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The Hilbert Transform and Empirical Mode

Decomposition

Powerful Tools for Data Analysis

- Suz Tolwinski
- University of Arizona
- Program in Applied Mathematics
- Spring 2007 RTG

Deriving the Hilbert Transform

For f(z) u(x,y) iv(x,y) analytic on the upper

half-plane, and decays at infinity

Cauchys Integral Formula Decay of function

Clever rearrangement of terms

Relationship between u(x,y) and v(x,y) on R

Define the Hilbert transform in similar spirit

Another View of the Hilbert Transform.

Looks like minus the convolution of f(t) with

1/pt. Apply Convolution Theorem for something

more manageable in frequency space?

Take away message The Hilbert transform rotates

a signal counter-clockwise by p/2 at every point

in its positive frequency spectrum, and clockwise

at all its negative frequencies.

Real Signals.

A signal is any time-varying quantity containing

information.

Signals in nature are real.

Real signals have even frequency spectra.

This makes them difficult to analyze. We would

like to know, how is the signal energy

distributed in time and/or frequency space?

For a signal with even frequency spectrum,

-Mean frequency? -Spread of

signal in frequency space?

(Energy of s(t) ? s(t)2 dt ? S(?)2 d?)

Electrocardiographic signal in time and frequency

domain

Analytic Signals.

Have positive frequency spectrum only, so lt?gt,

spread in ? are meaningful quantities.

We can construct an analytic signal

corresponding to any real signal (takes only

the positive frequencies)

Frequency spectrum of a real signal.

FANALYT.(?) 1/2(F(?) sgn(?)F(?))

Then in the time domain, the analytic signal is

given by

Frequency spectrum of the corresponding analytic

signal.

fANALYT.(t) 1/2(f(t) iHf(t))

Analytic signal can be represented as

time-varying frequency and amplitude

fANALYT.(t) A(t)ei?(t)

A(t) (1/2f(t))21/2 (Hf(t))2)1/2

?(t) tan-1(Hf(t)/f(t))

Empirical Mode Decomposition of Real Signals.

(Method due to N. Huang, 1998.)

- Creates an adaptive decomposition of signal
- Result is a generalized Fourier series
- (Modes with time-varying amplitude and phase)
- Components are called Intrinsic Mode Functions

(IMFs) - IMFs satisfy two criteria by designs
- -Have only one zero between successive extrema
- -Have zero local mean
- EMD separates phenomena occurring on different

time scales. - Residue shows overall trend in data.

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) (residue--gt) i i 1 end

IMFk(t) Ii(t) Residue

Residue - IMFk k k1

end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) (residue--gt) i i 1 end

IMFk(t) Ii(t) Residue

Residue - IMFk k k1

end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) (residue--gt) i i 1 end

IMFk(t) Ii(t) Residue

Residue - IMFk k k1

end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

Huangs Sifting Process.

Residue s(t) I1(t) Residue i 1 k 1

while Residue not equal zero or not monotone

while Ii has non-negligible local

mean U(t) spline through local maxima of

Ii L(t) spline through local minima of

Ii Av(t) 1/2 (U(t) L(t)) Ii(t) Ii(t) -

Av(t) i i 1 end

IMFk(t) Ii(t) Residue Residue -

IMFk k k1 end

The Hilbert Spectrum

Now that we have a set of IMFs construct their

analytic counterparts using the Hilbert transform.

Now we can look at f(t) in time and frequency

space simultaneously. Instantaneous frequency is

given by the derivative of the phase angle. The

Hilbert spectrum H(?,t) gives the instantaneous

amplitude (energy) as a function of frequency. A

plot of H(?,t) provides an intuitive, visual

representation of the signal in time and

frequency.

Example EMD for a Test Signal with Known

Components.

Hilbert Spectrum for EMD of Example Signal.

Real-World Example Temperature Data from

Amherst, MA from 1988 - 2005.

(No Transcript)

Physical Interpretation of IMFs for Amherst

Temperature Data.

IMF 8

IMF 5

Huang-Hilbert Spectrum for EMD of Amherst

Temperature Data

-Most of the signal information contained in the

bottom 10 of all frequencies represented -Can we

make a correspondence between what we see here

and the IMFs of the decomposition? -This would

help identify which modes are important in

determining the character of the data.

Hilbert Spectrum for IMF 8

Hilbert Spectrum for IMF 7

Increasing Frequencies Decreasing Information!

IMF 4

IMF 5

IMF 6

(No Transcript)

Interpretation of Residue (Shows Overall Trend).

Bibliography.

- Langton, C. Hilbert Transform, Anayltic Signal

and the Complex Envelope Signal Processing and

Simulation Newsletter, 1999.

http//www.complextoreal.com/tcomplex.htm - Electrocardiograph signal graphic

http//www.neurotraces.com/scilab/scilab2/node61.h

tml - MATLAB code used for all computations, and

sifting graphics - Rilling, G. MATLAB code for computation of

EMD and illustrative slides. - http//perso.ens lyon.fr/patrick.flandrin/emd.h

tml - Temperature data from Amherst, MA
- Williams, C.N., Jr., M.J. Menne, R.S. Vose, and

D.R. Easterling. 2006. United States - Historical Climatology Network Daily

Temperature, Precipitation, and Snow Data. - ORNL/CDIAC-118, NDP-070. Online at

http//cdiac.ornl.gov/epubs/ndp/ushcn/usa.html

site MA190120 - Burning Globe graphic http//river2sea72.wordp

ress.com/2007/03/24/its-not-all-about-carbon-or-is

-it/

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