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 請把握加分的良機 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