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=sin(ωt+φ)''.
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)''.
The cross phase ''&phi;'' can be determined from the delay ''&Delta;'' of the maxima of the cross correlation &gamma; (modulo 2&pi;): ''&phi; = -&Delta;<sub>max</sub>/&omega;''. In the presence of noise, ''&gamma;(&Delta;)'' will decay as a function of ''&Delta;'', typically exponentially.
The cross phase ''&phi;'' can be determined from the delay ''&Delta;'' of the maxima of the cross correlation &gamma; (modulo 2&pi;): ''&phi; = -&Delta;<sub>max</sub>/&omega;''.  


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 &gamma;<sub>XY</sub> 'inherits' the spectral properties of the original time series.
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 &gamma;<sub>XY</sub> 'inherits' the spectral properties of the original time series.