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 ($⟨X_i⟩=0$) and unit standard deviation ($⟨X_i^2⟩=1$).

The standard cross spectrum is the Fourier transform of the correlation

$C_1(t_1)=⟨X_1(t)X_2(t+t_1)⟩$

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

$C_2(t_1,t_2)=⟨X_1(t)X_2(t+t_1)X_2(t+t_2)⟩$

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:

$b^2({\omega }_1,{\omega }_2)=\frac{|⟨B({\omega }_1,{\omega }_2)⟩{|}^2}{⟨|{\hat{X}}_1(\omega ){|}^2⟩⟨|{\hat{X}}_2({\omega }_1){\hat{X}}_2({\omega }_2){|}^2⟩}$

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 (as occurs with a driving mode and a slave mode; energy transfer in a single direction).
• '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

$C_{22}(x_1,x_2,t_1,t_2)=⟨X_1(x,t)X_2(x+x_1,t+t_1)X_2(x+x_2,t+t_2)⟩$

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 $k=k_1+k_2$.