On Level-Crossing Excursions of Gaussian Low-Pass Random Processes

Level-crossing problems have been the subject of much study for centuries. Practical applications can be found in many fields of engineering, e.g., predicting flood levels, random mechanical stress analysis, electrical excursions that overload equipment, and disruptive effects in signal processing. In this paper, we focus on stationary Gaussian low-pass random processes and, in particular, of the discrete-time first-order Markov type. We derive new results for discrete-time processes and links to known results for continuous-time processes to arrive at a general understanding of the excursion statistics. Expressions are derived for the crossing rate, shape of large excursions, peak distribution, and sojourn probability. Simulation results are compared with the various theoretical formulations. Along the way, we obtain a new continuous-time sojourn density approximation that extends the useful range of Rice's formula to longer intervals.