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Shearing

Shearing. Name : Yi- wei chen Student number : r02942096. Shearing. Sheep shearing? To remove (fleece or hair) by cutting or clipping. Motions in the time-frequency distribution Multiply chirp function Generalized shearing phase is a polynomial HOW to do?.

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Shearing

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  1. Shearing Name : Yi-weichen Student number : r02942096

  2. Shearing • Sheep shearing? • To remove (fleece or hair) by cutting or clipping. • Motions in the time-frequency distribution • Multiply chirp function • Generalized shearing • phase is a polynomial • HOW to do?

  3. HIGHER ORDER MODULATION AND THE EFFICIENT SAMPLING ALGORITHM FORTIME VARIANT SIGNAL Jian-Jiun Ding, Soo Chang Pei, and Ting Yu Ko Department of Electrical Engineering, National Taiwan University 20th European Signal Processing Conference(EUSIPCO 2012)

  4. Abstract

  5. Abstract • Higher order modulation scheme • High order modulation with the fractional Fourier transform • Minimize the area of a signal in the time-frequency domain • Much reduce the number of sampling points • Efficient for sampling a time variant signal (ex : voice of an animal and the speech signal)

  6. Introduction

  7. Shannon’s sampling theory • fs: the sampling frequency • △ = 1/fs: the sampling interval • F : the total bandwidth • The sampling frequency should be larger than the Nyquistrate • suppose that the support of a signal is T(x(t )~= 0 for t < t0 and t > t0+T), its bandwidth is F: • TF valuedetermines the lower bound of sampling points

  8. Fundamental harmonic part of a whale voice signal STFT : TF value = 2.1*1000 = 2100 conventional modulation : TF value = 2.1* 100 = 210 (f1 = 440Hz)

  9. Conventional modulation • analytic signal form : • Xa(f) = X(f) for f > 0 and Xa(f) = 0 for f < 0 • the conventional modulation operation: • f1 is chosen as 440 Hz

  10. Sampling algorithm The innovation is…… • the higher order exponential function is adopted for modulation • reducing the aliasing effect before sampling • the fractional Fourier transform • the signal segmentation technique • the pre-filter

  11. Higher order modulation

  12. Generalized modulation operation • m(t) is an nth order polynomial, and the instantaneous frequency: • The STFT relations between x(t) & y(t)

  13. Central frequency The central frequency (varies with time) of the whale voice signal Using a 5th order polynomial to approximate the central frequency of the whale voice(P5(t))

  14. Approximation • Legendre polynomial expansion: central frequency of the signal is h(t) • [t0, t0+T] is the support of h(t) • {Lk(t) | k = 0, 1, 2, …} is the Legendre polynomial set • For this example :

  15. X2(t) • The STFT of x2(t) where x2(t) is the result of proposed high order modulation of the analytic signal of the whale voice • TF value = 35*2.1 = 73.5

  16. Combining higher order modulation with the fractional fourier transform

  17. Fundamental harmonic part of a whale voice signal conventional modulation (f1 = 400Hz) After performing the FRFT and the scaling operation, the STFT is rotated(x3)

  18. FRFT • “signal segmentation” and “bandwidth reduction” • Definition : • performing the Fourier transform 2α/π times • placing a separating line : where H(u) = 1 for u < u0 and H(u) = 0 for u > u0 • Scaling + rotating

  19. Fundamental harmonic part of a whale voice signal The 5th order polynomial (black line) to approximate the central frequency after the scale FRFT + proposed high order modulation (TF value = 21*2.4 = 50.4) (x4)

  20. X4(t) • according to the 5th order polynomial that can approximate the central frequency of x3(t)

  21. TF value • (a) The original sampling algorithm. • (b) Analytic signal conversion + modulation. • (c) Analytic signal conversion + FRFT + modulation. • (d) Analytic signal conversion + FRFT + proposed higher order modulation

  22. Results

  23. Reconstruction • sinc function interpolation is the inverseof the sampling operation • removing the imaginary part is the inverse of the analytic function generation operation

  24. Other simulations

  25. Another whale voice signal

  26. Speech signal : “for” STFT of the first harmonic part of the speech signal The STFT of the analytic signal conversion + conventionalmodulation + scaled FRFT operations

  27. Speech signal : “for” A 5th order polynomial (black line) to approximate the central frequency The STFT of the signal after high order modulation

  28. Speech signal : “for”

  29. Conclusion

  30. Conclusion • A new signal sampling algorithm : • the higher order modulation operation • the STFT • the FRFT filter • The number of sampling points is very near to the area of the nonzero region • much fewer number of sampling points to represent a signal • Other applications : • data transmission • communication

  31. Thank you

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