On Gaussian noise envelopes

Abstract

The first-passage time problem for a continuous one-dimensional Markov process is reviewed, and upper bounds are obtained for both the probability of failure (or passage and the moments of the time to failure, in terms of the mean time to failure. In addition, stationary Gaussian variables arising from systems with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> degrees of freedom that have autocorrelation functions of the form \begin{equation} R(r) = e^{-b \mid \tau \mid} \sum_{k=1}^{N} d_k^2 \cos \omega_k \tau \end{equation} are shown to be derivable from a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</tex> -dimensional (or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</tex> - 1, if one of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_k</tex> is zero) Markov process that possesses a "pseudoenvelope," which is itself the result of a one-dimensional Markov process. This pseudo-envelope can be used as a bound on the magnitude of the Gaussian variable, and its first-passage time problem can be solved explicitly or utilized to obtain convenient bounds for the probability of failure of the Gaussian process.