Some analytical and experimental phase-locked loop results for low signal-to-noise ratios

This paper is concerned with the nonlinear behavior of the second-order phase-locked loop (PLL) in the presence of noise. The loop filter is of the proportional-plus-integral control type. This filter corresponds to the one generally employed for carrier tracking purposes in the implementation of phase-coherent communication systems. The paper is composed essentially of two parts: the first part presents analytical results which pertain to the probability distribution of the phase-error. Since these analytical results are approximations, valid only for certain regions of signal-to-noise ratio, they are complemented by experimental results obtained from simulation of the PLL system in the laboratory. The experimental techniques used to measure the statistical properties of the loop behavior and the corresponding results comprise the second part of the paper. Approximate analytical expressions for the distribution of the system phase-error are first obtained by using the Fokker-Planck apparatus and, secondly, by assuming that the PLL behaves as a very narrow band-pass filter. The range of signal-to-noise ratios for which these approximations are valid is obtained by graphically comparing the analytical expressions to experimentally derived phase-error distributions. In addition, measurements relative to the variance of the phase-error are compared to those predicted by the linear PLL theory and the variance as computed from the approximate solutions. Finally, experimental results relative to the probability distribution of the time intervals between cycle-slipping events are given for signal-to-noise ratios in a range where the linear PLL theory does not apply. In particular, the maximum length of time the loop may be expected to remain in-lock is illustrated graphically as a function of signal-to-noise ratio in the loop bandwidth.