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

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

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

s(t) s(t) = Combining time and frequency Fourier transform |S(f)| S(f) = Blind to non stationnarities! u θ

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

= Q(t,f) Combining time and frequency Short-time Fourier Transform = FfFf TtTt

Combining time and frequency Short-time Fourier Transform

frequency time

Combining time and frequency Wavelet Transform time frequency = O(t,f = f 0 /a) Ψ 0 (u) Ψ 0 ( (u–t)/a ) DaDa TtTt

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) Combining time and frequency Wavelet Transform STFT: Constant bandwidth analysis STFT: redundant decompositions (Balian Law Th.) Good for: compression, coding, denoising, statistical analysis Computational Cost in O(N) (vs. O(N log N) for FFT) Good for: Regularity spaces characterization, (multi-) fractal analysis

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

Readability versus Properties Trade-off time frequency

Readability versus Properties Trade-off time frequency

Readability versus Properties Trade-off Cohen Class Affine Class Covariance: time-frequency shifts Covariance: time-scale shifts Energy

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

Empirical Mode Decomposition N. E. Huang et al. (98) 1.Adaptive non-parametric analysis 2.Quasi-orthogonal decomposition 3.Invertible decomposition 4.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

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

Empirical Mode Decomposition Multi-component signal Ideal Time-Frequency representationTime series

Empirical Mode Decomposition Multi-component signal IMF1 IMF2 IMF3 IMF4

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

Empirical Mode Decomposition A Real World IMF6 IMF7 IMF5 IMF4 IMF1 IMF2 IMF3 timefrequency

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