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HHT and Applications in Music Signal Processing

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?.

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HHT and Applications in Music Signal Processing

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  1. HHT and Applications in Music Signal Processing 電信一 R01942128 陳昱安

  2. About the Presenter • Research area: MER • Not quite good at difficult math

  3. About the Topic • HHT : abbreviation ofHilbert-Huang Transform • Decided after the talk given byDr. Norden E. Huang

  4. Why HHT? • Fourier is nice, but not good enough • Clarity • Non-linear and non-stationary signals

  5. Hilbert-Huang Transform Hilbert Transform Empirical Mode Decomposition

  6. Hilbert Transform • Not integrable at τ=t • Defined using Cauchy principle value

  7. Dealing with 1/(τ-t) =0 -∞ ∞ τ=t

  8. Quick Table

  9. I know how to compute Hilbert Transform

  10. That’s cool… SO WHAT?

  11. exp(jz) =cos(z) + jsin(z) • exp(jωt) =cos(ωt) + jsin(ωt) • θ(t) = arctan(sin(ωt)/cos(ωt)) • Freq.=dθ/dt

  12. 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)

  13. Frequency Analysis with HT • 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

  14. Congrats!!! Forgot something?

  15. Hilbert-Huang Transform Hilbert Transform Empirical Mode Decomposition

  16. 0 8 F = 1Hz

  17. 0 8 F = 1Hz

  18. 0 8 F = 1Hz

  19. 0 8 F = 1/8Hz 1Hz

  20. = +

  21. To make instantaneous frequency MEANINGFUL

  22. Need to decompose signals into “BASIC” components

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

  24. Criteria of IMF • 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.

  25. TO BE SHORT

  26. IMFs are signals

  27. Oscillate

  28. Around 0 0

  29. Review: EMD • Empirical Mode Decomposition • Used to generate IMFs EMD

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

  31. Application I:Source Separation

  32. Problem

  33. Problem Source Separation

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

  35. Problem Solved? No!

  36. Gabor Transform of piano

  37. Gabor Transform of organ

  38. Gabor Transform of piano + organ

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

  40. How to win Doraemon in paper-scissor-stone? Easy. Paper always win.

  41. The tip is to know the answer first!

  42. Single-MixtureAudio Source Separationby Subspace Decompositionof Hilbert Spectrum Khademul Islam Molla, and Keikichi Hirose

  43. Approximation of sources Desired result

  44. PHASE I: Construction of possible source model

  45. IMFs EMD HilbertTransform HilbertSpectra Original Signal IMF 1 IMF 2 IMF 3 ∶ Spectrum of

  46. Spectrum of original signal Spectrum of IMF1 Spectrum of IMF2 X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6 ... frequency

  47. Original Signal Projection 2 IMF1 Projection1 IMF2

  48. AFTER SOME PROCESSING

  49. RESULTS IN

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