The Optimum Closed-Loop Transfer Function of a Phase-Locked Loop Used for Synchronization Purposes

This paper presents an expression for the optimum closed-loop transfer function of a phase-locked loop (PLL) which estimates the phase of an unmodulated sinuosoid corrupted by an additive random disturbance. This matter has been extensively treated (e.g., [1, ch. 4]) in the conventional PLL theory, where the disturbance is a stationary noise process. However, for an important class of synchronizers operating on a PAM waveform corrupted by stationary noise, the disturbance at the input of the PLL cannot be treated as a stationary process. In this case we show that the PLL transfer function which is optimized according to the conventional PLL theory is far from optimum. Therefore, we derive the closed-loop transfer function which is optimum for a more general disturbance. This extension of the conventional PLL theory applies to most synchronizers.