Characterization of Hall Effect Thruster Plasma Oscillations based on the Hilbert-Huang Transform

G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005

Characterization of Hall Effect Thruster Plasma

Oscillations based on the Hilbert-Huang TransformGérard BONHOMME

Cédric Enjolras, LPMIA, UMR 7040 CNRS - Université Henri Poincaré, Nancy

Stéphane Mazouffre, Luc Albarède, Alexey Lazurenko, and Michel Dudeck, Equipe PIVOINE, Laboratoire d’Aérothermique, UPR 9020 CNRS-Université d’Orléans

Jacek Kurzyna, IPPT-PAN, Varsovie

and Collaborators:


Outline

1. Introduction• Plasma thruster oscillations• Signal processing

2. Experimental set-up3. The EMD or Hilbert-Huang Transform

• Hilbert transform and instantaneous frequency• EMD or Decomposition into Intrinsic Mode Functions (IMF)• Hilbert spectra

4. Analysis of SPT-100 plasma fluctuations• Filtering of different frequency ranges• Time-frequency analysis• Investigating poloidal propagation of HF oscillations

5. Perspectives


Introduction

•Hall thrusters plasma oscillations 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


Solutions : Wavelets?

• Short-time (or windowed) Fourier transform

(DFT of sub-series) Pb : frequency resolution = 1/T

the time resolution is the same at all frequencies

• Wavelet tranform = generalization of the Fourier analysis

change for an other analysis function giving a time resolution depending on the frequency

find an orthogonal basis localised in time and frequency


Wavelet Analysis

• Principle of the wavelet transform:

replace the sine waves of the Fourier decomposition by orthogonal basis functions localised in time and frequency

• Aim : Decomposition of a signal into components (small waves, i.e. wavelets) corresponding to :

scales or levels (i.e., frequencies) and

localisations for each of these scales

Two different approaches :

• Continuous Wavelet Transform (e.g. Morlet) time-frequency analysis

• Discrete Wavelet Transform orthogonal decomposition (filtering)

dtttfT

tTW tTf )()(

1),( *

,0 0

T

ttttT

0, )(

0


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)


The Hilbert-Huang Transform or 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 specific temporal behaviour of the signal.

• Subsequently, the Hilbert Transform can be used to compute the instantaneous frequency 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)]([

)]([()( txHty

)(

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

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

22

tx

tyttytxtA

titAtiytxtz

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

But in most cases the instantaneous frequency

has no physical meaning

Example

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

dt

tdt

)()(

ttx sin)(

Empirical Mode Decompositionset of IMF : (1) equal number of extrema

and 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=1

2. Extract the j-th IMF:

a) Initialize h0(t) = rj(t), k=1

b) 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+1

3. 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 finished

5. rj(t) is the residue of x(t)

The Empirical Mode Decomposition (sifting process)

A typical

IMF

from Huang et al


Analysis and Reconstruction (PIVOINE data)

Analysis Synthesis


The Hilbert spectrum

After having obtained the IMF, generated the Hilbert transform of each IMF, x(t) is

represented by: This time-frequency distribution of the

amplitude is designated as the Hilbert spectrum

dttitAtX j

n

jj )(exp)()(

1

),( tH Two possible representations: global (as for wavelets) or for each IMF

Integration in the time domain Hilbert marginal spectrum

(to be compared to Fourier spectrum


ExampleThe Hilbert transform compared to the continuous wavelet transform (Morlet)


A1, A2

A5

A3, A4

A6

A8 A7

A4 A3

Br, R=50 mm

z

Electron driftAzimuthal instabilities

BEz

B

Experimental set-up

Laboratory version of a SPT 100 Hall effect Thruster

Operating conditions: Ud=300 volts, Id=4.2 A

Xe 5.42 mg/s

Diagnostics: - Langmuir probes A7 and A8

location: thruster exit plane

coaxial, unbiased, 2π/3 angular

separation, 1MΩ load (Vfloat fluct.)

