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Bicoherence: Difference between revisions
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:<math>X_i(t)\,</math> | :<math>X_i(t)\,</math> | ||
For convenience and simplicity of notation, the data can be taken to have | |||
''zero mean'' (<math>\langle X_i \rangle = 0</math>) and | |||
''unit standard deviation'' (<math>\langle X_i^2 \rangle = 1</math>). | |||
The standard cross spectrum is the Fourier transform of the correlation | The standard cross spectrum is the Fourier transform of the correlation | ||
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== Bispectrum == | == Bispectrum == | ||
The Fourier transforms of the signals ''X<sub>i</sub>(t)'' are denoted by | |||
:<math>\hat X_i(\omega)</math> | :<math>\hat X_i(\omega)</math> | ||
the bispectrum | Thus, the bispectrum, computed as the Fourier transform of the bicorrelation ''C<sub>2</sub>'', becomes: | ||
:<math>B(\omega_1,\omega_2) = \hat X_1^*(\omega)\hat X_2(\omega_1) \hat X_2(\omega_2)</math> | :<math>B(\omega_1,\omega_2) = \hat X_1^*(\omega)\hat X_2(\omega_1) \hat X_2(\omega_2)</math> | ||
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:<math>\omega = \omega_1 + \omega_2</math> | :<math>\omega = \omega_1 + \omega_2</math> | ||
Hence, the bispectrum is interpreted as a measure of the degree of three-wave coupling. | |||
== Bicoherence == | == Bicoherence == | ||
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statistically equivalent realizations, and normalizing the result: | statistically equivalent realizations, and normalizing the result: | ||
:<math>b^2(\omega_1,\omega_2) = \frac{\left \langle | :<math>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}</math> | {\left \langle |\hat X_1(\omega)|^2\right \rangle\left \langle | \hat X_2(\omega_1) \hat X_2(\omega_2)|^2\right \rangle}</math> | ||
The normalization is such that 0 ≤ ''b<sup>2</sup>'' ≤ 1. | The normalization is such that 0 ≤ ''b<sup>2</sup>'' ≤ 1. | ||
The bicoherence is symmetric under the | The bicoherence is symmetric under the transformations ''(ω<sub>1</sub>,ω<sub>2</sub>) → (ω<sub>2</sub>,ω<sub>1</sub>)'' and | ||
''(ω<sub>1</sub>,ω<sub>2</sub>) → (-ω<sub>1</sub>,-ω<sub>2</sub>)'', so that only one quarter of the plane ''(ω<sub>1</sub>,ω<sub>2</sub>)'' contains independent information. | ''(ω<sub>1</sub>,ω<sub>2</sub>) → (-ω<sub>1</sub>,-ω<sub>2</sub>)'', so that only one quarter of the plane ''(ω<sub>1</sub>,ω<sub>2</sub>)'' contains independent information. | ||
Additionally, for discretely sampled data all frequencies must be less than the | Additionally, for discretely sampled data all frequencies must be less than the | ||
Nyquist frequency: ''ω<sub>1</sub>,ω<sub>2</sub>,ω ≤ ω<sub>Nyq</sub>''. These restrictions define a polygonal subspace of the plane, which is how the bicoherence is usually represented. | [[wikipedia:Nyquist frequency|Nyquist frequency]]: ''|ω<sub>1</sub>|,|ω<sub>2</sub>|,|ω| ≤ ω<sub>Nyq</sub>''. 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 | The summed bicoherence is defined by | ||
:<math>\frac{1}{N} \sum_{\omega_1+\omega_2=\omega}{b^2(\omega_1,\omega_2)} </math> | :<math>\frac{1}{N(\omega)} \sum_{\omega_1+\omega_2=\omega}{b^2(\omega_1,\omega_2)} </math> | ||
where ''N'' is the number of terms in the sum. | where ''N'' is the number of terms in the sum. | ||
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the phase between the wave at ω and the sum wave | the phase between the wave at ω and the sum wave | ||
ω<sub>1</sub>+ω<sub>2</sub> is nearly constant over a significant number of realizations. | ω<sub>1</sub>+ω<sub>2</sub> 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<ref>Y. Nagashima et al, ''Observation of coherent bicoherence and biphase in potential fluctuations around geodesic acoustic mode frequency on JFT-2M'', [[doi:10.1088/0741-3335/48/5A/S38|Plasma Phys. Control. Fusion '''48''' (2006) A377]]</ref>) - 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.<ref name="milligen1995"></ref><ref>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'', [[doi:10.1088/0029-5515/52/1/013006|Nucl. Fusion 52 (2012) 013006]]</ref> | |||
== Notes == | == Notes == | ||
* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements '' | * The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref name="milligen1995">B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', [[doi:10.1063/1.871199|Phys. Plasmas '''2''', 8 (1995) 3017]]</ref> | ||
* | * 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. <ref>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'' , [[doi:10.1063/1.3429674|Phys. Plasmas '''17''' (2010) 052313]]</ref> | |||
Starting from the spatio-temporal bicorrelation | |||
:<math>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 </math> | |||
the spatio-temporal bispectrum is | |||
:<math>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)</math> | |||
where <math>\omega = \omega_1 + \omega_2</math> and <math>k=k_1+k_2</math>. | |||
== References == | == References == | ||
<references /> | <references /> | ||