Bicoherence

The following applies to the analysis of data or signals

$X_i(t)\,$

For convenience and simplicity of notation, the data can be taken to have zero mean ($\langle X_i \rangle = 0$) and unit standard deviation ($\langle X_i^2 \rangle = 1$).

The standard cross spectrum is the Fourier transform of the correlation

$C_1(t_1) = \left \langle X_1(t)X_2(t+t_1) \right \rangle$

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

$C_2(t_1,t_2) = \left \langle X_1(t)X_2(t+t_1)X_2(t+t_2) \right \rangle$

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

$\hat X_i(\omega)$

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

$B(\omega_1,\omega_2) = \hat X_1^*(\omega)\hat X_2(\omega_1) \hat X_2(\omega_2)$

where

$\omega = \omega_1 + \omega_2$

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:

$b^2(\omega_1,\omega_2) = \frac{| \left \langle B(\omega_1,\omega_2) \right \rangle |^2} {\left \langle |\hat X_1(\omega)|^2\right \rangle\left \langle | \hat X_2(\omega_1) \hat X_2(\omega_2)|^2\right \rangle}$

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

$\frac{1}{N(\omega)} \sum_{\omega_1+\omega_2=\omega}{b^2(\omega_1,\omega_2)}$

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

$\frac{1}{N_{tot}} \sum_{\omega_1,\omega_2}{b^2(\omega_1,\omega_2)}$

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

$C_{22}(x_1,x_2,t_1,t_2) = \left \langle X_1(x,t)X_2(x+x_1,t+t_1)X_2(x+x_2,t+t_2) \right \rangle$

the spatio-temporal bispectrum is

$B_2(k_1,k_2,\omega_1,\omega_2) = \hat X_1^*(k,\omega)\hat X_2(k_1,\omega_1) \hat X_2(k_2,\omega_2)$

where $\omega = \omega_1 + \omega_2$ and $k=k_1+k_2$.

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. 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