Probability distributions of peaks and troughs of non-Gaussian random processes

This paper deals with the development of probability density functions applicable for peaks, troughs and peak-to-trough excursions of a non-Gaussian random process where the response of a non-linear system is represented in the form of Volterraʹs second-order functional series. The density functions of peaks and troughs are derived in closed form and presented separately. It is found that the probability density function applicable to peaks (and troughs) is equivalent to the density function of the envelope of a random process consisting of the sum of a narrow-band Gaussian process and sine wave having the same frequency. Furthermore, for a non-Gaussian random process for which the skewness of the distribution is less than 1.2, the density function of peaks (and troughs) can be approximately presented in the form of a Rayleigh distribution. The parameter of the Rayleigh distribution is given as a function of parameters representing the non-Gaussian characteristics. The results of comparisons between newly derived density functions and histograms of peaks, troughs and peak-to-trough excursions constructed from data with strong non-linear characteristics show that the distributions well represent the histograms for all cases.