Time-Frequency Tools: a Survey

1. Time-Frequency Tools: a Survey Paulo Gonçalvès INRIA Rhône-Alpes, France & INSERM U572, Hôpital Lariboisière, France 2nd meeting of the European Study Group of Cardiovascular Oscillations Italy, April 19-22, 2002

2. Time-Frequency Tools: a Survey Paulo Gonçalvès INRIA Rhône-Alpes, IS2, France & Pascale Mansier Christophe Lenoir INSERM U572, Hôpital Lariboisière, France Séminaire U572 - 28 mai 2002

3. Combining time and frequency Classes of energetic distributions Readability versus properties: a trade-off Empirical Mode Decomposition Outline

4. s(t) s(t) = < s(.) , δ(.-t) > s(t) = < S(.) , ei2πt. > |S(f)| S(f) = < s(.) , ei2πf.> S(f) = < S(.) , δ(.-f) > Blind to non stationnarities! Combining time and frequencyFourier transform u θ

5. frequency time time frequency Musical Score Combining time and frequencyNon Stationarity:Intuitive Fourier x(t) X(f)

6. < s(.) , δ(. - t) > Tt Ff Combining time and frequencyShort-time Fourier Transform < s(.) , δ(. – f) > < s(.) , gt,f(.) > = Q(t,f) = <s(.) , TtFf g0(.) >

7. Combining time and frequencyShort-time Fourier Transform

8. Combining time and frequencyShort-time Fourier Transform frequency time

9. Tt Ψ0( (u–t)/a ) Da Ψ0(u) Combining time and frequencyWavelet Transform frequency time < s(.) , TtDa Ψ0 > = O(t,f = f0/a)

10. Combining time and frequencyWavelet Transform • Frequency dependent resolutions (in time & freq.) (Constant Q analysis) • Orthonormal Basis framework (tight frames) • Unconditional basis and sparse decompositions • Pseudo Differential operators • Fast Algorithms (Quadrature filters) STFT: Constant bandwidth analysis STFT: redundant decompositions (Balian Law Th.) Good for: compression, coding, denoising, statistical analysis Good for: Regularity spaces characterization, (multi-) fractal analysis Computational Cost in O(N) (vs. O(N log N) for FFT)

11. Quadratic class: (Cohen Class) Quadratic class: (Affine Class) Wigner dist.: Combining time and frequencyQuadratic classes

12. Readability versus PropertiesTrade-off frequency time

13. Readability versus PropertiesTrade-off frequency time

14. Affine Class Cohen Class Readability versus PropertiesTrade-off Covariance: time-scale shifts Covariance: time-frequency shifts Energy Energy

15. Readability versus PropertiesAdaptive schemes • Adaptive radially gaussian kernels • Reassignment method • Diffusion (PDE’s, heat equation) • … R. G. Baraniuk, D. Jones (92) Kodera, Gendrin, Villedary (80) - P.Flandrin et al. (98) P. Goncalves, E. Payot (98)

16. Empirical Mode Decomposition N. E. Huang et al. (98) • Adaptive non-parametric analysis • “Quasi-orthogonal” decomposition • Invertible decomposition • Local time procedure self contained (no a priori choice of analyzing functions) intrinsic mode functions – non-overlapping narrowband components Perfect reconstruction ( by construction! ) Efficient for non linear and non stationnary time series

17. Empirical Mode Decomposition Sifting Scheme Signal = residu R(0) Local minima and maxima extraction R(k)=R(k-1)-C(k) S(j+1) = S(j) - M Upper and Lower Envelopes fits Compute mean envelope M If E(M) ~ 0 No Yes C(k) Component C(k) = S(j)

18. Empirical Mode Decomposition Multi-component signal Ideal Time-Frequency representation Time series

19. Empirical Mode Decomposition Multi-component signal IMF2 IMF1 IMF3 IMF4

20. Empirical Mode Decomposition A Real World RR time series (rat, Wistar)

21. Empirical Mode Decomposition A Real World time frequency IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7

22. Non stationarities Time-varying spectra (time-frequency) Transients (singularities, shifts,…) Component-wise analysis (EMD) Complex analysis Fractal analysis (Wavelets) Multiresolution structures (Markov models,…) Concluding remarks