On the Discrete Spectral Densities of Markov Pulse Trains

An important problem related to first-order Markov pulse trains is that of calculating the discrete and continuous power spectral densities of such processes. The spectral formulation first given by Huggins 1 and later extended by Zadeh 2 is perhaps the most appropriate and straightforward solution of this problem, the results being conveniently expressed in terms of the customary flow diagrams and recurrent event relations associated with Markov systems. As regards discrete spectra, however, their formulation lacks complete generality in two respects: (i) the limit notions of distribution theory, although essential for discrete components, are not incorporated; (it) discrete components do not appear explicitly. In this paper we reformulate the Huggins-Zadeh result on a distribution theoretic basis, and derive both explicit relations and existence criteria for the discrete spectral densities. It is intended also that the analysis illustrate the distribution theoretic techniques required in cases involving more general spectral formulations.