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Fusion Engineering and Design 88 (2013) 2767– 2772

Contents lists available at ScienceDirect

Fusion Engineering and Design

journa l h om epa ge: www.elsev ier .com/ locat e/ fusengdes

ime–frequency analysis of nonstationary complexagneto-hydro-dynamics in fusion plasma signals using the

hoi–Williams distribution

.Q. Xu, L.Q. Hu ∗, K.Y. Chen, E.Z. Linstitute of Plasma Physics, Chinese Academy of Science, Hefei 230031, China

i g h l i g h t s

Choi–Williams distribution yields excellent time–frequency resolution for discrete signal.CWD method provides clear time–frequency pictures of EAST and HT-7 fast MHD events.CWD method has advantages to wavelets transform scalogram and the short-time Fourier transform spectrogram.We discuss about how to choose the windows and free parameter of CWD method.

r t i c l e i n f o

rticle history:eceived 31 January 2012eceived in revised form 4 April 2013ccepted 4 April 2013vailable online 14 May 2013

ACS:2.35.−g

a b s t r a c t

The Choi–Williams distribution is applied to the time–frequency analysis of signals describing rapidmagneto-hydro-dynamic (MHD) modes and events in tokamak plasmas. A comparison is made with SoftX-ray (SXR) signals as well as Mirnov signal that shows the advantages of the Choi–Williams distributionover both continuous wavelets transform scalogram and the short-time Fourier transform spectrogram.Examples of MHD activities in HT-7 and EAST tokamak are shown, namely the onset of coupling tearingmodes, high frequency precursors of sawtooth, and low frequency MHD instabilities in edge localizedmode (ELM) free in H mode discharge.

7.05.Kf6.30.Ft

eywords:hoi–Williams distributiononstationary signals

© 2013 Elsevier B.V. All rights reserved.

agneto-hydro-dynamic

. Introduction

Macroscopic magneto-hydro-dynamics instability plays anmportant role from the viewpoint of maximum achievable plasmaarameters associated with tokamak safety operation [1]. It is nooubt that the study of MHD instabilities is important in the field ofagnetic confinement plasma. Mode frequency is one of the most

mportant parameter characterize an MHD instability [2]. Clearime–frequency representation [3–5] is well-known powerful tool

o reflect the time evolution of MHD instability.

In fusion plasma signals, the spectrogram may not always be theest tool to analyze some nonstationary MHD instability. Wavelets

∗ Corresponding author at: PO Box 1126, Hefei, Anhui 230031, China.el.: +86 551 5591347; fax: +86 551 5591310.

E-mail address: lqhu@ipp.ac.cn (L.Q. Hu).

920-3796/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.fusengdes.2013.04.017

are well known in fusion research, particularly the Morlet wavelet,the scalogram constituting an alternative to the spectrogram [6,7].In simple terms, going from the spectrogram to the scalogram is amatter of using smaller windows with higher frequencies and viceversa. Recently, the Choi–Williams distribution [8–12] has beeneffectively used to analyze nonstationary phenomena in fusionplasmas for which the spectrogram did not produce the best pos-sible result. Choi–Williams distribution with very good resolutionbut comes the existence of artifacts, of which the spectrogram andscalogram are practically free of. Still, the Choi–Williams distribu-tion allows artifact reduction, at the cost of some time–frequencyresolution.

Here, such a comparison is made and the advantages of using the

Choi–Williams distribution over wavelets and short-time Fouriertransform are shown. Examples of MHD activities in HT-7 andEAST tokamak are shown, namely the onset of coupling tearingmodes [13,14], high frequency precursors of sawtooth [15], and low

2 ng and Design 88 (2013) 2767– 2772

fd

siChiS

2

suT

P

0.525 0.530 0.535 0.540 0.545 0.5500

2

4

6

8

10

12

a.u

.

time/s

SXA27_111635

r=5 .66cm

Fig. 1. Complex MHD oscillations in the core region of plasma in Soft X-ray signals

F2

768 L.Q. Xu et al. / Fusion Engineeri

requency MHD instabilities in edge localized mode free in H modeischarge.

