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Long-range correlation: Difference between revisions
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== Coherent states == | == Coherent states == | ||
Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y= | Coherent system states (regular oscillations or 'modes') lead to oscillatory behaviour of the correlation function, as is easily checked by setting ''X = sin(ωt)'' and taking, e.g., ''Y=sin(ωt+φ)''. | ||
Note also that the correlation function is a convolution, hence its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | The cross phase ''φ'' can be determined from the delay ''Δt'' of the maxima of the cross correlation γ (modulo 2π): ''φ = -Δt<sub>max</sub>/ω''. In the presence of noise, ''γ(Δt)'' will decay as a function of ''Δt'', typically exponentially. | ||
Note also that the correlation function is a [[:Wikipedia:convolution|convolution]], hence by the [[:Wikipedia:convolution theorem|convolution theorem]] its spectrum is the product of the spectra of ''X'' and ''Y'', so that γ<sub>XY</sub> 'inherits' the spectral properties of the original time series. | |||
== Turbulence == | == Turbulence == | ||