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An Introduction to Time-Frequency Analysis

An Introduction to Time-Frequency Analysis. Speaker: Po-Hong Wu Advisor: Jian-Jung Ding Digital Image and Signal Processing Lab GICE, National Taiwan University. Outline. Introduction Short-Time Fourier Transform Gabor Transform Wigner Distribution Function Spectrogram S Tranform

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An Introduction to Time-Frequency Analysis

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  1. An Introduction to Time-Frequency Analysis Speaker: Po-Hong Wu Advisor: Jian-Jung Ding Digital Image and Signal Processing Lab GICE, National Taiwan University NTU, GICE,MD531, DISP Lab

  2. Outline • Introduction • Short-Time Fourier Transform • Gabor Transform • Wigner Distribution Function • Spectrogram • S Tranform • Cohen’s Class Time-Frequency Distribution • Fractional Fourier Transform • Motion on Time-Frequency Distributions • Hilbert-Huang Transform • Conclusion • Reference NTU, GICE, MD531, DISP Lab

  3. Introduction Fourier transform (FT) t varies from ∞~∞ Time-Domain  Frequency Domain [A1] Why do we need time-frequency transform? NTU, GICE, MD531, DISP Lab

  4. Example: x(t) = cos( t) when t < 10, x(t) = cos(3 t) when 10 t < 20, x(t) = cos(2 t) when t  20 [B2] NTU, GICE, MD531, DISP Lab

  5. Short Time Fourier Transform w(t): mask function 也稱作 windowed Fourier transform or time-dependent Fourier transform NTU, GICE, MD531, DISP Lab

  6. When w(t) is a rectangular function w(t) = 1 for |t| B , w(t) = 0 , otherwise [B3] NTU, GICE, MD531, DISP Lab

  7. Advantage: less computation time • Disadvantage: worse representaion • Application: deal with large data Ex: real time processing NTU, GICE, MD531, DISP Lab

  8. Gabor Transform A specail case of the STFT where Other definition [B4] NTU, GICE, MD531, DISP Lab

  9. Why do we choose the Guassian function? • Among all functions of w(t), the Gaussian function has area in time-frequency distribution is minimal than other STFT. • Gaussian function is an eigenfunction of Fourier transform, so the Gabor transform has the same properties in time domain and in frequency domain. NTU, GICE, MD531, DISP Lab

  10. Approximation of the Gabor Transform Because of when |a|>1.9143 Because of when |a|>4.7985 NTU, GICE, MD531, DISP Lab

  11. Generalization of the Gabor Transform • For larger σ: higher resolution in the time domain but lower resolution in the frequency domain • For smaller σ: higher resolution in the frequency domain but lower resolution in the time domain NTU, GICE, MD531, DISP Lab

  12. Resolution • Using the generalized Gabor transform with larger σ • Using other time unit instead of second NTU, GICE, MD531, DISP Lab

  13. Wigner Distribution Function Other definition [B5] NTU, GICE, MD531, DISP Lab

  14. Signal auto-correlation function Spectrum auto-correlation function Ambiguity function (AF) [B6] IFTf FTt Cx(t, ) IFTf FTt Ax(, ) Wx(t, f ) Sx(, f ) FTt IFTf NTU, GICE, MD531, DISP Lab

  15. Modified Wigner Distribution • Wigner Ville Distribution For compressing inner interference Analytic signal NTU, GICE, MD531, DISP Lab

  16. Pseudo Wigner Distribution For surpressing outer interference where [B7] NTU, GICE, MD531, DISP Lab

  17. Gabor-Wigner Distribution • [B8] NTU, GICE, MD531, DISP Lab

  18. Spectrogram Another form [B9] NTU, GICE, MD531, DISP Lab

  19. S-Transform • Original S-Transform Where w(t)= [B10] NTU, GICE, MD531, DISP Lab

  20. Generalized S-Transform Another definition Ristriction NTU, GICE, MD531, DISP Lab

  21. Novel S-Transform with the Special Varying Window Restriction When , it becomes the Gabor transform. When , it becomes the original S-trnasform. NTU, GICE, MD531, DISP Lab

  22. FTt IFTf FTt IFTf FTt IFTf Cohen’s Class Time-Frequency Distribution Ambiguity function [B11] NTU, GICE, MD531, DISP Lab

  23. For the ambiguity function • The auto terms are always near to the origin. • The cross terms are always from the origin. [B12] NTU, GICE, MD531, DISP Lab

  24. Kernel function • Choi-Williams Distribution [B13] NTU, GICE, MD531, DISP Lab

  25. Cone-Shape Distribution NTU, GICE, MD531, DISP Lab

  26. Fractional Fourier Transform How to rotate the time-frequency distribution by the angle other than /2, , and 3/2? NTU, GICE, MD531, DISP Lab

