Phase jitter dynamics of first-order digital phase-locked loops with frequency-modulated input

Inherent to digital phase-locked loops is frequency quantization in the number-controlled oscillator which prevents the loop from locking exactly onto its reference signal and introduces unwanted phase jitter. This paper investigates the effect of frequency quantization in a first-order loop with a frequency-modulated input signal. Using tools of nonlinear dynamics, we show that, depending on the modulation amplitude, trajectories in the phase space eventually fall into either an invariant region or a trapping region, the boundaries of which give useful bounds on the steady-state phase jitter excursion. We also derive a sufficient condition for the maximum modulation amplitude to prevent loop cycle slipping.