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 U572 - 28 mai 2002
Outline
Combining time and frequency
Classes of energetic distributions
Readability versus properties: a trade-off
Empirical Mode Decomposition
Combining time and frequency
Fourier transform
s(t)
|S(f)|
δ(u-t)
δ(θ-f)
u
θ
s(t) = < s(.) , δ(.-t) >
S(f) = < s(.) , ei2πf. >
s(t) = < S(.) , ei2πt. >
S(f) = < S(.) , δ(.-f) >
Combining time and frequency
Non Stationarity: Intuitive
frequency
x(t)
Fourier
X(f)
-5
-1
-1
time
-2
-2
time
frequency
Musical Score
Combining time and frequency
Short-time Fourier Transform
< s(.) , δ(. – f) >
Tt
Ff
Δt2 Δ f2  1 2
4π
< s(.) , δ(. - t) >
< s(.) , gt,f(.) > = Q(t,f) = <s(.) , TtFf g0(.) >
Combining time and frequency
Short-time Fourier Transform
Combining time and frequency
Short-time Fourier Transform
frequency
time
Combining time and frequency
Wavelet Transform
Tt
frequency
Ψ0( (u–t)/a )
Da
Ψ0(u)
time
< s(.) , TtDa Ψ0 > = O(t,f = f0/a)
Combining time and frequency
Wavelet Transform
• Frequency dependent resolutions (in time & freq.)
(Constant Q analysis)
STFT: Constant bandwidth analysis
• Orthonormal Basis framework (tight frames)
STFT: redundant decompositions (Balian Law Th.)
• Unconditional basis and sparse decompositions
Good for: compression, coding, denoising, statistical analysis
• Pseudo Differential operators
Good for: Regularity spaces characterization,
(multi-) fractal analysis
• Fast Algorithms (Quadrature filters)
Computational Cost in O(N) (vs. O(N log N) for FFT)
Combining time and frequency
Quadratic classes
2
 |s(t)|
dt


Ex
2
 |S(f)|
df
||
 |  s , g
t,f
 |2 dt df
tf
Ex 2) { gt,f (t1) gt,f (t2) } dt1 dt2
| s , gt,f  |2  s(t1) s(t
Π (t1,t2 ; t,f)
u--tt,,θaθ
Quadratic class: Ω
Css(t,af ; Π)  W
W
s(u,
 Π(u
-f)) du dθ
s(u,
θ)θΠ(
a
(Cohen Class)
(Affine
Class)

Wigner dist.: Ws(t,f) : s(tσ/2) s(t-σ/2) exp{-i2π fσ } dσ
Readability versus Properties
Trade-off
Ws(t,f)   s(tσ/2) s(t-σ/2) exp{i2πσf} dσ
0.5
0.45
0.4
0.35
0.3
frequency
0.25
0.2
0.15
0.1
0.05
0
0
50
100
150
time
200
250
γs (t) : 1 d Arg (zs(t))   f.Ws(t,f) df
2π dt
Readability versus Properties
Trade-off
Cs(t,f;Π)  Ws(u,θ) Π(u-t , θ-f) du dθ
frequency
time

s
γ (t) :  f.Cs(t,f; Π) df
Readability versus Properties
Trade-off
Affine Class
Cohen Class
Cs(t,f;Π)   Ws(u,θ) Π(u-t , θ-f) du dθ Ωs(t,a;Π)   Ws(u,θ) Π(ua-t , aθ) du dθ
Δ(t, a)
Δ(t,f)
Covariance: time-frequency shifts
Covariance: time-scale shifts
s(t)
s(t-t0).e- i 2 π f t
s(t)
1 s(t-t0)
a0
a0
Cs(t,f)
Cs(t-t0 , f-f0)
Ωs(t,a)
Ωs(t-t0, a.a0)
a0
0
Energy
d μ (t,f)
Es   Cs (t,f) dt df
Energy
d μ (t,a)
Es   Ωs(t,a) dt da
a2
Readability versus Properties
Adaptive schemes
• Adaptive radially gaussian kernels
R. G. Baraniuk, D. Jones (92)
• Reassignment method
Kodera, Gendrin, Villedary (80) - P.Flandrin et al. (98)
• Diffusion (PDE’s, heat equation)
P. Goncalves, E. Payot (98)
•…
Empirical Mode Decomposition
N. E. Huang et al. (98)
1. Adaptive non-parametric analysis
self contained (no a priori choice of analyzing functions)
2. “Quasi-orthogonal” decomposition
intrinsic mode functions – non-overlapping narrowband components
3. Invertible decomposition
Perfect reconstruction ( by construction! )
4. Local time procedure
Efficient for non linear and non stationnary time series
Empirical Mode Decomposition
Sifting Scheme
Signal = residu R(0)
Local minima and
maxima extraction
R(k)=R(k-1)-C(k)
Upper and Lower
Envelopes fits
S(j+1) = S(j) - M
Compute mean envelope M
If E(M) ~ 0
No
Yes
Component C(k) = S(j)
C(k)
Empirical Mode Decomposition
Multi-component signal
Ideal Time-Frequency representation
Time series
0
10
4
3
-1
10
2
-2
10
1
0
-3
10
-1
-4
10
0
500
1000
1500
2000
-2
500
1000
1500
2000
Empirical Mode Decomposition
Multi-component signal
1
1
IMF1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
200
400
600
800
1000
1200
1400
1600
1800
IMF2
0.8
2000
-1
200
400
600
800
1000
1200
1400
1600
1800
2000
1.2
1
IMF3
0.8
IMF4
1
0.6
0.8
0.4
0.6
0.2
0.4
0
-0.2
0.2
-0.4
0
-0.6
-0.2
-0.8
-1
-0.4
200
400
600
800
1000
1200
1400
1600
1800
2000
200
400
600
800
1000
1200
1400
1600
1800
2000
Empirical Mode Decomposition
A Real World
RR time series (rat, Wistar)
0
20
40
60
80
100
120
140
160
180
Empirical Mode Decomposition
time
A Real World
frequency
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
50
100
150
0
2
4
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,…)