The Hilbert Transform and Empirical Mode Decomposition:

1
The Hilbert Transform and Empirical Mode
Decomposition
Powerful Tools for Data Analysis

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

2
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
3
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.
4
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
5
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))
6
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.

7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
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
26
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
27
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
28
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
29
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
30
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
31
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
32
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
33
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
34
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
35
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.
36
Example EMD for a Test Signal with Known
Components.
37
Hilbert Spectrum for EMD of Example Signal.
38
Real-World Example Temperature Data from
Amherst, MA from 1988 - 2005.
39
(No Transcript)
40
Physical Interpretation of IMFs for Amherst
Temperature Data.
IMF 8
IMF 5
41
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.
42
Hilbert Spectrum for IMF 8
43
Hilbert Spectrum for IMF 7
44
Increasing Frequencies Decreasing Information!
IMF 4
IMF 5
IMF 6
45
(No Transcript)
46
Interpretation of Residue (Shows Overall Trend).
47
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/