The Wigner distribution as a tool for time-frequency analysis of fusion plasma signals: application to broadband reflectometry data

The Wigner distribution is introduced as an appropriate tool for processing data from fusion plasma diagnostics whose signals have a time varying frequency spectrum, and is thus presented as a particularly suited form for the time-frequency analysis of transients and other non-stationary phenomena. Its effectiveness is illustrated by applying it to the problem of electron density profile measurement through broadband microwave reflectometry in fusion devices, its advantages being demonstrated via a detailed comparison with the well known short time Fourier transform spectrogram. In particular, the Wigner distribution is used in a novel application as a means not only to retrieve from reflectometry data the instantaneous frequency needed for profile inversion, but also to provide an accurate representation of reflectometry signals in the time-frequency plane. Further, in a careful discussion stressing its benefits relative to more standard approaches based on the quadrature signal, the analytic signal is proposed as being appropriate to routinely extract from reflectometry data an unambiguously defined set of instantaneous amplitude, phase and frequency. The problems associated with the fact that digitized reflectometry data give rise to discrete time signals are properly handled, and the different estimates for the instantaneous frequency obtained from the Wigner distribution and the spectrogram by calculating their first frequency moments and mean frequencies, as well as by locating their peaks, are also compared.

[1]  Bizarro.  Weyl-Wigner formalism for rotation-angle and angular-momentum variables in quantum mechanics. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[2]  T. Claasen,et al.  The aliasing problem in discrete-time Wigner distributions , 1983 .

[3]  W. A. Peebles,et al.  Advances in reflectometric density profile measurements on the DIII‐D tokamak (abstract)a) , 1995 .

[4]  J. Y. Hong,et al.  Spectral analysis of nonstationary plasma fluctuation data via digital complex demodulation , 1988 .

[5]  Benjamin A. Carreras,et al.  Wavelet bicoherence: A new turbulence analysis tool , 1995 .

[6]  Plasma fluctuation diagnostics via maximum entropy spectral estimation , 1988 .

[7]  Arnold Migus,et al.  Space–time Wigner functions and their application to the analysis of a pulse shaper , 1995 .

[8]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.

[9]  F. Simonet,et al.  Measurement of electron density profile by microwave reflectometry on tokamaks , 1985 .

[10]  M. Manso,et al.  Reflectometry in fusion devices , 1993 .

[11]  Hidalgo,et al.  Nonlinear phenomena and intermittency in plasma turbulence. , 1995, Physical review letters.

[12]  T. Dudok de Wit,et al.  The biorthogonal decomposition as a tool for investigating fluctuations in plasmas , 1994 .

[13]  Thomas S. Huang,et al.  Image processing , 1971 .

[14]  William H. Press,et al.  Numerical Recipes: FORTRAN , 1988 .

[15]  L. Mandel.  Interpretation of Instantaneous Frequencies , 1974 .

[16]  Kon Max Wong,et al.  Estimation of the time-varying frequency of a signal: the Cramer-Rao bound and the application of Wigner distribution , 1990, IEEE Trans. Acoust. Speech Signal Process..

[17]  David Vakman,et al.  On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for defining amplitude and frequency , 1996, IEEE Trans. Signal Process..

[18]  Madhu Gupta.  Definition of instantaneous frequency and frequency measurability , 1975 .

[19]  J. D. Bell,et al.  Signal analysis of fluctuations in toroidal fusion plasmas , 1993 .

[20]  A. W. Rihaczek,et al.  Hilbert transforms and the complex representation of real signals , 1966 .

[21]  M. Scully,et al.  Distribution functions in physics: Fundamentals , 1984 .

[22]  Bernard C. Picinbono,et al.  On instantaneous amplitude and phase of signals , 1997, IEEE Trans. Signal Process..

[23]  M. Hirsch,et al.  Heterodyne methods in millimetre wave plasma diagnostics with applications to ECE, interferometry and reflectometry , 1997 .

[24]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals , 1992, Proc. IEEE.

[25]  G. H. Danielson.  Frequency selective circuits for a microelectronic FSK receiver , 1966 .

[26]  Sen,et al.  Characterization of coherent structures in tokamak edge turbulence. , 1994, Physical review letters.

[27]  R. Benner,et al.  Chemical composition of dissolved organic nitrogen in the ocean , 1997, Nature.

[28]  W. A. Peebles,et al.  X‐mode broadband reflectometric density profile measurements on DIII‐D , 1990 .

[29]  H. Zohm,et al.  Wavelet analysis of fusion plasma transients , 1997 .

[30]  Steven W. McDonald,et al.  Phase-space representations of wave equations with applications to the eikonal approximation for short-wavelength waves , 1988 .

[31]  E. Bedrosian.  A Product Theorem for Hilbert Transforms , 1963 .

[32]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. II. A/lgorithms and applications , 1992, Proc. IEEE.

[33]  E. Mazzucato,et al.  Microwave reflectometry for magnetically confined plasmas , 1998 .

[34]  P. Loughlin,et al.  On the amplitude‐ and frequency‐modulation decomposition of signals , 1996 .

[35]  D. E. Vakman,et al.  METHODOLOGICAL NOTES: Amplitude, phase, frequency---fundamental concepts of oscillation theory , 1977 .

[36]  A. Donné,et al.  REVIEW ARTICLE: Reflectometry techniques for density profile measurements on fusion plasmas , 1996 .

[37]  P. Loughlin,et al.  Comments on the interpretation of instantaneous frequency , 1997, IEEE Signal Processing Letters.

[38]  W. H. Holmes,et al.  A new method of gyrator-RC filter synthesis , 1966 .

[39]  Michael Martin Nieto,et al.  Phase and Angle Variables in Quantum Mechanics , 1968 .

[40]  Atsushi Mase,et al.  Measurement of plasma density using wavelet analysis of microwave reflectometer signal , 1998 .

[41]  A. Nuttall,et al.  On the quadrature approximation to the Hilbert transform of modulated signals , 1966 .

[42]  L. Meneses,et al.  Ultrafast broadband frequency modulation of a continuous wave reflectometry system to measure density profiles on ASDEX Upgrade , 1996 .

[43]  D. Brower,et al.  Advanced plasma fluctuation analysis techniques and their impact on fusion research (invited) , 1988 .

[44]  M. E. Manso,et al.  Initialization of plasma density profiles from reflectometry , 1995 .

[45]  E. Doyle,et al.  Development of a fast solid-state high-resolution density profile reflectometer system on the DIII-D tokamak , 1997 .

[46]  T. Witt.  Enhancement of multichannel data in plasma physics by biorthogonal decomposition , 1995 .