An Upper Bound on the Zero-Crossing Distribution

Let Q(T) equal the probability that a random process, x(t), does not cross the zero axis in a given interval of length T. The problem of determining Q(T) (and related functions) has important applications in communications theory and has been investigated by many authors. 1-0 Reference 5 gives an extensive bibliography of most of the related work on this subject prior to 1962. Despite all this effort, Q{T) is known only when x(t) is a simple nongaussian process (such as a process whose zero-crossings obey the Poisson distribution) or a stationary gaussian zero-mean process with one of four explicit correlation functions. 5 - 6 Most of the rest of the results obtained are either approximate or form upper or lower bounds. 5 In this paper, we develop a whole family of upper bounds on Q{T). For computational purposes, however, only one member of the family has been found to provide useful results for most cases of interest. II. DERIVATION OF AN UPPER BOUND ON Q ( T ) Consider the transformation T