Statistics of level crossing intervals

Nobuko Fuchikami; Shunya Ishioka

doi:10.1117/12.544594

25 May 2004 Statistics of level crossing intervals

Nobuko Fuchikami, Shunya Ishioka

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/τ^c, 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/f^α noise has the form 1/f^β 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/f^α noise should be given by L(τ) = 1/τ^c, 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.

Citation Download Citation

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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