Spectral analysis of nonstationary plasma fluctuation data via digital complexdemodulationE. J. Powers, H. S. Don, J. Y. Hong, Y. C. Kim, G. A. Hallock, and R. L. Hickok Citation: Review of Scientific Instruments 59, 1757 (1988); doi: 10.1063/1.1140102 View online: http://dx.doi.org/10.1063/1.1140102 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/59/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spectral analysis of full field digital mammography data Med. Phys. 29, 647 (2002); 10.1118/1.1445410 Time-series analysis of nonstationary plasma fluctuations using wavelet transforms Rev. Sci. Instrum. 68, 898 (1997); 10.1063/1.1147715 Spectral estimation of plasma fluctuations. II. Nonstationary analysis of edge localized mode spectra Phys. Plasmas 1, 501 (1994); 10.1063/1.870939 Digital complex demodulation applied to interferometry Rev. Sci. Instrum. 57, 1989 (1986); 10.1063/1.1138762 Real time digital spectral analysis as a plasma fluctuation diagnostic Rev. Sci. Instrum. 51, 1151 (1980); 10.1063/1.1136397

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

128.123.44.23 On: Thu, 18 Dec 2014 15:30:47

Spectral analysis of nonstationary plasma fluctuation data via digital complex demodulation

E. J. Powers, H. S. Don,a) J. Y. Hong,b) and Y. C. Kime)

The University o/Texas at A ustin, Austin. Texas 78712

G. A. Hallockd) and R. l. Hickok

Rensselaer Polytechnic Institute, Troy, New York 12180

(Presented on 16 March 1988)

Digital complex demodulation techniques are used to generate time-varying auto- and cross­power spectra which, in turn, are useful in characterizing the time-frequency characteristics of nonstationary piasma fluctuation data. The approach is illustrated with fluctuation data measured with a heavy-ion beam probe on the TMX experiment.

INTRODUCTION

Spectral analysis (power spectra, cross-power spectra, hi­spectra) has proven to be a powerful technique with which to analyze plasma fluctuation data. For such an approach to yield satisfactory results the data must be sufficiently sta­tionary over the time duration of the data record. Fortunate­ly, this requirement is often met, e.g., during the current plateau of tokamak experiments. On the other hand, it must be recognized that there are situations in plasma fluctuation diagnostics when the requirement of stationarity is not suffi­ciently satisfied, but it is still desirable, if not necessary, to quantify the statistical properties and effects of the fluctu­ations. For instance, one class of nonstationary data is asso­ciated with transient events such as gas puffing, impurity injection, and so on. Another class of nonstationary data results when a relatively coherent mode drifts in frequency or amplitude. In this case, the application of classical spec­tral analysis techniques results in a smearing of amplitude and frequency information. It is this latter class of nonsta­tionary data that this article is concerned with.

To the extent that the time variation of the amplitude, frequency, and/or phase is sufficiently slowly varying, one can represent this behavior in the frequency domain with complex Fourier coefficients, the amplitude and phase of which are slowly time varying. Knowledge of the time~vary­ing Fourier coefficients then allows one to compute time­varying auto- and cross~power spectra. It is the objective of this article to point out that digital complex demodulation techniques, when appropriately applied, can be used to de­termine the time-varying Fourier coefficients and, hence, the time-varying auto- and cross-spectra.

I. DIGITAL COMPLEX DEMODULATION

Digital complex demodulation is, in essence, the digital equivalent of homodyne detection. To illustrate the ap­proach, we assume that we have one channel of time series fluctuation datax(t) residing in a computer. For the sake of this explanation, we assume that xU) can he expressed in

terms of an amplitude- and phase-modulated wave as fol­lows:

x(t) = a (t)cos [UJot + p(t)] , (1)

where UJo denotes the carrier frequency, and a(t) and p(t) denote the relatively slowly time-varying amplitude and phase, respectively.

Next we express cos[aJo! + p(t)] in terms of complex exponentials to get

xU) = [a(t)/2j{exp[iUJot + i'p(t)]

+ exp[ - imot - ip(t)]} . (2)

We then "demodulate" the time series data by multiplying (in the computer) xU) by 21/2 exp( - iUJot). The result of this "mixing" operation is then passed through a low-pass digital filter of bandwidth tJ,m, the cutoff frequency being suitably adjusted to reject the 2UJo frequency component. The output of the low-pass filter will be the so-called com­plex demodulate c(t),

ex (mo,t) = ax (mo,t)exp [ip x «(Un,t) ]121/2 , (3)

where the amplitUde ax (mo,t) and the phase Px (UJo,t) corre­spond to the slowly time-varying amplitude and phase mod­ulation of the carrier UJo in ( 1 ). The subscript x refers to the time series x ( t) .

