Mathematical Analogies Between First-Order Digital and Analog Phase-Locked Loops

This concise paper provides an exact analysis of the phase error statistics of a first-order digital phase-locked loop (DPLL) by soloing the chapman-Kolmogorov equation using the method of moments. Both time independent and time dependent solutions are presented. In addition, the parameters which characterize the performance of a DPLL are identified with those of an analog phase-locked loop (APLL). It is Shown under what design parameter conditions the solution provided herein for a DPLL is equivalent to that obtained by applying the Fokker-Planck equation to the analysis of an APLL. Numerical comparisons are provided for specific parameter ranges of interest in practice.