- Current probe

Acquisition: 1GHz bandwidth digital scope

250 Msamples/s, time series 50 ksamples


Results of the EMD

Empirical Mode Decomposition of 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 correlated with low frequency

oscillations


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

with maximum amplitude ~ inflexion point minimum of Vfl and 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 strong nonlinearity of HF oscillations the Fourier spectrum exhibits many peaks All these peaks do not correspond to actual modes

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


Filtering before cross-correlating

Pivoine data from probes A1, A3, and A6(from A. Lazurenko)

A1

A3

A6


Cross-correlation functions for the HF components

probe A7

probe A8

HF oscillations propagate azimuthally at EB velocity v~2106 m/s Frequency peaks ~6-11-22 MHz correspond to azimuthal wavenumber m =1,2,4A (non explained) discrepancy is observed between the phase shift measured for 120° angular separation and the period non purely azimuthal waves? Asymetry?


Morlet Time-frequency analysis after EMD filtering

probe A7 probe A8


1. The Hilbert-Huang Transform method:• Has proven to be a promising and attractive method to analyze non

stationary and nonlinear time-series because of:- a very efficient ability in filtering different physical phenomena- accurate time-frequency representation - moderate cpu time consumption and ability to analyse long time series

• Some improvements would be useful, e.g., Hilbert spectra representation

2. Application to Hall thruster plasma oscillations:• Clear separation of three typical time and physical scales (~25 kHz,

100-500 kHz, and ~10MHz) - unambiguous identification of bursts in the transit time inst. freq. range - analysis of HF bursts azimuthal propagation and correlation with Id

• More reliable experimental data are needed to go further into the physics of the observed plasma fluctuations, in particular to get an accurate access to azimuthal and axial propagation.

(J. Kurzyna et al., submitted to PoP)

Conclusions and Perspectives


Drawbacks of Fourier methods

Fourier transform definition

F () complex lnformation on time localisation contained in the phase

difficult access

• Example 1 A musician playing either successively two notes, or

simultaneously these two notes same amplitude

spectra

deFtfdtetfF titi )(2

1)()()(

2)()( FS ff

50)(5.1sin

05)(sin

0

0

tt

tt

with = 1. ; 0 =10

)(2cos 20

/ 22

ttfe t

with =10 ; =2. ; f0=1

chirp

Spectral density


The Continuous Wavelet Transform

Principle : mother-wavelet (t)

T

ttttT

0, )(

0 Translation

+ dilatation

Wavelet Transform dtttfT

tTW tTf )()(

1),( *

,0 0

normalisationT

1

• compact support

• orthogonal basis in L2(R)Pb : find a "good " mother-

wavelet Necessary conditions (admissibility) :

onlocalisati)(

),(1

)()(

0 00*

,0

0

2

0

dtt

dtT

ttTtTWdT

ctfdt

t

tc tT

f

(reconstruction)

Parseval’s theorem 0

0

2

0

2),(

1)( dttTWdT

cdttf f

Morlet wavelets )()(22

02/π2 ttdti eceCet 22

20dec

with: Time resolution Frequency resolution

d4ω/ω Tdt 2(with ) T

2

with

πω t


Morlet Wavelet

T

ttik

T

tt

TttT

0

2

0, 2

1exp

1)(

0

222

0, 2

1exp2)(

0TkTktiTtT

Maximum at T = k

Mother-wavelet

k=2, t0=0, T=1

dtT

ttik

T

tt

Ttf

Tdtttf

TtTW tT

f

0

2

0*,0 2

1exp

1)(

1)()(

1),(

0

Morlet transform:

convolution product f*T )()(1

),( 10 T

f TFfTFTFT

tTW

with

222

2

1exp2)()( TkTkTFT T

d = 1


The Discrete Wavelet Transform

Drawbacks of the continuous wavelet transform : redondancy, CPU time, admissibility conditions non completely fullfilled (Morlet)

Solution ? Discrete Wavelets (similar to the DFT)

