# 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 *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

- $ \hat X_i(\omega) $

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

- $ \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 *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

- $ 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

- ↑ 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