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 Group of Cardiovascular

OscillationsOscillationsItaly, April 19-22, 2002

Time-Frequency Tools: a Time-Frequency Tools: a SurveySurvey

Paulo Gonçalvès

INRIA Rhône-Alpes, IS2, France

&

Pascale MansierChristophe 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 statio

nnarities!

t)-δ(u

f)-δ(θ

u

θ

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

4π1 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 )

D

a

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

0

0 at-ts

a1

a)(t,Ωs )( 00

0s a.a ,a

t-tΩ

Energy Energy

df dt f)t,( C E ss 2ss ada dt a)(t,Ω E

f)(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 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

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 Decomposition

Multi-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,

…)