II. IMPLEMENTATION: TIME~VARYING SPECTRA

First we consider a single channel of data xU), It is assumed that xU) is of duration T s, and consists of N = Tit, samples, where t, is the sampling interval. In or­der that the sampling theorem be satisfied, we require that the highest frequency B present in the data be equal to or less than the Nyquist frequency, iN = 1/2 ts' For the sake of example, we take B = It,,.

If Am is the frequency resolution desired, then we divide the region between zero frequency and the Nyquist frequen­cy into L frequency bands, where L = UJNlb.UJ. Then the digital complex demodulation technique of Sec. I can be ap­plied to each of the L frequency bands. Consider the k th

1157 Rev. Sci.lnstrum. 59 (8), August 1988 0034-6148/86/081757-03$01.30 @ 1988 American Institute of Physics 1757

•••• ' •••••.••••• , •••••••• -.~.-•• -•• ? ••••• .-••••••••• :.:..-•••• ~ ••••• ,

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

128.123.44.23 On: Thu, 18 Dec 2014 15:30:47

frequency band centered at ()Jk = kA()J, where k = O,1, ... ,L - 1. We multiply the time series data residing in the computer by 21/2 exp( - lcUkt), pass it through a low­pass filter of bandwidth A()J, to recover the complex demodu­late Cx«()Jk,t). This process is repeated for k = 0, 1,2, ... ,L - 1.

If we have a second time seriesy(t), then similar analy­sis can be carried out to yield cy (rvk,t) for k = 0,1,2, ... ,L - 1.

Then from knowledge of the time-varying complex de­modulates, we can compute the respective time-varying auto- and cross-power spectra as foHows:

P xx «()Jk' t) = Ic, (()Jk'!) 12 = a~ ({Vk ,1)/2,

PI'Y «()Jk,t) = Jcy «()Jk,t) 12 = a; «(Uk ,t) 12 ,

PYX «()Jk,t) = Cy «()Jk,t)C; «()Jk,t)

= [Or (O)k,t)ax «()Jk,t)/2]

Xexp[zpy((Uk,t) -(Px«()Jk,t)] ,

( 4a)

(4b)

(4c)

where ()Jk = kA()J, k = O,1,2, ... ,L - 1. Examples of time­varying spectra, calculated using this approach, are to be found in Sec. III. Note that the spectra of Eq. (4) are not "true" power spectra in that they are time varying and do not involve the operation of expectation.

In using complex demodulation to estimate time-vary­ing spectra, the choice of an appropriate digital filter is criti­caI. l Specifically, the filter must have appropriate amplitude response (fiat), phase response (no distortion), and transi­tion-band (for good spectral resolution) characteristics. The "MAXFLAT" filter described by Kaiser and Reed2 is suit­able for this application.

III. EXPERIMENTAL EXAMPLE

To illustrate the practicality of this article's approach, we utilize heavy-ion beam probe (HIBP) data collected dur­ing a study of the radial electrostatic potential profile in the central cell of the TMX experiment. 3 Although the primary focus was on static measurements, certain portions (not the

o 1-'" :::J6 ~_~or.O-O-----'lO-.-O-O----2TO-.O-O----~30~.-0-O----4'O-.-OO----~50.00

NAVE FREQUENCY [KHZ]

FIG. 1. Classical auto-power spectrum of density [i.e., nF( Te) J fluctu­ations.

1758 Rev. Scl.lnstrum., Vol. 59, No.8, August 1988

entire trace) of the raw HIBP data time trace suggested the presence of a relatively coherent mode. When the classical auto-power spectrum of the trace was computed, little evi­dence ofthis relatively coherent mode was found. This spec­trum is shown in Fig. 1.

Next we proceeded to utilize the digital complex de­modulation approach ofthis article to generate time-varying auto-power spectra for the density n [actually nF( Te)] and potential ¢I channels, and the cross-power spectrum between nand 1;. The auto-power spectra for nand ¢ are similar in appearance to the amplitude of the time-varying cross-pow­er spectrum iPntb (()Jk,t) i shown in Fig. 2. Figure 2 was com­puted from two time series records consisting of 2048 sam­ples each. The sampling frequency was 100 kHz, thus the Nyquist fi-equency is 50 kHz, and the duration of each time series record is 20.48 ms. The size of the time-frequency ar­ray of Fig. 2 is 32 X 40, respectively. Thus the frequency reso­lution t..()J/2rr is 50 kHz/40 = 1.25 kHz/division and the frequency axis extends from 0.0 Hz to 48.75 kHz( = 39X 1.25 kHz).

