The axis-crossing intervals of random functions--II

This paper considers the intervals between axis crossings of a random function . Following a previous paper, continued use is made of the statistical properties of the function and the output after is infinitely clipped. Under the assumption that a given axis-crossing interval is independent of the sum of the previous intervals, where takes on all values, , , an integral equation is derived for the probability density 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 and the correlation coefficient 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, is large. For low-pass spectra, is small, yet the statistical dependence between successive intervals may be strong even when the correlation is nearly zero.