Identifying Arnold's tongue for digital oscillators through event-based control in phase-locked loops

Digital phase-locked loops (PLLs) are essential feedback circuits for synchronizing signals in digital communication systems. While amplitude and phase vary continuously in analog oscillators, the amplitude remains constant in digital oscillators with dynamical variations manifesting exclusively through changes in the timing of signal transitions. In this work, we introduce a novel analytically solvable event-based model for phase-locking in digital PLLs that leverages the discrete nature of digital signals. By employing a sampled control strategy, we demonstrate one-to-one and higher ratios of frequency locking under positive and negative feedback. By discretizing the continuous control signal, we drive a discrete iterative map, which we then use to derive analytical expressions for bifurcation curves, analogous to Arnold's tongue in analog oscillators. This mathematical framework provides an analytical approach for the analysis of synchronization and phase-locking in digital oscillators. Furthermore, the event-based control presented in this work for digital oscillators paves the way for energy-efficient circuit design and optimized control strategies for future digital communication systems.