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HHT and Applications in Music Signal Processing. 電信一 R01942128 陳昱安. About the Presenter. Research area: MER Not quite good at difficult math. About the Topic. HHT : abbreviation of Hilbert-Huang Transform Decided after the talk given by Dr. Norden E. Huang. Why HHT?. - PowerPoint PPT Presentation
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HHT and Applications in Music Signal Processing
電信一 R01942128 陳昱安
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Research area: MER
Not quite good at difficult math
About the Presenter
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HHT : abbreviation ofHilbert-Huang Transform
Decided after the talk given byDr. Norden E. Huang
About the Topic
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Fourier is nice, but not good enough Clarity Non-linear and non-stationary signals
Why HHT?
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Hilbert-Huang Transform
Hilbert Transform Empirical Mode Decomposition
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Hilbert Transform
dttutuHt
)(1)}({)(
Not integrable at τ=t Defined using Cauchy principle value
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Dealing with 1/(τ-t)
-∞ ∞τ=t
=0
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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)
Quick Table
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I know how tocompute
Hilbert Transform
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That’s cool…SO WHAT?
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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
Frequency Analysis with HT
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Congrats!!!Forgot something?
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Hilbert-Huang Transform
Hilbert Transform Empirical Mode Decomposition
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0 8
F = 1Hz
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0 8
F = 1Hz
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0 8
F = 1Hz
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1Hz
0 8
F = 1/8Hz
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=+
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To makeinstantaneous frequency
MEANINGFUL
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Need to decomposesignals
into “BASIC” components
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Decompose the input signal Goal: find “basic” components Also know as IMFIntrinsic Mode Functions BASIC means what?
Empirical Mode Decomposition
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1) num of extrema - num of zero-crossings≤ 1
2) At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.
Criteria of IMF
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TO BESHORT
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IMFsare signals
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Oscillate
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Around 00
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Empirical Mode Decomposition Used to generate IMFs
Review: EMD
EMD
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Empirical Mode Decomposition Used to generate IMFs
Review: EMD
EMD
Hint:Empirical means
NO PRIOR KNOWLEDGES
NEEDED
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Application I:Source Separation
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Problem
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Problem
Source Separatio
n
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What if…We apply STFT, then
extract different componentsfrom different freq. bands?
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Problem Solved?No!
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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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How to win Doraemonin paper-scissor-stone?Easy. Paper always win.
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The tip is to know the answer first!
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Single-MixtureAudio Source Separation
by Subspace Decompositionof Hilbert Spectrum
Khademul Islam Molla, and Keikichi Hirose
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Approximation of sources
Desired result
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PHASE I:Construction of
possible source model
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HilbertSpectra
IMFsEMDHilbert
TransformOriginal Signal
IMF 1IMF 2IMF 3
∶
Spectrum of
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X1
X2
X3
X4
X5
X6
Spectrum of original signal
X1
X2
X3
X4
X5
X6
Spectrum of IMF1
X1
X2
X3
X4
X5
X6
Spectrum of IMF2 ..
.frequ
ency
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Original Signal
IMF1
IMF2Projection 1
Projection 2
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AFTERSOME
PROCESSING
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RESULTSIN
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INDEPENDENTBASIS
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Frequency Band I
Freq
uenc
y Ba
nd II
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Frequency Band I
Freq
uenc
y Ba
nd IIHint:
Data points are different
observations
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Frequency Band I
Freq
uenc
y Ba
nd IISo…
What does this basis
mean?
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Frequency Band I
Freq
uenc
y Ba
nd II
7F1 +2F2
3F1 +4F2
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Gabor Transform of piano
F(piano) = 10F1 + 9F2 + F3
3F1 + 4F2
7F1 + 2F2
3F2 + F3
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FindSourceModels
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ClusterBasis
Vectors
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Cluster
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Raw data
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Clustered
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Approximated SourcesIn hand!
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The “figure” of sources obtained We have been through
1) EMD : Obtain IMFs2) Hilbert Transform : Construct spectra3) Projection : Decompose signal in frequency space4) PCA and ICA : Independent vector basis5) Clustering : Combine correlated vectors together6) Voila!
Finally
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PHASE II:Reconstruction of
separated source signals
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Spectrum of each source is a linear combination of the vector basis generated
Reconstruction
]... [];... [ , 2121
1
aaaAyyyYYAH
ayH
T
i
Tii
SignalSpectrum
Combinationof sources’
spectra
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Let the clustered vector basis to be Yj
Then the weighting of this subspace is
Reconstruction
1Tj j jA Y H
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Tjjj AYH
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Why HHT?◦EMD needs NO PRIOR KNOWLEDGE◦Hilbert transform suits for non-linear and non-stationary condition
However, clustering…
Conclusions
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Application II:Fundamental
Frequency Analysis
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ProblemSTFT of C4(262Hz)
Music Instrument Samples of U. Iowa
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FUNDAMENTAL FREQUENCY
ESTIMATION FOR MUSIC SIGNALS WITH
MODIFIED HILBERT-HUANG TRANSFORMEnShuo Tsau, Namgook Cho and C.-C. Jay Kuo
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Basic Idea
EMD
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Mode mixingExtrema finding
◦Boundary effect◦Signal perturbation
Problems of EMD
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1. 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
2. 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
3. EnShuo Tsau; Namgook Cho; Kuo, C.-C.J.; , "Fundamental frequency estimation for music signals with modified Hilbert-Huang transform (HHT)," Multimedia and Expo, 2009. ICME 2009. IEEE International Conference on , vol., no., pp.338-341, June 28 2009-July 3 2009
4. 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
References
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Q&A請把握加分的良機
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Thank you for your attention!
THE END
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Recycle bin
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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)
Quick Table
Insight:Hilbert transform
rotate input by π/2on complex plane
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EMD
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~!@#$%︿&*
Spectrum of original signal
*&︿%$#@!~
Spectrum of IMF1
~@!#$︿%&*
Spectrum of IMF2
…
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Original Signal
IMF1
IMF2Projection 1
Projection 2
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Data Spread on Vector Space
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PCA ICA
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PCAICA
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PCAICA
Fact:PCA & ICA are
linear transforms
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FAQ
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Q: Why Hilbert Transform?