The axis-crossing intervals of random functions

For an arbitrary random processxi(t)there exists a functionx(t)which may be obtained by infinite clipping. The axis crossings ofx(t)are identical with those ofxi(t). This paper relates the probability densityP(tau)of axis-crossing intervals togamma(tau), the autocorrelation function ofx(t), i.e., the autocorrelation after clipping. It is shown that the expected number of zeros per unit time is proportional togamma prime (0+), i.e., the right-hand derivative ofgamma (tau)attau = 0. Next a theorem is proved, stating thatP(tau) = 0over a finite range0 leq tau < Tif and only ifgamma(tau)is linear inmid tau midover the corresponding range ofmid tau mid. Ifgamma (tau)is nearly linear for smalltau, then the initial behavior ofP(tau)is likegamma prime prime (tau). These results are illustrated by some simple, random square-wave models and by a comparison with Rice´s results for Gaussian noise.