#### Transcript Hilbert-Huang Transform and Applications in Music Signal Processing

```電信一 R01942128 陳昱安
1

Research area: MER

Not quite good at difficult math
2
 HHT
: abbreviation of
Hilbert-Huang Transform
 Decided
after the talk given by
Dr. Norden E. Huang
3

Fourier is nice, but not good enough

Clarity

Non-linear and non-stationary signals
4
Hilbert Transform
Empirical Mode
Decomposition
5
1

u (t )
 (t )  H {u (t )}  
d
  t  


Not integrable at τ=t
Defined using Cauchy principle value
6
=0
-∞
∞
τ=t
7
Input u(t)
Output H{u}
sin(t)
-cos(t)
cos(t)
sin(t)
exp(jt)
-jexp(jt)
exp(-jt)
jexp(-jt)
8
9
10
 exp(jz)
=
cos(z) + jsin(z)
 exp(jωt)
=
cos(ωt) + jsin(ωt)

θ(t) = arctan(sin(ωt)/cos(ωt))
 Freq.=dθ/dt
11
 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)
12

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
13
14
Hilbert Transform
Empirical Mode
Decomposition
15
0
8
16
0
8
17
0
8
18
0
8
19
20
21
22

Decompose the input signal

Goal: find “basic” components

Also know as IMF
 Intrinsic

Mode Functions
BASIC means what?
23
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.
24
25
26
27
0
28

Empirical Mode Decomposition

Used to generate IMFs
EMD
29
Hint:

Empirical Mode Decomposition

Used to generateEmpirical
IMFs
means
NO PRIOR
KNOWLEDGES
EMD
NEEDED
30
31
32
Source
Separation
33
What if…
We apply STFT, then
extract different components
from different freq. bands?
34
35
Gabor Transform of piano
36
Gabor Transform of organ
37
Gabor Transform of piano + organ
38
I see…
So how to make sure we do it right?
39
40
41
The tip is to know the answer first!
42
Single-Mixture
Audio Source Separation
by Subspace Decomposition
of Hilbert Spectrum
Khademul Islam Molla, and Keikichi Hirose
43
Approximation of sources
Desired result
44
45
EMD
IMFs
Hilbert
Transform
Spectrum of
Original Signal
IMF 1
IMF 2
IMF 3
∶
Hilbert
Spectra
46
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
47
Projection 2
Original
Signal
IMF1
Projection 1
IMF2
48
49
50
51
Frequency Band II
Frequency Band I
52
Frequency Band II
Hint:
Data points are
different
observations
Frequency Band I
53
Frequency Band II
So…
What does this
basis mean?
Frequency Band I
54
Frequency Band II
3F1 +4F2
7F1 +2F2
Frequency Band I
55
Gabor Transform of piano
F(piano) = 10F1 + 9F2 + F3
3F1 + 4F2
7F1 + 2F2
3F2 + F3
56
57
58
59
60
61
62

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!
63
64

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 ]
65

Let the clustered vector basis to be Yj

Then the weighting of this subspace is
1
j
A Y Hj
T
j
66
H j  Y j Aj
T
67
 Why
HHT?
◦ EMD needs NO PRIOR KNOWLEDGE
◦ Hilbert transform suits for non-linear
and non-stationary condition
 However,
clustering…
68
69
STFT of C4(262Hz)
Music Instrument
Samples of U. Iowa
70
FUNDAMENTAL FREQUENCY
ESTIMATION FOR MUSIC
SIGNALS WITH
MODIFIED HILBERT-HUANG
TRANSFORM
EnShuo Tsau, Namgook Cho and C.-C. Jay Kuo
71
EMD
72
 Mode
mixing
 Extrema
finding
◦ Boundary effect
◦ Signal perturbation
73
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
74

75
THE END
76
77
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)
78
EMD
79
Spectrum of
original signal
Spectrum of
IMF1
Spectrum of
IMF2
～
！
＠
＃
＄
％
︿
＆
＊
＊
＆
︿
％
＄
＃
＠
！
～
～
＠
！
＃
＄
︿
％
＆
＊
80
Projection 2
Original
Signal
IMF1
Projection 1
IMF2
81
82
83
84
Fact:
PCA & ICA are
linear transforms
85
86
87
```