Bicoherence: Difference between revisions

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== Notes ==
== Notes ==


* The bicoherence can of course be defined in wavenumber space instead of frequency space by applying the replacements ''t &rarr; x'' and ''&omega; &rarr; k''. Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 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]</ref>
* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', Phys. Plasmas '''2''', 8 (1995) 3017]</ref>
* The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. <ref>[http://link.aip.org/link/?PHPAEN/2/3017/1 B.Ph. van Milligen et al, ''Wavelet bicoherence: a new turbulence analysis tool'', 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 &rarr; x'' and ''&omega; &rarr; k''.
* Combined temporal-spatial studies are also possible. <ref>[http://link.aip.org/link/PHPAEN/v17/i5/p052313/s1 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]</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 />

Revision as of 21:03, 28 September 2010

The following applies to the analysis of data or signals

Xi(t)

For convenience and simplicity of notation, the data can be taken to have zero mean (Xi=0) and unit standard deviation (Xi2=1).

The standard cross spectrum is the Fourier transform of the correlation

C1(t1)=X1(t)X2(t+t1)

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

C2(t1,t2)=X1(t)X2(t+t1)X2(t+t2)

The signals Xi can either be different or identical. In the latter case, one speaks of the autocorrelation, autospectrum, auto-bicorrelation or auto-bispectrum.

Bispectrum

Denoting the Fourier transforms of the signals Xi(t) by

X^i(ω)

the bispectrum is defined as

B(ω1,ω2)=X^1*(ω)X^2(ω1)X^2(ω2)

where

ω=ω1+ω2

Bicoherence

The bicoherence is obtained by averaging the bispectrum over statistically equivalent realizations, and normalizing the result:

b2(ω1,ω2)=|B(ω1,ω2)|2|X^1(ω)|2|X^2(ω1)X^2(ω2)|2

The normalization is such that 0 ≤ b2 ≤ 1.

The bicoherence is symmetric under the transformations 12) → (ω21) and 12) → (-ω1,-ω2), so that only one quarter of the plane 12) contains independent information. Additionally, for discretely sampled data all frequencies must be less than the Nyquist frequency: 1|,|ω2|,|ω| ≤ ωNyq. 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

1N(ω)ω1+ω2=ωb2(ω1,ω2)

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

1Ntotω1,ω2b2(ω1,ω2)

where Ntot 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 ω12 is nearly constant over a significant number of realizations.

Notes

  • The bicoherence can be computed using the (continuous) wavelet transform instead of the Fourier transform, in order to improve statistics. [1]
  • 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. [2]

Starting from the spatio-temporal bicorrelation

C22(x1,x2,t1,t2)=X1(x,t)X2(x+x1,t+t1)X2(x+x2,t+t2)

the spatio-temporal bispectrum is

B2(k1,k2,ω1,ω2)=X^1*(k,ω)X^2(k1,ω1)X^2(k2,ω2)

where ω=ω1+ω2 and k=k1+k2.

References