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Presenter : 施 信 毓 Date : 2007/11/29

Another Point of View : FT - JTFA From Fourier Transform to Joint Time-Frequency Analysis. Presenter : 施 信 毓 Date : 2007/11/29. Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 106, Taiwan. Outline. Fourier Transform Joint Time-Frequency Analysis

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Presenter : 施 信 毓 Date : 2007/11/29

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  1. Another Point of View : FT - JTFAFrom Fourier Transform to Joint Time-Frequency Analysis Presenter : 施 信 毓 Date : 2007/11/29 Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 106, Taiwan

  2. Outline • Fourier Transform • Joint Time-Frequency Analysis • Linear Time-Frequency Method • Quadratic Time-Frequency Method • Short-time Fourier Transform & Spectrogram • Wavelet Transform & Scalogram • Conclusion • Reference

  3. Time Signal Frequency Spectrum Fourier Transform • Fourier Transform (FT) : Convert the time-domain signals into frequency-domain spectrum. • Example : Linear chirp signal

  4. “Zoom” Fourier Transform Analysis (1/3) • Zoom FT : Concentrates (“zooms”) FFT on a narrow band of frequencies. • Pros : • Improves frequency resolution • Distinguishes between closely-spaced frequencies • Cons : • Baseband analysis requires longer acquisition time for better resolution – requires more computation

  5. “Zoom” Fourier Transform Analysis (2/3) Baseband FT Analysis “Zoom” FT Analysis

  6. LPF “Zoom” Fourier Transform Analysis (3/3) • How to implement ?

  7. Do not appear at all times Magnitude Magnitude Time (s) Frequency (Hz) Limitation of FT (1/2) • It’s not suitable for time-varying signals. • Example : non-stationary signals T = 0.0 ~ 0.4s : Freq = 2 Hz T = 0.4 ~ 0.7s : Freq = 10 Hz T = 0.7 ~ 1.0s : Freq = 20 Hz

  8. Magnitude Magnitude Magnitude Magnitude Time (s) Frequency (Hz) Time (s) Frequency (Hz) Limitation of FT (2/2) • Different time-domain signals  Identical frequency spectrum • Example : original v.s reversed signals Frequency = 2Hz  20Hz Frequency = 20Hz  2Hz

  9. Examples : Non-stationary Signals • In real world, most interesting signals contain numerous non-stationary or transitory characteristics.

  10. Joint Time-Frequency Analysis • Joint Time-Frequency Analysis (JTFA) : Give a good time-frequency representation of the non-stationary signal.

  11. Different Analysis Tools : JTFA v.s FT

  12. Time-Frequency Plane • Visualize time-frequency location/concentration of time-domain signal x(t)

  13. Linear Time-Frequency Method (1/2) • Linear TF analysis : Measure contribution of TF point to signal x(t) • General approach : Inner product of x(t) with “test signal” or “sounding signal” located about • Linear TF Representation (LTFR) :

  14. Linear Time-Frequency Method (2/2) • Linear TF synthesis : Recover or synthesize signal x(t) from • General approach : where x(t) is represented as superposition of TF localized signal components, weighted by “TF coefficient function” • Problem : How to construct test (analysis) functions and synthesis functions ?

  15. Quadratic Time-Frequency Method (1/2) • Quadratic TF analysis : Measure “energy contribution” of TF point to signal x(t) • Simple Approach : Calculate the square function of the LTFR magnitude • Quadratic TF Representation (QTFR) :

  16. Quadratic Time-Frequency Method (2/2) • TF energy distribution : Use QTFR to distribute signal energy over TF plane. • Problem : How to construct test (analysis) functions ?

  17. Construction of Analysis/Synthesis Functions • Problem : Construct family of analysis functions such that is localized about TF point . • Systematic approach : derived from “prototype function” via unitary “TF displacement operator” : • Same for synthesis functions :

  18. Two Classical Definitions of Operator U • TF shift : (STFT) • TF scaling + time shift : (WT)

  19. Short-Time Fourier Transform (STFT)

  20. STFT and Constant-BW Filter-bank : Analysis • STFT analysis as convolution: • Filter-bank interpretation/implementation:

  21. STFT and Constant-BW Filter-bank : Synthesis • STFT synthesis as convolution: • Filter-bank interpretation/implementation:

  22. Spectrogram Analysis as Constant-BW Filter-bank • Spectrogram analysis as convolution: • Filter-bank interpretation/implementation:

  23. STFT / Spectrogram : Example

  24. Wavelet Transform (WT)

  25. WT and Constant-Q Filter-bank : Analysis • WT analysis as convolution: • Filter-bank interpretation/implementation:

  26. WT and Constant-Q Filter-bank : Synthesis • WT synthesis as convolution: • Filter-bank interpretation/implementation:

  27. Scalogram Analysis as Constant-Q Filter-bank • Scalogram analysis as convolution: • Filter-bank interpretation/implementation:

  28. WT / Scalogram : Example

  29. Spectrogram v.s Scalogram

  30. Comparison of Different Analysis Tools

  31. Conclusion • Fourier Transform : (stationary signals) • Analyzes the frequency components in the time-domain signals. • Joint Time-Frequency Analysis : (non-stationary signals) • Short-Time Fourier Transform (STFT) : • Maps a signal into a two-dimensional function of time and frequency. • Precision is determined by the size of the window. • Window is always the same for all frequencies. • Wavelet Transform (WT) : • Uses a windowing technique with variable-sized regions. • Does not use a time-frequency region, but rather a time-scale region. • Higher computation complexity

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