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電信一 R01942128 陳昱安 1

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Research area: MER Not quite good at difficult math 2

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HHT : abbreviation of Hilbert-Huang Transform Decided after the talk given by Dr. Norden E. Huang 3

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Fourier is nice, but not good enough Clarity Non-linear and non-stationary signals 4

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5 Hilbert Transform Empirical Mode Decomposition

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6 Not integrable at τ=t Defined using Cauchy principle value

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7 -∞-∞∞ 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) 8

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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 11

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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) ? 12 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)) f u (t)= d θ(t) /dt 13

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15 Hilbert Transform Empirical Mode Decomposition

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Decompose the input signal Goal: find “basic” components Also know as IMF Intrinsic Mode Functions BASIC means what? 23

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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. 24

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Empirical Mode Decomposition Used to generate IMFs 29 EMD

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Empirical Mode Decomposition Used to generate IMFs 30 EMD Hint: Empirical means NO PRIOR KNOWLEDGES NEEDED

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33 Source Separation

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34 What if… We apply STFT, then extract different components from different freq. bands?

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36 Gabor Transform of piano

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Gabor Transform of organ 37

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Gabor Transform of piano + organ 38

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39 I see… So how to make sure we do it right?

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42 The tip is to know the answer first!

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43 Single-Mixture Audio Source Separation by Subspace Decomposition of Hilbert Spectrum Khademul Islam Molla, and Keikichi Hirose

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44 Approximation of sources Desired result

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Hilbert Spectra IMFs 46 EMD Hilbert Transform Original Signal IMF 1 IMF 2 IMF 3 ∶ Spectrum of

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47 X1X2X3X4X5X6X1X2X3X4X5X6 X1X2X3X4X5X6X1X2X3X4X5X6 Spectrum of original signal X1X2X3X4X5X6X1X2X3X4X5X6 X1X2X3X4X5X6X1X2X3X4X5X6 Spectrum of IMF1 X1X2X3X4X5X6X1X2X3X4X5X6 X1X2X3X4X5X6X1X2X3X4X5X6 Spectrum of IMF2 frequency

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48 Original Signal IMF1 IMF2 Projection 1 Projection 2

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52 Frequency Band I Frequency Band II

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53 Frequency Band I Frequency Band II Hint: Data points are different observations

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54 Frequency Band I Frequency Band II So… What does this basis mean?

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55 Frequency Band I Frequency Band II 7F 1 +2F 2 3F 1 +4F 2

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56 Gabor Transform of piano F(piano) = 10F 1 + 9F 2 + F 3 3F 1 + 4F 2 7F 1 + 2F 2 3F 2 + F 3

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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! 63

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Spectrum of each source is a linear combination of the vector basis generated 65 Signal Spectrum Combination of sources’ spectra

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Let the clustered vector basis to be Y j Then the weighting of this subspace is 66

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Why HHT? ◦ EMD needs NO PRIOR KNOWLEDGE ◦ Hilbert transform suits for non-linear and non-stationary condition However, clustering… 68

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70 STFT of C4(262Hz) Music Instrument Samples of U. Iowa

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71 FUNDAMENTAL FREQUENCY ESTIMATION FOR MUSIC SIGNALS WITH MODIFIED HILBERT-HUANG TRANSFORM EnShuo Tsau, Namgook Cho and C.-C. Jay Kuo

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72 EMD

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Mode mixing Extrema finding ◦ Boundary effect ◦ Signal perturbation 73

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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 74

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請把握加分的良機 75

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THE END 76

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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) 78 Insight: Hilbert transform rotate input by π/2 on complex plane

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79 EMD

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80 Spectrum of original signal Spectrum of IMF1 Spectrum of IMF2

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81 Original Signal IMF1 IMF2 Projection 1 Projection 2

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85 Fact: PCA & ICA are linear transforms

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