In a stationary Gaussian Markov process (or Ornstein-Uhlenbeck process) the expected number of axis crossings per unit time, the probability density of the lengths of axis-crossing intervals, and the probability of recurrence at zero level do not exist as ordinarily defined. In this paper new definitions are presented and some asymptotic formulas are derived. Certain renewal equations are approximately satisfied, thereby suggesting an asymptotic approach to independence of the lengths of successive axis-crossing intervals. Mention is made of an application to the filter-clip-filter problem.

A new method is given for obtaining the probability density of the output of an RC filter when the input is a stationary binary random process. The axis-crossing intervals of the input are assumed to be statistically independent and identically distributed, but with an arbitrary density function. The method involves a linear integral equation which can be reduced by Laplace transforms. A new family of solutions is given which includes two previously known cases: the random square wave of Poisson type, and the periodic square wave with random time origin. The general result of this family is given in terms of tabulated functions. For other solutions, a recursive technique may be necessary.

By the method of Darling and Siegert, a differential equation is obtained for the characteristic function of the output of a linear system when the input is a random telegraphic signal, i.e., a random square wave with zeros obeying the Poisson distribution. For the ideal integrator with finite memory and for the RC low-pass filter, the solutions agree with previous results. For a truncated exponential weighting function, the characteristic function is obtained in terms of Bessel functions. For a particular ratio of the constants, the probability density of the output is rectangular except for\delta- functions at the edges. The applicability of this rectangular solution to other systems is investigated,

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.

This paper presents a simple formula for the correlation function of an extremely clipped signal when the input is Gaussian noise plus a sine wave of small amplitude.

For an arbitrary random process\xi(t)there exists a functionx(t)which may be obtained by infinite clipping. The axis crossings ofx(t)are identical with those of\xi(t). This paper relates the probability densityP(\tau)of axis-crossing intervals to\gamma(\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 to\gamma \prime (0+), i.e., the right-hand derivative of\gamma (\tau)at\tau = 0. Next a theorem is proved, stating thatP(\tau) = 0over a finite range0 \leq \tau < Tif and only if\gamma(\tau)is linear in\mid \tau \midover the corresponding range of\mid \tau \mid. If\gamma (\tau)is nearly linear for small\tau, then the initial behavior ofP(\tau)is like\gamma \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.