The rest of this paper is organized as follows. The compari-on between the Choi–Williams distribution and the scalograms reported in Section 2. In Section 3, typical examples of thehoi–Williams distribution application performed of magneto-ydro-dynamic (MHD) modes are presented. A discussion

s present in Section 4. Finally, the summary is given inection 5.

. The Choi–Williams distribution

The time–frequency spectrum of a discrete time signal Sig(n)ampled at frequency fs can be usually obtained by the square mod-lus of its short-time Fourier transform with the time window ω(n).he spectrogram is expressed as [12]:

∣∣ l−1 ( ) ( ) ∣∣2

(tn, fm) = 12�

∣∣∣∑k=0

Sig n + k − l − 12

ω k − l − 12

e−i2�km/l∣∣∣ (1)

ig. 2. Comparison between (a) the spectrogram with l = 127, (b) the Choi–Williams distri0 s2 of Soft X-ray signals as shown in Fig. 1.

with r = 5.66 cm (the minor radius of HT-7 is 27 cm) in discharge #111635 on HT-7tokamak.

bution with l� = 511, l� = 511, � = 25 and (c) the scalogram with Fc = 0.5 Hz, Fb =

ng and Design 88 (2013) 2767– 2772 2769

wIt

ı

a

bt

�ato

S

wr

0.459 0.460 0.461 0.462 0.463 0.464 0.465 0.466 0.467-0.4

-0.2

0.0

0.2

SXA27

r=5.66cm

(a.u

.)(a

.u.)

crash

High frequency

time/s

0.459 0.460 0.461 0.462 0.463 0.464 0.465 0.466 0.467

18

21

24

MT9

Fig. 3. Rapid sawtooth crash in HT-7 discharge with lower hybrid wave heating,absence of any discernible precursor oscillation in plasma core region (SXA27 sig-nal), and high frequency mode appear just before sawtooth crash detected by edged

Fl

L.Q. Xu et al. / Fusion Engineeri

here tn = t0 + n/fs, fm = mfs/l and the window ω(n) has length l.t is zero except for |n| ≤ (l − 1)/2, whereby the time resolution ofhe spectrogram is

tp = l − 12fs

The complex Morlet mother wavelet is basically a sinusoid with Gaussian envelope [8]

(�) = 1√�Fb

exp(i2�Fc�) exp

(−�

2

Fb

)

eing often preferred for its harmonic character and goodime–frequency localization. Fc is the center frequency of wavelet

=√

ln(103Fb) is the effective half length of (�); the translatednd scaled wavelet is a,b(�) = [(� − b)/a]/

√a where b is the

ranslation and a is the scale. The scalogram S(a, b) is the squaref the continuous wavelet transform

∣∣∣∫ +∞∗

∣∣∣2

(a, b) = ∣−∞

Sig(�) a,b(�) d�∣ (2)

here time and frequency are given by t = b and f = Fc/a, the timeesolution of the scalogram is ıts(f ) = Fc�/f .

ig. 4. Analysis of high frequency mode just before sawtooth crash of MT9 signal by u� = 511, l� = 127, � = 1 and (c) the scalogram with Fc = 0.5 Hz, Fb = 100 s2.

Mirnov probes (MT9 signal).

sing (a) the spectrogram with l = 127, (b) the Choi–Williams distribution with,

2 ng and Design 88 (2013) 2767– 2772

c

C

w

t

I

ra

3.3 3.4 3.5 3.6 3.7 3.80.2

0.4

0.6

0.8

1.0

a.u

.