With respect to the time domain, 88 points of data were effectively truncated at the beginning and end of each record as a result of the difficulty of accurately evaluating (at the beginning and end regions) the convolution operation asso­ciated with the digital filter. Therefore, the effective record length is 18.72 ms, and the time resolution in Fig. 2 is equal

Amplitude (a)

(b)

C>

~ ()

TIME (SEC)

O.O,::,::-................. ~~~ ........ ~ ......................... ~~ 0.0 FREQUENCY (kHz) 48.75

FlG. 2. Amplitude of the time-varying cross-power spectrum between den­sity [nF( Te) J aild potential fluctuations: (a) perspective view, (b) contour plot. The range of time and frequency in (a) is the same as that in (b). In (b) the contour level ranges from zero to 300.00 (in arbitrary units) with a contour interval of 60.

Data acquisition and analysis 1758

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

128.123.44.23 On: Thu, 18 Dec 2014 15:30:47

FREQUENCY (kHz)

FiG. 3. Contour plot of the phase of the time-varying cross-power spectrum. The phase is plotted from O· to 320·, with each contour level representing 80·, Note that the phase is constant in that region of the time-frequency plane occupied by the "coherent" mode. Outside this region the calculated values of phase are meaningless, since there is no signal present.

to 18.72 ms/32 = 0.59 ms. Thus the time interval extends from 0 to 18.29 ms( = 0.59 msX 31) in Fig. 2.

We note that there is a relatively coherent mode present at about 10 kHz, although its amplitude appears to be ran­domly modulated. At approximately 14 ms, we observe that the time-frequency spectrum indicates that the frequency of the mode suddenly decreases in frequency. This decrease in frequency is what results in the "smearing" of the classical auto-power spectrum and, hence, the inability to detect the presence of the mode in Fig. 1. The decrease in frequency is associated with a shutting-off of the neutral beams, which modified the electrostatic potential profile, which in turn changed the EX B drift velocity, and, thus, the observed fre­quency of the mode. A detailed description of such oscilla­tions appears in Ref. 4.

In Fig, 3 we have plotted the phase of the time-varying cross-power spectrum. Note that Eq, (4c) indicates that the phase of the time-varying cross-power spectrum preserves phase information in the form of a phase difference, thus Fig.

1759 Rev, Sci. Instrum., Vol. 59, No.8, August 1988

3 represents the time-frequency characteristics of the phase difference between density and potential fluctuations. Of particular importance is the fact that the phase difference is very constant over those regions ofthe time~frequency plane occupied by the relatively coherent mode. Such information is extremely important in ultimately extending the results of this article to a time-varying fluctuation-induced particle transport diagnostic.

IV. CONCLUSION

In summary, we have demonstrated that judicious ap­plication of digital complex demodulation techniques is ca­pable of providing fresh insight into time-varying spectra associated with modulation and transient phenomena in plasmas. The principal "cost" associated with the complex demodulation approach is the computer time required to carry out the convolution operation associated with the low­pass digital filtering.

ACKNOWLEDGMENTS

This work is supported by the U.S. DOE. The digital complex demodulation technique utilized in this article was originally developed under the auspices of the DoD Joint Services Electronics Progam, AFOSR Contract F49620-86-C-0045 , The fluctuation data used in this article were mea­sured on the TMX experiment at Lawrence Livermore Na­tional Laboratory.

a) Now at State University of New York at Swny Brook. h) Now at Intelligent Signal Processing, Austin, Texas, c) Now at Exxon Production Research, Houston, Texas. d) Now at The University of Texas at Austin. 'U-S Don, M.S. thesis, The University of Texas at Austin, 1.981. 2J. Po Kaiser and W. A. Reed, Rev. Sci. lnstrum. 48, 1447 (1977). 'G. A. Hallock, Lawrence Livermor~ National Laboratory Report UelD· 19759, 1983.

4E. B. Hooper, Jr., G. A. Hallock, and J. H. Foote, Phys. Fluids 26,314 (1983),

Data acquisition and snalysis 1759

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

128.123.44.23 On: Thu, 18 Dec 2014 15:30:47