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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?