L modeHA18A

3.3 3.4 3.5 3.6 3.7 3.80.5

1.0

1.5

2.0

a.u

.

time(s)

sxc8u

r=-2.66cm

Low frequency mode

3.57 3.58 3.59 3.60 3.61 3.62 3.631.0

1.5

2.0

Fig. 5. Time evolution of low frequency MHD mode during edge localized modefree in core region of plasma in discharge #33055 in EAST tokamak (minor radiusis 45 cm). The top raw is the hydrogen alpha-ray radiation signal and the bottom is

F1

770 L.Q. Xu et al. / Fusion Engineeri

The Choi–Williams distribution of the discrete time signal Sig(n)an be given as follows [12]:

WD(tn, fm, �) = 2

+(l�−1)/2∑�=−(l�−1)/2

l�−1∑�=0

n,m,�,�(�) e−i2��m/l� (3)

here fm = mfs/(2l�) and n,m,�,�(�) is given by

n,m,�,�(�) = Sig(n + � + � − l� − 1

2

)× Sig∗

(n + � − � + l� − 1

2

)

× ω(�)ω�(� − l� − 1

2

)× I

(�, � − l� − 1

2, �

)× ei�m(l�−1)/l�

The windows ω�(n) and ω�(n) having lengths l� and l� , respec-ively. The auxiliary function I(�, �, �) is given by

(�, �, �) ≈√

�e−�

2�/(4��2)

4��2

For large � values, CWD(tn, fm, �) has very good time–frequencyesolution but little artifact reduction. Reduction of artifacts,long with some loss of resolution, is achieved by decreasing

ig. 6. Analysis of low frequency mode during ELM-free of sxc8u signal by using (a) the s27, � = 10 and (c) the scalogram with Fc = 0.5 Hz, Fb = 10 s2.

SXR signals.

pectrogram with l = 1023, (b) the Choi–Williams distribution with, l� = 1023, l� =

ng and

�gban1

3

whH

3

doiarwFCca

F1

L.Q. Xu et al. / Fusion Engineeri

. Intermediate values of � yield a low level of artifacts andood time–frequency resolution. Choi–Williams distribution willecome Wigner–Ville distribution [16,17] when � → ∞. To avoidliasing, Sig(n) is replaced with the corresponding analytic sig-al. The effective time resolution of CWD(tn, fm, �) is ıtcwd = (l� −)/2fs.

. Experimental results

The spectrogram, scalogram, and Choi–Williams distributionill now be applied to some strongly nonstationary magneto-ydro-dynamic instabilities SXR as well as Mirnov signals in bothT-7 and EAST tokamaks.

.1. Coupling tearing modes in HT-7

Coupling tearing mode was observed during the soft internalisruption in HT-7 discharge with shot number #111635. MHDscillations behavior was observed in the core region as shownn Fig. 1. The coupling MHD modes are identified as m/n = 1/1nd 2/1, where m and n are poloidal and toroidal mode number,espectively. Windows ω(n) and ω�(n) are of the Hanning type,hereas ω�(n) are rectangular windows. The signals represented in

ig. 2(a)–(c) has been studied previously using the spectrogram, thehoi–Williams distribution and the scalogram respectively. Signalomponents below 1.5 kHz have been removed by filtering to avoidrtifacts caused by low-frequency components. The frequency of

ig. 7. Analysis of low frequency mode during ELM-free of sxc8u signal by using the C27, � = 10, (c) l� = 1023, l� = 127, � = 100, and (d) l� = 1023, l� = 127, � = ∞ (Wign

Design 88 (2013) 2767– 2772 2771

complex MHD mode is 2–4 kHz. The time evolution of frequency isclearly shown in Fig. 2(b) by means of Choi–Williams distribution.

3.2. High frequency precursors of sawtooth

The rapid collapse of a sawtooth oscillation [15] which is charac-terized by the absence of any discernible precursor oscillation (SXRsignals) in the HT-7 tokamak has been observed in detail on a fasttime scale as shown in Fig. 3. There is almost no any discernible pre-cursor oscillation in the core region of plasma, but high frequencymode (m/n = 2/1) appears just before sawtooth crash in the outerregion of plasma. High frequency mode can be shown more detailedby means of Choi–Williams distribution in Fig. 4(b).

