Linear dynamical systems under random trains of non-overlapping, arbitrary-shape pulses: generalized approach

The excitation considered is a train of non-overlapping pulses, defined as a train of alternate random pulse durations and time gaps between the consecutive pulses. The time gaps and durations are assumed to be independent, continuous random variables. In this model the pulses are not only non-overlapping, but they are also non-adjoining, i.e. they can only adjoin with probability zero. All durations are assumed to be characterized by one common, Erlang (or integer parameter gamma) type, probability distribution. Likewise, the time gaps are all identically, Erlang distributed, but the probability distributions of the durations and of the time gaps have different parameters. Pulse shapes are given by an arbitrary, deterministic function and pulse heights are given by independent, identically distributed random variables. The mean value and autocorrelation function of the pulse train are evaluated analytically and are shown to be expressed in terms of the renewal densities of the renewal process governing the pulse arrivals. It is shown that for rectangular pulses with equal, deterministic heights these expressions reduce to the ones obtained independently with the aid of another approach, where the mean value and autocorrelation function are given in terms of Markov ‘on’ state probabilities. The mean value and variance of the response of a linear oscillator are obtained via a time domain analysis. Illustrative numerical analysis is carried out for example trains of pulses with parabolic and rectangular pulse shapes.