWT

    EMD automatically estimates number of modes whereas inEWT, we set number of modes.

    EMD overestimates number of modes but EWT givesdifferent components which are closer to original signal .

    Gec Thrissur Empirical Wavelet Transform

  • Conclusion

    Wavelets built adapted to signal being analyzed.

    Wavelet filter bank based on Fourier support detected frominformation.

    Dilation factors dont follow a prescribed scheme but aredetected empirically.

    Further research can be done to segment spectrum even moreefficiently .

    A matlab toolbox is available in matlab central website.

    Gec Thrissur Empirical Wavelet Transform

  • Conclusion

    Wavelets built adapted to signal being analyzed.

    Wavelet filter bank based on Fourier support detected frominformation.

    Dilation factors dont follow a prescribed scheme but aredetected empirically.

    Further research can be done to segment spectrum even moreefficiently .

    A matlab toolbox is available in matlab central website.

    Gec Thrissur Empirical Wavelet Transform

  • Conclusion

    Wavelets built adapted to signal being analyzed.

    Wavelet filter bank based on Fourier support detected frominformation.

    Dilation factors dont follow a prescribed scheme but aredetected empirically.

    Further research can be done to segment spectrum even moreefficiently .

    A matlab toolbox is available in matlab central website.

    Gec Thrissur Empirical Wavelet Transform

  • Conclusion

    Wavelets built adapted to signal being analyzed.

    Wavelet filter bank based on Fourier support detected frominformation.

    Dilation factors dont follow a prescribed scheme but aredetected empirically.

    Further research can be done to segment spectrum even moreefficiently .

    A matlab toolbox is available in matlab central website.

    Gec Thrissur Empirical Wavelet Transform

  • Conclusion

    Wavelets built adapted to signal being analyzed.

    Wavelet filter bank based on Fourier support detected frominformation.

    Dilation factors dont follow a prescribed scheme but aredetected empirically.

    Further research can be done to segment spectrum even moreefficiently .

    A matlab toolbox is available in matlab central website.

    Gec Thrissur Empirical Wavelet Transform

  • References

    J. Gilles Empirical wavelet transform, IEEE Trans. SignalProcess., vol. 61, no. 16, pp.3999 -4010 2013 .

    I. Daubechies, J. Lu, and H.-T. Wu, Synchrosqueezed wavelettransforms: An empirical mode decomposition-like tool, J. Appl.Computat.Harmon. Anal., vol. 30, no. 2, pp. 243261, 2011.

    N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q.Zheng, N.-C. Yen, C. C. Tung, andH.H.Liu, The empirical modedecomposition and the Hilbert spectrum for nonlinear andnon-stationary time series analysis, Proc. Roy. Soc. London A, vol.454, pp. 903995, 1998..

    ECE 804 - Spring 2012 - Lecture 005 with Dr. Patrick Flandrin,CNRS & Ecole Normale Superieure de Lyon, France - Mar. 16,2012.

    Gec Thrissur Empirical Wavelet Transform

  • Thanks for your patience

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX A (orthogonality of IMF)

    By virtue of the decomposition, the elements should all belocally orthogonal to each other, for each element is obtainedfrom the difference between the signal and its local meanthrough the maximal and minimal envelopes.

    X (t) =n+1j=1

    Cj(t)

    X 2(t) =n+1j=1

    C 2j (t) + 2n+1j=1

    n+1k=1

    Cj(t)Ck(t)

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX B (orthogonality of IMF)

    Figure: orthogonal IMFs

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX

    (limit for IMF iteration in EMD)

    SD =Tt=0

    [|h1(k1)(t) h1k(t)|2

    h21(k1)(t)

    ]

    SD can be set between 0.2 - 0.3

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX C (Detecting value for N)

    Detect {Mi}Mk=1, set of M detected maximas in magnitude ofspectrum so that, M1 M2 ... MM .keeping all the maximas larger than thresholdMM + (M1 MM), where is relative amplitude ratio. around 0.3 - 0.4 gives consistent result.

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX D (function)

    (x) =

    0 if x 0and(x) + (1 x) = 11 ifx 1eq : (x) = x4(35 84x + 70x2 20x3)

    Gec Thrissur Empirical Wavelet Transform

  • APPENDIX E (time frequency representation)

    Hilbert-Huang transform.

    Hf (t) =1

    pip.v

    inf inf

    f ()

    t d.

    for f (t) = F (t)cos((t)), it provides fa(t) = F (t) expi(t)

    where fa(t) = f (t) + iHf (t), .

    extract instantaneous amplitude F(t) and freq (t).

    Gec Thrissur Empirical Wavelet Transform