• Octave scaling

Tj = 2j et t0 j,k = k/ 2j

• orthogonality

with

There are 2m base functions at the m level

• reconstruction

)2(2)( 2/, ktWtW mmkm

klmnlnkm WW ,, ,

dttgthgh )()(, *

)()()()( ,*, tWXtxdttWtxX km

k

kmmkm

mk

)()( ,,

tWXtx kmkm

km

from Newland [1]


Wavelet construction: Daubechies wavelet

(t) for r=2 (from Newland [1])

-1

1

1 1/2

t

12

0

)2(2

1)(

r

kk ktct

12

02

12

0

12

0

12

0

2

1,,2,1,for0

1,,0for0)1(

2,2

r

kmkk

r

kk

mk

r

k

r

kkk

rmmcc

rmck

cc

12

0

)122()1()(r

kk

k rktctW

• Haar

lack of regularity

• Daubechies wavelets

- must be determined by recurrence from a scaling function (t)

(Meyer, 1993)

- they are completely defined by the coefficients ck

2r+1 conditions must be satisfied:


Mallat tree (pyramid algorithm)

from Newland [1]

Solutions :

• Haar (r=1) c0 = c1 = 1

• Daubechies D4 (r=2)

• r > 3, (numerical computation) discrete transform (computed by using the Mallat algorithm) analysis, and reconstruction formula synthesis

)31(4

1,)33(

4

1

)33(4

1,)31(

4

1

32

10

cc

cc

)2(

)34(

)24(

)14(

)4(

)12(

)2()()()(

276543210 ktWa

tW

tW

tW

tW

aaaatW

tWaatWatatf j

kj

D4 D20


Practical considerations (1)

• The Continuous Wavelet Transform (Morlet):

FFT computation of

Practically f={fn} et N=2m

Oversampling of F() required

solution = zero padding of fn (for T=NTe (N-1)N zeros)

)

2

1

2

1exp()(),( 2221

0 TkTkfFFTFFTtTW f

F(

/Te /Te

/kTe /kTe

2pour

142 k

TkTNT ee

See for example, D. Jordan et al, Rev.Sci.Instrum. 68 (1997) 1484-1494


Practical considerations (2)

• Discrete Wavelet Transform: pyramid algorithm (no need for the W(t))

½L3 ½ H2 ½ H1 example : f=f(1:8) ——— f'(1:4) ——— f'(1:2) ——— f'(1)

½ H3 ½ H2 ½ H1

a[5:8] a[3:4] a(2) a(1)

Hn et Ln are matrices build directly from the ck coefficients (cf. Newland [1])

1032

3210

3210

3210

1032

3210

3120

cccc

cccc

cccc

cccccccc

cccccccc

3

2

1

L

L

L

2301

0123

0123

0123

2301

0123

0213

cccc

cccc

cccc

cccccccc

cccccccc

3

2

1

H

H

H

Low-pass filter High-pass filter


Analysis and reconstruction of a square wave

SynthesisAnalysis


)(2cos 20

/ 22

ttfe t

= 10 ;

= 0.5 ; f0 = 0.5

te t0

)( cos2

= 0.1 ; = 1. ; 0 = 10

• pulse

• chirp

Time-frequency analysis


Completeness and Orthogonality

Example (from Huang [1])

The orthogonality is satised in all practical sense, but it is not guaranteed theoretically

The completeness is established both theoretically and numerically

IO = overall index of orthogonallity

for this example IO = 0.0067

or for two IMF:


Degree of stationarity

The degree of stationarity DS is defined as:

with mean marginal spectrum

and the degree of statistic stationarity DSScan be is defined as:

If the Hilbert spectrum depends on time, the index will not be zero,

then the Fourier spectrum will cease to make physical sense.

The higher the index value, the more non-stationary is the process.


Application to experimental time-series (PIVOINE)

IMF 9

IMF 1-4

Spectre de Hilbert marginal

Signal analysé

J. Kurzyna et al.,

submitted to Phys. of Plasmas

Documents

Characterization of Hall Effect Thruster Plasma Oscillations based on the Hilbert-Huang Transform