Hilbert-Huang Transform and Applications in Music Signal Processing

Report
電信一 R01942128 陳昱安
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
Research area: MER

Not quite good at difficult math
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 HHT
: abbreviation of
Hilbert-Huang Transform
 Decided
after the talk given by
Dr. Norden E. Huang
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
Fourier is nice, but not good enough

Clarity

Non-linear and non-stationary signals
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Hilbert Transform
Empirical Mode
Decomposition
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1

u (t )
 (t )  H {u (t )}  
d
  t  


Not integrable at τ=t
Defined using Cauchy principle value
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=0
-∞
∞
τ=t
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Input u(t)
Output H{u}
sin(t)
-cos(t)
cos(t)
sin(t)
exp(jt)
-jexp(jt)
exp(-jt)
jexp(-jt)
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 exp(jz)
=
cos(z) + jsin(z)
 exp(jωt)
=
cos(ωt) + jsin(ωt)

θ(t) = arctan(sin(ωt)/cos(ωt))
 Freq.=dθ/dt
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 S(t)
= u(t) + jH{u(t)}
 θ(t) = arctan(Im/Re)
 Freq.=dθ/dt
 What
happen if u(t) = cos(ωt) ?
Hint:
H{cos(t)} = sin(t)
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
Input : u(t)

Calculate v(t) = H{u(t)}

Set s(t) = u(t) + jv(t)

θ(t) = arctan(v(t)/u(t))

fu(t)= d θ(t) /dt
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Hilbert Transform
Empirical Mode
Decomposition
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0
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0
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0
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0
8
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
Decompose the input signal

Goal: find “basic” components

Also know as IMF
 Intrinsic

Mode Functions
BASIC means what?
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1)
2)
num of extrema num of zero-crossings
≤1
At any point, the mean value of the
envelope defined by the local
maxima and the envelope defined
by the local minima is zero.
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0
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
Empirical Mode Decomposition

Used to generate IMFs
EMD
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Hint:

Empirical Mode Decomposition

Used to generateEmpirical
IMFs
means
NO PRIOR
KNOWLEDGES
EMD
NEEDED
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Source
Separation
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What if…
We apply STFT, then
extract different components
from different freq. bands?
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Gabor Transform of piano
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Gabor Transform of organ
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Gabor Transform of piano + organ
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I see…
So how to make sure we do it right?
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The tip is to know the answer first!
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Single-Mixture
Audio Source Separation
by Subspace Decomposition
of Hilbert Spectrum
Khademul Islam Molla, and Keikichi Hirose
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Approximation of sources
Desired result
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EMD
IMFs
Hilbert
Transform
Spectrum of
Original Signal
IMF 1
IMF 2
IMF 3
∶
Hilbert
Spectra
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Spectrum of
IMF1
Spectrum of
IMF2
X1
X1
X1
X2
X2
X2
X3
X3
X3
X4
X4
X4
X5
X5
X5
X6
X6
X6
frequency
Spectrum of
original signal
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Projection 2
Original
Signal
IMF1
Projection 1
IMF2
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Frequency Band II
Frequency Band I
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Frequency Band II
Hint:
Data points are
different
observations
Frequency Band I
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Frequency Band II
So…
What does this
basis mean?
Frequency Band I
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Frequency Band II
3F1 +4F2
7F1 +2F2
Frequency Band I
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Gabor Transform of piano
F(piano) = 10F1 + 9F2 + F3
3F1 + 4F2
7F1 + 2F2
3F2 + F3
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
The “figure” of sources obtained

We have been through
1) EMD : Obtain IMFs
2) Hilbert Transform : Construct spectra
3) Projection : Decompose signal in frequency space
4) PCA and ICA : Independent vector basis
5) Clustering : Combine correlated vectors together
6) Voila!
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
Spectrum of each source is a linear
combination of the vector basis generated
Signal
H
Spectrum


T
Combination
yi ai spectra
of sources’
i 1
H  YA
T
, Y  [ y1 y2 ...y  ]; A  [a1 a2 ...a ]
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
Let the clustered vector basis to be Yj

Then the weighting of this subspace is
1
j
A Y Hj
T
j
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H j  Y j Aj
T
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 Why
HHT?
◦ EMD needs NO PRIOR KNOWLEDGE
◦ Hilbert transform suits for non-linear
and non-stationary condition
 However,
clustering…
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STFT of C4(262Hz)
Music Instrument
Samples of U. Iowa
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FUNDAMENTAL FREQUENCY
ESTIMATION FOR MUSIC
SIGNALS WITH
MODIFIED HILBERT-HUANG
TRANSFORM
EnShuo Tsau, Namgook Cho and C.-C. Jay Kuo
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EMD
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 Mode
mixing
 Extrema
finding
◦ Boundary effect
◦ Signal perturbation
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1.
2.
3.
4.
Kizhner, S.; Flatley, T.P.; Huang, N.E.; Blank, K.; Conwell, E.; ,
"On the Hilbert-Huang transform data processing system
development," Aerospace Conference, 2004. Proceedings.
2004 IEEE , vol.3, no., pp. 6 vol. (xvi+4192), 6-13 March 2004
Md. Khademul Islam Molla; Keikichi Hirose; , "Single-Mixture
Audio Source Separation by Subspace Decomposition of Hilbert
Spectrum," Audio, Speech, and Language Processing, IEEE
Transactions on , vol.15, no.3, pp.893-900, March 2007
EnShuo Tsau; Namgook Cho; Kuo, C.-C.J.; , "Fundamental
frequency estimation for music signals with modified HilbertHuang transform (HHT)," Multimedia and Expo, 2009. ICME
2009. IEEE International Conference on , vol., no., pp.338-341,
June 28 2009-July 3 2009
Te-Won Lee; Lewicki, M.S.; Girolami, M.; Sejnowski, T.J.; ,
"Blind source separation of more sources than mixtures using
overcomplete representations," Signal Processing Letters, IEEE ,
vol.6, no.4, pp.87-90, April 1999
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請把握加分的良機
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THE END
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Input u(t)
sin(t)
Output H{u}
Insight:
-cos(t)
Hilbert transform
cos(t) rotate input by
sin(t)
π/2
on
complex
plane
exp(jt)
-jexp(jt)
exp(-jt)
jexp(-jt)
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EMD
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Spectrum of
original signal
Spectrum of
IMF1
Spectrum of
IMF2
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Projection 2
Original
Signal
IMF1
Projection 1
IMF2
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Fact:
PCA & ICA are
linear transforms
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