General method for computing the power spectrum density of irreducible periodic Markov chains

Many signals occurring in data communications can be described by Markov models. The formula for the power spectrum density (PSD) of a signal driven by an irreducible aperiodic Markov chain with uniformly spaced transitions was first published in 1961 by Titsworth and Welch and in 1973 by Lindsey and Marvin. There are, however, a number of systems in which the aperiodicity assumption is not correct and the formula is not applicable. A formula for the PSD of periodic Markov chains is derived and discussed in this paper. We show that the continuous part of the PSD is given by the same formula stated by Titsworth et al. which was known to be valid for a stationary irreducible aperiodic finite state Markov chain. The discrete part of the PSD, which is due to the periodic component of the Markov chain, has extra terms that do not appear in the aperiodic case. The necessary and sufficient condition of absence of spectral spikes is given, too. An application of this result to the calculation of the PSD of a binary continuous phase frequency shift keying (CPFSK) signal, which is a periodic Markov chain, is given. Unlike the use of the formula of Titsworth et al., which will result in no spectral spikes at all, we have shown that there are some extra spectral spikes. The computer simulation result of the PSD of a binary CPFSK signal through the periodgram method is given, which verifies the existence of the spike spectrum.