An efficient and robust technique for tracking amplitude and frequency envelopes in oscillators

Envelope-following methods face special challenges when applied to oscillators because of their fundamental property of dynamically-changing frequencies. In this paper, we present a novel and robust approach for oscillator envelope following. Our method combines, unifies and extends ideas from two prior oscillator envelope-following approaches, Petzold\´s method and the WaMPDE. Our technique uses two extra system unknowns, as well as two extra "phase condition" equations, to track quantities related to dynamical frequency/time-period changes. These advances confer significant robustness without appreciable computational overhead. We validate our method on LC, ring and crystal oscillators, predicting frequency and amplitude modulations as well as transient startup envelopes accurately. Speedups of 1-2 orders of magnitude are obtained over traditional alternatives.