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25 May 2004 Statistics of level crossing intervals
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Proceedings Volume 5471, Noise in Complex Systems and Stochastic Dynamics II; (2004) https://doi.org/10.1117/12.544594
Event: Second International Symposium on Fluctuations and Noise, 2004, Maspalomas, Gran Canaria Island, Spain
Abstract
We present an analytic relation between the correlation function of dichotomous (taking two values, ± 1) noise and the probability density function (PDF) of the zero crossing interval. The relation is exact if the values of the zero crossing interval τ are uncorrelated. It is proved that when the PDF has an asymptotic form L(τ) = 1/τc, the power spectrum density (PSD) of the dichotomous noise becomes S(f) = 1/fβ where β = 3 - c. On the other hand it has recently been found that the PSD of the dichotomous transform of Gaussian 1/fα noise has the form 1/fβ with the exponent β given by β = α for 0 < α < 1 and β = (α + 1)/2 for 1 < α < 2. Noting that the zero crossing interval of any time series is equal to that of its dichotomous transform, we conclude that the PDF of level-crossing intervals of Gaussian 1/fα noise should be given by L(τ) = 1/τc, where c = 3 - α for 0 < α < 1 and c = (5 - α)/2 for 1 < α < 2. Recent experimental results seem to agree with the present theory when the exponent α is in the range 0.7 ⪅ α < 2 but disagrees for 0 < α ⪅ 0.7. The disagreement between the analytic and the numerical results will be discussed.
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Nobuko Fuchikami and Shunya Ishioka "Statistics of level crossing intervals", Proc. SPIE 5471, Noise in Complex Systems and Stochastic Dynamics II, (25 May 2004); https://doi.org/10.1117/12.544594
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