Line spectrum of series of time-correlated pulses having random shape and amplitude

Abstract

The most general case is examined of series of pulses having a line spectrum besides a continuous one, characterized by the following conditions. \begin{enumerate} \item Pulses are distributed in time according to an arbitrary distribution function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(x), x</tex> being the time interval between two consecutive pulses. \item The time intervals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> relative to different pairs of pulses are uncorrelated. \item Pulse shape is random and no correlation exists between shapes of different pulses. However, an arbitrary correlation can exist between a pulse amplitude and the time interval separating this pulse from the preceding one. \end{enumerate} It is shown that under these conditions a line component can be present only in the spectrum of series of pulses whose distribution function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(x)</tex> has the form: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(x) = \sum_{m} b_{m} \cdot \delta (x-x_{m})</tex> . A simple expression giving the intensity of the lines is derived by means of an integration method in the complex plane, which can be applied to similar calculations in more general cases. As an application it is shown that the power spectrum of series of pulses which have a distribution function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(x)</tex> of the said form, and are moreover characterized by a random pulse position modulation, can be easily obtained from the derived general results.