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 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 (ω₁,ω₂) → (ω₂,ω₁) and (ω₁,ω₂) → (-ω₁,-ω₂), so that only one quarter of the plane (ω₁,ω₂) contains independent information. Additionally, for discretely sampled data all frequencies must be less than the Nyquist frequency: |ω₁|,|ω₂|,|ω| ≤ ω_Nyq. 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 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 ω₁+ω₂ 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) - 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.

Notes

• The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics.
• 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.

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 $k=k_{1}+k_{2}$.