traitement du signal appliqué au cas des mesures de sonde

G. BonhommeAtelier CNRS « Analyse de plasmas par sondes électrostatiques », Sarcenas 14-16 juin 2006

Traitement du signal appliqué au casdes mesures de sonde

Gérard Bonhomme LPMIA - UMR 7040 CNRS -

Université Henri Poincaré, [email protected]


Plan• Introduction

– Problèmes à résoudre– Méthodes disponibles

• Méthodes basées sur Fourier– Interpolation– Systèmes linéaires– Filtrage numérique ; filtres FIR, Savitzky-Golay– Régression linéaire et non linéaire

• Méthodes avancées– Transformées en ondelettes– Décomposition en modes empiriques

• Conclusions


Systèmes linéaires

∫+∞

∞−−= duutxuhty )()()(

∫+∞

∞−−= duutuhth )()()( δ

)()()( ωωω XHY =

h(t), H(ω)x(t) y(t)

Réponse d’un système linéaire à unePerturbation dépendant du temps :avec h(t) réponse impulsionnelle etH(ω) fonction de transfert

Intérêt de la transformée de Fourier :le produit de convolution se transformeen produit simple

Systèmes à temps discret ∑+∞

−∞=−=

kknkn hxy

hn est la réponse à⎩⎨⎧

=≠

=0100

nn

nδ

)(~)(~)(~ zXzHzY =

avec ∑= kk zhzH )(~

Convolution discrète

ta 0sinω )sin( 0 ϕω +tA

ωiez −=et


Interpolation


Interpolation

∑+∞

−∞=−=

kknkn hxy

⎪⎩

⎪⎨

⎧

==

−−=−

∫

∑∞+

∞−e

eti

nen

TtTteHh

nTtftf

//sin)(

21)(

)()(

ππω

πτ

τδτ

ω∑+∞

−∞=−=

kknkn hxy

Interpolation naturelle dite reconstruction deWittaker

Fp Transformée de Fourier de la suite discrète d‘échantillons ∑ −= eTin

np efF ωω)(

On filtre dans le domainefréquentiel

eTpFF /2* ).()( πωω Π=

∫+∞

∞−−= τττ dtfhtf )().()(*

∑∑−−

−−

=−=1

0

1

0

*

/)(/)(sin)()(

N

ee

een

N

en TnTtTnTtfnTthftf

ππ


Interpolationmise en oeuvre très simple par insertion de zérosInterpolation naturelle


Filtrage numérique (1)Exemple


Filtrage numérique (2)Exemple


Filtrage numérique: applications (1)

Signal brut Moyenne pondérée sur trois points

Dérivation


Filtrage numérique : applications (2)

Dérivée du signal filtréfiltre passe-haut

Filtrage passe-haut


Filtres de Savitzky - Golay (1)

Les coefficients peuvent être calculésune fois pour toute quel que soit le signal

D‘après Numerical Recipes, Cambrige Univ. Press


Filtres de Savitzky - Golay (2)

Applications à des caractéristiques de sondeFiltre utilisé


Méthode des moindres carrésExemple N=3


Détermination des paramètres plasma

efloatp TVV )ln8.2(61.0 μ−+=


Détermination du potentielflottant et calcul du potentielplasma : Vfloat~ 15,3 volts

Détermination de la températureélectronique :on „fit“ la région de transitionpar une exponentielle

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−=

e

floate

TUVTSnI exp196.0 0 μVp~ 17,85 volts

Te~ 2,94 eV


Modélisation non linéaire : méthode de Levenberg -Marquardt


⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−=

e

floate

TUVTSnI exp196.0 0 μ

exemple


Méthodes avancéesData collected from a turbulent plasma → very large frequency

range, many ≠ physical mechanisms⇒ How can we extract reliable informations from time series ?

"Classical“ Methods (Statistical Analysis, Fourier methods)drawbacks of Non stationarity, non linearity ⇒

Pivoine data

Experimental data from tokamaks


• Wavelet transform = generalization of the Fourier analysis→ change for an other analysis function

⇒ find an orthogonal basis localised in time and frequency

→ Two different approaches :Continuous Wavelet Transform (e.g. Morlet) → time-frequency analysisDiscrete Wavelet Transform → orthogonal decomposition (filtering)

• Hilbert-Huang transform• Decomposition of a non stationary time-series into a finite sum of orthogonal eigenmodes, or Intrinsic Mode Functions (IMF).• Self adapative approach in which the eigenmodes are derived from the specifictemporal behaviour of the signal.• Subsequently, the Hilbert Transform can be used to compute the instantaneousfrequency and a time-frequency representation of each mode as well as a global marginal Hilbert energy spectrum.

