# Bicoherence

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 *X _{i}* 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 *X _{i}(t)* are denoted by

Thus, the bispectrum, computed as the Fourier transform of the bicorrelation *C _{2}*, 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 ≤ *b ^{2}* ≤ 1.

The bicoherence is symmetric under the transformations *(ω _{1},ω_{2}) → (ω_{2},ω_{1})* and

*(ω*, so that only one quarter of the plane

_{1},ω_{2}) → (-ω_{1},-ω_{2})*(ω*contains independent information. Additionally, for discretely sampled data all frequencies must be less than the Nyquist frequency:

_{1},ω_{2})*|ω*. These restrictions define a polygonal subspace of the plane, which is how the bicoherence is usually represented (for an example, see TJ-II:Turbulence).

_{1}|,|ω_{2}|,|ω| ≤ ω_{Nyq}The summed bicoherence is defined by

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

where *N _{tot}* 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
ω_{1}+ω_{2} 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

- ↑ 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 - ↑
^{2.0}^{2.1}B.Ph. van Milligen et al,*Wavelet bicoherence: a new turbulence analysis tool*, Phys. Plasmas**2**, 8 (1995) 3017 - ↑ 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 - ↑ 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