مرکز منطقه ای اطلاع رساني علوم و فناوري - 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, $^1$ continued use is made of the statistical properties of the function $x(t)$ 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, where $m$ takes on all values, $m =0$ , $1, 2, \\cdots$ , an integral equation is derived for the probability density $P_0(\\tau )$ 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 $\\sigma^2 (\\tau )$ and the correlation coefficient $\\kappa$ between 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, $\\kappa$ is large. For low-pass spectra, $\\kappa$ is small, yet the statistical dependence between successive intervals may be strong even when the correlation $\\kappa$ is nearly zero.