All of spectrogram, the scalogram and the Choi–Williams dis-tribution show high frequency mode appears (about 40 kHz) justbefore sawtooth crash and it is clear that the advantages of theChoi–Williams distribution over both scalogram and the spectro-gram (Fig. 4).

3.3. Low frequency MHD instabilities in edge localized mode freein H mode discharge

High-confinement mode (H-mode) [18] with type-III edge local-

ized modes has been obtained with about 1 MW lower hybrid wavepower on the EAST superconducting tokamak. As shown in Fig. 5,low frequency MHD mode was observed during ELM-free, the fre-quency of the mode is about 1 kHz, more detailed information will

hoi–Williams distribution with (a) l� = 1023, l� = 127, � = 1, (b) l� = 1023, l� =er–Ville distribution).

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4

iFm�otωtts

5

pboenwmdttCtoi

[

[[[[

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772 L.Q. Xu et al. / Fusion Engineeri

e given by time–frequency analysis using Choi–Williams distribu-ion as shown in Fig. 6(b).

Time–frequency analyses of low frequency mode are presentedn Fig. 6. The mode have two branches in frequency and f2 = 2f1,2 ≈ 1.2 kHz and f1 ≈ 0.6 kHz at t∼3.6 s. The mode of f2 is the sec-nd harmonic. The coherent mode of f ∼0.9 kHz is a typical artifactenerated by Choi–Williams distribution. But also it is clear that thedvantages of the Choi–Williams distribution over both scalogramnd the spectrogram (Fig. 6).

. Discussion

Good time–frequency resolution can be achieved by choos-ng a larger � but along with high level of artifacts as shown inig. 7(a)–(c). CWD will become Wigner–Ville distribution with opti-al time–frequency resolution when � → ∞ (Fig. 7d). For smaller, artifacts can be reduced but accompanied some loss of the res-lution. Therefore, the factor � controls the compromise betweenhe resolution and the artifacts suppression. The weighting window�(n) is asymmetrical window whose length and shape determines

he frequency resolution and ω�(n) determines the duration of theime indexed autocorrelation function corresponding to the innerummation of Eq. (3).

. Summary

Time–frequency analysis is one of the most important datarocessing techniques for MHD instabilities. A comparisonetween the Choi–Williams distribution and the scalogram basedn the continuous wavelet trans-form has been done for realxamples of nonstationary fusion plasma signals. For complete-ess, the spectrogram based on the short-time Fourier transform,hich remains the most widely used tool to analyze the afore-entioned signals, has also been calculated. The Choi–Williams

istribution has always yielded better time–frequency resolutionhan the spectrogram and the scalogram. In addition, the fluctua-ion characteristics can be identified more directly and clearly by

hoi–Williams distribution when applied it to the MHD instabili-ies for instance, coupling tearing modes, high frequency precursorsf sawtooth, and low frequency MHD instabilities in edge local-zed mode (ELM) free in H mode discharge. So, the Choi–Williams

[[

[

Design 88 (2013) 2767– 2772

distribution can offer an improved time–frequency view of sometypes of events, provided � and the length of windows can be cho-sen to avoid masking of signal components by artifacts.

Acknowledgements

One of the authors (L.Q. Xu) wants to thank Prof. A.C.A.Figueiredo (Portugal) for his disinterested help about theChoi–Williams distribution method. This work was partially sup-ported by the JSPS-NRF-NSFCA3 Foresight Program in the field ofPlasma Physics (NSFC no. 11261140328). This work was partiallysupported by the National Nature Science Foundation of Chinathrough Grant nos. 10935004, 11205007, 10975155, 10990212, andwas partially supported by the CAS Key International S&T Cooper-ation Project collaboration with Grant no. GJHZ1123.

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