# Bicoherence

The following applies to the analysis of data or signals

${\displaystyle X_{i}(t)\,}$

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

The standard cross spectrum is the Fourier transform of the correlation

${\displaystyle C_{1}(t_{1})=\left\langle X_{1}(t)X_{2}(t+t_{1})\right\rangle }$

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

${\displaystyle C_{2}(t_{1},t_{2})=\left\langle X_{1}(t)X_{2}(t+t_{1})X_{2}(t+t_{2})\right\rangle }$

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

${\displaystyle {\hat {X}}_{i}(\omega )}$

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

${\displaystyle B(\omega _{1},\omega _{2})={\hat {X}}_{1}^{*}(\omega ){\hat {X}}_{2}(\omega _{1}){\hat {X}}_{2}(\omega _{2})}$

where

${\displaystyle \omega =\omega _{1}+\omega _{2}}$

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:

${\displaystyle b^{2}(\omega _{1},\omega _{2})={\frac {|\left\langle B(\omega _{1},\omega _{2})\right\rangle |^{2}}{\left\langle |{\hat {X}}_{1}(\omega )|^{2}\right\rangle \left\langle |{\hat {X}}_{2}(\omega _{1}){\hat {X}}_{2}(\omega _{2})|^{2}\right\rangle }}}$

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

${\displaystyle {\frac {1}{N(\omega )}}\sum _{\omega _{1}+\omega _{2}=\omega }{b^{2}(\omega _{1},\omega _{2})}}$

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

${\displaystyle {\frac {1}{N_{tot}}}\sum _{\omega _{1},\omega _{2}}{b^{2}(\omega _{1},\omega _{2})}}$

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

${\displaystyle C_{22}(x_{1},x_{2},t_{1},t_{2})=\left\langle X_{1}(x,t)X_{2}(x+x_{1},t+t_{1})X_{2}(x+x_{2},t+t_{2})\right\rangle }$

the spatio-temporal bispectrum is

${\displaystyle B_{2}(k_{1},k_{2},\omega _{1},\omega _{2})={\hat {X}}_{1}^{*}(k,\omega ){\hat {X}}_{2}(k_{1},\omega _{1}){\hat {X}}_{2}(k_{2},\omega _{2})}$

where ${\displaystyle \omega =\omega _{1}+\omega _{2}}$ and ${\displaystyle k=k_{1}+k_{2}}$.

## 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