g. bonhomme iepc-2005, princeton, oct. 31 – nov. 4, 2005 characterization of hall effect thruster...
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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:
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
Discrete wavelets: Analysis and reconstruction
Analysis
Synthesis
Daubechies wavelets
Efficient algorithmn,But:- physical meaning of the filtering?- not well suited to time frequency analysis
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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)
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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).
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
Analysis and Reconstruction (PIVOINE data)
Analysis Synthesis
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
ExampleThe Hilbert transform compared to the continuous wavelet transform (Morlet)
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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)
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
Comparison with wavelet time-frequency analysis
Morlet freq. = 375/scale 33 11.4 MHz
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
Filtering before cross-correlating
Pivoine data from probes A1, A3, and A6(from A. Lazurenko)
A1
A3
A6
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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?
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
Morlet Time-frequency analysis after EMD filtering
probe A7 probe A8
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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]
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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:
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
Analysis and reconstruction of a square wave
SynthesisAnalysis
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
)(2cos 20
/ 22
ttfe t
= 10 ;
= 0.5 ; f0 = 0.5
te t0
)( cos2
= 0.1 ; = 1. ; 0 = 10
• pulse
• chirp
Time-frequency analysis
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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:
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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.
G. BonhommeIEPC-2005, Princeton, Oct. 31 – Nov. 4, 2005
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