Power spectral density of pulse train over random time scaling

This study analyses power spectral density (PSD) of a pulse train where the pulses take from a prototype pulse but randomly take an independently and identically distributed time scaling and an independent stationary amplitude. A closed-form expression of PSD is obtained, which is an implicit function of the Fourier transform of the prototype pulse without time scaling, the probability distribution of time scaling, and the first and the second moment means of amplitude. In the special case when the time scaling and amplitude are fixed with probability one, the PSD is degenerated to the well-known PSD of a periodic signal. Results of numerical evaluation and simulation for pulse trains with three rates as well as with Gaussian rates demonstrate that the analytical formula well predicts the data PSD.