论文信息 - The distribution of the number of crossings of a Gaussian stochastic process

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.

Carl W. Helstrom

[1] W. Wasow,et al. On the Duration of Random Walks , 1951 .

[2] D. Middleton. Spurious Signals Caused by Noise in Triggered Circuits , 1948 .

[3] A. Siegert. On the First Passage Time Probability Problem , 1951 .

[4] P. Schultheiss,et al. Short‐Time Frequency Measurement of Narrow‐Band Random Signals by Means of a Zero Counting Process , 1955 .

[5] John E. Freund,et al. Some Results on the Analysis of Random Signals by Means of a Cut‐Counting Process , 1956 .

[6] G. Uhlenbeck,et al. On the Theory of the Brownian Motion II , 1945 .

[7] Arnold J. F. Siegert,et al. A systematic approach to a class of problems in the theory of noise and other random phenomena-III: Examples , 1957, IRE Trans. Inf. Theory.