Nonlinear Analysis of Correlative Tracking Systems Using Renewal Process Theory

A new method is presented which describes the behavior of an
(N + 1)
th-order tacking system in which the nonlinearity is either periodic [phase-locked loop (PLL) type] or a nonperiodic [delay-locked loop (DLL) type]. The cycle slipping of such systems is modeled by means of renewal Markov processes. A fundamental relation between the probability density function (pdf) of the single process and the renewal process is derived which holds in the transient as well as in the stationary state. Based on this relation it is shown that the stationary pdf, the mean time between two cycle slips, and the average number of cycles to the right (left) can be obtained by solving a single Fokker-Planck equation of the renewal process. The method is applied to the special case of a PLL and compared with the so-called periodic-extension (PE) approach. It is shown that the pdf obtained via the renewal-process approach can be reduced to agree with the PE solution for the first-order loop in the steady state only. The reasoning and its implications are discussed. In fact, it is shown that the approach based upon renewal-process theory yields more information about the system's behavior than does the PE solution.