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README

README. Lecture notes will be animated by clicks. Each click will indicate pause for audience to observe slide. On further click, the lecturer will explain the slide with highlighted notes. STFT as Filter Bank. Introduction to Wavelet Transform Yen-Ming Lai Doo-hyun Sung.

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README

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  1. README • Lecture notes will be animated by clicks. • Each click will indicate pause for audience to observe slide. • On further click, the lecturer will explain the slide with highlighted notes.

  2. STFT as Filter Bank Introduction to Wavelet Transform Yen-Ming Lai Doo-hyun Sung November 15, 2010 ENEE630, Project 1

  3. Wavelet Tutorial Overview • DFT as filter bank • STFT as filter bank • Wavelet transform as filter bank

  4. Discrete Fourier Transform

  5. DFT for fixed w_0

  6. DFT for fixed w_0 fix specific frequency w_0

  7. DFT for fixed w_0 pass in input signal x(n)

  8. DFT for fixed w_0 modulate by complex exponential of frequency w_0

  9. DFT for fixed w_0 summation = convolve result with “1”

  10. Why is summation convolution?

  11. Why is summation convolution? start with definition

  12. Why is summation convolution? Let

  13. Why is summation convolution?

  14. Why is summation convolution? convolution with 1 equivalent to summation

  15. DFT for fixed w_0 summation = convolve result with “1”

  16. DFT for fixed w_0 output X(e^jw_0) is constant

  17. DFT for fixed w_0

  18. DFT for fixed w_0 input signal x(n)

  19. DFT for fixed w_0 fix specific frequency w_0

  20. DFT for fixed w_0 modulate by complex exponential of frequency w_0

  21. DFT for fixed w_0 summation = convolve with “1”

  22. DFT for fixed w_0 Transfer function H(e^jw)

  23. DFT for fixed w_0 summation = convolution with 1

  24. DFT for fixed w_0 i.e. impulse response h(n) = 1 for all n

  25. DFT for fixed w_0

  26. DFT for fixed w_0 output X(e^jw_0) is constant

  27. Frequency Example

  28. Frequency Example Arbitrary example

  29. Frequency Example modulation = shift

  30. Frequency Example convolution by 1 = multiplication by delta

  31. DFT as filter bank

  32. DFT as filter bank fix specific frequency w_0

  33. DFT as filter bank one filter bank

  34. DFT as filter bank w continuous between [0,2pi)

  35. DFT as filter bank … uncountably many filter banks

  36. DFT as filter bank … Uncountable  cannot enumerate all (even with infinite number of terms)

  37. DFT as filter bank … bank of modulators of all frequencies between [0, 2pi)

  38. DFT as filter bank … bank of identical filters with impulse response of h(n) = 1

  39. Short-Time Fourier Transform

  40. Short-Time Fourier Transform two variables

  41. Short-Time Fourier Transform frequency

  42. Short-Time Fourier Transform shift

  43. Short-Time Fourier Transform shifted window function v(k)

  44. Short-Time Fourier Transform let dummy variable be n instead of k

  45. Short-Time Fourier Transform fix frequency w_0 and shift m

  46. Short-Time Fourier Transform pass in input x(n)

  47. Short-Time Fourier Transform multiply by shifted window and complex exponential

  48. Short-Time Fourier Transform summation = convolve with 1

  49. Short-Time Fourier Transform output constant determined by frequency w_0 and shift m

  50. Short-Time Fourier Transform

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