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a seminar on empirical wavelet transform
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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