Performance of a First-Order Transition Sampling Digital Phase-Locked Loop Using Random-Walk Models

A new mechanization of a first-order all digital phaselocked loop (ADPLL) is discussed and analyzed. The purpose of the loop is to provide continuous tracking of the incoming waveform corrupted by the presence of white Gaussian noise (WGN). Based on a random-walk model, solutions are obtained for the steady-state timing-error variance and mean time to slip a cycle. As a result of the mean first-time-to-slip analysis, a difference equation and its solution are obtained that generalize a result of Feller [1]. Using a procedure that appears new, an upper bound on the timing-error bias due to a Doppler shift of the synchronized waveform is also derived. An example, for which the results presented here are applicable, is considered in some detail.