# 14 THE FIRST PASSAGE TIME

## 14.1 INTRODUCTION

For signals defined by a random process *X*(Ω, *t*) on an interval [0,*T*], the characteristics, illustrated in Figure 14.1, can be defined and, in an appropriate context, provide valuable information. These characteristics include the maximum and minimum level; the first passage time to, or first level crossing time of, a set level; the time spent above a set level; the number of level crossings of a set level; the signal path length; etc.

This chapter provides an introduction to first passage time theory. In general, useful ways of characterizing the first passage time for signals defined by a random process are analytically difficult with results obtainable for a few important cases including the random walk, one-dimensional Brownian motion, monotonically increasing signals, and the linear signal plus noise case under limiting, but useful, conditions. These cases are considered along with general approaches including a Rice series approach and an integral equation approach. Finally, the relationship for a random process, between the maximum level on an interval and the first passage time on the same interval, is defined.

## 14.2 FIRST PASSAGE TIME

When the system performance, in some manner, is affected by a system ...

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