Statistical Properties of Ramplike Random Processes

The work of Lampard and Redman [5] on the statistical properties of the integral of a binary random process is generalized and extended to include more general ramplike random processes
z(t)
. These ramplike processes are defined to be ramps of random slope that switch slope at random times. Both the intervals between switching times and the slopes of the ramp segments are assumed to be independent random variables. The general properties of the one-dimensional and multidimensional probability density functions of
z(t)
are discussed. A Laplace transform technique is employed to obtain the Laplace transform of the characteristic function of these probability density functions in terms of the Laplace transform of the characteristic function of the ramp slopes multipled by the probability functions of the switching times. For Poisson switching times, these expressions are relatively compact. However, the inversion problem is formidable, and closed form results are presently available only for a binary slope probability density function (a previously known result). Due to the properties of characteristic functions, the moments of
z(t)
are obtained by a single Laplace inversion of the derivative of the Laplace transform expressions. These moments may be used to characterize the probability density functions of
z(t)
with a series expansion involving the Gaussian density function and its derivatives. Examples are given for the cases of Gaussian and rectangular slope probability density functions with Poisson switching times.