Time-Frequency Tools: a Survey
Paulo Gonçalvès
INRIA Rhône-Alpes, France&
INSERM U572, Hôpital Lariboisière, France
2nd meeting of the European Study 2nd meeting of the European Study Group of Cardiovascular OscillationsGroup of Cardiovascular Oscillations
Italy, April 19-22, 2002
Time-Frequency Tools: a Time-Frequency Tools: a SurveySurvey
Paulo GonçalvèsINRIA Rhône-Alpes, IS2, France
&Pascale Mansier
Christophe Lenoir
INSERM U572, Hôpital Lariboisière, France
Séminaire U572 - 28 mai 2002
Outline
Combining time and frequencyClasses of energetic distributions
Readability versus properties: a trade-offEmpirical Mode Decomposition
s(t)
s(t) = < s(.) , δ(.-t) >
s(t) = < S(.) , ei2πt. >
Combining time and frequencyFourier transform
|S(f)|
S(f) = < s(.) , ei2πf. > S(f) = < S(.) , δ(.-f)
>
Blind to non stationnaritie
s!
t)-δ(uf)-δ(θ
uθ
time
frequency
Combining time and frequencyNon Stationarity: Intuitive
x(t) X(f)Fourier
Musical Score
-25
-20
-15
-10
-5
time
frequency
< s(.) , gt,f(.) > = Q(t,f)
< s(.) , δ(. - t) >
Combining time and frequencyShort-time Fourier Transform
< s(.) , δ(. – f) >
= <s(.) , TtFf g0(.) >
Ff
Tt
222 4π
1 f Δ Δt
Combining time and frequencyShort-time Fourier Transform
Combining time and frequencyShort-time Fourier Transform
frequency
time
Combining time and frequencyWavelet Transform
time
frequency
< s(.) , TtDa Ψ0 > = O(t,f = f0/a)
Ψ0(u)
Ψ0( (u–t)/a )
Da
Tt
• 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 frequencyWavelet 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 frequencyQuadratic classes
xE
xE dt |s(t)| 2 df |S(f)| 2
| |
df dt | g , s | 2 ft, t f
212ft,1ft,212ft, dt dt } )(t g )(t g { )s(t )s(t | g , s |
f)t, ; t,(t Π 21
dθ du f)-θt,-Π(u θ(u,W Π) ; f(t, C ss Quadratic class: (Cohen Class)
dσ } f σ exp{-i2π σ/2)-s(t σ/2)s(t : f)(t,WsWigner dist.:
Quadratic class: (Affine Class)
dθ du ) aθ , at-u Π( θ)(u,W Π) ; a(t,Ω ss
0 50 100 150 200 2500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Readability versus PropertiesTrade-off
time
frequency
dσ σf}exp{i2π σ/2)-s(t σ/2)s(t f)(t,Ws
df f)(t,f.W (t))(z dtd 2π
1 : (t) sss Argγ
Readability versus PropertiesTrade-off
time
frequency
dθ du f)-θ , t-Π(u θ)(u,W Π)f;(t,C ss
df Π) f;(t,f.C : (t) ss γ
Readability versus PropertiesTrade-off
dθ du f)-θ , t-Π(u θ)(u,W Π)f;(t,C ss dθ du aθ , at-uΠ θ)(u,W Π)a;(t,Ω )(ss
Cohen Class Affine Class
Covariance: time-frequency shifts Covariance: time-scale shifts
f)(t,Cs
s(t) t f π 2 i -0 0).et-s(t
)f-f , t-(tC 0 0s
s(t) )(0
00 at-ts a
1
a)(t,Ωs )( 00
0s a.a ,at-tΩ
Energy Energy
df dt f)t,( C E ss 2ss ada dt a)(t,Ω Ef)(t, μ d a)(t, μ d
f)Δ(t, a)Δ(t,
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)
Empirical Mode DecompositionN. 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 DecompositionSifting 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 DecompositionMulti-component signal
500 1000 1500 2000-2
-1
0
1
2
3
4
Ideal Time-Frequency representation Time series
0 500 1000 1500 200010
-4
10-3
10-2
10-1
100
Empirical Mode DecompositionMulti-component signal
200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
200 400 600 800 1000 1200 1400 1600 1800 2000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
200 400 600 800 1000 1200 1400 1600 1800 2000-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
IMF1 IMF2
IMF3 IMF4
Empirical Mode DecompositionA Real World
0 20 40 60 80 100 120 140 160 180
RR time series (rat, Wistar)
50 100 150 0 2 4
Empirical Mode DecompositionA Real World
IMF6
IMF7
IMF5
IMF4
IMF1
IMF2
IMF3
time frequency
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,…)
