Phase-locked loops, demodulation, and averaging approximation time-scale extensions
Among the many applications of the phase-locked loop (PLL), a device used extensively in telecommunications and electronics, is the demodulation of modulated carrier signals. The PLL contains a voltage controlled oscillator (VCO) that tracks a reference signal whose frequency may be changing. This is accomplished through a feedback mechanism-the VCO's frequency is adjusted by a control signal that, after filtering, depends mostly on the phase difference between the reference and VCO output. Phase-lock describes an operating state for which this phase difference remains constant. During nearly phase-locked operation, the filtered signal controlling the VCO approximates the demodulation of the reference. A standard model is used to give a rigorous mathematical explanation of the described operation of the PLL in a physically realistic operating regime. While the model does not allow strict phase-locking, a theorem is formulated and proved that predicts operation near an attracting torus with quasi-periodic flow in the state space. The proof uses high-order averaging, a new result on extension of the averaging estimate to the forward infinite time-scale, and continuation theory for invariant manifolds. For the averaged system (equivalent to a simplified model that assumes ideal filtering), we obtain an approximation for solutions on an attracting invariant torus (for quasi-periodic reference signal modulation of sufficiently small amplitude and frequency), in which the dominant response of the filtered control signal is the demodulation of the reference signal, up to a rescaling and constant shift. Furthermore, we show that the full model (allowing nonideal filtering) also has an attracting torus, on which solutions exhibit the same dominant response. In addition, some results on continuation of invariant manifolds, which may have applications beyond the PLL, are proved.