The distribution of the number of crossings of a Gaussian stochastic process

It is shown how filtered Gaussian noise having a power spectrum which is a rational function of the square of the frequency can be represented as one component of a multidimensional Markov process. Methods are studied for obtaining the distribution of the number of times such a noise process crosses a given amplitude level in a fixed time interval. The generating function of this distribution is the solution of a Fokker-Planck type differential equation with appropriate boundary conditions. Integral equations are found for the generating function from which all the moments of the distribution can be calculated by iteration.