The axis-crossing intervals of random functions--II

Abstract
This paper considers the intervals between axis crossings of a random function\xi(t). Following a previous paper,^1continued use is made of the statistical properties of the functionx(t)and the output after\xi(t)is infinitely clipped. Under the assumption that a given axis-crossing interval is independent of the sum of the previous(2m + 2)intervals, wheremtakes on all values,m =0,1, 2, \cdots, an integral equation is derived for the probability densityP_0(\tau)of axis-crossing intervals. This equation is solved numerically for several examples of Gaussian noise. The results of this calculation compare favorably with experiment when the high-frequency cutoff is not extremely sharp. Under the assumption that the successive axis-crossing intervals form a Markoff chain in the wide sense, infinite integrals are found which yield the variance\simga^2 (\tau)and the correlation coefficient\kappabetween the lengths of two successive axis-crossing intervals. These parameters are obtained numerically for several examples of Gaussian noise. For bandwidths at least as small as the mean frequency,\kappais large. For low-pass spectra,\kappais small, yet the statistical dependence between successive intervals may be strong even when the correlation\kappais nearly zero.

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