Solutions : Wavelets and Hilbert-Huang transforms

⎟⎠⎞

⎜⎝⎛ −

=T

ttttT0

, )(0

ϕϕ dtttfT

tTW tTf )()(1),( *

,0 0ϕϕ ∫

+∞

∞−

=

)()()(1

trtimftXn

jnj∑

=

+=


Discrete wavelets: Analysis and reconstruction

Analysis

Synthesis

Daubechies wavelets

Efficient algorithmn,But:- physical meaning of the filtering?- not well suited to time frequency analysis


Continuous wavelets: Time-frequency analysis

Pivoine data(from A. Lazurenko)

Time-frequency representation obtained with Morlet wavelets

Fourier spectrum

Drawback → cpu time demanding (because high level of redundancy)


Continuous wavelets: Time-frequency analysis

Time-frequency representation obtained with Morlet wavelets

Drawback → cpu time demanding (because high level of redundancy)


The Hilbert-Huang Transformor Empirical Mode Decomposition

• Decomposition of a non stationary time-series into a finite sum of orthogonal eigenmodes, or Intrinsic Mode Functions (IMF).

• Self adapative approach in which the eigenmodes are derived from the specifictemporal behaviour of the signal.

• Subsequently, the Hilbert Transform can be used to compute the instantaneousfrequency and a time-frequency representation of each mode as well as a global marginal Hilbert energy spectrum.

N. E. Huang et al., The Empirical Mode Decomposition and Hilbert Spectrum for Nonlinear and Non-Stationary Time Series Analysis, Proc. R. Soc. London, Ser. A, 454, pp. 903-995 (1998).

T. Schlurmann, Spectral Analysis of Nonlinear Water Waves based on the Hilbert-Huang transformation, Transactions of the ASME Vol.124 (2002) 22.

J. Terradas et al, The Astrophys. Journal 614 (2004) 435.P. Flandrin, G. Rilling, P. Gonçalves, Empirical Mode Decomposition as a Filter Bank,

IEEE Sig. Proc. Lett., Vol.11, N°2, pp. 112-114 (2004).


Hilbert Transform and instantaneous frequency

duut

uxvptxH ∫+∞

∞− −=

)()(..1)]([

π)]([()( txHty =

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

=+=

)()(arctan)(and)()()(with

))(exp()()()()(

22

txtyttytxtA

titAtiytxtz

θ

θ

Hilbert transform of a data series x(t) is defined by:

But in most cases the instantaneous frequencyhas no physical meaning

Example

By substituting we can definez(t) as the analytical signal of x(t)

dttdt )()( θω =

ttx sin)( +=α

⇒ Empirical Mode Decompositionset of IMF : (1) equal number of extremaand zero crossings; (2) mean value of the minima and maxima envelopes = 0

from Huang et al


IMF = Intrinsic Mode Functions

)()()(1

trtimftXn

jnj∑

=

+=

1. Initialize : r0(t) = X(t), j=12. Extract the j-th IMF:

a) Initialize h0(t) = rj(t), k=1b) Locate local maxima and minima of hk-1(t)c) Cubic spline interpolation to define upper and lower

envelope of hk-1(t)d) Calculate mean mk-1(t) from upper and lower enve-

lope of hk-1(t)e) Define hk(t) = hk-1(t) - mk-1(t)f) If stopping criteria are satisfied then imfj(t) = hk(t)

else go to 2(b) with k=k+13. Define rj(t) = rj-1(t) - imfj(t) 4. If rj(t) still has at least two extrema then go to 2(a) with

j=j+1, else the EMD is finished5. rj(t) is the residue of x(t)

The Empirical Mode Decomposition (sifting process)

⇒A typicalIMF

from Huang et al


Empirical Mode Decomposition: Analysis and Reconstruction

Analysis Synthesis


Empirical Mode Decomposition: Hilbert spectrum


Results of the EMD

Empirical Mode Decompositionof time-series from probe A7

HF bursts: 6 – 22 MHz

Evidences of osc. in the Ion transit time inst. freq. domain

Breathing oscillations: ~ 25 kHz

Strongly correlatedwith low frequencyoscillations →

⇐

⇐


Hilbert spectra

HF bursts : ~ 6 – 11 – 22 MHz

Low frequency oscillations:- breathing mode (top)- bursts in the ion transit time frequency range

HF bursts start mainly on a negative slope of Id withmaximum amplitude ~ inflexion point ↔ minimum of Vfland maxima of ne and Ez (A. Lazurenko)


Comparison with wavelet time-frequency analysis

Morlet → freq. = 375/scale ⇒ 33 ↔ 11.4 MHz


Marginal Hilbert spectrum vs Fourier spectrum

Because of the strongnonlinearity of HF oscillations the Fourier spectrum exhibits manypeaksAll these peaks do not correspond to actual modes

A peak in the marginal Hilbert spectrumcorresponds to a whole oscillation around zero


Conclusions

traitement du signal appliqué au cas des mesures de sonde

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