Functional techniques for the analysis of the nonlinear behavior of phase–locked loops

In this paper we consider the analysis of a nonlinear feedback system. The purpose of the paper is twofold. The first objective is to demonstrate the efficiency of the Volterra functional expansion technique as a method of analyzing nonlinear feedback systems. The techniques we demonstrate are valid for a large class of nonlinear systems. Several important advantages of the functional approach are as follows: 1) Random and deterministic inputs and disturbances are included. 2) All input-output relationships are explicit. One does not have to solve complicated differential equations. 3) Once one becomes facile with the properties of the expansion, the analysis of any particular nonlinear system is rapid and straightforward. The second objective is to obtain some new and useful results for a device of practical importance. The particular nonlinear system that we will use as an example represents a phase-locked loop whose input signal is a phase-modulated sinewave which has been corrupted by additive noise. Two interesting cases of phase modulation are considered. In the first case the phase θ1(t) is a deterministic function. In the second case the phase θ1(t) is a sample function from a random process. The results are presented as closed form analytic expressions. Several interesting cases are plotted as a function of the significant parameters.