  27. Zero rotation: • Consistency with Fourier transform: = FT • Additivity of rotation: • rotation: NTU, GICE, MD531, DISP Lab

  28. [A3] NTU, GICE, MD531, DISP Lab

  29. Application Decomposition in the time-frequency distribution NTU, GICE, MD531, DISP Lab

  30. f-axis Signal noise noise noise Signal Signal FRFT FRFT t-axis cutoff line cutoff line NTU, GICE, MD531, DISP Lab

  31. Modulation and Multiplexing NTU, GICE, MD531, DISP Lab

  32. Time domain Frequency domain fractional domain • Modulation Shifting Modulation + Shifting • Shifting Modulation Modulation+ Shifting • Differentiation j2f Differentiation and j2f • −j2f Differentiation Differentiationand −j2f NTU, GICE, MD531, DISP Lab

  33. Motion on Time-Frequency Distributions • Horizontal Shifting • Vertical Shifting NTU, GICE, MD531, DISP Lab

  34. Dilation • Shearing NTU, GICE, MD531, DISP Lab

  35. Rotation If F{x(t)}=X(f), then F{X(t)}=x(-f). We can derive: NTU, GICE, MD531, DISP Lab

  36. Hilbert-Huang Transform • Introduction Most of distribution are designed for stationary and linear signals, but, In the real world, most of signals are non-stationary and non-linear. HHT consists two parts: • empirical mode decomposition (EMD) • Hilbert spectral analysis (HSA) NTU, GICE, MD531, DISP Lab

  37. Empirical decomposition function Any complicated data can be decomposed into a finite and small number of intrinsic mode functions (IMF) by sifting processing. • Intrinsic mode function (1)In the whole data set, the number of extrema and the number of zero-crossing must either equal or differ at most by one. (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. NTU, GICE, MD531, DISP Lab

  38. IMF 1; iteration 0 IMF 1; iteration 0 2 2 1 1 0 0 -1 -1 -2 -2 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100 100 110 110 120 120 • Sifting Process • First, find all the local maxima extrema of x(t). (2) Interpolate (cubic spline fitting) between all the maxima extrema ending up with some upper envelope . NTU, GICE, MD531, DISP Lab

  39. IMF 1; iteration 0 IMF 1; iteration 0 2 2 1 1 0 0 -1 -1 -2 -2 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100 100 110 110 120 120 (3) Find all the local minima extrema. (4) Interpolate (cubic spline fitting) between all the minima extrema ending up with some lower envelope . NTU, GICE, MD531, DISP Lab

  40. IMF 1; iteration 0 2 1 0 -1 -2 10 20 30 40 50 60 70 80 90 100 110 120 residue 1.5 1 0.5 0 -0.5 -1 -1.5 10 20 30 40 50 60 70 80 90 100 110 120 (5) Compute the mean envelope between upper envelope and lower envelope. (6) Compute the residue NTU, GICE, MD531, DISP Lab

  41. (7) Repeat the above procedure (step (1) ~ step (6)) on the residue until the residue is a monotonic function or constant. The original signal equals the sum of the various IMFs plus the residual trend. NTU, GICE, MD531, DISP Lab

  42. EX: NTU, GICE, MD531, DISP Lab

  43. NTU, GICE, MD531, DISP Lab

  44. NTU, GICE, MD531, DISP Lab

  45. NTU, GICE, MD531, DISP Lab

  46. Hilbert Spectral Anaysis NTU, GICE, MD531, DISP Lab

  47. NTU, GICE, MD531, DISP Lab

  48. Conclusion • We introduce many distributions here and put most attention on computation time and representations. We can find that the representation with higher clarity cost more computation time for all methods. Resolution Computation time • The Hilbert-Huang transform is the most power method to deal with non-linear and non-stationary signals but lacks of physical background. NTU, GICE, MD531, DISP Lab

  49. Reference [1][A]J. J. Ding, “Time-Frequency Analysis and Wavelet Transform,” National Taiwan University, 2009. [Online].Available: http://djj.ee.ntu.edu.tw/TFW.htm. [2][B]W. F. Wang, “Time-Frequency Analyses and Their Fast Implementation Algorithm,” Master Thesis, National Taiwan University, June, 2009. [3]Luis B. Almeida, Member, IEEE, “The Fractional Fourier Transform and Time-Frequency Representations,” IEEE Transaction On Signal Processing, vol. 42, no. 11, November 1994. [4]M. R. Spiegel, Mathematical Handbook of Formulas and Tables, McGraw-Hill, 1990. [5]N. E. Huang, Z. Shen and S. R. Long, et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time Series Analysis＂, Proc. Royal Society, vol. 454, pp.903-995, London, 1998. NTU, GICE, MD531, DISP Lab

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