Statistics of level crossing intervals

摘要

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/f~β 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 + l)/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 approx < α < 2 but disagrees for 0 < α approx < 0.7. The disagreement between the analytic and the numerical results will be discussed.