Short Time Fourier Transform (STFT)

1. Short Time Fourier Transform (STFT) CS474/674 – Prof. Bebis

2. Fourier Analysis (inverse DFT) where: (forward DFT) Fourier analysis expands signals or functions in terms of sinusoids (or complex exponentials). It reveals all frequency components present in a signal.

3. Examples

4. Examples (cont’d) F1(u) F2(u) F3(u)

5. Fourier Analysis – Examples (cont’d) F4(u)

6. Limitations of Fourier Analysis (cont’d) 1. Cannot not provide simultaneous time and frequency localization. 2. Not useful for analyzing time-variant, non-stationary signals. 3. Not efficient for representing discontinuities or sharp corners (i.e., requires a large number of Fourier components to represent discontinuities).

7. Fourier Analysis – Examples (cont’d) Provides excellent localization in the frequency domain but poor localization in the time domain. F4(u)

8. Limitations of Fourier Analysis (cont’d) 1. Cannot not provide simultaneous time and frequency localization. 2. Not useful for analyzing time-variant, non-stationary signals. 3. Not appropriate for representing discontinuities or sharp corners (i.e., requires a large number of Fourier components to represent discontinuities).

9. Stationary vs non-stationary signals • Stationary signals: time-invariant spectra • Non-stationary signals: time-varying spectra

10. Stationary vs non-stationary signals (cont’d) Stationary signal: Three frequency components, present at all times! F4(u)

11. Stationary vs non-stationary signals (cont’d) Non-stationary signal: Perfect knowledge of what frequencies exist, but no information about where these frequencies are located in time! F5(u)

12. Limitations of Fourier Analysis (cont’d) 1. Cannot not provide simultaneous time and frequency localization. 2. Not useful for analyzing time-variant, non-stationary signals. 3. Not appropriate for representing discontinuities or sharp corners (i.e., requires a large number of Fourier components to represent discontinuities).

13. Representing discontinuities or sharp corners

14. Representing discontinuities or sharp corners (cont’d) FT

15. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 1

16. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 2

17. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 7

18. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 23

19. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 39

20. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 63

21. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 95

22. Representing discontinuities or sharp corners (cont’d) Original Reconstructed 127

23. Short Time Fourier Transform (STFT) • Need a local analysis scheme for a time-frequency representation (TFR). • Windowed F.T. or Short Time F.T. (STFT) • Segmenting the signal into narrow time intervals (i.e., narrow enough to be considered stationary). • Take the Fourier transform of each segment.

24. Short Time Fourier Transform (STFT) (cont’d) • Steps : (1) Choose a window function of finite length (2) Place the window on top of the signal at t=0 (3) Truncate the signal using this window (4) Compute the FT of the truncated signal, save results. (5) Incrementally slide the window to the right (6) Go to step 3, until window reaches the end of the signal

25. Short Time Fourier Transform (STFT) • For each time location where the window is centered, we obtain a different FT • Each FT provides the spectral information of a separate time-slice of the signal, providing simultaneous time and frequency information

26. Short Time Fourier Transform (STFT) (cont’d) Frequency parameter Time parameter Signal to be analyzed STFT of f(t): computed for each window centered at t=t’ Windowing function centered at t=t’

27. Short Time Fourier Transform (STFT) (cont’d) • STFT maps 1-D time domain signals to 2-D time-frequency signals (i.e., in u and t’ )

28. Example f(t) [0 – 300] ms  100 Hz sinusoid [300 – 600] ms  50 Hz sinusoid [600 – 800] ms  25 Hz sinusoid [800 – 1000] ms 10 Hz sinusoid

29. Example f(t) W(t) scaled: t/20

30. Choosing Window W(t) • What shape? • Rectangular, Gaussian, Elliptic…? • How wide? • Window should be narrow enough to make sure that the portion of the signal falling within the window is stationary. • Very narrow windows do not offer good localization in the frequency domain.

31. STFT Window Size W(t) infinitely long: STFT turns into FT, providing excellent frequency localization, but no time information. W(t) infinitely short: gives the time signal back, with a phase factor, providing excellent time localization but no frequency information.

32. STFT Window Size (cont’d) • Wide window good frequency resolution, poor time resolution. • Narrow window good time resolution, poor frequency resolution.

33. Example different size windows (four frequencies, non-stationary)

34. Example (cont’d) scaled: t/20

35. Example (cont’d) scaled: t/20

36. Heisenberg (or Uncertainty) Principle Time resolution: How well two spikes in time can be separated from each other in the transform domain. Frequency resolution: How well two spectral components can be separated from each other in the transform domain

37. Heisenberg (or Uncertainty) Principle (cont’d) • One cannot know the exact time-frequency representation of a signal. • We cannot precisely know at what time instance a frequency component is located. • We can only know what interval of frequencies are present in which time intervals.