Analysis of a digital PLL Markov model (Q923515)

This paper presents the Markov model of the circular random walk which may very well represent the DPLL (digital phase-locked loop). There is recently a great interest in the study of DPLL's. The qualitative results include the existence conditions of a recursive state class or cyclic subclasses (for a periodic Markov chain). Then the general form of the phase error asymptotic distribution at \(t=1\) and t greater than 1 is presented, for frequency offsets that correspond to rational values of the control period cumulative phase difference. The types of DPLL stationary and periodic operating modes which appear with time are discussed. It is shown that a system which is described by the same transition matrix may have a stable stationary distribution if a Markov chain is used for its model but will not have one if it is modeled by a Markov chain family.