Bicoherence

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The following applies to the analysis of data or signals

For convenience and simplicity of notation, the data can be taken to have zero mean () and unit standard deviation ().

The standard cross spectrum is the Fourier transform of the correlation

where the square brackets imply averaging over t. Similarly, the bispectrum is the Fourier transform of the bicorrelation

The signals Xi can either be different or identical. In the latter case, one speaks of the autocorrelation, autospectrum, auto-bicorrelation or auto-bispectrum.

Bispectrum

The Fourier transforms of the signals Xi(t) are denoted by

Thus, the bispectrum, computed as the Fourier transform of the bicorrelation C2, becomes:

where

Hence, the bispectrum is interpreted as a measure of the degree of three-wave coupling.

Bicoherence

The bicoherence is obtained by averaging the bispectrum over statistically equivalent realizations, and normalizing the result:

The normalization is such that 0 ≤ b2 ≤ 1.

The bicoherence is symmetric under the transformations 12) → (ω21) and 12) → (-ω1,-ω2), so that only one quarter of the plane 12) contains independent information. Additionally, for discretely sampled data all frequencies must be less than the Nyquist frequency: 1|,|ω2|,|ω| ≤ ωNyq. These restrictions define a polygonal subspace of the plane, which is how the bicoherence is usually represented (for an example, see TJ-II:Turbulence).

The summed bicoherence is defined by

where N is the number of terms in the sum. Similarly, the total mean bicoherence is

where Ntot is the number of terms in the sum.

Interpretation

The bicoherence measures three-wave coupling and is only large when the phase between the wave at ω and the sum wave ω12 is nearly constant over a significant number of realizations.

The two-dimensional bicoherence graph tends to show mainly two types of structures:

  • 'Points': indicating sharply defined, unchanging, locked frequencies.
  • 'Lines': these are more difficult to interpret. It is often stated that 'lines' are due to single mode (frequency) interacting with a broad range of frequencies (e.g., a Geodesic Acoustic Mode and broad-band turbulence[1]) - but it is not evident that this is the only explanation. Particularly, two interacting oscillators (continuously exchanging energy) also produce lines in the bicoherence graph.[2][3]

Notes

  • The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. [2]
  • The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements t → x and ω → k.
  • Combined temporal-spatial studies are also possible. [4]

Starting from the spatio-temporal bicorrelation

the spatio-temporal bispectrum is

where and .

References

  1. Y. Nagashima et al, Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M, Plasma Phys. Control. Fusion 48 (2006) A377
  2. 2.0 2.1 B.Ph. van Milligen et al, Wavelet bicoherence: a new turbulence analysis tool, Phys. Plasmas 2, 8 (1995) 3017
  3. B.Ph. van Milligen, L. García, B.A. Carreras, M.A. Pedrosa, C. Hidalgo, J.A. Alonso, T. Estrada and E. Ascasíbar, MHD mode activity and the velocity shear layer at TJ-II, Nucl. Fusion 52 (2012) 013006
  4. T. Yamada, S.-I. Itoh, S. Inagaki, Y. Nagashima, S. Shinohara, N. Kasuya, K. Terasaka, K. Kamataki, H. Arakawa, M. Yagi, A. Fujisawa, and K. Itoh, Two-dimensional bispectral analysis of drift wave turbulence in a cylindrical plasma , Phys. Plasmas 17 (2010) 052313